Unit 15 negative numbers lesson plans and notes PDF

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Unit 15 negative numbers lesson plans and notes...

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Mathematics Enhancement Programme

Y7

UNIT 15 Negative Numbers

Lesson Plan 1 Addition and Subtraction 1

Activity 1

Notes Introduction T: We've looked at negative numbers before, but now we're going to learn how to add and subtract them. But first, I'd like to check that you can add and subtract positive numbers! Let's start! 8

7

10

20

300

11

6

80

30

700

8 5.

3

7 6.

4 1.

9

200 500 6 8.

27

34

71

28

11 2.

Mental work as a warming-up activity; everyone contributing. T asks a question, nominates P, P answers, other Ps agree or not, T praises. As the calculations become more difficult, stronger Ps should be asked.

7 9.

etc. 6 mins 2

Calculations using a thermometer T: We've sometimes used a sketch of a thermometer to do simple additions and subtractions. Let's see if you can remember how it's done. OS 15.1

Whole class activity. Task appears on OHP. T asks, volunteer Ps come to OHP, show and explain the solution. T makes Ps draw a thermometer (perhaps just a vertical number line) from –12 to +12 in their Ex.Bs to follow volunteer Ps' answers agree or not. Praising.

12 mins 3

Further work with a number line Now use the vertical number line you have just drawn in your Ex.Bs to do the following calculation. Water gauges are placed at certain points along rivers to measure the water level. Last week, the level of the Danube in Budapest increased by 4 cm every day. What was the water level on Monday and on Saturday if it was –3 cm on Wednesday?

Individual work. Task appears on OHP, Ps work in Ex.Bs. T monitors Ps' work and sees if they are able to use the number line to make simple additions and subtractions; T helps slower Ps. Checking at BB. T sketches a vertical number line on BB for Ps to come and show their solution on it. Agreement, feedback, selfcorrection. Praising.

18 mins 4

T: Use a number line to do the following additions and subtractions: 4

6

(10)

4

6

(–2)

4

2

(2)

4

6

(–10)

4 2 (–2) 4 6 (2) T: In a quiz game you score 4 marks in one round and 6 marks in the next round. What is your total number of marks? (10) T: What mathematical operation is used here? (Addition) T: If you win points on one round and lose points on the next, will you get the maximum score? (No) (continued) © CIMT, University of Exeter

Whole class activity. Tasks are written on BB by T, and a horizontal number line (from –10 to +10) is also drawn on BB. Volunteer Ps come out and write results on BB, using the number line if necessary. Other Ps work in Ex.Bs; agree or not. Then T leads Ps to find a rule for the results, so that a number line is not needed. The aim is for Ps to be able to handle additions and subtractions competently.

Mathematics Enhancement Programme

Y7

UNIT 15 Negative Numbers

Lesson Plan 1 Addition and Subtraction 1

Activity 4 (continued)

Notes What mathematical operation is used when a score is to be reduced? (Subtraction) T: Tim gains 4 marks, then loses 6 marks. Tom gains 4 marks, then loses 2 marks. What happens to the scores as a result of these two losses? (Both losses reduce the scores. Tim's loss is greater than the number of marks he had gained. Tom loses fewer marks than he had gained) T: So?

(The result is given as '+' or '–' according to whether the gain or the loss is the larger)

Some discussion might be helpful here. Agreement. Praising.

T: Let's see who can explain the next one: 4 6 ? (Here the loss becomes a greater loss, as the two subtractions together make a larger negative number) T: So do we use addition or subtraction to work this out? (Addition, as we are adding two negative numbers to make a larger negative number) T: And the result? ( 4 6 10 ; a '–' sign is added, to give the answer –10) etc. 28 mins 5

Further practice 8

(a)

6

(b) (c)

10

5

7

30

20

9 9

(d)

10 30

2

80

50 70 60

22

46

34

13

23

58

67

25

38 mins 6

Individual work (a)

12

(b)

8

19 11

9

8

5

7

7

4

19

3

For stronger Ps: (c) (d)

112 3 4.

209

77

89

4

4 4.

8 3.

96 9 26 .

38 130

Whole class activity. Task appears on OHP. T calls Ps to BB to answer and explain. Slower Ps should be given easier calculations, perhaps using number line still shown on BB. During explanations, all Ps listen attentively - either agree or give alternative explanation. Agreement. Praising. Individual work. Task appears on OHP. T monitors Ps' work, and helps slower ones, suggesting they use number line and reason as in Activity 4. Checking and explanation on BB. Tasks (c) and (d) are also shown on BB but slower Ps need not write them down.

45 mins 7

Set homework PB 15.1, Q1 (b) - (d), (h) - (j) PB 15.1, Q2 (b) - (e) PB 15.1, Q9

© CIMT, University of Exeter

.

Mathematics Enhancement Programme

Y7

UNIT 15 Negative Numbers

Lesson Plan 2 Addition and Subtraction 2

Activity 1

Notes Checking homework PB 15.1, Q1 (b) 3 (c) 2 (d) 3 (h) 0 (i) –3 (j) –4 PB 15.1, Q2 (b) – 2

(c) – 2

Detailed checking at BB. T draws a number line on BB; volunteer Ps come out and explain the solutions both ways showing on the number line and reasoning without the number line.

(d) – 2 (e) –5

PB 15.1, Q9 42 cm P (e.g. at Q1 (b)): Says, while writing on BB: Starting at –5, we have to move 8 to the left, so

5

8

3

Continues explanation: but we can calculate this in another way: 5 and –8 are numbers with opposite signs, so we can calculate the difference between them and add the sign of the larger number. The difference is 3, and 8 is the larger of the two numbers, so the answer is 3 , or 3. 10 mins 2

Sequences T: I'm going to give you a sequence, and you have to continue it. You won't need to write anything down. (a) What are the terms of the sequence which begins with 8 and where 3 is always subtracted to get the next term? (8, 5, 2, – 1, – 4, – 7 ...)

Mental work. T says the question and then points to a P who gives the next term, then T points to another P for the next term, etc. Agreement. Praising.

(b) What are the terms of the sequence which has –10 as its first term, with 4 added to get the next term? (–10, – 6, – 2, 2, 6, 10, ...) (c) What are the terms of the sequence with –2 as its first term, and which has the rule subtract 5 to get the next term? (–2, –7, –12, –17, –22, ...) 15 mins 3

Problems in context T: Two days before pay day, Jim has £90 in his wallet. He knows he must pay his landlord's bill of £50 for rent. T: What effect will this have on the amount of money in Jim's wallet? (Reduce it) T: How much will he have left when he has paid the rent? (£40) T: Is it correct to use the '+' sign for money Jim receives and the '–' sign for money he pays from his wallet? (Yes) T: OK. The postman brings Jim a telephone bill for £30. Did Jim give the bill or did he receive it? (Received it

Whole class activity. T tries to give Ps a practical example of

1 1. If the reasoning for this causes problems for some Ps, they will have to accept the rule for its own sake.

)

T: When Jim pays the bill, does this have a '+' or a '–' effect on the money in his wallet? (–) T: We'll write it like this (writes on BB): £30 T: What is the effect of paying this bill on the money in Jim's wallet? (– £30) T: Now, any time you see 30 , you can write it as 30. (continued)

T: How much money will Jim have left when he has paid his rent? (£10) © CIMT, University of Exeter

Ps write it in their Ex.Bs.

Mathematics Enhancement Programme

Y7

UNIT 15 Negative Numbers

Lesson Plan 2 Addition and Subtraction 2

Activity 3 (continued)

Notes T: The next day Jim gives his son £5 to buy a book. How do we write this?

(–)

Ps write in their Ex.Bs.

T: What is the effect of this on the money in Jim's wallet? (– £5) T: So £5 of the money Jim had was given to his son; we can write this as

5 (T writes on BB).

5

T: Jim is feeling depressed: yesterday ago he had £90 and now he has only £5 to last him until he is paid.

Praising.

That evening, Jim's father comes to see him. He says that he will take the bill for Jim's rent and pay it himself. T: So, is Jim giving or receiving, money or a bill? (He's giving a bill) T: How can this be written in figures? P (at BB):

£50

T: And what is the effect on the amount of money Jim has? (+£50) T: That's right. The effect is positive:

50

Ps volunteer, T chooses one of them to write on BB. Agreement. Praising. Praising. Ps write in Ex.Bs.

50.

Write it down in your Ex.Bs. T: Using Jim's money as an example, we've seen that: 30

30,

5

5,

50

50

T: What will happen to finish this story? (Jim is paid the following day) T: Let's say that he is paid £800. T: Who would like to write on BB the effect this will have on Jim's money? ( 800 800 ) 27 mins 4

Further examples T: Now, write these statement in mathematical terms: (a) I have £30; I give £10 to my son. ( 30

10

30

(b) I have £30 and I receive a gas bill for £60. 60 30 ( 30

10

60

20 )

30 )

(c) I have £2 and then I win £100 000 on the lottery. 100 000 2 100 000 100 002 ) (2 (d) After paying tuition fees of £300 I have £120 left. The next day I find out that my local council will pay the tuition fees. 300 120 300 420 ) ( 120

35 mins

© CIMT, University of Exeter

Whole class activity. T reinforces the concepts just explained by giving similar examples. T asks, Ps volunteer, one of them comes to BB, writes and explains; the others agree or not. Correction. Praising. Here Ps may voice doubts that someone can have – £30 in money, but T stresses that it's not the actual money but the financial position that is meant.

Ps have to understand why the total increases.

Mathematics Enhancement Programme

Y7

UNIT 15 Negative Numbers

Lesson Plan 2 Addition and Subtraction 2

Activity 5A

Notes Summarising the rule T encourages Ps to draw up and state the rule.

T: Let's summarise the rule:

T: Write the rule in your Ex.Bs with a box around it to highlight it. 5B

Simplifying using the rule T: Now simplify what I say: 3

(–3)

200

( 200 )

8

( 8)

1000

( 1000 )

12

( 12 )

3 .4

( 3.4 )

Mental work. T asks, points to, P answers, T agrees (or makes first this P and then other Ps correct) and praises, question by question.

etc. 40 mins 6A

Applying the rule when adding and subtracting T: Let's practise adding and subtracting whole numbers: (a)

6B

5

(5

8

8

3)

(b)

7

6

(

7

6

(c)

3

9

(

3

9

(d)

8

7

(

8

7

1) 6) 15)

Individual work T: Now try some on your own: (e)

7

9

(

7

9

16)

(f)

8

3

(

8

3

5)

45 mins Set homework PB 15.1, Q1 (e), (f), (l) PB 15.1, Q2 (f), (h), (l) PB 15.1, Q10

© CIMT, University of Exeter

Whole class activity. Task appears on BB. T suggests method, and leads Ps to make the additions and subtractions in two stages, as shown in Activity 4. In the first step they apply the rule they have just learnt, then they apply the rule they learnt in Lesson 1.

Individual work, monitored, helped. Tasks appear on BB, Ps write in Ex.Bs. Checking and repeating the steps at BB. Agreement, feedback, selfcorrection. Praising.

Mathematics Enhancement Programme

Y7

UNIT 15 Negative Numbers

Lesson Plan 3

Activity 1

Multiplication and Division 1 Notes

Check homework PB 15.1, Q1

(e) – 1

PB 15.1, Q2

(f) – 3 (h) 11 (l) 0 64 284 348 years (There was no year 0, so 284 64 does give the correct answer.)

PB 15.1, Q10

(f) – 5

(l) 0

When checking Q1 and Q2, T calls Ps to BB to repeat the steps and rules they have learnt in the previous lesson. For Q 10, verbal checking. Agreement, feedback, selfcorrection. Praising.

5 mins 2A

Revision of previous topics and of coordinates T: I have a hat. Earlier today its coordinates were A (– 3, – 1), B (7, – 1), C (4, 0), D (3, 1), E (1, 1), F (0, 0) but someone has taken 4 from the second coordinate of each point and now I can't find it. Please help me look for it - can you give me the coordinates of the new A', B', C', D' E' and F' points? Ps (aloud at BB): Since

1

5,

4

(writes) A' (– 3, – 5)

0

4

4

B' (7, – 5) C' (4, – 4)

1

4

3

D' (3, – 3) E' (1, – 3) F' (0, – 4)

2B

Individual work T: Thanks everyone! But now, it's gone again! This time the first coordinate of each point has been reduced by 7. Can you write down the new coordinates and then plot the new points A'', B'', C'', D'', E'' and F", all on the same grid ?

Whole class activity for 'warming up'. Ps can practice new topic and recall information on plotting points. T writes the original coordinates on BB, draws a set of axes (bearing in mind that more space will be needed as the activity develops) and calls different Ps to plot the points. Each P does the same in Ex.B. Praising. Then T calls different Ps to find and write the new coordinates on BB and explain what they are doing. Agreement. Praising. Another group of Ps then come to BB to plot the final set of points. This task is not difficult and can be an effective start to the lesson if conducted at pace, but not rushed. Individual work, monitored, helped. Checking at BB; Ps write down new coordinates and plot the points. Other Ps agree or not, T praises, self-correction. Then T and Ps discuss how the hat was affected by the changes.

20 mins 3A

Multiplication and division T: We've looked at adding and subtracting negative numbers, now we'll go on to multiplication and division. Can you remember how we introduced multiplication? For example, what is meant by 3 4? Ps: 3 3 3 3 , we have to add the number 3 four times. T: So how can we describe the process?

(continued)

Ps: Multiplication is a repeated addition of the same number.

© CIMT, University of Exeter

Whole class activity. T uses discussion to give the reasons for the rules of multiplication and division of negative numbers. Ps may be allowed to answer in chorus.

Mathematics Enhancement Programme

Y7

UNIT 15 Negative Numbers

Lesson Plan 3

Activity 3A

Multiplication and Division 1 Notes

T: Fine! So what does this mean?

(continued) Ps:

3

4 ?

3

3

3

3

T: What does that come to? Who'd like to write it on BB? ( 3 3 3 3 12) 3 4 3 4 . What do you notice? T: Compare the answers to and

From now on, T waits for individual answers from volunteer Ps.

(Only the signs are different)

3B

T: Can you put the calculation into a real-life context involving money and bills to be paid? (I receive four amounts of £3, giving me a total of £12; I receive 4 bills to be paid, each of £3, so I have to pay a total of £12) Multiplication with different signs T: Let's look at some simple multiplications: 4

3

(–12)

8

5

10

(–50)

100

2

(–200)

(–200)

3.2

2

(–6.4)

20

10

0.46

4

finally,

3

T: Why does 4

(–56)

etc.

(–4.6)

10

7

T writes questions on BB, points to Ps to answer. T agrees, praises, writes the answers on BB, Ps in Ex.Bs.

(–12)

3

12 ?

Ps: Because, as we've just seen, a positive number multiplied by a negative number gives a negative number. T: No, we've seen that a negative number multiplied by a positive number gives a negative number. Ps: It's the same!

After the final multiplication T puts the question to the whole class and they may answer in chorus. T should not accept this immediately, but should ask for explanations.

T: Not exactly - can you give me a reason for saying this? Ps: ?? T: OK, we'll leave this for now and just agree that 3

4

4

3 .

30 mins 4A

Further rules with signs T: Look at the ways the products and quotients change in these calculations: (a)

3

4

12

2

4

8

4

4

1

0

(continued)

4

0

(b)

12

4

8 4

0

3

4

2

4

1

4

0

1

4

4

4

4

1

2

4

8

8

4

2

3

4

12

12

4

3

© CIMT, University of Exeter

Whole class activity, making Ps observe the changes to the quotients and deduce the answers. T leads Ps to see what results when a negative number is divided by a positive number. Tasks appear on OHP, Ps dictate, T writes on OHP, Ps in Ex.Bs.

Mathematics Enhancement Programme

Y7

UNIT 15 Negative Numbers

Lesson Plan 3

Activity 4A (continued)

Multiplication and Division 1 Not...