2
Number knowledge
Unit objectives
Website links
• Understand and use decimal notation and place value; compare and order
decimals in different contexts
• 2.1 Planet statistics
• Multiply/divide integers and decimals by 10, 100, 1000; explain the effect
• Know that comparative measurements must be in the same units
• Understand negative numbers as positions on a number line; order, add and
subtract positive and negative integers in context, multiply and divide integers
• Consolidate rapid recall of number facts, including positive integer
complements to 100, multiplication facts to 10 × 10 and quickly derive
associated division facts
• Use standard column procedures to add and subtract whole numbers and
decimals, including a mixture of large and small numbers with differing
numbers of decimal places
• Extend written methods to: HTU × U, TU × TU, HTU ÷ U and mental methods
to work with squares and square roots
• Use known facts to derive unknown facts
• Know how to use the laws of arithmetic and inverse operations including to
check results in addition to considering the right order of magnitude
• Know squares to at least 10 × 10, recognise to at least 12 × 12 and the
corresponding roots; use the square root key, squares and positive and
negative square roots
• Divide three-digit by two-digit numbers and round up/down after division
18
Number knowledge
• 2.5 Squares on a
chessboard
• To view websites
relevant to this unit
please visit
www.heinemann.co.uk/
hotlinks
Notes on context
LiveText resources
Spitfires were single-seat fighter planes and were produced in greater numbers
than any other Allied planes during the Second World War. The Spitfire was used
by the RAF between 1938 and 1955.
•
Scale models can also be used to represent other things, such as dolls’ houses,
model villages or toy cars. The scale of the model can vary; for example dolls’
houses are usually built to a scale model of 1 : 12.
Games
Elements of
numbers
• Use It!
Audio glossary
Skills bank
Discussion points
• Discuss other things that are built to scale, such as architects’ models of new
developments and railroads.
• Discuss what a suitable scale to use for scale models is. Is making things 100
times smaller always the most practical approach?
• Discuss what pitfalls you could fall into if you choose the wrong scale?
• Discuss the practical applications of making scale models, such as in
architecture. Why is it important for all of the sizes to be accurate?
Activity A
Length – 9.12 cm or 91.2 mm
Wing span – 11.23 cm or 112.3 mm
• Extra questions – There
are extra questions for
each lesson on your
LiveText CD.
Level Up Maths Online
Assessment
The Online Assessment
service helps identify
pupils’ competencies and
weaknesses. It provides
levelled feedback and
teaching plans to match.
• Diagnostic automarked tests are
provided to match this
unit. Select Year 7.
Choose to Assign
a Test, then select
Medium Term Plans.
Select Autumn Term
Unit 2 Number 1
Tail span – 3.2 cm or 32 mm
Activity B
Tyre diameter – 0.6 cm
Max wing thickness – 2.54 cm
Propeller diameter – 3.3 cm or 33 mm
Answers to diagnostic questions
1 a) 70
b) 40
c) 34
2 a) 361
b) 155
c) 73
3 a) 6, 3
b) 8, 32
c) 42, 42, 6
4 −1°C
Opener
19
2.1 Decimal know-how
Objectives
• Understand and use decimal notation and
place value
• Multiply and divide integers and decimals by
10, 100, 1000, and explain the effect
• Compare and order decimals in different contexts
• Know that when comparing measurements
they must be in the same units
Starter (1) Oral and mental objective
Ask pupils to begin with 1 and double it repeatedly.
How many numbers in the sequence can pupils write
in two minutes?
Starter (2) Introducing the lesson topic
Challenge pupils to a counting activity. Start at zero
and count on in steps of 0.2 or 0.02, or start at 10
and count down in steps of 0.1, 0.3 or 0.01.
Main lesson
– Display a place value table and write in it 0.03 and
0.89.
How do you say 0.03? 0.89?
Remind pupils that decimals may be described differently depending on the
context. For example 1.56 is ‘one point five six’ and £1.56 is ‘one pound fiftysix’. Other examples include measurements.
How do you write 0.2 in words? 52.5 m? £3.02?
Ask pupils to place other numbers in the place value table: 0.50, 320, 0.2, 7,
52.5, 7.0. Continue until the table is full. Q1–3
– How can you decide which numbers are bigger or smaller?
–
1 Comparing and ordering decimals
Reinforce the fact that the further to the left a digit lies, the greater its value;
pupils should work from left to right.
–
Resources
Starter: mini whiteboards
(optional)
Activity A: dice
Plenary: mini whiteboards
(optional)
Intervention
Level Up Maths 2–3,
Lesson 2.1
Functional skills
Challenge pupils to put these decimals in order: 0.543, 0.342, 0.35, 0.5,
0.053, 0.305, 0.53.
Examine patterns and
relationships Q7, 9
Next consider decimal measurements.
Framework 2008 ref
2 Ordering decimals in different units
How would you order 45 cm, 1.23 m and 0.96 m?
Why might some people think that 1.25 is greater than 1.4?
Are they correct?
By considering 1.25 and 1.4, discuss the misconception that the greater the
number of digits, the greater the value of the number. Q4–6
The following activity can be completed as a group exercise.
–
3 Multiplying by 10, 100, 1000
–
4 Dividing by 10, 100, 1000
20
Number knowledge
Process skills in bold type:
1.3 Y7/8, 2.1 Y7/8
PoS 2008 ref
Process skills in bold type:
2.2l, 2.2m, 2.2p, 3.1a, b
Website links
www.heinemann.co.uk/
hotlinks
Choose a starting number and ask pupils to
multiply and divide the number by 10, 100 and
1000, using a calculator. Ask pupils to consider the
following questions.
What happens to the value of a positive number
when you multiply by 10? by 100? by 1000?
What happens to the value of a positive number
when you divide by 10? by 100? by 1000?
What happens to the digits when you multiply by 10?
by 100? by 1000?
What happens to the digits when you divide by 10?
by 100? by 1000?
Collect feedback from pupils and ensure any
incorrect ideas are discussed by the class. Q7–9
– Introduce ascending decimals. Consider ordering
decimals with up to four or five significant figures.
Q10
– What do < and > mean? Which symbol should
come between 15.45 and 15.54? Q11
Activity A
This activity could also be used as a plenary. Roll
the dice six times and challenge pupils to complete
the place value table to make the highest number by
adding each number to a column.
Related topics
Activity B
This activity promotes strategic thinking and involves further work on multiplying
and dividing integers and decimals by 10, 100.
Plenary
Display the following numbers: 3.8, 380, 0.38, 26, 52, 260, 52000, 0.008, 0.26,
2.05, 205, 80, 0.0205. Challenge pupils to find related pairs by multiplying and
dividing the numbers by 10, 100 and 1000.
Homework
Homework Book section 2.1.
Challenging homework: Does division always make a number smaller? Does
multiplication always make a number larger?
Common difficulties
2.3 × 10 = 2.30? 20.3?
Use the place value table
to reinforce the concept
that 2.3 means 2 + 0.3
and 10 multiplies both 2
and 0.3 giving a value of
23.
LiveText resources
Answers
1 a) three tenths
b) three thousandths
c) three hundredths
d) three hundredths
2 a) three point one
b) four point two three six
c) thirty-five point zero eight
d) zero point one nine five
3 largest 532.0
smallest 0.235
4 a) 4.56, 4.8, 6.02, 6.17, 6.3
b) 0.032, 0.09, 0.45, 0.48, 0.5
c) 4.05 mm, 4.15 mm, 4.50 mm = 4.5 mm, 4.54 mm
5 a) 7.64
b) 1.432
c) 1.78
d) 0.07
6 859 cm, 8470 mm, 8.32 m, 8310 mm, 827 cm, 8.25 m
7 a) 432
b) 4200 420
c) 2650 265
8 2.31
9 a) 10
b) 1000
c) 10
d) 100
e) 1000
f)
10 a) 14.38, 14.999, 15.016, 15.23, 15.7
b) 4.354, 4.386, 4.4, 4.402, 4.632
c) 0.05263, 0.052631, 0.05326, 0.0536, 0.05362
11 a) >
b) <
Investigation of the use
of decimals in context by
comparing the properties
of planets in the solar
system.
Explanations
Extra questions
Worked solutions
1000
Decimal know-how
21
2.2 Negative numbers
Objectives
• Understand negative numbers as positions on
a number line
• Order, add and subtract positive and negative
integers in context
• Add, subtract, multiply and divide integers
Starter (1) Oral and mental objective
Ask ‘What number am I?’ questions. For example:
I am in the 5 and 9 times tables. What number am I?
If you multiply me by 5, I am 14 more than if you
multiply me by 3. What number am I?
Starter (2) Introducing the lesson topic
Display a vertical number line from −10 to 10. Ask
questions such as: A lift starts at floor 2. It goes up
three floors. Where is it now? End by giving several
movements at a time before asking where the lift is.
Main lesson
– Display a vertical number line from −10°C (at the
bottom) to +10°C (at the top). Challenge pupils to
mark temperature points on the line.
–
Which is colder −3°C or −8°C? Ensure pupils understand that the lower the
position on the ‘thermometer’, the colder the temperature.
Resources
What temperature is 10 degrees lower than 4°C? If it is −7°C and it warms up
by 12°C, what is the new temperature?
Activity B: scrap paper or
card
Demonstrate how the number line can be used to answer addition and
subtraction questions.
Intervention
1 Addition and subtraction 1 Q1–3
– The following activity can be completed as a group exercise. Ask pupils
(using a calculator) to work out: −6 + 10, 8 − 10, −6 − 10 and 8 − −10 and
consider the following questions.
What happens when you add a positive number? What happens when you
subtract a positive number?
What happens when you add a negative number? What happens when you
subtract a negative number?
Collect feedback from pupils. If pupils have difficulty understanding the
concept of adding a negative number, ask them to imagine adding ice to a
warm drink. What happens to the temperature of the drink?
Summarise: Adding a negative number is the same as subtracting a positive
number; subtracting a negative number is the same as adding a positive
number.
–
2 Addition and subtraction 2 Q4–5
– Investigate multiplying and dividing positive and negative numbers.
I owe £10. How would you write this? +£10 or −£10? If I owe three people
£10 each, how much do I owe in total?
Display −10 × 3 = −30.
22
Number knowledge
Level Up Maths 2–3,
Lesson 2.2
Functional skills
Examine patterns and
relationships Q7, 9
Framework 2008 ref
Process skills in bold type:
1.1 Y7/8, 1.3 Y7/8, 1.4 Y7
2.2 Y7/8
PoS 2008 ref
Process skills in bold type:
1.2b, 2.1a, 2.2b, 2.2c,
2.2d, 2.2l, 2.2p, 2.3e,
3.1a,b
I owe a total of £45 to nine people. If I owe an equal
amount to each, how much do I owe each person?
Display −45 ÷ 9 = −5.
Ask pupils to investigate the answers to the following.
2 × 2, 2 × 1, 2 × 0, 2 × −1, 2 × −2, 2 × −3; and −2 × 3,
−2 × 2, −2 × 1, −2 × 0, −2 × −1, −2 × −2, −2 × −3.
What happens when you multiply two negative
numbers together?
What happens when you divide a negative number
by a negative number?
Summarise: If the signs are different, the answer is
negative; if the signs are the same, the answer is
positive.
–
3 Multiplication and division Q6–8
–
4 Ordering negative decimals
– Display: 5.1, 4.45, 6.48, 4.54, 4.04, 5.38; −5.1,
−4.45, −6.48, −4.54, −4.04, −5.38. Ask pupils to
order the groups in ascending order.
Using 5.1, 4.45, −5.1 and −4.45 discuss the possible
misconception that −4.45 is smaller than −5.1. Q9
Activity A
Pupils work with negative numbers in context.
a) two H
b) two Ar
c) one H + one He
d) two Ar + one He
Related topics
Activity B
Encourage pupils to use a mixture of temperature values and negative decimals
in the creation of their ‘follow-me’ games.
Heights above and below
sea level, using contour
lines on maps.
Plenary
Discussion points
Try some of the ‘follow-me’ games pupils made in Activity B.
Discuss the use of
negatives in golf (below
par scores) and banking
(overdrafts and loans).
Homework
Homework Book section 2.2.
Challenging homework: Find the mean, median and range of 2, −7, −2, 8, 6, −4.
Answers
1
2
3
4
5
a)
a)
c)
a)
a)
−3
b) −2
c)
−7°C, −4°C, 3°C, 6°C, 8°C
−14°C, −8°C, −1°C, 3°C, 5°C
2°C
−31
b) 13
c)
8
2
6
⫺1
3
3
7
⫺8
11
⫺8
1
b)
b)
−2
d) −90
e) −10
−9°C, −5°C, −1°C, 2°C, 4°C
25°C
d) −20
e)
−29
f)
60
f)
−5
3
4
⫺12
23
⫺31
6
7
8
9
a)
b)
a)
a)
c)
a)
c)
i) 4
ii) 0
iii) −4
i) 5
ii) –5
iii) −3
−9
b) −5
c) −5
yellow and red
brown and yellow
−6.4, −5.4, −4.6, −4, −3.9
−5.63, −5.43, −5.34, −3.65, −3.06
Common difficulties
10 − −8 = 2? Emphasise
that subtracting a negative
number is the same
as adding a positive
number, so ‘minus minus’
becomes ‘plus’. Use a
practical example to aid
understanding: Imagine
taking ice out of a drink;
what happens to the
temperature of the drink?
LiveText resources
iv) −8
iv) −3
d) 2
e) −3
f)
b) yellow and blue
d) blue and red
b) −3.26, −3.15, −2.5, −1.95, −1.8
Explanations
12
Extra questions
Worked solutions
Negative numbers
23
2.3 Addition and subtraction
Objectives
• Consolidate the rapid recall of number facts,
including positive integer complements to 100
• Use standard column procedures to add and
subtract whole numbers and decimals with up
to two places
• Use standard column procedures to add and
subtract integers and decimals of any size,
including a mixture of large and small numbers
with differing numbers of decimal places
• Understand addition and subtraction as they
apply to whole numbers and decimals
Starter (1) Oral and mental objective
Start a spider diagram by displaying 2, 3, 7, 3, 1, 9, 8,
4 in a circle. Ask pupils to choose three of the digits
and add them mentally. Write the sum at the end of a
spider’s leg.
Can you make all the numbers between 6 and 24?
22 and 23 are not possible, but can be made using
four of the digits. Can pupils see how?
Starter (2) Introducing the lesson topic
Display 3, 7, 2, 5 and a decimal point. Tell pupils that the decimal point must be
placed between two of the digits.
Resources
What is the smallest number you can make? What is the largest number you can
make? What is the smallest even number you can make? What is the largest
odd number you can make?
Activity B: digit cards
0–7 (optional)
Main lesson
Level Up Maths 2–3,
Lesson 2.3
– What do you add to £30 to make £100?
If I subtract a number from 100, I get 51. What is the number that I subtracted?
Functional skills
What number when added to 36 gives 100?
Use appropriate
mathematical procedures
Q6, 9
Discuss pupil methods (counting on and subtracting from 100) and
emphasise the importance of checking answers.
36 + 74 = 100. Is this correct? Q1
– How would you work out 541 + 882 using a written method? Ask a volunteer
to complete the calculation.
–
1 Adding whole numbers
Pay particular attention to the lining up of columns with the same place
value, and the fact that the addition must start from the right. Discuss how to
deal with ‘carry over’, in particular how it should be recorded.
–
2 Subtracting whole numbers
Display (in column form): 541 − 382 = 241. Tell pupils that this calculation is
incorrect. Why is it incorrect? Emphasise that they should subtract the bottom
number from the top number (not subtract the smaller digit from the larger digit).
Complete the calculation correctly and discuss how to deal with ‘borrowing’.
Q2–3
24
Intervention
Number knowledge
Framework 2008 ref
Process skills in bold type:
1.2Y7/8, 1.3, 1.4 Y7/8, 2.2
Y8, 2.5Y7, 2.6 Y7/8
PoS 2008 ref
Process skills in bold type:
1.1a, 2.2d, e, j, o, 2.3a,
b, c
3.1a, b
– Mica is 0.8 m tall. Kevin is 0.4 m taller than Mica.
How tall is Kevin?
3 Adding decimals
–
Display (in column form): 0.8 + 0.4 = 0.12. Is this
correct?
Remind pupils of the place value of each digit and
discuss the importance of considering the answer in
context. 0.8 has 8 tenths and 0.12 has 1 tenth. Kevin is
taller than Mica so 0.12 cannot be the correct answer.
Work through 36.45 + 21.8. Emphasise that lining
up the decimal points ensures that the place value
of the digits in each column is correct. Q4
4 Subtracting decimals
–
How would you work out 1.45 − 0.3? Remind pupils
that 0.3 is the same as 0.30 and therefore, when
subtracting, a zero place holder can be used to fill
in any ‘blanks’. Q5–9
(Q7–8 include integers and decimals with varying
numbers of digits. Q9 includes decimals with up to
three decimal places.)
Activity A
Pupils investigate the sum of the tens and units digits
in pair complements to 100.
a) 10 or 0
b) 9 or 10
each multiples of 10.
c) 0 and 100, and pairs of numbers that are
Discussion points
Use of addition and
subtraction in daily life.
Activity B
Pupils find the sum and difference of two four-digit numbers.
Explain to pupils that each digit can only be used once per pair of numbers.
Answers: (total and differences can be made several ways)
a) 1340 + 2657 = 3997
b) 5301 − 4276 = 1025, 5276 − 4301 = 975
Plenary
Display (in column form): 1.45 − 0.3 = 1.42, 541 − 382 = 923, 1.7 + 4.6 = 5.13,
6.7 − 0.8 = 6.1.
Ask pupils to identify and correct the mistakes that have been made.
Discuss the context
photo and caption in the
Pupil Book. Explain how
in 1879 the Tay Bridge
(in Scotland) collapsed
(killing 75 people) because
wind load had not been
added in to the design
calculations.
Common difficulties
Homework
3 512
– 25
Homework Book section 2.3.
Challenging homework: Find the missing digits: 5.䊐79 + 2.64䊐 = 8.027 and
27.32䊐 − 9.5䊐7 = 䊐7.776 (3, 8; 3, 4, 1).
Answers
1
2
3
4
5
6
7
8
9
a)
a)
a)
a)
a)
a)
a)
a)
i) 44
3523
2046
22.5
21.6
£97.84
831
675
16.622
10.252
22.992
14.074
ii)
b)
b)
b)
b)
78
82 338
3203
25.81
45.88
b)
b)
73.32
12 469
b)
12.8
24.266
6.43
20.444
8.978
17.896
15.348
21.718
25.54
11.526
19.17
7.704
b)
c)
c)
c)
c)
82
6725
6867
123.86
17.68
c)
c)
41.36
12.72
£252.16
337
Make sure pupils take one
from the tens as well as
adding one to the units
when they ‘borrow’. If they
try to remember to deal
with the tens when they
come to them, they are
likely to forget.
LiveText resources
Explanations
Extra questions
Worked solutions
Addition and subtraction
25
2.4 Multiplication
Objectives
• Extend written methods to: HTU × U, TU × TU
• Consolidate the rapid recall of number facts,
including multiplication facts to 10 × 10
• Use known facts to derive unknown facts,
including products such as 0.7 and 6, and 0.03
and 8
• Know how to use the laws of arithmetic and
inverse operations
• Check a result by considering whether it is of
the right order of magnitude and by working
the problem backwards
Starter (1) Oral and mental objective
Which pair of numbers multiply together to give 72?
Can pupils find all six factor pairs?
Repeat for the six factor pairs of 96.
Starter (2) Introducing the lesson topic
What is the product of 6 and 8? What do you get
if you multiply 6 and 0.8? What is 0.06 times 8?
Continue asking pupils to derive multiplication facts,
varying the language as much as possible.
Main lesson
– Display 842 × 7. What would be an approximate calculation? Why is it
important to estimate the answer first?
Discuss possible estimates, e.g. 800 × 10, 840 × 10, 800 × 7. Draw out that
it is best to use numbers that are as close as possible to the given numbers,
but that the calculation must be easy to do mentally.
How can we work this out using a written method?
–
1 Written multiplication: the grid method 1
Explain that the numbers are split into units, tens, hundreds and so on (as
appropriate). Ask for volunteers to complete a box each, explaining which
multiplication they are doing.
How do we complete the calculation? Remind pupils that the separate
boxes have to be added together to give the total. Encourage pupils to write
‘Answer = …’ to remind themselves of this step. Q1
–
2 Written multiplication: the standard method 1
Display 528 × 9. Work through the calculation, detailing each step.
Ask a volunteer to attempt a similar question, explaining the method in their
own words to the rest of the group. Q2–3, 6–7
– Move on to ask a series of mental multiplication questions, up to 10 × 10, in
many different ways, e.g. What is the product of 9 and 8? What is the cost of
seven books priced at £6 each?
Illustrate to pupils that they know twice as many facts as they think they
know. If you know 9 × 5 = 45, you also know 5 × 9 = 45. Q4
26
Number knowledge
Resources
Plenary: mini whiteboards
(optional)
Intervention
Level Up Maths 2–3,
Lesson 2.4
Functional skills
Use appropriate
mathematical procedures
Q4, 10
Framework 2008 ref
Process skills in bold type:
1.1 Y7/8, 1.2 Y7/8, 1.3,
2.2Y8, 2.6Y7, 2.8Y7/8
PoS 2008 ref
Process skills in bold type:
1.2b, 1.3c, 2.1d, 2.2g,h, l,
p, 3.1b
– Starter (2) could be used at this point to discuss
mental multiplication with decimals. Display
6 × 8 = 48. 6 × 0.8 = ? 6 × 0.08 = ?
How can you use the first fact to solve the other
two? Q5
3 Wriiten multiplication: the grid method 2
–
Finish by considering TU × TU calculations.
Display 68 × 24. Work through the calculation using
the grid method, allowing pupils to dictate each step.
What is a good estimate for this calculation? How
many boxes do we need in the grid? Why? Q8
4 Written multiplication: the standard method 2
–
Display (in column form) 32 × 21 = 62.
How could you check if this calculation was correct?
How has this calculation been done? Complete the
calculation correctly. Q9–10
Activity A
This activity involves ‘trial and error’. However,
encourage pupils to consider the relative size of the
product of two numbers.
Answers: 125 × 4 = 500, 3 × 8 = 24, 6 × 9 = 54,
18 × 7 = 126, 42 × 5 = 210.
Activity B
Pupils use and apply their maths to solve a ‘missing digits’ multiplication
problem.
Answers: square = 5, triangle = 0, circle = 7, pentagon = 4.
Plenary
Display 826 × 6 = 4䊐56. What’s the missing digit? (9) Repeat for other
calculations, e.g. 123 × 9 = 11䊐7 (0), 49 × 94 = 4䊐06 (6).
Homework
Homework book section 2.4.
Challenging homework: Box 1: 18, 45, 62, 27, 54, 81, 36, 76, 98, 16.
Box 2: 1296, 1368, 1620, 1458, 6076. Use the numbers in Box 1 to
make five multiplications with a product from Box 2.
Answers
1 a)
e)
2 a)
e)
3 a)
4
5
6
7
8
9
10
2457
3927
2905
1887
£1080
c)
g)
c)
g)
b)
2816
4115
2448
4705
€765
d)
h)
d)
h)
3241
4960
2695
6237
Discussion points
Discuss the use of
multiplication in real life.
Examples: calculating
areas (refer to the context
photo in the Pupil Book),
stock-taking in shops,
calculating wages (pay
per hour times number of
hours worked).
Common difficulties
Pupils may start by
making an estimate, but
forget to use it to check
their answer.
Try using ‘Eat Creepy
Crawlies’ to help them
remember to ‘Estimate.
Calculate. Check’.
b)
f)
b)
f)
2864
956
2692
5808
7
2
LiveText resources
×
3
5
8
24
40
56
16
Explanations
9
27
45
63
18
4
12
20
28
8
Extra questions
6
18
30
42
12
Worked solutions
a) 2.7
b) 0.48
c) 50
b) and c) are incorrect. Correct answers are 2244 and 2884, respectively.
a) and c) are correct.
a) 364
b) 1932
c) 2484
£1105
a) 368
b) 10 080
c) 324
Multiplication
27
2.5 Squares and square roots
Objectives
• Know squares to at least 10 × 10
• Work out squares of numbers beyond 10 × 10
and the corresponding roots
• Use the square root key
• Know how to use the laws of arithmetic and
inverse operations
• Consolidate and extend mental methods of
calculation, working with squares and square roots
• Use squares and positive and negative square
roots
Starter (1) Oral and mental objective
Display a vertical number line from −20 m to +20 m.
Indicate the sea from −20 m to 0 m.
Indicate a bird at 13 m. How far is the bird above the
bottom of the sea? (33 m)
Indicate a dolphin at −8 m. How far is the dolphin from
the bottom of the sea? (12 m) How far below the bird
is the dolphin? (21 m)
Write a selection of depths on the board. Which is
nearest to sea level?
Starter (2) Introducing the lesson topic
What is three times 8? 9 multiplied by 7? four 4s? Continue practising tables,
varying the language as much as possible.
Resources
Activity B: ICT –
spreadsheet (optional)
Intervention
Main lesson
–
1 Square numbers
What is a square number? Demonstrate the square number 9 by drawing
dots.
Show pupils how a square multiplication, for example 3 × 3, is written in
shorthand (32). Emphasise that this is read as ‘3 squared’.
What is 102? 72? Is 1 a square number? Is 63 a square number? Q1–4
Framework 2008 ref
2 Square roots
Remind pupils about inverse operations. What is the inverse of multiplying?
adding?
Explain
that finding the square root is the inverse of squaring and introduce
__
the √2 notation. What is the square root of 49? Why?
Demonstrate how to use the square root key on a calculator. What is the
square root of 998 001?
Display: 3 × 3, 3 × −3, −3 × 3, −3 × −3. Ask pupils for the answers. Point out
that 3 × 3 and −3 × −3 both have the same answer, 9.
What is the square root of 9? Draw out that there are two possible answers,
3 and −3. Conclude that all positive integers have a positive and a negative
square root.
28
Functional skills
Decide on the methods,
operations and tools,
including ICT, to use in a
situation Q6, 9
How would you write the square number 9 as a multiplication?
–
Level Up Maths 2–3,
Lesson 2.5
Number knowledge
Process skills in bold type:
1.1Y7/8, 1.2 Y7/8, 1.3
Y7/8,1.4Y7, 2.2Y7/8, 2.5
Y7/8, 2.7 Y7/8
PoS 2008 ref
Process skills in bold type:
1.1b, 1.2b, 2.1a, 2.2d,
2.3d,e, h, i, l, 3.1b
Website links
www.heinemann.co.uk/
hotlinks
__
Get pupils to work out √9 on a calculator. Say that
the calculator only gives the positive square root,
__
because, by convention, when you use the √2
notation it always means the positive square root.
Q5–12 (Q9–11 require pupils to use mental methods
with squares and square roots.)
Activity A
Pupils find a pattern in the units digit of square
numbers up to 20 × 20.
a) The last digits of the square numbers up to 20 × 20,
repeat the sequence 1, 4, 9, 6, 5, 6, 9, 4, 1, 0.
b) Neither 5327 nor 6423 can be square, as they do
not end in numbers from the above sequence.
Activity B
Pupils add pairs of square numbers to find sums less
than 100. Pupils could use an addition table with
square numbers along the top and side. This could be
done on a spreadsheet.
a) 2, 5, 8, 10, 13, 17, 18, 20, 25, 26, 29, 32, 34, 37, 40,
41, 45, 50, 52, 53, 58, 61, 65, 68, 72, 73, 74, 80, 82,
85, 89, 90, 97.
b) 50 = 12 + 72 or 52 + 52; 65 = 12 + 82 or 42 + 72;
85 = 22 + 92 or 62 + 72.
c) 52 = 32 + 42; 102 = 62 + 82
Related topics
Plenary
Investigate square
numbers in context by
exploring the number of
squares on a chessboard.
Display several squares with the length of one side labelled. Explain how area is
calculated then ask pupils for the area.
Display a few more squares, this time with the area written inside the square.
Ask pupils for the side length.
Homework
Homework Book section 2.5.
Challenging homework: Can every square number up to 12 × 12 be expressed
as the sum of two prime numbers?
Answers
1
2
3
4
5
6
7
8
9
10
11
12
4, 9, 16, 25, 100.
a) 9
b) 25
a) 121
b) 169
3600
a) 18
b) 2.5
c) 5678
a) It added 1 to the correct answer.
b) It added 10 to the correct answer.
c) It doubled the correct answer.
a) False. The greatest square number less than 100 is 81.
b) False. There are two square numbers between 101 and 160 (121 and 144).
c) False. 152 = 225.
d) False. 102 is four times 52.
a) 3
b) 6
c) 9
d) 11
a) 56
b) 75
c) 17
d) 49
a) 6
b) 10
c) 8
d) 8
e) 28
f) 4
g) 13
h) 225
a) 7
b) 81
c) 4
d) 8
a) +4, −4
b) +10, −10
c) +5, −5
d) +12, −12
e) +6, −6
f) +7, −7
g) +9, –9
h) +25, −25
Discussion points
It is only possible to
arrange a group of objects
in a square if the number
of objects is a square
number.
Common difficulties
Not understanding why
a square root can be
negative. Remind pupils
that a ‘negative’ times a
‘negative’ is a ‘positive’.
Or, if the signs are the
same, the answer is
positive.
LiveText resources
Explanations
Extra questions
Worked solutions
Squares and square roots
29
2.6 Division
Objectives
• Extend written methods to HTU ÷ U
• Divide three-digit by two-digit whole numbers
• Round up or down after division, depending on
the context
• Consolidate the rapid recall of number facts,
including multiplication facts to 10 × 10, and
quickly derive associated division facts
• Know how to use the laws of arithmetic and
inverse operations
• Check a result by considering whether it is of
the right order of magnitude and by working
the problem backwards
Starter (1) Oral and mental objective
Ask pupils to multiply and divide by 10, 100, 1000 in
context. For example:
A pencil costs 28 pence. How much do 10 pencils
cost? The owner of a sports shop buys 100 hockey
balls for £776. How much is that per ball?
Starter (2) Introducing the lesson topic
Display this multiplication grid.
×
5
2
7
30
No special resources
required
8
49
9
Resources
27
Intervention
Pupils take turns to fill in an empty square of their choice.
Level Up Maths 2–3,
Lesson 2.6
Main lesson
Functional skills
– Display 512 ÷ 4. Refer to the context photo in the Pupil Book and say that
512 mg of Drug X has to be divided into four doses per day.
Use appropriate
mathematical procedures
Q4, 7
What would be an approximate calculation? Why is it important to estimate
the answer first?
–
1 Division: written method
How can you work this out using a written method?
Demonstrate using repeated subtraction. Highlight to pupils that they need
to look for the highest multiple of 4 and that choosing multiples of 10, 100,
etc. makes the calculation easier.
What is the answer? Emphasise that they must work out how many lots of
the divisor that they have subtracted altogether.
Check the answer against the estimate. Q1
–
2 Division with remainders
Display 328 ÷ 9. Work through the calculation allowing pupils to dictate each
step. Prompts may be necessary: What is the highest multiple of 9 you can
use?
30
Number knowledge
Framework 2008 ref
Process skills in bold type:
1.1Y7/8, 1.2Y7, 1.3,
2.2Y8, 2.6Y8, 2.8 Y7/8
PoS 2008 ref
Process skills in bold type:
2.1a, 2.2h, j, l, m, 3.1b
When the point is reached where further subtraction
is impossible ask: What do you do now?
Discuss how the answer may be written as 36
remainder 4, or 36_49.
What is the answer to 328 ÷ 9? How many boxes
are needed to package 328 gifts if each box can
hold nine gifts? How many nine-litre bottles will be
full if there is 328 litres of water?
Why are the answers different? Emphasise that
the context of the problem is vital and that each
question must be read carefully. Q2–4
– Display 6 × 4 = 24. What other multiplication fact
can you make?
What is the inverse of multiplication? What division
fact can you derive? How could you check if
32 × 9 = 298 was correct? Q5–6
– Finish by considering HTU ÷ TU calculations.
Display 420 ÷ 12. Ask for a volunteer to complete
the calculation, explaining each step. Q7–10
Activity A
This activity promotes strategic thinking whilst enabling
pupils to practise basic division.
Activity B
Pupils may use a process of ‘trial and elimination’ to find the missing digits in
HTU ÷ TU calculations.
Discussion points
Plenary
Discuss how divisibility
tests can be used to see if
there will be a remainder.
Link to the Plenary
activity.
The answers to these calculations are incorrect. How can you tell?
Common difficulties
249 ÷ 2 = 124 (249 is odd, so there should be a remainder.)
Pupils may write 12.4
when they mean 12
remainder 4. Emphasise
that the correct shorthand
is 12 R 4.
Answers:
a) 476 ÷ 17 = 28
b) 405 ÷ 15 = 27
c) 434 ÷ 31 = 14
d) 432 ÷ 16 = 27
e) 999 ÷ 37 = 27
f) 756 ÷ 21 = 36
435 ÷ 19 = 45 (The answer should be about 400 ÷ 20 = 20.)
835 ÷ 5 = 170 R 6 (835 is divisible by 5, so there should not be a remainder.)
Homework
LiveText resources
Homework Book section 2.6.
Challenging homework: Find a HTU ÷ TU calculation that gives the answer 23_45.
Explanations
Extra questions
Worked solutions
Answers
1 a) 77
b) 61
c) 235
d) 82
e) 123
f) 215
g) 164
h) 258
2 a) 235 remainder 1
b) 58 remainder 3
c) 117 remainder 6
d) 88 remainder 5
3 39
4 a) and c) are incorrect. The correct answers are 41 remainder 7 and 89 remainder 2, respectively.
5 a) 72 ÷ 8 = 9, 72 ÷ 9 = 8
b) 180 ÷ 10 = 18, 180 ÷18 = 10
c) 552 ÷ 12 = 46, 552 ÷ 46 = 12
6 c) is correct.
7 a) 39
b) 15
c) 24
d) 8
60
2
7
2
__
__
8 a) 17__
b)
13
c)
49
d) 38__
29
61
11
19
6
9 £21__
= £21.50
12
10 a) 15
b)
14
c)
210
Division
31
Animating Shaun
Notes on plenary activities
The activities consider model construction and the
use of scale drawings. Pupils may need a quick recap
on scale and how it is used to represent ‘life-size’
measurements.
The problems are intended to be solved using
written methods. However, some pupils may use a
mental method as they can ‘see’ the mathematical
connection between the numbers used. These
activities could be used to assess which pupils
‘look’ at the numbers given and the context in which
they are given (and those pupils who simply identify
the calculation to be performed and perform it
automatically).
Part 6: As an interesting aside, an average of
7 seconds of footage per day was captured while
filming the series. Based on this fact, additional
questions could be asked:
How many days filming were needed for one
episode?
How many days filming were needed for the complete
series?
If one animator captured 7 seconds of footage per
day, how many animators would be required if the
whole series had to be filmed in 6 months?
Part 9: In arranging the decimals, pupils only need to consider the whole
number and the tenths. How quickly do pupils recognise this?
Introduce Frank (3.38), Gwen (4.0) and Harold (4.300) and ask pupils to place
these sheep in the correct positions.
Solutions to the activities
1
Length on sketch
Real-life length
20 cm
200 cm or 2 m
8 cm
80 cm
102 cm
1020 cm or 10.2 m
2 20_57 weeks or 20.7 weeks (to 1 decimal place)
3 126 days (or 18 weeks)
4 Not sensible; the scale model of the sheep would be 0.65 cm
Real-life height (m)
Real-life length (m)
Tractor
2.66
3.85
Jeep
1.88
4.83
5
6 1500 movements per minute
7 7 minutes
8 18 ounces ≈ 504 g; so the tub does not contain enough plasticine
9 3.256, 3.386, 4.099, 4.354, 4.632
(Amelia, Dorinda, Evie, Bertie, Callum)
32
Number knowledge
Answers to practice SATs-style questions
1 a) 5
b) 300
c) 14
(1 mark per correct answer)
2 Vikram is incorrect
4.6 + 5.5 = 10.1 (1 mark)
27.4 – 18.2 = 9.2 (1 mark)
3 a) −14
b) 0, 10 (or other correct answer)
(1 mark per correct answer)
4 Sami: £4.80 × 3 = £14.40 (1 mark)
Maniche: £8.10 × 2 = £16.20 (1 mark)
Maniche pays £1.80 more than Sami (1 mark)
5 Pack of 6 batteries is better value for money
Pack of 9 batteries = £3.90, so 3 batteries = £1.30
(1 mark)
Pack of 6 batteries = £2.50, so 3 batteries = £1.25
(1 mark)
(or similar argument)
6 23.5, 25.01, 25.6, 25.11, 27.9, 28.27, 28.72
(2 marks for all correct, 1 mark for 4 correct)
Functional skills
The plenary activity practises the following functional
skills defined in the QCA guidelines:
• Select the mathematical information to use
• Use appropriate mathematical procedures
Plenary
33