Designing tasks so that all learners can engage with hard

maths

Anne Watson

Toulouse, 2010

Decimals!

• 10% of 232.3

• 20% of 234.6 or 0.23 !!

Teaching context

• All learners generalise all the time• It is the teacher’s role to organise

experience• It is the learners’ role to make sense of

experience

Sorting

2x + 1 3x – 3 2x – 5

x + 1 -x – 5 x – 3

3x + 3 3x – 1 -2x + 1

-x + 2 x + 2 x - 2

Sorting processes

• Sort into two groups – not necessarily equal in size

• Describe the two groups• Now sort the biggest pile into two groups• Describe these two groups• Make a new example for the smallest

groups• Choose one to get rid of which would

make the sorting task different

Sorting grids

+ve y-intercept

-ve y-intercept

Goes through origin

+ve gradient

-ve gradient

Zero gradient

Make your own

• In topics you are currently teaching, what examples could usefully be sorted according to two categories?

Comparing

• In what ways are these pairs the same, and in what ways are they different?

• 4x + 8 and 4(x + 2)• Rectangles and parallelograms

• Which is bigger?• 5/6 or 7/9• A 4 centimetre square or 4 square centimetres

Make your own

• Find two very ‘similar’ things in a topic you are currently teaching which can be usefully compared

• Find two very different things which can be usefully compared

Ordering

• Put these in increasing order:

6√2 4√3 2√8 2√9 9 4√4

Make your own

• What calculations do your students need to practise? Can you construct examples so that the size of the answers is interesting?

Arguing about

• Anne says that when a percentage goes down, the actual number goes down

- Is this always, sometimes or never true?

• John says that when you square a number, the result is always bigger than the number you started with

- Is this always, sometimes or never true?

Make your own

• What assumptions do your students make? What statements could they argue about?

Characterising

• Which multiples of 3 are also square numbers?

• Which quadratic curves go through (0,0)?

• What cubics have coincident roots?

• What angles have interesting trig ratios?

Make your own

• By asking non-standard questions about standard topics, can you get students to practise, and fiddle around with ideas, but with a further purpose?

Construct a ... polygon with

1 2 3 4 5 6

1

2

3

4

5

6

pairs of parallel sides

right

ang

les

Constructing

• Unexpected objects• Unusual objects• Impossible objects

– Brings students face-to-face with the limitations and possibilities of concepts

Make your own

Enlargement (1)

Enlargement (2)

Enlargement (3)

Enlargement (4)

Make your own

• What techniques are you currently teaching? Can you lead your students to understand when they need to give up intuitive methods and adopt more powerful techniques?