Building more math success in Grades 7 – 10

Marian SmallApril 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?

$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?

• You raise your hand about where you see the 3 and the multiplication.

X X X 5 26

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