Discovering Slope Using Similar Right Triangles



8.4A Discovering Slope Using Similar Right Triangles

Step 1: Working in pairs, each student will randomly select any point they wish [works better if they select a “clean” point such as (3, 4) or (9, 12)]

Step 2: Draw a right triangle - have one student draw in the horizontal base while the other draws in the vertical height. [Highlighting the three sides of the right triangle in different colors helps]

Step 3: What is the horizontal length? _____ How did you find it? ______________________________

What is the vertical length? _______ How did you find it? ______________________________

Step 4: What is the ratio of the vertical length to the horizontal length in reduced form? ________ Show this ratio in fraction form: _________

Step 5: Read the following sentence and then try to develop the formula for slope (m) using these parts:

“Slope is defined as the comparison between two numbers: the vertical length to the horizontal length or it is also known as the difference in the y values versus the difference in the x values.”

Step 6: Do you think other groups, using different points, got the same answer for slope as you? ______

Why is that the case? ____________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

Step 7: Using 1 sentence, summarize, in your own words, what is slope:

______________________________________________________________________________

8.4A Discovering Slope Using Similar Right Triangles KEY

To show parallel lines:

Step 1: Working in pairs, each student will randomly select any point they wish [works better if they select a “clean” point such as (4, 3) or (12, 9)]

Step 2: Draw a right triangle - have one student draw in the horizontal base while the other draws in the vertical height. [Highlighting the three sides of the right triangle in different colors helps]

Step 3: What is the horizontal length? 12 How did you find it? by counting along the base of triangle

What is the vertical length? 9 How did you find it? by subtracting the big # minus small #

Step 4: What is the ratio of the vertical length to the horizontal length in reduced form? 9:12 ( 3:4 Show this ratio in fraction form: [pic]

Step 5: Read the following sentence and then try to develop the formula for slope (m) using these parts:

“Slope is defined as the comparison between two numbers: the vertical length to the horizontal length or it is also known as the difference in the y values versus the difference in the x values.”

or

Step 6: Do you think other groups, using different points, got the same answer for slope as you? Yes

Why is that the case? Because each triangle is just enlarged or reduced; they are similar triangles and similar figures have the same ratios of sides; it’s like scaling up Pythagorean triples (3,4,5 times two is 6,8,10 and times three is 9,12,15); [draw in parallel lines to show the same slope]

Step 7: Using 1 sentence, summarize, in your own words, what is slope:

Slope is how much you go up versus how much you go across!

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m

y2

x1

=

x2

y1

m

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x1

=

x2

y1

y2 -

x2 -

m

=

y1

x1

y1 -

m

=

y2

x1 -

X2

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