Building more math success in
Grades 7 – 10
Marian Small
April 2013
Focusing math instruction on what
students need to know, not just
what they can do
The outcomes
• , as they are presented in the curriculum
document, speak more to what students
need to do than what they need to know.
For example, Grade 7
• Solve percent problems involving
percents from 1% to 100%.
So what do you think matters most?
Poll:
• A: Recognizing the three different types
of percent problems and realizing what’s
different about solving them
• B: Being able to estimate the answer to a
percent problem.
• C: Recognizing the equivalent fractions
and decimals for a percent.
What do you think of this?
• That students realize that every percent
problem involves renaming a ratio in the
form []/100 in another way.
For example…
• Why is finding the sale price actually
renaming 40/100 as []/70?
For example…
• I paid $30 on sale.
• What was the original price?
• Why is renaming the original price really
renaming 40/100 as 30/[]?
For example…
• I paid $30 instead of $80.
• What was the percent paid?
• Why is finding the percent paid really
renaming 30/80 as []/100?
Another important idea…
• That students realize that knowing any
percent of a number tells you about any
other percent.
For example…
• I tell you that 20% of a certain number is
42.
• What other percents of that number do
you know even before you figure out the
number?
Write the percents you know
• on the blank screen.
If I tell you…
• 15% of a number, how could you figure
out 75% of that number?
• Raise your hand to respond.
If I tell you…
• 15% of a number, how could you figure
5% of that number?
• Raise your hand to respond.
If I tell you…
• 15% of a number, how could you figure
out 50% of that number?
• Raise your hand to respond.
Another important idea
• That renaming a percent as a fraction or
decimal sometimes helps solve
problems.
For example…
• I want to figure out how much I can
withdraw from my bank account if Mom
says 25% maximum.
• I have $424.
• How could I figure out the amount in my
head?
Which of these problems also makes
the point?
Poll
• A: estimating 35% of 612
• B: calculating 10% of 417
• C: calculating 43% of 812
• D: two of the above
Let’s look at Grade 8
• Model and solve problems concretely,
pictorially and symbolically, using linear
equations of the form:
• ax=b
• x/a = b,a≠0
• ax+b=c
• x/a+b=c,a≠0
• a(x+b)=c where a, b and c are integers.
What matters
• besides just solving?
I think that most important is…
POLL:
• A: Using more than one strategy.
• B: solving symbolically
• C: Estimating solutions
• D: Checking that a solution is correct by
substituting
Maybe estimation
• We want students to come up with a
reasonable estimate for a solution
without solving first.
So I might ask…
• Is the solution to 3x/4 – 12 = 6 closer to
0, 10, or 20? How do you know?
So I might ask…
OR
• The solution to an equation is close to
40, but not exactly 40. What might the
equation be?
So I might ask…
OR
• How might I estimate the solution to
5x – 80 = 300 without actually solving it?
So I might ask…
OR
• Can you estimate the solution to the
equation 3x + [] = 90 by ignoring the []
and just saying 30?
Another important idea
• That the same equation could represent
very different problems.
So I might ask…
• Write a real-life problem that might be
solved using the equation x/4 – 12 = 10.
• How are our problems alike? Different?
• But let’s start with something just a tad
simpler since we’re online.
Which problem relates to
x/4 – 12 = 10?
POLL
• A: There were 4 kids sharing a prize. They
gave $12 away and there was $10 left. What
was the amount of the prize?
• B: 4 kids shared a prize. One kid gave $12
away and still had $10 left. What was the
amount of the prize?
• C: 4 kids shared a prize. One kid gave $12
away and still had $10 left. What was each
kid’s share?
Or…
• Represent each problem on the next
slide with an equation. What do you
notice? Why does that make sense?
Or…
• Problem 1: The perimeter of a regular
hexagon is 90 cm. What is each side
length?
• Problem 2: A rectangle’s length is twice
its width. The total perimeter is 90 cm.
What is the width?
• Write your response on the whiteboard
or raise your hand.
Problem 1: The perimeter of a
regular hexagon is 90 cm. What
is each side length?
Problem 2: A rectangle’s length
is twice its width. The total
perimeter is 90 cm. What is the
width?
And the flip side…
• You can always represent a problem with
an equation in more than one way.
For example…
• Write an equation to represent this
problem : Jennifer had twice as many
apps as Lia. Together, they had 78 apps.
How many did each have?
• What other equation could you have
written instead?
• Write one equation on the whiteboard.
Or…
• A problem is represented by the
equation 2x + 18 = 54.
• What could the problem have been?
• What other equation could have
represented the problem?
• Which equation do you like better? Why?
Another important point
• That modelling an equation to help you
solve it always involves some sort of
“balance”.
For example….
• Where are the 3, the multiplication
inside of 3x, the x, the 5, the 26 and the =
in the picture below that represents the
equation 3x + 5 = 26?
X
X
X
26
5
• You raise your hand about where you see
the 3 and the multiplication.
Or…
• Some people say that an equation (e.g.
4x – 5 = 19) describes a balance. What do
they mean?
Or…
• How could you use a pan balance to
model the equation 3x + 8 = 29?
• Why does the model make sense?
Another important point…
• Solving an equation means writing an
equivalent equation that is easier to
interpret. (e.g. We rewrite 3x – 8 = 19 as
3x = 27 or x = 9 since they say the same
thing but it’s quicker to see what x is.)
So we might ask…
• Why might it be useful to rewrite the
equation 4x + 18 = 66 as 4x = 48 in order
to solve it?
• Why are you allowed to do that?
Or we could ask…
• Why might someone call the equations
3x – 5 = 52 and 3x = 57 equivalent?
• Which would you rather solve? Why?
Do we have
• Grade 10 teachers on line?
• If so, we will continue with the
presentation.
• If not, we will engage in a conversation
about how to do this work with other
expectations.
Grade 10
• Develop and apply the primary
trigonometric ratios (sine, cosine,
tangent) to solve problems that involve
right triangles.
Of course…
• Students need to learn the definitions of
sine, cosine, tangent. These matter, but
they are not ideas.
• But what ideas do they need to learn?
What do you think is important?
• POLL:
• A: to predict whether sine, cosine or
tangent is greater for particular angles
• B: to learn that sin2 + cos2 = 1
• C: to learn that the size of sine or cosine
is independent of triangle size
Maybe…
• The size of the trig function has nothing
to do with the size of the triangle, i.e. a
big triangle and little triangle can have
the same sine, cosine and/or tangent.
So you might ask…
• The sine of an angle in a right triangle is
0.42.
• Is it more likely that the hypotenuse is 1
cm, 10 cm or 100 cm, or don’t you know?
Explain.
You might want them to know..
• That a bigger angle (in a 90° triangle) has
a bigger sine and a smaller cosine and
why, but that the change in angle size is
not proportional to the change in sine or
cosine.
So you might ask…
• The sine of <A is 0.2 greater than the sine
of <B.
• Do you know which angle is greater?
Explain.
• Do you know how much greater the
bigger angle is? Explain.
You might want students to realize…
• That even though sines and cosines have
to stay 1 or less, tangents can get really
big and why.
So you might ask….
•
•
•
•
Which statements below are true?
Explain.
The sine of an angle can never be 2.
The tangent of a small angle can be 2.
The tangent of a large angle can be 5.
There is no greatest possible tangent.
You might want students to…
• Have a sense of trig ratio relationships,
e.g. when sine > cosine, that tan > sine,
etc.
So you might ask…
Consider each statement. The angles are all less
than 90°. Is the statement always, sometimes
or never true? Use the pen tool to write a check or x to indicate
your thoughts.
•
•
•
•
•
sin A > sin B when A < B
cos A > cos B when A < B
tan A > sin A
cos A < tan A
sin A = cos B
To conclude
• The work we are talking about involves
looking deeply at outcomes to focus on
the ideas that are critical to really
understanding what is going on.
To conclude
• It is not about the complexity of
questions students can answer.
Download
• Download these slides at
• www.onetwoinfinity.ca
(Alberta7-10 webinar)
• ERLC wiki at http://goo.gl/LxO95