1. Enhancing Your Subject Knowledge
#mathsconf8

2. Types of Subject Knowledge
The Imperatives
A* / 9
Teaching Experience
Spotting Misconceptions
Explanatory
Making Sense
Anecdotal
Narrative
Links
Application

3. The Imperatives
You should as a GCSE maths teacher be able to confidently solve the GCSE questions put in front of you.
e.g. “Find the nth term of this sequence:
3, 6, 11, 18, 27”

4. Teaching Experience “Find the nth term of this sequence:
3, 6, 11, 18, 27”
Likely errors:
Not halving the second difference
Only using the first two terms and creating an arithmetic nth term
Forgetting to find the second part of the nth term

5. Basic Explanations

6. But why does it work? Are we able to make maths make sense?
Do we want it to make sense to students?
Does it all make sense to us?
You may be surprised at how little we interrogate the methods we use

7. A whistle-stop tour… Some of this you may know Some may be new
Some you may have simply forgotten
Maybe you won’t see anything enlightening at all!

8. Counting We have a base-10 number system The decimal system
Decimal means ‘tenth’
Why do we have a base-10 number system?

9. Counting Does everyone count in base-10? No
Papua New Guinea for example, has many different languages, and many different number systems!
Oksapmin is my favourite.
It has a base-27 number system

10. Counting 27 for 27 body parts
The words for each number are identical to the body parts

12. Oksapmin

13. Addition / Subtraction Algorithms
Our addition and subtraction algorithms are built around the decimal system, although they can be used for other bases with a little adjustment.

14. “Carrying and Borrowing”
‘Carrying’ is the exchange of a group of one unit to a unit of a higher power
e.g. ten units -> one ten

16. Carrying
The process of carrying therefore is simply to ensure that the answer is in decimal format
How can we promote deeper understanding of the column method?

19. -913

20. Different Bases Try using the column method for these: (base 4)

21. Actual sums in base 10 are 15+7=22 and 111 – 39 = 72

22. Vocabulary

23. Commutative Property 3 + 5 + 7 = 7 + 3 + 5 = 5 + 3 + 7 = …
2 x 3 x 4 = 3 x 4 x 2 = 4 x 2 x 3 = ...
Subtraction?
Division?

24. So what’s going on here? 10 – 3 – 2 – 1 10 – 2 – 3 – 1 10 – 1 – 2 – 3
200 ÷ 2 ÷ 5 ÷ 10
200 ÷ 10 ÷ 2 ÷ 5
200 ÷ 2 ÷ 10 ÷ 5

25. Commutative? Subtrahends are commutative Divisors are commutative
Can help with mental maths:
280 – 27 – 50
144 ÷ 36 ÷ 2

26. Why do these work?

29. Is this more straight forward?

30. Prime Factors, Factors and Multiples
List the factors of negative 20

31. Prime Factorisation

32. Venn Diagram

33. Prime Factorisation Factors of 40? 1 2 4 (2 x 2) 5 8 (2 x 2 x 2)

34. Prime Factorisation
LCM = 40 x 11

35. Prime Factorisation
LCM = 88 x 5

36. Finding Integer Roots

37. Student Question Ideas
160 as a product of its prime factors is
25 x 5
Use this information to show that
160 has 12 factors
2800 as a product of its prime factors is
24 x 52 x 7
How many square numbers are factors of 2800?

38. Exponents (Indices)

39. When we stack exponents, we work right-to-left, so the answer is 3.
If we performed it left-to-right, we are essentially replicating (ab)c rather than creating something new.

40. Why does 3-1 = 1/3 ?
Why does 3 ½ = √3 ?
Why does 30 = 1?

41. Embed these examples into your questioning
a1 ÷ a1 = a0
a0 ÷ a1 = a-1
1 ÷ a = a-1
a-1 ÷ a1 = a-2
1/a ÷ a = 1/a2 … and so on
a2 x a2 = a4
a1/2 x a1/2 = a1

42. Geometry
Can you name these shapes?

43. Ellipse An oval has no mathematical definition
An oval is (literally) egg shaped
Ovals often only have one line of symmetry
Ellipses always have two.

44. Square I of course mean parallelogram … I mean rectangle
… I mean rhombus
… I mean kite
... I mean trapezium
All of the things.

45. Rhombus Diamonds are jewellery.
If we’re going to call them diamonds, we may as well call trapeziums ‘little tables’, and kites … kites.

46. Stadium Probably called a stadium because stadiums are shaped like it.
Strange how we don’t learn the name of this shape but it crops up in GCSE questions all the time!
Also called an obround and a discorectangle!

47. Quadrilateral This shape has 4 sides.
It is called a ‘crossed quadrilateral’
A side is a straight line connection between two points.

48. How many sides does a circle have?

49. “the set of points equidistant from a fixed point”

50. The Pythagorean Theorem
Try and prove it to yourself right now.

51. There are hundreds of different proofs for the Pythagorean Theorem.
We often rely on the ‘Bride’s Chair’ proof, but neglect to actually prove it!
The Bride’s Chair proof is actually one of the most complicated proofs of the Pythagorean Theorem.

52. Slightly more straightforward proofs
By dissection

53. PythagoreanProof

54. Which of these is “The Pythagorean Proof”?

55. Aryabhatta

56. Trigonometry

58. Tan(gent)

60. Which of these cannot be a function?

61. A function matches each x value with only one y value.

62. Quadratic Sequences… 5, 13, 25, 41, … What is the nth term?
Second difference is 4
2n2 + 3?

63. Find the first 4 terms of this sequence: an2 + bn + c

64. an2 + bn + c