maths frameworking

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TEACHERS PACK 1YEAR 7

HarperCollinsPublishersLtd 2002


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Oral and mental starter

Simple tests of divisibility Start with, say, the number 105 and ask who knows which whole

numbers (integers) divide exactly into this number. Many will offer theanswer 5. Ask them why. Bring out the fact that any number ending in 5

or 0 will divide exactly by 5. It is divisible by 5. Then ask for other numbers divisible by 5. Next, ask if the class know any number that will divide exactly into 106.

Lead them to 2. Request some more numbers that are divisible by 2. Ask what is common

to these numbers, bringing out the fact that any number ending in aneven number or 0 is divisible by 2.

Finish by holding up some number cards and asking which numbers aredivisible by 5 or 2, or both.

Main lesson activity

Ask the class for the next numbers in the sequence (on the board)2, 5, 8, 11, 14,

You should get some correct answers of 17, 20, 23, etc. Ask how they could tell. Get from the class the idea that there is a rule

here of adding on 3 each time. You could introduce the term differencehere.

Ask whether this rule always gives the same sequence. It does not, sinceif we change the starting number, we will get a different sequence.

Get someone to suggest a rule. Keep it simple and accept only add ormultiply to start with. Subtraction and division are a little trickier andwill be left till later (unless you feel like introducing them here).

With the same rule, ask for some different starting points and get the classto tell you what the different sequences are.

Do a few of these, making sure that you choose a variety of additions andmultiplications. But do keep the numbers within the scope of the class.

Write 1 and 2 on the board and ask: What comes next? Get some answers and rules from the class, or prompt them. Examples

are:1, 2, 3, 4, Add on 1 each time

1, 2, 4, 8, Double the number each time1, 2, 4, 7, Add on 1, then 2, then 3,

The same starts can lead to different sequences with different rules.

2 HarperCollinsPublishersLtd 2002

Algebra 1CHAPTER

1

LESSON1.1

Framework objectives Sequences and rulesGenerate and describe simple integer sequences.

Maths Frameworking Year 7 Teachers Pack 1 ISBN 0 00 713862 8


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Write 1, , 10 on the board and ask: What sequence of numbers can goin between?

Get some answers from the class, or prompt them. Examples are:1, 2, 3, 4, 5, 6, 7, 8, 9, 10 Add on 1 each time1, 4, 7, 10 Add on 3 each time1, 5, 10 Add on 4 then 5

The class can now do Exercise 1A from Pupil Book 1.

Plenary

Write on the board the sequence O, T, T, F, F, . Ask for the next twoletters in the sequence. They are S, S (One, Two, Three, Four, ).

Finish by reminding the class that there are very many differentsequences found within nature and life in general, and that part ofmathematics is to help find these patterns.

Extension Answers

1 Answers depend on the two sequences chosen2 Answers depend on the sequences chosen

HarperCollinsPublishersLtd 2002 3

Exercise 1A Answers

1 a 2, 5, 8, 11 b 1, 3, 9, 27 c 4, 9, 14, 19 d 2, 20, 200, 2000 e 6, 15, 24, 33f 2, 10, 50, 250 g 3, 10, 17, 24 h 5, 10, 20, 40

2 a 8, 10, add 2 b 12, 15, add 3 c 1000, 10000, multiply by 10d 250, 1250, multiply by 5 e 21, 28, add 7 f 19, 24, add 5 g 36, 45, add 9h 48, 60, add 12

3 a 2, 5, 8, 11, 14, 17, 20 b 1, 6, 11, 16, 21, 26 c 1, 10, 100, 1000, 10000d 5, 7, 9, 11, 13, 15 e 9, 14, 19, 24, 29, 34 f 3, 6, 12, 24, 48, 96

Homework

Homework

Answers

Write the next two terms in each of these sequences. Describe the term-to-term rule you have used.

a 3, 7, 11, b 5, 10, 20, c 30, 25, 20, d 2, 4, 8, e 1, 8, 15,

f 7, 9, 11, g 5, 14, 23, h 7, 13, 19, i 13, 26, 39,

a 15, 19, add 4 b 40, 80, multiply by 2 c 15, 10, subtract 5 d 16, 32, multiply by 2 e 22, 29, add 7f 13, 15, add 2 g 32, 41, add 9 h 25, 31, add 6 i 52, 65, add 13

integer sequence

term rule starting point difference

Key Words



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Oral and mental starterDouble then multiply by 4

Possible starter: I was on holiday in Tango Land, where to find the priceof things in pence, I had to double their dollar prices.

So, for example, a plain ice cream costs $24. What is the price in pence?Give some more examples of easy doubles. For example, Mars Bar $32,Rolos $25, Coke $43, morning paper $35.

Then introduce a more difficult one. For example, a lolly costs $17. Whatis its price in pence?

Show a possible strategy on the board:Double 17 = 2 10 + 2 7 = 20 + 14 = 34

Get the students to share other strategies for doubling such numbers.Make up some more prices and get them doubled.

At some point, move this on to 4 by doubling and doubling again. Givesome examples and then ask the class to give answers to your numbersmultiplied by 4.


The local park is having a new path laid to look like this:It will have black slabs when complete.

How many white slabs will be needed to go with them tocomplete the pattern?

Lets break down the pattern.

Number of black slabs 1 2 3 4Number of white slabs 4 7 10 13

Look at the differences. We add on 3 white slabs each time. Notice how

many 3s we add on each time.For 2 black slabs, we add on 1 three.For 3 black slabs, we add on 2 threes.For 4 black slabs, we add on 3 threes, and so on.Hence, for 10 black slabs, we add on (10 1) threes = 9 threes to

the first term of 4.That is, the number of white slabs to go with 10 black slabs is

4 + (10 1) 3 = 4 + 27 = 31 Ask the class: If we have a sequence, say, 7, 12, 17, 22, , how do we

find out what the 10th term is? Talk about the terms: 1st term is 7, 2nd term is 12, 3rd term is 17, etc.

We again look at the differences. Here it is 5 each time. To get to the10th term, we will need to add on (10 1) fives to the first term of 7.This gives

9 5 + 7 = 45 + 7 = 52


LESSON

1.2

Framework objectives Finding missing terms

Generate terms of a simple sequence, given a rule. (For example, find a term fromthe previous term; find a term given its position in the sequence.)



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The class can now do Exercise 1B from Pupil Book 1.

Plenary

Ask the class to explain the words term, difference. Remind the class that today they have been looking at simple sequences,

but there are many other different sequences that they will meet. Put a sequence on the board, say, 8, 10, 12, 14, Ask them: What is the 10th term? What is the 21st term? What is the 51st

term? What is the 101st term, the 100th term, ?

Extension Answers

54 and 549


Exercise 1B Answers

1 a 12, 22 b 21, 46 c 31, 66 d 17, 32 e 17, 37 f 34, 74 g 60, 110h 46, 91 i 27, 57

2 a 7, 25 b 3, 48 c 7, 43 d 5, 86

3 Sequence A 9, 11, 13, 15, 17, 19, 21, 23, and 27Sequence B 4, 9, 14, 19, 24, 29, 34, 39, and 49Sequence C 2, 9, 16, 23, 30, 37, 44, 51, and 65Sequence D 5, 15, 25, 35, 45, 55, 65, 75, and 95Sequence E 2, 5, 8, 11, 14, 17, 20, 23, and 29Sequence F 8, 10, 12, 14, 16, 18, 20, 22, and 26

4 1015 1256 2747 544

Homew

ork

Homework

Answers

In each of the following sequences, find the missing terms and the 10th term.

Term 1st 2nd 3rd 4th 5th 6th 7th 8th 10th

Sequence A 14 16 18 20

Sequence B 9 19 29 39

Sequence C 18 26 42 50

Sequence D 25 47 80

Sequence E 11 23 41

Sequence A 6, 8, 10, 12, 14, 16, 18, 20, and 24Sequence B 4, 9, 14, 19, 24, 29, 34, 39, and 49Sequence C 2, 10, 18, 26, 34, 42, 50, 58, and 74Sequence D 3, 14, 25, 36, 47, 58, 69, 80, and 102Sequence E 5, 11, 17, 23, 29, 35, 41, 47, and 59

sequence term difference

Key Words



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Oral and mental starterHalf of

Tell the story: Last week I went shopping at . One shop that was closingdown was having a half-price sale. They were selling shirts that usuallycost 8 at the new price of ? Get the students to give you the answer, 4.Continue with other examples, such as suits at 60, trainers at 24, booksat 4.

They were easy to do, but what about some pictures at 7? What is halfof that? Continue with a few similar problems. For example, tapes at15, DVDs at 21.

Discuss strategy with the class. How did they work out half of 21?Probably of 20 added to of 1, which is 10 + 50p = 10.50.

Ask for some more halves: half of 27, 45, 89. Then what about half of 38? How do we work this out? (Be ready for

lots of different strategies.) We could work out of 30, which is 15,and add it to of 8, which is 4, giving us 19.

Try a few more halves: half of 32, 54, 56, 94. You could then extend this to half of amounts such as 37:

of 30 + of 7 = 15 + 3.50 = 18.50


Show on the board the rule

Explain that today we are looking at functions. A function is a rule whichgives a unique result for each different starting number or input.

We can think of this as a machine a function machine, which has aninput number and an output number.

If 2 goes into this machine, we say the input is 2. What comes out is

2 + 3, so the output is 5.Show this on the diagram, so that it looks like this:

Ask for the outputs from some other input numbers and add them to thediagram. Create a list under Input and Output.

Demonstrate that we can go backwards. Ask: What input is needed toget 12 as the output? Extend the function diagram, illustrating that wecan work from either side.

The class can now do Exercise 1C from Pupil Book 1.

12

12

12

12

12

12


LESSON

1.3

Framework objectives Function machines

Express simple functions in words, then using symbols. Represent them inmappings.


+ 3

input 2 + 3 5 output


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Plenary

What is a function? Can someone give me a simple function? Put it on the board.

Can someone give me an input? What is the output? Ask for an output. What is the input? Then ask what would be the function that took outputs and created inputs? Tell the class that this is called the inverse function.


Exercise 1C Answers

1 a output {7, 8, 11, 14} b output {2, 3, 6}, input 11 c output {20, 25, 55}, input 8d output {10, 8, 6}, input 50

2 a add 2 b multiply by 3 c divide by 2 (or halve) d multiply by 83 a {3, 5, 9}, 19 b {3, 6, 9}, 20 c input {5, 10}, output {12, 16}

d input {1, 3, 10}, output 32

Extension Answers

Variety of answers depending on the starting inputs

Homework

Homework

Answers

Draw a diagram to show each of the following functions.

a b

c d

Answers will depend on the starting inputs

function input

output

Key Words

add 5 multiply by 4

multiply by 3 add 11


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Mental and oral starterMultiply by 5

Ask for a few numbers from the five times table. Remind the class thatthey can use their hands to help them.

Notice that all the products end in either 5 or 0. Ask for the answer to 14 5. Discuss the different strategies for this, such as 5 10 + 5 4 or

(twice 5) (14 2). Put a number grid on the board, such as that on the right. Start by asking

for five times those numbers that start with 2, 4, 6 and 8, since they willhalve easily.

Then ask for the other multiples of 5.


Remind the class about functions. Ask about the rule

Start with some inputs and some outputs.

Then show a combined function or double function, for example:

Go through the input to the output with say {2, 3, 4 and 5} to give thefollowing diagram:

Go through another example, such as:

The class can now do Exercise 1D from Pupil Book 1.


LESSON

1.4

Framework objectives Double function machines

Express simple functions in words, then using symbols. Represent them inmappings.


24 62 18 70

48 12 36 86

54 28 72 66

30 44 58 82

5

2 + 3

2 + 3

+ 4 2

2

3

4

5

2 2 + 3

3 2 + 3

4 2 + 3

5 2 + 3

7

9

11

13


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Plenary

Ask: Can you think of a double function that would keep the numbers asthey are? (For example, 3 3.)

Talk about their suggestions such as:


Exercise 1D Answers

1 a 9, 11, 13, 15 b 7, 13, 19, 25 c 9, 12, 15, 18 d 18, 20, 22, 242 A variety of diagrams depending on the starting inputs3 a multiply by 2; 18, 20 b subtract 1; 14, 17 c subtract 1; 15, 19

d multiply by 4; 7,84 a multiply by 2 add 1 b multiply by 3 subtract 1 c multiply by 5 add 2

d add 3 multiply by 2 or multiply by 2 add 6e add 1 multiply by 3 or multiply by 3 add 3f add 2 multiply by 5 or multiply by 5 add 10

function input output

Key Words

Extension Answers

a {1, 3, 4, 7} b {2, 3, 5, 8} c {3, 5, 9, 10}

Homework

Homework

Answers

Draw a diagram to show each of the following functions.

a b

c d

A variety of diagrams depending on the starting inputs

multiply by 4 add 5

multiply by 3add 1

multiply by 4add 5

multiply by 3 add 1

+ 2 2

1 1


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Plenary

Give the students some starting numbers and finishing numbers (inputsand outputs). Ask if anyone can write on the board a function that will

map the first number onto the second.For any pair there are a number of different functions available, but onlytwo obvious ones. For example, for an input and output of 2 6,possible functions include n + 4 and 3 n.

Extension Answers

Variety of answers depending on the starting point


Exercise 1E Answers

1 a {7, 8, 12, 15} b {12, 15, 24, 33} c {12, 24, 30, 42} d {2, 6, 11, 17}2 a n + 5 b n 2 c n 2 d n 83 a n + 7 b n 5 c n 1 d n 4 e n 3 f n + 5 g n 10 h n 34 Diagrams will depend on their starting numbers. Encourage the students to use

simple but different starting sets

Homework

Homework

Answers

Draw a mapping diagram to show each function.

a n + 8 b n 6 c n 5 d n 2

Diagrams will depend on their starting numbers

function mapping

Key Words



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Oral and mental starterComplements of 100

You could use a number grid here. For example:

Explain that the complements of 100 are any two numbers which add

together to give 100. For example, the complement of 27 is (100 27), which is 73. Get the students to explain their different strategies for doing this. Then

you can use the number grid to point to numbers at random in order toask them for the complements. They need to be encouraged to find astrategy which enables them to do this quickly in their heads.(These grids can easily be made and used throughout the department.They allow you to have quick questions ready for all sorts of problem,not just complements.)


The activity in Pupil Book 1 needs introducing as it is an investigation. Explain what is meant by a two-digit whole number with examples. The

students will gladly give you many more. Then introduce this function

Illustrate how it works using the examples given in Pupil Book 1. Go through the example in Pupil Book 1 of how to create a chain, or

choose any other number. This is an investigation with some suggested structure, but should a

student suddenly ask What if ?, take this idea and encourage him/herto pursue it after he/she has seen some pattern in the main startingactivity. Numbers starting or ending with nine always repeat themselvesand go on for ever. The chain either repeats or stops after no more thansix links. Since there is a lot of arithmetic for some students, do checkwhether any students have chains that seem to go on for more than sixlinks. If so, they have the wrong function or have made mistakes in thearithmetic or have continued on an infinite loop where they should havestopped.

Use of a calculator could help some of the weaker students. Question 6 is where the students are encouraged to ask their own What

if ? questions. This is a good activity for creating display material.


58 92 71 43 26

37 19 64 85 29

17 34 62 73 91

88 67 46 39 24

LESSON

1.6

Framework objectives A function investigation

Suggest extensions to problems by asking What if ? Begin to generalise and tounderstand the significance of a counter example.

multiply the two digits add the result to the sum of the two digits


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Plenary Get students to explain verbally what it is they have discovered and to

suggest other similar functions that they could try.


SATs Answers

1 30, 25, 202 a double b 1 5 25 125 625 (multiply by 5),

1 5 9 13 17 (add 4)1 5 10 16 23 (add 4, then 5, then 6, )There will be many other answers. Check that the rules work

3 a 5400 b 5.4

Activity Answers

2 They should complete at least six chains3 All numbers starting or ending with 9 go on for ever, as well as 34, 43, 45, 47, 54,

57, 64, 74, 75 and 844 3 is the most common stop number5 6 is the longest chain: 66, 67, 76 and 77

Homework

Try the investigation with some three-digit numbers and see what happens.

complement two-digit number three-digit

number chain

Key Words


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Use a number line drawn on the board or a counting stick with 10segments marked on it. State that one end is the number 0 and the otherend is the number 10. Mark 0 and 10.

Ask students to identify the middle number then fill in the rest, notnecessarily in order.

As a group or with an individual student, do some counting-on activities.For example, start at 0 and count on in steps of 2. The positions can bepointed out on the line or the stick until 10 is reached. Then the studentshave to continue without prompts.

Start at 10 and count down in steps of 1. Repeat the activity with the line marked with 10 segments but ending at 1.

Discuss what each decimal place means, using the term tenths.


This can be done with students working in groups orindividually. Calculators will be needed: one per individual,or one/two per group.

Ask one member of the class to choose a number, or startwith 52, say.

Write the number in the middle of the board and eitherask students to give you the answers or get one student towork them out using an OHP calculator. Using a spiderdiagram, show what happens when you multiply or divideby 1 and 10.

Now ask the students, either working alone or in groups, to repeat theactivity in their books or on sheets with the following numbers (or othersimilar numbers):

7 78 0.2 341 203 0.056 After this is done, ask for the rules when multiplying or dividing by

1 or 10.(1 has been included as it is an important concept that is often missed.100 could be included if appropriate.)

0 1

0 10


52

1

52

10 52010

152

5.2

Number 1CHAPTER

2

LESSON2.1

Framework objectives DecimalsUnderstand and use decimal notation and place value. Multiply and divide integersby 10 and 100.


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Students are likely to talk about moving the decimal point, but theyshould be encouraged to think of the digits moving to the left or right.The chart on the right will prove helpful, particularly when doingExercise 2A.


Plenary

Write numbers on the board (or have prepared cards available), such as34 and 340, 0.72 and 7.2, 0.05 and 0.5. Ask the students to put themtogether with a multiplier or divisor. For example:

34 10 = 340 Make sure that both the multiplier and divisor are identified. Ask the students if they can explain the connection. Make it clear that

multiplication is the inverse operation to division.


Ho

mework

Homework

Answers

1 Without using a calculator, write down the answers to:

a 57 10 b 32 10 c 1.9 10 d 1.3 10

e 0.2 10 f 2.37 10 g 2.37 10 h 6.09 10

2 Fill in the missing operation in each case.

a b

c d

3 Find the missing number in each case.

a 6 10 = b 6 = 60 c 6 10 =

d 6 = 0.6 e 0.6 10 = f 0.6 = 6

4040062.36.23

34.5345777.7

1 a 570 b 3.2 c 19 d 0.13 e 0.02 f 23.7 g 0.237 h 60.92 a 10 b 10 c 10 d 103 a 60 b 10 c 0.6 d 10 e 6 f 10

decimal number decimal place digit order place value tenth zero place holder inverse operation

Key Words

Exercise 2A Answers

1 40, 20, 50, 110, 10, 500, 370, 6902 a 340 b 45 c 6 d 890 e 53 f 0.3 g 40 h 58 i 3.4 j 0.45

k 0.06 l 8.93 a x 10 b 10 c x 10 d 10

4 a 30 b 10 c 0.3 d 10 e 3 f 10 g 0.03 h 10 i 0.3 j 300k 3000 l 30

5 Chews 0.30, mints 2.30, pop 9.90, total 12.506 800, 1200, 600, 6200, 300, 4000, 1000, 250000

52 10

52 10

5 2

5 2

5 2 0

Hundreds

Tens

Units

Tenths

Hundredths


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Oral and mental starter Use a target board such as the one shown on the right. Randomly select

students and ask them to double the number pointed at. Discuss the strategies for doubling. For example:

2 38 = 2 30 + 2 82 28 = 2 30 2 2

Randomly select students and ask them to halve the number pointed at. Discuss the strategies for halving. For example:

half of 26 = half of 20 + half of 6 Ask students to pick out pairs on the board that are half of or double each

other. Make them say the relationship in full. For example:Eight is half of sixteen Fifty-two is double twenty-six

Pairs on the target board above are:(22, 44) (15, 30) (8, 16) (26, 52) (16, 32) (14, 28) (12, 24)


Draw on the board (or an OHT) the table on the right. Write on the board (or have prepared cards available) the following (or

similar) numbers:320 7 40 78 4 325 44

Ask students to come to the board, select a number and put it in thetable, using the appropriate place-value columns.

Alternatively, ask each student to select the biggest (or smallest) numberand fill in on the top line and continue to select the next biggest (orsmallest) as appropriate.

When the table is completely filled in, discuss how we can decide whichnumber is biggest.

When comparing numbers, ensure that the concept of working from theleft until the largest digit is encountered is understood.

Repeat with:345 342 35 3 39 307 38 Introduce the symbols < and > and discuss what they mean. Which symbol should come between 35 and 347? Which symbol should come between 4111 and 4118? How can we tell easily? Discuss places values. What does this mathematical expression mean: 318 < 325 < 340? This can be read as 318 is less than 325, which is less than 340; or as

325 is between318 and 340.



LESSON

2.2

Framework objectives Ordering numbers

Compare and order numbers in different contexts.

26 38 8 20 22

8 28 14 25 44

30 52 16 6 18

32 15 12 24 34

Thousands

Hundre

ds

Tens

Units


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Plenary

Write numbers on the board (or have prepared cards available), such as340 and 345, 72 and 70, 50 and 45. Ask the students to put themtogether with a greater than or less than sign. For example:

340 < 345


Exercise 2B Answers

1 b 4, 5, 45, 405, 450, 457, 40572 a 29, 47, 69, 70, 75 b 92, 98, 203, 302, 9073 a 450, 403, 400, 54, 45 b 2531, 513, 315, 153, 1354 Edinburgh (by 178 miles)5 Fort William (by 82 miles)6 a Nottingham b 5 miles

7 a Joe Bloggs b Fred Davies8 a < b > c > d < e < f >9 a 0.07, 56p, 0.60, 1.25, 130p b 0.04, 35p, 0.37, 101p, 1.04

Extension Answers

1 25min, 0.5h, 1h 10 min, 1.25h2 0.32, 0.34, 2.69, 2.70, 6.25m3 0.055, 0.056, 0.467, 0.500, 1.260kg4 a Three point one-four is between three point one and three point one four two

b Seven pence is less than thirty-two pence, which is less than fifty-six pence

Homework

Homework

Answers

1 Using a table with place-value headings, fill in the following numbers. Thenuse your table to write the numbers in order from smallest to largest.

65 70 56 602 622 60 8

2 Write each of the following sets of numbers in order from the smallest to the largest.

a 205 190 210 223 199 b 56 50 62 502 60

3 Put the correct sign, > or 40 b 132 < 140 c 80 > 78

less than greater than between order compare most least

significant digit

Key Words


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Oral and mental starter This is best done using a number hoop, but can also be done using a

number line marked with 10 segments drawn on board, or a countingstick marked with 10 segments.

Point at one marker on the hoop and say: This is 20. Point at the nextmarker and say: This is 18. The class can then count down in steps of 2until some students get it wrong. (This can also be done individually.)

Repeat with different starting numbers and different jumps, but alwayscount into the negative integers.


Draw on the board (horizontally or vertically) a number line and mark 21divisions on it. Then, starting at the mid-point, number the right (or top)half of the line from 0 to 10.

Ask the class how we could use the number line to calculate 7 3.Establish that we start at zero and move first in the positive direction for 7,and then in the negative direction for 3. Mark the number line as below.

Repeat if necessary with other similar examples but make sure that ineach case the students will obtain a positive answer.

Now ask the students how we could use the same idea to find the answerto 3 7.

Using the same procedures, the students will quickly grasp the idea ofextending the line in the negative direction.

Repeat with other examples and encourage the mental (or actual) use ofthe number line.


9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1010

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10


LESSON

2.3

Framework objectives Directed numbers

Understand negative numbers as positions on a number line. Order, add andsubtract positive and negative numbers in context.


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Plenary

Draw a blank number line on the board. Mark it with zero and divisionsfor positive and negative numbers.

Give the class some mental questions on directed numbers. For example:2 + 5, 4 9, 2 3 + 4

Answers can be written on white boards or held up on number fans,although the latter do not usually include the minus sign.

Encourage the visualisation of the process as counting on or back,starting at zero, by drawing sketches after each answer. For example:2 + 5 2 3 + 4

1 0

+4

23

250 +3

+5

2


Exercise 2C Answers

1 a Circle 1, 2, 3, 4, 5, 6 b Circle 3, 2, 1, 0, 12 a T b T c F d F e T3 a < b < c > d > e = f >4 a < b > c > d b > c = d < e = f > g < h < i > j = k < l =


Exercise 4B Answers

1 a 0.1 b 0.7 c 2.1 d 2.6 e 0.9 f 4.7

2 a b 1 c 3 d e 9 f3 a 0.07 b 0.61 c 2.17 d 2.01 e 0.01 f 4.274 a b 2 c 3 d e 5 f 95 0.3 = , 0.9 = , 0.75 = , 0.03 = , 0.22 = , 0.08 = , 0.8 = , 0.4 = ,

0.09 =6 a 6 b 7 c 1 d 8.5 e 2.5 f 11.57 a 3 b 1 c 6 d 7.25 e 2.25 f 11.258 a 1 b 6 c 4 d 5.75 e 2.75 f 3.7534

34

34

14

14

14

12

12

12

9100

410

810

8100

22100

3100

75100

910

310

2100

18100

6100

67100

21100

8100

4

10

3

10

9

10

7

10

2

10

5

10

Homework

Homework

Answers

1 Convert each of the following decimals to a fraction.

a 0.3 b 0.1 c 0.38 d 0.75 e 0.85

2 Convert each of the following fractions to a decimal.

a b c d e 1472

10011

1007

10310

1 a b c d e2 a 0.3 b 0.7 c 0.11 d 0.72 e 0.25

85

100

3

4

38

100

1

10

3

10

top-heavyfraction

cancelling factor mixed number decimal place value tenth hundredth

Key Words



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Oral and mental starter Relate to the previous oral and mental starter (on page 36). The last

activity was counting on in quarters. Ask the class to count on using Figure 1. Fill in to get Figure 2. Ask them to study it for a few moments. Cover up the grid. Have

prepared cards or OHTs of diagrams like those shown in Figure 3,which are fragments of the whole grid. The students need to fill thesein with the missing numbers. This can be done by copying into theirbooks or coming to the board to fill in. Finally, reveal the grid to checkthe answers.

Figure 1 Figure 2 Figure 3

This can be repeated with other grids, such as


The grids from the oral and mental starter will be useful for this. What is + ? How can we use the grids to do this? It is also possible to use a number line or a ruler marked in eighths of an

inch.

What about + ? Other examples are:1 + = 2 2 + = 31 = 2 = 1

Make sure the students can count up and down a number line or use thefraction chart above. This is reproduced in Pupil Book 1, page 44.

Look at the diagram. What does this show?

+ =

+ = 1 Can you find the missing number in this statement?

+ = 1711

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10 18 14 38 12 58 34 78 181 141 381 121

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581 341 781582 342 782587 347 787

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LESSON

4.3

Framework objectives Adding and subtracting fractions

Begin to add and subtract simple fractions and those with common denominators.



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Is there an easy way of doing this? Discuss the method and do more examples, such as

+ = 1 + = 1 The idea of a top-heavy fraction, such as , has been met before. Recall

the methods for changing this to a mixed number (3 ). Do other examples:

(= 2 = 2 ) (= 1 ) Reverse the process. For example, change 3 to a top-heavy fraction (= ). Repeat with other examples:

1 (= ) 4 (= )


Plenary

Using the fraction charts or the number lines from the start of the lesson,ask the students to explain how they would solve addition andsubtraction problems such as:

1 + 2 2 1 + 1

Discuss methods and rules. What about + ? How could we do this? Discuss the method.2919

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Extension Answers

1 a b c d e f g h 12 a 45 b 723 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 3604 40 9 = 360, so 40 = of a full turn.19

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Exercise 4C Answers

1 a 1 b 1 c 1 d 2 e 2 f 2 g 2 h 12 a b c d e 1 f g h3 a 1 b c d e f g h4 a b c d 4 e f g h i j k l 15 a 2 b 2 c 1 d 1 e 1 f 3 g 3 h 4 i 2 j 2 k 3 l 46 a b c d e f g h i j k l 119199

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Homework

Homework

Answers

1 Work out each of these. Convert to mixed numbers or cancel down to lowest terms if necessary.

a 1 + b + 1 c 1 d 2 1

2 Find the missing fraction.

a + = 1 b + = 1 c + = 1 d 1 = 492

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1 a 2 b 1 c d2 a b c d 591113

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denominator numerator fraction cancelling lowest terms simplest form top-heavy

fraction mixed number

Key Words



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Oral and mental starter Use a target board, such as one shown on the right, two cards marked

with and and three cards marked with 10, 100 and 1000.

Select a student, an operation, a power of 10 and a number. Ask the student for the result of the number multiplied (or divided) by 10,

100 or 1000. Repeat with other numbers. Recall the rules for multiplying and dividing by powers of 10.


Ask the class to give as many equivalent fractions,decimals and percentages as they know. They should atleast come up with

50% = = 0.5 25% = = 0.25 10% = = 0.1 Now create a spider diagram based on 10%, showing

equivalence and linked values, which should include

those given on the right. Use these to complete the following equivalence chart.

Percentage 0% 5% 10% 15% 20% 25% 30% 33.3% 35% 40% 45% 50%

Decimal 0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.333 0.35 0.4 0.45 0.5

Fraction 0

Some students may need to be reminded about 33%. Discuss how to find other equivalences, such as 60%, 65%, 66%, 75%,

80%, etc. Leave the chart on the board for the plenary.


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100010010


LESSON

4.4

Framework objectives Equivalences

Understand percentage as the number of parts per 100. Recognise theequivalence of percentages, fractions and decimals.


28 38 7 22 60

8 16 14 26 48

30 52 36 9 13

32 15 12 24 34

20% = 0.2 = = 15210

25% = 0.25 = 14

10% = 0.1 = 110

5% = 0.05 =120

75% = 0.75 = 34

50% = 0.5 = 12

Exercise 4D Answers

1 a 30%, b 40%, c 65%, d 80%, e 75%, f 20%, g 5%,h 15%, i 95%, j 25%,

2 a 0.35, b 0.7, c 0.45, d 0.4, e 0.65, f 0.3, g 0.05,h 0.75, i 0.2, j 1.1, 1

3 a 15%, 0.15 b 70%, 0.7 c 90%, 0.9 d 85%, 0.85 e 25%, 0.25f 60%, 0.6 g 20%, 0.2, h 175%, 1.75 i 10%, 0.1 j 95%, 0.95

4 a 33.3%, 0.333 b 66.6%, 0.666

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Plenary

Have a set of Follow me cards (30 are suggested below) that useequivalent fractions, percentages and decimals. Record the time taken tocomplete. Later, revisit the activity as an oral and mental starter to see ifthis time can be beaten.

1 START. You are as a decimal 2 I am 0.4. You are 0.6 as a percentage

3 I am 60%. You are 0.5 as a percentage 4 I am 50%. You are 80% as a fraction

5 I am . You are 75% as a fraction 6 I am . You are 95% as a decimal

7 I am 0.95. You are 0.45 as a fraction 8 I am . You are 0.3 as a fraction

9 I am . You are 0.2 as a percentage 10 I am 20%. You are 20% as a fraction

11 I am . You are as a decimal 12 I am 0.25. You are as a percentage

13 I am 33.3%. You are 0.25 as a percentage 14 I am 25%. You are 35% as a fraction

15 I am . You are 0.05 as a fraction 16 I am . You are 0.1 as a fraction

17 I am . You are 0.1 as a percentage 18 I am 10%. You are 15% as a fraction19 I am . You are 0.50 as a fraction 20 I am . You are 85% as a decimal

21 I am 0.85. You are 0.666 as a fraction 22 I am . You are 40% as a fraction

23 I am . You are 65% as a fraction 24 I am . You are as a decimal

25 I am 0.35. You are as a percentage 26 I am 70%. You are 70% as a decimal

27 I am 0.7. You are 0.6 as a fraction 28 I am . You are 100% as a decimal

29 I am 1. You are 0.8 as a percentage 30 I am 80%. END

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Homework

Homework

Answers

1 Work out the equivalent percentage and fraction to each of the following decimals.

a 0.5 b 0.6 c 0.95 d 0.8 e 0.33

2 Work out the equivalent decimal and fraction to each of the following percentages.

a 45% b 90% c 30% d 20% e 5%

3 Work out the equivalent percentage and decimal to each of the following fractions.

a b c d e 3412

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25

710

1 a 50%, b 60%, c 95%, d 80%, e 33%,2 a 0.45, b 0.9, c 0.3, d 0.20, e 0.05,3 a 70%, 0.7 b 40%, 0.4 c 30%, 0.3 d 50%, 0.5 e 75%, 0.75

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equivalence decimal fraction percentage denominator factor

Key Words

Extension Answers

1 a 12%. 0.12 b 36%, 0.36 c 48%, 0.48 d 18%, 0.18 e 34%, 0.34f 82%, 0.82

2 a 0.64, b 0.28, c 0.84, d 0.22, e 0.62, f 0.86,3 a 16%, b 44%, c 32%, d 38%, e 66%, f 94%,

4750

3350

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Oral and mental starter A mental test covering previous work and all work in this chapter.

1 Convert to a mixed number.2 How many quarters are there in 1 ?3 Write down any fraction that is equivalent to .4 Write down as a decimal.5 Write down as a percentage.6 Write down 75% as a decimal.7 Write down 25% as a fraction.8 Write down 10% as a fraction.

9 Put the fractions , and in order, smallest first.10 12 cm of a plant is below the ground. This is of the length of the

whole plant. How tall is the plant altogether?

Answers 1 1 2 7 3 Any fraction, other than , where the numerator is half of the denominator 4 0.33(33)5 20% 6 0.75 7 8 9 , , 10 36cm


How can we calculate of 75 m?First find of 75 = 25. So, of 75 = 2 25 = 50 m.

Give other examples, such asFind of 30. (5 5 = 25)Find of 45kg. (3 4.5 = 13.5kg)

How can we calculate 5 ? This is the same as+ + + + = = (cancel by 5)

Find a simple percentage. Base it on 10% and show how to use this tofind other percentages that are multiples of 5 and 10. For example:

35% of 700 (245) 5% of 68 (3.4) 15% of 24 (3.6) Extend these ideas to simple increase/decrease problems.

Problem 1 The price of a jacket is reduced by 20% in a sale. The

original price of the jacket is 45. How much will the jacketcost after the price reduction?

20% = 9 so new price is 36Problem 2 Freds sports shop is selling some Airflow trainers for 48 but

is offering a off. Berts sports shop is selling the same Airflowtrainers for 44 but is offering a off. Which shop is sellingthe cheaper trainers?

of 48 = 12 so Freds cost 36.00of 44 = 8.80 so Berts cost 35.20

The class can now do Exercise 4E from Pupil Book 1.

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LESSON

4.5

Framework objectives Solving problems

Calculate simple percentages.



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Plenary

Explain that as a decimal, a fraction and a percentage are all differentways of writing the same thing, we can sometimes make a calculationeasier by using one of them instead of the other.

Example 1 20% of 35. As 20% is , this is the same as

35 = 7.Example 2 0.3 340. As 0.3 is 30%, this the same as 30% of 340.10% of 340 is 34. So, 30% of 340 is 3 34 = 102.

Example 3 of 40. As is 0.12, this is the same as 0.12 40 = 4.8. Ask the class to rewrite each of the following, using an alternative to the

percentage, decimal or fraction given. Then work out the answer.a 20% of 75 b of 60 c 25% of 19 d 60% of 550

e of 90 f 0.125 64 g of 735320

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SATs Answers

1 a 1.06, 3.94 b 16p2 a 120 b 11.60 c 2.90 d 5 boxes3 a 1 , 123, 54, 108 b any pair that work, e.g. 3 9, 2 13.5; 54 2, 270 104 a 3.20 b 102 c 14 packets5 a 30% b 4 triangles shaded, 40%6 a Tuesday, Friday b 0.25cm c 1.5cm 15mm7 b 1, 4, 48

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Exercise 4E Answers

1 a 6 b 18 c 5 d 35 e 1 f 202 a 3 b 9 c 8 d 16 e 10 f 303 500g4 600 pupils5 80 pages6 a 20 b 12 c 3.50 d 10p e 4 f 75

7 a 14.40 b 54 p c 5.22 d 48.60 e 24.66 f 18.728 a 12 b 12 c 11 d 3 e 7 f 59 a 24 m b 33p c 15 d 28kg e 15cm f 18km g 16mm h 20

10 a 24 b 161 c 90 d 12.8 e 81 f 19 g 144 h 3.5

Homework

Homework

Answers

1 Find a of 34 b 25% of 44 c 10% of 6

2 Which of these is the greater? a of 35 or of 20 b of 108 or of 70

3 Which of these shops is giving the better value?

Dereks Fashions: Armani suits reduced by from 650

Marys Modes: Armani suits reduced by from 68014

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1 a 17 b 11 c 60p2 a of 20 b of 703 Dereks 520, Marys 510

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increase decrease percentage equivalence

Key Words



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Write a list of positive numbers on the board. For example:7, 3, 9, 2, 10, 8, 5, 4, 14, 12

Ask the students to put them in numerical order with the smallest first.Ask them if they have any strategy for doing this.

Repeat for another list of numbers. For example:34, 67, 38, 19, 44, 57, 24, 31, 62, 20

What about any strategies now? Identify the tens column first? Write a list of positive and negative numbers on the board. For example:

2, 3, 1, 4, 4, 0, 5, 2 Ask the students to put them in numerical order with the smallest first. What about any strategies now? Remind the class of the number line.


Explain the term average by giving everyday examples where the word isused. For example, average rainfall, average examination mark, averageheight.Ask the class to give their own examples.An average is a single value that represents a set of data.

Explain how to find the mode for a set of data.The mode is the value that occurs most often in a set of data. For example:

5, 6, 8, 2, 4, 5, 3, 5

The mode is 5. Explain that for some sets of data there is no modebecause either all the values are different, or no single value occurs moreoften than other values.

Explain how to find the range for a set of data.The range is the largest value minus the smallest value.Explain that range is not an average. It shows how data is spread out.A small range shows that the values in a set of data are similar in size.A large range shows that the values in a set of data differ considerably.


5 4 3 2 1 0 1 2 3 4 5


Handling Data 1CHAPTER

5

LESSON5.1

Framework objectives Mode and rangeFind the mode and range, and the modal class for grouped data.



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Plenary

On the board, write two short lists of numbers. Ask the class to explainhow to find the mode and the range for each set of data.

Ask the class if they know of another average. Explain that the mean isthe most common average used and will be discussed in a later lesson.


Exercise 5A Answers

1 a red b sun c E d 2 a 5 b 34 c 13 d 1013 a 19 b 6 c 27 d 144 a 1.80, 2.40 b no mode, 14kg c 132cm, 19cm d 32, 65 a 2.20 b 1.606 a 2 b 3

7 a 5, 10, 11, 7 b 6180 c 798 a For example: 7, 8, 9, 10, 10, 10, 19 b For example: 7, 8, 9, 10, 10, 10, 12, 17

c For example: 4, 5, 6, 10, 12, 12, 12, 12, 12, 12

Homework

Answers 1 a 12, 3 b 78, 15 c 1, 1 d 24, 152 a 300g b 410g c 320g

3 a 14 b 161cm c 39kg d Anna, closest to all three modes

average data mode modal class

range

Key Words

Homewo

rk 1 Find the mode and range of each of the following sets of data.

a 11, 12, 13, 12, 14, 11, 12 b 66, 72, 78, 75, 78, 68, 63 c 1, 0, 1, 0, 1, 0, 1, 0, 1, 1

d 21, 24, 26, 29, 34, 32, 27, 25, 24, 19

2 David is taking part in a fishing competition. At the end of the competition, the weight of each fishin his keep net is as follows: 300 g, 450 g, 620 g, 300 g, 550 g, 300 g, 410g.

a Find the modal weight of the fish. b Find the range for the weight of the fish.

3 Given below are the age, height and weight of each of seven girls in a netball team.

Anna Claire Chloe Beth Lauren Martha Sarah

Age (yr) 14 16 13 14 12 16 14

Height (cm) 160 164 161 157 153 161 168Weight (kg) 39 41 36 31 39 41 39

a Find the modal age of the team.

b Find the modal height of the team.

c Find the modal weight of the team.

d Who would you choose as the average player in the team? Give a reason for your answer.


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Plenary

Ask the class to give examples of all the different types of table and chartthat can be used to represent data.


Homework

Homework

Answers

1 The timetable shows the times that trainsdepart in the morning from Kings Cross stationin London and the times they are due to arrivein Doncaster and York.

Kings Cross Doncaster York

Depart Arrive Arrive

07 00 08 55

07 30 09 02 09 25

08 00 09 51

08 30 10 13 10 37

09 00 10 51

09 30 11 06 11 30

a i At what time does the 0830 train fromKings Cross arrive in York?

ii How long does the journey take?

b How many trains in the timetable do notstop at Doncaster?

c Stephen lives in London and wants to be inDoncaster before 10.30am. Which is thelatest train he can catch?

d Which train from Kings Cross to York takesthe shortest time?

2 The two-way table shows the average dailytemperature in C for four months in five cities.

January April July October

Athens 10 16 28 19

London 4 9 18 11

Los Angeles 13 15 22 18

Madras 24 30 31 28

Tokyo 3 13 25 17

a What is the daily average temperature in LosAngeles in July?

b Which city has the highest daily averagetemperature in October?

c What is the difference in the daily average

temperature in Athens and Tokyo in April?3 Peter asks some of his friends how many

brothers and sisters they have. He writes downtheir answers in a list as follows.

1 3 1 0 2 4 0 2

0 1 1 2 3 2 1 1

1 2 0 1 0 0 3 4

a Copy and complete the survey sheet belowfor his data.

No. of brothers Tally Frequency

and sisters

0

1

2

3

4

Total

b How many friends did Peter ask?

c What is the mode for the number of brothers

and sisters his friends have?

d Draw a line graph for his data.

1 a i 1037 ii 2 hours 7 minutes b 3 c 08 30 d 08 00 or 09 002 a 22C b Madras c 3 C3 a frequencies: 6, 8, 5, 3, 2 b 24 c 1

chart frequency line graph survey sheet table tally two-way table

Key Words


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Oral and mental starter Ask the students to explain the following statistics terms:

mode, range, modal class Ask them to make up their own small sets of data and find the mode,

range and modal class for each.


Explain that data is often displayed on diagrams to make it easier to

understand and interpret. Introduce the following three types of statisticaldiagram.

Pictogram Used for visual impact particularly in advertising andwhere there is a need for data to be easily understood. A pictogram isessentially a frequency table in which the frequency is represented by arepeating symbol. The symbol itself usually represents a number of items,as shown in a key.

Bar chart Used for visual impact and for comparison.The bars can be vertical or horizontal. Mention that bar charts used toshow temperatures in holiday brochures, for example, are usuallyhorizontal.

For data with single categories, such as colour of hair, number of pets,gaps should be left between the bars.

For grouped discrete data, such as age groups, examination marks, nogaps should be left between the bars.

Dual bar charts are used to compare two sets of similar data. Forexample, temperatures in two cities, favourite sports of boys and girls.

Composite bar charts are used to display data in a single bar to save space.

For example, amount of time spent in different lessons and activities in aschool day.

Line graph Used to show patterns and trends, mainly for continuousdata. For example, profit of a company over a number of years, dailycurrency conversions, recording body temperature in hospital.

Beware Statistical diagrams can sometimes be misleading. Forexample, wrong scales, bars with different widths, bar charts drawnin 3-D. Ask the students to look for misleading diagrams in newspapersand magazines and school textbooks!



LESSON

5.3

Framework objectives Statistical diagrams

Interpret diagrams and graphs.



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Plenary

Ask the students to sketch an example of each type of statistical diagramthey have met in the lesson.


Exercise 5C Answers

1 a 60 b 70 c Year 7 d 3802 a Ceri b 16 c 18 d 3 e 693 a 3 b bus c 304 a London b July for London, August for Edinburgh c 7 hours for London,

6 hours for Edinburgh5 a blue b 10% c green and purple d 10 e Shows all the information in one

bar and takes up little space6 a 32 b 6180 marks c 157 a 15C b 17C c 9 C d Shows the trend and the approximate temperature

at any time can be read off

Homework

Homework

Answers

1 The pictogram shows the number of studentsin Year 7 who were absent from school duringone week.

Monday

Tuesday

Wednesday

Thursday

Friday

Key: represents 4 students

a How many students were absent on Monday?

b How many students were absent on Tuesday?

c How many students were absent on Thursday?

d Suggest a reason why so many students wereabsent on Friday.

2 The bar chart shows the number of studentswho were late for school in each year groupon a particular day.

a How many students were late in Year 10?

b How many students were late altogether?

c Which year group had the least number oflate students?

3 The dual bar chart shows the lengths of 100words in two different newspaper passages.

a What is the modal word length for eachnewspaper?

b How many words have more than six lettersin i Guardian ii Mirror?

c Compare the length of words for the twonewspapers.

0

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1 2 3 4 5 6 7 8+

Number of letters

Numberofw

ords Guardian

Mirror

02468

10121416

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Year7

Year8

Year9

Year10

Year11

Year group

Numberoflatestude

nts

1 a 12 b 14 c 9 d For example, some students out on a school trip, taking a religious holiday or kept athome by bad weather.2 a 13 b 60 c Year 83 a both 3 b i 21 ii 12 c Word lengths do not differ greatly. The Guardianuses more long words.

bar chart line graph pictogram

Key Words


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For this you will need a bag containing 50 counters in three differentcolours, in different amounts. For example, 10 red, 15 blue and 25 green.

Ask a student to take a counter from the bag without looking and replaceit in the bag.

Ask the class to name the colours of the counters in the bag. Shake the bag and repeat the activity about ten times. Tell the class that there are 50 counters in the bag.

Ask the students to write down how many counters of each colour theythink are in the bag.

Show the class the counters.


For this lesson, it would be helpful to have: coins, dice, a pack of cards,cards numbered 1 to 10, and various spinners.

Ask the class to explain the term chance by giving everyday examples.Write them on the board. For example, the chance of getting a six whenthrowing a dice, the chance of rain tomorrow, the chance of winning thefootball pools.

Draw the probability scale on the board:

Impossible Very unlikely Unlikely Evens Likely Very likely Certain

Ask the class to give examples of events which are described by any ofthe above terms.

Explain that probability is the mathematical way of describing thechance that an event will happen.

What other terms do they know which describe probability? Forexample, fair chance, 5050 chance, uncertain.

To describe probability more accurately, we use a scale from 0 to 1:

0 1

Throw a coin. Ask the class to write down all the possible ways the coincan land.Each way is called an outcome for the event. For throwing a coin, theoutcomes are:

Head (H) or Tail (T).Each outcome is equally likely to happen since the coin is fair.

Since there are two equally likely outcomes, the probability of getting aHead is 1 out of 2. This is written as:

P(Head) = or P(H) = 1212

12


LESSON

5.4

Framework objectives Probability

Use vocabulary and ideas of probability, drawing on experience.

Understand and use the probability scale from 0 to 1. Find and justify probabilitiesbased on equally likely outcomes in simple contexts. Identify all the possiblemutually exclusive outcomes of a single event.



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Probability is usually written as a fraction or a decimal, althoughpercentages can be used, as in weather forecasts. Probability fractions arealways cancelled down.

Throw a dice. Ask the class to write down all the equally likely outcomes(1, 2, 3, 4, 5, 6). Hence:

P(6) = and P(1 or 2) = =


Plenary

Use the bag of counters from the start of the lesson to ask the class to findthe probability of choosing any of the colours in the bag.

13

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Exercise 5D Answers

1 a likely b very unlikely c certain d evens e impossible f very likelyg unlikely

2 a b c d3 a b c d e 14 a b c d 0 e5 a b c d6 a b c d 1611213512

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411

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Homework

Homework

Answers

All answers should be given as fractions.

1 A card is chosen at random from a set of cards numbered from 1 to 20. Find the probability ofchoosing a card that is:

a an even number b a number containing 5

c a two-digit number d a number less than six

2 Ella is playing a game using this fair five-sided spinner. What is the probabilitythat she scores:

a 7 b 1 or 9 c an odd number d an even number?

3 A letter is chosen at random from the word

Find the probability that the letter chosen is:

a M b E c a vowel d a consonant

4 A set of snooker balls consists of 15 red balls and one each of black, blue, brown, green, pink,white and yellow. If a ball is chosen at random, find

a P(choosing a red ball) b P(choosing a yellow ball)

c P(choosing a blue or green ball) d P(choosing a black or a red ball)

SCITAMEHTAM

1 a b c d2 a b c 1 d 03 a b c d4 a b c d 8111111221522

711

411

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certain chance even chance equally likely event outcome fair likely

unlikely possible impossible probability probability scale

Key Words


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Oral and mental starter A revision starter to check spellings and meanings of the terms used in

the present chapter. Ask a student to write on the board one of the words given below and

then explain briefly its meaning. Repeat the activity with other students.average moderange frequencyprobability pictogramevent outcome


This lesson will be mainly practical work. The following equipment willbe needed: coins, dice, packs of cards, card and cocktail sticks formaking spinners, Plasticine, drawing pins.

Explain that, in the previous section, the probability of an event could beworked out because we knew the outcomes were equally likely tohappen, such as when throwing a fair dice.

In other cases, it may not be possible to use equally likely outcomes.For example, the outcomes for throwing an unfair dice or a biased dicewill have different probabilities for each outcome. In cases like these, weneed to carry out an experiment to estimate the probability of an eventhappening. Here, we need to repeat the experiment a number of times inorder to find the experimental probability. Each separate experiment isknown as a trial. The results of all the trials can be recorded in afrequency table. The experimental probability can then be written as afraction, as in Lesson 5.4

Experimental probability is usually written as a fraction, but as before,this can be written as a decimal or sometimes a percentage.

The experiments in Exercise 5E show how experimental probabilities can

be calculated. The class can work in pairs or groups. The class can now do Exercise 5E from Pupil Book 1.


LESSON

5.5

Framework objectives Experimental probability

Collect data from a simple experiment and record in a frequency table. Estimateprobabilities based on this data.



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Plenary

Ask the students to explain experimental probability. Make sure they understand the terms equally likely outcomes and

trials.

SATs Answers

1 a 20 miles b 53 miles c 9 miles2 a 7 b Madonna c 6 d Abba and Spice girls3 a 60 b 45 c 25 d 4 symbols drawn e 1 symbol and less than symbol

f hot on Monday and cooler on Tuesday4 a It occurs the most b frequencies: 7, 10, 4, 3 c Yes, most people spent that

d They spend more later in the evening, because they are hungry or older5 a It is an evens chance; certain, likely

b For example, even than odd; higher than 5, more higher numbers

12


Exercise 5E Answers

3 c Probably get a different experimental probability4 b No. For example, shape of toast, weight of butter may have an effect

Homework

Homework

Answers

1 The frequency table shows the results of throwing a dice 30 times.

Score 1 2 3 4 5 6

Frequency 10 6 4 3 5 2

a Calculate the experimentalprobability for each score.

b Comment on whether you think the dice is a fair one.

2 Pauline throws two fair coins 40 times. Her results are shown in the following frequency table.

Outcome 2 Heads 2 Tails 1 Head and 1 Tail

Frequency 10 12 18

a Calculate the experimentalprobability of getting i 2 Heads ii 2 Tails iii 1 Head and 1 Tail

b What do you think is the actual probability of getting 1 Head and 1 Tail when throwing twocoins?

3 A biased spinner has four coloured sections.

Douglas wants to find the experimental probability of getting each colour.He spins it 100 times and records his results in a frequency table.

Colour blue red green yellow

Frequency 30 40 10

a Copy and complete the frequency table.

b Calculate the experimental probability for each colour.

1 a , , , , , b Probably not. Would not expect so few 6s or so many 1s2 a i ii iii b3 a yellow 20 b blue , red , green , yellow 1511025310

12

920

310

14

1

15

1

6

1

10

2

15

1

5

1

3

biased experimental

probability

fair unfair

Key Words



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Play the trial-and-improvement game of Guess my number. Think of anumber (less than 100 is best to start with) and get the students to discoverit in as few times as possible, by asking questions which can only beanswered Yes or No. For example:

Is it less than 50? Is it even? Is it larger than 25? You can then include a number over 100, but dont make it too big. Allow the class to choose a number to see whether you can get it in

fewer goes than their best attempt. You can have the numbers on prepared cards, which you stick facing the

wall. This demonstrates that you have not changed your mind during thegame.


Ask the class: Last night I was sorting out the cutlery at home. I foundthat I had eight more knives than forks. Altogether I had 32 items ofcutlery, so how many knives did I have?

After a short spell of speculation, tell the class that this is the sort ofproblem that algebra can be used to solve. Using algebra to solvepuzzles and problems is going to be explored in the next few lessons.(If they want an answer, tell them you had 12 forks and 20 knives.)

Tell the class that you are going to start with straightforward, simplepuzzles. Put on the board

+ 5 = 9

Ask them what stands for. You will find that a few will immediately give you 4. But let the class

discuss how they know that. Make sure that everyone does understandthat simply represents 4.

Give the class a few more like that, all simple additions (where they haveto subtract to obtain the answers).

If anyone says he/she knows a quick way to find the answer, ask him/herto explain it to the class. Help them to see that it always works.

Move on to, for example, 3 + = 10. The students need to recognisethat whenever addition is involved, it is essentially the same type ofproblem, whichever way round it is presented.

Then fire a subtraction problem, such as 8 = 2. Ask the class whatthe value of is here.Again, some will immediately get 6. Get them to discuss how they knewor how they found out.


Algebra 2CHAPTER

6

LESSON6.1

Framework objectives Finding unkown numbersUse letter symbols to represent unknown numbers or variables. Know the meaningsof the words term, expressionand equation.



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Do a few more subtractions this way round before putting up one theother way round, such as 4 = 1. Help the class to see the differencewith this type of problem.

Do a few more like this. Then put on the board one of each type and askthe class to solve them.

Discuss their answers. Help those who still cannot see the difference.

Finally, show them a problem 3 = 12. What does represent? Help the class to recognise that they may need to go through their tables

before spotting the answer. If anyone does spot a quick way to find theanswer, get him/her to explain it to the class.

Practise a few more multiplications. Division is not introduced here as it puts an extra tier of difficulty in for

the students. If any student asks about division problems, show them, forexample, 20 = 5. Help them to see how they can get to the answerby going through their tables.


Plenary

Have a game of Hangman, where the word is ALGEBRA or EQUATION.

Extension Answers

Many different correct answers


Exercise 6A Answers

1 a 6 + 2 = 8 b 5 + 6 = 11 c 2 + 7 = 9 d 7 + 2 = 9 e 4 + 3 = 7 f 3 + 5 = 8g 4 + 7 = 11 h 5 + 4 = 9 i 1 + 7 = 8 j 4 + 8 = 12 k 4 + 6 = 10l 2 + 9 = 11

2 a 2 3 = 6 b 5 2 = 10 c 3 5 = 15 d 4 3 = 12 e 3 3 = 9f 4 5 = 20 g 6 2 = 12 h 3 4 = 12 i 6 3 = 18 j 7 3 = 21k 8 3 = 24 l 6 5 = 30

3 a 9 1 = 8 b 10 3 = 7 c 9 6 = 3 d 12 7 = 5 e 8 1 = 7 f 8 3 = 5g 14 5 = 9 h 10 7 = 3 i 8 7 = 1 j 15 8 = 7 k 15 4 = 11l 12 8 = 4

4 a 3 b 3 c 5 d 10 e 8 f 11 g 8 h 6 i 5 j 5 k 7 l 85 a 4 b 5 c 4 d 1 e 5 f 2 g 11 h 4 i 8 j 9 k 6 l 7

Homework

Homework

Answers

In each sum below, find the number that the shape represents.

a + 5 = 9 b 10 = 1 c 5 = 20 d 7 + = 11e + 8 = 14 f 6 = 18 g 3 = 4 h 5 = 25i 14 = 12 j 4 = 28 k 9 + = 14 l 8 = 40

1 a 4 b 9 c 4 d 4 e 6 f 3 g 7 h 5 i 2 j 7 k 5 l 5

algebra

equation solving

Key Words



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Oral and mental starter Remind the class that a factor is a number that will divide exactly into

another number. Ask for the factors of 12. Accept them in any order (1, 2, 3, 4, 6, 12 ).

Help the students to realise that the simplest factors of a number are thenumber itself and one.

Use number cards, about 6cm square, with the following numbers on:1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Give them out so that each student has onecard. (You will probably need three sets.)

Then either call out a number or write it on the board. Ask the students tohold up their card if they show factors of that number.

Do encourage discussion. For example:Which number is always being shown?How do we know if 5 is a factor?How do we know if 10 is a factor?

You could end by asking a few students to hold up their numbers, say 2, 3and 5. Then say to the rest of the class: Give me a number that has all ofthese as factors In this case, the answers could be 30 or 60 or 90 .

Main lesson activity If I took you all to the bowling alley [or other suitable local place], how

much would it cost? Draw from the students that we would need to know the charge per

person and how many people. We would then use the rule:

Cost is * multiplied by the number of people.(* Put in the appropriate rate.)

Rules like this are used everyday in many different situations. Forexample:

A window cleaner, charging a price per window.The cost of meat in a supermarket.Pocket money paid by parents.

Go through these examples, showing how the rules would be used.The window cleaner, in Example 6.3 in Pupil Book 1, is paid at 40pper window. What does he charge Marge for cleaning her eightwindows?Meat in a supermarket, priced at 2.50 per kilogram. What wouldbe the cost of 3 kg?Childrens pocket money paid monthly, in pounds, at their age plus 5.

How much per month would Davina get when she is 12? The class can now do Exercise 6B from Pupil Book 1.


LESSON

6.2

Framework objectives Calculating using rules

Use simple formulae from mathematics and other subjects.

Identify the necessary information to solve a problem.



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Plenary

Tell the students that they have been working on simple formulae, whichare a quick way of writing rules for making calculations.

Show how the window cleaner in Example 6.3 might write his rule asC= 40W.

Ask the students how the other rules might be written. Note that aformula may be written in different ways depending on the letters chosento represent the variables. We usually try to use letters that remind us ofwhat the formula is working out.


Exercise 6B Answers

1 a 140 miles b 350 miles2 a 5 b 7.50 c 12.503 a 12 b 17 c 204 a 15cm2 b 42cm2

5 a 15cm b 24cm c 33 m6 a 10 minutes b 17 minutes c 24 minutes

7 a 80 b 1408 a 50mph b 40mph c 50mph

Homework

Homework

Answers

Wendy, a window cleaner, charges for cleaning windows according to the following rule:

Charge = 3 plus 1.50 for every Large Window Cleaned plus 50p for every Small WindowCleaned

How much does Wendy charge each of these people to clean his or her windows?

a Harry, who has 2 large windows and 8 small ones.b Padmini, who has 3 large windows and 7 small ones.c Sir Geoff, who has 8 large windows and 19 small ones.

a 10 b 11 c 24.50

formula rule

Key Words


Extension Answers

The outcome is always 5, which gives E


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Oral and mental starter Ask the class: How many factors does the number 6 have? Invite them

to tell you all four {1, 2, 3, 6}. Ask them for another number that has exactly four factors. Invite them to

check their suggestions. Remind them that one is always a factor, as wellas the number itself.

Then ask for any numbers that have exactly two factors. Again, get theclass to check them. Write them on the board as they aresuggested/ verified.

When the list has about ten numbers, ask who knows the name of thesespecial numbers. They are the prime numbers. (Make sure that 1 is not inthe list, and that it has been discussed so that the students are aware that1 is not a prime number.)


Remind the students that in Chapter 1 of Pupil Book 1, letters were usedfor numbers their introduction to algebra.

Go through the following ways of writing terms and expressions to show

the students what they are.3n means 3 multiplied by the variable n (Explain the term

variable again.)n + 7 means 7 added to the variable n8 n means subtract n from 8n 1 means subtract 1 from n (Explain the difference between

these last two.)

means n divided by 2

Each of these terms and expressions assumes a different value as the

value of the variable changes. For example, when n is 6:3n = 18 n + 7 = 13 8 n = 2 n 1 = 5 = 3

You will need to carefully explain to some students exactly what is beingdone here. Use plenty of examples to make certain that they realise thatthe variable n can stand for millions of different numbers.


n

2

n

2


LESSON

6.3

Framework objectives Algebraic terms and expressions

Use letters to represent unknown numbers or variables. Know the meanings of thewords term, expressionand equation.



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Plenary

Go through all the names in algebra to which the class have so far beenintroduced. Ask them to tell you what each means.

You could introduce this with a quick hangman-type game using one ofthe more difficult terms.

The students do need to leave the lesson with a clear idea of themeanings of the names used.

Extension Answers

Many different correct answers


Exercise 6C Answers

1 a n + 4 b 8n c 9 n d 3t e f 7 t g 3x h 6x i 5y j 7y2 a i 3 ii 6 iii 12 b i 11 ii 19 iii 13 c i 10 ii 13 iii 14

d i 9 ii 7 iii 11 e i 7 ii 2 iii 17 f i 4 ii 3 iii 7g i 16 ii 46 iii 96

3 a i 9 ii 12 iii 36 b i 40 ii 70 iii 55 c i 8 ii 14 iii 2d i 20 ii 50 iii 70 e i 12 ii 16 iii 32 f i 6 ii 30 iii 54

g i 36 ii 27 iii 634 a i 5 ii 6 iii 7 b i 13 ii 14 iii 15 c i 2 ii 1 iii 0

d i 15 ii 16 iii 17 e i 11 ii 10 iii 9

t5

Homework

Homework

Answers

1 Write down the values of these expressions for each different value of n.

a n + 3 where i n = 3 ii n = 5 iii n = 8

b n 1 where i n = 5 ii n = 1 iii n = 3

c 17 n where i n = 4 ii n = 3 iii n = 1

2 Write down the values of ecah of these terms for each different value of n.

a 3n where i n = 2 ii n = 3 iii n = 8

b 5n where i n = 4 ii n = 6 iii n = 7

c where i n = 8 ii n = 12 iii n = 18n

2

1 a i 6 ii 8 iii 11 b i 4 ii 0 iii 2 c i 13 ii 14 iii 162 a i 6 ii 9 iii 24 b i 20 ii 30 iii 35 c i 4 ii 6 iii 9

algebra variable substitute term expression

Key Words



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Miles and kilometres Ask if anyone knows approximately how many kilometres are in 5 miles.

It would help if there were a specific place 5 miles from school that youcould focus on.

By trial and discussion, let the class arrive at approximately 8 km. Then offer a variety of multiples of 5 miles and ask for their kilometre

equivalents. For example:Approximately how many kilometres are in 10 miles, 15 miles,

25 miles, ? Then ask: Approximately how many kilometres are in 18 miles? Lead the discussion along the line 15 miles is 24 km, 20 miles is 32km,

so 18 miles is in between, but closer to 32 than 24, say about 29 or30km.

Ask a few more similar questions. Use the following table to help.

Miles 18 22 38 44 53 59 62 72 98

Kilometres 29 35 61 70 85 94 99 115 157

Discuss different speeds. What is equivalent to 30 mph, 40mph, ?


Tell the class a story, something like this.I was talking to my window cleaner the other day, and I asked himhow he worked out how much to charge for cleaning the windowsin different-sized houses.

Easy, he says, I use a rule of 70p for every window cleaned. I thoughtthats clever. Ill write the rule down. But being a mathematician, I wrote

it down as a formula. What do you think I wrote down? Lead the class to something like C= 70W. (Let the class choose the letter

for the variable, but do give guidance on suitable letters.) Explain howuseful it is to write down rules as formulae.

Get them to work out a few charges. For example:Mrs Smith: 4 windows give

70 4 = 280 penceMr Jordan: 6 windows give

70 6 = 420 pence



LESSON

6.4

Framework objectives Formulae

Use simple formulae from mathematics and other subjects. Substitute positivewhole numbers into simple linear expressions and formulae and, in simple cases,derive a formula.



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Multiplication facts Put on the board 25. Around it put 3, 4 and 7. Ask: How can I make 25, using these three small numbers? Answer:

3 7 + 4. Then ask: Is there another way? Were now looking for 7 4 3. Then give them three different small numbers, 2, 3 and 5, but keep the 25. Ask: How many different ways can we now see of making 25? There is

only one: (2 + 3) 5. Change the numbers to 4, 7 and 8, and the target number to 24.

Ask: How many now? Again there is only one: (7 4) 8. Now add another small number, 6, and ask: How many different ways

can we now find of making 24? [4 6 (8 7)], [(4 6) (8 7)] Finally ask: How many different numbers can we make in this way using

all four numbers, 4, 6, 7 and 8?


An excellent start will be made if you can demonstrate thisfirst part with a pair of scales and some bags of marbles orcoins. (But do practise this to make sure the scales aresensitive enough.)

Show a set of scales as illustrated.The left-hand pan has 3 bags and 2 marbles.The right-hand pan has 17 marbles.The scales balance when the two sides are equal.

Each bag has the same number of marbles in it. How manymarbles are in one bag?

At this point, introduce the What is it we want to know? Lets call itx.

(It could be m or any other letter except o.) What do the scales show us? Both sides are equal. That is:

Left-hand side = Right-hand sideThe left-hand side is 3x+ 2. (Discuss why.) The right-hand side is 17. So:

3x+ 2 = 17 Show how, when two marbles are taken from each side, the scales are

still balanced:3x+ 2 2 = 17 2

3x= 15 This next stage needs to be explained carefully to many students. Class

discussion is invaluable for everyone to recognise that 3xmeans 3timesx, and so we are looking at 3 times what = 15?, leading tox= 5.


LESSON

6.5

Framework objectives Equations

Identify the necessary information to solve a problem. Represent problemsmathematically, making correct use of symbols, words, diagrams and tables.

(This lesson plan starts to address this objective, but does not complete it. )



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Invite the class to imagine a square with its two diagonals drawn in. Ask them to describe the shapes that are formed inside the square.

Now ask them to imagine a rectangle with its two diagonals drawn in. Ask them to describe the shapes that are formed inside the rectangle.


A line segment has a fixed length with two end points, A and B.

A B Any two distinct lines either must be parallel or must intersect at a point.

Explain that when two lines meet at a point, they form an angle. An angleis a measure of rotation and is measured in degrees ().

Explain that a compass is used to measure the amount of turn from one

direction to another.Draw the main compass points on the board or OHT.Use the diagram to explain that to give a specific direction we need: direction of the turn clockwise or anticlockwise amount of turn this is usually in multiples of turns.

Draw on the board the following angles.

Draw on the board a number of different angles. Explain how to estimatethe sizes of the angles.

Describing anglesThe angle at B can be written as B.

Half turn 180 Full turn 360 Right angle 90 Acute angleless than 90

Obtuse anglebetween 90 and 180

14

Parallel lines never meet. These two lines intersect at apoint.

These two lines intersect at apoint when extended.

These two lines areperpendicular.


Shape, Space and Measures 2CHAPTER

7

LESSON7.1

Framework objectives Lines and anglesUse correctly the vocabulary, notation and labelling conventions for lines, anglesand shapes.

Identify parallel and perpendicular lines.

Begin to identify and use angle, side and symmetry properties of triangles andquadrilaterals.

Use angle measure. Distinguish between and estimate the size of acute and obtuseangles.


N

S

W E


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Describing trianglesThe triangle ABC can be represented by ABC.It has three corners, A, B and C; three angles, A, B and C;and three sides, AB, BC and AC.

Geometrical properties of shapesExplain the arrow and bar notation for parallel sides and equal sides

respectively. Illustrate the use of the notation on the parallelogram ABCD.


Plenary Ask individual students to draw on the board: parallel lines, intersecting

lines, perpendicular lines. Draw on the board various acute and obtuse angles. Ask some students

to estimate the sizes of the angles. Draw a rectangle on the board and ask other students to describe the

geometrical properties of this shape.


Exercise 7A Answers

1 a turn anticlockwise b turn clockwise c turn clockwise d turnanticlockwise e turn clockwise f turn anticlockwise

2 a East b South c North d West e West fNorth3 a acute, 60 b obtuse, 100 c acute, 20 d obtuse, 1404 a two of BC or CD or DE b DE and BC c DE and CD or BC and CD

d They intersect at 905 a AB = BC = AC, A = B = C = 60

b AB = BC = CD = AD, A = B = C = D = 90, AB //CD, AD// BCc AB = BC = CD = AD, AD// BC, AB // CD, A = C and B = D

34

34

12

12

14

14

Extension Answers

Home

work

Homework

Answers

1 State whether the angles below are acute or obtuse. Estimate the size of each angle.

a b c

2 Copy and complete the following.

a FG is parallel to

b BC is parallel to

c DE is perpendicular to

d AG is perpendicular to

1 a obtuse, 160 b obtuse, 100 c acute, 402 a DE b AG c EF d AB

acute angle obtuse angle right angle degree compass points

clockwise anticlockwise diagonal intersection line segment parallel perpendicular

Key Words

A

B C

A B

CD

A C

D

EF

GB

PARALLEL

L I N E S

PERPENDICULAR

RIGHT A N G L E

A CU T E

T R I AS N

E GL

R E C TS AE L G N


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Invite the class to imagine a square. Now cut off a corner. Ask some students to describe the shape that is left. Invite the class to imagine another square. Now cut off two corners. Ask other students to describe the shape that is now left. Repeat by cutting off three or four corners.


Explain that it is not always necessary to measure angles on diagrams.Angles can be calculated on diagrams by using given geometricalinformation. Angles whose values are not given are denoted by the lettersa,b,c . These are called unknown angles.

Draw the following diagrams on the board and explain how to calculatethe sizes of unknown angles by giving various examples.

Sum of the angles around a point a + b + c = 360

For example, calculate the size of angle a.

a = 360 160 120a = 80

Sum of the angles on a straight line a +b +c = 180

For example, calculate the size of angle b.b = 180 80 40b = 60



LESSON

7.2

Framework objectives Calculating angles

Use correctly the vocabulary, notation and labelling conventions for lines andangles.

Know the sum of angles at a point and on a straight line.


a b

c

120

a160

40 b80

a b

c

maths frameworking

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