100% coverage of New Zealand Mathematics and Statistics Curriculum for Phases 1-3
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Strand(s): Number
Vocabulary
Objectives Materials Resources
• CB: pp. 1–2 • Math Pro: Recall
• Read and write a numeral within 1,000,000 given the corresponding number word
• Read and write a number word within 1,000,000 given the corresponding numeral
• Write a 6-digit number in terms of hundred thousands, ten thousands, thousands, hundreds, tens and ones
• Identify the values and place values of digits in a 6-digit number
• Compare numbers within 1,000,000
• Compare and order numbers within 1,000,000
• Complete a number pattern by counting forwards and backwards by 3s within 10,000
• Round a whole number to the nearest ten, hundred or thousand
• Find all the factors of a number that results from multiplying any two whole numbers between 1 and 10
Scheme of Work Unit
Let’s Remember
Unit 1: Numbers to 1,000,000,000
New Zealand Curriculum Year 6: Number — Number structures
1.1 Knowledge ( ★ ) and Practices ( ◆ )
★ The base 10 number system extends infinitely in two directions.
• billion • million
◆ Reading, writing, comparing, and ordering any whole number and representing them using base 10 structure
• CB: pp. 3–5
• PB: p. 1
• Math Pro: Practice
• Read and write a number within 1,000,000,000 — the numeral and the corresponding number word
Reading and writing numbers
Knowledge ( ★ ) and Practices ( ◆ )
1.1
1.2
★ The base 10 number system extends infinitely in two directions.
◆ Reading, writing, comparing, and ordering any whole number and representing them using base 10 structure
• CB: pp. 5–7
• PB: p. 2
• Math Pro: Practice
• Identify the values and place values of digits in a 7-digit, 8-digit or 9-digit number
1.2 Identifying values of digits
• Write a 7-digit, 8-digit or 9-digit number in terms of hundred millions, ten millions, millions, hundred thousands, ten thousands, thousands, hundreds, tens and ones
1.3 Knowledge ( ★ ) and Practices ( ◆ )
★ The base 10 number system extends infinitely in two directions.
◆ Reading, writing, comparing, and ordering any whole number and representing them using base 10 structure
• CB: pp. 8–9
• PB: p. 3
• Math Pro: Practice
• Compare and order numbers within 1,000,000,000
1.3 Comparing and ordering numbers
Knowledge ( ★ ) and Practices ( ◆ )
1.4
★ Negative numbers are to the left of 0 on a horizontal number line and below 0 on a vertical number line.
★ Negative numbers are represented symbolically with a negative sign (−) and named ‘negative’ along with the numeral (e.g. -4 is named negative four).
★ Zero is neither positive nor negative.
★ Negative numbers arise in a range of situations (e.g. debt, temperature).
◆ Counting forwards and backwards with positive whole numbers, including working with negative numbers (e.g. starting at -6 and counting backwards in 2s)
• CB: pp. 10–12
• PB: p. 4
• Math Pro: Practice
• Count forwards and backwards with positive whole numbers, including working with negative numbers
Number patterns
1.4
• Describe and complete a number pattern involving positive and negative numbers by counting forwards and backwards by ones, twos, threes, fours, fives or tens
Unit 2: Rounding Numbers New Zealand Curriculum Year 6: Number — Number structures
Knowledge ( ★ ) and Practices ( ◆ )
2.1
★ Rounding is based on identifying the nearest place value or unit (ten, hundred, thousand) for a given number; a number line supports this.
◆ Rounding whole numbers to the nearest million, hundred thousand, ten thousand, thousand, hundred, or ten
• CB: pp. 13–15
• PB: pp. 5–6
• Math Pro: Practice
• Round a whole number to the nearest million, hundred thousand, ten thousand, thousand, hundred or ten
2.1 Rounding whole numbers
Unit 3: Factors, Square Numbers and Cube Numbers
New Zealand Curriculum Year 6: Number — Number structures
• factor
• factor pair
3.1 Practices ( ◆ )
◆ Finding factor pairs for numbers that result from multiplying any two whole numbers between 1 and 12
• CB: pp. 16–17
• PB: p. 7
• Math Pro: Practice
• Finding factor pairs for numbers that result from multiplying any two whole numbers between 1 and 12
3.1 Finding factors of a whole number
• square number
• Find all the factors of a number that results from multiplying any two whole numbers between 1 and 12
3.2 Knowledge ( ★ ) and Practices ( ◆ )
★ Square numbers are produced by multiplying a number by itself.
◆ Recognising square and cube numbers and the notation for squared (2) and cubed (3)
◆ Memorising the square numbers to 144 and cube numbers to 125
• cube number
• CB: pp. 18–19
• PB: p. 8
• Math Pro: Practice
• Counters
• Identify square numbers up to 144
3.2 Identifying square numbers
3.3 Knowledge ( ★ ) and Practices ( ◆ )
★ Cube numbers are produced by multiplying a number by itself twice (e.g. 4 × 4 × 4).
◆ Recognising square and cube numbers and the notation for squared (2) and cubed (3)
◆ Memorising the square numbers to 144 and cube numbers to 125
• CB: pp. 20–22
• PB: p. 9
• Math Pro: Practice
• Connecting cubes
• Identify cube numbers up to 125
3.3 Identifying cube numbers
• CB: pp. 23–25
• Math Pro: Assessment
• 1 copy of Maths Journal (BM1.1) per student
• Solve a non-routine problem involving whole numbers using the strategy of making a table
• Solve a non-routine problem involving whole numbers using the strategy of logical reasoning
The suggested duration for each lesson is 1 hour.
Whole Numbers
Let's Remember Remember
1. Write eight hundred and sixty thousand, four hundred and twelve in numerals.
2. Write 982,015 in words.
3. Fill in the missing numbers.
631,420 = 600,000 + + 1,000 + + 20
4. Fill in the blanks.
7. Count forwards or backwards to complete the number pattern.
6,224, 6,227, , 6,233,
8. Round 4,358 to the nearest a) ten.
b) hundred. c) thousand.
9. List all the the factors of 24.
EXPLORE
a) In 702,144, the digit 0 is in the place, and its value is b) In 417,388, the digit is in the place and its value is 400,000.
5. Write > or < a) 66,912 606,912 b) 125,211 122,511 c) 132,402 123,240 d) 357,999 375,339
6. Arrange the numbers in order. Begin with the greatest.
nine hundred and eighty-two thousand and fifteen 30,000 0 483,505400,85543,855 4 400 ten thousands hundred thousands < > > <
483,505, 43,855, 400,855 (greatest)
Chapter 1 Whole Numbers
Chapter Overview
Let’s Remember
Unit 1: Numbers to 1,000,000,000
Unit 2: Rounding Numbers
Unit 3: Factors, Square Numbers and Cube Numbers
Unit 4: Problem Solving
Let's Remember
Recall:
1. Reading and writing a numeral within 1,000,000 given the corresponding number word (CB5 Chapter 1)
2. Reading and writing a number word within 1,000,000 given the corresponding numeral (CB5 Chapter 1)
3. Writing a 6-digit number in terms of hundred thousands, ten thousands, thousands, hundreds, tens and ones (CB5 Chapter 1)
4. Identifying the values and place values of digits in a 6-digit number (CB5 Chapter 1)
5. Comparing numbers within 1,000,000 (CB5 Chapter 1)
6. Comparing and ordering numbers within 1,000,000 (CB5 Chapter 1)
a) Which region protected more trees? b) Round the number of trees protected in Region A to the nearest hundred thousand and to the nearest million. 6,230 6,236
Last year, 6,482,913 native trees were protected and restored in Region A through various conservation projects. In a nearby Region B, 6,905,240 native trees were protected through similar conservation work.
How can we solve this problem? Discuss in your group and fill in columns 1 and 2.
1. What I already know that will help me solve the problem 2. What I need to find out and learn 3. What have learnt
Answer varies.
Compare and order numbers within 1,000,000. Round a whole number to the nearest hundred thousand.
Compare and order numbers within 1,000,000,000. Round a whole number to the nearest million.
7. Completing a number pattern by counting forwards and backwards by 3s within 10,000 (CB4 Chapter 1)
8. Rounding a whole number to the nearest ten, hundred or thousand (CB5 Chapter 1)
9. Finding all the factors of a number that results from multiplying any two whole numbers between 1 and 10 (CB5 Chapter 1)
EXPLORE
Have students read the word problem on CB p. 2. Discuss with students the following questions:
• Why do countries like New Zealand invest in protecting and restoring native trees?
• What might happen to the environment and wildlife if large areas of native forest are not conserved?
• How can individuals, schools or communities contribute to conservation efforts in everyday life?
Have students form groups to complete the tasks in columns 1 and 2 of the table. Let students know that they do not have to solve the word problem. Ask the groups to present their work.
Tell students that they will come back to this word problem later in the chapter.
Unit 1: Numbers to 1,000,000,000 1.1 Reading and writing numbers
Let's Learn Let's Learn
Objective:
• Read and write a number within 1,000,000,000 — the numeral and the corresponding number word
Resources:
• CB: pp. 3–5
• PB: p. 1
Vocabulary:
• billion
• milion
(a) Stage: Abstract Representation
Refer students to (a) on CB p. 3. Remind them that they have previously learnt that the number 1,000,000 is read as one million.
Write: one million
Say: To get to one million, we can count on from 100,000 in steps of 100,000. Get students to count on in steps of 100,000 from 100,000 until they reach one million. Write the numbers as they count on.
Say: 100,000 more than 900,000 is 1,000,000. Explain that 1,000,000 can also be seen as 1,000 thousands because it is made up of 1,000 groups of 1,000. Point out that the word ‘thousand’ represents three zeros, so 1,000 thousands can be written as 1,000,000.
Write: 1,000 thousands = 1,000,000 1 million = 1,000 thousands
Say: So, there are 1,000 thousands in 1 million.
Ask: How many thousands are there in 2 million? (2,000)
Write: 2 million = 2,000 thousands
You
(b) Stage: Abstract Representation
Have students look at the place value cards in (b) on CB p. 3.
Say: We have partitioned the number using the values of its digits. Let us look at another way to partition the number.
Guide students to see that since 400,000, 60,000 and 7,000 make 467,000, and 500, 80 and 3 make 583, the number can also be partitioned as ‘2,000,000 + 467,000 + 583 = 2,467,583’.
Write: 2,000,000 + 467,000 + 583 = 2,467,583
Point out to students that we can partition a number in different ways.
Say: We read 2,467,583 as two million, four hundred and sixty-seven thousand, five hundred and eighty-three.
(c) Stage: Abstract Representation
Refer students to the example shown in (c) on CB p. 4.
Write: One million = 1,000,000
Ask: How many zeros are there in one million? (6) How do we write ten million in numerals? (10,000,000)
Have a student write ten million in numerals on the board.
Write: Ten million = 10,000,000
Guide students to see that we write ‘10’ and append ‘000,000’ to it to write ten million in numerals.
Ask: How do we write one hundred million in numerals? (100,000,000)
Have another student write one hundred million in numerals on the board. Guide students to see that we write ‘100’ and append ‘000,000’ to it to write one hundred million in numerals.
Ask: How do we write one thousand million in numerals? (1,000,000,000)
Have another student write one thousand million in numerals on the board. Guide students to see that we write ‘1,000’ and append ‘000,000’ to it to write one thousand million in numerals.
Say: We read one thousand million as one billion.
Write: billion
Highlight to students that 1 billion is equal to 1,000 million. So, to write 1 billion in numerals, we write 1,000,000, and then append another set of ‘000’ to indicate thousand.
Write: 1,000 million = 1,000,000,000 = 1 billion
(d) Stage: Abstract Representation
Have students look at the place value cards in (d) on CB p. 4.
Say: We have partitioned the number using the values of its digits.
Guide students to see that since 700,000,000, 70,000,000 and 8,000,000 make 778,000,000, and 500,000, 40,000 and 7,000 make 547,000, the number can also be partitioned as ‘778,000,000 + 547,000 + 200 = 778,547,200’.
Write: 778,000,000 + 547,000 + 200 = 778,547,200
Say: We read 778,547,200 as seven hundred and seventy-eight million, five hundred and forty-seven thousand and two hundred.
c) According to the 2026 census, the population of India is more than 1 billion How large is 1 billion?
One thousand million is also known as one billion 1,000 million = 1,000,000,000 = 1 billion
d) The distance between the Sun and Jupiter is about 778,547,200 kilometres.
778,000,000 + 547,000 + 200 = 778,547,200
778,547,200 is read as seven hundred and seventy-eight million, five hundred and forty-seven thousand and two hundred
Let's Do Let's
1. Write the numerals.
a) ninety million, eight hundred and fifty thousand and thirty-six
b) three hundred and fifty-two million, seven hundred and forty-nine thousand and eighty
2. Write the numerals in words.
43,210,123
904,086,470
forty-three million, two hundred and ten thousand, one hundred and twenty-three nine hundred and four million, eighty-six thousand, four hundred and seventy
90,850,036 352,749,080
Let's Do Let's
Task 1 requires students to read and write numerals within 1,000,000,000 given the corresponding number words.
Task 2 requires students to read and write number words within 1,000,000,000 given the corresponding numerals.
Let's Practise Let's Practise
Task 1 requires students to read and write numerals within 1,000,000,000 given the corresponding number words.
Task 2 requires students to read and write number words within 1,000,000,000 given the corresponding numerals.
1.2 Identifying values of digits
Let's Learn Let's Learn
Objectives:
• Identify the values and place values of digits in a 7-digit, 8-digit or 9-digit number
• Write a 7-digit, 8-digit or 9-digit number in terms of hundred millions, ten millions, millions, hundred thousands, ten thousands, thousands, hundreds, tens and ones
Resources:
• CB: pp. 5–7
• PB: p. 2
(a) Stage: Abstract Representation
Copy the place value chart in (a) on CB p. 5 on the board without filling in the numbers. Say: Let us write 2,149,637 in the place value chart.
Write 2,149,637 in the place value chart. Say: In 2,149,637, the digit 2 is in the millions place. So, there are 2 millions. The value of the digit 2 is 2,000,000.
Ask: Which place is the digit 1 in? (Hundred thousands)
Say: Since the digit 1 is in the hundred thousands place, its value is 100,000.
Ask: Which place is the digit 9 in? (Thousands) What is its value? (9,000)
Say: We can now write this number using its place values.
Write: 2,149,637 = 2 millions 1 hundred thousand 4 ten thousands 9 thousands 6 hundreds 3 tens ____ ones
Elicit the number of ones from students. (7) Point out that the group of thousands, which is 1 hundred thousand, 4 ten thousands and 9 thousands, together make 149 thousands. Highlight also that 2 millions is the same as 2,000 thousands. So, we can write the number in another way.
a) thirty-four million, eight hundred and fifty-nine thousand, two hundred and three
b) six hundred and forty-eight million, seven hundred and fifty thousand and two c) one billion
2. Write the numerals in words.
(b) Stage: Abstract Representation
Copy the place value chart as shown in (b) on CB p. 6 on the board without filling in the numbers. Say: Let us write 120,539,264 in the place value chart.
Write 120,539,264 in the place value chart. Say: In 120,539,264, the digit 1 is in the hundred millions place. So, there is 1 hundred million. The value of the digit 1 is 100,000,000. The digit 2 is in the ten millions place.
Ask: What is its value? (20,000,000)
Point to the digit in each place value in the place value chart, starting from the hundred millions. Ask: In 120,539,264, how many hundred millions are there? (1) How many ten millions are there? (2)
Ask similar questions about the other place values.
Write: 120,539,264 = 1 hundred million 2 ten millions 5 hundred thousands 3 ten thousands 9 thousands 2 hundreds 6 tens ____ ones
Elicit the number of ones from students. (4)
Guide students to see that since 1 hundred million and 2 ten millions make 120 millions and 5 hundred thousands, 3 ten thousands and 9 thousands make 539 thousands, 120,539,264 can also be written as 120 millions 539 thousands 2 hundreds 6 tens 4 ones.
Reiterate to students that there are different ways to partition a number.
Let's Do Let's Do
Task 1 requires students to identify the values and place values of digits in a 7-digit, 8-digit or 9-digit number.
Task 2 requires students to write a 7-digit, 8-digit or 9-digit number in terms of hundred millions, ten millions, millions, hundred thousands, ten thousands, thousands, hundreds, tens and ones.
Write the missing numbers or words.
Let's Practise Let's Practise
Task 1 requires students to identify the values and place values of digits in an 8-digit number.
Task 2 requires students to recognise if a 7-digit, 8-digit or 9-digit number is correctly represented in terms of hundred millions, ten millions, millions, hundred thousands, ten thousands, thousands, hundreds, tens and ones and write true or false.
Task 3 requires students to write a 7-digit, 8-digit or 9-digit number in terms of hundred millions, ten millions, millions, hundred thousands, ten thousands, thousands, hundreds, tens and ones.
Task 4 requires students to identify the value of a chosen digit in various 7-digit, 8-digit or 9-digit numbers.
1. Write the missing numbers or words.
In 68,047,293: the digit is in the ten millions place. the digit 7 is in the place. the value of the digit 8 is the value of the digit 0 is the digit 4 is in the place and its value is
b) 675,089,743 = 600,000,000 + 75,000,000 + 890,000 + 743
c) 41,905,027 = 41 millions 95 thousands 2 tens 7 ones
d) 3,806 thousands 15 ones = 3,806,015
3. Write the missing numbers.
a) 2,085,050 = 2,000,000 + + 5,000 + 50
b) 7,500,000 + 19,000 + 630 = c) 9,105,953 = thousands 9 hundreds 5
d) 200,830,960 = 2 hundred millions
4. What is the value of the digit 7 in each of the following numbers?
a) 3,720,100
b) 87,204,500
c) 840,632,179
d) 736,015,928
1: Whole Numbers
1.3 Comparing
and ordering numbers
Let's Learn Let's Learn
Objective:
• Compare and order numbers within 1,000,000,000
Resources:
• CB: pp. 8–9
• PB: p. 3
Stage: Abstract Representation
Write: 30,058,700, 309,580,070, 30,850,007
Copy the place value chart as shown on CB p. 8 on the board without filling in the numbers.
Say: Let us compare these three numbers using the place value chart.
Have three students fill in the place value chart to show the numbers 30,058,700, 309,580,070 and 30,850,007.
Ask: Which place value should we compare first? (Hundred millions)
Say: Look at the three numbers in the place value chart.
Ask: Which is the only number that has value in hundred millions? (309,580,070) So, what can we say about this number? (It is the greatest.)
Say: Let us now compare the two remaining numbers, 30,058,700 and 30,850,007.
Ask: What do you notice about the ten millions of the two numbers? (They are the same.) How about the millions? (They are the same as well.)
Say: Since they have the same value in ten millions and millions, we compare the next place value, the hundred thousands place.
Ask: How many hundred thousands are there in 30,058,700? (0) What about 30,850,007? (8) Which is greater, 0 hundred thousands or 8 hundred thousands? (8 hundred thousands)
Say: Since 8 hundred thousands is greater than 0 hundred thousands, 30,850,007 is greater than 30,058,700.
Write: 30,850,007 > 30,058,700
Ask: Which number is the least? (30,058,700)
Guide students to arrange the numbers in order, beginning with the least.
Compare 30,058,700, 309,580,070 and 30,850,007. H MilT
1
2 Then, compare the ten millions and millions in 30,058,700 and 30,850,007. They have the same value in ten millions and millions.
3 Then, compare the hundred thousands in 30,058,700 and 30,850,007. 8 hundred thousands is greater than 0 hundred thousands.
30,850,007 is greater than 30,058,700. 30,850,007 > 30,058,700 30,058,700 is the least number.
Arranging the numbers in order, beginning with the least, we get: 30,058,700, , (least)
30,850,007309,580,070
Let's Do Let's
Task 1 requires students to compare two numbers within 1,000,000,000 using the symbols ‘>’ and ‘<’.
Task 2 requires students to compare and order three numbers within 1,000,000,000, beginning with the least.
Let's Practise Let's Practise
Task 1 requires students to compare two numbers within 1,000,000,000 using the symbols ‘>’ and ‘<’.
Task 2 requires students to compare and order four numbers within 1,000,000,000, beginning with the greatest.
Task 3 requires students to compare and order three numbers within 1,000,000,000, beginning with the least.
2. Compare 1,008,960, 1,008,690 and 100,869,000. a) is the greatest number. b) is the least number. c) Arrange the numbers in order. Begin with the least.
(least)
2. Arrange the numbers in order. Begin with the greatest. 378,069,141, 387,090,411, 378,096,114, 387,096,114 , , , (greatest)
3. Arrange the numbers in order. Begin with the least. 6,038,597, 6,380,597, 6,038,975
1.4 Number patterns
Let's Learn Let's Learn
Objectives:
• Count forwards and backwards with positive whole numbers, including working with negative numbers
• Describe and complete a number pattern involving positive and negative numbers by counting forwards and backwards by ones, twos, threes, fours, fives or tens
Resources:
• CB: pp. 10–12
• PB: p. 4
(a) Stage: Abstract Representation
Write: 5
Say: Let us make a number pattern by counting forwards by threes from 5.
Ask: What number is 3 more than 5? (8)
Write: 5, 8
Repeat the above question to form the pattern: 5, 8, 11, 14, 17.
Ask: What number is 3 more than 17? (20) Continue the above pattern and write 20.
Say: The number that comes next is 20.
(b) Stage: Abstract Representation
Write: 7, 3, –1, –5, –9, ?
Say: We want to find the next number in this pattern. Let us look at how the numbers change. Refer students to the number line shown in (b) on CB p. 10.
Ask: What do we do from 7 to 3? (Subtract 4) What do we do from 3 to –1? (Subtract 4)
Say: The pattern is decreasing by 4 each time. Ask: What is the rule of this pattern? (Start at 7, then count backwards by fours.)
Say: The last given number is in the pattern is –9. –9 is a negative number. We read it as negative 9.
Ask: What is 4 less than –9? (–13)
Write: 7, 3, –1, –5, –9, –13
Say: The number that comes next is –13.
Reiterate to students that negative numbers are to the left of 0 on a horizontal number line and below 0 on a vertical number line. They are represented with a negative sign (−) and named ‘negative’ along with the numeral.
Ask: Is 0 a positive or negative number? (Neither)
1.4
(c) Stage: Abstract Representation
Write: –1, –4, –7, ?, –13
Say: We continue working with number patterns, but now we are starting with negative numbers. Let us find the pattern rule. Refer students to the number line shown in (c) on CB p. 10.
Ask: What do we do from –1 to –4? (Subtract 3) What do we do from –4 to –7? (Subtract 3)
Say: So, the pattern is decreasing by 3 each time. We can also say: Start at –1. Count backwards by threes.
Emphasise that although the numbers may look like they are getting bigger in size, they are actually decreasing. On a number line, –1 is greater than –4, and –4 is greater than –7.
Ask: What is –7 minus 3? (–10)
Write: –1, –4, –7, –10, –13
Say: The missing number is –10.
2. Complete each number pattern. Then, describe the number pattern.
a) 8, 5, 2, , –7,
Rule: Start at . Count
b) –15, –11, –7, , , 5,
8 backwards by threes forwards by fours
Rule: Start at . Count
1. Complete each number pattern. Then, describe the number pattern.
a) –26, –21, –16, , –6, , 4
Start at –26. Count forwards by fives.
Rule:
b) –21, –18, , –12, , –6, –3
Start at –21. Count forwards by threes.
Rule:
c) , 3, 1, , –3, –5,
Start at 5. Count backwards by twos.
Rule:
d) 37, 27, 17, 7,
Start at 37. Count backwards by tens.
Rule:
I have learnt to... read and write numbers within 1,000,000,000 identify the values of digits in a 7-digit, 8-digit or 9-digit number compare and order numbers within 1,000,000,000 count forwards and backwards with whole numbers, including working with negative numbers describe and complete a number pattern involving positive and negative numbers by counting forwards and backwards
Point out to students that we can also find the answer by comparing the pattern with the positive number pattern.
Write: 1, 4, 7, 10, 13,…
Highlight that in the above positive number pattern, we add 3 each time and can see that 10 is between 7 and 13. The negative pattern follows the same steps but in the negative direction: –1, –4, –7, –10, –13
Say: So, the missing number is –10.
(d) Stage: Abstract Representation
Write: –16, –14, ?, –10, –8, –6
Say: We want to find the missing number in this number pattern. The numbers are increasing in this pattern, so we count forwards to find the missing number.
Ask: How much more is each number than the number before it? (2)
Say: The number pattern is formed by starting from –16 and counting forwards by twos. So, the missing number is 2 more than –14.
Ask: What is 2 more than –14? (–12)
Write –12 in the blank in the number pattern.
Ask: What is 2 more than –12? (–10)
Say: The missing number is –12.
Let's Do Let's
Task 1 requires students to complete number patterns involving positive and negative numbers by counting forwards and backwards.
Task 2 requires students to describe and complete number patterns involving positive and negative numbers by counting forwards and backwards.
Let's Practise Let's
Task 1 requires students to describe and complete number patterns involving positive and negative numbers by counting forwards and backwards.
EXPLORE
Have students go back to the word problem on CB p. 2.
Ask: Can you solve the problem now? (Answer varies.) What else do you need to know? (Answer varies.)
Students are not expected to be able to solve the problem now. They will learn more skills in subsequent lessons and revisit this problem at the end of the chapter.
Unit 2: Rounding Numbers
2.1 Rounding whole numbers
Let's Learn Let's Learn
Objective:
• Round a whole number to the nearest million, hundred thousand, ten thousand, thousand, hundred or ten
Resources:
• CB: pp. 13–15
• PB: pp. 5–6
(a) Stage: Pictorial Representation
Draw the number line in (a) on CB p. 13 on the board but do not label ‘187,325’.
Guide students to see that there are 10 equal intervals between 180,000 and 190,000 and each interval stands for 1,000.
Ask a student to mark 187,325 on the number line. Say: 187,325 is between two ten thousands, 180,000 and 190,000.
Ask: Is 187,325 nearer to 180,000 or to 190,000? (Nearer to 190,000)
Stage: Abstract Representation
Say: 187,325 is nearer to 190,000 than to 180,000. So, 187,325 is 190,000 when rounded to the nearest ten thousand.
Write: 187,325 ≈ 190,000
Say: We read this statement as ‘187,325 is approximately 190,000’.
Remind students that when we round a number to a place value, instead of drawing a number line, we can also look at the digit to the right of the place value stated. If the digit is 5 or greater, we round up. If the digit is less than 5, we round down. Ask: What place value are we rounding this number to? (Ten thousand) What is the place value to the right of this? (Thousand) What is the digit in this place value? (7) Is it greater or smaller than 5? (Greater)
Say: The digit to the right of the ten thousands place value is greater than 5. So, we round up the number to the higher ten thousands number, which is 190,000.
Unit
2 Rounding Numbers
You will learn to...
• round a whole number to the nearest million, hundred thousand, ten thousand, thousand, hundred or ten
2.1 Rounding whole numbers
Let's Learn Let's Learn
a) The number of visitors to ski fields in central Otago in July on a particular year was 187,325.
187,325 is between 180,000 and 190,000. It is nearer to 190,000 than to 180,000. So, we round up.
187,325 is 190,000 when rounded to the nearest ten thousand.
187,325 ≈ 190,000
To round a number to a place value, look at the digit to the right of the place value stated. If the digit is 5 or greater, round the number up. If the digit is less than 5, round the number down. For example, the digit to the right of the ten thousands place in 187,325 is 7. It is greater than 5. So, we round up the number to the higher ten thousands number, which is 190,000.
b) Round 4,325,680 to the nearest hundred thousand. 4,300,000 4,350,000 4,400,000 4,325,680
4,325,680 is 4,300,000 when rounded to the nearest hundred thousand.
4,325,680 ≈ 4,300,000
1: Whole Numbers
The digit to the right of the hundred thousands place is 2. So, we round down.
(b) Stages: Pictorial and Abstract Representations
Follow the procedure in (a) to guide students to round 4,325,680 to the nearest hundred thousand. Show that on the number line we can see that the number is nearer to the lower hundred thousand number, so we round down to that number. We can also see that the digit to the right of the hundred thousands place is 2. This is less than 5. So, we round down the number.
c) Round 8,500,000 to the nearest million. 8,000,000 9,000,000 8,500,000
8,500,000 is 9,000,000 when rounded to the nearest million. 8,500,000 ≈
1. Round 183,260 to the nearest ten thousand and to the nearest hundred thousand.
Complete the table.
Rounded to the nearest
Complete the table.
2. Round 5,540,986 to the nearest million.
5,540,986 is when rounded to the nearest million.
3. Round 6,375,819 to the nearest a) ten. b) hundred. c) thousand. d) ten thousand. e) hundred thousand. f) million.
3. About 3,480,000 visitors travelled to attractions in the Auckland region last year. Round this number to the nearest million.
(c) Stages: Pictorial and Abstract Representations
Follow the procedure in (a). Point out that when a number is halfway between two millions, we round up and the greater million is to be taken as the nearest million. In this case, the greater million is 9,000,000.
Let's Do Let's
Task 1 requires students to round a whole number to the nearest ten thousand or hundred thousand.
Task 2 requires students to round a whole number to the nearest million.
Task 3 requires students to round a whole number to the nearest million, hundred thousand, ten thousand, thousand, hundred or ten.
Let's Practise Let's
Task 1 requires students to round whole numbers to the nearest ten, hundred or thousand.
Task 2 requires students to round whole numbers to the nearest ten thousand, hundred thousand or million.
Task 3 requires students to round a whole number to the nearest million.
Task 4 requires students to round a cost to the nearest hundred thousand dollars.
Unit 3: Factors, Square Numbers and Cube Numbers
3.1 Finding factors of a whole number
Let's Learn Let's Learn
Objectives:
• Finding factor pairs for numbers that result from multiplying any two whole numbers between 1 and 12
• Find all the factors of a number that results from multiplying any two whole numbers between 1 and 12
Resources:
• CB: pp. 16–17
• PB: p. 7
Vocabulary:
• factor
• factor pair
(a) Stage: Abstract Representation
Say: Let us find all the factors of 12. To do so, we can first find the factor pairs of 12.
Remind students that they have previously learnt that factor pairs are two whole numbers that multiply together to give another whole number.
Write: factor × factor = product factor pair
Ask students which pairs of numbers can be multiplied to get 12. Guide students to write the multiplication sentences with products of 12 systematically.
Ask: What whole number and 1 can be multiplied to get 12? (12)
Write: 1 × 12 = 12
Ask: What whole number and 2 can be multiplied to get 12? (6)
Write: 2 × 6 = 12
Continue to write multiplication sentences with products of 12. Explain that the factors in the multiplication sentence 4 × 3 = 12 is a repeat of those in 3 × 4 = 12. So we can stop writing further multiplication sentences as the numbers will now repeat.
Say: From the multiplication sentences, we can see that (1, 12), (2, 6) and (3, 4) are the factor pairs of 12.
Ask: So, what are the factors of 12? (1, 2, 3, 4, 6, and 12)
Point out that when listing the factors, we should rearrange the numbers in the factor pairs and list them in ascending order. Highlight also that a number will always have 1 and the number itself as a factor. So, the least number of factors a number will have is 2.
Unit 3 Factors, Square Numbers and Cube Numbers
You will learn to...
• find factor pairs and all the factors for numbers that result from multiplying any two whole numbers between 1 and 12
• identify square numbers up to 144
• identify cube numbers up to 125
3.1 Finding factors of a whole number
Let's Learn Let's Learn
a) Find all the factors of 12.
1 × 12 = 12
2 × 6 = 12
3 × 4 = 12
factor × factor = product
The factor pairs of 12 are (1, 12), (2, 6) and (3, 4).
We can rearrange the numbers in the factor pairs to get the list of factors. So, the factors of 12 are 1, 2, 3, 4, 6 and 12.
b) We can find the factors of a large number by systematically dividing it by numbers starting from 1. If the answer is a whole number, record it and form the factor pair.
Find all the factors of 72. factor pair
72 divided byQuotientFactor pairs
1 72 (1, 72)
2 36 (2, 36)
3 24 (3, 24)
4 18 (4, 18)
56 12 (6, 12)
78 9 (8, 9)
9 8 (9, 8)
The factors of 72 are 1, 2, 3,
Year 6: Number — Number structures
72 ÷ 1 = 72 1 × 72 = 72 → (1, 72)
The next pair (9,8) is a repeat of this. So, we stop here.
4, 6, 8, 9, 12, 18, 24, 36 and 72
(b) Stage: Abstract Representation
Say: Factors are whole numbers that divide another number exactly. So, we can also find factors of large numbers systematically by listing and dividing by all the numbers, starting from 1. Let us find all the factors of 72 using systematic listing.
Make a table such as the one in (b) on CB p. 16. Starting from 1, divide 72 by each number and write down the quotient.
Say: The divisor and the quotient forms a factor pair of 72 since they give 72 when multiplied with each other.
Continue dividing and listing the quotient and factor pairs until you reach 5.
Say: We cannot find a whole number that gives a product of 72 when multiplied by 5. So, 5 is not a factor of 72.
Ask: Do we need to divide by all the numbers up to 72? (No) How do you know when to stop? (When the numbers in the factor pairs start to repeat)
Have students observe that when we divide 72 by 9, we get the factor pair of (9, 8) which is a repeat of the factor pair (8, 9). So, we can stop listing factors pairs after (8, 9).
Ask: From the factor pairs, what are the factors of 72? (1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72)
(c) Stage: Abstract Representation
Say: Let us find the factors of 121.
Ask students which pairs of numbers can be multiplied to get 121 and write the multiplication sentences on the board. Guide students to write the multiplication sentences with product of 121 systematically.
Write: 1 × 121 = 121 11 × 11 = 121
Say: The factor pairs of 121 are (1, 121) and (11, 11).
Ask: So, what are the factors of 121? (1, 11 and 121)
Say: 121 has only three factors as one of the factors is repeated.
Let's Do Let's
Task 1 requires students to find the factor pairs for a number that results from multiplying any two whole numbers between 1 and 12, and then list all its factors.
Let's Practice Let's Practise
Task 1 requires students to find the unknown factor of a number, given another factor of the number.
Task 2 requires students to find the factor pairs for a number that results from multiplying any two whole numbers between 1 and 12, and then list all its factors.
3.2 Identifying square numbers
Let's Learn Let's Learn
Objective:
• Identify square numbers up to 144
Materials:
• Counters
Resources:
• CB: pp. 18–19
• PB: p. 8
Vocabulary:
• square number
Stages: Concrete Experience and Pictorial Representation
Have students work in groups. Distribute counters to each group. Have students follow each step of your demonstration.
Stick 1 counter on the board.
Ask: How many rows of counters are there? (1) How many counters are there in the row? (1) What is the multiplication sentence we can write to find the number of counters? (1 × 1 = 1)
Write: 1 × 1 = 1
Draw a box around the counter on the board. Point out to students that the shape of the box around the counter is a square. Have students relate this to the first picture of a counter on CB p. 18.
Next, stick 2 rows of 2 counters beside the first counter on the board. Point out that there are 2 rows of 2 counters.
Write: 2 × 2 = 4
Draw a box around the 2 rows of 2 counters on the board.
Ask: What is the shape of the box around the second group of counters? (Square)
Have students relate this to the second picture of counters on the page.
Follow the same procedure for the third and fourth pictures of counters on the page. Lead students to see that the boxes drawn around the group of 9 counters and the group of 16 counters are also squares.
Ask: What is the multiplication sentence we can write to find the number of counters in the third picture of counters? (3 × 3 = 9) What is the multiplication sentence we can write to find the number of counters in the fourth picture of counters? (4 × 4 = 16)
Write: 3 × 3 = 9 4 × 4 = 16
Stage: Abstract Representation
Point to the single counter on the board.
Say: When we multiply a number by itself, we get the square of the number. Since 1 × 1 = 1, the square of 1 is 1. The square of a whole number is called a square number.
Write: square number
Highlight to students that all square numbers can be arranged in equal groups to form a square.
Write: 12 = 1
Explain to students that we use the exponent notation ‘2’ to represent 1 multiplied by itself, that is 1 × 1. Tell them that 12 is read as 1 squared. Point to the second group of counters on the board.
Say: Since 2 × 2 = 4, the square of 2 is 4. 2 squared is 4.
Write: 22 = 4
Follow the same procedure for ‘3 × 3 = 9’ and ‘4 × 4 = 16’ and lead students to see that the square of 3 is 9 and the square of 4 is 16.
Say: 1, 4, 9 and 16 are square numbers.
Let's Do Let's Do
Task 1 requires students to model square numbers. Students are required to interpret the diagrams and identify the square numbers represented by the diagrams.
1. Write the square number for each diagram.
Let's Do Let's
Task 2 requires students to understand the exponent notation of square numbers.
Task 3 requires students to identify square numbers up to 144.
Let's Practise Let's Practise
Task 1 requires students to model square numbers. Students are required to colour the squares to represent a square number.
Task 2 requires students to understand the exponent notation of square numbers.
Task 3 requires students to identify square numbers up to 144.
Task 4 requires students to identify square numbers up to 144 that are odd or even.
3.3 Identifying cube numbers
Let's Learn Let's Learn
Objective:
• Identify cube numbers up to 125
Materials:
• Connecting cubes
Resources:
• CB: pp. 20–22
• PB: p. 9
Vocabulary:
• cube number
Stages: Concrete Experience and Pictorial Representation
Have students work in groups. Distribute connecting cubes to each group. Have students follow each step of your demonstration. Place 1 connecting cube on a table. Ensure that students can see the cube.
Ask: How many rows of cubes are there? (1) How many cubes are there in the row? (1) How many layers of cubes are there? (1) What is the multiplication sentence we can write to find the number of connecting cubes? (1 × 1 × 1 = 1)
Write: 1 × 1 × 1 = 1
Have students relate this connecting cube to the first picture of a cube on CB p. 20. Next, join 8 connecting cubes such that they form a large cube as shown in the second picture on the page.
Ask: How many rows of cubes are there? (2) How many cubes are there in each row? (2) How many layers of cubes are there? (2) How many connecting cubes are there altogether? (8)
Write: 2 × 2 × 2 = 8
Have students relate this large cube to the second picture of a cube on the page. Follow the same procedure for the third and fourth pictures of cubes on the page. Guide students to see that the third cube has 27 connecting cubes altogether, is made up of 3 layers of cubes with 3 rows of cubes in each layer, and with 3 cubes in each row. Similarly, have students see that the fourth cube has 64 connecting cubes altogether, is made up of 4 layers of cubes with 4 rows of cubes in each layer, and with 4 cubes in each row.
Ask: What is the multiplication sentence we can write to find the number of connecting cubes in the third picture? (3 × 3 × 3 = 27) What is the multiplication sentence we can write to find the number of connecting cubes in the fourth picture? (4 × 4 × 4 = 64)
Write: 3 × 3 × 3 = 27
4 × 4 × 4 = 64
Stage: Abstract Representation
Say: When we multiply a number by itself two more times, we get the cube of the number. Since 1 × 1 × 1 = 1, the cube of 1 is 1. The cube of a whole number is called a cube number.
Write: cube number
Point out to students that any cube number, when represented by blocks or cubes, can be arranged to form a perfect cube.
Write: 13 = 1
Explain to students that we use the exponent notation ‘3’ to represent 1 × 1 × 1. Tell them that 13 is read as 1 cubed.
Hold up the first large cube.
Say: Since 2 × 2 × 2 = 8, the cube of 2 is 8. 2 cubed is 8.
Write: 23 = 8
Follow the same procedure for ‘3 × 3 × 3 = 27’ and ‘4 × 4 × 4 = 64’ and lead students to see that the cube of 3 is 27 and the cube of 4 is 64.
Say: 1, 8, 27 and 64 are examples of cube numbers.
Let's Do Let's
Task 1 requires students to identify and write the cube number that is represented by the cubes formed in the diagram.
2.
Task 2 requires students to understand the exponent notation of cube numbers.
Task 3 requires students to identify cube numbers up to 125.
Let's Practise Let's Practise
Task 1 requires students to model cube numbers. Students are required to draw a diagram to show that 27 is a cube number.
Task 2 requires students to understand the exponent notation of cube numbers.
Task 3 requires students to identify cube numbers up to 125.
Task 4 requires students to identify cube numbers up to 125 that are odd or even.
THINK ABOUT
Sarah relates cube numbers to square numbers.
A cube number cannot be a square number.
are both
and
I
Think of a time in your daily life when knowing square or cube numbers can be useful.
I have learnt to... find factor pairs and all the factors for numbers that result from multiplying any two whole numbers between 1 and 12 identify square numbers up to 144 identify cube numbers up to 125
THINK ABOUT IT
Have students work in groups to discuss the tasks. Ask the groups to present their answers.
Have students list some square numbers and cube numbers. Guide them to observe that some numbers, such as 1 and 64, are both square and cube numbers.
Conclude that Sarah is not correct because a number can be both a square number and a cube number.
Reiterate to students that square numbers are formed by multiplying a number by itself once, while cube numbers are formed by multiplying a number by itself two more times.
Encourage students to share situations in daily life where square numbers or cube numbers may be useful. Make use of the examples presented by the groups to help students understand the importance and usefulness of square and cube numbers in real-life situations.
Unit 4: Problem Solving
4.1 Mind stretcher
Let's Learn Let's Learn
Objectives:
• Solve a non-routine problem involving whole numbers using the strategy of making a table
• Solve a non-routine problem involving whole numbers using the strategy of logical reasoning
Materials:
• 1 copy of Maths Journal (BM1.1) per student
Resource:
• CB: pp. 23–25
Have students read the problem on CB p. 23.
1. Understand the problem. Pose the questions in the thought bubble in step 1.
2. Plan what to do.
Say: We can make a table to list the number of blocks required for each tower and keep a running total of the number of blocks used so far.
3. Work out the Answer
Guide students to organise the information in a table showing the tower number, the number of blocks in each tower and the total number of blocks used. Have students find the number of blocks in each tower by squaring the tower number.
Lead students to find the running total of blocks used after each tower is built. Guide students to observe that 385 blocks are needed to build 10 towers, while 506 blocks are needed to build 11 towers.
Ask: Can Lucas build 11 complete towers with 500 blocks? (No) Why not? (506 is more than 500)
Conclude that Lucas can build only 10 complete towers. Then, have students find the number of blocks left over by subtracting 385 from 500.
Write: 500 − 385 = 115
Conclude that Lucas will have 115 blocks left over.
4. Check if your answer is correct. Have students check their answers by comparing the total number of blocks used with the number of blocks Lucas has.
Write: 385 + 115 = 500
385 < 500
385 + 121 = 506 > 500
Guide students to see that 10 towers are possible because 385 is less than 500, but 11 towers are not possible because 506 is greater than 500. Conclude that the answer is correct.
5. + Plus Solve the problem in another way. Have students try to solve the problem in a different way. Have 1 or 2 students share their methods. If students are unable to solve the problem in a different way, explain the method shown on CB p. 24.
Ask: Which method do you prefer? Why? (Answers vary.)
Let's Do Let's
Students are required to find a mystery number given some clues about the number. They should recognise that since the number is between 4,000,000 and 5,000,000, the digit in the millions place is 4. The digit in the ten thousands place is 3 less than 4, so it is 1. The digit in the tens place must be greater than 1 and a factor of 9, so the possible digits are 3 or 9. Students should test these possibilities: if the tens digit is 9, then the thousands digit would be 12, which is not possible, so eliminate 9 and conclude the tens digit is 3 and the thousands digit is 6.
Next, students should identify that the digit in the hundreds place must be a cube number, so it can only be 1 or 8. The digit in the hundred thousands place, hundreds place and ones place are the same, so these three positions must all be 1 or all be 8. Students should then combine the valid conditions to obtain two possible numbers: 4,116,131 or 4,816,838, and conclude that these are the only possible mystery numbers.
EXPLORE
Have students go back to the word problem on CB p. 2. Get them to write down in column 3 of the table what they have learnt that will help them solve the problem, and then solve the problem. Have a student present his/her work to the class.
Ariki is trying to find a mystery number using the following clues:
• The number is between 4,000,000 and 5,000,000.
• The digit in the ten-thousands place is 3 less than the millions digit.
• The digit in the thousands place is 3 more than the tens digit.
• The digit in the tens place is greater than 1 and is a factor of 9.
• The digit in the hundreds place is a cube number.
• The digit in the hundred thousands place, hundreds place and ones place are the same.
What is the mystery number? Let's Do
Strategy: Use logical reasoning
The mystery number is between 4,000,000 and 5,000,000.
So, the digit in the millions place is 4.
The digit in the ten-thousands place is 3 less than 4, so it is 1.
The digit in the tens place is greater than 1 and is a factor of 9.
The possible digits are 3 or 9.
If the tens digit is 9, then the thousands digit would be 9 + 3 = 12, which is not possible.
So, the tens digit is 3, and the thousands digit is 6.
The digit in the hundreds place is a cube number.
Cube digits are 1 and 8 (since 13 = 1 and 23 = 8).
The digit in the hundred thousands place, hundreds place and ones place are the same.
So, these three digits must all be 1 or all be 8.
So, the mystery number is either 4,116,131 or 4,816,838.
Maths Journal Maths Journal
Have students work on the tasks in Maths Journal (BM1.1) independently to check and reinforce their understanding.
Use the rubric provided on page 305 of the blackline masters to score students’ work.
Practice Book Chapter 1: Answers
Exercise 1.1
1.
a) 8,064,020
b) 29,580,915
c) 760,354,800
2. a) six million, four hundred and five thousand, nine hundred and sixty
b) forty-seven million, six hundred and eighteen thousand, two hundred and fifteen
c) eight hundred million, three hundred and seven thousand, six hundred and thirteen
Exercise 1.2
1. a) 3
b) ten thousands; 0
c) 5; 500,000,000
2. a) 600,000
b) 500,000,000; 10
c) 760,000,000
d) 507
e) 2,900,160
f) 800,000,903
3. a) 200
b) 2,000,000
c) 200,000,000
d) 2,000
Exercise 1.3
1. a) < b) >
2. a) 3,506,597, 3,506,975, 3,508,196
b) 83,705,219, 83,750,291, 907,352,108
3. a) 479,911,585, 473,911,588, 79,911,588
b) 64,281,509, 6,482,915, 6,428,951
Exercise 1.4
1. a) 30; 20; 10; 0; –10; –20
b) –9; –5; –1; 3; 7; 11
2. a) –3; 3; Start at –9. Count forwards by twos.
b) –2; –11; Start at 4. Count backwards by threes.
c) 7; –8; Start at 17. Count backwards by fives.
d) –14; 16; Start at –24. Count forwards by tens.
e) –35; –26; –20; Start at –35. Count forwards by threes.
Exercise 2.1
1. a) 4,370 b) 58,910
c) 703,490 d) 9,246,730
e) 125,680 f) 8,004,260
2. a) 3,500 b) 68,000
c) 804,300 d) 5,319,800
e) 249,700 f) 7,003,500
3. a) 6,000 b) 93,000
c) 705,000 d) 4,673,000
e) 319,000 f) 9,010,000
4. a) 250,000 b) 440,000
c) 8,050,000 d) 6,280,000
e) 5,730,000 f) 1,000,000
5. a) 500,000 b) 900,000
c) 6,300,000 d) 1,900,000
e) 4,200,000 f) 14,000,000
6. a) 3,000,000 b) 2,000,000 c) 5,000,000 d) 5,000,000 e) 6,000,000 f) 30,000,000
7. a) 54,572,000 b) 54,570,000 c) 54,600,000 d) 55,000,000
