Building a Richer Math
Environment
Marian Small
October, 2016
What matters?
• Most fundamental is a classroom
environment where students:
• Are engaged
• Talk a lot (about math)
• Work collaboratively
• Figure things out
• Have lots of questions
• Constantly explore ideas
• Willingly take risks
• But the focus today is on what the math
feels and looks like
Learning goals
• Learning goals are about ideas, and not
just performances, so that a teacher can
stay focused on the IDEAS s/he is trying to
get at.
For example…
• Instead of a learning goal being about
calculating the volumes of cylinders, it
might be about how some measures of a
cylinder are related but others are not.
For example…
• I might ask:
• You want to figure out the volume.
• What measures might be enough to let
you figure it out?
• Think of options.
Or…
• Instead of a learning goal being about
subtracting polynomials, it is about how
subtractions of polynomials work just like
subtracting of numbers.
For example…
• You are calculating 4x2 –3x + 7 –
(–2x2 +8x +4).
• How could you get the answer by
modelling (–2x2 +8x +4) first?
Or
• How can you use substitution to prove that
4x2 –3x + 7 – (–2x2 +8x +4) is NOT
• 2x2 –11x +11?
Sample lesson
• This lesson helps students relate fractions
to percents.
Task
• 2
3
4
5
8
10
• Using these values as numerators and
denominators, create four fractions less
than 1.
• Shade that fraction of the 10 x 10 grid to
tell what percent that fraction is. Be ready
to explain.
Consolidate
• Which of your fractions were close to 1?
What sorts of values were those percents?
• Which of your fractions were close to 0?
What sorts of values were those percents?
• Which of your answers were exact number
of percents and which estimates? Why?
Consolidate
• Why does it make sense that 1/6 is about
16%? Is it a bit more or less than 16%?
• How does knowing that 1/6 is about 16%
help you figure out the percent for 5/6?
Success criteria
1. I create four fractions less than 1 using
combinations of 2, 3, 4, 5, 8 and 10 as
numerators and denominators.
2. I use the grid to explain why each one is
the percent it is.
Success criteria
3. I realize that
• fractions close to 1 have percents close to
100
• fractions close to 0 have percents close to
0
Success criteria
• Some criteria should be discussed before
the task is begun, but some should be
discussed after consolidation.
Question asking
• I believe improving our question asking is
essential whether to
• Pose as tasks
• Scaffold
• Extend
• Probe
• Build connections
Manipulatives/visuals
• There is a lot of use of manipulatives and
visuals not in procedural ways but in
service of “seeing” ideas.
For example
• How could this picture help you figure out
the 100th term of 7, 10, 13, 16,…?
For example
• How could this picture help you figure out
the 100th term of 7, 10, 13, 16,…?
50 x 3 + 4
Or
• How could this picture help you figure out
what 4/5 – 1/3 is?
Or
• How could this picture help you figure out
what 4/5 – 1/3 is?
Or
• How could this picture help you figure out
what 4/5 – 1/3 is?
Estimation/mental math
• Number sense is essential.
• There needs to be significant attention to
mental math and estimation, whether
through number talks or in other ways.
Might be….
• How many dots are there?
Or
• Describe different ways to calculate 26%
of 480 without a calculator.
Using problems that make you
wonder
• About how many times is your fridge door
opened each week?
Or
• About how many cookies could you buy
for $100?
Or
• How much more or less is a kilogram of
quarters worth than a kilogram of nickels?
Differentiation
• There should be significant use of open
questions to allow for differentiation as
well as parallel tasks.
• This is true in both the tasks assigned as
well as assessment.
For example
For example
For example
For example
For example
For example
For example
Kinds of tasks you use
• You need a good blend of concept based
lessons, game days, etc.
• You need a good blend of very focused
tasks to reveal very particular math ideas
and bigger, thinking tasks that apply what
has already been learned.
• You need engagement.
Problems need to beThoughtful,
not complicated
• Compare the two:
Complicated
• I bought 3 shirts that each cost $12.45.
• I bought 4 sweaters that each cost $39.95.
• I bought 5 pairs of pants that cost $9.95,
$12.95, $22.95, $17.95 and $18.95.
• How much of the $500 I had budgeted for
clothes do I have left?
Thoughtful
• I bought 7 items that cost under $20 and
bought 5 items that cost more than $20. I
spent almost $300.
• What are possible prices for the 12 items?
Explain your thinking.
What it might look like
• You multiply two fractions: a/b and c/d.
• The result is a lot more than a/b, but a bit
less than c/d.
• What could the fractions be?
• Maybe 9/10 x 95/2
What it might look like
• You start with the integer a.
• What integer b could you use so that a + b
is:
• 12 more than a – b
• 12 less than a – b
• 3 times as much as a – b
What it might look like
•
•
•
•
You buy a jacket at 40% off.
You buy shoes at 20% off.
You pay the same for both items.
What do you know about the relationship
between the two original prices?
What it might look like
• If 60% of A = 80% of B,
• then B = 3/4 of A.
• 0.6x = 0.8y means 6x = 8y or y = 3/4 x
OR
What it might look like
0
10
20
30
40
50
60
70
80
0 10 20 30 40 50 60 70 80 90 100
90 100
What it might look like
0
10
20
30
40
50
60
70
80
0 10 20 30 40 50 60 70 80 90 100
90 100
What it might look like
• An algebraic expression is worth 20 more
when x = 5 than when x = 3.
• How much more will it be worth when
x = 20 than when x = 12?
Maybe 10x + 3
What it might look like
• The quotient of two fractions is 4 times
greater the product. What might the
fractions be?
• 5 ÷ ½ = 10 but 5 x ½ = 2.5
• 4 ÷1/2 = 8 but 4 x 1/2 = 2
What it might look like
• The quotient of two fractions is 2 ¼ times
greater than the product. What might the
fractions be?
Just a taste
• I’ve only provided a “taste” of each idea.
• But hope it was helpful to you.
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