Nine task based strategies for dealing with diversity while offering

experiences covering common content

Common to all 9 approaches• Building classroom community• Task based, considering the trajectory of tasks

(what comes next!)• Explicit pedagogies• Different ways of solving the tasks, and the

different approaches are themselves educative• Representing solutions in different ways is

both engaging and important mathematically

What these approaches are not!

• Asking questions that are so easy that everyone can do them

• Setting up groups that allow some students to hide

• Excessive repetition (of course, some is needed)

• …

Strategy 1

• Asking questions with multiple entry points and multiple exit points

• Such questions will nearly always be open-ended

Write a sentence with 5 words, with the mean of the number of letters in the words being 4. Do not use any words of 4 letters.

Draw some rectangles with a perimeter of 20 cm. Work out the area of each of your rectangles.

A set of 36 cubes is arranged to form a rectangular prism.

What might the rectangular prism look like?

What is the surface area of your prisms?

Strategy 2

• Using enabling and extending prompts• These apply to any type of challenging task

Suppose we posed this task:

Seven people went fishing. The mean number of fish they caught was 5, the median was 4 and the mode was 3. How many fish might each of the people have caught? (Give at least 3 answers)

Some enabling prompts

• Seven people went fishing. The median number of fish caught was 4. How many fish might each of the people have caught?

• Seven people went fishing. The mode number of fish they caught was 3. How many fish might each of the people have caught?

What are enabling prompts?• Enabling prompts can involve slightly varying an

aspect of the task demand, such as – the form of representation, – the size of the numbers, or – the number of steps,

so that a student experiencing difficulty, if successful, can proceed with the original task.

• This approach can be contrasted with the more common requirement that such students – listen to additional explanations; or – pursue goals substantially different from the rest of the

class.

Extending prompt

• Seven people went fishing. The mean number of fish they caught was 5, the median was 4, the mode was 3, and the range is 6. How many fish might each of the people have caught? (Give all possible answers)

What might be enabling and extending prompts for these tasks?

A brick weighs the same as 3 kg plus half a brick. What does the brick weigh?

Represent your answer in two different ways, one of which involves drawings

A brick weighs the same as 3 kg plus half a brick. What does the brick weigh?

3 kg

SA coaches day 5 September 12

A brick weighs the same as 3 kg plus half a brick. What does the brick weigh?

3 kg

SA coaches day 5 September 12

3 kg

• I used 1 metre of string to tie up this box. The bow takes 300 mm. What might be the dimensions of the box?

• Race to 5x + 5y

Strategy 3

• Realistic investigations that are multi faceted, take time and are meaningful for collaborative group work

Which fits better: a round peg in a square hole or a square peg in a

round hole?

1mm of rain on 1 sq m of roof is 1 L ofwater.Design a tank for this building that captures all of the rain that usually falls this month.

Design a cola can that has the same volume as the current cans but requires less aluminium

Calculate the volume of the cylinder that is made by bending an A4 sheet of paper vertically.

Now calculate the volume if the sheet was horizontal.

How many people can we fit into this room?

• A chameleon has a tongue that is half as long as its body ...

• … how long would your tongue be if you were a chameleon?

In what ways are the arch at St Louis and the Sydney Harbour Bridge similar to or different from a parabola, circle, ellipse, hyperbola, sine curve, catenary?

The school is considering building an arch over the front gate. What curve would you recommend? Write the equation (using actual measurements) for your curve.

Strategy 4:

• Using a text book in different ways

NB September 12 difference

Some examples

• Read the last question first. What do you need to learn to be able to do that question? Which of the earlier questions look like they might help?

• Work in pairs. One of you does the odd questions. The other does the even ones. Then each of you can explain your working to the other.

• In what ways are the questions different from each other?

Strategy 5

• Asking questions that emphasise connections and are challenging (but at the right level for the curriculum)

• At the end of the season, the coach noticed that the mean and median of the number of goals kicked by the 20 players was 10. He also noticed that ¼ of the players kicked less than 5 goals, ¼ of the players kicked 5 or more but less than 10, ¼ of the players kicked 10 or more but less than 15, and ¼ of the players kicked 16 or more. How many goals might each of the players have kicked?

CEOM 2012

To give us something to discuss

• On a train, the probability that a passenger has a backpack is 0.6, and the probability that a passenger as an MP3 player is 0.7.

• How many passengers might be on the train?• How many passengers might have both a

backpack and an MP3 player?• What is the range of possible answers for this?• Represent each of your solutions in two different

ways.

Some sample questions

• Find two objects with the same mass but different volumes

• Draw some closed shapes with 6 right angles• Draw a line 1 m long on this page• The perimeter of a rectangle is 20 cm. What might be

the area?• Draw (on squared paper) as many different triangles as

you can with an area of six square units• A number has been rounded off to 5.3. What might be

the number?

Strategy 6

• Creating task sequences where there is no expectation that all students can do the first one(s), but for which all can do the last one(s)

A “constructing” task

• In a tank there are 200 fish, 99% of which are guppies.

• How many guppies do I need to take out to end up with 98% guppies?

A consolidating task

• At the football there are 50 000 spectators, 55% of whom are Collingwood supporters.

• How many Collingwood supporters do I need to expel from the stadium to end up with 50% Collingwood supporters?

A possible sequence of tasks..

Write a sentence with 5 words, with the mean of the number of letters in the words being 4. Do not use any words of 4 letters.

Seven people went fishing. The mean number of fish they caught was 5, the median was 4 and the mode was 3. How many fish might each of the people have caught? (Give at least 3 answers)

• At the end of the season, the coach noticed that the mean and median of the number of goals kicked by the 20 players was 10. He also noticed that ¼ of the players kicked less than 5 goals, ¼ of the players kicked 5 or more but less than 10, ¼ of the players kicked 10 or more but less than 15, and ¼ of the players kicked 16 or more. How many goals might each of the players have kicked?

After the brick task

A box weighs the same as 12 kg plus a quarter of a box. What does the box weigh?(do this in two different ways)

Strategy 7

• Creating task sequences that proceed from simple (to engage students in the task) and which progressively become more difficult

Getting to schoolJohn Hindmarsh

20 km

How much does it cost John to get to work and back home again?

Assume that it costs $2 per km for the full costs of the

journey

Getting to schoolJohn Hindmarsh

73 km

How much does it cost John to get to work and back home again?

Assume that it costs $1.37 per km for

the full costs of the journey

Getting to school

John Mark

5 kmHindmarsh

20 km

How much should Mark give John if he picks him up and takes him home?

Assume that it costs $2 per km

for the full costs

Getting to school

John Mark

17 kmHindmarsh

57 km

How much should Mark give John if he picks him up and takes him home?

Assume that it costs $1.37 per km for the full

costs

Getting to school

John Mark

z kmHindmarsh

x km

How much should Mark give John if he picks him up and takes him home?

Assume that it costs $y per km

for the full costs

Getting to school

John Mark

5 kmHindmarsh

20 km

How much should Mark and Susan give John if he picks them up and takes them home?

Assume that it costs $2 per km

for the full costs

10 km

Susan

Getting to school

John Mark

15 kmHindmarsh

72 km

How much should Mark and Susan each give John if he picks them up and takes them home?

Assume that it costs $1.35 per km for the full

costs

23 km

Susan

Strategy 8:

• Games that are a mix of skill and luck

In turn, players roll a 10 sided die (numbered 0 to 9)

and, after each roll, write the number rolled in one of

the rectangles on their own board that looks like

. ÷ 0 .

The winner has the answer closest to 100 (for example).

Strategy 9

• “Create your own experience” activities on which subsequent questions might build

10 20 30 40 50 60 70 80 90 100

110 120 130 140 150 160 170 180 190 200

210 220 230 240 250 260 270 280 290 300

310 320 330 340 350 360 370 380 390 400

410 420 430 440 450 460 470 480 490 500

510 520 530 540 550 560 570 580 590 600

610 620 630 640 650 660 670 680 690 700

710 720 730 740 750 760 770 780 790 800

10 20 30 40 50 60 70 80 90 100

110 120 130 160 170 180 190 200

210 220 230 240 250 260 270 280 290 300

310 320 330 340 350 370 380 390 400

410 420 430 440 450 470 480 490 500

510 520 530 540 550 560 570 590 600

610 630 640 650 660 670 680 690 700

710 720 730 740 750 760 770 780 790 800

Which numbers are missing?

10 20 30 40 50 60 70 80 90 100

110 120 140 140 150 160 170 180 190 200

210 220 230 240 250 260 270 280 290 300

310 320 330 340 350 370 370 380 390 400

410 420 430 440 450 460 470 480 490 500

510 520 530 530 550 560 570 580 580 600

610 620 630 640 650 660 670 680 690 700

710 720 730 740 750 760 770 780 790 800

Which ones are wrong?

100 200 300 400 500 600 700 800 900 1000

1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

2100 2200 2300 2400 2500 2600 2700 2800 2900 3000

3100 3200 3300 3400 3500 3600 3700 3800 3900 4000

4100 4200 4300 4400 4500 4600 4700 4800 4900 5000

5100 5200 5300 5400 5500 5600 5700 5800 5900 6000

6100 6200 6300 6400 6500 6600 6700 6800 6900 7000

7100 7200 7300 7400 7500 7600 7700 7800 7900 8000

100 200 300 400 500 600 700 800 900 1000

1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

2100 2200 2300 2400 2500 2500 2700 2800 2900 3000

3100 3200 3300 3400 3500 3600 3700 3800 3900 4000

4100 4200 4300 4400 4500 4600 4700 4900 4900 5000

5100 5200 5300 5400 5500 5600 5700 5800 5900 6000

6100 6200 6200 6400 6500 6600 6700 6800 6900 7000

7100 7200 7300 7400 7500 7600 6700 7800 7900 8000

CEO cohort 6 2012

Which numbers are wrong?

10 20 30 40 50 60 70 80 90 100

110 120 130 140 150 160 170 180 190 200

210 220 230 240 250 260 270 280 290 300

310 320 330 340 350 360 370 380 390 400

410 420 430 440 450 460 470 480 490 500

510 520 530 540 550 560 570 580 590 600

610 620 630 640 650 660 670 680 690 700

710 720 730 740 750 760 770 780 790 800

What is the hardest question you could use this to answer?