y7 arithmetic: revision: unit 20 fractions lesson plan 1 ... · 4 multiplying and dividing decimals...

Mathematics Enhancement Programme

Y7

© CIMT, University of Exeter

Activity Notes

1 Revision - warming up

T: (a) 5 3+ ( = 8) 5 30+ ( = 35) 5 300+ ( = 305)

26 32+ ( = 58) 2 6 3 2. .+ (= 5.8) 260 320+ ( = 580)

18 49+ ( = 67) 1 8 4 9. .+ (= 6.7) 3 2 0 17. .+ ( = 3.37)

(b) If you spend £1.35 in one shop and £2.52 in another shop,how much do you spend altogether? (£3.87)

(c) 64 31− ( = 33) 640 310− ( = 330) 6 4 3 1. .− ( = 3.3)

5 7 3 6. .− ( = 2.1) 4 9− ( = –5) 2 7 4 9. .− ( = –2.2)

(d) How much change should you get from a £10 note if you buya book for £4.30 ? (£5.70)

7 mins

2 Further revision

PB 20.1, Q1 (c) (1901), (h) (17.61), (l) (14.41),

PB 20.1 Q2, (c) (512), (f) (1.19), (h) (7.78), (k) (45.1)

14 mins

3 Rules of signs and powers of 10T: (a) 5 3× ( = 15 ) −( ) ×6 10 ( = –60)

−( ) × −( )7 100 ( = 700) 6 2 10. × ( = 62)

8 9 100. × ( = 890 ) 200 30× ( = 6000 )

(b) PB 20.1, Q6 ( 6 34 204 2 04× = =p p £ . )

(c) 12 3÷ ( = 4) 340 10÷ −( ) ( = –34)

500 100÷ ( = 5) 280 100÷ ( = 2.8)

0 35 10. ÷ ( = 0.035) 800 40÷ ( = 20)

(d) How many coaches will be needed to carry 450 ManchesterUnited fans to Liverpool, if there are 50 seats on each coach?

(9 coaches)21 mins

4 Multiplying and dividing decimals

PB 20.1, Q3 (c) (441), (f) (897), (i) (12.87), (l) (2.272)

PB 20.1, Q4 (d) (69), (g) (1.71)

49 2 12. ÷ (4.1)

146 4÷ (36.5)

4 0 08÷ . ( 400 8 50÷ = )

15 6 1 3. .÷ (156 13 12÷ = )

31 mins

5 Working with fractions

T: So now you can do almost any calculations with whole numbersand decimals. Next we're going to learn how to add, subtract,multiply and divide fractions.

T: What do we mean when we write 2

5 ?

(The unit is divided into five equal parts and we have 2 of them)

UNIT 20 Lesson Plan 1Arithmetic:Fractions

Revision:Addition and Subtraction

Fractions 1

Mental warm-up activity to startrevision with some easycalculations. T asks Ps question-by-question, encouraging slowerPs.Agreement. Praising.

(continued)

Whole class activity. T callsvolunteer Ps to come to front togive solutions, and explain ifnecessary. Other Ps agree or notand help. T monitors discussion;agrees and praises.

Mental work; revision. Recallingrules of signs and multiplicationand division by powers of 10.Asking, agreeing, praising,question-by-question.

Whole class activity.Volunteer Ps show at BB themethods already learnt (longmultiplication, rule for positioningof decimal point, dividing withdecimals).Agreement. Praising.

Whole class activity; somerevision of the meaning offractions, mixed numbers andtheir improper forms.


Y7


Activity Notes


Revision:Addition and Subtraction

Fractions 1

5 T: Can you tell me the other meaning?

(2

5 can also mean that we have divided 2 units into

5 equal parts and are referring to one of the parts).

T: Let's see if you really understand what this means!

OS 10.1, (a), (b), (d), (e)

OS 10.9, (a), (b), (c)

36 mins

6 Addition and subtraction of fractions

OS 20.1 (A) (B)

43 mins

7 Mental work to practice calculating with fractions

T:2

7

3

7+ = (

5

7)

1

4

2

4+ = (

3

4)

5

9

4

9− = (

1

9)

7

10

4

10− = (

3

10)

3

5

1

5+ = (

4

5)

5

6

4

6− = (

1

6)

4

3

2

3− = (

2

3)

5

8

3

8+ = (

8

81= )

45 mins

Set homework

PB 20.1, Q1 (k)

PB 20.1, Q2 (i), (l)

PB 20.1, Q3 (e), (k)

PB 20.1, Q4 (f), (i)

PB 20.2, Q1 (a), (c), (f), (i), (k), (l)

.

Task appears on OHP, T pointsto P, P comes to front and readsthe fraction represented by theshaded part of the diagram.

Discussion of the mixednumbers and their improperforms.

Agreement. Praising.

(continued)

Task appears on OHP.T lets Ps guess the answer to(A), then T explains what thediagram shows and calls avolunteer P to write in themissing numbers.A volunteer P then completesand explains the solution to (B)at OHP.All Ps write the problems andsolutions in their Ex.Bs.Then T makes Ps state the ruleof adding and subtractingfractions with the samedenominator.

Mental work to check that all Pshave understood the work.T asks, agrees, praises.


Y7


Activity Notes

1 Checking homework

PB 20.1, Q1 (k) (3.51)

PB 20.1, Q2 (i) (1.55) (l) (0.19)

PB 20.1, Q3 (e) (364), (k) (9.1)

PB 20.1, Q4 (f) (327), (i) (1.32)

PB 20.2, Q1 (a) (4

7), (c) (

8

9), (f) (

6

7),

(i) (2

9), (k) (

5

11), (l) (

2

15)

4 mins

2A Mental work with mixed numbers

T: In the last lesson we talked about mixed numbers. Now we'regoing to convert some mixed numbers into their improper forms.

T: 12

3= Ps:

5

3

11

4= 5

4

21

4= 9

4

21

5= 11

5

31

2= 7

2

15

6= 11

6

T: Now we'll do some the other way round!

T:4

3= Ps: 1

1

3

7

5= 1

2

5

12

6= 2

7

3= 2

1

3

− =8

5−1

3

5

13

4= 3

1

4

4

6= ?

Arithmetic:FractionsUNIT 20 Lesson Plan 2

T asks Ps to give the answers.Agreement/correction andexplanation if there are problems.When checking answers to PB20.2, Q1, T makes Ps state therule of adding and subtractingfractions with the samedenominator. T must makes surethat all Ps understand this!Self-correction. Praising.

Addition and SubtractionFractions 2

Mental work.Warm-up activity and revision.T asks, points to P, P answers.T agrees or makes P self-correct.Praising.


Y7


Activity Notes

2B Other ways of writing fractions

There are other ways in which we can write this last fraction. Whatidea do we use when writing fractions in several forms?(The value of a fraction stays the same if we multiply or divide bothits numerator and denominator by the same non-zero number)

T: Give me some other forms of 4

6. (

40

60

8

12

2

3

12

18, , , , etc.)

T: Which of these do we call the simplest form of 4

6 ? (

2

3)

T: Give me the simplest form of each of these fractions:

T:2

4Ps:

1

2

2

6

1

3

3

9

1

3

4

16

1

4

5

10

1

2

8

6

4

31

1

3=

12 mins

3 Fractions in their simplest form

T: Do the following additions and subtractions, writing the answersin their simplest form.

PB 20.2, Q1 (b) (4

8

1

2= )

PB 20.2, Q1 (d) (10

101= )

PB 20.2, Q1 (e) (4

5)

PB 20.2, Q1 (h) (2

8

1

4= )

PB 20.2, Q1 (j) (2

10

1

5= )

PB 20.2, Q1 (o) (4

25)

17 mins

4 Sizes of fractions

T: Which of these fractions is the larger, and why?

4

5

6

5 or


Further revision and mental work.

(continued)


Individual work.Here T can check that slower Ps(who failed in PB 20.2, Q1homework) have understood themethod. T calls these Ps to BB toshow and explain solutions.Agreement, feedback, self-correction. Praising.

Whole class activity.This prepares Ps for work tofollow.Ps may use different methodswhen explaining their answers.


Y7


Activity Notes

4 (<, because for fractions with the same denominator,the one with the larger numerator is larger)

3

5

3

7 or

(>, because, if we divide 3 units into 5 equal parts, we getlarger pieces than if we divide 3 units into 7 equal parts)

− 6

8 or

7

9

(<, because for fractions with the same denominator,the one with the larger numerator is larger)

2

3 or

3

4(?)

T: Are these the only forms of these fractions? (No)

T: Give some other forms of 2

3. (

2

3= = = = =4

6

6

9

8

12

10

15... )

T: And other forms of 3

4 ? (

3

4

6

8

9

12

12

16= = = = ...)

T: Can you see similar forms in these equivalent fractions?

Can you compare them?

(Yes, 2

3

8

12

9

12

3

4= < = )

T: What is the difference between them? (1

12)

T: How did we find the difference?(We had to make changes to both fractions sothat each of them had the same denominator)

T: Good - we call this the common denominator.

25 mins

5 Fractions with different denominators

T: So now we've reached the main subject of today's lesson; how toadd and subtract fractions with different denominators.

For example, calculate (T writes on BB):

1

4

2

5+

(and puts OS on OHP).

OS 20.2 (A) (B)

Arithmetic:Fractions

(continued)

Ps call out fractions, T writesthem on BB.

Agreement. Praising wheneverpossible.

Task appears on OHP.T waits for Ps to reason forthemselves from the OS, firstlyfor diagram (A) and then diagram(B). Volunteer Ps come to OHPto fill in the gaps and give thesolution.Agreement. Praising.

30 mins

Addition and SubtractionFractions 2UNIT 20 Lesson Plan 2


Y7


Activity Notes


Whole class activity.T helps Ps understand how toform common denominators, andthen writes on BB the fractionsthey dictate.Agreement. Praising.Ps write these examples in Ex.Bs.

Individual work, monitored,helped.Detailed checking at BB. (For Q2,slower Ps are called to BB tocalculate.)Agreement, feedback, self-correction. Praising.

6 Finding common denominators

(a)2

3

3

5+ = 10

15

9

15

19

15+ =

(b)5

8

1

4− = 5

8

2

8

3

8− =

(c)5

12

3

8− = 10

24

9

24

1

24− =

(d)4

6

1

3+ = 2

3

1

3

3

31+ = =

35 mins

7 Practice with common denominators

PB 20.2, Q2 (a) (5

10

2

10

7

10+ = ), (e) (

5

6

4

6

1

6− = )

PB 20.2, Q3 (g) (38

35), (i) (

6

10

3

5= ), (n) (

2

21)

41 mins

8 Adding and subtracting mixed numbers

T: And what about mixed numbers? How do we add or subtractthem?

Ps: We have to change them into their improper forms first.

T: Let's look at an addition (writes on BB):

21

21

1

3+ =

Who can show us how to do this?

P1 (at BB):

= + = + = =5

2

4

3

15

6

8

6

23

63

5

6

T: Before we started the addition we could see that 2 1 3+ = .Can anyone suggest a quicker way to do the calculation?

P2 (at BB):

= +( ) + +

=2 1

3

6

2

63

5

6

T: Who would like to try the next one?T writes on BB:

21

41

2

3− =

P3 (at BB):

(continued)


Whole class activity.Now T leads Ps to work withmixed numbers for addition andsubtraction.


Y7


Activity Notes


= −( ) + −

= + − = − =2 1

1

4

2

31

3

12

8

121

5

12

7

12

8 T: That was a long method! Can anyone show us how to do it morequickly?

P4 (at BB):

= − = − =9

4

5

3

27

12

20

12

7

12

T: Which method is quicker? You'll find that it depends on thequestion.

45 mins

Set homework

PB 20.2, Q2 (b), (c), (d)

PB 20.2, Q3 (b), (e), (k), (o)

PB 20.2, Q8 (b), (f)



(continued)


Y7


Activity Notes

1 Checking homework

PB 20.2, Q2 (b) (12

15

10

15

22

15+ = ), (c) (

5

30

24

30

29

30+ = ),

(d) (12

21

7

21

5

21− = )

PB 20.2, Q3 (b) (17

12), (e) (

7

8), (k) (

2

6

1

3= ), (o) (

1

20)

PB 20.2, Q8 (b) ( 41

4

17

4 or ), (f) (

33

201

13

20 or )

5 mins

2A Practice: addition and subtraction of fractions

T:2

5

1

5+ = (

3

5)

3

4

2

4− = (

1

4)

7

9

2

9+ = (

9

91= )

5

11

3

11− = (

2

11) 1

2

3

2

3− = (1) 3

1

42

2

4+ = (5

3

4)

22

33

1

3+ = ( 5

3

36= ) 1

1

7− = (

6

7)

21

7− = (

14

7

1

7

13

71

6

7− = = )

12

6

3

6− +

= (1

5

6

1

6− = ) 2

1

4

3

4− = (

9

4

3

4

6

4

1

2− = = )

2B A question in context

PB 20.2, Q4 (a) 8

12

2

3= (b) 8 pieces

14 mins

3A Practice: common denominators

T writes on BB:

(a)3

4

2

3− = P

1:

9

12

8

12

1

12− =

(b)2

14

3

7+ = P

2:

1

7

3

7

4

7+ =

(c)3

20

2

10+ = P

3:

3

20

4

20

7

20+ =

(d)5

6

2

9− = P

4:

15

18

4

18

11

18− =

3B (a) 33

41

1

3+ = 3 1

9

12

4

124

13

125

1

12+( ) + +

= + =

or

15

4

4

3

45

12

16

12

61

125

1

12+ = + = =


(continued)

Practising:Multiplying Fractions

by Whole Numbers

Two calculations for fourvolunteer Ps to explain andshow solutions in two differentways at BB.

T has asked a P to write solutionson BB as soon as P arrives,Detailed discussion if necessary.Agreement, feedback, self-correction. Praising.

Mental work, all Ps contributing.It is important that Ps practicethis topic.

This question tests Ps'understanding of the topic andgives practice on it. T reads textout slowly, waits for Ps tocalculate and then asks foranswers.

Whole class activity.T asks four volunteer Ps toexplain how to obtain commondenominators in each example,and to write the calculation onthe BB.Agreement. Praising. Ps writein Ex.Bs.


Y7


Activity Notes

3B (b) 23

51

1

3− = 2 1

9

15

5

151

4

15−( ) + −

=

or

13

5

4

3

39

15

20

15

19

151

4

15− = − = =

22 mins

4 Individual practice

PB 20.2, Q5

P1:

3

4

3

5

15

20

12

20

27

201

7

20+ = + = =

P2: 2 1

7

201

7

20

13

20− = − =

28 mins

5A Whole class activity

T: Do these calculations:

T writes on BB, one at a time:

2

7

2

7

2

7+ + = Ps:

6

7

3

10

3

10

3

10

3

10+ + + = Ps:

12

10

4

9

4

9

4

9

4

9

4

9+ + + + = Ps:

20

9

T: This is very tiring! Can you suggest a quicker way to write theseadditions?

P1:

2

73

6

7× =

P2:

3

104

12

10× =

P3:

4

95

20

9× =

T: We've been multiplying fractions by whole numbers.Can you find a rule for this?

What hasn't changed in each fraction? What has changed, andhow?

Ps: The denominator has stayed the same, while the numerator haschanged as it has been multiplied by the whole number.

T: Can you show this with letters? (writes on BB):

a

bc

a c

b× = ×

(finished by a volunteer P)

(continued)


Ps write in Ex.Bs.



by Whole Numbers

Whole class activity. T writestasks on BB, one at a time, andcalls (mainly slower) Ps to doadditions. All Ps write inEx.Bs.

Then T asks Ps to find a shortermethod, and they discover therule of multiplying fractions bywhole numbers.

Individual work, monitored,helped.Detailed checking at BB bytwo volunteer (slower) Ps.Agreement, feedback, self-correction. Praising.T reminds Ps to answer with awhole sentence.


Y7


Activity Notes

5B Further practice

T: Now do these calculations:

2

7 of 3 Ps: = ÷( ) × = × =3 7 2

3

72

6

7

3

10 of 4 = ÷( ) × = × =4 10 3

4

103

12

10

4

9 of 5 = ÷( ) × = × =5 9 4

5

94

20

9

T: What do you notice about these solutions?

Ps: They are all the same as before!

T: So?

Ps:2

7 of 3 = × = ×2

73 3

2

7

3

10 of 4 = × = ×3

104 4

3

10

4

9 of 5 = × = ×4

95 5

4

9

T: OK, this isn't really a very surprising result, but don't forget that

a fraction of something = something × the fraction

38 mins


T: Now try some on your own!

(a) 2

34× = (b)

2

217× = (c)

2

510× =

(d) 65

12× = (e) 5

3

5× = (f) 8

4

7× =

Solutions:

(a) 8

3(b)

14

21

2

3= (c)

20

54=

(d)30

12

5

2= (e)

15

53= (f)

32

7

44 mins

7 Quick mental question

T: Calculate the area of the rectangle with sides of length 3 m and 4

7 m.

(4

73

12

71

5

7×

= = m m m2 2 2)

45 mins

Set homework

PB 20.3, Q1 (a), (c), (h), (i)

PB 20.3, Q4

PB 20.3, Q5



by Whole Numbers

Whole class activity continues.T writes questions on BB,volunteer Ps come out to explainand write solutions.

Other Ps agree or suggestcorrections. T praises.

Individual work, monitored,helped.

Detailed checking at BB. Psdictate solution, T agrees orwaits for correction, and writeson BB.

Self-correction. Praising.

(T might suggest that stronger Pslook at the answers to parts (b),(d) and (e) at home.)

Quick mental question to finishlesson.

Volunteer P gives explanationand answer.

T agrees and praises.


Y7


Activity Notes


Dividing Fractionsby Whole Numbers;

Multiplying Fractionsby Fractions

1 Checking homework

PB 20.3, Q1 (a) 3 (c) 14 (h) 15 (i) 10

PB 20.3, Q4 (a) 3

4 m2 (b)

5

2 m2 or 2

1

2 m2

PB 20.3, Q5 (a) 11

2 kg (b) 4

1

2 kg (c) 7

1

2 kg

6 mins

2 Operations with fractions

T: Which operations can we now do with fractions?

Ps: We can add, subtract and multiply fractions by whole numbers.

T: Let's solve this problem, which involves all these operations.

On Saturday I am going to put a fence around my herb garden.

The sides of the garden are of length 21

3 m and

5

2 m, and the

width of the gateway is 3

4 m. What length of fence will I need?

T (after agreeing, writes what Ps dictate):

a = =21

3

7

3 m m

b = 5

2 m

g = 3

4 m

F = ? (length of fence needed)

P a b= +( ) × 2

= +

×7

3

5

22

= +

×14

6

15

62

= ×29

62

= 58

6

29

3= (m)

F p g= −

= −29

3

3

4

Verbal checking.Before checking starts, T asks aP to write on BB the formula formultiplying fractions by whole

numbers, a

bc

a c

b× = ×

.

While checking Q1, Ps recall that' x × fraction' is equal to

' fraction × x' , and see that thequickest way to solve this is, e.g.

7

816

16

87 14× = × =

not using the formula.

Whole class activity.

T sketches arectangle onBB and saysthe problemslowly.A volunteer P comes to BB andwrites the data on the rectangle,while repeating the text.

Then Ps instruct T what to writeon BB to solve problem. Ps writein Ex.Bs.

a

b

g

5

2 m

21

3 m

3

4 m

(continued)


Y7


Activity Notes

2 = −116

12

9

12

= =107

128

11

12(m)

So I will need an 811

12 m length of fence.

12 mins

3 Multiplying fractions by fractions

T: You can see that it's a very small herb garden. Can you work outits area? Who'd like to write the calculation on the BB for us?

P: A a b= ×

A = ×

7

3

5

2 m2

T: So now we have to multiply a fraction by a fraction. But wehaven't yet learnt how to do this. Can you think of anything we'vecovered already that might help us here?

Ps: Multiplying something be a fraction is equal to finding a fractionof it.

T: Can you tell me what to do here? (Writes what Ps say):

7

3

5

2

5

2

7

32 5× = ÷

× of =

7

3

T: Well done ... but there's still a problem here. Now we have todivide a fraction by a whole number, and we haven't learnt howto do that yet ... Let's go on to that next.

16 mins

4 Dividing fractions by whole numbers

PB 20.4, Worked Example 1

T: What hasn't changed in the fraction, what has changed, and how?

Ps: The numerator has stayed the same but the denominator haschanged - it has been multiplied by 3.

T: Can anyone write it in letters on BB?

P (writes on BB):

a

bc

a

b c÷ =

×

20 mins

5A Practice: dividing fractions by whole numbers

PB 20.4, Q1 (a) 1

6, (b )

3

8, (d)

3

16, (h)

4

45, (i)

1

24

5B Further practice

PB 20.4, Q1 (c) 1

16, (e)

5

24, (f)

4

10= 2

5 , (g)

6

21= 2

7




Praising.

(continued)

Whole class activity.T makes Ps discover the key tomultiplication of fractions byfractions..

T writes 1

43÷ and sketches the

figure on BB, leading Ps to

solution of 1

43

1

12÷ = , as

shown on p 131 of PB Y7B.The T makes Ps discover the rule.After finding and agreeing on therule, a volunteer P comes out andwrites the formula with letters.Praising. Ps write it in Ex.Bs.

Whole class activity.First, volunteer Ps come to BB tosolve and explain solutions, usingthe formula. Then slower Ps haveto be encouraged to do the same.Agreement. Praising.

Individual work, monitored,helped. Checking at BB: Ps writesolutions on BB.Agreement. Praising.

(continued)


Y7


Activity Notes




5B

28 mins

6 Further practice multiplying fractions by fractions

T: We'll return to finding the area of my herb garden. What do wedo next?

(T writes on BB, Ps dictate):

7

3

5

2

7

32 5

7

65

35

6× = ÷

× = × =

So the area is 55

6 m2 .

T: Let's do some other multiplications.

P1:

3

4

3

2

3

42 3

3

83

9

81

1

8× = ÷

× = × = =

P2:

3

5

7

4

3

54 7

3

207

21

201

1

20× = ÷

× = × = =

T: Now let's examine the results (writes on BB):

7

3

5

2

35

6× =

3

4

3

2

9

8× =

3

5

7

4

21

20× =

Ps: When we multiply a fraction by a fraction, we get a new fractionin which the numerator is the product of the factors' numeratorsand the denominator is the product of the factors' denominators.

T: Can you give me a formula for this?

P:a

b

c

d

a c

b d× = ×

×

36 mins

7A Whole class practice with the formula

PB 20.3, Q2 (a) 1

6, (e)

12

35, (f)

9

32

PB 20.3, Q3 (a) 9

8, (c)

7

10, (i) 2

7B Individual practice

PB 20.3, Q2 (c) 1

12, (g)

8

63

PB 20.3, Q3 (d) 5

7, (j)

25

83

1

8=

(T can suggest that stronger Psexamine the solutions to parts (f)and (g) at home and compare withActivity 6 of Lesson 3.)

(continued)

T writes two other multiplicationson BB and chooses two volunteerPs to solve them.

Then T writes the multiplicationsand their results in their improperforms on a separate part of BBand leads Ps to discover the ruleand write down the formula.


Whole class activity.

At first, volunteer Ps, thenslower Ps, are asked to comeand solve the problems at BB.

Individual work, monitored,helped.

Checking at BB: Ps dictatesolution, T writes on BB afteragreement. Self-correction.Praising.

45 mins


Y7


Activity Notes

Set homework

PB 20.4, Q1 (j), (k), (l)

PB 20.3, Q2 (b), (d), (k)

PB 20.3, Q3 (f), (h), (l)





Y7


Activity Notes


Detailed checking at BB.Volunteer Ps come to BB to solveproblems and explain how to usethe formulae they have learnt.


Then T makes Ps return toquestions Q1 (k), Q2 (d) and Q3(l) and suggests (or leads Ps todiscover) a quicker method withcancelling down beforemultiplying the numerators/denominators of the factors.

(continued)

Dividing by Fractions

Whole class activity.T makes Ps practice the quickermethod. They recall what theyhave learnt so far aboutmultiplication and division offractions.

T asks 3 volunteers to write theformulae on BB, then writesproblems on BB and asks Ps touse both the formulae and thequicker method using cancelling.Volunteer Ps come to BB, writeand explain; other Ps agree orcorrect. T praises.

1 Checking homework

PB 20.4, Q1 (j)5

64

5

6 4

5

24÷ =

×=

(k)9

106

9

10 6

9

60

3

20÷ =

×= =

(l)4

57

4

5 7

4

35÷ =

×=

PB 20.3, Q2 (b)1

2

1

2

1 1

2 2

1

4× = ×

×=

(d)2

3

3

4

2 3

3 4

6

12

1

2× = ×

×= =

(k)1

2

3

19

1 3

2 19

3

38× = ×

×=

PB 20.3, Q3 (f) 31

7

1

3

22

7

1

3

22 1

7 3

22

211

1

21× = × = ×

×= =

(h) 11

21

1

2

3

2

3

2

3 3

2 2

9

42

1

4× = × = ×

×= =

(l) 11

42

1

5

5

4

11

5

5 11

4 5

55

20

11

42

3

4× = × = ×

×= = =

8 mins

2 Practice of multiplication and division with fractions

T: What are the formulae we have learnt for multiplication anddivision of fractions?

P1:

a

bc

a c

b× = ×

P2:

a

bc

a

b c÷ =

×

P3:

a

b

c

d

a c

b d× = ×

×

T:5

84× = Ps:

5 4

8

5

2

× =

4

96× = 4 6

9

4 2

3

8

3

× = × =


Y7


Activity Notes

UNIT 20 Lesson Plan 5Arithmetic:Fractions Dividing by Fractions

210

75÷ = Ps:

10

7 5

2

7×=

6

56÷ = 6

5 6

1

5×=

3

8

5

3× = 3 5

8 3

5

8

××

=

4

7

3

2× = 4 3

7 2

2 3

7

6

7

××

= × =

12

118÷ = 12

11 8

3

11 2

3

22×=

×=

6

5

5

4× = 6 5

5 4

3 1

1 2

3

2

××

= ××

=

4

33× = 4 3

3

4 1

14

× = × =

16 mins

3A Revision

T: This problem should be revision for you. Listen while I readit out ...

I thought of a number, multiplied it by 5 and the product was 45.

What was the number?

P1 (writes on BB): x × =5 45

x = ÷45 5

x = 9

T: Here's another one ...

I thought of another number, multiplied it by 3 and got 4

5.

P2 (writes on BB): x × =3

4

5

x = ÷4

53

x = 4

15T:

Lisa thought of a number, multiplied it by 2

3 and got 5.


35

x = ÷52

3 x = ?

(continued)

Whole class activity revisingtopic learnt in Unit 16, andleading to the discovery of themethod of dividing by fractions.T asks, volunteer Ps come to BBto write and solve the equationsas far as they can.


Y7


Activity Notes

T: It's too difficult for us now. Never mind, let's try a number that

Mary thought of. She multiplied her number by 3

4 and got

2

5.


4

2

5

x = ÷2

5

3

4 x = ?

3B Dividing by fractions

T: Now we have to learn how to divide by fractions.

Let's translate this from English to mathematics; how can we say

' x multiplied by 3

4' in another way?

Ps: We can say 3

4 of x .

T: So the question can now be reworded as,

"What is x if 3

4 of x is

2

5?"

T: If we know that 3

4 of x is

2

5, how can we find

1

4 of x ?

Ps: That will be one third of 2

5.

T: So?

Ps:1

4

2

53

2

5 3

2

15 of is x ÷ =

×=

T: And how do we find the whole of x if we know what 1

4 of it is?

Ps: We must multiply 1

4 of it by 4.

T: That's right. Tell me what to write on the BB.

Ps:2

154

2 4

15

8

15× = × =

T: Good. Now let's work through the other one. This time I won'tsay anything - I'll just write what you tell me to.

Ps (dictate, T writes):

If 2

3

1

35 2

5

2 of is 5, of is x x ÷ =

If 1

3

5

23

5 3

2

15

2 of is

5

2, x x = × = × =

T: Can you see a pattern in these results?

I'll write them down again:

2

5

3

4

2 4

5 3× = ×

×

52

3

5 3

2÷ = ×


(continued)

T leads Ps to discover forthemselves how to divide byfractions.

Ps can answer in chorus.

T writes on BB what Ps say.

T writes on BB.

T writes on BB, Ps in Ex.Bs


Y7


Activity Notes

3B T: What has happened to the numbers 2

5 and 5 ?

Ps:2

5 has been multiplied by

4

3, while 5 has been multiplied by

3

2.

T: So will we need another formula for dividing by fractions?

Ps: We don't need another formula, but we must remember that

dividing by a

b is the same as multiplying by

b

a.

T: So who would like to complete these calculations on the BB?

T: ab

c÷ = Ps: a

c

b

a c

b× = ×

T:a

b

c

d÷ = a

b

d

c

a d

b c× = ×

×

32 mins

4A Practice: dividing by fractions

PB 20.4, Q2 (a) (12)

PB 20.4, Q3 (a) (3

2)

PB 20.4, Q4 (c) (91

40)

4B Using 'cancelling'

PB 20.4, Q2 (h) (7)

PB 20.4, Q3 (d) (6

5 or 1

1

5)

PB 20.4, Q4 (a) (6

13)

4C Further practice

PB 20.4, Q2 (f) (16

3)

PB 20.4, Q3 (c) (9

81

1

8 or )

PB 20.4, Q4 (e) (33

4)

40 mins


PB 20.4, Q2 (c) (24), (j) (9)

PB 20.4, Q3 (b) (3

4), (i) (

9

16)

PB 20.4, Q4 (b) (14

5)


(continued)

Volunteer Ps write on BB; all Pswrite in Ex.Bs.


Whole class activity; practiceusing the rule for dividing byfractions.

Individual work, monitored,helped.Detailed checking at BB;volunteer Ps explain and writetheir solutions.Agreement, feedback, self-correction. Praising.

Part 4C is a warning to cancelonly after the divider fractionhas been turned upside-down.

T writes questions on BB, Psdictate calculations, T agrees (orwaits for corrections) and writeseach solution on BB, Ps write inEx.Bs.

45 mins


Y7


Activity Notes

Set homework

PB 20.4, Q2 (b), (e), (i)

PB 20.4, Q3 (e), (g), (h)

PB 20.4, Q4 (d), (f)



Y7


Activity Notes

(Checking homework is shown in Activity 1D)

1A Revision of addition and subtractions of fractions

T: For the final part of this unit we'll revise all the operations withfractions we've learnt.

Addition and subtraction of fractions

(a)4

7

3

7− ( = 1

7)

2

5

3

5+ (= =5

51)

9

8

3

8− (= =6

8

3

4) 1

2

9

4

9− ( = − =11

9

4

9

7

9)

11

6− ( = − =6

6

1

6

5

6) 3

1

52

2

5− (= − =16

5

12

5

4

5)

1

6

2

6

3

6+ + ( = =6

61)

5

4

3

4

2

4− − (= − =5

4

5

40 )

T: How do we add or subtract fractions with the same denominator?

(We have to add or subtract their numerator and leavethe denominator as a common denominator)

(b)2

7

1

3+ ( = + =6

21

7

21

13

21)

5

6

3

4− ( = − =10

12

9

12

1

12)

3

40

1

20− ( = − =3

40

2

40

1

40)

8

16

1

2+ ( = + = =8

16

8

16

16

161)

1B Multiplying and dividing fractions by whole numbers

T: What rules have we learnt for multiplying and dividing fractionsby whole numbers?(When multiplying a fraction by a whole number, the numeratorhas to be multiplied by the whole number, and the denominator

is unchanged)

(When dividing, the denominator has to be multiplied by thewhole number and the numerator is unchanged)

T: What are the formulae for this?

P1 (writes on BB):

a

bc c

a

b

a

bc

a c

bof = × = × = ×

P2 (writes on BB):

a

bc

a

b c÷ =

×

(a)3

7 of 2 ( = × =3

72

6

7)

3

54÷ ( = × =3

5

1

4

3

20)

−

×2

34 ( = − × = − = −2 4

3

8

32

2

3)

5 11

3× ( = × = =5

4

3

20

36

2

3)


Revision of Operationswith FractionsUNIT 20 Lesson Plan 6

(continued)

Addition and subtraction offractions with the samedenominator is suggested asrevision and mental practice.

Examples where fractions arechanged to obtain a commondenominator should be repeatedat BB, with Ps writing in Ex.Bs.

T may also get Ps to practicemultiplications/divisions withdifferent signs.The second set of calculations isfor discussing and practisinghow to cancel out fractionsbefore starting themultiplication process.Agreement. Praising.

Whole class activity.T helps Ps to draw up the rules,ensuring that the spokenmathematics is correct at alltimes.

Then practice takes place at BB(and in Ex.Bs) with all Pscontributing.


Y7


Activity Notes

1B −

÷ −( )1

32 ( = −

× −

= =1

3

1

2

2

6

1

3)

21

42÷ ( = × =5

4

1

2

5

8)

(b)3

42× ( = =6

41

1

2)

−

÷14

37 ( = −

× = −

= −14

3

1

7

14

21

2

3)

52

5× ( = =10

52 )

6

79÷ ( = × = × =6

7

1

9

2

7

1

3

2

21)

11

38÷ −( ) ( = × −

= × −

= −4

3

1

8

1

3

1

2

1

6)

11

6

9

3× ( = × = =1

1

63 3

3

63

1

2)

1C Multiplying fractions by fractions

T: How do we multiply fractions by fractions?(We divide the product of the numerators

by the product of the denominators)T: And the formula?

P (writes on BB):

a

b

c

d

a c

b d× = ×

×

(a)3

4

1

2× ( = 3

8)

1

7

1

3× ( = 1

21)

2

5

1

3× −

( = − 2

15)

13

4

1

5× ( = × =7

4

1

5

7

20)

−

× −

2

1

2

3

2 ( = −

× −

=5

2

3

2

15

4)

21

31

1

4× ( = × = =7

3

5

4

35

122

11

12)

(b)2

3

3

5× ( = × =2

1

1

5

2

5)

3

4

2

7× ( = × =3

2

1

7

3

14)

5

6

9

8× ( = × = =5

2

3

4

15

81

7

8)

4

3

9

10× ( = × = =2

1

3

5

6

51

1

5)

11

3

3

4× −

( = × −

= − = −4

3

3

4

12

121)

21

21

1

5× ( = × = × =5

2

6

5

1

1

3

13)

Revision of Operationswith Fractions


(continued)

In Activity 1C, Ps can use twosteps to tackle the problems.

Volunteer Ps and othersencouraged by T, show at BBhow to use the rule/formula;others agree or help to correct.All Ps write in Ex.Bs.

Ps can be encouraged to usecancelling where appropriate.

UNIT 20 Lesson Plan 6


Y7


Activity Notes

1D Dividing by fractions - checking homework

T: The final process we looked at was how to divide by fractions.Can you tell me how to do this?

(We can divide either a whole number or a fraction by a fractiona

b in the same way: we have to multiply by

b

a)

T: Let's use letters to show this.

P1: a

b

ca

c

b

a c

b÷ = × = ×

P2:

a

b

c

d

a

b

d

c

a d

b c÷ = × = ×

×

T: Let's see how you did with your homework.

Checking homework

PB 20.4, Q2 (b) (27), (e) (15

2), (i) (21)

PB 20.4, Q3 (e) (6), (g) (25

14), (h) (

9

7)

PB 20.4, Q4 (d) (7

3), (f) (

147

25)

24 mins

2 Mental work T: So now we can do almost anything with whole numbers,

decimals or fractions. Let's see how we get on here!

MT 20.2

28 mins

3 Calculations with fractionsOS 20.7

e.g.

P1:

1

22+ =x

P2: x = −2

1

2

x = 11

2 35 mins

4 Final test1. (a) 14 3 5 68. .− ( = 8 62. ) (b) 5 4 4. ÷ ( = 1 35. )

2. (a)3

4

5

8− ( = − =6

8

5

8

1

8)

(b) 12

32

3

4+ ( = + = =1

8

122

9

123

17

124

5

12)



In Activity 1D, Ps review thedivision of fractions and checksin detail the progress they madewith their homework, ensuringthat they applied the rulecorrectly.

(continued)

Mental work with MT 20.2.

T asks, points to P, P answers,T agrees/waits for correctionand praises, question byquestion.

Task appears on OHP.

T may make Ps write each ofthe four problems as anequation in order to see what todo.Volunteer Ps write down theequations on BB, then T pointsto other Ps to solve it. All Pswrite in Ex.Bs.Agreement. Praising.

T asks Ps to give their answersverbally; agreement (or not),discussion if necessary, self-correction. Praising.

Individual work.

Ps have the opportunity to testthemselves.T writes problems on BB oreach P has a copy of the test.T gives 8 minutes to solve thetest, then puts on OHP(prepared in advance).


Y7


Activity Notes

4 3. (a)2

3 of 7 ( = × = =2

37

14

34

2

3)

(b)4

36÷ ( = × = =4

3

1

6

4

18

2

9)

(c)6

7

2

9× ( = × =2

7

2

3

4

21)

(d) 9 11

2÷ ( = × = × =9

1

2

3

3

1

2

16 )

45 mins

Homework

OS 20.8 (each P has a copy)

PB 20.4, Q7

PB 20.4, Q9 (b)

Ps correct their own work,noting their weak areas.

Praising and encouraging.



or more

(continued)

y7 arithmetic: revision: unit 20 fractions lesson plan 1 ... · 4 multiplying and dividing decimals...

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