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? $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 sin 2 + cos 2 = 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 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 (Alberta7-10 webinar) ERLC wiki at