1. science
2. mathematics
3. algebra

Plus à penser
Marian Small
April, 2015
Ordre du jour
•  répondre à des questions à réponse construite
•  modification d'une culture de le faire à une
culture de réflexion
Let’s start with understanding
questions
•  Connaissance ≠ compréhension
•  Teachers need help creating compréhension
questions.
For example
Grade 7:
•  additionner et soustraire dans divers
contextes des fractions positives en utilisant
une variété de stratégies
What might a comprehension question be?
Maybe
•  a/b + c/d = []/15.
•  Think of 6 different possibilities for b and d.
•  Show that your values make sense.
I make sure they know
•  6 PAIRS of numbers
•  No pairs are the same
•  You proved each pair worked.
A good answer might be
• 
• 
• 
• 
• 
3 and 5 since 1/3 + 1/5 = 8/15
15 and 3 since 1/15 + 1/3 = 6/15
15 and 5 since 1/15 + 1/5 = 4/15
15 and 30 since 1/15 + 2/30 = 2/15
3 and 6 since 1/3 + 2/6 = 10/15
Grade 8
•  reconnaître en quoi l’ajout ou la suppression
d’une ou de plusieurs valeurs affecte les
mesures de tendance centrale
A comprehension question might be
•  You sent out surveys and your percent
responses over five different weeks was:
42%
18%
63%
22%
35%
• Will the percent go up a lot, up a little, down a
lot or down a little if you didn’t include the
18% ? Predict without actually calculating.
You need to
•  Predict before you do it and tell why . (The
prediction can’t happen afterwards.)
•  The “why” can’t refer to the answer.
•  The “why” explains either how means are
calculated OR what they mean.
Possible solution
•  Since 18% is the lowest value and the average
is always between the lowest and highest, the
average will go up.
•  If the values were 40, 20, 60, 20, and 40, the
average would be more than 30ish, so getting
ready of a 20ish number will help some, but
not a lot. Grade 9
•  expliquer les premières lois des exposants
I might ask
• 
• 
• 
• 
ab ÷ cd = 2-3.
What could a, b, c and d be?
List at least two possibilities where a = c.
List at least two possibilities where a ≠ c.
You need to
•  Use the same base twice.
•  Use different bases twice.
•  Show your calculations.
Maybe
• 
• 
• 
• 
25 ÷ 28 = 2-3
210 ÷ 213 = 2-3
25 ÷ 44 = 2-3
83 ÷ 46 = 2-3
Grade 10
•  comparer les caractéristiques d’une
fonction affine et d’une fonction du
second degré d’après
….
I might ask
•  Fill in missing values to make this a table of
values for a linear. Then a quadratic
x y
x y
1 1
1 1
2 10 2 ? 3
?
3 ? 4 40 4 40
You need to •  Put in values for the three missing numbers.
•  You need to explain why the values make it
quadratic on the left and linear on the right. Solution
•  Fill in missing values to make this a table of
values for a linear. Then a quadratic
x y
x y
1 1
1 1
2 10 2 14
3 23 3 27
4 40 4 40
Solution
•  I am right because for the linear, the values of
y keep going up by 13.
•  I am right because the values of the ys go up
by 9, then 13, then 17, so the second
differences are all 4.
Application questions
•  estimer et calculer le volume de prismes
droits dans divers contextes.
I might ask
•  A prism has this base:
12 cm 10 cm •  What is the height of the prism if the volume
is 100 cm3?
You need to •  Show all the necessary dimensions of the
prism.
•  Show how you calculated the height.
•  Verify that your height is correct.
Data
•  lire, décrire et interpréter des données
présentées dans un diagramme circulaire
et utiliser ces données pour résoudre des
problèmes.
I might ask..
•  People were surveyed about the summer
holidays.
•  The four responses were Ontario, elsewhere
in Canada, the U.S., and somewhere else.
•  If 295 people said Ontario, how many said the
U.S.?
Holidays
Somew
here else 12% Other 15% Canada 14% U.S. Ontario 295 You need to •  Show how you figured out the percent for
Ontario.
•  Show how you used the 14% percent for the
U.S. to help you figure out the U.S. number.
Grade 9
•  déterminer la dimension manquante d’une
figure plane d’une aire ou d’un périmètre
donnés, y compris les situations faisant
appel aux valeurs exactes
Thinking question
•  I might ask:
•  The area of a circle with a circumference of
30π cm is identical to the area of a square.
•  What is the side length of that square?
You need to •  Show all of your work.
•  Explain why your answer makes sense without
just doing the calculations again.
Solution
•  30π = π x d, so d = 30, so r = 15.
•  Area = πr2 = 225π
•  The square’s area is 225π, so the side length
is √225π = 15√π is about 26.59 cm.
Solution
•  That makes sense since 26.59 is just a little
less than 30.
•  A 30 cm square would be a little too much
area, so going down a little seems right.
So let’s talk
•  What do you think gets in students’ ways the
most when answering constructed response
questions?
•  How do we get them over some of these
problems?
A culture of thinking
•  For most of us, math is mostly about being
“busy” doing.
•  Instead, shouldn’t students be busy thinking?
Talking about decisions
•  Math is clearly full of problem solving
•  There is often decision making in how to
proceed on a problem.
•  We can, and should, bring in decision making
into interpretation too.
For example…
•  Which bank account grew the most last year?
Ben’s Lea’s January 1 $1000 $100 Dec 31 $1200 $250 Or…
A car goes 280 km in 3 hours.
Which would be easiest for you to figure out?
How far it goes in
•  9 hours?
1 hour?
1.5 hours?
Or…
•  A shape does not have much area but it has
lots of perimeter.
•  What might it look like?
Maybe
Did you know?
•  A medium pizza is about 730 cm2.
•  In 2013/2014, in Ontario, there were
1 352 965 K – 8 students and 662 458
secondary students.
So I could ask…
•  Estimate the number of square centimetres of
pizza that all of the kids in Ontario eat in one
week.
Or I could ask…
You are going to spin a spinner.
•  You are twice as likely to get red as blue.
•  You are half as likely to get blue as green.
•  What could the spinner look like?
Red Green Green Red Blue Pink Red Pink Red Pink Blue Green Green Or I could ask…
•  for both reasonable and unreasonable possible
equations for the line connecting these two
points.
Asking the right questions
•  This is the heart of the issue.
•  We need to ask questions that encourage or
even demand critical thinking behaviours.
•  You could make it the “normal” way you
teach.
Compare
•  What is 10% of 300?
TO
•  Is 10% a lot or not? OR
•  If you know 10% of a number, what other
percents do you know?
Compare
•  What is 2/3 ÷ 1/4?
TO
•  How could you use a 1/4 cup measure to
measure out thirds of a cup?
Compare
•  What is the solution to 3/4 x – 2 = 5/8 x + 9?
TO
•  Do equations with fractions in them usually
have whole number solutions or fraction
solutions?
OR
•  What percentage would you call a LOT of a
whole?
OR
•  Someone says that ½ of the design is yellow
and someone else says that 2/3 is yellow.
•  What do you think?
OR
•  A number is a bit more than 3.8 fives. What
might it be?
OR
•  Would you use the same strategy or different
strategies to decide which number in each pair
is greater? Explain.
•  Pair 1: 3π/2 and 5π/8
•  Pair 2: √20 and 3√50
OR
•  I subtracted a fraction from another one and
the answer was less than but close to what I
subtracted. What could the fractions have
been?
•  Maybe 1/2 – 51/100
OR
•  Is it necessary to learn the formula for the
area of a trapezoid or is that a waste of time if
you know some other formulas?
OR
•  Is there value in thinking about what
measurements of this prism are IRRELEVANT
when calculating its volume?
OR
•  What picture could you draw to show why it
makes sense that π is around 3?
Maybe
2r 2r OR
Which of these doesn’t belong and why?
3x – 4 = 2x – 7
6/x = –2
2x = –8
5x + 8 = –7
OR
•  A rectangle has a length triple
the width.
•  Another rectangle with the
same perimeter has a length
five times the width.
• Which rectangle has a greater
area? Are you sure? Why?
OR
•  You know that a line goes through the point
(1, -1) and that it slants down to the right.
•  Tell other things you know about that line.
Maybe
• 
• 
• 
• 
• 
The y-intercept is greater than -1.
The x-intercept is less than 1.
The slope is negative.
The point (1, -3) is not on the line.
The line goes through only one of quadrants I
and III.
OR
•  Graph a number of lines of the form y = mx +
m.
•  What do you notice? •  Why does it make sense?
•  What other group of lines would act in a
similar way?
OR
Which of the following is most like
y = 6x2 + 5x + 1?
•  y = 7x2 + 5x + 1
•  y = 6x2 + 6x + 1
•  y = 6x2 + 5x + 2
Or we might ask
Is it more useful to:
•  know a shape’s area or perimeter?
•  be able to solve a system of equations
graphically or algebraically?
Or we might ask
Is it easier to construct:
•  An equilateral triangle?
•  An isosceles triangle with one angle of 40°?
•  A scalene triangle with side lengths of 4 cm, 5
cm and 6 cm
•  A right triangle with a 35° angle
Many more examples
•  What might be reasonable values for the dots
on the number line?
1000
20 000 2x 5x Fractions
•  Is it usually or always or seldom true that if a
and b are closer together than c and d, that
means a/b > c/d?
Is it true????
•  If you can add and subtract integers, it should
be easy to add and subtract polynomials.
Is it true????
•  That two negatives make a positive?
Is it true????
•  That if you know a relationship between any
two of L, W and P for a rectangle, you know
the relationship between any other two of
them?
Or
What other numbers (or expressions) go with
the given ones?
•  114, 99, 57
•  59, 2, 101
•  3x + 8, 16 – x, 10 + x2
Asking for thinking
•  is more inviting.
•  tells students that you have faith in their
abilities.
•  leads to sense making, and that is always a
good thing.
Reflection time
•  Spend some time talking with your colleagues
about how much you see critical thinking (or
thinking) questions play in the curriculum.
•  Anticipate their concerns about them and
think about how you would respond.
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