SPM Form 4 Add Maths Chapter 6 Linear Law

1. CHAPTER 6
Linear Law
By Mr. Jack
Additional Mathematics

2. SPM Analysis
Paper 1 1 Question
Paper 2 Section B - 1 Question

3. Intro - Things you will need:
• 20cm/30cm transparent ruler • Lots of graph papers
Release It a few days before class
How many? 7

4. Intro - Things you need to know:
• Recognising linear / non-linear relations from graphs
Linear Non-linear
• Straight line
• Relation of x and y is to the
power of 1
• Curved line
• Relation of x and y is not to
the power of 1

5. Intro - Things you need to know:
• Recognising linear / non-linear relations from graphs
Linear Non-linear

6. Quiz 1 ( ? minutes)
1.Go to https://socrative.com/
2.Click on “Login” and select “Student
Login”
3.Enter Room Code: JACK2304

7. Quiz 1

8. Overview of the Chapter
Stage 1 • involves linear relations
Stage 2 • involves non-linear relations
Stage 3 • application in real life

9. Stage 1

10. Steps to score:
Scale Label
DrawFind

11. • Notes:
• Pass through as many points as possible
• Above & below the line must have almost same number of points
which didn’t pass through
Stage 1: A) Drawing best fit line

12. • Notes:
• Pass through as many points as possible
• Above & below the line must have almost same number of points
which didn’t pass through
Stage 1: A) Drawing best fit line
𝑦 𝑦𝑦
𝑥 𝑥𝑥
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x

13. Stage 1: A) Drawing best fit line
Example 1:
Given the table below, plot the graph of y against x and draw the best fit line.
x 10 20 30 40 50 60 70 80
y 16.5 20.0 23.5 27.5 31.5 35.0 38.5 42.5

14. Stage 1: A) Drawing best fit line
Example 2:
The following table shows the data of two corresponding variables, 𝑝 and 𝑣,
obtained from an experiment. Plot the graph of 𝑣 against 𝑝 and draw the best
fitted line.
p 1.8 2.5 3.6 4.9 5.3 6.4
v 4.8 6.2 8.4 11.2 11.8 13.8

15. Stage 1: A) Drawing best fit line
Example 3:
The above diagram shows an electrical circuit used to find the electromotive
force, 𝐸 volts, and the resistance r ohms of a cell. The rheostat is adjusted to
obtain a few readings of the voltmeter, 𝑉 volts and the ammeter, 𝐼 amperes, as
recorded in the following table. It is known that the quantities 𝐸, 𝑉, 𝐼 and 𝑟 are
related to the formula 𝐸 = 𝑉 + 𝐼𝑟. Draw the graph of 𝑉 against 𝐼.
Reading of ammeter,
I amperes 0.2 0.3 0.4 0.5 0.6
Reading of voltmeter,
V volts 1.20 1.11 1.00 0.89 0.80

16. Stage 1: B) Forming equation of best fit line
𝑦 = 𝑚𝑥 + 𝑐
constants
variables
𝑦
𝑥
x
x
x
x
x
x
x

17. Example 1:
From your graph, find the equation of best fit line.
x 10 20 30 40 50 60 70 80
y 16.5 20.0 23.5 27.5 31.5 35.0 38.5 42.5
Stage 1: B) Forming equation of best fit line

18. p 1.8 2.5 3.6 4.9 5.3 6.4
v 4.8 6.2 8.4 11.2 11.8 13.8
Example 2:
a) From your graph, find the value of the gradient and of 𝑦-intercept.
b) Find the value of v when p = 2.0
Stage 1: B) Forming equation of best fit line

19. Example 3:
a) From your graph, find the value of 𝐸 and of 𝑟 .
b) Find the current in the circuit when 𝑉 = 0.75 volts.
c) Use the equation that you have formed to calculate the value of 𝑉 when 𝐼 =
0.95 amperes.
Reading of ammeter,
I amperes 0.2 0.3 0.4 0.5 0.6
Reading of voltmeter,
V volts 1.20 1.11 1.00 0.89 0.80
𝐸 = 𝑉 + 𝐼𝑟
Stage 1: B) Forming equation of best fit line

20. Quiz 2 ( ? minutes)

21. Quiz 2
1.
Ans:
𝑎 ≈ 0.55
𝑏 ≈ −0.9

22. Quiz 2
2.
Ans: (b)(i) 𝑎 ≈ 2.0 (b)(ii) 𝑏 ≈ 4.0 (c) 10s

23. Stage 2

24. Steps to score:
Transform Table Scale
LabelDrawFind

25. Stage 2: A) Changing non-linear to linear
Number of ways:
a) Divide
b)Multiply
c) Log
d)Square / square root
Main goal:
𝑌 = 𝑚𝑋 + 𝑐
𝑚 & 𝑐 -> constants
𝑋 & 𝑌 -> variables

26. Stage 2: A) Changing non-linear to linear
a) Divide
𝑦 = 𝑎𝑥2
+ 𝑏𝑥
[𝑌 = 𝑚𝑋 + 𝑐]
b) Multiply
𝑦 = 𝑎𝑥 +
𝑏
𝑥
[𝑌 = 𝑚𝑋 + 𝑐]

27. Stage 2: A) Changing non-linear to linear
c) Log
[𝑌 = 𝑚𝑋 + 𝑐] [𝑌 = 𝑚𝑋 + 𝑐]
𝑦 =
𝑎 𝑥
𝑏
d) Square root
𝑦 =
4
𝑎2
(𝑥 + 𝑏)2

28. Quiz 3 ( ? minutes)
• If 𝑦/𝑥 means
𝑦
𝑥
• If x^2 means 𝑥2
• If 𝑦/(𝑥^2), means
𝑦
𝑥2
• If sqrt(𝑥) , means 𝑥
• If y*sqrt(𝑥), means y 𝑥
• If lg(𝑥), means 𝑙𝑜𝑔10 𝑥

29. Quiz 3
1. 𝑦 = 𝑝𝑥2 − 𝑞
2. 𝑦 = 𝑎𝑥3 + 𝑏𝑥2
3. 𝑦 = 𝑎𝑏 𝑥
4. 𝑦 = 𝑎 𝑥 +
𝑏
𝑥
5.
𝑎
𝑦
=
𝑏
𝑥
+ 1
Answers:
1
Question Y X m c
1 𝑦
𝑥2 / 𝑦
1
𝑥2 / 𝑥2 −𝑞 / 𝑝 𝑝 / −𝑞
2 𝑦
𝑥3 /
𝑦
𝑥2
1
𝑥
/ 𝑥 𝑏 / 𝑎 𝑎 / 𝑏
3 lg(𝑦) 𝑥 lg(𝑏) lg(𝑎)
4 𝑦
𝑥
/𝑦 𝑥 𝑥/𝑥 𝑏 / 𝑎 𝑎 / 𝑏
5 1
𝑦
1
𝑥
𝑏
𝑎
1
𝑎

30. • Hands on activity:
The table below shows the experimental values of two variables, 𝑥 and 𝑦.
It is known that 𝑥 and 𝑦 are related by the equation 𝑦 = 𝑎 𝑥 +
𝑏
𝑥
, where 𝑎 and 𝑏 are constants
a) Draw the graph of 𝑦 𝑥 against 𝑥.
b) From your graph, find
(i) the value of 𝑎 and 𝑏
(ii) the value of 𝑦 when 𝑥 = 2.4
(iii) the value of 𝑥 when 𝑦 𝑥 = 3
x 1 2 3 4 5
y 1.7 3.3 4.7 5.8 6.5
Stage 2: B) Forming equation of best fit line (P2)
Transform Table Scale
LabelDrawFind

31. Example 1:
Stage 2: B) Forming equation of best fit line (P1)

32. Example 2:
Stage 2: B) Forming equation of best fit line (P1)

33. Example 3:
Stage 2: B) Forming equation of best fit line (P1)

34. Quiz 4 ( ? minutes)
Consists of
- 3 P1 questions
- 1 P2 question

35. Quiz 4
1.
Ans: y =
𝑥−1
𝑥
2
+1

36. Quiz 4
2.
Ans: p=1000000, q=100

37. Quiz 4
3.
Ans: p=10, q=2

38. Quiz 4
4.
Ans:
b)i) ℎ ≈ 0.2, 𝑘 ≈ −0.32
ii) 𝑦 ≈ 4

39. Stage 3

40. Hands on activity:
Stage 3: Solving real life problems

41. Recap

42. Recap: Overview of the Chapter
Stage 1
(involves linear relations)
• Draw best fit lines
• Forming equation of best fit line (𝑦 = 𝑚𝑥 + 𝑐)
Stage 2
(involves non-linear relations)
• Changing non-linear to linear form
• Forming equation of best fit line (Y = 𝑚𝑋 + 𝑐)
Stage 3
(application in real life)
• Solving real life situations.

43. Recap: Steps to score
Transform Table Scale
LabelDrawFind

44. Completed Chapter 6!