Transcript Michaels corporsate master

MMP Day 2008
Teaching algebra for
meaning
Michael Drake
Algebra
• What is algebra?
Think…
Discuss in pairs…
• If you ask parents (or Y11
students), what would they say?
How do we use letters in
mathematics?
• Have a chat about it…
• Tell me what you have come up with…
How do we use letters in
mathematics?
1)
•
A letter can be used to name
something
In the formula for the area of a
rectangle, the base of the rectangle
is often named b
The development of formulae
Area = base × height
A=b×h
3 cm
A = bh
5 cm
A=5×3
A = 15cm2
How do we use letters in
mathematics?
2)
A letter can be used as a
variable
• This happens in patterning
Developing patterning
• Write the next three terms of the
sequence 3, 6, 9, 12, …
1 3

26
39
Down
rule
4  12

Across rule 
Write a sentence
explaining how to
get from one term
to the next. Start
with “To get the
new term …”
• Write a sentence for the across rule.
Start it with “the number of matches is
found by taking the number of triangles
and…”
Number of
triangles
Number of
matches
1

3
2

6
3

9
4

12
How many terms?
• What is the next term of this
sequence?
1, 2, _
• What is the next term of this
sequence?
1, 2, 4, _
Differentiating…
• Write the next four terms of the
sequence
10 507, 10 627, 10 747, 10 867,
• Write a sentence that explains how
to get from one term in the sequence
to the next
• Can we write this more briefly, and
still know what we mean?
How do we use letters in
mathematics?
3)
A letter can be used to stand for a
specific unknown number that needs
to be found
In a triangle, x is often used for the
angle students need to find
• This happens when we solve equations
When do we need to use letters
to solve problems?
• Jemima has some sweets. She eats 5
and has 1 left. How many did she start
out with?
• Rewi has 47 super 14 rugby cards. 19 of
them are repeats that he can swap. How
many different cards does he have?
• In numeracy schools, student may have
seen an algebraic form of recording
problems since Year 1…
6 +  = 10
…and may have even learned to record
problems in this format
How do we use letters in
mathematics?
4)
A letter can be used as a variable
that can take a variety of possible
values
(This also happens with equations)
+
= 10
How do we use letters in
mathematics?
5)
A letter is really a number
c = speed of light
e
π
Or if you are a student, in
substitution a = 1…
How do we use letters in
mathematics?
6)
In the equation y = 2x + 1, x can
be any number, a whole number,
a fraction, integer or decimal
7)
In y = mx +c, m and c act as
parameters
Lessons
• Letters mean different things in
different situations
• Algebra is not something that should be
taught in a single block…
• Algebra often makes sense when it
arises from a context…
The algebra domains
Number
property
generalisation
Patterns and
relationships
Equations
Formula
development
Introduction to the conventions of algebra
• Time to do some mathematics…
Starters
• Think of a number
• Add 3 to it
• Double it
• Subtract two
• Halve it
• Subtract the number you started with
• Add 6
• Multiply the number by 3
Think of a number…
• How do we write down that we are
thinking of a number – without giving
away the number we are thinking of?
• What would one more than the number
look like?
• What would 3 less than the number look
like?
• What would double the number
look like?
Starters
What is?
33
24
52
64
66
57
77
86
102
9  11
Generally generalising
• 6+6+6+6+6+6+6=76
• Explain why the sentence is true
• Give me another example
Thinking time: discuss in pairs
• How can we show that this always
works – no matter the number?
• Did we always have to use 7 of them?
• What other generalisations could we
have made?
• How can we show this always works –
regardless of the number of things we
are multiplying?
The development of algebraic
thinking
First order
abstraction
Using and
exploring number
properties
Generalising
number
properties
Second order
abstraction
Generalising
algebra
properties
What is algebra?
• What are your answers now?
• Generalised number
A way of describing trends and patterns
we find in sets of similar problems
• The structure of the number system
A way of looking behind (underneath) a
set of number problems to see what is
really going on
Write me some…
• Write me a story problem which has the
answer x = 4. Try using a context that
tells us something about your culture or
your family
Sharing time…
• So how does this algebra work?
• Why does it help us to work out what
is going on?
Teaching volume
• How do we each teach volume?
• How might these resources be useful?
• How can we use learning about solids
as a context for algebra development?
• What prior knowledge is needed for
this approach to be useful?
What is 4  4?
• What about 16  16?
• 250  250?
• Does this always work?
• Explain why this works using a drawing
or some equipment from the box
• Can you write this as a rule in words?
• Write a sentence with symbols to
show your rule