Open-ended
problems with
purpose
May, 2015
Marian Small
Agenda
•  Why open-ended?
•  What does it mean for problems to have
purpose?
Start with straightforward ones
•  Choose two two-digit numbers.
•  Multiply them.
Becomes
•  You multiply two two-digit numbers and the
product has a 5 in the ones place.
•  What numbers? What product?
Becomes
•  You multiply two two-digit numbers and the
product has a 5 in the tens place.
•  What numbers? What product?
You learn
•  Where values in the product “come from”.
•  Why it’s so much easier to predict what value
will be in the ones place than in the tens
place.
Or…
Who doesn’t belong?
•  4 x 15
•  3 x 20
•  4 x 5 x 3
•  4 x 12
You could ask
•  The area of a rectangle is 30 cm2.
•  What could the length and width be?
Purpose
•  Understanding the relationship between
length, width, and area
Or
•  The number of square centimetres in the area
of a rectangle is less than the number of
centimetres in the perimeter.
•  What could the dimensions be?
What do you think…
•  you get out of the last question?
Alike and different
•  How is comparing 4/9 and 3/9 like comparing
9/4 and 9/3? How is it different?
What do you think..
the purpose of the last question was?
Let’s start with this problem
•  Build one or more rectangles where the
perimeter is three times the length.
•  What do you notice about the length and
width relationship?
Possibilities
•  I’ll choose a length of 4.
•  The perimeter must be 12.
4
4
Possibilities
•  The 4 that is left for the 2 widths means each
width is 2.
4
2
2
4
Possibilities
•  I’ll choose a length of 10.
•  The perimeter must be 30.
10
10
Possibilities
•  The 10 that is left for the 2 widths means each
width is 5.
10
5
5
10
Now you try one
•  Use a different length– not 4 or 10.
•  I bet I can predict your width.
An alternate approach
•  If the length is
•  Then the perimeter is
So
•  Put one stick on top, one on bottom, and split
the last stick in half to make the sides.
This leads to a much bigger
idea-•  That WHENEVER there is a relationship
between any two of L, W, and P for a
rectangle, there is an automatic relationship
between any other two.
You try
•  A length is triple the width of a rectangle.
•  What do you notice about the relationship
between perimeter and width?
It turns out
• 
• 
• 
• 
If w = 1 and l = 3, then P = 8.
If w = 3 and l = 9, then P = 24.
If w = 30 and l = 90, then P = 240.
Every time: P = 8w (and 8/3 L).
I can see that
Why open-ended?
•  To be accessible
•  To challenge
•  To evoke rich conversations
A problem I tried in Grade 4
Notice
•  Some properties were easier to use than
others (e.g. a very small angle OR angle
bigger than a right angle OR symmetry) to
make the problem accessible.
•  There was challenge for kids who wanted to
try lots of combinations to see which worked
and which didn’t.
•  Very rich conversation, e.g.
We ..
•  Showed some created shapes and other kids
guessed what properties were used.
For example
•  What properties for this one?
We ..
•  Could talk about what combos didn’t work
•  Could talk about what automatically happened
when certain combos were chosen
The Big Idea
•  Certain properties don’t mix (e.g. some, but
not all, equal side lengths AND four equal side
lengths).
•  Some properties automatically come together
(e.g. 4 equal side lengths and some parallel
sides)
We tried in Grade 6:
Use pattern blocks.
1. Build a design where:
•  there are three times as many red blocks
as yellow ones and
• twice as many green blocks as blue ones
2. Tell what fraction of the blocks are each
colour.
3/7
1/7
1/7
2/7
red
yellow
blue
green
We tried in Grade 6:
3. Add another copy of your design to your first
copy.
4. a) Are there still three times as many red
blocks as yellow ones?
b) Are there still twice as many green blocks
as blue ones?
c) What fraction of the blocks are each colour
now?
3/7
1/7
1/7
2/7
red
yellow
blue
green
We tried in Grade 6:
Use pattern blocks.
1. Build a design where:
•  The red area is three times the yellow
area
• The green area is twice the blue area.
2. Tell what fraction of the area is each
colour.
3/5 red
1/5 yellow
2/15 green
1/15 blue
We tried in Grade 6:
3. Add another copy of your design to your first
copy.
4. a) Is there still three times as much red
area as yellow area?
b) Is there still twice as much green area
as blue area?
c) What fraction of the area is each colour
now?
3/5 red
1/5 yellow
2/15 green
1/15 blue
Did you notice?
•  Accessible
•  But challenging
•  Evoked lots of conversation about the
difference between fraction of area and
fraction of set, e.g.
Is the yellow ½ or 2/3 or both?
Probability
• 
• 
• 
• 
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.
Draw possible spinners.
The point is
•  The relationship between twice and half.
•  The assumptions you can make and how they
influence your solution.
Or this question
•  Consider numbers that take exactly four words
to say (e.g. 4023: four thousand twenty three)
•  What sort of number can it be?
•  What sort of number can it not be?
What does it do for me?
•  Students get practice in reading numbers,
but…
•  They gain a better understanding of what 0 as
a placeholder means.
Number
•  You model a number with EXACTLY 15 base
ten blocks.
•  What could the numbers be?
•  What property does the number have to have?
Possible numbers
• 
• 
• 
• 
• 
• 
681
555
366
87
1275
1833
•
•
•
•
•
•
15
24
321
600
330
1500
What the point is
•  Understanding how the place value system
works is why you realize what the digit sums
can and cannot be
Elapsed time
•  Three activities take place between 10:30 and
11:30.
•  Activity 2 is twice as long as Activity 1.
•  Activity 3 is twice as long as Activity 2.
•  There are 5 minute breaks between activities.
•  Write a schedule showing the start and end
time for each activity and each break.
What happens?
•  It can be simple for kids if they use 5, 10 and
20 minutes.
•  It can be challenging for kid figuring out the
maximum time lengths possible.
•  It brings out the point that there is a max
time, but not a min time.
•  It brings out the idea that if you don’t say
whether the time is am or pm, there are more
options.
The game is how to create good
open questions
• 
• 
• 
• 
• 
• 
• 
There are strategies you can use:
Start with the answer.
How are these alike and different?
Who doesn’t belong?
Choose your own values.
Use flexible language.
But then you still have to have purpose.
So let’s try
•  Start with a “mundane” question and jazz it up
to make it open and have purpose.
One example
•  Continue the pattern for four more terms.
4, 7, 10, 13,…
•  Could become
•  The 30th term of a pattern is 91. What could
the pattern be?
•  A pattern is a lot like 4, 7, 10, 13,…. What
could it be?
•  To increase from the 10th term of a pattern to
the 20th, you add 30. What could the pattern
be?
You try
•  Now you try jazzing up a couple of mundane
questions.
•  But consider purpose.
Your questions?
In summary
•  Open questions work so well.
•  But we don’t use them just to use them.
•  We use them to make problems accessible to
all.
•  We use them to challenge students who need
challenge.
•  We use them to bring out important
mathematical ideas.
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