La réflexion sur l'enseignement
des mathématiques- 9eannée
Marian Small
avril2015
Sudbury, ON
Agenda
•  Parallel Tasks
•  Asking Better Questions
2
Parallel Tasks
•  Sometimes the task you planned is just a
little too much for some kids.
•  But if you changed it a little, it would be
fine.
•  One way to do this is using parallel tasks.
For example:
•  Choice 1: A jacket is on sale for 40% off. It
costs $96. What was the original price?
•  Choice 2: A jacket is on sale for 40% off. It
costs $92. What was the original price?
Common questions
•  Before you calculate- Was the original
price more or less than $100? Why?
•  Before you calculate- More or less than
$200? Why?
•  What kind of picture could you draw to
help you figure out the answer?
•  How did you figure out the answer?
Picture
0%
10%
20%
30%
40%
50%
$96
$92
60%
70%
80%
90% 100%
Picture
0%
10%
$0 $16
20%
$32
$0 $15.33
30%
40%
$48 $64
50%
$80
$92
$96
60%
70%
80%
90% 100%
$112 $128 $144 $160
$153.33
Or it might be:
•  Choice 1: The solution to an equation of
the form []x + [] = []x + [] with integer
coefficients is x = –3. What could the
equation be?
•  Choice 2: The solution to an equation of
the form []x + [] = [] is x = –3. What could
the equation be?
Common questions
•  Does at least one coefficient of x have to
be negative or not? Explain.
•  Could one of the constants be 100 or not?
Explain.
•  Could both constants be positive? Explain.
•  Could one of the coefficients of x be -1?
•  How did you create your equation?
Or maybe
•  Choice1:You are building a skateboard
ramp whose ratio of height to base must
be 2:3.Write and solve a proportion that could be
used to determine the base if the height is
4.3 m.
•  Choice 2:You are building a skateboard
ramp whose ratio of height to base must
be 2:3.Write and solve a proportion that could be
used to determine the base if the height is
3 m.
Common questions
•  What is a proportion?
•  Could your proportion have been written
2/[] = 3/[] or did it have to be 2/3 = []/[]?
•  How did you know your base was more
than 3 m?
•  How did you calculate your base?
Or
•  Choice 1: Come up with two video rental monthly
plans. Plan A charges a flat fee for unlimited
rentals. Plan B charges a smaller flat fee but also
charges some for each video. Set up values to
that Plan B is better for 5 videos, but not 6 videos.
•  Choice 2: A video rental company has two monthly
plans. Plan A charges a flat fee of $30 for
unlimited rentals; Plan B charges $9, plus$3 per
video. Use a graphical model to
determine the conditions under which
you should choose Plan A or Plan B.
Common questions
•  How is it possible for one plan to be better
sometimes and the other other times?
•  What values can you “play with” when
setting up your two plans?
•  What would your graphs look like if you
compared your two plans?
•  How do you know you answered your
question correctly?
Or
•  Choice 1: Two rectangles have a
perimeter of 80 cm. How could one area
be 312 cm2 more than the other area?
•  Choice 2: Two rectangles have the same
perimeter. One rectangle has a length
three times the width. One rectangle has a
length five times the width. Which has
more area?
Common questions
•  How can two rectangles with the same
perimeter have different area?
•  Will the area be bigger or smaller if it’s
more square-shaped? Why?
•  Which of your two rectangles is more
square shaped? How do you know?
•  How did you solve your problem?
You try
•  Now you list a problem you would normally
give your 9 applied students.
•  How will you create a parallel task?
•  What will the common questions be?
Asking good comprehension
questions
•  reconnaître deux types de fonctions
affines (de variation directe ou de variation
partielle)
So I might ask
•  A line goes through only 2 quadrants.
•  What do you know about the equation of the
line? Why?
•  OR
•  Y = 3x + 2.
•  Is the value of y usually exactly triple the
value of x, close to triple or far from triple the
value of x?
So I might ask
•  Is it possible that Line a has a greater
slope than Line b? Explain.
b
a
Asking good comprehension
questions
décrire l’effet sur le graphique et sure
l’équation d’une fonction affine lorsque l’on
change certaines données
So I might ask
•  I drew a graph and wrote an equation to
describe this situation: I had $150 in the
bank. I put in $24 each week. How much
will I have after different numbers of
weeks?
•  If the graph was a lot steeper, which
number changed and how?
•  If the graph was a lot lower down, which
number changed and how?
Asking good comprehension
questions
•  Déterminer l’aire de prisms, de pyramides
et de cylindres
So I might ask
•  Is it possible for a prism and a cylinder to
have EXACTLY the same surface area?
Explain.
•  OR
•  Is it possible for a pyramid to have a lot
less area than a prism? How?
Asking good comprehension
questions
Utiliser des rapports, des pourcentages et
des proportions dans différentes situations
So I might ask
•  I solved a percent problem and knew the
sale price. I figured out that the original
price was $48. What could the problem
have been?
•  OR
•  I solved a percent problem and the original
price was exactly $45 more than the sale
price. What could the problem have been?
It could be
•  résoudre des équations du premier
degré dont les coefficients sont non
fractionnaires
I could ask
•  WITHOUT SOLVING THE EQUATION,
explain why it makes sense that the
solution to
100x + 6 = 87x + 2 HAS to be negative.
Asking good comprehension
questions
additionner et soustraire des polynômes
I might ask
•  You subtracted two trinomials and ended
up with a binomial.
•  What could the two trinomials have been?
Now you try
•  Choose 3 different contenus.
•  Create compréhension questions for those
contenus.
Thinking questions
•  Let’s consider some good thinking
questions in 1P.
Contenu
•  décrire l’effet sur le graphique et sur
l’équation d’une fonction affine lorsque
l’on change certaines données
I could ask
•  Create descriptions of two similar
situations which result in linear relations.
•  In the first situation, the value of y when
x=20 has to be 40 more than the value of
y when x = 20 in the second situation.
Contenu
•  déterminer l’aire de prismes, de pyramides
et de cylindres.
I could ask
•  You change the two dimensions of a
rectangular prism.
•  The surface area increases by a little less
than 100 cm2.
•  What could the dimensions be?
Contenu
•  résoudre des problèmes portant sur des
rapports, des taux, des pourcentages et des
proportions tirés de situations réelles
I could ask
•  The sale price of an item that was 40% off
was equal to the sale price of a different
item that was 60% off.
•  How did the original prices compare?
Contenu
•  résoudre, à l’aide du théorème de
Pythagore, des problèmes portant sur le
périmètre et l’aire de figures simples et
composées et le volume de solides simples
So I might ask
•  A shortcut across the diagonal of a field is
800m.
•  What could its area be?
Contenu
•  déterminer le volume de solides simples et
composés
So I might ask
•  The bottom of the figure below has 3 times
the volume of the top. What could all the
dimensions be?
Contenu
•  résoudre des problèmes pouvant être
modélisés par des équations et comparer
cette méthode de résolution à d’autres
méthodes
I could ask
•  Jennifer’s mom is 24 years older than
Jennifer.
•  Right now, Mom is 4 times as old as Jennifer.
•  When will mom be twice as old?
• 
• 
• 
• 
Solve the problem using linking cubes.
Solve the problem using algebra.
Solve the problem using a graph.
Which method do you prefer? Why?
Now your turn
•  Choose three different contenus.
•  Create some HP problems that you think
would be good ones.
•  What do you like about them?
Teaching to Ideas
•  One of the most important changes I think
a teacher can make is to change his or her
learning goals to focus on ideas and not
skills.
•  It changes the tasks you give.
•  It changes the questions you ask about
the task.
For example
•  Suppose I was teaching about graphing
lines.
•  My learning goal might be that you can
change the steepness of a line by
changing the scale on an axis of a graph
(and that means you can’t estimate the
slope by looking).
My task could become
•  Use each of the graphs below.
•  Graph y = 10x + 100 on each.
•  What do you notice?
Questions to consolidate
•  What defines the slope of a line?
•  How can you tell the slope using a graph?
•  If a line is steep, what do you know about
the slope?
Or it might be
•  I want kids to understand that you are
likely to solve different proportions
different ways.
So my task might be…
•  How would you solve for x in 3/x = 21/49?
•  Create a different proportion involving 3/x
that you would probably solve using a
different strategy.
•  Tell why you would use a different
strategy?
Consolidation questions
•  What are different ways you could solve
the proportion 3:a = b:x?
•  What would you do if b were 30?
•  What would you do if a were 12?
•  What would you do if b were 7 and a were
11?
•  Why are there always different ways to
solve a proportion?
It might be about
•  Solving equations
I want students to realize
•  That solving an equation involves writing
an equivalent equation with the same
solution. The goal is to write the simplest
equation possible.
My task might be
•  Don’t solve.
•  Tell or draw pictures to show why all of
these equations HAVE To have the same
solution.
•  3x + 8 = 2x + 17.
•  6x + 16 = 4x + 34
•  2x = x+ 9
•  x = 9.
Consolidating questions
•  Why can you add the same amount to
both sides of an equation without changing
the solution?
•  Why can you multiply both sides of an
equation by the same amount without
changing the solution?
•  What’s the point of isolating a variable to
solve an equation?
Your turn
• 
• 
• 
• 
Think of something you teach as a “skill”.
How could you turn it into an idea?
What task would you give?
What would your consolidating questions
be?
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