Asking the right questions

How to reach every student!

November, 2007

What do we know about questioning in classrooms?There are lots of teacher questions.

It’s estimated they typically take up 30% or more of instructional time.

Used primarily to check understanding

Primarily convergent and low level

They are often “rapid-fire”.

Why would that be?

Maybe it is a management technique.

Maybe it is a way to ensure that the conversation doesn’t lead to misinformation.

Maybe it is because that’s all teachers are used to seeing.

Some of the issues

We don’t pre-plan our questions.We check rather than initiate.We focus on rules, rather than underlying concepts.

Some of the issues

We focus on details, rather than the big picture.We over-scaffold.We “bully”.

Some of the issues

We don’t always make sure the question is broad enough to allow multiple entry points.

Think about the difference between these

You saved $6.30 on a pair of jeans during a 15% off sale.How much did you pay for the jeans? ORYou saved $6.30 on a pair of jeans during a sale. What might the percent discount have been? How much might the jeans have originally cost?

Possible discounts and payments

15% off and a payment of $35.70

25% off and a payment of $18.50

10% off and a payment of $56.70

12% off and a payment of $46.20

And even more open

You saved some money on a sale of jeans.Choose your savings: $5, $7.50 or $8.20 and choose a percent as the discount.What would the original price have been?How much would you have paid?

Possible solutions

Choose a $5 savings and a 20% discount. You would have paid $20 instead of $25.

Choose an $8.20 savings and a 10% discount. You would have paid $73.60 instead of $82 (very pricey jeans).

Choose a $5 savings and a 12% discount. You would have paid $36.67 instead of $41.67.

Think about the difference between these

What is (-23)2 - (-22)3? OR

Write the number 128 as the sum or difference of powers of negative integers.

128 is

(-23)2 - (-22)3

(-1)3- (-5)3 - (-2)2 - (-2)3

[-8]2 - [-82](-10)2 + [(-2)2]3 - (-6)2

- [ (-13) + (-13) + (-13) +….. + (-13) ]

How might you open these up to a broader audience?

What is the sin of this angle ?

What is 1 ÷ 5–3?

What is the sum of the interior angle measures of a 5-sided polygon?

We asked teachers:

What sort of question would you choose to start a lesson?

Determining prerequisite knowledge

Determining if students already know what you’re planning to teach

Piquing curiosity (hooking them in)

Starting a lesson

On the next slide I’ll list some questions that could be used to start a lesson on rate in grade 9 applied.

What would (or wouldn’t) each accomplish and which would you value most?

Your choice

Write 3 sets of equivalent fractions for each: 3/4, 25/6, and 120/3. Six cookies cost $3.99 at a bakery. How much would the bakery charge for 8 cookies? The Olympic record for the men’s 100m butterfly is 51.25 s. The women’s 100m butterfly record is 56.61 s. What would you predict for the two records for the 200 m butterfly?

And with a new topic…

On the next slide I’ll list some questions that could be used to start a lesson on adding and subtracting polynomials in grade 9.

What would (or wouldn’t) each accomplish and which would you value most?

Your choice

Show me 3x2 + 2x + 1 and 2x2 + 3x +2 with your algebra tiles. Here are some algebra tiles. How might you subtract 2x2 + x – 2 from 6x2 + x + 2? When you calculate 3x2 + 2x + 1 – (2x2 – 3x +1), you start with an expression you can model with 6 tiles, subtract an expression you can model with 6 tiles, and end up with an expression you can model with 6 tiles. Does that usually happen?

How can we use questions to focus on the important ideas?

Linear Relations

Consider 2x + 4 and 4x + 2

For how many values are the expressions worth the same? How do you know?

Why is the value for which they are worth the same not far from 0?

What is that value? Why?

How do you know that both of these relations are linear?

Exponential functions

What about dividing rational numbers?

What do you think is the most important idea about dividing rational numbers?

What would you ask to get there?

Questions to End the Lesson

The last thing you hear often sticks with you.

How could you end the class on dividing rational numbers we just talked about?

How about these?

When you divide two rationals, how can you predict whether the answer will be positive or negative? How can you decide whether it will be greater than 1 or less than – 1?

Some people divide by a rational by multiplying by its reciprocal? Why does that make sense?

What would you ask instead?

How would you end these lessons?

A grade 9 applied lesson on the volume of pyramidsA grade 9 academic lesson on the meaning of m in y = mx + bA grade 10 lesson introducing sine and cosineA grade 11 university prep functions course lesson on annuitiesA grade 12 data management lesson on combinations and permutations

Creating engaging questions/tasks

Using interesting contexts

Possible sources:

The student’s personal world

Facts and figures

Make it personal

You are buying something on-line that costs $39 U.S. How much will you pay in Canadian dollars?

How can you use trig to figure out the height of the tree outside our classroom?

Estimate the number of meals you have eaten in your life. Use two different radical expressions to name that number.

Make it personal

What is a fair price for car insurance for a 16 year old male?You and three friends line up for a photo. What is the probability that you and your best friend end up standing next to each other?Make up an arithmetic sequence where the 8th term is the sum of the ages of everyone who lives in your house.

The revolving restaurant in the CN tower completes 5/6 of a revolution every hour.

If your dinner takes 2.5 hours, through how many radians have you rotated?

People and places

People facts

Most people lose about 80 scalp hairs each day.

How long would it take to lose 1012 hairs?

Records

The record for a person with the longest hair is a Chinese woman whose hair is 5.627m long. It took her 31 years, beginning at age 13, to grow it that long.

How many centimetres would her hair grow each day?

Natural phenomena

Did you know that if you pour gravel into a pile, the shape will form a cylindrical cone with a slope of about 30°?

Suppose there is room for a pile that is 90 m wide. About how tall could the pile be? About how much gravel could be in the pile?

For the curious

For the curious

For those with mathematical curiosity

Choose 3 consecutive numbers, square them and add. Divide by 3 and calculate the remainder. What happens? Why?

Making connections

Consider the expressions x2 + 2x + 1.

Evaluate it for different values of x.

What do you notice?

What does it tell you about the expression?

x2 + 2x + 1

x 1 2 3 4 5

x2 +2x + 1 4 9 16 25 36

x2 +2x + 1 (factored)

2x2 3x3 4x4 5x5 6x6

2x2 - x - 1

x 1 2 3 4 5

2x2 – x – 1 0 5 14 27 44

2x2 – x – 1 (factored)

2x7 3x9 4x11

Questions for practice

Inequalities

The common solution to both inequalities is x > 3. What could the values for the coefficients and constants be?

ax + b > c

dx2 + ex < f

Possible solutions

x > 3 and – x2 < – 9

2x > 6 and 5 – x2 < – 4

6x + 7> 25 and – 2x2 – 6x < – 36

Slope

Place the digits 0-9 into the right spots.

A line with slope []/[] goes through (9,[]) and ([],1)

A line with slope 3/4 goes through ([],2) and ([],[])

A line with slope 5/7 goes through ([],6) and ([],[])

Slope

Place the digits 0-9 into the right spots.

A line with slope 7/3 goes through (9,8) and (6,1)

A line with slope 3/4 goes through (0,2) and (4,5)

A line with slope 5/7 goes through (9,6) and (2,1)

Creating questions

Turn-it-around

One side of a right triangle is 5 cm long. What could the other side lengths be?

Possibilities

3 and 4

5 and 5√2

12 and 13

Use blanks

The tenth term of an arithmetic sequence is 6[] (between 60 and 70).

What could the sequence be?

Possible sequences

55, 56, 57, 58, 59, 60, 61, 62, 63, 64

46, 48, 50, 52, 54, 56, 58, 60, 62, 64

37, 40, 43, 46, 49, 52, 55, 58, 61, 64

28, 32, 36, 40, 44, 48, 52, 56, 60, 64

19, 24, 29, 34, 39, 44, 49, 54, 59, 64

10, 16, 22, 28, 34, 40, 46, 52, 58, 64

Relationships

The graph of y = a sin (k(x - d)) + c goes through (180°,9). What are possible values of a, k, d, and c?

Some possibilities

y = 9 sin (1(x - 90°)) + 0

y = 1 sin (1(x - 90°)) + 8

y = 3 sin (2(x - 135°)) + 6

Similarities and differences

Tell how the graph of y = x2 is like the graph of y = 2x2 + 4 and how is it different.

Tell all you can about…

Tell everything you can about the tangent of angles.

Building sentences

Build a sentence to use ..

sine, rational, 0.5, amplitude

50%, 84, 22, less

6!, divided, 1440, 2

Build a sentence to use ..

The amplitude of the graph of y = 0.5 sin x is a rational number.

50% of 84 is 22 less than 64.

If you divided 1440 by 6!, you would get 2.

What if not…

What if the slope of a line were defined to be the change in x divided by the change in y? (e.g. Would a line with a greater slope be steeper or less steep?)

Pre-planning

Create a chart of questions before the lesson, with a reminder to yourself of why you might be asking each.

Consider questions to start the lesson, probing questions to use as work progresses, helping questions for students with difficulties, and closing questions.

Pre-planning

Anticipate student responses to your questions in order to prepare your follow-up questions.

Nice article in Mathematics Teaching in the Middle School, May 2007 by Ann Wallace

How do you start?

Choose to focus on one aspect at a time and then build:

Starting a lesson

Ending on a main idea

Opening up to a broader audience

Choosing interesting contexts

Download

You can download this presentation at www.onetwoinfinity.ca

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