Exploring Similarity and Algebra Marian Small February, 2016

Agenda Moving from similarity to trig Factoring polynomials

Solve this. Two triangles are similar. One has side lengths of 8, 10 and 12. Another has a side length of 40. What could the other two side lengths be?

Why … did I not draw a picture? How would your students react? How might you react to what they ask?

And one more task.. Draw a triangle. Choose a vertex. Extend the sides that meet at the vertex using a factor of 3 (final side is 3 times as long as original). Join the end points. Make as many comparisons as you can between the measurements of the new shape and the original triangle.

Notice I didn’t tell you which measurements to look at. Could have been side lengths, BUT Could have been angle measures Could have been perimeter Could have been area Could have been heights

Did anyone…. Compare ratios? What ratios could you have compared? What do you notice about those?

Let’s talk about similarity Where does it come from?

Let’s go back to multiplying numbers What does 4 x 8 mean? One meaning is that something that is 8 units long becomes 4 times as long. 8 32

Students learn this in Grade 7 How do you dilate this shape to make the side lengths 4 times as big?

And what did they learn about similarity? Similarity is the result of dilatation. and Similarity is about enlarging or reducing shapes so they look the same.

Let’s talk more about similarity How can you physically (not numerically) test for it?

Which pair of triangles is similar?

Did you notice? The angles were equal. Is equal angles enough to guarantee similarity with triangles? With quadrilaterals?

Why not with quads?

Why with triangles?

Let’s look

There was a dilatation.

And now, the focus is on proportions By what factor did each side length change What does that scale factor tell you? For example, a triangle is enlarged using a scale factor of 2.

And now, the focus is on proportions Show the original and the final version What changed by a factor of 2? What did NOT change by a factor of 2?

As well as.. The notion that whenever two angles of one triangle match two angles of another, the triangles are similar.

So… Let’s consider this problem.

So… Look for dilatation factor If 18 turns into 30, the scale factor must have been 30/18 = 5/3, so 5/3 x 15 = 25

So… Or look for a relationship within the triangle ; sometimes that’s better. For example: Height is half of base 18 ` 30 9 x

Changing gears just a little

Let’s look at right triangles Look at these triangles. 40° a b c e d f AC B D F E

Some questions How do I know the triangles are similar? What measurements in them are the same? How do I know? What ratios are there within each triangle? What ratios do I know are equal when I compare the two triangles? Do you think it depends on using a 40° angle? Let’s have some of you try a 25°, some a 72°, some a 45°.

Let’s look at right triangles Look at each pair of similar triangles. 60° 40° 60°

I wonder Which ratio is greater? Vertical/hypotenuse in blue or in yellow? Horizontal/hypotenuse in blue or in yellow? Vertical/horizontal in blue or in yellow? What do you think would happen with an 80° angle? What do you think would happen with a 10° angle?

With all of this in the background.. We can define sine, cosine and tangent. We can reinforce that the ratios are the same when the angles are equal. Go back to the ratios for 40° and 60° and see how it matches what the calculator shows for those ratios.

Eventually need to use this It’s only because all right triangles with a 75° angle are similar that we can solve this.

Critical to see that Angles and sines are not proportional, i.e. if double angle, DO NOT double sine (or cosine or tangent). How might we get at that?

Moving into acute triangle trig Sine law Again, want to focus on the ratio of sines matching the ratios of the side lengths. What activity might you propose?

Factoring Suppose you have 4x 2 +8x Could you organize them into equal groups? How many groups? What size? 2 sets of (2x 2 +4x + 6) OR 4 sets of (x 2 + 2x + 3) Could you have organized them into a rectangle? What are the length and width?

Factoring Suppose you have 4x 2 +8x. Could you organize them into equal groups? How many groups? What size? 2 sets of (2x 2 +4x) OR 4 sets of (x 2 + 2x) Could you have organized them into a rectangle? What are the length and width?

For some kids Reinforce with number x 4x 2 +8x

For some kids Reinforce with number x 4x 2 +8x = 4 x = 4 x = 4 x 15

For some kids Reinforce with number x 4x 2 +8x = 4 x (1 + 2) 2 32 = 4 x (4 + 4) 3 60 = 4 x 15 (9 + 6) ????

Common factors are really about Dimensions of rectangles Sometimes need to get “pictorial”, e.g. 4x 3 + 8x 2 + 4x 4x x2x2 2x1

Factoring Trinomials Create a rectangle with area x 2 + 3x + 2 Create a rectangle with area 2x 2 + 4x + 2 Create a rectangle with area 6x x + 10

Factoring Trinomials Create a rectangle with area x 2 – x – 2

Factoring numerically E.g. x 2 + 3x + 2 x x 2 + 3x

Factoring numerically E.g. x 2 + 3x + 2 x x 2 + 3x = 3 x = 4 x 5

Factoring numerically E.g. x 2 + 3x + 2 x x 2 + 3x = 1 x 2 16 = 2 x = 3 x = 4 x 5 = (x +1)(x + 2)

Mise en train Create three rectangles using at least some of the +x and/or +x 2. You can also use +1 tiles if you choose.. Tell what the length, width and area of your rectangles are. Write each as an equation.

For example Length = 2x Width = 3 Area = 6x 3 (2x) = 6x

For example Length = 3x Width = 2x Area = 6x 2 (3x)(2x) = 6x 2

For example Length = 2x Width = x+2 Area = 2x 2 +4x 2x (x +2) = 2x 2 + 4x

Your task Use these areas and build rectangles. Find the lengths and widths and write the equations. x 2 + 5x + 6 x 2 + 5x + 4 x 2 + 9x + 20

What do you notice? Why were your x terms not all together? Were your 1s all together? What shape did they make? What sorts of polynomials (monomials, binomials, trinomials, first degree, second degree) were the length and the width?

What do you notice? How did the constants relate to the terms in the area? How did the coefficients of x relate to the terms in the area?

What do you notice? Can you predict what the length and width will be for this area? x 2 + 9x + 18 Try together. What did the 18 tell you? What did the 9 tell you?

At this point Some will not see the relationship, but they will learn from the others and have enough background to make sense of what they hear.

So.. What do you think my learning goal was?

My learning goal Factoring a quadratic polynomial is about determining the two measurements of a rectangle with an area represented by that quadratic. (e.g. x 2 + 3x + 2 = (x +1)(x +2) since there is a rectangle with length x +2 and width x + 1 and area x 2 + 3x + 2)

I would follow up by Using polynomials with different coefficients of x 2, e.g. 2x 2 + 5x + 2 3x x + 6 6x 2 + 5x + 1 And ask very similar questions.

What do you think? Is it good enough if students can answer A and not B? A: What is x if 3/x = 14/30? B: How do you know that x has to be more than 6 without actually solving?

So I might ask A ratio is equivalent to 4:10. Can its terms be 40 apart? Can they be 36 apart?

So I might ask 5/36 = 27/x Without setting up any equations, suggest a good estimate and say why it’s good.

So I might ask Why might you solve these two proportions using different strategies? 12/39 = 4/x 14/42 = 8/x

So I might ask Without solving, tell which of these two proportions have the same solution and why. 14/22 = x/39 OR 7/11 = x/39 OR 15/23 = x/39

So I might ask How might this picture help you estimate the solution to the proportion 45/100 = 80/x?

So I might ask You have to describe a quadratic relationship. When might you rather have the table of values? the graph? The equation? Factored form? standard form?

So I might ask Describe four ways that make a quadratic different from a linear relationship.

So I might ask Create a situation involving a quadratic relationship where the maximum value is 22 when the independent variable has a value of 4. Tell one other fact about the situation.

So I might ask Describe a situation that is quadratic that has a maximum. Why is there no minimum?

So I might ask I am figuring out the volume of a cone. What measurements do I need to know? Which don’t I care about? Are there choices?

So I might ask I know Volume cone = 1/3 π r 2 h. Volume sphere = 4/3 πr 3. How many measurements of the cone do I need to figure out its volume? The sphere? Why does that make sense?

So I might ask How could the volume of a sphere be both 7.8 cubic units and 7800 cubic units? Can every measurement be made to sound bigger? Smaller?

So I might ask How could the volume of a sphere be both 7.8 cubic units and 7800 cubic units? Can every measurement be made to sound bigger? Smaller?

Questions? Other questions you want to raise

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