Differentiating Mathematics Instruction Session 4:Questioning to Evoke and Expose Thinking

Adapted from Dr. Marian Small’s presentation August, 2008

Goals for Session 4

• Become familiar with and practise opening up questions and creating parallel tasks

• Practise adapting lessons to be more inclusive

• Practise creating diagnostics

Sharing Thoughts

Stand Up/Hands Up/Pair Up

With your partner share:• something you learned that would be useful.• something with which you disagree or about

which you have doubts.• the role diagnostic assessment plays in

differentiating learning and planning for instruction.

Differentiating instruction… rather than consolidation

Your answer is….?

A Problem:

A graph goes through the point (1,0). What could it be?

• What makes this an accessible or inclusive question?

Opening up Questions

A conventional question:

You saved $6 on a pair of jeans during a 15% off sale. How much did you pay?

Opening up the question…

You saved $6 on a pair of jeans during a sale.

What might the percent off have been?

How much might you have paid?

Another Example

You saved some money on a jeans sale.

• Choose an amount you saved: $5.00, $7.50, or $8.20.

• Choose a discount percent.• What would you pay?

Another Example

A conventional question:

What is 52 + 62 + 33?

Opening up:

Represent 88 as the sum of powers.

How can you open up these questions?

Add: 3/8 + 2/5. A line goes through (2,6) and has a

slope of -3.

What is the equation?

Graph y = 2(3x - 4)2 + 8.

Add the first 40 terms of

3, 7, 11, 15, 19,…

Using Parallel Tasks

Provide 2-3 similar tasks designed to meet different students’ needs, but make sense to discuss together.

Parallel Questions

• Task A:

1/3 of a number is 24. What is the number?• Task B:

2/3 of a number is 24. What is the number?• Task C:

40% of a number is 24. What is the number?

Reflection Questions:How do you know the number is more than 24?Is the number more than double 24?How did you figure out your number?

Parallel Questions• Task A:

One electrician charges an automatic fee of $35 and an hourly fee of $45. Another electrician charges no automatic fee but an hourly fee of $85. What would each company charge for a 40 minute service call?

• Task B:

An electrician charges no automatic fee but an hourly fee of $75. How much would she charge for a 40 minute service call?

Reflection Questions:How do you know the charge would be more than $40?How did you figure out the fee?

Parallel Questions• Task 1:

Find two numbers where:- the sum of both numbers divided by 4 is 3.- two times the difference of the two numbers is -36.

• Task 2:Solve: (2x + y) / 4 = 3 and 2(x – y) = -36

Reflection Questions:

How did you use the first piece of information?

The second piece?

How did you know the numbers could not both be negative?

Making it more inclusiveUnit 4: Day 1: Going Around the Curve (Part 1) Grade 10

Applied

75 min

Math Learning GoalsCollect data that can be modelled by a quadratic relation, using connecting cubes, and calculate first and second differences.Draw the curve of best fit on chart paper.Realize that the shape of the graphs are curves rather than lines.

Materialslinking cubeschart papergrid chart paperBLM 4.1.1, 4.1.2, 4.1.3, 4.1.4, 4.1.5

AssessmentOpportunities

Minds On…

Groups of 3 PlacematStudents complete a placemat with the phrase “linear relationship” in the centre. They reflect on everything they recall about the characteristics of linear relations in their own section and share their results within their groups. They write the characteristics they agree upon in the centre. Repeat the process using the phrase “non-linear relationship.”Summarize characteristics on chart paper and post.Recall that first difference implies a linear relationship.Show students how to find second differences, using an example.Curriculum Expectations/Demonstration/Observation/Checklist: Observe what characteristics students recall about linear and non-linear relations.

These experiments provide “clean data” from which a constant second difference can be determined.

Action! Groups of 3 ExperimentsEach group completes an assigned experiment (BLMs 4.1.1–4.1.5). They record their data in a table on chart paper, and plot the data on grid chart paper.Groups who do not complete their experiment can be given some time during the next lesson.Learning Skills/Teamwork/Observation/Checklist: Observe how well students work as a productive team to complete the task.

Consolidate Debrief

Groups of 3 SharingGroups who complete the activity post their graphs and tables of values and plan how they will present their work to the class.

Concept PracticeHome Activity or Further Classroom ConsolidationComplete the practice questions.

Provide students with appropriate practice questions.

Making it more inclusive

Another example

Bill has two part-time jobs. At the store he earns $9/h. At the recreation centre he earns $12/h. He would like to earn $240 to purchase a DVD player.

What is the fewest number of hours he needs to work to save this amount of money?

Another exampleA farmer wants to build an enclosure for pigs,

chickens and ducks. He has 50m of available fencing to build three identical, adjacent enclosures.

a) Write an equation to represent the amount of fencing required.

b) Rearrange your equation to isolate one of the variables.

c) Graph the relationship.d) Identify possible dimensions for the farmer’s

enclosure.

You try …

• Form grade groups.

• Work together with a TIPS or textbook lesson and make it more inclusive.

• Include one suggestion for differentiating assessment.

• Consider YOUR four students.

• Post your work for sharing with the group.

Making lessons more accessible

Open up tasksCreate parallel

questions

Interview Paper-and-pencil

items Graffiti exercise Anticipation guide

Home Activity

Journal prompt:• How did you differentiate your lesson?

• What was the hardest thing for you to deal with?

• How did you consider your four students?• How much did it help to do it with

colleagues?