Download Enhancing Your Subject Knowledge

Enhancing Your Subject Knowledge
#mathsconf8
Types of Subject Knowledge
The
Imperatives
A* / 9
Teaching
Experience
Spotting
Misconceptions
Explanatory
Making
Sense
Anecdotal
Narrative
Links
Application
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”
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
Basic Explanations
But why does it work?
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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
A whistle-stop tour…
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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!
Counting
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We have a base-10 number system
The decimal system
Decimal means ‘tenth’
Why do we have a base-10 number system?
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
Counting
• 27 for 27 body parts
• The words for each number are identical to
the body parts
OKSAPMIN
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.
“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
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?
-913
Different Bases
• Try using the column method for these:
(base 4)
(base 5)
Vocabulary
Commutative Property
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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?
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
Commutative?
• Subtrahends are commutative
• Divisors are commutative
• Can help with mental maths:
• 280 – 27 – 50
• 144 ÷ 36 ÷ 2
Why do these work?
Is this more straight forward?
Prime Factors, Factors and Multiples
• List the factors of negative 20
Prime Factorisation
Venn Diagram
Prime Factorisation
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Factors of 40?
1
2
4 (2 x 2)
5
8 (2 x 2 x 2)
10 (2 x 5)
20 (2 x 2 x 5)
40 (2 x 2 x 2 x 5)
Prime Factorisation
LCM = 40 x 11
Prime Factorisation
LCM = 88 x 5
Finding Integer Roots
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
2 4 x 52 x 7
How many square numbers are factors of 2800?
Exponents (Indices)
• When we stack exponents, we work right-toleft, so the answer is 3.
• If we performed it left-to-right, we are
essentially replicating (ab)c rather than
creating something new.
• Why does 3-1 = 1/3 ?
• Why does 3 ½ = √3 ?
• Why does 30 = 1?
Embed these examples into your
questioning
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a 1 ÷ a1 = a 0
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
Geometry
• Can you name these shapes?
Ellipse
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An oval has no mathematical definition
An oval is (literally) egg shaped
Ovals often only have one line of symmetry
Ellipses always have two.
Square
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I of course mean parallelogram
… I mean rectangle
… I mean rhombus
… I mean kite
... I mean trapezium
All of the things.
Rhombus
• Diamonds are jewellery.
• If we’re going to call them diamonds, we may
as well call trapeziums ‘little tables’, and kites
… kites.
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!
Quadrilateral
• This shape has 4 sides.
• It is called a ‘crossed quadrilateral’
• A side is a straight line connection between
two points.
How many sides does a circle have?
“the set of points equidistant from a fixed point”
The Pythagorean Theorem
• Try and prove it to yourself right now.
• 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.
Slightly more straightforward proofs
• By dissection
PythagoreanProof
Which of these is “The Pythagorean
Proof”?
Aryabhatta
Trigonometry
Tan(gent)
Which of these cannot be a function?
A function matches each x value with only one y
value.
Quadratic Sequences…
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5, 13, 25, 41, …
What is the nth term?
Second difference is 4
2n2 + 3?
Find the first 4 terms of this sequence:
an2 + bn + c
an2 + bn + c