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ISBN 978-1-80230-288-2
Design and layout: Design!mage
Illustrations: Nadene Naude, Andrew Pagram
Copy editor: Donna Garvin
Proofreaders: Jane Rogers, Eric Pradel and Clodagh Burke
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Introduction
The Maths and Me Pupil’s Book consolidates learning by bringing the Primary Maths Curriculum (2023) to life through engaging, playful and interactive activities. It links maths to daily life through real-life pictures, problems and tasks, so children will appreciate the relevance and significance of maths in their everyday experiences.
The relatable Maths and Me characters, Lexi, Dara, Mia, Jay and Monty the Dog help children to understand that we are all mathematicians, and model a positive disposition to maths.
Pupil’s Book Features
Colour coding – Pages are colour-coded so you can see what the main strand is at a glance:
Number
Measures
Shape and space
Let’s talk! Let’s play!
Let’s investigate!
Maths eyes
Try this!
Data and chance
Algebra
Let’s play! – Incorporates playfulness into maths through engaging games and interactive activities, making maths a fun and enjoyable adventure for children.
Let’s talk! – Provides opportunities for children to share their strategies and ideas, helping them to reflect on their current knowledge and identify emerging concepts.
Let’s investigate! – Encourages children to develop creative strategies through active participation and exploration.
Maths eyes – Encourages children to look around them and recognise maths in the real world.
Try this! – Provides optional, cognitively challenging tasks that offer an enriching learning opportunity for children.
Self-assessment – Gives children an opportunity to reflect on their work and colour in up to three stars at the end of each page, depending on how they felt they performed on a certain task.
Additional resource icons – Indicates if a photocopiable is needed to complete an activity.
Digital resources – Allows easy access for teachers using the ebook to the extensive menu of interactive resources provided for each unit.
Clear signposting – Allows easy navigation across the programme through direct correlation of the Lesson Title and the footer information to the Teacher’s Planning Book and the digital resources.
Orange Blue Purple Green Red
PCM 1
Numbers Beyond 10,000
Let’s talk!
With a partner, take turns to read these numbers aloud:
What is the value of the digit 4 in each of the numbers?
In which number does the digit 5 have (a) the greatest value, and (b) the least value?
Write each number using digits (standard form).
1. Thirty-two thousand, six hundred and fifteen
2. Eleven thousand, two hundred and thirteen
3. Fifty-one thousand, two hundred and sixty
4. Sixty thousand and seventeen
5. Ten thousand, one hundred and one
6. Fifteen thousand and forty-one
7. Thirty-nine thousand and nineteen
8. Seventy thousand and thirty-eight
9. Fifty-six thousand, five hundred and two
10. Eleven thousand, one hundred and twelve
Build it! Sketch it! Write it!
Choose a number from above Build it using materials. Sketch it in your copy. Write it in different ways. Now choose two more numbers and do the same.
Don’t forget to use a comma. Also, use zero as a placeholder where necessary.
Try this!
2. A number that is not a multiple of 5 Example:
Standard form: 13,759
Word form: thirteen thousand, seven hundred and fifty-nine Expanded
Write a 5-digit number to fit in each section of the Carroll Diagram to suit the given rule.
Rule
1. A number that is a multiple of 5
(a) Even number (b) Odd number
D What number is represented? Write the answer in different ways.
With a partner, take turns to read these numbers aloud. Where might you come across numbers like these?
Estimating and Rounding Whole Numbers
Look at the table and answer the questions.
1. Round the population of each county to the nearest thousand.
2. Which county has…
(a) roughly half the population of Roscommon?
(b) roughly double the population of Longford?
(c) a population of roughly one hundred thousand?
(d) a population of nearly 50 thousand?
Estimate the population of these counties.
1. Sligo:
2. Offaly:
3. Carlow: 4. Monaghan:
Let’s talk!
Which point on the number line below is the best estimate for 34,432? Explain why.
D Round the numbers and complete the table.
Leitrim 35,199
Fermanagh 63,585
Try this! Two 5-digit numbers have a difference of 5. When they are both rounded to the nearest thousand, the difference is 1,000. What could the numbers be? and
Carlow Monaghan
Thousandths
What (a) decimal fraction, and (b) fraction (thousandths) is coloured?
In above, what (a) decimal fraction, and (b) fraction of each shape is uncoloured?
1. (a) (b)
3. (a) (b)
5. (a) (b)
In above, what is the…
2. (a) (b)
4. (a) (b)
6. (a) (b)
1. largest decimal fraction that is (a) coloured? (b) uncoloured?
2. smallest decimal fraction that is (a) coloured? (b) uncoloured?
Let’s talk!
If you add the number of coloured parts to the number of uncoloured parts in each shape in above, what do you notice? Can you explain why this is so?
Write each answer as a fraction and a decimal number.
1. There are 1,000m in 1km. If Mia’s dad has run 234m of a 1km race, what fraction of the race has he (a) run? (b) left to run?
2. There were 1,000 paper clips in a box. If 750 paper clips have been taken out, what fraction of the paper clips…
(a) has been taken out?
(b) is still in the box?
1,000
Place Value in Decimal Numbers
Dara used base ten blocks to represent decimals. Write the numbers he built in (a) fraction form, and (b) decimal form in the table below.
form
Fraction form (b) Decimal form
Mia also used base ten blocks to represent numbers. Write the numbers she built in decimal form. 1. 2 tenths and 5 hundredths
6 tenths and 8 thousandths 3. 9 hundredths and 2 thousandths
15 hundredths
12 tenths
27 thousandths
Try this! Jay used six base ten blocks to build a decimal number. If he did not use any red blocks, but used at least one of each of the other block colours, what was the…
1. greatest decimal number that he could have built?
2. smallest decimal number that he could have built?
0829 3. 0702 4. 0.028
Let’s talk!
Read aloud each of the expanded numbers in above What is the value of the digit 2 in each of the numbers?
In which number does the digit 8 have (a) the greatest value, and (b) the least value?
E Let’s play! Wipeout!
Number of players: 2–6
You will need: calculator, mini-whiteboard and marker per player
+ 0·08 + 0·003
Read aloud each of the expanded numbers. For example, 2 tenths, 8 hundredths and 3 thousandths.
● To start, each player inputs a 5-digit number (any digits except 0) with 3 decimal places (to thousandths) on their calculator and writes the number on their mini-whiteboard.
● A caller (perhaps the teacher) calls out a digit, for example 6.
● Any player with a 6 on their calculator display wipes out only this digit, so that there is a 0 in its place. They record how they did it on their mini-whiteboard.
● Only one digit can be wiped out each time.
● The player who wipes out all of their digits first wins the game.
Let’s talk!
With a partner, take turns to read these numbers aloud: Where might you come across numbers like these?
Try this!
Using only the digits above, make a number with three decimal places in which: 1. 6 has
Estimating and Rounding Decimal Numbers
For each number line below, write (a) the number the arrow is pointing to, and (b) the whole number to which it rounds.
For each number line below, write (a) the number the arrow is pointing to, and (b) the tenth to which it rounds.
Let’s talk!
1. Which of the numbers in A and B were trickiest to work out? Explain why.
2. If a number is halfway between two numbers, to which one does it round?
Hint: To round to the nearest whole number, look at the digit in the place.
3. How might you round decimal numbers without using a number line?
4. What does it mean to ‘round a number to one decimal place’?
5. Which of the answers in A and B have only one decimal place? Which of them have two decimal places?
(a) (b)
(a) (b)
3.
(a) (b)
(a) (b)
(a) (b)
(b)
D Round each price to (a) the nearest whole euro, and (b) the nearest 10 cent.
Use these digits to make a number with three decimal places that…
8 0 4 2
1. when rounded to the nearest whole number is 5.
2. when rounded to the nearest tenth is 0·3. ·
3. when rounded to one decimal place is 0·5. ·
Try this! For each clue, write a number with three decimal places that is (a) the least number possible, and (b) the greatest number possible.
When I round my number to the nearest whole number, I get 4. When I round it to the nearest tenth, I get 3.7.
My number is between 5 and 9 When I round it to the nearest tenth, I get the same number as when I round it to the nearest whole number
My number contains the digit 2. When rounded to one decimal place it is 8.3.
Comparing and Ordering Whole and Decimal Numbers
Write each decimal number marked by an arrow on the number line.
(a) (b) (e) (c) (e) (d) ( )
(a) (b) (c) (d) ( ) 7·97
Write <, > or = to make these true.
1. 8433 844 2. 3307 34 3. 10,006 9,987
4. 39,460 39,640 5. 5015 4,301 6. 4115 4 15 100
7. 65306 5,356 8. six thousand and six 6,060
In your copy, write each group of numbers in order, starting with the smallest.
1. 0.986, 0.989, 0.99 2. 52,011, 52,110, 52,101
3. 042, 07, 0368 4. 3 10 , 003, 33 1,000
5. 6 10 , 0.06, 66 1,000 6. 3.07, 3 15 100, 3.017
D Look at the table and answer the questions.
1. Who was the fastest in the race?
2. Who was the second fastest?
Try this!
1. Read the clues below. What could the mystery number be?
● It is an odd number.
● The sum of its digits is 20.
● When rounded to the nearest thousand, it is 63,000.
3. Write a number to make this true. 60,740 < < 65,970
2. What number comes next in each of these patterns? (a) 6, 65, 7, 75, (b) 375, 35, 325, 3, (c) 1.125, 1.25, 1.375, 1.5, 1.625, 1.75,
4. Write <, > or = to make these true. (a) 515m 515cm (b) 4,506m 4.61km
1. (a) (b) (c) (d) (e) (f)
2. (a) (b) (c) (d) (e) (f)
Estimating and Checking Calculations
A Look at these calculations. Which answer looks most reasonable?
1. 56 + 48 =
2. 96 28 =
3. 6,125 + 1,349 =
4. 6,264 − 285 =
B Let’s talk!
=
=
.
Use your calculator, if needed, to help you check.
Compare your answers for above with those of a partner Take turns to explain which answer was most reasonable and why. Then, look at the unreasonable answers. Take turns to suggest how you might arrive at those answers.
Complete this table.
(a) Estimate using front-end (b) Estimate using rounding (c) Calculator answer
1. 23,875 + 5,946
2. 3,189 + 42,204
3.
4.
5.
D Let’s talk!
In above, which estimate (a) was faster? (b) was closer? (c) do you prefer?
Try this! Look at the table and answer the questions.
1. Do you think the figures are approximate or exact amounts? Why?
2. About what number is the population of Bangor and Bray in total?
3. About how many more people live in Drogheda than in Tralee?
4. What is the approximate difference in population between Newbridge and Ennis?
Populations (2022 Census) Town
Bangor, Co Down
Ennis, Co Clare27,923 Bray, Co Wicklow33,512 Newbridge, Co Kildare24,366
5. With a partner, write three more questions about the information in the table. Then, swap with another pair and solve.
A Let’s
talk!
Adding Using the Column Method
With a partner, take turns to estimate a reasonable answer for each of the calculations in and below. Explain your reasoning
B Solve these.
Write in the unnecessary zeros to help.
In your copy, rewrite these using columns, and solve.
1. 73,854 + 15,367
D Let’s talk!
Review your answers to and above Do they look reasonable? Check the answers using a calculator, but without using the + button.
Look at the table and answer the questions.
1. Anna is buying a City Hatchback and a single charger About how much money will she need? €
2. The Kelly family are buying a 7-seater and a double charger About how much money will they need? €
3. One weekend, a salesperson sold one of each type of car and charger Approximately, what was the total value of her sales that weekend? €
4. Is it better to round up or down when estimating how much money is needed? Why?
5. Now work out the exact cost of (a) question 1, (b) question 2 and (c) question 3 above (a) € (b) €
€
Solve these.
1. Jay bought the following items at the shop: butter (0.227kg), cereal (0.75kg) and lasagne (1.5kg). If the cloth bag weighed 009kg, what was the total weight of the bag and its items? kg
2. Mia bought the following items: orange juice (1.75l), shampoo (035l) and fabric softener that was double the capacity of the shampoo. What was the total capacity of the three items? l
Try this! Re-write these in your copy. Insert the missing digits.
I think there might be more than one correct answer for number 3.
G Let’s play!
Number of players: 2–6
Chance Calculations – Addition
You will need: mini-whiteboard and marker per player, 0–9 spinner, pencil and paper clip
● Each player should draw these boxes for a calculation on their mini-whiteboard.
● Each player, in turn, spins the 0–9 spinner and writes the digit spun into one of their empty boxes.
● When all the boxes are full, each player calculates their answer
● The player with the greatest total scores a point.
● The player with the highest score when the time is up wins the game.
Variation
● At the beginning, choose one of these alternative layouts instead, or agree on your own chosen layout.
Adding Using Other Strategies
A Complete these to arrive at an answer.
1. 171 + 225 = 2. 23,995 + 15,146 = 24,000 15,141
3. 15.45 + 1.25 =
B Solve these without using the column method.
1. Mia has three parcels weighing 125kg, 164kg and 1.81kg Jay has three parcels weighing 164kg, 181kg and 135kg Whose parcels weigh more?
2. A factory made 15,140 phones in January, 13,000 phones in February and 17,000 phones in March. How many phones did the factory produce in total over the three months?
3. What is the total weight of two pieces of meat if one weighs 0.697kg and the other weighs 1248kg? kg
4. A concert venue has 34,780 seats and standing room available for 2,000 more people. What is the total capacity of the venue?
5. Three planks of wood measure 17m, 23m and 14m. What is their total length? m
Try this! Total any two of the numbers shown.
Show the total to a partner Without doing a calculation, can they work out which two numbers you totalled?
Swap roles.
Variation:
Total any three of the numbers shown.
Subtracting Using the Column Method
A Let’s talk!
With a partner, take turns to estimate a reasonable answer for each of the calculations in and below. Explain your reasoning
B Solve these.
Write in the unnecessary zeros to help.
In your copy, rewrite these using columns, and solve.
D Let’s talk!
Review your answers to and above. Do they look reasonable? Check the answers using a calculator, but without using the – button.
Try this! Look at the table and answer the questions.
1. What was the difference between the fastest lap and the slowest lap? secs
2. What was the total time for laps 5 and 6? secs
3. What was the time difference between laps 1 and 2? secs
4. Approximately how many minutes did the six laps take? mins
5. With a partner, make up two more questions about the information in the table. One must be addition and one must be subtraction. Then, swap with another pair and solve.
E Let’s play! Play ‘Chance
– Subtraction’ on page 38.
Subtracting Using Other Strategies
A Complete these to arrive at an answer.
B Let’s talk!
In what other ways could the questions in A above be solved?
C Solve these without using the column method.
1. Over Saturday and Sunday, 82,457 passengers travelled through Dublin Airport. If 31,956 passengers travelled through the airport on Saturday, how many travelled through on Sunday?
2. The distance from Ballybeg to Athmore is 18.65km. Sarah has cycled part of the way, and she has 5925km left to go. What distance has she cycled so far? km
3. The total weight of a box of cereal bars is 0.78kg If 003kg of this weight is made up of wrappers, what is the weight of the actual cereal bars? kg
4. 48,783 people attended an open-air concert this year This was 3,560 more than attended the previous year. What was the total attendance for the two years?
Try this! Write the answers in your copy.
1. Each letter to the right represents a different digit: 0, 1, 2, 3, 6, 7 or 9. What is the calculation?
2. Make up a story to match this model and solve it.
A plane started a flight with 89,000 litres of fuel. If the plane used up 13,580 litres of fuel for every hour of the flight, how much was left after two hours? litres
Try this!
–
2. Explain to a partner how you thought of an answer 7,000 200 80 26,000 ?
I will use rounding to the nearest thousand.
1. Calculate these mentally and record the answer. (a) 800 375 = (b) 7,000 5,650 = (c) 784 599 = (d) 457 298 = (e) 6 278 = (f) 62 19 = (g) 74 28 = (h) 81 685 =
Properties of Operations
A Use one of the phrases below to make each statement true. Give an example for each. subtracted multiplied added subtracted from multiplied by divided added to divided by
(a) Statement
A When two whole numbers are , the order does not affect the answer
B When two whole numbers are , the order does affect the answer.
C When three whole numbers are in a given order, the way I group them does not affect the answer
D When a number is zero, the answer is the same whole number
E When a number is 1, the answer is the same whole number
F When a number is zero, the answer is zero.
G When a number is the same whole number, the answer is 1.
H When a number is the same whole number, the answer is zero.
I A number cannot be zero.
I think there might be more than one correct answer to some of these.
(b) Example
B Write the letter of a statement(s) in A that describes…
1. the commutative (turnaround) property of addition/multiplication?
2. the associative property of addition/multiplication?
3. the identity property of multiplication?
4. the zero property of addition?
5. the zero property of multiplication?
6. a characteristic of subtraction?
7. a characteristic of division?
C Solve these.
If is any number, place a symbol (+, −, ×, ÷) in each to make these true.
D Let’s talk!
I think some of the questions in both and could have more than one correct answer.
Use a statement from A to describe each of the expressions in C above Explain why the commutative or associative property does not apply to subtraction and division.
Try this! Find the value for each symbol below. Each time, choose only one of these numbers: 0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12.
Factors
A What is the missing factor in each of these? 1. 420 2. 432 3. 856
B Build it! Sketch it! Write it!
A factor is a number that divides evenly into another number.
Express each number below as multiplication sentences of its factors. Example: To express 12 as multiplication sentences of its factors, we would write: 1 × 12, 2 × 6, 3 × 4 1. 25: 2. 32: 3. 44: 4. 45: 5. 54: 6. 72:
C Dara was asked to list the factors of each of the numbers below. Ring the mistake in each group.
1. Factors of 14: 1, 2, 4, 7, 14 2. Factors of 10: 1, 2, 5, 10, 20
3. Factors of 18: 2, 3, 8, 9, 18 4. Factors of 32: 1, 2, 3, 4, 8, 16, 32
D Let’s talk!
Discuss as a class or in groups. Are these always, sometimes or never true?
● Every number is a factor of itself.
● 1 is a factor of every number
● The factors of a number are smaller than the number
● 2 is a factor of numbers with 2, 4, 6, 8, or 0 ones.
● 3 is a factor of numbers with 3, 6 or 9 ones.
Try this! Sometimes factors can be factorised further to create a multiplication sentence with more than two factors. Look at the examples for 12 and 16. Then, in your copy, factorise each of the numbers below.
This is often called a factor tree. Why do you think this is so?
Multiples
A Mia was asked to list multiples of each of the numbers below. Ring the mistake(s) in each group.
1. Multiples of 10: 30, 55, 90, 240, 671, 890
2. Multiples of 2: 36, 74, 83, 130, 421, 758
3. Multiples of 5: 25, 54, 80, 195, 552, 915
4. Multiples of 3: 42, 57, 68, 144, 313, 468
B Let’s talk!
How could you use your calculator to check your answers to A above?
C Answer these.
1. Identify and list the first five multiples of each number below.
(a) 2: 3: (b) 4: 5: (c) 6: 9: (d) 8: 12:
A multiple is the result of multiplying a whole number by another whole number
2. Then, find the lowest common multiple (LCM) for (a) 2 and 3: (b) 4 and 5: (c) 6 and 9: (d) 8 and 12:
D Use your understanding of the lowest common multiple to help you solve these.
1. Using cubes, find out the smallest number that can be arranged in equal groups of 2, equal groups of 3 and equal groups of 4.
2. What is the smallest number that is divisible by (a multiple of) 4, 9 and 12?
3. Burgers come in packs of 4. Burger buns come in packs of 6. If buying these packs, what is the least number of individual burgers and buns needed to ensure that every burger has a bun, with no leftovers? burgers and buns
4. Jay, Mia and Lexi went cycling on the first day of the month. If Jay then cycled every second day, Mia cycled every third day and Lexi cycled every fifth day, what was the next date on which they all went cycling? The day of the month
Try this! Dara was arranging his books on shelves. If he arranged the books in groups of 4, there were 3 left over If he arranged them in groups of 5, there were 4 left over
1. Exactly how many books could Dara have had, if there were… (a) fewer than 20? (b) between 20 and 50?
2. Using the answer to (b) above, in what way could he organise that amount of books into groups with exactly the same number in each? groups of
Prime and Composite Numbers
A prime number is a number that has exactly two factors – itself and one.
Which number is neither prime nor composite?
A composite number is a number with more than two factors.
A Read and follow the instructions.
This activity is called the Sieve of Eratosthenes, after the Greek mathematician who used it to identify the prime numbers up to 100.
1. Cross out 1.
2. Ring 2, but colour in all of the other multiples of 2.
3. Ring 3, but colour in all of the other multiples of 3.
4. Ring 5, but colour in all of the other multiples of 5.
5. Ring 7, but colour in all of the other multiples of 7.
6. Continue with this process for the remaining uncoloured numbers.
7. The ringed numbers are all numbers.
8. The coloured numbers are all numbers.
B Let’s talk!
Look at A above
● Why do you think it is called a sieve?
● What patterns do you notice?
● Why do we cross out 1?
● Why do we ring some numbers?
● Why do we colour other numbers?
C What is...
1. What is the only even prime number?
2. What is the smallest odd prime number?
3. What is the smallest odd composite number?
Try this! Twin primes are pairs of prime numbers that have a difference of exactly 2. For example, 3 and 5 are twin primes.
1. Find all the other twin primes up to 100: , , , , , , , ,
2. List the in-between even numbers: , , , , , , , , ,
3. Complete this sentence: The numbers listed in question 2 are all of . 5 6 7
Let’s talk!
Without referring to the Sieve of Eratosthenes, can you use another strategy to work out…
● which of these are not prime numbers?
● which of these are not composite numbers?
E Is each of these numbers prime or composite? (✓) 1. 311 2. 352
You can use your calculator to help you.
635
778
829
977
Solve this.
In a school, there is a class in which the teacher cannot divide the children into equal groups on days when everyone is present. If the amount of children in this class is greater than 20 and less than 30, what are possible amounts?
Mystery number! Use the clues to work out Jay and Dara’s numbers.
I’m thinking of a prime number greater than 50 and less than 100. When this number is divided by 10, there is a remainder of 9. When this number is divided by 9, there is a remainder of 8.
I’m thinking of a prime number between 20 and 70. When this number is divided by 5, there is a remainder of 3. When this number is divided by 4, there is a remainder of 1.
1. Jay's number is
2. Dara's number is
Multiplying and Dividing by 10, 100, and 1,000
A Use the moving digits strategy to help you solve these.
1. 0.82 × 100 =
B Let’s talk!
Look at Lexi and Dara. What do you think?
Explain why.
C Solve these. 1. 2,873 × 100 =
I notice a pattern based on the number of zeros in the multiplier or divisor
D Use the moving digits strategy to help you solve these.
Try this!
1. A toy shop sold 100 toy cars at €225 each. What was the total sales amount? €
2. A farmer has harvested 2,500 apples. He wants to divide them equally into 100 boxes. How many apples will there be in each box?
3. How many kilometres are there in a 10,000m race? km
4. A factory produces 1,000 batteries and each battery weighs 0·015kg What is the total weight of the batteries produced in kg? kg
I think this strategy could be used to convert measures, such as cm to m, or g to kg.
Multiplying and Dividing with Multiples of 10, 100 and 1,000
Multiplication is the inverse of division, so I can solve the division sentences by thinking about the opposite multiplication sentences.
1. 160 ÷ 2 = 2. 360 ÷ 4 = 3. 4,000 ÷ 8 =
4. 5,400 ÷ 9 = 5. 480 ÷ 6 = 6. 1,500 ÷ 5 =
7. 6,300 ÷ 7 = 8. 270 ÷ 3 =
D Solve these.
1. Newtown Primary School has about 300 pupils. Newtown Secondary School has about five times that amount. About how many pupils attend the two schools altogether?
2. A tablet costs €400. How much would it cost for 40 such tablets? €
3. If a grant of €35,000 for new IT equipment was divided among seven schools, how much would each school get? €
4. The Camera Club bought six new cameras, and had €200 left out of €2,000. How much did each camera cost? €
5. Dara has a collection of 2,000 stamps. He has four times as many international stamps as Irish stamps. How many Irish stamps does he have?
Try this!
n erna ional rish
1. If a 3-digit number is multiplied by a 2-digit number, how many digits will there be in the answer? digits
2. If a 4-digit number is multiplied by a 2-digit number, how many digits will there be in the answer? digits
I think there is more than one answer for these.
Estimating and Checking Products and Quotients
A Let’s talk!
Sometimes I use front-end estimation, and just look at the digit at the front.
I prefer to round the numbers to the most significant place value.
If there is an obvious friendly number, I will use that.
What estimation strategy do you think was used for each of the calculations below?
Do you think it was the most efficient strategy for the numbers involved? Explain why.
B Estimate a reasonable product for each of these.
C Estimate a reasonable quotient for each of these. 1. 536 ÷ 6
D Calculate the exact answers to B and C above using a calculator.
Let’s talk!
With a partner, compare the estimation strategies you used for and above
Do you think you used the most efficient strategy each time? Explain why.
Try this! Look at the table. Approximately what would be the total cost of…
1. 16 laptops? €
2. 16 laptops, a networked printer and a laptop trolley? €
3. 16 tablets? €
4. 16 tablets, a networked printer and a tablet charging cart? €
With a partner, take turns to estimate a reasonable answer for each of the calculations in C below.
Explain your reasoning.
C Solve these in your copy. 1. 7 × 9,758 =
How could you check your answers?
6,867 × 3 =
5 × 7,006 = 4. 4.63 × 8 = 5. 9 × 8.71 =
7 × 26.18 =
D Model and solve these.
1. A teaspoon holds 0005l. What is the capacity, in litres, of nine teaspoons? l
2. Moran’s Printing Services is buying three new printers for their business, each costing €1,325. What will be the total cost of the printers? €
3. Regan's Hotel is buying new couches for the hotel lobby, each costing €2,135. If they have a budget of €10,000…
(a) how many can they afford to buy?
(b) how much money will be left? €
4. A jar of jam weighs 0454kg How much do 8 jars weigh? kg
5. How much change did a school get out of €3,000 when it bought 7 new laptops at €385 each? €
6. Each lane in a running track is 1·22m wide. What is the total width of the running track if there are 8 lanes? m
18.67 × 3 =
E Use a calculator to check your answers to the questions on this page, but without using the × button.
Dividing by 1-digit Numbers
A Try to solve these mentally. 1. 32 10 ÷ 8 =
B Let’s talk!
With a partner, take turns to estimate a reasonable answer for each of the calculations in below. Explain your reasoning
C Solve these in your copy.
D Model and solve these.
1. The children are painting a mural on a school wall that is 852m long. If the wall is divided into six sections, one for each group, what length of wall is allocated to each group? m
2. Lexi and her mum walked from her house to the end of the town and back, every day for a week. This was a total distance of 18.2km.
(a) What distance did they walk each day? km
(b) What distance is it from Lexi’s house to the end of the town? km
(c) If they walk a further 05km past the end of the town every day the following week, what will be their total distance walked that week? km
E Use a calculator to check your answers to the questions on this page, but without using the ÷ button.
Try this! What digit does the letter in each of these calculations represent?
Expressions and Equations
A Ring the number sentences below that are equations.
An equation is a number sentence with an = symbol showing that both sides (expressions) have the same value.
B Which of the number sentences in A above are expressions? Write each expression and then use it to write a true equation.
1. Expression: True equation:
2. Expression: True equation:
C Complete these equations.
D Is each equation below true (T) or false (F)?
Try this!
How can you prove it?
1. If 48 is the answer, what is the question? Write four equations using a different operator (+, –, ×, ÷) each time.
2. The school needs some new equipment: 12 footballs, 9 basketballs and 1 pump If footballs cost €5 each, basketballs cost €8 each and pumps cost €6 each, write an equation that represents the money needed to purchase the new equipment.
Unknown Values and Variables
A Let’s talk!
At Dee’s Diner, the orders are written in code to make the ordering system more efficient.
Look at the menu. Can you work out what each table below ordered?
● Table 1: B + F + A
● Table 2: L + F + T
● Table 3: 2C + F + R + 2A
● Table 4: 3B + 3F + 3W
● Table 5: 3C + 2R + P + 3M
● Table 6: 4C + 2R + 2F
Estimate how many people were dining at each table.
B Using the menu in above, write each order below in code and calculate the cost.
Order
1. Table 1: burger, salad, tea
2. Table 2: potatoes, salad, milk
3. Table 3: 2 curries, rice, fries, 2 waters
4. Table 4: 2 lasagnes, 2 salads, 2 apple juices
C Write an equation to represent each of these.
(a) Code (b) Cost
1. Mia multiplied the length (l) of this rectangle by the width (w) of this rectangle to find out that it had an area of 48 sq cm.
2. Dara multiplied the side length (l) of this square by another side length (l) to find out that it had an area of 36 sq cm.
3. Jay totalled the length of the sides of this scalene triangle to find out that the perimeter was 24cm.
4. Lexi totalled the length of the sides of this equilateral triangle to find out that the perimeter was 24cm.
D Look at C and answer these in your copy.
1. Which of the unknowns are constants? Write ‘C’ beside them, and solve them to work out the unknown constant.
2. Which of the unknowns are variables? Write ‘V’ beside them, and work out one set of values that would suit in each.
A constant is a fixed value.
E Which equation below can be used to calculate the… (✓)
1. number of days (d) in a number of weeks (w)?
d= w÷ 7
d= w× 7
w= d× 7
w= d+ 7
2. number of hours (h) in a number of days (d)?
h= 24 × d
d= 24 × h
d= h× 24
d= 24 ÷ h
F Write an equation to represent each of these.
1. The cost of a hurley (h) is 9 times the cost of a sliotar (s).
2. Ben (B) is 2 years younger than Cáit (C).
3. In an input-–output table, the output number (o) is the input number (i) when doubled and 4 is added.
4. When sharing items, the amount (a) that each person gets is the total number of items (t) divided among the number of people(p).
5. Mo (M) is twice as old as Frank (F), who is three times as old as Emily (E).
A variable is an unknown value that can change or take on different values.
3. number of minutes (m) in a number of hours (h)?
m= h 60
m= 60 ÷ h
h= 60 × m
m= 60 × h
Try this! Calculate the value of the remaining unknowns in above, if…
1. the cost of a hurley (h) is €45
2. Cáit (C) is 13
3. the input number (i) is 5
4. the total number of items (t) is 21 and number of people(p) is 3
5. Emily (E) is 10
I think the word ‘is’ could be swapped with an = symbol in these sentences.
Solving Unknown Values
A Work out the value of each expression if z = 10.
B Now, work out the value of each expression from A above, if z = 15.
C Write two different equations for each of these and solve for y.
D Visualise or draw a branching bond to help you solve for w in each equation.
E Write two different equations for each and solve for z.
Model and solve for y in each equation.
Functions, Inputs and Outputs
A What are the missing inputs and outputs?
B In your copy, create an input/output table to represent each of these scenarios. Record the function rule clearly at the top. Choose three different inputs each time.
1. You are baking cookies, and each tray holds 12 cookies. Work out how many cookies (output) you will bake based on a certain number of trays (input).
2. A snail is crawling at a speed of 3cm per minute. Work out how far the snail has crawled (output) after a certain number of minutes (input).
3. You are saving €5 each week. Work out how much money you will have saved after a certain number of weeks.
4. You are ordering pizzas for a party, and you estimate that one pizza is needed for every 4 people. Work out how many whole pizzas are needed for a certain number of people.
Try this! In your copy, create a suitable input/output table to represent each of these scenarios. Record the function rule clearly at the top.
1. Tom’s Taxis charges €5 at the start of the journey, and then €2 per km thereafter Work out the cost of three different journeys measuring a certain number of km.
2. A plumber charges a standard €50 call-out fee, and then €30 per hour thereafter Work out the charge for three different jobs lasting a certain number of hours.
3. Mia is collecting cards, and she gets 4 new cards every day. Work out how many cards she will have after a certain number of days, if she already had 10 cards to begin with.
Number of players: 2–6
Spin and Place
You will need: mini-whiteboard and marker per player, 0–9 spinner, pencil and paper clip
● Each player draws this place value grid on their mini-whiteboard.
● Each player, in turn, spins the 0–9 spinner and writes the digit spun into one of the empty places on their place value grid.
● When all of the places are full, each player reads their number aloud.
● The player with the greatest number scores a point.
Play for the time allocated by your teacher.
● The player with the highest score when the time is up wins the game.
Variations
1. The player with the least number scores a point.
2. Instead of making a 5-digit whole number, make a 5-digit number with three decimal places: tenths, hundredths and thousandths.
3. Nice or Naughty: Each player, on their turn, writes the digit spun into one of the empty places on their place value grid OR instructs another player to write that digit into a specific place on their place value grid. In doing so, they can choose to be nice or naughty to their fellow players!
Put a 2 in the ten-thousands place on your grid.
Number of players: 2–6
Chance Calculations – Subtraction
You will need: mini-whiteboard and marker per player, 0–9 spinner, pencil and paper clip
● Each player draws these boxes for a calculation on their mini-whiteboard.
● Each player, in turn, spins the 0–9 spinner ten times, and then arranges the digits spun in their empty boxes.
● When all of the boxes are full, each player calculates their answer.
● The player with the greatest difference (answer) scores a point.
● The player with the highest score when the time is up wins the game.
Variations
1. The player with the smallest difference (answer) scores a point.
Don’t forget – the greater number needs to be on top.
2. Add a decimal point in both rows to create a number with three decimal places.
Number of players: 2–6
Chance Calculations – Multiplication
You will need: mini-whiteboard and marker per player, 0–9 spinner, pencil and paper clip
● Each player draws these boxes for a calculation on their mini-whiteboard.
● Each player, in turn, spins the 0–9 spinner five times, and then arranges the digits spun in their empty boxes.
● When all of the boxes are full, each player calculates their answer
● The player with the greatest product (answer) scores a point.
● The player with the highest score when the time is up wins the game.
Variations
1. The player with the smallest product (answer) scores a point.
2. Multiplication with Decimals: Agree in advance to create a number on the top row with two or three decimal places, and add a decimal point accordingly.
Chance Calculations – Division
Number of players: 2–6
You will need: mini-whiteboard and marker per player, 0–9 spinner, pencil and paper clip
● Each player draws these boxes for a calculation on their mini-whiteboard.
● Each player, in turn, spins the 0–9 spinner four times, and then arranges the digits spun in their empty boxes.
● When all of the boxes are full, each player calculates their answer
● The player with the greatest quotient (answer) scores a point.
● The player with the highest score when the time is up wins the game.
Variations
1. The player with the smallest quotient (answer) scores a point.
If a calculation results in a remainder, you can ignore it for the purposes of scoring.
2. Division with Decimals: Agree in advance to create a dividend with one or two decimal places, and add a decimal point accordingly.
Let’s Look Back 1
1. Write the number shown in expanded form.
3. Ring the number that is not a factor of 12. 1 4 8 12
5. Work out the value of 4aif a= 9.
7. Round 40,789 to the nearest thousand.
9. 3,879 + 84,209 = Estimate: Solve:
1. If Jay used 297g from a 1kg bag of sugar, what (a) decimal and (b) fraction has he used?
3. When two whole numbers are added, the order does not affect the answer What property of addition does this describe? ✓
(a) Associative property
(b) Commutative property
(c) Identity property
6. Round to the nearest euro. €
7. Mia totalled the length of the side(s) of this equilateral triangle to find that the perimeter was 36cm. Write an equation to represent this.
10. Estimate the answer to 24,556 + 2,374, and then solve using the column method.
Estimate: Answer:
2. Model and solve: Amy sold 88,703 coffees this year Last year, she sold 59,772. How many more coffees did she sell this year?
4. Which of these are expressions? ✓
(a) 4 × 5 (b) 3 + 7 = 10
(c) 20 – 7 (d) 12 – 4 = 2 × 4
6. Write and solve the matching number sentence. =
8. Always, sometimes or never true? 9 is a composite number.
10. Write <, > or = to make these true: (a) 4·27 47 (b) 1279 3 (c) 821·45 821·5
2. Tom and Anya picked 11·6 kg of potatoes on their uncle’s farm. They want to share the potatoes equally. How much should each of them get?
Estimate: Solve:
4. Write 3 tenths and 7 thousandths in decimal form.
5. Solve these related calculations.
(a) 4 × 60 =
(b) 240 ÷ 4 =
(c) 2,400 ÷ 4 =
8. Find the fifth multiple of 7.
9. Lexi’s printer is broken. When she tried to print 9 pages, it printed 3. When she tried to print 12 pages, it printed 4. When she tried to print 3 pages, it printed 1. Write a function rule to describe the error that the printer is making
1. 22,457 people attended a concert on Friday. 29,078 people attended on Saturday. Calculate the total number of people who attended the concert.
3. Write the missing number 7 + 20 + 40,000 + 900 + = 49,927
5. Write two different equations to match the branching bond. (a) = (b) =
y 44 12
2. Last year, Peter earned €35,909. This year, he earned €40,798. Calculate the difference in earnings between the two years.
4. ✓ the prime numbers. 12 7 21 29
6. Jay has made 2,807ml of apple juice. He wants to divide it equally between 7 bottles. How much juice will be in each bottle?
Estimate: Solve:
7. Put these in order from greatest to least: 3 10 , 33, 333 1,000, 300
8. Write if these statements are always (A), sometimes (S) or never (N) true.
(a) When you add three whole numbers, the answer is an even number (b) When a number is multiplied by zero, the answer is the same whole number. (c) When three whole numbers are added in a given order, the way they are grouped does not affect the answer.
Try this!
1. Fill in the missing digits in the answer boxes.
2. 123,000 < < 124,800
Insert the missing number, if…
● when rounded to the nearest thousand, the number is 125,000
● the sum of its digits is 13.
3. What number is 17 thousandths smaller than 18?
5. What 5-digit number might I be? Write three possible answers.
● I have 4 thousands.
● I have twice as many ones as tens.
● I have more ten thousands than ones.
● The sum of my digits is 25.
7. It costs €30 per hour for four players to rent a tennis court. The players can also rent racquets and balls for €2·50 per person, no matter how many hours they play for Complete the input/output table to show the cost to 4 friends of renting racquets and balls, and a court for 1 hour, 2 hours and 3 hours.
4. A marathon is 42195km in length. What is the distance of 100 marathons? km
6. Two 5-digit numbers have a difference of 8. When both are rounded to the nearest ten thousand, the difference between them is 10,000. What could the numbers be?
Community Centre Fundraiser
COMMUNITY CENTRE
Solve these. Use a calculator if necessary to help you.
I’m knitting toys to sell! To calculate the amount raised (R) from each toy, find the difference between the cost to make it (C) and the selling price (P).
We’re raising money to buy new equipment for the community centre.
1. Look at Lexi. Write the equation to calculate the amount raised (R) from each toy. R =
2. Calculate the amount raised from each toy.
(a) € (b) € (c) € (d) €
3. Lexi sold 3 mice, 5 lions, 2 pigs and 1 tiger How much did she raise in total? €
4. Lexi’s aunt Mei made a donation to the fundraiser that was 10 times the amount that Lexi raised. Write an expression to represent the value of Mei’s donation.
5. Calculate the value of Mei’s donation. €
6. Calculate the total that Lexi and Mei gave to the community centre. €
Solve these.
1. Dara and his mum collected €980 in donations from 7 shops, who each donated the same amount.
(a) Write an expression to represent this.
(b) Calculate the amount donated by each shop. €
Mum and I asked local businesses for donations in exchange for putting up an ad in the community centre.
2. TheDailyEchonewspaper donated €3 for every newspaper they sold that day.
(a) Write an expression to represent this.
(b) 461 newspapers were sold. Calculate the amount donated by the DailyEcho €
3. Use your preferred method to estimate the total donations from businesses. €
4. Calculate the total donations from businesses. €
5. Calculate the difference between your estimation and the actual amount. €
Let’s talk!
Discuss and compare with others: What estimation strategies were used for ?
If different strategies were used, which one was closest to the actual amount? Why do you think this was?
D Look at the table and answer the questions.
1. Write each amount below in digits.
Amounts raised in the first three months of the year
(a) January One thousand and forty-two euros and sixty-two cents
(b) February Five hundred and forty-six euros and twenty cents
(c) March Two thousand, four hundred and fifty-six euros and thirty-one cents €
2. Look at Mia and Jay. Complete the place value grids.
We made these place value grids to display the total amount raised each month.
3. Calculate the total amount raised over the first three months. €
4. The community centre hopes to raise €10,000 by the end of the year. How much more do they need to raise? €
5. Estimate, to the nearest hundred, how much needs to be raised each month for the reminder of the year to reach the target.
E Let’s create!
Design a model that the community centre could use to show the amount that they have left to raise.
I think an open number line would work. I think a bar model would be clearer I think I could draw a graph.
Identifying Angles
A Which of these shapes has…
CD
1. 2 obtuse angles and 2 acute angles?
4 right angles? 3. 6 obtuse angles?
B Let’s create!
In your copy, draw
1. a shape that has only acute angles
2. a shape that has only right angles
3. a shape that has only obtuse angles.
D Maths eyes
1. an acute angle?
2. an obtuse angle?
3. a reflex angle?
4. a right angle?
Let’s talk!
Let’s talk!
Look at Lexi. What do you think? Explain why.
Which angle marked on the photo is…
I found an acute angle. It’s a little bit less than 90°. I think it’s 80°.
I think it’s impossible to draw a shape with only reflex angles.
Find another angle in the photo in D above How would you describe the angle?
Estimate how many degrees it has.
Try this! Look at what Dara and Jay are saying. How can you work out the size of each angle?
Use the same approach that Jay used to work out the total.
The three angles all look the same size.
Each one can fit into a right angle.
Measuring Angles
A Look at the protractors below. What is the measure of each angle?
Let’s talk!
For each triangle in C below, take turns with a partner to…
● identify the smallest angle and the largest angle
● estimate the measure in degrees of each angle
● say what type of triangle it is.
C Use your protractor to measure the angles in each triangle.
D Measure the size of each angle.
Try this! Look at what Mia and Jay are saying
How many angles are there?
Find the sizes of all the angles. How many did you need to measure?
What do you notice about your answers?
I can see 2 angles. If I measure one, I can work out the size of the other without measuring.
see 3 angles.
Constructing Angles
A In your copy, use a protractor to construct the angles below. Label each angle with its size.
Let’s talk!
Which of the angles in A above did you find easiest to draw?
Which did you find hardest? Discuss your answers with a partner. What different strategies did you use?
C Maths eyes
For the reflex angles, I just drew a straight angle first and then added the second angle needed to create the total reflex angle.
Sometimes, I calculated the degrees in the smaller inner angle, drew that angle and then marked the outer angle instead.
Triangles in bridges
1. What types of triangles can you see in the photos above?
2. Design a bridge using triangles.
3. Use sticks or straws to build your bridge.
For more on angles in bridges, see page 163.
D In your copy, use a ruler, protractor and pencil to draw the triangles below. Include a 50° angle in each triangle. Measure and label the angles in each triangle.
1. Scalene triangle
2. Isosceles triangle
3. Right-angled triangle
E Let’s investigate!
Is it possible to draw a triangle with two obtuse angles?
Use a ruler and pencil or concrete materials to investigate.
Try this! Use a ruler, pencil and protractor to draw an equilateral triangle.
An isosceles triangle has two sides the same length and two angles the same size.
In an equilateral triangle, all the angles are 60° and all the sides are the same length.
Triangles and Quadrilaterals
A In your copy, construct triangles with the following angles.
B Write the letter for each shape in the correct section of the Venn diagram.
Has a right angle Has parallel sides
I've done one for you. I wrote D outside both circles, because it doesn’t have a right angle and it doesn’t have any parallel lines.
C Let’s create!
Create your own Venn diagram and use it to classify the shapes in B above. You could use type of angle, number of sides, or your own ideas.
D Are these always (A), sometimes (S) or never (N) true?
1. A triangle has 3 acute angles.
2. A rectangle has 4 right angles.
3. A square is a type of rhombus.
4. A parallelogram is a type of rectangle.
5. A square can be divided into 2 equilateral triangles.
6. A right-angled triangle can be divided into 2 smaller right-angled triangles.
E Which of these shapes can be cut into two identical triangles?
✓ the shapes below that can be cut into two identical triangles
Try this! Is it possible to draw a quadrilateral that has a reflex angle?
Draw or use materials to test your ideas.
Angles in a Triangle
A Use what you know about the angles in a triangle to calculate the size of the missing angle without measuring.
Let’s talk!
What is the same and what is different about the triangles in A above?
C A triangle has an angle of 30°. What might the other two angles be, if the triangle is… 1. isosceles? ° ° 2. right-angled?
3. scalene?
Let’s talk!
How many answers can you think of for C above? Which type of triangle has (a) the most and (b) the least possible answers? Explain why. Look at the triangle below. What could the missing angles be? Who do you agree with?
I think they are 90° and 40°.
I think they are both 65°. I think the missing angles are 100° and 30°.
E Let’s investigate!
1. Do you agree with Jay about the angles in a square? Explain why.
2. Investigate if the same is true for a rectangle.
If I join two corners of a square, I get two triangles. The total of the angles in each triangle is 180°. Therefore, the total of the angles in the square must be double 180°.
3. Use what you found in questions 1 and 2 to work out the total of the angles in a quadrilateral. Draw another quadrilateral to test your idea.
4. Use the same approach that Jay used to work out the total of the angles in a pentagon.
Calculating Angles
A ✓ the number sentence that describes each diagram.
1.
(a) 50° + w= 360°
(b) 50° + w= 180°
(c) 50° + 180° = w 2.
3.
(a) 70° + 40° +y= 180°
(b) 70° + 40° +y= 360°
(c) 70°– 40° –y= 360° 4.
(a) 180° – 45° = x
(b) 45° – 360° = x
(c) 360 – 45° = x
(a) 360° – 100°= z
(b) 360° – 100° – 110° = z
(c) 360° + 100°+ 110° = z
Write another number sentence to describe each diagram in A above.
C Find the missing angle in each diagram in A above.
Let’s talk!
There are 2 missing angles in this diagram. What strategy could you use to find both of them? Discuss your ideas with a partner
E Use the strategy you worked out to find the missing angles in D above. A B
Let’s talk!
The diagram is made of 3 straight lines. How could you find the missing angles? Discuss your ideas with a partner
G Use the strategy you worked out to find the missing angles in above.
Try this! Use what you know about parallelograms to work out the missing angles.
The Circle
A Maths eyes Parts of a circle
AB
Which photo above shows…
1. a centre? 2. an arc?
3. a sector? 4. a circumference?
Let’s talk!
Look at Jay. What do you think? Explain why.
I think all the images in A above show more than one of the parts of a circle.
C Use a ruler to measure the (a) radius and (b) diameter of each circle.
(a) Radius = cm (b) Diameter = cm
(a) Radius = cm (b) Diameter = cm
(a) Radius = cm (b) Diameter = cm
(a) Radius = cm
(b) Diameter = cm
Let’s talk!
Look at your measurements in above. What do you notice?
Discuss any patterns you notice with a partner.
Try this!
1. In your copy, use a compass and ruler to draw an animal made up of circles, sectors and segments.
2. ✓ the parts of a circle below that you can find in your drawing: centre arc radius diameter circumference
Circle Relationships
A Work out the diameter of each circle.
These circles haven’t been drawn to scale, so you will need to work out your answers.
Diameter: cm
Diameter: cm
B Work out the radius of each circle.
Diameter: cm
Diameter: cm
Let’s talk!
Look back at activities A and B above What did you need to do to work out the answers? Explain why you needed to do this. Estimate the circumference of each circle based on the information that you have
D Let’s create!
1. Construct a circle with a diameter of 4cm in the drawing box.
2. Estimate the circumference of your circle. cm
3. Use a piece of string to measure the circumference of your circle. cm
Let’s talk!
How close were your estimate and measurement in activity D above? Why were they not exactly the same?
The circumference is slightly more than 3 times the diameter.
Equivalent Fractions and Simplifying Fractions
A What fraction of each shape is coloured? Write it as (a) the fraction shown and (b) a fraction in its simplest form.
‘Simplest form’ means expressed using the smallest numbers.
B Calculate an equivalent fraction for these by multiplying by a fraction equivalent to 1.
C Simplify these by dividing by a fraction equivalent to 1.
To simplify a fraction, find an equivalent fraction with a lower numerator and denominator.
D Find the missing numerators or denominators in these equivalent fractions.
You can use your fraction wall to help with these!
Let’s talk!
Look at Jay and Lexi. Do you agree with Lexi? Explain why. Colour in 3 12 of this
That’s not possible!
Try this! Find the missing numerators or denominators in these equivalent fractions.
Improper Fractions and Mixed Numbers
A Express each amount as (a) an improper fraction and (b) a mixed number.
B Change these mixed numbers to improper fractions.
C Change these improper fractions to mixed numbers.
D Express each value, represented at the letter below, as a mixed number and as an improper fraction.
Try this!
1. A water dispenser contains 4 1 2 litres of water A glass holds 1 4 of a litre. How many glasses can be filled from the dispenser?
2. A cruise ship is 1 3 km long How many such cruise ships would be equivalent to 4km?
3. How many 1 8 kg weights would be needed to balance the box?
Comparing and Ordering Fractions
A Let’s talk!
What patterns do you notice in the number line below? Prove your reasoning
I notice a growing pattern in the numerators. I see a symmetrical pattern in the denominators.
B What values are missing from each number line? Write each value in its simplest form.
Let’s talk!
Look at Jay’s bar models. Do you agree with Jay? Explain why.
I sketched
I think the fraction wall would be useful for this.
Write <, > or = to make these true.
I might use my fraction wall for some of these.
I might use common denominators.
E Put each group of fractions in ascending order (from smallest to greatest).
F Model and solve these.
1. Carton A contains 5 8 l of juice and carton B contains 6 10 l of juice. Which carton contains more, A or B?
2. In the water relay, the Red Team filled 7 12 of their container, and the Blue Team filled 7 8 of an identical container Which team’s container was more full?
3. Mia and Jay checked out copies of the same book from the library. So far, Mia has read 8 9 of hers, and Jay has read 11 12 of his. Who has read more pages, Mia or Jay?
4. On the cross-country course, Ali has run 5 6 of the course, Chloe has run 3 4 of the course, and Seán has run 7 9 of the course. So far, who has run (a) the most? (b) the least?
5. Tim, Jan and Lee ordered a different pizza each. After they had finished eating, there was 3 12 of the veggie pizza left, 1 6 of the cheese pizza left and 2 9 of the ham pizza left. If Lee ate the most and Jan ate the least, what type of pizza did each person order?
Tim: Jan: Lee:
Try this! What values are missing below?
I think there is more than one answer for some of these.
Calculating Amounts Using Fractions
A For each of the bar models, write a matching fraction sentence and solve.
If = 250, then =
If = 210, then =
B Use the comparison bar models to help you solve these.
1. At an animal shelter, 4 7 of the animals are cats. The rest are dogs. If there are 12 more cats than dogs, how many
(a) dogs are there?
(b) animals are at the shelter altogether?
2. Jay has 1 6 of the collector cards. Mia has the rest. Mia gives 16 cards to Jay so that they can both have the same amount. How many
(a) more cards did Mia have than Jay?
(b) cards did they have altogether?
Animal Shelter
3. Dara has saved 1 4 of the money that Lexi has saved. Lexi has saved 1 3 of the money that Mia has saved. Together, they have saved a total of €170 How much has Mia saved?
C Use the bar models to help you solve these.
1. Of the 224 books in the library, 3 4 are fiction and the rest are non-fiction. How many fiction books are there?
224
2. Of the 369 pupils in a school, 1 3 are boys and the rest are girls. How many girls are there in the school? 3
3. At a concert, 5 12 of the audience were adults. If there were 105 adults, what was the total number in the audience?
105
4. In a survey, 7 9 of the people said that they preferred popcorn to nuts. If 58 people preferred nuts, how many people preferred popcorn?
Total boo s:
224
Non- iction Fiction?
Total pupils: Boys Girls?
Adults: Children
Total audience? Popcorn? Nuts:
People surveyed
Try this! Draw models in your copy to help you solve these.
1. 2 11 of the strawberries in a box had to be thrown away If there were 99 strawberries at first, how many… (a) were thrown away? (b) were left?
2. A farmer sold 2 9 of her 126 sheep How many did she have left?
3. Four fifths of the children in Mathstown NS walk to school. How many children are there in the school if 256 children walk to school?
4. There are 78 people in a cinema theatre. 1 6 of the people are 18 years or older How many people are under 18?
5. Dara is saving for a new gaming console and he has saved 7 10 of the money. If he has €280 saved, what is the price of the gaming console? €
Adding Fractions
A Write and solve the matching addition expressions.
B Model and solve these. Remember to simplify the answers fully.
1. 2 3 + 3
C Model and solve these.
1. Lexi spent 4 9 of her money on runners and 1 3 of her money on a pair of jeans. What fraction of her money has she spent?
2. Mia walked 2 3 4 km to the library and then a further 1 7 12 km to the playground. What distance did she walk altogether? km
3. Dara and Lexi ordered three pizzas between them. Dara ate 1 1 8 pizzas and Lexi ate 1 3 4 pizzas. How much of the pizzas did they eat altogether? pizzas
4. On Monday, Jay drank 1 4 5 litres of water On Tuesday, he drank 1 9 10 litres of water How much water did he drink over the two days? litres
5. One parcel weighs 2 5 12 kg and another weighs 3 5 6 kg What is their combined (total) weight? kg
Try this! For each equation, identify three proper fractions with different denominators that equal the given total.
Subtracting Fractions
A Write and solve the matching subtraction expressions.
The fractions marked with X are to be subtracted.
B Model and solve these.
1.
I could also think of subtraction as finding the difference.
C Model and solve these.
At Cathy’s Café, brownies are baked in a tray and then cut into 12 individual brownies, all equal in size. One Saturday morning, Cathy baked five trays of brownies.
1. By 12:00, 1 3 4 trays were sold. How many trays of brownies were left? trays
2. By 14:00, another 1 5 12 trays were sold. How many trays of brownies were left? trays
3. By 16:00, only 2 3 of a tray was left. How many trays of brownies were sold after 14:00? trays
4. Work out the number of individual brownies that were sold… (a) by 12:00 (b) between 12:00 and 14:00 (c) between 14:00 and 16:00 (d) in total at 16:00
Try this! Find the missing values that would make each of these true. Write each answer in ones and fractions.
Multiplying Fractions
A Write and solve the matching multiplication expressions.
B Write each of these as a multiplication expression and solve.
C Model and solve these. Remember to simplify wherever possible.
D Choose your own approach to model and solve these.
1. Four children each ate 5 8 of a pizza.
(a) How much pizza did they eat in total?
(b) If there were three whole pizzas at first, how much pizza is left?
2. Five children each drank 2 3 of a carton of apple juice.
(a) How much juice did they drink altogether?
(b) If there were four full cartons of apple juice at the beginning, how much juice is left?
Let’s talk!
Are these always, sometimes or never true?
● When any number is multiplied, the result will be a larger number
● A fraction with the same numerator and denominator is equal to one.
● In fractions, the numerator is smaller than the denominator.
Multiplying using Related Facts
A Calculate the answers for each group.
1.
(a) 320 × 10 =
(b) 320 × 20 =
(c) 320 × 5 =
(d) 320 × 25 =
2.
(a) 2 × 023 = (b) 4 × 0.23 =
(c) 10 × 023 =
(d) 14 × 023 =
In each group, use the previous product(s) to help calculate the next product.
3.
(a) 4 × 150 = (b) 10 × 150 =
(c) 14 × 150 = (d) 14 × 140 =
B Solve these using strings of related facts.
1. 16 × 021 =
2. 23 × 52 =
3. 34 × 1.52 =
4.
(a) 2 × 032 = (b) 10 × 0.32 =
(c) 13 × 032 = (d) 13 × 0032 =
For the first one, I’ll start with 6 × 0 21, then 10 × 0.21, then
C Solve these using strings of related facts.
1. How much for 18 pencils costing €065 each? €
2. If one lap of a running track measures 095km, what is the distance of 15 laps? km
3. A large tank is being filled with water at a rate of 62 litres per minute. How many litres will there be in the tank after 2.5 minutes? l
4. A box weighs 0.18kg. What is the weight of 35 of these boxes? kg
Try this!
1. The children have been asked to arrange these digits to make two 2-digit numbers, which when multiplied together result in the greatest product. ×
8 6 9 7
Look at the children. What do you think? Use related facts to prove your reasoning
I think 98 × 76 will result in the greatest product.
I think 97 × 86 will result in the greatest product.
I think those two products will be the same, because they are using the same digits. 3 1 4 2
2. Arrange each group of digits below to make two 2-digit numbers, which when multiplied together result in the greatest product.
6 4 7 5
4 0 6 2
Multiplying by 2-digit Whole Numbers
A Let’s talk!
Look at the calculations below. Which ones do not look correct? Explain your reasoning. Now, check them with a calculator.
B In your copy, use the column method to solve the incorrect calculations in A above.
C Let’s talk!
Estimate a reasonable answer for each of the calculations in D below. Explain your strategy.
E Calculate the answers to these. Choose your own strategy.
1. What is the total length of 18 planks of wood measuring 1.75 metres each? m
2. A car dealer makes €1,450 profit on each sale of a particular car How much profit would be made on the sale of 25 such cars? €
3. A school needs to purchase 27 books costing €885 each. What is the total cost of the books? €
4. What is the total weight of 34 packages if each one weighs 3125kg? kg
For question 4, I’ll multiply this as if they are both whole numbers. Then I’ll put the decimal in the answer so that there are two decimal places.
5. A theatre can seat 1,145 people. How much money would they make in ticket sales for a sold out show, if the tickets cost €32 each? €
F Use a calculator to check your answers to E and D without using the × button.
G What digit does each letter represent?
H Let’s play!
Number of players: 2–6
Chance Calculations – Multiplication
You will need: mini-whiteboard and marker per player, 0–9 spinner, pencil and paper clip
● To start, on their mini-whiteboard, each player draws these boxes for a multiplication calculation (3-digit × 2-digit).
● Each player, in turn, spins the spinner and writes the number spun into one of their boxes. When all of the boxes are full, each player calculates their product.
● The player with the highest product wins.
Variations
1. Play as above, but draw six boxes to start (4-digit × 2-digit).
2. Play as above, but agree in advance that the top number will have 1, 2 or 3 decimal places
Try this! Lexi and Jay were playing chance calculations. What might their decimal numbers have been?
I multiplied a decimal number with ones, tenths and hundredths by 15. The product, when rounded to the nearest ten, was 90.
I multiplied a decimal number with three digits, and one decimal place, by 25. The product, when rounded to the nearest ten, was 550.
Multiplying by Decimal Numbers
A Let’s investigate! Decimal estimation strategies
Which strategy gives a more accurate answer? Use your calculator to check.
Estimate a reasonable answer for each of the calculations in below. Explain your strategy.
C Use a calculator to solve these.
1. What is the area in square metres of a room measuring 48m by 32m? sq m
3. If fabric costs €1215 per metre, what is the cost for 26 metres? €
2. If a delivery worker is paid €1550 per hour, how much is he paid for 65 hours?
4. If the cost of postage is €220 per kg, how much would it cost to post a parcel weighing 3 and a quarter kg?
D Let’s play! Chance Calculations – Multiplication
Number of players: 2–6
You will need: mini-whiteboard, marker and calculator per player, 0–9 spinner, pencil and paper clip
● To start, on their mini-whiteboard, each player draws these boxes for a multiplication calculation.
● Each player, in turn, spins the spinner and writes the number spun into one of their boxes. When all of the boxes are full, each player uses their calculator to calculate their product.
● The player with the highest product wins.
A Solve
these.
Dividing by 2-digit Numbers
I can use a multiplication fact to solve the matching division fact.
That’s because they are the inverse of each other
1.
(a) 24 × 10 =
(b) 24 × 5 =
(c) 24 × = 360
(d) 360 ÷ 24 = 2.
(a) 10 × 21=
(b) 20 × 21 =
(c) 21 × = 315
(d) 315 ÷ 21 = 3.
(a) 25 × 10 =
(b) 25 × 4 =
(c) 25 × = 450
(d) 450 ÷ 25 = 4.
(a) 10 × 45 =
(b) 20 × 45 =
(c) 45 × = 855
(d) 855 ÷ 45 =
B Let’s investigate!
Investigate if factors will work to solve each of the division calculations in A above
C Let’s talk!
For each question in D below:
● What multiple of ten do you think the answer will be closer to: 10, 20 or 30?
● Do you think the answer will be greater than (>) or less than (<) that multiple of ten?
Factors can also be used to divide. In 1.(d), since 6 and 4 are factors of 24, I’ll divide 360 by 6, and then divide the quotient, or answer, by 4.
D Choose your own preferred way to model and solve these.
1. Eggs are being packed in cartons of 12. How many cartons are needed for 156 eggs?
2. 300 chairs are being placed in rows for an event. How many rows can be made if there are 25 chairs in each row?
3. This tray can hold 15 ice cubes. How many of these trays are needed for 345 ice cubes?
4. Pauric swam 28 lengths of a swimming pool, a total distance of 700 metres. What is the length of the swimming pool in metres?
5. A school purchased 16 wobble boards and had €92 change from €300. What was the cost of each wobble board? €
6. A minibus can take 14 people. How many are needed for 160 people?
7. It is estimated that 130 parents will come to the parents’ evening, and that they will eat 3 biscuits each. How many packets of biscuits are needed if each packet holds 12 biscuits?
Use a calculator in more than one way to check your answer to each calculation in D above.
Dividing by 2-digit Numbers: Estimating and Calculating
Do you agree with Lexi? If yes, what factor pairs would be the most efficient to use?
C Estimate and ring the multiple of ten that you think will be closest to the answer for these.
I’ll quickly multiply the divisor by 10, then 20, and so on, to see which is closest to the dividend.
1. Yoghurt is sold in packs of 12 pots. How many packs of 12 can be made from 236 pots?
2. A medicine bottle holds 400ml. If a dose is 15ml, how many full doses does the bottle contain?
3. A bag contains 23 fun-size bars. If there are 450 bars, how many bags can be filled?
4. How many pieces of ribbon measuring 25cm can be cut from a length of ribbon measuring 4·5 metres?
5. 334 pupils and staff are going on a school tour How many 52-seater buses are needed to carry them all?
6. Catriona is saving for a new games console and games. How many weeks will it take her to save €780 if she plans to save €15 every week?
I think factors could be used for the division calculations in
I will look at the tens: for question 5, 33 tens ÷ 5 tens = 6
D For each of the calculations in C above, use a different estimation strategy to refine the estimates further so that they are closer to the actual answer.
E Use your calculator to calculate the actual answers for C above, choosing the most appropriate whole number where necessary.
F Let’s talk!
Which of the answers in E above were between two whole numbers? Which whole number did you choose as the answer? Explain your reasoning
Division as Sharing using Concrete Materials
A Use base ten materials to model and solve these. 1. 182 ÷ 14 = 2. 165 ÷ 15 = 3. 132 ÷ 11 = 4. 144 ÷ 16 =
Let’s talk!
Use a calculator to check if the quotients (answers) to A above are correct. Can you do this without using the ÷ button? How? How might you use factors to solve the calculations in A ? Which one(s) cannot be easily solved using factors? Explain why.
We could factorise each divisor to make two single-digit divisors that are easier to divide.
Use base ten materials to model and solve these.
159 ÷ 11 = 2. 137 ÷ 12 =
171 ÷ 15 = 5. 187 ÷ 14 =
I don’t think that will work with all of the divisors.
170 ÷ 13 =
200 ÷ 16 = 7. 186 ÷ 15 = 8. 159 ÷ 12 =
Let’s talk!
What do you notice is different about the calculations in above in relation to the calculations in ?
How might you use a calculator to check the quotients (answers)?
E Use base ten materials to model and solve these.
1. A local lottery prize of €308 is being shared among 14 people. How much will each person get? €
2. A length of rope measuring 105m long is to be divided up into 15 equal pieces to make skipping ropes. What will be the length of each piece of rope? m
3. Kathy wants to share €960 equally among her 16 grandchildren. How much money will each grandchild receive? €
4. A class of 25 children are visiting an art gallery. Up to two adults are allowed to accompany them for free. If the total cost is €275, how much does each ticket cost? €
Read the questions carefully. Think –what do I need to do?
136 ÷ 11 =
Division as Repeated Subtraction
A Solve these using a calculator and repeated subtraction. 1. 210 ÷ 42 = 2. 342 ÷ 57 = 3. 324 ÷ 36 = 4. 688 ÷ 86 =
Let’s talk!
Some calculators have a shortcut for doing calculations. On a calculator, input 120 – 15 , and then press = = = = .
Do this again, but this time count how many times you have to press = until the display becomes 0 .
C Solve these using a calculator in the way described in above.
D Solve these using a calculator.
Let’s talk!
What do you notice about the calculations in D above? Is there a more efficient way to solve them? Could chunks of the divisor be subtracted each time?
For number 1, I subtracted ten chunks of 18 as 180 in one go.
F Solve the calculations in D above again, but this time subtract chunks of the divisor each time. In your copy, show what you did.
Try this!
1. A painter needs 500 litres of paint to paint a new house. How many 15-litre containers does he need to buy?
2. The Chip Shop has received a delivery of 150kg of potatoes. If they use about 17kg of potatoes to make chips each day, for how many full days should the delivery last?
3. A baker baked 96 brown scones and 72 fruit scones, which he packed in boxes with a mixture of 12 scones in each. How many boxes did he fill?
4. A florist has a 5m roll of ribbon. How many pieces of ribbon measuring 23cm long will she be able to cut from the roll?
Use a calculator to check to check your answers, but without using – or ÷ .
Mixed Operations
Maths Choice Board: Choose 3 of these tasks to explore and solve. Present your solutions in different ways.
Years and Years
Run for It!
Patrick 156 months old
Sadie 168 months old
Raheem 540 months old
Ocean Town School is holding its annual crosscountry competition. The plan is to have 14 runners on each team. 230 pupils have signed up to take part.
1. How many teams will there be?
Willow 75 months old
1. Work out each person’s age in years.
2. Work out your age in months.
Show Business
FOR ONE NIGHTONLY! EPIC THEATRE PRESENTS… ROBIN HOOD
TICKETS:ADULT€28, CHILD €16
CAPACITY:1,214 SEATS
If the show was sold out and there were 356 children in the audience, how much money was made from…
1. child ticket sales?
2. adult ticket sales?
3. total ticket sales?
Big Spender!
A motorbike costs €2,350. A car costs 19 times as much. How much would the motorbike and the car cost in total?
2. Will there be some participants who are not on those teams? How many?
3. What could the school do to make sure that every runner is on a team?
Emma’s Dilemma
6 for €1·70 12 for €2·90 18 for €3·90
I’ll need 170 eggs to bake my cakes next week.
1. How many boxes will Emma need to buy if she chooses boxes of…
(a) 6 eggs? (b) 12 eggs? (c) 18 eggs?
2. Should she buy boxes of 6, 12 or 18 to…
(a) have the least number of eggs left over?
(b) get the best value option?
3. Special offer! Three boxes of 12 eggs for €8. Which is the best value option now?
School Trip
595 pupils and 23 adults need transport for a school trip If each coach holds 52 passengers, how many…
1. coaches are needed?
2. spare seats will there be?
Eighths spinner
Ninths spinner
Tenths spinner
Twelfths spinner
Who Has More? – Fractions
Number of players: 2
You will need: mini-whiteboard and marker per player, pencil and paper clip to use with the spinners above
● Each player, in turn, spins the eighths spinner and writes the fraction spun on their mini-whiteboard.
● The player who thinks they have the larger fraction in that round must prove it by sketching models; if they are correct they score a point.
● Repeat, using the next spinner, and so on.
● The player with the highest score when the time is up wins the game.
Variations
Play for the time allocated by your teacher.
1. Comparing Using Different Spinners: Option A – Each player chooses a spinner at the beginning, and spins that spinner throughout the game. Option B – The players choose two spinners at the start, and then swap spinners for each round. (Tip: Play both options to see which one the players prefer.)
2. Adding Fractions: Each player spins twice (using two different spinners), and totals the two numbers spun. The player with the greatest total each time scores a point.
3. Subtracting Fractions: Each player spins twice (using two different spinners), and finds the difference between the two numbers spun. The player with the greatest difference each time scores a point.
4. Multiplying Fractions: Each player spins both a fraction spinner of their choosing and the 0–9 spinner (or else throws a dice), and finds the product of both amounts. The player with the greatest product each time scores a point.
Target 100,000
Number of players: 2–6
You will need: mini-whiteboard and marker per player, 0–9 spinner, pencil and paper clip
● Each player draws these boxes for a calculation on their mini-whiteboard.
● Each player, in turn, spins the 0–9 spinner five times, and then arranges the digits spun in their empty boxes.
● When all of the boxes are full, each player calculates their answer
● Each player keeps a running total from their turns.
● The first player to reach or pass 100,000 (or the highest score when the time is up) wins the game.
Variations
To keep a running total, add the product from each turn to the total of your previous products.
1. Target Zero from 100,000: Play as above, but on each turn, the product is subtracted from 100,000, and so on. The first player to reach or pass zero (or the lowest score when the time is up) wins the game.
2. Target 50,000: Play as above, but the first player to reach or pass 50,000 (or the highest score when the time is up) wins the game.
Number of players: 2–6
You will need: mini-whiteboard and marker per player, 0–9 spinner, pencil and paper clip
● Each player draws these boxes for a calculation on their mini-whiteboard.
● Each player, in turn, spins the 0–9 spinner five times, and then arranges the digits spun in their empty boxes.
● When all of the boxes are full, each player calculates the answer using a strategy of their choosing
● The player with the greatest quotient (answer) scores a point.
● The player with the highest score when the time is up wins the game.
Variation
● The player with the smallest quotient (answer) wins.
If a calculation results in a remainder, you can ignore it for the purposes of scoring.
6. What fraction of this shape is coloured? ✓ all that apply. 2 3 9 12 3 4 4 5
8. The total value of two cars is €58,478. If the more expensive one is valued at €31,098, what is the value of the other car? €
10. Look at the shapes. ✓ the shape that can be cut into two identical triangles. (a) (b)
2. Write this improper fraction as a mixed number 14 4 =
4. Solve this using a strategy of your choosing: 896 ÷ 64 =
5. A triangle has one angle of 70° and another of 55°. How many degrees in the third angle? °
7. Write as a multiplication sentence and solve: 3 5 + 3 5 + 3 5 + 3 5 + 3 5 × =
9. Which of the angles marked on the football pitch is a reflex angle?
1. A laptop weighs 1.098kg How much would 24 of these laptops weigh? kg
3. Estimate and then measure the angle.
Estimate: ° Measure: °
5. Write >, < or = to make this true. 3 5 7 12
6. Solve using a calculator: A rectangular playground has a length of 21.2m and a width of 9.8m. What is its area?
sq m
9. Look at the diagram. Match each letter to the correct part of the circle.
centre
diameter C circumference
2. Solve these related calculations.
(a) 2 × 0.12 = (b) 10 × 0.12 =
(c) 5 × 0.12 = (d) 17 × 0.12 =
4. True or false? The equation below matches this diagram.
310° + 50° = 360°
7. Model and solve: 204 ÷ 17 =
8. Mia and Dara were running laps of the park. Mia ran 3 3 4 laps. Dara ran 2 1 2 laps. How many more laps did Mia run? laps
10. Look at the bar model. Write and solve the matching number sentence. of =
1. Identify the measure of the angle shown on the protractor °
3. Solve this: 3245 × 14 =
5. A factory makes 24 lunchboxes every hour How long would it take to make 864 lunchboxes? hrs
7. Find the missing angles (A and B) in the diagram. A: ° B: °
9. Consider the yellow fractions below. Write and solve the matching multiplication sentence for these: × =
Try this!
1. Use a ruler, pencil and protractor to draw a quadrilateral with exactly one pair of parallel lines and exactly one right angle.
2. What is the diameter of this circle? cm
4. Write the missing denominator to make these equivalent fractions. 3 5 = 9
6. A farmer sells apples in bags of 14. How many full bags can she get from 542 apples? Ring the best estimate below.
10 20 30 40 50
8. In a triangle… ✗ or ✓
(a) the angles always add up to 180°.
(b) there may be an obtuse angle.
(c) the angles are always the same size.
(d) there may be a right angle.
10. Last week, Jay drank 7 2 5 l of water from Monday to Friday. He drank 4 1 2 l of water at the weekend. How many litres of water did he drink last week? l
3. Dara’s mum is doing a 500km cycle challenge for charity. If she cycles 42km each day, on what day will she finish the challenge? Day
5. The circles in this diagram are identical. If each has a diameter of 7cm, what is the distance between points A and B? cm
6. Complete the number line.
2. Write the missing values to make these true. (a) 4 4 5 – = 2 1 10 (b) 7 1 2 – = 5 7 8 (c) 3 3 4 – = 1 1 12
4. 2 5 < A < 1 Write two possible solutions for A above that have a denominator of 4.
Let’s talk!
A Trip to Forest Adventure Park
We got to choose from three activities: climbing wall, obstacle course or bike trail.
There were 64 children on the trip to Forest Adventure Park. The table below should show the fraction and number of children who chose each activity.
Work out a strategy for completing the table.
Use your strategy to complete the table in above. Write the fractions in their simplest form.
Look at the children and answer the questions below. Jay, Mia and Lexi have different methods to calculate the fraction of children who chose combinations of activities in above
1. Use Jay’s method to calculate the fraction of children who chose the climbing wall or the obstacle course.
2. Use Mia’s method to calculate the fraction of children who chose the obstacle course or the bike trail.
3. Use Lexi’s method to calculate the fraction of children who chose the bike trail or the climbing wall
Use a different method to check each of your answers.
I’m going to add the fractions from the table.
I’m going to add the numbers, then calculate what fraction of 64 the answer is.
I’m going to subtract the fraction for the missing activity from 1.
Fraction
2. Number 64
(a) Climbing wall (b) Obstacle course (c) Bike trail (d) Total
D Look at this image, of an aerial view of a ro obstacle course, and answer the questions.
For part of the obstacle course, we had to climb through all of these triangles without touching the ropes!
1. Measure each of these angles in the image above. A C D J L
2. Calculate the size of each of the remaining angles B E F G H I K M N O
Look at the image and answer the questions.
What angle rules do you need to use in number 2?
We had a picnic in the forest. I saw log piles where trees had been cut down to clear space. I measured the diameter of three logs.
1. Calculate the radius of each log
2. Estimate the circumference of each log. A
3. There were 123 log piles with 32 logs in each. Calculate the total number of logs.
4. If the park sells the logs at €0.52 each, calculate the selling price of…
(a) one log pile € (b) all of the logs €
The circumference is slightly more than 3 times the diameter.
The park needs to replace a 320m section of fencing. There are three fencing companies they could use. Complete the table.
Fractions as Decimals and Decimals as Fractions
A Express each coloured amount as a fraction and a decimal.
B Use your thousandths grids (PCM 1) to represent the fractions below. Write the equivalent decimal form for each.
C Write the position of each letter as (a) a decimal and (b) a fraction.
D Complete the table.
(a) Decimal (b) Decimal expanded form (c) Fraction (d) Fraction expanded form
Try this! Jay made a 4-digit decimal number: . 0
The other digits he used were 2, 4 and 6.
Write in (a) decimal and (b) fraction form…
1. the greatest number that he could have made (a) (b)
2. the least number that he could have made. (a) (b)
Converting Fractions to Decimals and Decimals to Fractions
A Complete these in order to express the fractions as decimals.
B In your copy, do these in a similar way to the ones in A above.
C Complete these in order to express the decimals as fractions.
D In your copy, do these in a similar way to the ones in C above.
E Match each decimal to its most likely equivalent fraction. Use a calculator to check your answers.
F Let’s talk!
Not all fractions convert easily to decimals Some become recurring decimals.
1. Which fractions in E convert into recurring decimals? Why do you think this is?
2. Recurring decimals are often written with a dot over the repeating digit(s) to show they repeat. Why do you think this is?
Try this!
1. In your copy, express each test result on this report card as...
(a) a fraction in its simplest form (b) a decimal number
2. In which subject did this pupil perform... (a) best? (b) worst?
Report card: test results
26 out of 40
40 out of 50
18 out of 20
out of 24
17 out of 20
3. When Lexi converted a specific fraction to a decimal her, answer was 0·6. Write down three possible fractions.
A recurring decimal has infinite (neverending) digits
What model could you use to prove your answers?
Percentages
A Write the amount that is coloured as per cent and hundredths (fraction form and decimal form).
B In your copy, write the amount that is uncoloured in A above as per cent and hundredths (fraction form and decimal form).
C Solve these.
1. Jay has read 79 pages of a 100-page book. What percentage has he (a) read? (b) left to read?
2. There are 100 children in a club. 37 of them are boys. What percentage are (a) boys? (b) girls?
3. There were 100 trees planted, but 13 were damaged during a storm. What percentage of the trees were undamaged?
Try this!
1. Of the 10 bananas in a bunch, three of them are not yet ready to eat (unripened).
(a) What percentage are ready to eat?
(b) What percentage are not ready to eat?
2. 12% of the people surveyed wear glasses. How many people wear glasses, if the number of people surveyed was... (a) 100? (b) 200? (c) 50?
Let’s talk!
Read the statements. What do you think they mean? When might people use them? What other per cent phrases do you know?
I believe in you 100%. I’m 100% sure that’s right.
Percentages as Fractions and Decimals
A Write the amount that is coloured as a percentage and a fraction.
D Fill in one of the values below to make each expression true.
Solve this.
The children were practising taking shots. Out of 25 shots, Jay scored 15 points. Out of 20 shots, Mia scored 13 points. Out of 10 shots, Lexi scored 7 points.
Whose point-scoring was the most accurate?
Try this! Who has read the most of their book?
I’ve read 243 pages of a 300-page book.
I’ve read 210 pages of a 250-page book.
I’ve read 182 pages of a 200-page book.
A Solve these.
Calculating Percentages of Amounts
1. In a drama club, 60% of the members were boys and the rest were girls. If there were 120 members, how many were (a) boys?
(b) girls?
2. Of the cars in a car park, 25% were black, 35% were white and the rest were silver If 56 of the cars were silver, how many cars…
(a) were there altogether?
(b) were black?
(c) were white?
3. There are 80 members in a chess club If 30% of the members were absent last week, how many were
(a) absent?
(b) present?
4. Mia has saved 60% of the cost of a new computer game. If she has saved €30 so far, how much…
(a) does the game cost? €
(b) more does she need to save? €
5. Dara has read 40% of his book. If he has 96 pages left to read, how many pages are there in the entire book?
B Use a calculator to check your answers to A above.
Use the bar models to help you.
C Model and solve these. Use a calculator to check your answers.
1. In a class of 30 pupils, 60% said that their favourite sport is Gaelic football. How many pupils in the class chose Gaelic football?
2. James has saved €35, which is 50% of the money he needs to buy a new sports strip. What is the total cost of the sports strip? €
3. 75% of the players on a soccer squad played in the final. If 12 of the players played in the final, what is the total number of players on the squad?
4. 20% of the 185 flowers planted in a garden had bloomed. How many flowers were yet to bloom?
5. So far, a school has raised €200, which is 10% of their total fundraising goal. What is the school’s total fundraising goal? €
6. The O’Neill family are travelling from Cork to Dublin, a distance of 260km. When they have travelled 90% of the journey, how far have they still to go? km
7. In a test in which every question was worth the same marks, Toni scored 75%. If he answered 36 questions correctly, how many questions…
(a) did he get wrong?
(b) were there in the test altogether?
8. Zofia wants to send letters to 16 friends. 50% of the letters will need one sheet of paper each and the rest will need two sheets each. How many sheets will be needed altogether?
9. Sam’s smart watch is 80% charged. If the fully charged battery lasts 5 days, how many days are left?
Try this! Look at the image and answer the questions.
1. How long more until this device is fully charged? mins
2. At what time will that be?
3. If this device was allowed to completely discharge (i.e. for the battery to run out), for how many hours and minutes would it need to be plugged in to fully recharge? hrs mins
4. If this device was plugged in 1 hour and 10 minutes ago, what percentage was it at then?
Percentage Increase and Decrease
A Let’s talk!
Look at B below. Is each special offer an increase or a decrease? Could you solve it mentally? If yes, how? If no, how else might you solve it?
B Model and solve each of these (without a calculator) to find out the current value.
Was €260 Now 50% off Was €80 Now 50% off Was €300 Now 40% off Was €480 Now 50% off Was 1·5l Now with 10% extra free Was 0·5kg Now with 30% extra free Was 20l Now with 5% extra free
Was 800ml Now with 20% extra free
C Use a calculator to check your answers to B above in a different way. In your copy, record the inputs you made on the calculator.
Try this! Use a calculator to work out the…
1. total cost of a meal that was €95 before adding a 10% service charge. €
2. total cost of a holiday that was €1,785 before adding a 4% booking fee. €
3. total cost of cinema tickets that cost €48 before adding a 3% bank card fee. €
4. amount of fibre in an apple that contains 8% of the recommended daily fibre intake of 75g g
Round your answers to the nearest whole number as needed.
5. amount of sugar in a cookie that weighs 23g and is made up of 26% sugar. g
Millimetres
Estimate (E) and then measure (M) with a ruler the length or width shown.
What is the length of each line in mm?
Let’s talk!
Without using your ruler, say which of the measurements in and above are longer or shorter than a centimetre.
Explain your reasoning
What is the difference in mm between the longest and shortest measurements?
D Convert these from cm to mm or vice versa. 1. 1cm = mm 2. cm = 60mm 3. 4cm = mm
12cm = mm
75cm = mm
cm = 90mm
cm = 100mm
cm = 35mm
Do you remember how many millimetres are in a centimetre?
In your copy, draw a line of each length below. Write the length beside each line in both mm and cm.
30mm
Try this! The measure of fours ribbons is as follows: Ribbon A: 50mm Ribbon B: 5.5cm Ribbon C: 54mm Ribbon D: 5.2cm
1. Which ribbon is (a) the 3rd longest? (b) the shortest?
2. When Ribbon E is added to the group it then becomes the 3rd longest. What might be its length in mm? mm
3. Ribbon F is 96mm long It is cut into two pieces, one of which is 0.8cm longer than the other piece. What is the length of the longer piece in mm? mm
Measuring Perimeter
Let’s talk!
Estimate which sticker in below has (a) the longest and (b) the shortest perimeter Do you think you need to measure every side of each sticker to get its perimeter? Why or why not?
Measure with a ruler to work out the perimeter of each sticker.
To get the perimeter of any shape, total the lengths of the sides.
Write the perimeter of each sticker in above in mm.
Try this!
1. What was the perimeter of this rectangular sticker before it was torn? mm
2. Work out the perimeter of this sticker by measuring only 2 sides. mm
Calculating Perimeter
Calculate the perimeter.
Let’s talk!
How might you calculate the answers for above using operation(s) other than just addition?
Calculate the missing lengths in these.
1. Perimeter (square) = 24m
Perimeter (rectangle) = 150m
D Are these always (A), sometimes (S) or never (N) true?
Perimeter (rectangle) =
1. You can find the perimeter of a square by finding the length of one side and multiplying it by 4.
2. You can find the perimeter of a rectangle by finding the length of one side and multiplying it by 2.
3. If two identical squares are put together to make a rectangle, the perimeter of the rectangle is twice that of one of the squares.
Try this!
1. Find four possibilities for the length and width of a rectangular garden with a perimeter of 120m.
2. Jay has squares, each measuring 10cm in length. What is the perimeter of a rectangle that he could make using…
(a) 2 squares? cm
(b) 3 squares? cm
(c) 4 squares? cm
Calculating Area
Calculate the area of each of these shapes in cm2 .
Calculate the area of each of these shapes in cm2 .
Answer these.
The diagrams below show three gardens drawn to different scales. In reality, which garden has the… 1. greatest area? 2. least area?
Area is the amount of space covered by a 2-D shape.
Let’s talk!
In or , did you count all of the small squares? What other strategies could be used?
Calculate the area of each flower bed.
Calculate the missing length (L) or width (W).
Try this!
I think that if we calculate the area of these shapes it will be the same value.
In your copy, make
I don’t think they will be the same value.
Connecting Area and Perimeter
Look at the letters below. Which one has the…
1. greatest area?
3. greatest perimeter?
2. least area?
4. least perimeter?
Find the area (A) and perimeter (P) of each shape.
Each square in this grid represents 1cm2
Work out the missing area (A) or perimeter (P) for each of these.
Hint: The first two shapes are squares!
D Solve these.
1. A farmer has planted a square vegetable patch in the middle of a small square field. The vegetable patch has an area of 49m2 What is the area of the field? m2
The field is the light green area of grass in the picture.
2. Jay likes to go for a walk with his dad in the local park. The park is rectangular and has an area of 60m2 If one side of the park measures 12m, how far does Jay walk along the perimeter of the park? m
Try this!
1. Mia has 10 squares, each measuring 10cm in length. How might she arrange them so that at least one side is touching another side? In your copy, draw the arrangement that would result in… (a) at least one side of each square touching another side of another square. (b) the greatest perimeter (c) the least perimeter
2. Dara has 20 straws, each measuring 10cm in length. If he uses all of them to make a rectangle with a perimeter of 200cm, what might the length and width be? L cm, W cm
3. What would be the length and width of the rectangle that Dara could make with his straws with the…
(a) greatest area? L = cm, W = cm (b) least area? L = cm, W = cm
Surface Area of Cubes and Cuboids
Look at each cube and its net. Work out the surface area of each cube.
The surface area is the total area of all outer surfaces of a 3-D object.
Work out the surface area of each cube.
Work out the surface area of each cube.
D
Look at each cuboid and its net. Work out the surface area of each cuboid.
I can’t measure these with my ruler, because they’re not drawn to scale.
Work
the surface area of each cuboid.
1. 2.
1.
Calculating Mean
Find the mean in each of these.
1. Apples in bags.
Total = Mean =
2. Markers in cases.
Total = Mean =
3. Books in piles. Total = Mean =
4. People in cars. Total = Mean =
Find the mean in each set of numbers. Use a calculator to check your answers.
1. 10, 5, 6
2. 7, 9, 13, 15
3. 22, 18, 25, 10, 5
4. 3, 10, 11, 4, 17
5. 10, 15, 12, 11, 17
6. 50, 42, 55, 49
Total = Mean =
Total = Mean =
Total = Mean =
Total = Mean =
Total = Mean =
Total = Mean =
Try this! The children spent 5 weeks raising money for a class trip Here is how much they raised:
1. What was the total amount of money they raised? €
2. What was the mean amount of money raised per week? €
3. If they had continued fundraising for another week, what is a reasonable estimate for how much they would have raised in week 6? €
The mean is the total of the values in a set divided by the number of values in a set.
Use counters or cubes to model your thinking.
The word ‘average’ is often used when talking about the mean amount in a set.
Explain your answers to questions 3 and 4.
4. The cost of the trip was €12 per child, and there were 25 children in the class. Was enough money raised to pay for all 25 children?
Let’s talk!
Data Cards
Choose one data card and describe the person, using the information given.
1 hr/day
Hobby Dancing
Favourite subject Art 148cm
30 mins/day
Hobby Hurling
Favourite subject PE
35 mins/day
Hobby Musical instrument
Favourite subject Music
40 mins/day
Hobby Hurling
Favourite subject PE
35 mins/day
Hobby Dancing
Favourite subject PE
20 mins/day
Hobby Musical instrument
Favourite subject Music 163cm
25 mins/day
Hobby Camogie
Favourite subject Maths
156cm
1 hr/day
Hobby Dancing
Favourite subject Maths
9.
145cm
50 mins/day
Hobby Horse riding
Favourite subject Music
1. Faye
4. Fionn
7. Daniel
2. Amelia
5. Elliot
8. Leah 3. Ben 6. Libbi
Saoirse
Look at the data cards.
The median is the middle value in an ordered list.
1. Is it numerical data?
2. Is it categorical data?
3. Can you identify the mean?
4. Can you identify the median?
Hobbies (f) Favourite subjects 4th 5th 6th
Look at the data cards and answer the questions.
1. Who is the tallest?
2. Who reads the least?
3. What is the total time spent reading per day? mins
4. What is the most common favourite subject?
5. What is the most popular hobby?
6. What is the range of values for time spent reading? mins
7. Use a calculator to work out the mean time spent reading by children in 4th Class. mins
8. What is the range of values for height?
9. (a) Put the heights in order from shortest to tallest.
(b) What is the median height? cm
10. What is the median time spent reading for (a) boys? mins (b) girls? mins
(c) children in 6th Class? mins
D Look at the data cards. True or false?
1. More boys than girls were surveyed.
2. More boys play team sports than do other hobbies.
3. 6th Class children read the most.
The range is the difference between the highest and lowest values in a set.
Use a calculator for questions 4 and 5.
4. The mean time spent reading in 6th Class is greater than in 4th Class.
5. The mean height of children in 5th Class is greater than the mean height of children in 6th Class.
Let’s talk!
Explain your answers for D above, with proof.
Data Displays
Look at the bar chart and answer the questions.
1. What was the range?
2. What was the median score?
3. Find a way to calculate the mean without using your calculator. Use your calculator to check your answer.
In your copy, write 3 questions to ask about the data in .
Look at the pie chart and answer the questions.
In a survey, 36 children were asked what was their favourite book genre.
1. How many children preferred fantasy?
2. Twice as many children preferred graphic novels than adventure. How many preferred (a) graphic novels? (b) adventure?
3. Which genre got a quarter of the votes?
4. What fraction of the children preferred history?
5. Which two genres would you combine to sell the most books? and
D Look at the line plot and answer the questions.
These are the number of goals scored by players in a soccer competition. Each X represents a different player
1. What was the mode?
2. What was the median number of goals scored per player?
3. What is the mean number of goals scored per player?
4. If we only consider the data relating to the players who scored more than 1 goal, what would be the mean score then?
Let’s talk!
Which of the chart(s) above contain (a) numerical data and (b) categorical data? Which chart do you think is best for displaying each type of data? Explain why.
Amelia Daniel Fionn Saoirse
Multiple Bar Charts
Look at the multiple bar chart and answer the questions below.
1. How many pupils altogether preferred skateboarding?
2. Which event was the most popular overall?
3. (a) Which class preferred basketball more? Class (b) By how many more pupils?
4. What was the difference between the most and least popular events for 5th and 6th Classes combined?
Let’s talk!
What other examples of data can you think of that could be displayed using a multiple bar chart? Look at the table and answer the questions below.
1. How many people in total attended…
(a) a class on Wednesday?
(b) karate last week?
2. How many more people attended karate than yoga on Wednesday?
3. (a) Which was more popular overall, yoga or karate?
(b) By how many more people?
4. What was the mean number of people who attended karate?
Construct a multiple bar chart to display the data in the table in above. Write 3 questions in your copy about the data shown.
More Pie Charts
A Let’s create!
Complete the pie chart to display the data shown in the tally chart.
Favourite subject Tally
Maths
Art
Gaeilge
Science
PE
Look at the pie chart in above and answer these.
1. Which subject was the most popular?
2. What fraction of the pupils preferred maths?
3. What percentage of the pupils preferred Gaeilge? %
4. Which two subjects made up half of the total tally? and
Complete this table, which shows the results of a survey of 120 children.
Favourite things to do on a rainy day (a) Watch TV (b) Read (c) Play a game (d) Draw (e) Listen to music
1. Fraction:
2. Percentage: 25%
D Let’s create!
1. Draw a pie chart to display the data in above.
2. Write five questions based on the data to ask your partner or group
Try this! In a survey, a number of children were asked what was their favourite fruit. Use the clues and the pie chart to complete the table below.
● 12 children preferred banana.
● 5% of the children preferred kiwi.
● 1 10 of the children preferred grapes.
● 50% of the children preferred apple.
● 30 children preferred orange.
Votes: Fraction: Percentage:
Hint: First work out how many children were surveyed.
Orange Apple Banana Grapes Kiwi
Orange Apple
Banana
Kiwi Grapes
Line Graphs
Look at the line graph and answer the questions.
1. What was the temperature at 11:00? °C
2. At what time was the highest temperature recorded?
3. What was the highest temperature? °C
4. By how many degrees did the temperature increase between 09:00 and 12:00? °C
5. What was the mean temperature? °C
You can use a calculator for question 5.
Let’s talk!
Look at the line graph in above Do you think the temperature would have increased or decreased at 15:00? Why?
C
Let’s create!
Use your activity sheet (see PCM 2) to plot a line graph to display the data in the table below. Views
in the classroom throughout the day
What do you think the pupils were doing when the temperature began to decrease?
D Look at the data in above and answer the questions.
1. What was the overall range?
2. What is the median of this data set?
3. Between which two days was the greatest increase in views?
4. Calculate the mean number of views per day.
5. If the mean number of views rose to 800 on Saturday, how many times was the video viewed on that day?
Let’s talk!
For above, how could you explain the increase in views on Wednesday? Why do you think the most views recorded were on Friday?
Our class investigated how many views this video got online each day.
Use a calculator for question 5.
Data Investigations
Data Investigation Cycle (PPDAC)
1. Pose a question Ask an investigative question that is interesting
l What do we want to find out?
l What is the question we want to answer?
l Is it an interesting question?
l Is it an answerable question?
l How will we find out the answer?
2. Plan Discuss and decide what data needs to be collected and how it will be gathered.
l How can we find out?
l What should we do?
l What data do we need to collect?
l Who should we collect it from?
l Who will collect the data? Where? For how long?
l How will we record and organise the data?
l What do you think we will find out?
3. Data Collect the data for the chosen investigative question.
l Is the collected data organised?
4. Analysis Analyse the data.
l Look at the data carefully. What do you notice?
l What are the greatest and least values?
l Is the collected data clear and easy to read?
l How might we display the data for others?
l Which graph (and scale) would be most suitable?
5. Conclusion Share the findings and answer the original investigative question posed.
l What are the findings? What do they tell you?
l Have you answered the original question?
l How accurate and reliable are the findings?
l Do you have any unanswered or new questions?
A Let’s investigate!
How long does it take everyone in the class to get to school?
Let’s make a plan!
How could everyone measure their journey in the same way?
Maybe we should ask them to time their journey tomorrow using a stopwatch.
Or they could use a clock at home and at school, and record their time to the nearest minute.
Can you design and conduct a fair test to investigate this?
B Let’s investigate!
I wonder what kind of chart we should use to display the data.
Don’t forget to use the PPDAC Cycle to conduct your data investigation.
Maths Choice Board: Choose a topic below to investigate. Use the PPDAC Recording Sheet (PCM 3) to help you design and conduct a fair test.
1. Pose a question 2. Plan 3. Data 4. Analysis 5. Conclusion
Let’s talk!
Afterwards, discuss the following with your group:
● What is interesting about the findings?
● Are there any patterns?
● Who might be interested in these findings? Explain why.
● How could the findings be shared with a wider audience?
Were you surprised by your findings? Explain why
Number of players: 2–6
Four Spins to 1
You will need: mini-whiteboard and marker per player, place value counters for tenths, hundredths and thousandths, 0–9 spinner, pencil and paper clip
● Each player, in turn, spins the spinner and decides whether to collect that number in the form of tenths, hundredths or thousandths. For example, if a 5 was spun, the player can decide to collect 5 tenths, 5 hundredths or 5 thousandths.
● Continue until each player has spun four times.
● The player with the total closest to 1 (can be above or below 1) scores a point.
● The player with the highest score after four spins wins the game.
Variations
1. The player with the total closest to 1, that is also less than 1, scores a point.
2. Four Spins to 100,000: Play as above, but each time, choose whether to collect that number of ten thousands, thousands, hundreds, tens or ones. The player with an amount closest to 100,000 scores a point.
Spin and Keep – Decimals
Number of players: 2–6
You will need: mini-whiteboard and marker per player, place value counters for tenths, hundredths and thousandths, 0–9 spinner, pencil and paper clip
● Round 1: Each player, in turn, spins the spinner and decides whether to collect that number in the form of tenths, hundredths or thousandths.
● Round 2: Each player, in turn, spins the 0–9 spinner and decides to collect that number of whatever they did not collect in round 1.
● Round 3: Each player, in turn, spins the 0–9 spinner and decides to collect that number of whatever they did not collect in rounds 1 or 2.
● The player who has made the greatest amount scores a point.
● The player with the highest score when the time is up wins the game.
Variations
1. The player who has made the least amount scores a point.
Play for the time allocated by your teacher
2. In Round 1 players collect thousandths, in Round 2 they collect hundredths, and in Round 3 they collect tenths.
3. Spin and Shade Decimals: Each player needs a thousandths grid (see PCM 1) and (if available) a reusable dry-erase pocket. Play as above, except that when each player spins a number, they choose to shade in that number of tenths, hundredths or thousandths on their grid.
Number of players: 2–6
Mean Quest
You will need: calculator,mini-whiteboard and marker per player, 0–9 spinner, pencil and paper clip
● Each player draws these five boxes on their mini-whiteboard.
● Each player spins the spinner five times and writes the digits spun in their boxes.
● Each player calculates the mean (average) of their set (total ÷ 5) using a calculator.
● The player with the highest mean scores a point.
● The player with the highest score when the time is up wins the game.
Variations
1. Players find the median (or range) of their set, and the player with the highest median (or range) scores a point.
2. Spin each spinner twice and arrange the digits to create a series of five 2-digit numbers.
Capture the Area
Number of players: 2–3
You will need: marker of a different colour per player, dot grid (see PCM 4) or squared copy, 1–6 spinner or 0–9 spinner, pencil and paper clip
● Each player, in turn, spins the spinner twice to get two digits.
● Using their colour, the player draws a rectangle, sized according to the digits spun, on the dot grid, and writes the total number of square units inside the rectangle. For example, if 4 and 2 are spun, the player draws a rectangle that is 4 squares long and 2 squares wide, and writes 8 inside the rectangle.
● Play continues until time is up, or there is not enough space to add another rectangle for the next digits spun.
● The player who has ‘captured’ the greatest area of square units overall wins the game. (Tip: Players could keep a running total of the area of square units that they capture with each spin, e.g using addition at the side.)
Variation
● Instead of a dot grid, a mini-whiteboard and marker per player are needed.
● Each player, in turn, spins the spinner three times to get three digits, and arranges them to create a 2-digit dimension and a 1-digit dimension for an area model, which they draw on their mini-whiteboard. For example, if 6, 4 and 2 are spun, the player could draw an area model of 64 by 2, 24 by 6, 26 by 4, etc. The player calculates the area and writes this in the centre of their area model.
● The player with the largest area in each round scores a point.
● The player with the highest score when the time is up wins the game.
Let’s Look Back 3
1. Find the (a) area and (b) perimeter of this sticker
(a) cm2 (b) cm
5. Lexi and her friends had to guess the number of marbles in a jar Their guesses were: 77, 81, 84, 79, 68. What was the median guess?
7. Express 17 25 as a decimal.
8. True or false? 137% > 137
9. This bar chart shows the amounts of rainfall over three months. What was the difference between the wettest and driest months in (a) mm and (b) cm? (a) mm (b) cm
2. A teacher recorded the number of pupils who were absent each day last week. Find the mean number of absences.
3. If a rectangle with a perimeter of 14cm has a length of 6cm, what is its width? cm
4. Model and solve: 240 people attended a play. 15% of them were children. How many children were there?
6. Jay has run 48m of a 100m race. What percentage of the race is left? %
1. A square lawn has an area of 64m2 What is the length of one side? m
3. Write this percentage as a fraction in its simplest form: 22% =
5. The shoe sizes of players on a basketball squad are: 10, 9, 11, 11, 12, 14, 12, 13, 10, 11, 11, 13.
What is the mode of this data set?
7. Measure the perimeter of the pentagon. cm
9. True or false? The median of a data set cannot be greater than the mean.
10. Look at the tally chart. True or false?
(a) This is categorical data.
(b) Peach was the least popular flavour
(c) 28 people in total provided data.
2. Write 0.761 as fractions in expanded form.
4. Convert from cm to mm: 4.2cm = mm
6. Calculate the surface area of the cube. cm2
8. Write the coloured section of the shape as... (a) a fraction in its simplest form (b) a percentage %
1. A square has an area of 49cm2 What is its perimeter? cm
3. Express 0.98 as a fraction in its simplest form.
5. Last week, Dara did 80 skips with his skipping rope. This week, he increased his skips by 10%. How many skips did he do this week?
7. 80 children were asked what was their favourite pet. The results are displayed in the pie chart.
2. Put these in order starting with the least: 44%, 04, 1 4
4. True or false? It is impossible to find the median of categorical data.
6. 90% of the pupils in Ms Lee’s class were at school on Friday. If 18 pupils were in on Friday, what is the number of pupils in the class?
8. Find the surface area of the cuboid. cm2
9. What is the length of the paper clip in mm? mm
How many children said… (a) dog? (b) cat? (c) hamster? (d) rabbit?
10. Lexi’s mum is travelling a distance of 350km to visit a friend. When she has travelled 30% of the journey, how far has she left to travel?
Try this!
1. Look at the scores in the table. If the mean score was 84, what was Rob’s score?
2. Last year, Mia’s dad ran a marathon in 3 hours, 40 minutes. This year, he ran the same marathon, but his time improved by 20%. What was his time this year? hrs, mins
3. Dara cut out a rectangular piece of card with the dimensions shown. He then cut out two more identical rectangular pieces of card. He arranged the three rectangles to make a square.
(a) What was the perimeter of the square? cm
(b) What was the area of the square? cm2
4. Draw an arrow pointing to where 27 100 would be on the number line below.
The Wildlife Park
Lexi and Jay are visiting a Wildlife Park. Look at the image and answer the questions below.
1. Write the width (shorter side) of the Red Fox Zone in cm. cm
2. Write the length (longer side) of the Car Park in km. km
3. Calculate in both m and km the perimeter of the: (a) Car Park m km (b) Red Fox Zone m km (c) Lynx Zone m km
4. Calculate the area of the: (a) Gift Shop
(c) Small Mammals Zone
2 (b) Brown Bear Zone
5. The Wolf and Boar Zones each have a perimeter of 800m. Without calculating, which of these do you think has the larger area? Zone
6. Calculate the area of the: (a) Wolf Zone m2 (b) Boar Zone
7. Write how much of the park’s area the Brown Bear Zone is as (a) a fraction (b) a decimal (c) a percentage %
Solve these.
1. Write 15% as: (a) a fraction (b) a decimal
2. The park has 150 small mammals. 20% of these are pygmy shrews. How many pygmy shrews are there?
Calculate the surface area of each small-mammal hutch.
You could use magnetic tiles to build these and then count them to get the surface area.
D Answer these.
A park keeper measured the head and body lengths of the small mammals. The body lengths of the field mouse, hedgehog and red squirrel are shown in the pictures.
1. Complete the table.
(a) Field mouse (b) Hedgehog (c) Red squirrel
Length in cm
Length in m
2. What is the difference between the longest and shortest lengths above? cm
Look at the table and answer the questions below.
Number of boar piglets born in the park each year 2022 2023 2024 2025 2026 11 17 13 915
1. Find the median number of boar piglets born per year
2. How many boar piglets were born in total over the five years?
3. Calculate the mean (average) number of boar piglets born per year
F Let’s create!
Draw a bar chart to show the number of boar piglets born each year in . Remember to choose an appropriate scale.
At the park, 360 visitors were asked what was their favourite animal. The pie chart shows the results. Answer the questions.
1. How many visitors preferred the… (a) boar? (b) wolf? (c) brown bear? (d) lynx? (e) red fox?
2. How many degrees are there in the… (a) red fox sector? ° (b) wolf sector?
3. What percentage of visitors preferred the brown bear, red fox or boar? %
4. What fraction of visitors preferred the lynx or wolf?
Compass Directions
Look at the map and fill in the gaps below, using a compass as your guide.
1. Waterford is of Galway.
2. Westport is of Ballina.
3. Dublin Heuston is of Sligo
4. Cork is of Limerick.
5. Dublin Connolly is N of .
6. Sligo is NW of
7. Limerick is NE of
8. Galway is SW of
Let’s talk!
Look at Lexi. What do you think? Can you come up with a rule? Discuss as a class or in groups.
Use a compass to complete the table.
Rail network of Ireland
If Westport is NW of Waterford, then Waterford must be SE of Westport.
Start facing… (a) End facing… (b) Direction (c) Degrees turned (d) Number of
1.
2.
3.
D Use a compass to answer these questions.
1. Mia is using a compass. If she walks towards North and then makes a 45° turn clockwise, which direction is Mia facing now?
2. Jay is facing East. He makes a 90° turn anti-clockwise. Which direction is Jay facing now?
3. Lexi is facing South-West. She turns 135° clockwise. Which direction is Lexi facing now?
4. Dara is facing North. He turns 225° clockwise. What direction is Dara facing?
Belfast
Dublin Connolly Dublin Heuston
Rosslare Europort
Waterford Cork
Tralee
Limerick Galway
Westport
Ballina Sligo
Look at the map and answer the questions below.
To get from the bike racks to the ice cream van, I would go SW, then turn 45° anticlockwise, then go S.
Fill in the gaps to complete the sentences.
1. To get from the coffee truck to the bandstand I would go
2. To get from the oak tree to the bench I would go E, then turn ° clockwise, then go
3. To get from the fountain to the picnic table I would go NE, then turn ° clockwise, then go .
4. To get from the picnic table to the bins I would go NE, then turn ° anti-clockwise, then go .
Let’s talk!
Find a different route for each journey in above Describe your new route. Is it a longer or shorter journey than the one in ?
Try this! A bearing is an angle measured clockwise from North. Bearings are used for navigation at sea.
1. Work out which direction you’d be moving if you travelled on a bearing of: (a) 90° (b) 180° (c) 0°
2. Work out what bearing you’d be travelling on if you were moving: (a) NE (b) SW (c) NW
boat is moving SE. It’s travelling on a bearing of 135°.
The
Oak tree
Bins
Bench
Bandstand
Fountain
Picnic table
Coffee truck
Ice cream van
Bike racks
Coordinates
Look at the coordinate plane. Name a point…
1. on the vertical axis
2. on the horizontal axis
3. on both axes
4. with coordinates of (1, 7)
5. with coordinates of (4, 3)
6. on the x-axis
7. on they-axis
8. at the origin
Let’s talk!
Look at Mia. What do you think?
Discuss with a partner
In coordinates, the first number is on the x-axis, and the second is on the y-axis.
I think some parts of above have more than one answer
Find the coordinates of each point on the coordinate plane.
What do you notice about the answers?
Try this!
1. Draw a coordinate plane. Make the x-axis andy-axis 6 squares long
2. Plot these points: A (4, 2) B (5, 6) C (0, 1) D (3, 3) E (6, 0) F (2, 4) G (3, 5)
3. Plot the point that is halfway between (3, 3) and (3, 5). Write the coordinates of the point you plotted.
4. Repeat step 3 for two more pairs of points from step 2.
5. Look at the coordinates you wrote down for the halfway points. Can you find a way to predict the coordinates of the point that is halfway between two points?
D A team of park guides is based at the information booth at the origin (0, 0). They help visitors find their way around a large theme park.
1. A guide walks from (0, 0) 5 units East and 2 units North. What are the guide’s new coordinates? ( , )
2. Visitor A reported losing their backpack, and another staff member spotted it at the snack stand located at (4, 5). The guide is heading there from (0, 0) to pick it up
Fill in the gaps to complete the sentence: The guide needs to walk units and units to get to (4, 5).
3. From (4, 5), the guide turns to face NorthEast. Then the guide walks 2 units West and 2 units South. Where does the guide end up?
E Let’s create!
● Use your 0–9 spinner to create 3 coordinate points. Spin twice to create xandyvalues for your first point. Repeat this for your second and third points. Plot your 3 coordinate points on the grid.
● Join the lines together and colour the inside of the outline. What shape did you make?
● Use your spinner again to create 4 coordinate points. In a different coloured pencil/marker, plot your coordinate points on the grid, join the lines together and colour the inside of the outline. What shape did you make?
F Let’s investigate!
1. Choose a number between 5 and 10.
(a) Write all the coordinates you can think of, where the sum of the x-coordinate and they-coordinate is your number
I chose 7. The sum of the x-coordinate and the y-coordinate in (3, 4) is 7. It is also the sum of the x-coordinate and the y-coordinate in (5, 2).
(b) Plot all the coordinates on the coordinate plane on your activity sheet (see PCM 5).
(c) What do you notice?
2. Choose a number between 0 and 5.
(a) Write all the coordinates you can think of where the difference between the x-coordinate andy-coordinate is your number
(b) Plot the coordinates on the coordinate plane on your activity sheet.
(c) What do you notice?
3. What other patterns can you find?
Plotting Shapes
Let’s talk!
Look at the sets of points in below. You are going to plot these points on a coordinate plane. Can you predict what shape they will make? Explain your reasoning
Plot each set of points on the coordinate plane on your activity sheet (see PCM 5). Join the points in the order given and join the finish to the start. What shape does each set make?
1. (0, 0), (2, 0), (1, 3)
2. (2, 7), (4, 7), (3, 8), (1, 8)
3. (5, 6), (6, 5), (5, 4), (4, 4), (3, 5), (4, 6)
4. (5, 1), (3, 1), (3, 3), (5, 3)
For each coordinate plane, write the coordinates of two points that would make a square with the two points shown.
(a) ( , ) (b) ( , )
Let’s talk!
( , ) (b) ( , )
( , ) (b) ( , )
Look at the points in again. Discuss and compare with others. Are there any ways to make different squares in the grids?
Try this!
These four coordinates make the corners of a rectangle. What do you notice about them?
Without plotting anything, can you work out the missing coordinates for a…
( , ) (b) ( , )
I can find three different answers for question 1.
(2, 1), (5, 1), (5, 3), (2, 3)
1. rectangle with corners at (1, 2), (4, 2) and (4, 5)?( , )
2. rectangle with corners at (3, 1) and (6, 1)? ( , )
3. square with a corner at (1, 4)? ( , )
Compare your answers with those of a partner Are they the same?
Discuss why or why not.
Symmetry and Transformations
Are these always (A), sometimes (S) or never (N) true?
1. A shape with 4 sides has 4 lines of symmetry.
2. An equilateral triangle has rotational symmetry of order 3.
3. An irregular shape has no lines of symmetry.
4. A trapezium does not have rotational symmetry.
Look at your answers for the statements in above and answer these in your copy.
1. If you wrote ‘A’ (always), draw an example of where the statement is true.
2. If you wrote ‘N’ (never), draw an example of where the statement is not true.
3. If you wrote ‘S’ (sometimes), draw (a) an example of where the statement is true, and (b) an example of where it is not true.
Let’s talk!
Look at the circle below. Do you agree with Mia and Jay? Discuss as a class or in groups.
I think a circle has reflective and rotational symmetry.
Let’s talk!
Look at the shapes and answer the questions below.
A
Any line I draw that goes through the centre is a line of symmetry
Try this! Look at the shape and answer the questions below.
A
When you rotate a shape around a vertex (corner), that corner stays in the same place and all the other corners move.
B
1. Which two transformations are needed to transform shape A to shape B? translation reflection rotation
2. Does it matter which order you do the transformations in? Explain why.
1. Reflect shape A in the dashed line. Label the new shape B.
2. Reflect shape B in the solid line. Label the new shape C.
3. Rotate shape A 180° around its bottom right corner. Label the new shape D. What do you notice?
Translations
Look at the shapes and answer the questions.
1. Which two shapes are not a translation of shape A? and
2. Which shape is a translation of shape A by 5 squares up?
3. Which shape is a translation of shape A by 4 squares left and 2 squares down?
4. Which shape is a translation of shape A by 3 squares left and 3 squares up?
Look at the shapes in above. Complete these descriptions.
1. Shape A translated 3 squares left and 3 squares up is shape
2. Shape A translated 4 squares left and 3 squares down is shape
3. Shape translated 6 squares down is shape
4. Shape translated 3 squares right and 3 squares up is shape
Answer these.
Shape A translated 5 squares up is shape C.
Dara drew this shape on the coordinate plane. After he translates it, the coordinates of two of the corners will be (6, 2) and (3, 2).
The coordinates of the third corner will be (4, 5). No, I think they will be (5, 4).
1. Who do you agree with, Dara or Mia?
2. Draw the triangle in the correct place.
Let’s talk!
With your partner, describe the translation in above Is there more than one way to do this?
Fill in the gaps.
1. When you translate a shape to the left, the x-coordinate gets .
2. When you translate a shape up, the -coordinate gets bigger
3. When you translate a shape to the right, the -coordinate gets bigger
4. When you translate a shape down, they-coordinate gets .
Let’s talk!
Look at Lexi. Do you agree? Why or why not?
Write the new coordinates of each shape if it is translated…
When you translate a shape, the coordinates of all the corners always change.
3 squares down and 2 squares to the right. ( , ), ( , ), ( , ), ( , )
2 squares down and 2 squares to the left. ( , ), ( , ), ( , )
so that the vertex at (7,1) moves to (3, 5). (7, 1) ( , ), ( , ), ( , )
H Let’s create!
so that the vertex at (3, 2) moves to (6, 3). (6, 3) ( , ), ( , ), ( , )
1. Use software and/or shapes to recreate these logos made from translated shapes. (a) (b) (c)
2. Create your own logo made from translated shapes.
Reflections
For each of these, write which shape is the reflection of shape A in the blue line.
For each shape in above, write (a) the coordinates of one vertex of the original shape and (b) the coordinates of that vertex after it is reflected in the line.
Let’s talk!
Look at the children. Is what they say always, sometimes or never true? Discuss as a class or in groups.
When you reflect a shape, its dimensions change.
When you reflect a shape, its position changes.
If I reflect a shape in a line, then reflect the new shape in the same line, the resulting shape will have the same coordinates as the original shape.
1. (a) ( , ) (b) ( , )
(a) ( , ) (b) ( , )
3. (a) ( , ) (b) ( , )
(a) ( , ) (b) ( , )
D Draw the reflections described below and write the coordinates of the vertices of each image (new shape).
1. Reflect shape A in line L. The coordinates of the image are:
( , ), ( , ), ( , ), ( , ).
2. Reflect shape B in line L.
The coordinates of the image are:
( , ), ( , ), ( , ), ( , ).
3. Reflect shape C in line M.
The coordinates of the image are:
( , ), ( , ), ( , ),
( , ), ( , ), ( , ).
What do you notice about the reflected shapes?
An image is the resulting new shape(s) after an original shape is transformed.
Draw the reflections described below and write the coordinates of the vertices of each image.
1. Shape F has been reflected in line P.
Label the image G.
The coordinates of shape G are:
( , ), ( , ),
( , ), ( , ).
2. Reflect shape F in line Q.
Label the image H.
The coordinates of shape H are:
( , ), ( , ), ( , ), ( , ).
3. Reflect shape G in line Q.
Label the image J.
The coordinates of shape J are:
( , ), ( , ),
( , ), ( , ).
Let’s talk!
Look at the coordinates you have written for your answers to . What do you notice?
What do you think would be the coordinates of the image of shape H reflected in line P? Explain your reasoning.
How might you check your prediction?
Representing and Recording Patterns
A Build it! Sketch it! Write it!
Model each pattern and complete the matching T-chart. Pattern 1Pattern 2
Pattern 3Pattern 4
B Look at the patterns in A above and answer these.
1. Which pattern does the graph represent? Pattern
2. Use your activity sheet (see PCM 5) to draw a graph representing each of the other three patterns.
Try this!
1. Look at A above. For each pattern, work out these term values: 5th term 10th term 20th term Pattern 1Pattern 2Pattern 3Pattern 4
I think sketching a T-chart would help.
2. At the library, books are stored in stacks. The first stack has 2 books. If each new stack has 3 more books than the previous stack, how many books will there be in the (a) 5th stack? (b) 10th stack? (c) 20th stack?
3. A new online video channel gets 5 subscribers on its first day. Every day after that it gets double the amount of subscribers it got the day before. How many new subscribers did it get on the... (a) 5th day? (b) 8th day? (c) 10th day?
Square Numbers
A Answer the questions and then fill in the blanks to complete the pattern.
Explain what is happening using concrete materials or pictorial representations.
E Without adding, write the total of the first…
1. five odd numbers 2. eight odd numbers
3. ten odd numbers
For ‘one squared’ we can also say ‘1 to the power of 2’. It means 2 ones multiplied by each other. ‘Indices’ is another name for numbers with a raised number. 1
Explain your strategy
Try this! Look at the display of cans. In layer 1 (the highest layer), there is 1 can. In layer 2, there are 4 cans. In layer 3, there are 9 cans.
Layer 1 1. One squared = 12 = 1 × 1 = 2. Two squared = 22 = 2 × 2 = 3. Three squared = 32 = = 9 4. Four squared = = 4 × 4 = 16 5. Five squared = 52 = 5 × 5 = 6. Six squared = 62 = = 36
1. How many cans are there in (a) layer 5? (b) layer 10?
2. Which layer has (a) 49 cans? (b) 81 cans?
3. How many cans are there in total in the 10 layers?
4. How many cans would there be in the (a) 12th layer? (b) 15th layer? (c) 20th layer? Layer 10
Rectangular Numbers
A Match each array to the correct term number. Complete the matching T-chart.
1 4 5 2 3
B Ring the numbers that are not rectangular numbers.
1. 42, 56, 64, 72, 81, 90
Let’s talk!
2. 84, 96, 100, 110, 121, 132
Look back at the table in A on the previous page. Find the sequence of rectangular numbers. Explain how it is that they are there. Come up with a rule to explain how to find any rectangular number
D Solve these.
The children are arranging chairs in rectangular arrays, where each array is one seat longer than it is wide.
The 1st array has 1 row with 2 seats.
The 2nd array has 2 rows with 3 seats.
The 3rd array has 3 rows with 4 seats.
1. If the pattern continues, how many seats will there be in the…
(a) 5th array?
(b) 10th array?
(c) 20th array?
2. Which array will have 56 seats?
Try this! Are these always (A), sometimes (S) or never (N) true?
1. Rectangular numbers are even numbers.
2. Rectangular numbers are multiples of 3.
The 12th rectangular number is 12 multiplied by Term Array Value 11 × 22 22 × 3 × 4 × 5 ×
3. The difference between any two rectangular numbers is an odd number
4. The 3rd rectangular number is double the sum of the first three counting numbers; the 4th rectangular number is double the sum of the first four counting numbers, and so on.
Triangular Numbers
Let’s talk!
Look at the shapes in these images. What do you notice? What do you wonder? Can you think of any other shapes like these?
B Look at image 3 in and answer these.
1. How many paper cups would be used if the stack was… (a) 4 rows high? (b) 6 rows high? (c) 10 rows high?
2. If a similar stack of cups has a total of 28 cups, how many rows high is it?
C Ring the numbers that are not triangular numbers.
1. What triangular number is represented by the red dots?
2. What triangular number is represented by the blue dots?
3. How many dots altogether?
4. What number shape do they each make?
5. Does this work for other triangular numbers? Show proof in your copy.
Look at the dots and answer these.
1. What triangular number is represented by the green dots?
2. What triangular number is represented by the orange dots?
3. How many dots altogether?
4. What number shape do they each make?
5. Does this work for other triangular numbers? Show proof in your copy.
Try this! Are these always (A), sometimes (S) or never (N) true?
1. The sum of two prime numbers is a square number.
2. Double each triangular number is the corresponding rectangular number
3. The total of two consecutive triangular numbers is a square number.
4. The second triangular number is equal to the sum of the first two counting numbers; the third triangular number is equal to the sum of the first three counting numbers, and so on.
5. The second square number is equal to the sum of the first two even numbers; the third square number is equal to the sum of the first three even numbers, and so on.
Consecutive numbers come one after the other.
Brackets
A Read the word problems and ✓ the one that matches this expression: 6 × ( + )
1. Lexi had six packs of collector cards with ten cards in each. She lost sixteen cards. How many does she have left?
2. Dara’s dad bought six bags of apples. In each bag, there were five red apples and four green apples. How many apples did he buy altogether?
3. The principal bought four boxes, each containing six packs of markers. If there were five markers in each pack, how many markers did she buy in total?
4. At a fruit and veg shop, four hundred pears were packed in bags of ten. If six bags have been sold so far, how many bags are left?
B Change each of the remaining word problems in A above to an expression with brackets and solve.
Word problem number:Expression:
C This sketch represents which one of the word problems in A above?
D In your copy, represent each of the remaining word problems in A above using a sketch of your choosing.
Let’s talk!
What is the same about these expressions? What is different?
20 + (10 + 2)
20 – (10 – 2)
(20 + 10) + 2
(20 – 10) – 2
Why does the location of the brackets make a difference in some, but not in others?
Try this! In your copy, using the same numbers and operator symbols each time, write two pairs of number sentences that will have
1. the same answer no matter where the brackets are positioned
2. different answers depending on where the brackets are positioned.
Order of Operations
A Use the order of operations to solve these.
Remember to use: BIMDAS: Brackets Indices
Multiplication / Division Addition / Subtraction
I’ll rewrite these and add brackets to remind me which part to solve first.
B Let’s talk!
● Were there any expressions in A above that would have arrived at the same answer no matter what order? Which ones? Explain why.
● Use a calculator to check your answers. What do you notice?
● Consult with your classmates. Are their calculator answers the same or different?
Calculators
A calculator will give different answers to the number sentences depending on whether it is an arithmetic or algebraic calculator.
Is your calculator smart (algebraic) or not (arithmetic)?
● Arithmetic calculator: This calculates the operations as they are input, and in that order Therefore, it does not follow the order of operations.
● Algebraic calculator: Also called a ‘smart’ calculator. This records all the inputs (often at the top of the screen) and calculates an answer only when the equals button is pressed. Then, it calculates the answer in the order of operations, which is an algebraic rule.
If it is not a smart calculator, how might you use your calculator to check the answers?
2. In your copy, use only brackets, the numbers 2, 3 and 4 and any two of the operators (+, −, ×, ÷) each time to… (a) make up number sentences to equal the following numbers: 2, 3, 4, 5, 6, 9, 10, 11, 14 (b) make a number bigger than 20 (c) make a decimal number.
For example: (2 + 5) – 4 = 3
Numbers Above or Below Zero
A For each statement, write if it describes a value above (A) or below (B) zero.
The temperature is 9°C.
Max’s office is on the 6th floor.
Anna’s bank account is overdrawn by €20.
The temperature is 7° below zero.
The Caspian Sea is 28m below sea level.
Tom has €10 in his savings. 7.
The height of Croagh Patrick is 764m.
St. Brigid’s have scored 10 points.
B In A above, which of the values…
Oakfield FC have not scored and have let in 6 goals.
1. above zero is (a) closest to zero? (b) furthest from zero?
2. below zero is (a) closest to zero? (b) furthest from zero?
1. 3rd floor and go down 2 floors?
3rd floor
Use the number path below to help you.
)a) Try this! Look at the image. Which floor will you end up on if you start on the…
2. 2nd floor below ground and go up 4 floors?
3. 2nd floor and go down 4 floors?
4. 1st floor below ground and go up 2 floor
2nd floor
1st floor
1st floor below ground
2nd floor below ground
Positive and Negative Numbers
A Write each of the temperatures as a positive or negative number.
B Look at the images in A above and answer these.
1. Which temperature is… (a) the warmest? °C (b) the coldest? °C
2. Which temperature is the closest… (a) positive number to zero? °C (b) negative number to zero? °C
3. If the temperature of each of the thermometers in questions 1 to 4 dropped by 5 ° what would each new temperature be? (a) 1. (b) 2. (c) 3. (d) 4.
4. If the temperature of each of the thermometers in questions 5 to 8 rose by 5° what would each new temperature be? (a) 5. (b) 6. (c) 7. (d) 8.
Try this! In
Comparing and Ordering Positive and Negative Numbers
A What number is represented by each letter on the number line?
B Which of the answers in A above is the…
1. greatest?
3. closest to zero?
2. least?
4. furthest from zero?
C Write the missing symbol (<, >, =) to make these true.
D Put these in order starting from the lowest value to the highest.
1. 3°C, −5°C, −1°C
3. €25, −€30, −€15
5. −35°C, +12°C, –9°C
E Answer these.
2. +35m, +114m, −156m
4. −24, +15, −32
6. −213m, −14m, −29m
1. In Alaska, the average temperature in December was −16°C. In January, it was −21°C and in February, it was −15°C Which month was the warmest?
2. Look at the table. Of the four cities, which is the…
(a) highest?
(b) lowest?
(c) closest to sea level?
City, Country
New Orleans, USA −2m
Baku, Azerbaijan −28m
Dublin, Ireland +8m
Jericho, West Bank –258m
Try this! Are these always (A), sometimes (S) or never (N) true?
1. Numbers with no signs are neither positive nor negative.
2. Numbers to the left of zero are greater than numbers to the right of zero.
3. As you move right on a number line, the values are greater
4. −5 and +5 are equal.
5. −5 and +5 are equidistant from zero.
6. Positive numbers are greater than negative numbers.
In question 5, ‘equidistant’ means the same distance.
7. When comparing two negative numbers, the number furthest from zero is greater
Write the missing 12-hour and 24-hour times to complete the tables.
p.m. (c) 6:31 p.m. (d) (e) 19:33
Look at the bus timetable and answer the questions below.
2. If the bus to Limerick takes 2 hours and 30 minutes to reach its destination, at what time will it arrive in Limerick in... (a) 12-hour time? (b) 24-hour time?
3. The bus to Mallow usually takes 50 minutes, but because of heavy traffic it arrived 15 minutes late. At what time did it arrive in Mallow in... (a) 12-hour time? (b) 24-hour time? 12-hour time 24-hour time 9:53 p.m. (a)
Try this! For each of these, write in 24-hour time:
1. 10 past 7 in the morning: 2. 1 4 to 11 at night: 3. 5 to 4 in the afternoon 4. 30 mins after 20 to 7 in the evening
5. 20 mins before 1 4 past midnight.
Adding and Subtracting Time
A Use open number lines to help you model and solve these.
1. Mia is throwing a surprise party for her sister at 3 p.m. If it will take her 1 hour and 35 minutes to set up the food and drinks, what is the latest time at which she should start this?
2. Lexi’s aunt won the Fun Run in the park at the weekend. If the race started at 1:35 p.m. and she finished at 2:18 p.m., how long did it take her to complete the run? mins
3. Dara and his parents are flying from Dublin to London, and then from London to New York. If their plane lands in London at 17:05, how long do they have to wait before their New York flight departs at 18:40? hr mins
4. The overnight bus left Cork at 22:40 on Tuesday and arrived in Donegal at 04:25 on Wednesday. How long did the journey take? hrs mins
B Use your open number line to model and solve these.
1. A bus left Dublin at 13:30. It was delayed on the way and arrived in Galway 40 minutes late at 18:05.
(a) At what time should the bus have arrived in Galway?
(b) What should the duration the total bus journey have been? hrs mins
2. Mia was travelling by train from Cork to Galway, and she had to change trains at Limerick Junction.
Complete the cinema timetable by calculating the missing times.
Cineflix Timetable
Stroke of Midnight
The Lost Diary
Multiplying and Dividing Time
A Use multiplication or division to help you solve these. 1. 2 hrs = mins
hrs = mins 3. 4 hrs = mins
130 mins = hrs mins
250 mins = hrs mins
B Use the models to help you solve these.
1. A pupil has three 40-minute classes one after each other If the first one starts at 09:20, at what time will the third one end?
2 One tray of flapjacks takes 35 minutes to bake in the school oven. If there are 5 trays to be baked, and the oven can only fit one tray at a time, for how long will the oven be on? hrs mins
3. An Olympic swimmer swims for a total of 2 hours and 20 minutes every morning Her session is split into 4 equal ‘sets’. How long does each set last? mins
Model and solve these.
5 hrs = mins
09:00
1. Siobhán works 34 hours per week. If she works 4 days per week, how many hours and minutes does she work on average each day?
hrs mins
2. It takes Mary 4 mins 20 seconds to walk the perimeter of the pitch. If she continued at the same pace, how long would it take her to walk 5 laps?
mins secs
3. Aidan ran 6 laps of the running track in 7 minutes and 36 seconds. What was the mean (average) time of 1 lap?
mins secs
Try this!
1. How many (a) hours in a week? (b) minutes in a week? (c) seconds in a week?
2. Calculate how many (a) hours, (b) minutes and (c) seconds per week you spend in school if the school day is 5 hours and 40 minutes long. (a) (b) (c)
Schedules and Timetables
A Look at the image and answer the questions.
1. Jay spent the same amount of time playing guitar each day. How long did he spend playing guitar that week? (7 days) hrs mins
2. Jay spent the same amount of time with his family from Monday to Friday, and 8 hours more in total at the weekend. How long did he spend with his family that week? hrs mins
3. Jay worked out that he also spent a total of 7 hours and 35 minutes helping out around the house that week. What was the mean time he spent helping each day? hrs mins
B Let’s create!
Use A above as an example to help you make estimates about your daily routine. Represent your routine on a graph or chart. Write some questions for another person to solve.
C Look at the school timetable and answer the questions.
Time MondayTuesdayWednesdayThursdayFriday 09:10 English Maths English Maths Gaeilge 09:50 Gaeilge English Maths Gaeilge English 10:30 Small Break Small Break Small Break Small Break Small Break 10:45 Maths Gaeilge English Gaeilge Maths 11:25 History SPHE GeographySPHE Music 12:05 Religion Drama Science Religion Drama
12:45 Lunch Break Lunch Break Lunch Break Lunch Break Lunch Break 13:15 PE ArtPEArt Golden Time 13:55 Wellbeing Walk Coding Reading Maps Games/Puzzles
14:25 Pack up Pack up Pack up Pack up Pack up
1. For how many hours and minutes is English scheduled for the week? hrs mins
2. How many hours and minutes of lessons do the children have before lunch on a Thursday? hrs mins
3. Calculate the total amount of time spent on Art, Music and Drama each week. hrs mins
4. How many hours and minutes of Maths would be completed in a 4-week period? hrs mins
Try this! If the school had to reduce the school day by 30 minutes, how would you reorganise Friday’s timetable while keeping all the same subjects and activities on the timetable? Explain your reasoning
Number Path Race
Number of players: 2
You will need: counter and –15 to +15 number path (see PCM 6) per player, deck of playing cards with picture cards, 6s, 7s, 8s, 9s and 10s removed
● To start, each player places their counter on zero on their number path.
● Each player, in turn, turns over the top card of the deck.
● If the card is black, the player moves forwards by that amount on the number path. If the card is red, they move backwards by that amount on the number path.
● The game is won by being the first player to reach +15 or by not being the first player to reach –15!
Tug o’ War
Number of players: 2
You will need: one counter, one –15 to +15 number path (see PCM 6), 1–6 spinner, pencil and paper clip (or dice)
● To start, the players agree on who is playing positive and who is playing negative. The player who is playing negative is aiming to get the counter to –15, and the player who is playing positive is aiming to get the counter to +15.
● The counter is placed on zero on the number path.
● Each player, in turn, spins the spinner (or throws the dice) and then moves the counter back (for playing negative) or forwards (for playing positive) by that number of spaces.
● The first player to get the counter to –15 or +15 wins the game, depending on who is playing negative and who is playing positive.
You could use Rock, Paper, Scissors to decide who is playing positive.
Let’s Look Back 4
1. A bus left Kerry at 9:15 a.m. and arrived in Donegal at 3:00 p.m. What was the duration of the journey? hrs mins
4. Solve using the order of operations: 14 + 8 × 7 – 54 ÷ 9 =
7. Fill in the blanks.
2. Write 1:20 p.m. as 24-hour time.
3. True or false? A square has a rotational symmetry of order 2.
5. Write 17:53 as 12-hour time.
6. ✓ the square numbers. 16 25 42 81
8. Each episode of Dara’s favourite TV series lasts for 1 hour and 15 minutes. How much time does he spend watching 7 episodes? hrs mins
9. Plot these points on the coordinate plane:
(a) The library is of the bank.
(b) The school is of the shop
(c) The library is of the hotel.
(d) The bank is of the café.
10. ✓ the statements that describe a value above zero.
A (4, 2)
B (0, 8)
C (7, 6)
D (6, 6)
(a) The temperature was –3°. (b) Mia scored 9 goals.
(c) My bank balance is €43.
1. Model as arrays and solve:
(a) 12 = rows of
(b) 20 = rows of
3. An artist spends 5 hours and 30 minutes painting every Sunday. How long does she spend painting over 5 weeks? hrs mins
5. Which of the shapes is the reflection of D in the red line?
2. Put these in order from the lowest value to the highest: +35m, +114m, −156m
4. Write as an expression with brackets: A farmer had 58 cattle. After selling 4 of them, he divided the remaining cattle equally between 6 fields
6. True or false? 9 is the second square number
7. Write the time shown in the image as 24-hour time.
8. Without adding, write the total of the first six odd numbers.
9. Gymnastics begins at 10:35 and ends at 11:10. How long does it last? mins
10. Fill in the blank. When you translate a shape up, the y-coordinate gets
1. What is the fourth triangular number?
3. Calculate the total time that it will take to travel the three journeys shown in the timetable.
Connolly Station 05:40 08:20 06:35 08:49 07:00 09:28
Tuam
6. To which shape will A be transformed if you…
(a) translate it 3 up? (b) translate it 4 to the right?
2. Put these in order starting with the greatest: –€45, €12, €38
4. Solve each of these indices: (a) 62 = (b) 72 = (c) 32 = (d) 12 =
5. Lexi’s mum teaches 2 piano lessons per week for a total of 1 hour and 20 minutes. If each lesson is the same duration, how long does a lesson last? mins
7. Write ‘S’ beside the square number, ‘R’ beside the rectangular number and ‘T’ beside the triangular number 30 49 15
8. True or false? If you were facing NW and you turned 90° clockwise, you would be facing NE.
9. Write the coordinates of these points:
10. A drama workshop started at 1:45 p.m. and finished at 3:25 p.m. For how long did it last? hr mins
Try this!
1. Complete the T-chart for this pattern.
2. Complete the equations using the missing operators (+, −, ×, ÷) and brackets if necessary.
(a) 15 9 6 4 = 0 (b) 2 3 4 6 = 4
3. Jay slept for 7 hours and 42 minutes last night. If he were to sleep for the same duration every night, how much sleep would he get in a week? hrs mins
4. Jay is following a pizza dough recipe. After making the dough, he leaves it in the fridge at 4:15 pm on Tuesday and takes it out on Friday at 1:40 pm. How many hours and minutes did the dough spend in the fridge? hrs mins
A Trip to Galway City
The Williams family are on holiday in Ireland. They want to rent a car and travel to Galway. Look at the information below and answer the questions.
Car rental options
● Elite Cars: €75 per day
● Drive Star: €100 plus €60 per day
● Top Rental: €499 per whole week plus €80 for each extra day
1. Which rental company does the expression below match? Total price = + ( × )
2. They need to rent a car for 2 weeks and 3 days. How many days is that?
3. Which company offers the best value option?
4. Calculate the difference between the least and most expensive options. €
B Let’s investigate!
Would your answer to question 3 in above change if the family wanted to rent a car for a different number of days? Complete the table to investigate.
Number of days (a) Elite Cars (b) Drive Star (c) Top Rental
Explain your conclusion.
Look at this map showing the route of a boat trip that the Williams family will take. Answer the questions.
1. Write the coordinates of…
(a) Galway City Docks
(b) Inis Mór
(c) Cliffs of Moher
2. A group of tourists leave the boat at the Cliffs of Moher to go hiking
(a) They go 3 units East then 2 units North. What are their new coordinates?
(b) When facing North, they then turn 135° anti-clockwise. Which direction are they facing now?
Galway City Docks
Inis Mór
Cliffs of Moher
D Look at the boat trip schedule and answer the questions.
1. Convert each time to 12-hour time.
Boat trip schedule
time
Depart Galway City Docks
time
09:30 (a)
Arrive at Inis Mór 11:00 (b)
Depart Inis Mór 15:30 (c)
Arrive at Cliffs of Moher 16:15 4:15 p.m.
Depart Cliffs of Moher 16:30 (d)
Arrive at Galway City Docks 18:00 (e)
2. How long will they spend on Inis Mór?
3. How long is the boat trip, from departing Galway City to arriving back again?
4. If they return to their hotel 2 hours and 15 minutes after they arrive back at Galway City Docks, at what time will that be in... (a) 12-hour time? (b) 24-hour time?
E Maths eyes
This one has been done for you!
Mrs Williams would like to buy an Aran jumper on Inis Mór She hopes to find one in this pattern. Look at the pattern and answer the questions.
1. How many lines of symmetry does this section of the pattern have?
2. What is the order of rotational symmetry of the pattern? Order Answer these.
Draw the lines of symmetry on the image to help you!
1. Look at the sticks used to model the pattern in above Complete the T-chart for this pattern.
2. What type of number are the numbers in the Rhombuses column?
Try this! Create a similar T-chart in your copy to represent the number of sticks in each term. Work out how many there will be in the 4, 5, and 10 terms.
Spending and Budgeting
Jay’s brother has a banking app on his phone that helps him keep track of money he receives and money he spends. Look at his phone app statement for December and answer the questions below.
1. How much did he spend in total for the month?
2. How much money came into his account in total in December?
3. What was the difference between the amounts in question 1 and question 2?
4. If Jay’s brother’s balance at the end of the month is €35, how much did he have before Granny sent him money on Dec 3rd?
Let’s talk!
Discuss the options below with your group. What are the advantages and disadvantages of each? Which option would you choose? Explain why.
1. Colm wants to buy a popular computer game. Digital download. Pay €70 with
2. Ava wants to buy a cinema ticket and treats. Buy ticket and treats in cinema. Pay €19 with
Buy second-hand disc with from savings.
Buy ticket and treats in cinema. Pay €19 with
(Buy now, pay later.)
Plan a class party for 24 children. Your budget is €100 and you can buy any of the items below. Discuss the budget and make a plan with your group. You should try to keep some money for unexpected costs.
Value for Money
A Calculate the unit price for each. ✓ the one that is the best value for money.
A 4-pack for €6
B 12-pack for €15
C 18-pack for €18
Unit price: €
Build it! Sketch it! Write it!
Unit price: €
Unit price: €
Model these to work out which option in each group is the best value for money, A, B or C.
A 1 packet raisins for €065
B 6-pack raisins for €320
C 12-pack raisins for €600
A 1l milk for €125
B 2l milk for €240
C 2 × 2l milk for €460
A 1 bottle water for €1
B 4 bottles water for €320
C 8 bottles water for €600
Best value option: Best value option: Best value option:
C Look at Dara. Then compare the prices in two supermarkets below and ✓ the better-value option for each ingredient.
6kg chicken
2.25kg onions
200g ginger
2.5kg chopped tomatoes 3.2l coconut milk
curry powder
My dad and I are making a special family lunch. Here are the ingredients we need to make chicken curry with rice for 30 people.
1. Chicken
2. Onions
3. Ginger
4. Chopped tomatoes
5. Coconut milk
6. Curry powder
7. Rice
Savings and Loans
A Find the interest and amount using the T-charts.
In what other ways could the interest be calculated?
Amount at end of year + €500
B These are interest rates offered by local banks on savings. Complete the table to identify the value of the savings at the end of 1 year and 3 years.
Savings: Money you lend to the bank.The bank pays you interest.
These are interest rates offered by local banks on loans. Complete the table to identify the value of the loan owed at the end of 1 year and 3 years.
Loan: Money the bank lends to you. You pay them interest.
Let’s talk!
What strategies did you use to calculate the answers to and above?
Use a calculator to check your answers. How did you do it? Compare the interest rates on savings and loans. What do you notice? Why do you think this is?
For number 4 in , I multiplied 2,000 × 0.09 to work out the interest. I multiplied 2,000 x 1 27 to work out the amount after 3 years.
E Solve these.
1. Sophie borrowed €8,000 to pay for a new kitchen. The interest rate is 7% per year, and she will pay back the loan over 4 years. How much…
(a) interest will she pay? €
(b) will she pay back in total? €
2. Liam got a gift of €5,500 from his granny. He put it in a savings account with an annual interest rate of 3% for 3 years. How much…
(a) interest will he earn? €
(b) will he have in total after 3 years? €
3. Aoife borrowed €10,000 to help pay for college. The interest rate is 8% per year, and she will pay back the loan over 6 years. How much…
(a) interest will she pay? €
(b) will she repay altogether? €
Solve this.
Amy’s parents are planning a family holiday. It will cost €10,000. They looked at three different options to pay for it which may suit them.
1. Complete the table to compare their options.
Sunny Florida
Family holiday for €10,000 Includes flights, hotel, airport bus and theme park tickets
Loan/Payment Options
Total interest to pay
Total amount to repay
2. How long would it take to save up for the price of the holiday if two people were able to save €50 each per week?
Let’s talk!
Look at above and discuss the following with your group:
● Which loan do you think is the best option? Why?
● Do you think the price of the holiday would still be the same after that length of time? Why or why not?
● Do you think getting a loan is the best way to pay for a holiday? Why or why not?
A Let’s talk!
Look at Dara talking about B below.
Describe the other images in B below in a similar way.
In the first one, for every 4 bikes, there are 3 cars.
B For each image, express as a ratio of...
1. bikes to cars
C Model and solve these.
2. bees to butterflies
We also use a colon to compare and represent the relationship between quantities.
1. Dara is making a pattern with shapes. For every 3 circles, he has 4 squares. Write the ratio of circles to squares. :
2. Lexi is making a chain. For every 5 red beads, she has 2 purple beads. Write the ratio of purple beads to red beads. :
3. Jay is diluting orange squash. For every part of squash, he adds 8 parts of water. Write the ratio of squash to water. :
4. On the school tour, groups were made of 6 children with every adult. Write the ratio of adults to children. :
Try this! Build it! Sketch it! Write it!
Build or sketch models to match each sentence. Write each as a ratio
1. In a tower of cubes, for every yellow cube, there will be 5 green. :
2. 3 4 of the balloons in a pack are silver The rest are gold. :
3. There are twice as many blue marbles as red marbles in the jar :
4. One third of the scones in a basket are brown scones. The rest are fruit. :
5. There are 50% more apples than bananas in the box. :
3. basketballs to soccer balls
4. blue flowers to red flowers
strawberries to oranges
circles to squares
Equivalent Ratios
A Express as a simplified ratio of…
Always use a simplified ratio, which uses the smallest possible numbers, e.g. 2 : 6 – 1 : 3
1. circles to squares : 2. forks to knives : 3. puppies to kittens :
4. chairs to stools :
B Model and solve these.
5. red counters to yellow counters :
6. blue cubes to yellow cubes :
1. For every 8 correct answers in his test, Amir answered 2 incorrectly. What was the ratio of correct to incorrect answers? :
2. There are 24 pencils and 30 pens in a box. What is the ratio of pencils to pens? :
3. Eoin has €50. Irina has €70. What is the ratio of Eoin’s money to Irina’s money? :
4. There are 56 ash trees and 49 oak trees in a wood. What is the ratio of oak trees to ash trees? :
5. In a box of 60 apples, there are 18 red apples and the rest are green. What is the ratio of red apples to green apples? :
C The ratio of bikes to cars in a car park is 2 : 1. True, false or possible (T, F, P)?
There are…
1. more bikes than cars.
2. 3 times as many bikes as cars.
3. 30 bikes and 15 cars.
Try this!
1. Look at the bar chart. What is the ratio of girls to boys in…
(a) 3rd Class? :
(b) 4th Class? :
(c) 5th Class? :
(d) 6th Class? :
2. What is the ratio of boys in 3rd Class to boys in 5th Class? :
3. What is the ratio of girls in 3rd Class to girls in 5th Class? : 1 3
Calculating Ratios Using Bar Models
A Use the bar models to help you solve these.
1. The ratio of the capacity of the bucket to the sink is 3 : 4. If the capacity of the sink is 12 litres, what is the capacity of the bucket? l
2. The ratio of a length of a rectangle to its width is 5 : 3. If the width is 15cm, what is the length? cm
3. Jay and Lexi shared some books between them in the ratio of 1 : 3. If there were 36 books, how many…
(a) did Jay get?
(b) more did Lexi get than Jay?
B Model and solve these.
1. Grandad gave Marie and James €35, which they divided between them in the ratio of their ages. If Marie is 12 and James is 9, how much did they each get? Marie: € James: €
2. The Hogan and Cole families were out for dinner together They divided their bill of €126 in proportion to the number of people in each family. If there are 4 people in the Hogan family and 5 people in the Cole family, how much did each family pay? Hogans: € Coles: €
3. The length of a rectangle is 8cm longer than its width. If the ratio of the length to the width is 3 : 2, calculate the (a) length cm (b) width cm (c) perimeter cm
Try this!
A prize fund for an art competition was divided in the ratio of 6 : 3 : 1 for first, second and third places, respectively. If the prize for first place was worth €90 more than the prize for second place, what was the prize for…
1. (a) first place? €
(b) second place? €
(c) third place? €
2. total value of the prize fund? €
Calculating Ratios Using Ratio Tables
A Calculate the missing figures in each ratio table.
1. On a tour, the ratio of adults to children is 1 : 8.
2. A car can travel 55km for every 5 litres of petrol.
3. A map scale shows that 3cm on the map represents 10km on land.
B Let’s talk!
Look at number 1 in A above How many adults would be needed if there were…
● 60 children on the tour?
● 90 children on the tour?
Explain why.
C Create ratio tables to help you solve these.
1. Mia is diluting blackcurrant squash. The instructions say to mix squash and water in the ratio of 1 : 4.
How much water is needed for
(a) 20ml squash? ml (b) 50ml squash? ml
(c) 150ml squash? ml
How much squash is needed for (d) 200ml water? ml (e) 100ml water? ml
(f) 1 litre water? ml
2. Jay’s dad said that for every €4 Jay raises for a charity, he will donate €9. How much will Dad have to donate if Jay raises…
(a) €16? € (b) €48? € (c) €120? €
How much has Jay raised if Dad has to donate… (d) €27? € (e) €81? € (f) €180? €
Use a ratio table to help you.
Try this! Look at the recipe. How much of each ingredient would be needed to make…
1. 24 cookies?
2. 36 cookies?
3. 6 cookies?
4. 30 cookies?
Cookies recipe (makes 12) 150g flour 80g sugar 100g butter 1 medium egg
Number of players: 2–4
Price Tag Deck
You will need: mini-whiteboard and marker per player, deck of cards with picture cards and 10s removed
● The players decide on a target amount of money (e.g €50).
● Each player, in turn, draws four cards and arranges them to create a ‘price’ For example, if a player draws 2, 5, 1 and 9, they could make €25·19 or €12·59.
● Each player writes their chosen price on their mini-whiteboard.
● After three rounds, each player totals their three prices to get their total spend.
● The player whose total spend is closest to the target amount (without going over) wins the game.
Variations
1. Players may swap one card with an opponent. For example, on their turn, a player can force an opponent’s total amount up by swapping a higher-value card for one of the opponent’s cards.
2. Players must add 20% (rounded, if necessary) or a fixed ‘service charge’ of €5 to their total spend.
Number of players: 3–6
Write-Hide-Show Equivalent Ratios
You will need: mini-whiteboard and marker per player, 1–6 spinner
● One player is the caller The caller spins the spinner and calls out the target ratio based on the number spun. For example, if a 1 is spun, the ratio is 1 : 1; if a 2 is spun, it is 1 : 2; if a 3 is spun, it is 1 : 3, and so on.
● Each player writes a ratio that is equivalent to the target ratio on their mini-whiteboard. The player then turns their mini-whiteboard upside down to hide the answer
● When ready, the players flip over their mini-whiteboards at the same time to show their answers to the caller.
● Each player who has written a correct answer scores a point. Any player who has written a correct answer that no one else has written scores 3 points.
● The player with the highest score when time is up wins the game.
Variation
● Play as above using the 0–9 spinner If zero is spun, spin again until you get a different digit.
Play for the time allocated by your teacher.
Introducing Volume
What is the volume of each shape in unit cubes?
Complete the T-chart to show the volume of each shape.
Volume is the amount of space something takes up.
Explain how you got your answers.
If the pattern in above continues, calculate the volume of cubes in…
1. shape 4
2. shape 5
3. shape 6
I can see a pattern in the shapes in
Try this! Two of the shapes in above have the same volume of cubes, even though the arrangements are different. Draw 2 other shapes that have the same volume as these with different arrangements.
One cube has been drawn to show how to start your drawing.
Volume of Cubes and Cuboids
Each side of this cube is 1cm so its volume is 1 cubic centimetre. I can work out some of these without counting all the cubes I see. 1cm 1cm 1cm
What is the volume of each shape in cubic centimetres?
Let’s talk!
What strategy did you use to work out the volume for each shape in above?
Is there a way to measure the volume of each of the shapes that does not require counting all the blocks? If yes, explain how. What strategy could be used to work out the volume for each transparent shape in below?
Answer these.
1. What is the volume of each transparent shape in cubic centimetres?
2. Which of the shapes above has the… (a) greatest volume? (b) least volume?
Try this! The volume of the stack of boxes below is 1 cubic metre. Each dimension measures 1m.
1. Complete these facts: (a) 1 cubic m = boxes (b) 2 cubic m = boxes
2. If there was a stack of boxes 4 boxes long, 4 boxes wide and 2 boxes high, what would the volume of the stack be in cubic metres? Use isometric grid paper (PCM 7) to sketch it out. cubic m
Volume of Other 3-D Shapes
The children have built models to estimate the volume of these 3-D shapes. Match each 3-D shape to its model.
Record the estimated volume of each of the models in above.
(a) cubic units (b) cubic units (c) cubic units (d) cubic units (e) cubic units
Estimate the volume of the transparent shapes in cubic cm.
Cubes in a layer:
Number of layers:
Estimated volume: cubic cm
Cubes in a layer:
Number of layers: Estimated volume: cubic cm
Cubes in a layer:
Number of layers: Estimated volume: cubic cm
Let’s talk!
Cubes in a layer:
Number of layers: Estimated volume: cubic cm
Think of a different 3-D shape for which the volume could not be estimated in the same ways as done in above Explain why. How might we estimate the volume of those shapes?
C
Complete this chart.
Capacity
Fraction (b) Decimal (c) Millilitres (ml)
Fill in the blanks. 1. 4,000ml = l 2. ml = 325l 3. 2,500ml = l 4. ml = 5 3 4 l 5. 6,100ml = l 6. ml = 8.2l
9,300ml = l 8. ml = 7 13 1,000l 9. 35ml = l
Let’s investigate! Express each answer as a decimal of a litre.
1. (a) Estimate the capacity of your water bottle. l (b) Measure the actual capacity. . l
2. With a partner, calculate the total capacity of both your water bottles. . l
3. (a) Using your calculator, calculate the mean (average) capacity of a water bottle based on your group . l
I will use a graduated container to help me measure.
(b) Based on this information, estimate the total amount of water required to fill the water bottle of every child in your class. . l
D Maths eyes Match each item to its most likely capacity.
How much liquid is there in each container?
Estimate the amount of liquid that would be in each container if filled to the dotted line.
Try this!
1. Look at the jugs below.
Some water has now been poured from jug A into jug B.
2. There were 2 litres of juice in the jug before Mia completely filled the 5 glasses. How much juice is there in each glass? ml
Explain your thinking.
If the capacity of jug B is 1l, what is the capacity of jug A? ml
3. Dara is using a bottle to fill Monty’s bathtub So far, he has poured in 6 full bottles of water, and the bathtub is a quarter full. When one bottle of water is poured into two jugs, it looks like this:
Calculate the capacity of Monty’s bathtub in litres. l
Complete this chart. (a) Fraction (b) Decimal (c) Grams (g) 1. 1kg kg g
Remember: Every 1,000g = 1kg, and every 1kg = 1,000g.
Fill in the blanks. 1. 2,000g = kg 2 g = 525kg 3. 4,500g = kg 4. g = 7 3 4 kg 5. 8,100g = kg 6. g = 3.6kg 7. 10,700g = kg 8. g = 6 17 1,000kg 9. 105g = kg
C Let’s investigate! Express each answer as a decimal of a kilogram.
1. (a) Estimate the weight of your pencil case. . kg (b) Measure the actual weight. kg
2. With a partner, calculate the total weight of both your pencil cases. kg
3. (a) Using your calculator, calculate the mean (average) weight of a pencil case based on your group. kg
(b) Based on this information, estimate the total weight of pencil cases of every child in your class. kg
D Maths eyes Match each item to its most likely weight.
Answer these.
1. Complete the table below.
Material
for 1 bed
We’re planting vegetables at school. These are the materials we need to make beds.
for 4 beds
(a)
175kg (b)
1 4 kg (c)
2. The children have 10kg of soil, 8kg of compost, and 2kg of fertiliser If they make beds using the amounts listed above, which material will run out first?
3. If 60% of the compost is used, what weight will be left? Write the answer in two different ways.
Complete the chart.
Ingredient (a) Fraction (b) Decimal (c) Grams (g)
1. Butter kg 02kg g
2. Flour kg 0 kg 250g
3. Carrots 3 10 kg 0 kg g
4. Sugar kg 015kg g
Answer these.
1. This pack of mandarins weighs 600g What is the weight of 8 of these packs in kg? kg
2. This pack of lettuce leaves weighs 75g What is the weight of 12 of these packs in kg? kg
3. This scoop can hold 150g of flour The bowl can hold 4 times as much. How many kg of flour can 5 of these bowls hold? kg
Try this!
1. Out of a 5kg container of carrots, 1 7 10 kg has been sold. What is the weight of carrots left in grams?
I can use these ingredients to make a carrot cake!
2. An amount of potatoes was placed on the scales and this was on the screen: 0·93kg . What weight of potatoes must be added to the scales to bring it to 2kg?
3. If Jay buys 2,500g of potatoes and 2,000g of carrots, how much change will he get from €10? €
Units of Measurement
Match each metric prefix to its meaning.
1. kilo- thousandth
2. milli- hundredth 3. centi- thousand
In your copy, complete as many of these as you can:
1,000 = 1 100 = 1
For example, 1,000g = 1kg You can use all types of measurements.
In pairs, match each item to its estimated size and unit of measurement.
D List a suitable item to measure for each category, and select the most appropriate unit of measurement for that item.
Length (mm, cm, m or km) (b) Area (mm2 , cm2 , m2 or km2)
Item
2. Unit of measurement
Let’s talk!
With a partner, explain your answers to D above What might the units of measurement below mean, and when might they be used? milligram (mg) kilowatt (kW)centilitre (cl)
Converting Between Units of Measurement
Remember, there are only 100cm in 1m!
Convert the units of measurement using the branching bonds. 1. 2. 3. 4.
Convert the units of measurement.
1. 1,052m = km 2. 8.493kg = g 3. 2,004mm = m 4. 271l = ml 5. 637m = km 6. 0.45kg = g
I’ll use the ‘moving digits’ strategy for some of these.
Complete the tables.
D Write <, > or = to make these true.
Try this!
1. Lexi was diluting blackcurrent squash. For every 1 10 l of water, she added 15ml of squash. How much squash did she add to 2 litres of water? ml
2. In a shopping trolley, there was a 456g jar of jam, a 0.75kg box of breakfast cereal and a 4 1 2 kg bag of potatoes. What was the total weight of the items in the trolley? kg
3. Mia had to take medicine for 7 days. She took 75ml twice a day for 3 days, followed by 5ml twice a day for 4 days.
(a) How much medicine did she take altogether? ml
(b) How much medicine was left over from a 1 4 l bottle? ml (a) (b) Fraction (c) Decimal 1. 3,700ml l l 2. 30cm m m 3. g g2.9kg 4. m 13 100km km (a) (b) Fraction (c) Decimal 5.
Let’s talk!
Measuring Likelihood
We can use words and phrases to describe the chance of an event happening.
Discuss where you might place the following statements on the scale above, according to the chance of the event happening:
(a) Tomorrow will be a Tuesday.
(b) The principal will visit your classroom today.
(c) Ireland will win the football World Cup within the next 50 years.
(d) The next person to walk into the classroom will have black hair
(e) You flip a coin and the coin will land on Heads.
Let’s talk!
As well as words, we can assign fractions and percentages within the scale to describe chance. We call this ‘probability’.
Look at the scale in above In pairs, take turns to identify the missing fraction and percentage for each of the points on the scale.
Look at the image and answer the questions below.
Using the scale above, express as (a) a fraction and (b) a percentage, the probability of Dara pulling out a…
1. marble of any colour (a) (b)
2. black marble (a) (b)
3. yellow marble (a) (b)
4. green marble (a) (b)
5. blue marble (a) (b)
6. yellow or red marble (a) (b)
7. marble that is not yellow. (a) (b)
Try this!
1. Draw 10 marbles in the jar so that the following statements are true.
● The probability of pulling out a yellow marble is 50%.
● You are more likely to pull out a yellow marble than a red marble.
● You are more likely to pull out a red marble than a blue marble.
● You are unlikely to pull out a colour that is not yellow, blue or red.
2. With a partner, write as a (a) fraction and (b) percentage the probability of pulling out a marble of each colour from your jar.
There are 10 marbles in total.
Answer these.
Possible Outcomes
1. Imagine randomly picking a ball from bag A and another ball from bag B below. Complete the branching to show all of the possible combinations.
AB Bag ACombinations
2. How many different combinations can be made in total?
3. What is the probability of getting red and white? Express this as a fraction.
4. If a third bag containing one black and one purple ball was added, how many possible combinations could be made with one ball from each bag?
B Let’s investigate!
Use your miniwhiteboard to help work out the possible combinations!
Look at Mia. She has forgotten the code to open her money box.
I know that it’s a 3-digit code, and each digit is on a different row. The first digit is either 1 or 2, the second is either 3 or 4, and the third is either 5 or 6.
1. On your mini-whiteboard, record all of the possible combinations for the code. How many different combinations can be made in total?
2. Express, as a fraction, the probability that Mia will guess the correct code using the possible combinations.
Let’s talk!
Look at above.
Do you agree with Dara? Why or why not?
Try this! Three children were sitting in the backseat of a car In your copy, write out all of the possible combinations for the order of boys (B) and girls (G) in the backseat.
I think there’s a 50% chance of guessing Mia’s code, because each digit has a 50% chance of being correct.
Hint! A possible combination could be B-B-G.
Chance Investigations
Before you do any of these investigations, consider the possible outcomes and make your best predictions for these outcomes.
A Let’s investigate! Mystery Bag
You will need: mini-whiteboard and marker per person, 10 cubes – 3 blue and 7 yellow, non-transparent bag
1. In theory, express as a fraction the chance of pulling out a…
(a) blue cube
(b) yellow cube
2. If you pulled out a cube at random a total of 30 times, how many times would you expect to get a…
(a) blue cube?
(b) yellow cube?
B Let’s investigate! Card Battle!
You will need: mini-whiteboard and marker per person, two sets of three playing cards – Set A containing 2, 4 and 9, and Set B containing 3, 5 and 7
Each player has a set of cards. The players shuffle their cards and place them face down on the table. Each player draws one card at random from their set and turns it over The player whose card has the higher number scores a point.
1. In theory, the chance of scoring a point if I draw…
(a) 2 is % (b) 9 is %
2. If I played 20 rounds of this game with my partner, I would expect… ✓
(a) Set A to score the most points
(b) Set B to score the most points
(c) Sets A and B to score an equal number of points
C Let’s investigate! Triple Coin Toss
You will need: mini-whiteboard and marker per person, three coins
1. In theory, the chance of tossing…
(a) three heads is
(b) three tails is (c) two heads and one tail is (d) one head and two tails is .
2. If you tossed the coins a total of 40 times, how many times would you expect to get… (a) three heads? (b) three tails?
(c) two heads and one tail?
(d) one head and two tails?
Let’s talk!
After you have made your predictions, discuss and agree on the following with your group:
● How will you conduct these investigations?
● How will you make sure they are fair?
● How many turns for each person?
● How will you record your results?
● How will you share your findings?
E Let’s investigate!
Once you have discussed and agreed upon the questions in D above with your group, conduct the investigations (on page 156).
Let’s talk!
After each investigation, discuss the following with your group:
● How did you do it?
● How did you make sure it was fair?
● What were your findings?
● Were you surprised by your findings? Explain why.
Try this! Lexi and Jay are taking turns to toss a coin and throw a dice. For the coin toss, heads is worth a value of 1, and tails is worth a value of 2. They add the value of the coin toss to the number on which the dice lands. If the total is an odd number, Lexi scores a point. If the total is an even number, Jay scores a point.
What are the possible outcomes for this game?
1. In your copy, use branching to work out all of the possible outcomes.
2. Is this a fair game? How can you prove this?
G Let’s play! Car Race
Number of players: 2-6
You will need: Car race game (see PCM 8), two dice
● Each car in the race is assigned a number, from 2 to 12.
● The race begins by throwing two dice, and adding the two outcomes. The car with the number that matches this total then moves forward.
● Cars may only move forward on the track, one space at a time.
● This is repeated until the winning car has crossed the finish line.
Which car would you choose to give yourself the best chance of winning the race?
● Throw the dice 10–20 times and keep track of the movements of each car.
1. Write the ratio of dogs to cats. :
3. Complete these.
(a) 3·4l = ml
(b) 1 1 4 kg = g
(c) 1,700ml = l
(d) 950g = kg
5. What is the probability of pulling a green sock from the drawer? %
7. What is the volume of this shape in unit cubes? unit cubes
9. Express as a fraction the probability of throwing an even number with a 6-sided dice. Let’s Look Back 5
1. Convert the unit of measurement.
1 4 kg = g
3. A bar of chocolate weighs 180g
There are 16 bars of chocolate in a box. How much does one box of chocolate weigh in kilograms? kg
5. Mia’s dad borrowed €6,000 from the bank. The interest rate is 8% per year, and he will pay back the loan over 4 years. How much in interest will he pay…
(a) in year 1? €
(b) by the end of year 4? €
2. There are 2·85kg of soil, 800g of stones and 1 3 4 kg of plants in a wheelbarrow What is the total weight of the items in the wheelbarrow? kg
4. ✓ the reasonable measurements.
(a) Capacity of a cup = 0·2l
(b) Area of a classroom = 15cm2
(c) Length of a maths copy = 200mm
6. ✓ the one that is better value for money.
(a) (b)
8. Lee orders all three items on the menu in a meal deal for €12. How much does he save? €
10. There are 12 children and 32 adults on a bus. What is the ratio of children to adults? :
2. A bag contains 4 yellow cubes and 6 green cubes. Express as a fraction the probability of pulling out a cube that is…
(a) yellow (b) green
4. Complete the branching to work out the possible combinations that Lexi could get by flipping a coin twice.
6. A multipack of 6 pens costs €12. Find the unit price for one pen. €
7. 0·304m = mm
8. Jay currently has €89·40 in savings. If he buys a jacket that costs €64·99, how much will he have left in his savings? €
9. On Friday, a deli sold more rolls than wraps at a ratio of 3 : 2. 24 wraps were sold. Use the bar model to work out the sales of…
(a) rolls
(b) rolls and wraps altogether
1. True or false? 4 packets of popcorn for €3·20 is better value than 6 packets for €5·10.
3. If you throw a 6-sided dice, express as a fraction the probability of throwing…
(a) a number greater than 4
(b) a 1
5. For the numbers 1 to 10, what is the ratio of composite numbers to prime numbers? :
7. Calculate the missing figures in the ratio table for this: A gardener was planting sunflowers He planted one sunflower every 30cm.
2. What is the probability of picking an odd number between 1 and 10? %
4. What is the volume of the cube?
cubic cm
6. A bedroom window of a house is 120cm wide. If the bathroom window is 1 2 m narrower, what is the width of the bathroom window in cm? cm
8. If all of the water from this jug is poured into a bowl already containing 1·75l of water, how much water will the bowl contain then? ml
9. Three farmers sold tomatoes at a market. Sam sold 2 3 4 kg, Cara sold 2,698g and Tom sold 2·8kg Who sold the most tomatoes?
1. Lexi used base ten blocks like the one shown to build a cube with sides of 6cm. What was the volume of the cube that she built? cubic cm
2. The petrol tank in a car holds 72l, but currently it is only 1 4 full.
(a) How much would it cost to fill up the tank at the price shown? €
(b) How much cheaper would it be to fill up the tank if the car ran on diesel? €
3. Mystery number! Look at Jay.
(a) List all of the possible numbers.
I’m thinking of a 2-digit number that is greater than 20 and less than 60. Both of the digits are even.
(b) Express as a fraction the probability of Jay’s number being 24.
4. Look at the recipe. How much of each ingredient would be needed to make…
(a) 16 pancakes?
(b) 4 pancakes?
(c) 32 pancakes?
Answer these.
Sports at the Stadium
I’ve taken my dad to a rugby match for his birthday!
Ticket prices
Adult train ticket (return) €15
Child train ticket (return) €10
Adult match ticket €27
Child match ticket €20
1. Write an expression for the total cost of tickets (match and train) for:
(a) 1 adult and 1 child
(b) 3 adults
(c) 2 adults and 3 children
2. Calculate each expression in question 1.
Look at the price list and answer the questions.
1. Write the capacity in litres of the…
(a) small bottle of water l
(b) regular bottle of water l
2. Which is the best-value option for (a) water? (b) popcorn?
Let’s talk!
If you were Mia, which options for buying water and popcorn would you choose in above? Explain why.
D Answer these.
1. Use these ratios to complete the table:
● Rockhill FC lost to won = 3 : 8
● Emerald Athletic won to drew = 2 : 1
● Clover United drew to lost = 1 : 3
● Blairstown FC won to lost = 2 : 9
● Oakfield United lost to drew = 9 : 1
A Large water (1l) €2·70
B Regular water (750ml) €1·70
C Small water (500ml) €1·00
D 1 packet of popcorn €0·50
E 6-pack of popcorn
F 12-pack of popcorn
Rockhill FC 16 0 (a)
Emerald Athletic (b) 57
Clover United 10 3 (c) Blairstown FC (d) 018
Oakfield United 22 (e)
2. Write, in simplest form, the ratio of goals scored to goals conceded (let in) for each of these.
(a) Harptown Rovers scored 18 goals and conceded 60. :
(b) West Isle Wanderers scored 40 goals and conceded 28. :
Solve these.
1. Jay went to a Gaelic football match between Cork and Kerry. He sat behind a row of 15 Cork supporters. Of these, 4 wore a hat, 5 wore a scarf and 6 wore neither If one of these people was chosen at random from the row, what is the probability, as a fraction, that they would be wearing…
(a) a scarf?
(b) a hat?
(c) neither a scarf nor a hat? (d) either a scarf or a hat?
2. Jay passed 3 Kerry supporters on the way out. Their names were Henry (H), Sam (S) and Nick (N). Complete the branching diagram to work out the possible orders they were sitting in.
Answer these.
1. The GAA pitch where Mei works measures 137m by 82m. Convert these values to (a) km and (b) cm.
(a) km by km
(b) cm by cm
This is my aunt Mei. She works as a groundskeeper at her local GAA club.
2. Below is some equipment that is used to look after a pitch. Match each item to its most likely weight.
3. Convert the kilogram weights in question 2 above to grams, and the grams to kilograms. (a) (b) (c) (d)
4. The GAA club took out a loan of €25,400 to re-turf part of the pitch. They paid the loan back over 5 years and paid 6% interest.
(a) Calculate the amount they paid in interest. €
(b) Calculate the total amount they repaid. €
CORK
End-of-year Challenge
Your mission is to organise an end-ofyear maths quiz for 5th Class. Here’s what your group must do.
Maths Quiz
● Create a questions bank.
● Plan teams and seating
● Create a scoreboard and rules.
● Design a maths trophy.
Questions bank
Your group must create a question bank for three rounds of questions.
A great quiz needs variety, so each round should include:
● six quick-fire mental maths questions.
For example: How many millimetres are there in a metre?
● two estimation questions using classroom objects.
For example: Estimate the length of the teacher’s desk.
● two word problems.
For example: If you read 49 pages of a 100-page book, what percentage of the book have you read?
If two or more teams finish with the same score, you will need to ask a tie-breaker question. Create at least one question that is…
● maths-based (but can be creative)
● challenging enough to break the tie
● not going to take too long to answer
Use this Pupil’s Book to help you come up with the questions!
Tie-breaker ideas
Mini-puzzle
Maths riddle
Quick-fire calculation
Physical maths challenge (e.g create an irregular octagon with match sticks!)
Teams and seating
● Count how many children are in the class. (If you’re hosting the quiz with your own class, don’t include your own group).
● Decide on the number of teams.
● Draw a diagram of the classroom showing where each team will sit so that they cannot see each other’s answers.
Scoreboard and rules
Consider how to make the quiz as fair as possible if the numbers are uneven.
Design a chart or table to record the points, and come up with some rules. For example:
● bonus points given for the best model, sketch, etc.
● points taken away for shouting out answers.
Maths trophy
● Draw a sketch of your design for a maths trophy.
● Use 3-D shapes like cubes, spheres or pyramids in your design.
● Label the materials you would use (e.g cardboard, gold foil).
You can make the trophy, if you wish!
STEM Exploring
Look at the photo and answer the questions below.
1. What do you notice? What do you wonder?
2. Describe the bridge using maths words. Try to make five observations about lines, angles, shapes or position. (For example: ‘The towers are vertical.’)
3. How do you think the bridge stays strong and balanced?
4. What materials is it made of?
5. If you could measure part of it, what part would you choose and how would you do it?
Same but different! What is the same? What is different?
Answer these.
Height: 227m
Length: 2,737m
Date completed: 1937
Bridge type: suspension bridge
Height: 104m
Length: 987m
Date completed: 1919
Bridge type: cantilever bridge
1. List the bridges from tallest to shortest.
2. List the bridges from oldest to newest.
3. What do all these bridges have in common?
Height: 134m
Length: 1,149m
Date completed: 1932
Bridge type: arch bridge
4. Look for 2-D shapes in the bridges. Which 2-D shape appears most often? Why might this be?
5. Where have you seen bridges like these, and how were they alike?
SydneyHarbourBridge,Australia
QuebecBridge,Canada
STEM Challenges and Investigations
A Let’s investigate!
Which 2-D shape is most rigid?
Using materials such as AngLegs or geostrips, build various 2-D shapes using the least possible amount of pieces each time. Test the rigidity of your shapes. How might you do this?
Use the Investigation Planning Sheet (PCM 9) to help you.
I could apply pressure to the sides or corners to see which shapes stay the same and which ones change.
Which shape(s) were the most rigid?
Which shape(s) were the least rigid? Is it possible to alter less rigid shapes to make them become more rigid? How? Explain why this works/does not work.
B Let’s investigate!
Rigid shapes are those that keep their exact size and form even when pushed, pulled, or turned.
How can the way a sheet of card is folded affect its ability to bear weight?
Each group needs three identical sheets of card. One of the sheets should be left as it is. The other two should each be folded in different ways.
I think we could fold one into a ‘U’ shape.
We could fold another one in a zigzag pattern.
How will you investigate the ability of each sheet of card to bear weight? What weights will you use?
We could use counters, and keep adding more to each sheet of card until it collapses.
How will you ensure it is a fair test? What will you keep the same each time? What will be different each time?
Use the Investigation Planning Sheet (PCM 9) to help you.
Design and Make Cycle
1. Explore Think about what you could design and make as a solution.
l What do you need to do?
l Look at the available materials. What will you use?
l How will you do it?
2. Plan Plan and design your solution.
l Draw it out.
l Write a list of what will be needed.
l Discuss it with your group and explain your reasons.
3. Make Using the plan and criteria, make or build your solution.
4. Evaluate Test your solution.
l Does it satisfy the criteria?
l How well does it work?
l How could it be improved?
Design and make a bridge using classroom or found materials.
Look at this design and make cycle.
Criteria: It must be able to hold an object weighing 100g, and span a distance of 30cm.
Materials could include: foam blocks/boards, lollipop sticks, cardboard, PVA glue
D Design and make a fair test to identify the best bridge.
In engineering, the best bridge isn’t the one that holds the most weight. It’s the one that holds the most weight relative to its own weight.
A bridge’s efficiency can be calculated by dividing the maximum weight held, by the weight of the bridge itself.
How might you test the efficiency of your bridge?
How might you measure the weight of your bridge?
How might you measure the maximum weight that can be held by your bridge?
Other factors to consider:
In real life, environmental factors are considered in bridge design. For example, what would happen during high winds or an earthquake?
How could you test what might happen to your bridge due to environmental factors? Use the Investigation Planning Sheet (PCM 9) to help you.
Number of players: 2
Four in a Row
You will need: marker of a different colour per player, 10 × 10 grid (see page 166), 0–9 spinner, pencil and paper clip
● Both players use the same grid.
● Each player, in turn, spins the spinner twice. The first digit spun is the xcoordinate and the second is theycoordinate. The player marks that point on the grid.
● The first player to get four points in a straight line (horizontal, vertical or diagonal) wins the game.
Coordinates Battleships
Number of players: 2
You will need: two Battleship grids per player (see PCM 10), something to make a barrier
● Each player begins with two grids. To start, each player draws three ‘battleships’ on their top grid. These can be drawn anywhere, but each must measure as follows: a line of 2 points, a line of 3 points, and a line of 4 points.
● Each player, in turn, calls out a set of coordinates (e.g 3, 5) and marks this coordinate on their lower grid as a record for themselves of where they have targeted.
● If their opponent does not have a battleship on the called-out coordinate, the opponent says, ‘Miss’.
● If their opponent has a battleship on the called-out coordinate, the opponent says, ‘Hit’, and crosses out that point on their top grid.
● When the hit is also the last remaining coordinate of that battleship, the opponent says, ‘Battleship going down!’
● The player who is still afloat when their opponent’s battleships have all been sunk wins the game.
Cinderella
Number of players: 2–6
You will need: mini-whiteboard and marker per player, 1–6 spinner, pencil and paper clip (or a dice)
● Each player, in turn, spins the spinner four times, and then arranges the digits spun to make a time as close to midnight as possible (without going past 23:59) in 24-hour format. For example, if a player spins 6, 1, 5 and 2, they could make 21:56.
● If a player forms a non-existent time (e.g 26:15 or 25:61), and this is recognised by an opponent, the player misses their go
● If a player cannot make a 24-hour time using the digits spun, they miss their go
● The player with the time closest to, but before, midnight scores a point.
● The player with the highest score when the time is up wins the game.
Variations
1. The player with the time closest to, but after, midnight scores a point.
2. The player with the time closest to midnight (either before or after) scores a point.
Twenty-one
Number of players: 2–6
You will need: no equipment needed
● This game involves the players counting from 1 to 21, with each player saying up to three numbers each time.
● The first player can count 1, or 1, 2 or 1, 2, 3. The other players continue, in turn, with each player counting on one, two or three numbers. The round continues until a player must say 21, and that player is then out.
● The next round continues with the next player, who must start back at 1 again. At the end of this round, another player is eliminated, and the next round starts.
● The last player left wins the game.
Variations
1. For shorter games that do not involve eliminations, play one round only, and the player who says 21 (or an agreed end number) wins the game.
2. The players agree on start and end numbers at the beginning (e.g start at 95 and end at 120). The player who says the end number wins the game.
3. Counting in Multiples: Decide on a set of multiples. Start at 0 and count in the agreed multiple up to the 12th (or higher) multiple.
4. Counting in Fractions: Decide on a set of fractions (e.g. quarters) and an agreed start and end number
That’s My Number!
Number of players: 2–6
You will need: mini-whiteboard and marker per player, dice, pencil and paper clip
● To start, each player, in turn, chooses a number from 1–6. (The player whose birthday is next can choose first.) Each player records their chosen number at the top of their mini-whiteboard.
● Each player, in turn, throws the dice. If the number thrown has been chosen by a player, that player scores a point, which they record as a tally mark on their mini-whiteboard. For example, if Player 1 throws 6, and Player 2 has chosen 6 as their number, Player 2 scores a point and draws a tally mark on their mini-whiteboard.
● The player who gets 5 tally marks first, or who has the highest number of tally marks after an agreed number of rounds, wins the game.
Variation: That’s My Number! (Two Throws)
● Play as above, but, to start, each player, in turn, chooses a number from 1–12.
● On their turn, each player throws the dice twice and totals the numbers thrown. As above, if Player 1 throws 6 and 5 (totaling 11), and Player 2 has chosen 11 as their number, Player 2 scores a point and draws a tally mark on their mini-whiteboard.
● The player who gets 5 tally marks first, or who has the highest number of tally marks after an agreed number of rounds, wins the game.
Play for the number of rounds allocated by your teacher.
I think some numbers are more likely than others to equal the total of the two numbers spun.
Number of players: 2–4
Duration Dash
You will need: mini-whiteboard and marker per player, dice
● The players agree on a start time and a finish time (e.g 09:00 and 17:00), and write these on their mini-whiteboards.
● Each player, in turn, throws the dice. The number thrown determines a duration as follows:
● The player adds the duration for the number thrown to their current time.
● The first player to reach or pass the agreed finish time (e.g. 17:00) wins the game.
Variation
● Subtraction: Start at 12:00 and subtract durations to see who reaches 00:00 first.
Big Spender!
Number of players: 2–6
You will need: mini-whiteboard and marker per player, 0–9 spinner, pencil and paper clip
● Each player, in turn, spins the spinner four times to collect four digits.
● Each player rearranges their digits to record the largest possible amount up to €9999. For example, if 4, 7, 1 and 9 are spun, the digits are rearranged to make €97.41.
● When every player has made their first amount, they each spin four times again to make a second amount, which they add to their first amount.
● The player with the highest total wins the game.
Variations
1. Play three rounds and total the three amounts.
2. Big Saver!: Each player plays as if they have €100. The amount made from the digits spun is subtracted from €100 to calculate the change. The player with the greatest amount of change at the end wins the game.
3. Bankrupt!: Play like ‘Big Saver!’ as above, but each player starts at €200 In each round, the amount made from the digits spun is subtracted from what was left after the previous calculation. The player who is first to reach zero or an amount that they cannot subtract from wins the game.
Number of players: 2–6
What’s the Difference?
You will need: mini-whiteboard and marker per player, 0–9 spinner, pencil and paper clip
● Each player, in turn, spins the spinner five times and records the numbers spun. The highest number represents thousands, and the other four numbers have to be rearranged to make a 4-digit number to be subtracted.
● For example, if a player spun 8, 3, 6, 5 and 7, the 8 would become 8,000, and the other four numbers could become 7,653, 3,756 or 5,673, etc. This player then subtracts their 4-digit number from 8,000 to calculate the difference (answer).
● The player who has the smallest difference (answer) wins the game.
Factor Frenzy
Number of players: 2–6
You will need: mini-whiteboard and marker per player, 0–9 spinner, pencil and paper clip
● Each player, in turn, spins the spinner three times and chooses two of the digits to arrange as a 2-digit number
● The player then finds all of the factors of their 2-digit number
● For each factor they identify, they score a point, which they record on either their mini-whiteboard or a sheet of paper.
● Play continues until each player has identified their factors and recorded their points.
● The player with the most points after a set number of rounds or when time is up wins the game.
Fair Play Total
Number of players: 2
You will need: 20 counters (or other resources), mini-whiteboard and marker per player, 1–6 spinner, pencil and paper clip
● Before the game, one player is named Odd and the other is named Even. (The player whose birthday is next can choose first.)
● Each player starts with 20 counters. One of the players spins the spinner twice and totals the numbers spun. If the total is even, the player named Even takes that number of counters from the other player. If the total is odd, the player named Odd takes that number of counters from the other player
● Continue until one player has all of the counters. This player wins the round.
● Play a number of times and record what is happening.
Do you think this is a fair game? Does each player have an equal chance? Explain why.
Number of players: 2
You will need: paper and pencil per player, ruler, protractor
● The first player draws an angle of 15°, 30° or 45° and labels it with their name.
● Each player, in turn, then draws another angle of 15°, 30° or 45° at the same point, clockwise from the first angle, and labels it with their name.
● The player who labels the final angle to make 360° wins.
Variations
1. Before starting, choose a different set of three angles that make 360° for the players to draw
2. Don’t Make 360°: Play so that the player who labels the final angle loses the game.
Real or Not Real
Number of players: 2–4
You will need: triangle cards (see PCM 11)
● The triangle cards are shuffled and the stack is placed face down on the table.
● Each player, in turn, takes the top card and allows everybody to see it. The player says whether the triangle is real (i.e. the angles add up to 180°) or not real. The other players may challenge their answer
● If nobody challenges, the player keeps the card.
● If there is a challenge, the player who makes the challenge must justify it by adding up the angles to show that they do not add up to 180°. If their challenge is correct, they keep the card. If not correct, the original player keeps the card.
● When the time is up or all of the cards are gone, the player with the most cards wins the game.
Not real! 110°, 45° and 35° add up 190°.
Dara
acute angle: see‘angles’
algebra
constant:
Glossary
a + 4 = 10
a = b + 5 constants variables
a quantity or term that has a fixed value (does not change), for example, in the equation a+ 4 = 10, all of the terms are constant, and amust be 6 and cannot change or vary
equation: a number sentence with an equals symbol (=) showing that both sides (expressions) have the same value
variable: a quantity or term that changes or varies; for example, in the equation a= b+ 5, aand bare variables, whereas 5 is a constant
angles
acute angle obtuse angle reflex angle right angle straight angle
acute angle: an angle that measures less than a right angle (<90°)
obtuse angle: an angle that measures greater than a right angle (>90°), but less than a straight angle (<180°)
protractor: an instrument used to measure the size of an angle in degrees°
reflex angle: an angle that measures greater than a straight angle (>180°), but less than a full rotation (<360°)
right angle: an angle that measures 90°; wherever perpendicular lines meet, at least 1 right angle (also called a square corner) is formed
straight angle: an angle that measures exactly 180°; looks like a straight line
arc: see‘circle’
area: see‘measuring’ equations
associative property: see‘number rules and properties’ categorical data: see‘data’
chance
chance: the likelihood that a particular outcome will happen probability: the measuring of chance, often expressed as a fraction, decimal or percentage
Probability can be recorded on a scale of 0 to 1, showing the likelihood of chance that a particular outcome will occur, ranging between 0 (impossible) and 1 (certain).
sample space: the complete set of all of the possible outcomes of a chance event circle
arc: section of a curve, part of the circumference of a circle
circumference: the distance around a circle, its perimeter diameter: a straight line passing through the centre of a circle to touch both sides of the circumference
radius: the distance from the centre of a circle to its circumference
circumference: see‘circle’
common denominator: see‘fractions, decimals and percentages’
commutative: see‘number rules and properties’
composite number: see‘number’
constant: see‘algebra’
data
categorical data: information organised into groups or categories, describing qualities like colour, yes/no answers, or type, e.g favourite book genre: fantasy, sci-fi or thriller
mean (average): the total of the values divided by the number of values in a set median: the middle value in an ordered set of data values mode: the value that occurs the most often in a data set
numerical data: information that can be counted, measured and represented by numbers, e.g age, height, weight, or number of pets
range: the difference between the greatest value and the least value in a data set
Range from a bar graph Numbers o marine animals at Oceanarium
Highest score – lowest score = range
100 – 20 = 80
Range = 80
Range of a data set
Highest score – lowest score = range
Number of books read in a month: 3, 4, 5, 5, 5, 6, 6, 7, 8, 8, 9 9 – 3 = 6
Range = 6
survey: a way to collect data by asking people questions
decimal form: see‘numbers, writing’ degree (°): see‘measurement, units of’
common denominator: when two or more fractions have the same denominator (bottom number), for example: 3 12 , 8 12 ; to add or subtract fractions that are not alike, we need to convert them to ones with a common denominator
fraction form: expressing an amount using fraction names (e.g one fifth) and/ or the fraction line
improper fraction: a fraction equivalent to or larger than one whole; the numerator is larger than or equal to the denominator (e.g 2 2 )
mixed number: a value written as a whole number with a fraction (e.g. 1 1 2 )
percentage, per cent: a fraction expressed as a number out of 100 followed by the % symbol
proper fraction: a fraction smaller than one whole; the numerator is smaller than the denominator (e.g. 1 2 )
recurring decimal: a decimal number with a repeating pattern of infinite (neverending) digits
simplest form:
expressing a fraction using the smallest possible numbers for the numerator and denominator (e.g 5 10 1 2 )
fraction form: see‘fractions, decimals and percentages’
function: a mathematical relationship or rule from a set of inputs to a set of outputs
hundredth: one of a hundred equal parts
identity property: see‘number rules and properties’
improper fraction: see‘fractions, decimals and percentages’
interest: extra money that you earn for keeping your savings in a bank/ financial institution, or the fee paid to a bank/lender for borrowing money
interest rate: the percentage (%) used to work out how much extra money is added to a loan or savings account
inverse: to do the reverse/opposite calculation
isosceles triangle: see‘shapes’
lowest common multiple: see‘number’
mean: see‘data’
measurement, units of degree (°): used to measure the size of an angle, or a unit for measuring temperature
metric units: area
1 centimetre (cm) = 10 millimetres (mm)
1 metre (m) = 100cm or 1,000mm 1 kilometre = 1,000m
measuring
area: the size that a surface takes up; measured in square units (e.g cm2,m2)
perimeter: the distance around the outside of a shape; usually calculated by adding the length of all sides together
surface area: the total area of the surface of a three-dimensional object; measured in square units (e.g. cm2 , m2)
volume: the amount of space that something takes up; measured in cubic units (e.g cubic cm, cubic m)
median: see‘data’
metric units: see‘measurement, units of’
mixed number: see‘fractions, decimals and percentages’, or ‘number’
mode: see‘data’
multiple: see‘number’
negative number: see‘number’
number
composite number: a number with more than two factors factor: a number that divides evenly into another number, or numbers that are multiplied by each other to get a product
lowest common multiple: the smallest number that is the multiple of two or more numbers
mixed number: a value written as a whole number with a fraction (e.g. 1 1 2 ) multiple: a whole number that is the result of multiplying a whole number by another whole number
negative number: any number less than zero; written with a minus sign (e.g –4, –10, –37)
positive number: a number greater than zero; sometimes written with a plus sign (e.g +4, 10, +37)
prime number: a number that has exactly two factors: itself and one product: the result of multiplying numbers by each other (e.g 5 × 2 = 10)
number rules and properties
associative property: multiplication and addition are this, because when calculating with three or more numbers, it does not matter which numbers are calculated first, the answer is the same, e.g (2 x 3) x 4 = 2 x (3 x 4)
commutative: multiplication and addition are this, because the total/product is the same regardless of order; for example: 3 + 4 = 4 + 3 = 7; 3 × 4 = 4 × 3 = 12
identity property: in multiplication, any number multiplied by 1 is the number itself (e.g. 9 × 1 = 9)
order of operations: the order in which mathematical operations should be done; an acronym can remind us of the order, for example: BIMDAS (Brackets, Indices, Multiplication or Division, Addition or Subtraction)
rectangular number: a number that can be represented in the shape of a rectangle (but not a square), which is the product of two consecutive whole numbers; examples:
square number: a number that results from multiplying a whole number by itself, which can be represented in the shape of a square; examples:
triangular number: a number that can be represented in the shape of a triangle; examples:
zero property: in multiplication, any number multiplied by 0 is 0
numbers, writing
decimal form: expressing an amount using a decimal point (e.g 0.6, 1.35)
equivalent form: expressing an amount in different ways or forms, that are equal in value (‘same value, different appearance’)
expanded form: a way of writing numbers to show place value (e.g 3,876 = 3,000 + 800 + 70 + 6)
standard form: the usual way of writing numbers using digits (e.g 3,876)
word form: expressing amounts using words (e.g. three thousand, eight hundred and seventy-six)
numerical data: see‘data’
obtuse angle: see‘angles’
order of operations: see‘number rules and properties’ percentage, per cent: see‘fractions, decimals and percentages’ perimeter: see‘measuring’ positive number: see‘number’ prime number: see‘number’ probability: see‘chance’ product: see‘number’
equilateral triangle: a triangle with 3 equal sides and 3 equal angles
isosceles triangle: a triangle with 2 equal sides and 2 equal angles
quadrilateral: a polygon (2D shape with straight sides) with 4 lines and 4 angles
rhombus: a quadrilateral with 4 equal sides, opposite sides that are parallel, and opposite angles that are equal
right-angled triangle: a triangle with 1 right angle, and 2 perpendicular sides
scalene triangle: a triangle whose 3 sides are all different lengths
scalene triangle: see‘shapes’
simplest form: see‘fractions, decimals and percentages’
square number: see‘number rules and properties’
standard form: see‘numbers, writing’ straight angle: see‘angles’
surface area: see‘measuring’
survey: see‘data’
thousandth: one of a thousand equal parts
triangular number: see‘number rules and properties’
unit price: the price of a single item or quantity (e.g. the price per litre or kilogram); allows us to compare prices to find out which is cheaper/better value
