Michael Drake

MMP Day 2008

Teaching algebra for meaning

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 × heightA = b × hA = bh3 cm

5 cm A = 5 × 3A = 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 32 63 94 12

Down rule

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…”

1 3

2 6

3 9

4 12

Number of triangles

Number of matches

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 foundIn 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 numberc = speed of lighteπ

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

Patterns and

relationships

Formula developmen

t

EquationsNumber property

generalisation

Introduction to the conventions of algebra

•Time to do some mathematics…

• Think of a numberStarters

• 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?

What is?Starters

3 352

6 6

7 7

102

2 4

5 7

6 4

8 6

9 11

Generally generalising• 6 + 6 + 6 + 6 + 6 + 6 + 6 = 7 6

• How can we show that this always works – no matter the number?

• Give me another example• Explain why the sentence is

true

Thinking time: discuss in pairs

• 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

Using and exploring number

properties

Generalising number properties

Generalising algebra properties

First order abstraction

Second order abstraction

• What are your answers now?What is algebra?

• Generalised numberA way of describing trends and patterns we find in sets of similar problems

• The structure of the number systemA 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?

• Can you write this as a rule in words?

• Explain why this works using a drawing or some equipment from the box

• Write a sentence with symbols to show your rule