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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? OR
You 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
pyramids
A grade 9 academic lesson on the meaning of
m in y = mx + b
A grade 10 lesson introducing sine and
cosine
A grade 11 university prep functions course
lesson on annuities
A 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.
People and places
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 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
2x7
3x9 4x11
2x2 – x – 1
(factored)
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
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www.onetwoinfinity.ca
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