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Leading effective math
instruction
Marian Small
April 2015
Agenda
• Planning a classroom problem
• Going into the classroom
• Debriefing what happened
• Issues you raised:
2
Agenda
• Value of manipulatives
• Open-ended questions vs practice; how
many
• Where do they fit? Before or after direct
instruction.
3
Agenda
• Value of math talk/documenting/
translating to report cards and grades
• Keeping proportional reasoning going
• Consolidation
4
Possible problems
•  Jason has a box of books.
•  Trisha has 2 boxes.
•  Alyson has 3 boxes.
•  All the boxes hold the same number of
books.
•  Decide how many books you want in a
box.
•  Tell how many books there are altogether.
Possible problems
•  Repeat trying different numbers of books
in each box.
•  What do you notice?
Consolidation
•  Who had the most books: Jason, Trisha or
Alyson? Does it depend on how many
books are in the box?
•  Did Alyson have twice as many as Trisha
or not?
•  Were your total numbers of books even or
odd?
•  Why do you think that happened?
Consolidation
•  How did the total number compare to how
many books Trisha had?
•  Why do you think that happened?
•  Could the total have ended up with 60
books? How about 50?
Minds on
•  Andrew picked a number.
•  Zayden picked one twice as big. (We’ll
make sure they understand twice as big.)
•  If you add the two numbers, what do you
notice?
Another problem
•  A yellow pattern block is worth 10.
•  Build a design worth more than 30 but less
than 40.
•  It should be closer to 40 than 30.
Consolidate
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How do you know it’s worth more than 30?
How do you know it’s worth less than 40?
Why is it not all yellow?
How many colours could it be?
How did you know it’s closer to 40?
Consolidate
•  How do you know your design is worth
more than 30?
•  How do you know your design is worth
less than 40?
•  Why is it not all yellow?
•  How many colours could it be?
Consolidate
•  How did you decide whether it was closer
to 30 or to 40?
•  How would you change your design to be
worth between 60 and 80?
Minds On
•  If a yellow block were worth 12, what
would a red be worth? A blue? A green?
An alternate problem
•  Use linking cubes to make rectangles.
•  The perimeter of the rectangle has to be
three times as much as the length.
•  What do you notice about how the length
and width compare?
Consolidating
•  Was your perimeter an even or odd
number?
•  Did it have to be?
•  Was your length an even or odd number?
•  Did it have to be?
•  What did you notice about how the width
compared to the length?
•  Why does that make sense?
Minds on
•  One number is three times as much as
another number.
•  What could those numbers be?
Another choice
•  A number is represented using base ten
blocks.
•  There are twice as many rods as flats.
•  There are twice as many unit blocks as
rods.
•  What could the number be if it’s less than
1000?
•  Think of lots of possibilities.
Consolidating
•  Was your number of rods even or odd?
Why?
•  What’s the most number of flats you could
use? Why?
•  What’s the least number of units blocks
you could have used? Why?
•  Were your numbers even or odd? Why?
•  If a number works, does its double?
Minds On
•  A number is shown with 6 ones and a lot
of rods.
•  What could it be?
My last choice
•  Three events took place between 10:30
am and 11:30, one right after the other,
with 5 minutes between them.
•  The second event was twice as long as
the first one.
•  The third event was twice as long as the
second one.
•  When could each have started and
ended?
Consolidation
•  Could the shortest event have been 20
minutes? Why or why not?
•  Could the longest event have been 20
minutes? Why or why not?
•  Could the shortest event have been 10
minutes? Why or why not?
Consolidation
•  Is there a shortest amount of time for each
event?
•  Is there a longest?
•  When did your events start and end?
Minds On
•  How much time is your walk if you started
at 10:45 and ended at 11:30?
Manipulatives to support PR
•  How can we effectively use a variety of
manipulatives to support proportional
reasoning?
Using linking cubes
•  For perimeter/area problems
For example
•  Use a train of linking cubes to build the
width of a rectangle.
•  Then use a train to build a length that is
four times as long as the width.
•  Create the rectangle with that length and
width.
•  How does the perimeter relate to the
length? the width?
•  What if the length were five times as long
as the width?
For example
•  Build multiple “trains” of five cubes.
•  Use different numbers of those trains to
build polygons.
•  How are the perimeters changing?
•  Why does that make sense?
•  How are the areas changing?
•  Why does that make sense?
•  Measure the angles in the corners.
•  What do you notice?
Using linking cubes
•  Relating to mean and median and mode
For example
•  Make trains of 8, 6 and 13 cubes, each a
different colour.
•  How can you use your cubes to find the
mean of those three numbers?
Using linking cubes
•  To show ratios
For example
•  Build a linking cube train to meet each
condition.
•  It shows the ratio 4:5, but also the ratio
4:9.
•  It shows the ratio 4:5, but also the ratio
3:4.
•  It shows at least six ratios.
Using linking cubes
•  To show factors and multiples
For example
•  Use linking cubes to make it easy to see
that 4 x 5 = 20.
•  Use linking cubes to make it easy to see
that 4 is a factor of 28.
•  Use linking cubes to make it easy to see
that 24 is a common multiple of 3 and 8.
•  Use linking cubes to make it easy to see
that 3 is a common multiple of 21 and 15.
Cuisenaire rods
•  One rod is twice as long as another.
•  What could the rods be?
Cuisenaire rods
•  How could you used rods to show that 6 is
a factor of 12?
•  That 24 is a multiple of 4?
•  That 30 ÷ 6 = 5?
Cuisenaire rods
•  It takes 8 of one rod to make the same
length as 3 of another.
•  What could the rods be?
•  What other rods (of one colour) could
make the same length?
Cuisenaire rods
•  What is the probability that a random three
Cuisenaire rods could make a triangle?
Square tiles
•  How could you show 3 x 4 using the tiles?
•  How could you see that 3 x 4 = 6 x 2 using
the tiles?
•  How could you show the fraction 2/5 using
the tiles? How else?
•  How could you make a design more than
½ red and more than ¼ blue using tiles?
How can you tell you are right?
Open-ended questions
• vs practice; how many
• Where do they fit? Before or after direct
instruction.
Tracking math talk
• What are you listening for?
• How much do you have to document?
• Can you grade it?
Where is PR in other strands?
• Have a look at my presentation called
PVNCC
Consolidation
•  What should be involved?
•  You saw many examples in my sample
problems.
•  For every expectation, you could decide
what is important and focus the
expectation on those ideas.
Grade 8
•  solve problems involving percent that arise
from real-life contexts
What ideas do you want to
come out?
•  I want students to realize that solving
percent problems ALWAYS involves
coming up with an equivalent ratio.
•  I want students to use reasonableness to
check their answers.
Grade 7
•  model real-life relationships involving
constant rates (e.g., speed, heart rate,
billing rate), using algebraic equations
with variables to represent the changing
quantities in the relationship
What ideas do you want to bring
out?
•  I want students to know WHY graphs of
constant rates go through the origin; why
the rate is the y-coordinate when x = 1;
and why and how the size of the rate
affects the steepness of the line and the
numbers in the table
Grade 6
•  multiply and divide decimal numbers to
tenths by whole numbers, using concrete
materials, estimation, algorithms, and
calculators
What ideas do you want to bring
out?
•  I want students to understand why dividing
or multiplying with decimals is the SAME
process as with whole numbers; it’s about
a unit change.
Grade 5
•  demonstrate and explain the concept of
equivalent fractions, using concrete
materials
What ideas do you want to bring
out?
•  I want students to realize why and that the
multiplicative relationship between
numerator and denominator of equivalent
fractions are identical.
Grade 4
•  describe, through investigation, the
relationship between various units of
length
What ideas do you want to bring
out?
•  I want students to realize that knowing
how many of unit A makes unit B allows
the same process to be used to change
units no matter how many of the units
there are.
An expectation
Grade 1:
•  Demonstrate an understanding of the use
of non-standard units of the same size for
measuring
Possible learning goal
•  When you measure an item with a nonstandard unit, the measure depends both
on the item and the size of the unit.
So….
•  I might ask kids, How many baby steps
make a giant step?
Consolidation
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Who got a big answer?
Why do you think your answer was big?
Who got a small answer?
Why do you think your answer was small?
Consolidation
•  What else could you measure the length of
to get a big answer? How would you
measure it?
•  What else could you measure the length of
to get a small answer? How would you
measure it?
Grade 2
•  Add and subtract money amounts to 100¢
using a variety of tools and strategies
Possible learning goal
•  That subtraction might help us solve
different kinds of money problems.
My problem
•  I subtracted 35¢ from 75¢ to answer a
problem about money.
•  Create several problems that sound really
different that I might have been solving.
To consolidate
•  Let’s hear a problem.
•  What is it about your problem that led you
to subtract?
•  Who has one that is really different? What
makes it different?
To consolidate
•  Did anyone have a problem that really is a
subtraction but sounds like an addition?
•  Did anyone have a problem that involved
two people in it?
And then I might ask…
•  I had 37¢.
•  I bought something that cost 10¢.
•  Do I find out what’s left by adding or
subtracting?
Grade 3
•  represent, compare, and order whole
numbers to 1000, using a variety of tools
I might ask
•  Choose a 2-digit or 3-digit number.
•  Represent it as many ways as you can.
•  Tell one thing each representation of the
number helps you know about that
number.
For example
•  If I showed 7 in a ten-frame, I am helping
you see that 7 is 3 less than 10 and also
that it is 2 more than 5.
But
•  If I showed 7 this way, I am helping you
see that it is odd and that it is 4 + 3.
To consolidate
•  My entire focus is on what each
representation does or does not show.
Or you might ask
•  A number is shown with base ten blocks.
•  There are a lot more ones than tens.
There are a few more tens than hundreds.
•  What might the number be?
My consolidation
•  How many digits could the number have?
•  Do there have to be hundreds?
•  Are there more tens or more hundreds?
Why?
My consolidation
•  Could the number 71 work? Why or why
not?
•  Could the number 140 work?
•  How would you have to represent it to
make it work?
My consolidation
•  Give me another number that can’t work.
•  Give me another number that can.
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