Number knowledgeassets.pearsonschool.com/asset_mgr/current/201214/...† Multiply/divide integers and decimals by 10, 100, 1000; explain the effect † Know that comparative measurements

18 Number knowledge

Unit objectives

• Understand and use decimal notation and place value; compare and order decimals in different contexts

• 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

Website links

• 2.1 Planet statistics

• 2.5 Squares on a chessboard

• To view websites relevant to this unit please visit www.heinemann.co.uk/hotlinks

2 Number knowledge

Opener 19

Notes on context

Spitfi res were single-seat fi ghter planes and were produced in greater numbers than any other Allied planes during the Second World War. The Spitfi re 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.

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

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

LiveText resources

• Elements of numbers

• Use It!

Games

Audio glossary

Skills bank

• 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 auto-marked 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

20 Number knowledge

2.1 Decimal know-how

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 fi ve six’ and £1.56 is ‘one pound fi fty-six’. 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.

Challenge pupils to put these decimals in order: 0.543, 0.342, 0.35, 0.5, 0.053, 0.305, 0.53.

Next consider decimal measurements.

– 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

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

Resources

Starter: mini whiteboards (optional)

Activity A: dice

Plenary: mini whiteboards (optional)

Intervention

Level Up Maths 2–3, Lesson 2.1

Functional skills

Examine patterns and relationships Q7, 9

Framework 2008 ref

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

Decimal know-how 21

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 fi ve signifi cant fi gures. 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.

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 fi nd 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?

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-fi ve point zero eight d) zero point one nine fi ve 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) 100010 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, 0.0536211 a) > b) <

Related topics

Investigation of the use of decimals in context by comparing the properties of planets in the solar system.

Common diffi culties

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

Explanations

Extra questions

Worked solutions

22 Number knowledge

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

Resources

Activity B: scrap paper or card

Intervention


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


1.2b, 2.1a, 2.2b, 2.2c, 2.2d, 2.2l, 2.2p, 2.3e, 3.1a,b


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?


Display a vertical number line from −10 to 10. Ask questions such as: A lift starts at fl oor 2. It goes up three fl oors. 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.

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?

Demonstrate how the number line can be used to answer addition and subtraction questions.

– 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 diffi culty 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.

Related topics

Heights above and below sea level, using contour lines on maps.

Discussion points

Discuss the use of negatives in golf (below par scores) and banking (overdrafts and loans).


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

Explanations

Extra questions

Worked solutions

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

Activity B

Encourage pupils to use a mixture of temperature values and negative decimals in the creation of their ‘follow-me’ games.

Plenary

Try some of the ‘follow-me’ games pupils made in Activity B.

Homework


Challenging homework: Find the mean, median and range of 2, −7, −2, 8, 6, −4.

Answers1 a) −3 b) −2 c) 1 d) −90 e) −10 f) 602 a) −7°C, −4°C, 3°C, 6°C, 8°C b) −9°C, −5°C, −1°C, 2°C, 4°C c) −14°C, −8°C, −1°C, 3°C, 5°C3 a) 2°C b) 25°C4 a) −31 b) 13 c) −2 d) −20 e) −29 f) −55

6 a) i) 4 ii) 0 iii) −4 iv) −8 b) i) 5 ii) –5 iii) −3 iv) −37 a) −9 b) −5 c) −5 d) 2 e) −3 f) 128 a) yellow and red b) yellow and blue c) brown and yellow d) blue and red9 a) −6.4, −5.4, −4.6, −4, −3.9 b) −3.26, −3.15, −2.5, −1.95, −1.8 c) −5.63, −5.43, −5.34, −3.65, −3.06

Negative numbers 23

8 2 �1 7 3

6 3 �8 4

3 11 �12

�8 23

�31

24 Number knowledge

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

Resources

Activity B: digit cards 0–7 (optional)

Intervention


Functional skills

Use appropriate mathematical procedures Q6, 9

Framework 2008 ref


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


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?


Display 3, 7, 2, 5 and a decimal point. Tell pupils that the decimal point must be placed between two of the digits.

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?

Main lesson

– 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?

What number when added to 36 gives 100?

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

Discussion points

Use of addition and subtraction in daily life.

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.


3 512– 2 5

3 3 7

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

– 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 fi ll 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 c) 0 and 100, and pairs of numbers that are each multiples of 10.

Activity B

Pupils fi nd 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.

Homework


Challenging homework: Find the missing digits: 5.�79 + 2.64� = 8.027 and 27.32� − 9.5�7 = �7.776 (3, 8; 3, 4, 1).

Answers1 a) i) 44 ii) 78 b) 822 a) 3523 b) 82 338 c) 67253 a) 2046 b) 3203 c) 68674 a) 22.5 b) 25.81 c) 123.865 a) 21.6 b) 45.88 c) 17.686 a) £97.84 b) £252.167 a) 831 b) 73.32 c) 41.368 a) 675 b) 12 469 c) 12.72

9 16.622 12.8 8.978 25.5410.252 24.266 17.896 11.52622.992 6.43 15.348 19.1714.074 20.444 21.718 7.704

Addition and subtraction 25

26 Number knowledge

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

Resources

Plenary: mini whiteboards (optional)

Intervention


Functional skills


Framework 2008 ref


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


Which pair of numbers multiply together to give 72? Can pupils fi nd all six factor pairs?

Repeat for the six factor pairs of 96.


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 fi rst?

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

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).


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’.

LiveText resources

Explanations

Extra questions

Worked solutions

– 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 fi rst 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�56. What’s the missing digit? (9) Repeat for other calculations, e.g. 123 × 9 = 11�7 (0), 49 × 94 = 4�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 fi ve multiplications with a product from Box 2.

Answers 1 a) 2457 b) 2864 c) 2816 d) 3241

e) 3927 f) 956 g) 4115 h) 4960 2 a) 2905 b) 2692 c) 2448 d) 2695

e) 1887 f) 5808 g) 4705 h) 6237 3 a) £1080 b) €765

4 × 3 5 7 2

8 24 40 56 16

9 27 45 63 18

4 12 20 28 8

6 18 30 42 12

5 a) 2.7 b) 0.48 c) 50 6 b) and c) are incorrect. Correct answers are 2244 and 2884, respectively. 7 a) and c) are correct. 8 a) 364 b) 1932 c) 2484 9 £110510 a) 368 b) 10 080 c) 324

Multiplication 27

28 Number knowledge

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


• Consolidate and extend mental methods of calculation, working with squares and square roots

• Use squares and positive and negative square roots

Resources

Activity B: ICT – spreadsheet (optional)

Intervention


Functional skills

Decide on the methods, operations and tools, including ICT, to use in a situation Q6, 9

Framework 2008 ref


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


1.1b, 1.2b, 2.1a, 2.2d, 2.3d,e, h, i, l, 3.1b

Website links

www.heinemann.co.uk/hotlinks


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?


What is three times 8? 9 multiplied by 7? four 4s? Continue practising tables, varying the language as much as possible.

Main lesson

– 1 Square numbers

What is a square number? Demonstrate the square number 9 by drawing dots.

How would you write the square number 9 as a multiplication?

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

– 2 Square roots

Remind pupils about inverse operations. What is the inverse of multiplying? adding?

Explain that fi nding 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.

Squares and square roots 29

Related topics

Investigate square numbers in context by exploring the number of squares on a chessboard.

Discussion points

It is only possible to arrange a group of objects in a square if the number of objects is a square number.


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

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 fi nd 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 fi nd 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

Plenary

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


Challenging homework: Can every square number up to 12 × 12 be expressed as the sum of two prime numbers?

Answers 1 4, 9, 16, 25, 100. 2 a) 9 b) 25 3 a) 121 b) 169 4 3600 5 a) 18 b) 2.5 c) 5678 6 a) It added 1 to the correct answer. b) It added 10 to the correct answer. c) It doubled the correct answer. 7 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. 8 a) 3 b) 6 c) 9 d) 11 9 a) 56 b) 75 c) 17 d) 4910 a) 6 b) 10 c) 8 d) 8 e) 28 f) 4 g) 13 h) 22511 a) 7 b) 81 c) 4 d) 812 a) +4, −4 b) +10, −10 c) +5, −5 d) +12, −12 e) +6, −6 f) +7, −7 g) +9, –9 h) +25, −25

30 Number knowledge

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


• Check a result by considering whether it is of the right order of magnitude and by working the problem backwards

Resources

No special resources required

Intervention


Functional skills


Framework 2008 ref


1.1Y7/8, 1.2Y7, 1.3, 2.2Y8, 2.6Y8, 2.8 Y7/8

PoS 2008 ref


2.1a, 2.2h, j, l, m, 3.1b


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?


Display this multiplication grid.

× 2 75 30

849

9 27

Pupils take turns to fi ll in an empty square of their choice.

Main lesson

– 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.

What would be an approximate calculation? Why is it important to estimate the answer fi rst?

– 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?

Discussion points

Discuss how divisibility tests can be used to see if there will be a remainder. Link to the Plenary activity.


Pupils may write 12.4 when they mean 12 remainder 4. Emphasise that the correct shorthand is 12 R 4.

LiveText resources

Explanations

Extra questions

Worked solutions

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 4 _ 9 .

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 fi nd the missing digits in HTU ÷ TU calculations.

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

Plenary

The answers to these calculations are incorrect. How can you tell?

249 ÷ 2 = 124 (249 is odd, so there should be a remainder.)

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


Challenging homework: Find a HTU ÷ TU calculation that gives the answer 23 4 _ 5 .

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

8 a) 17 2 __ 29 b) 13 60 __ 61 c) 49 7 __ 11 d) 38 2 __ 19

9 £21 6 __ 12 = £21.5010 a) 15 b) 14 c) 210

Division 31

32 Number knowledge

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 fi lming the series. Based on this fact, additional questions could be asked:

How many days fi lming were needed for one episode?

How many days fi lming 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 fi lmed 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 5 _ 7 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

5 Real-life height (m) Real-life length (m)

Tractor 2.66 3.85

Jeep 1.88 4.83

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)

Plenary 33

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 defi ned in the QCA guidelines:

• Select the mathematical information to use

• Use appropriate mathematical procedures

Documents

Number knowledgeassets.pearsonschool.com/asset_mgr/current/201214/...† Multiply/divide integers and decimals by 10, 100, 1000; explain the effect † Know that comparative measurements