8.EE.B.6Common CoreMathExpressions and EquationsGrade 8
8.EE.B.6: Similar Triangles, Slope, and the Equations y = mx and y = mx + b
In plain English: 8.EE.B.6 is the Common Core grade 8 math standard that asks students to use similar triangles to explain why a line has the same slope between any two of its points. Students then use that fact to derive the equation y = mx for a line through the origin and y = mx + b for a line crossing the y-axis at b. It explains where the equation y = mx + b comes from in Grade 8 Math.
Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
Common Core State Standards for Mathematics · Domain: Expressions and Equations (EE) · Cluster: Understand the connections between proportional relationships, lines, and linear equations. Also written as 8.EE.6 · Official standard
In 8.EE.B.5, students found the slope of a line as rise ÷ run: the vertical change divided by the horizontal change between two points. This lesson answers a question they may not have asked: why do you get the same slope no matter which two points you pick? The answer uses similar triangles, triangles with the same shape, whose matching angles are equal and whose matching sides are in the same ratio. Any two slope triangles on a line (right triangles with a horizontal side, a vertical side and the line as the third side) have equal angles, so they are similar, and the ratio rise ÷ run is the same for both.
Students then use the same idea to derive equations, which means building a general rule step by step from facts they already know. A slope triangle from the origin to any point (x, y) gives y ÷ x = m, so y = mx. A slope triangle from the y-intercept (0, b), the point where the line crosses the y-axis, gives (y - b) ÷ x = m, so y = mx + b. The standard covers every line that is not vertical, so the lesson includes lines that go down from left to right (negative slope) and explains why vertical lines are left out.
Learning Objectives
By the end of this lesson, students will be able to:
Explain, using angles, why any two slope triangles on the same non-vertical line are similar
Use similar triangles to show that the slope between any two points of a line is the same
Derive the equation y = mx for a line through the origin with slope m
Derive the equation y = mx + b for a line that crosses the y-axis at (0, b), and write the equation of a line from its graph
Explain why a vertical line has no slope and no equation of the form y = mx + b
Prior Knowledge Required
Students should already be comfortable with:
Graphing proportional relationships and finding slope as rise ÷ run 8.EE.B.5
Similar figures: equal matching angles and matching sides in the same ratio 8.G.A.4
Angle facts: corresponding angles (angles in matching positions where one line crosses two parallel lines) are equal, and two triangles with two pairs of equal angles are similar 8.G.A.5
Plotting points with negative coordinates in all four quadrants 6.NS.C.8
Hand out grid paper. Students draw the line through (0, 0) and (8, 6).
Warm-Up Prompt
"Draw two right triangles under your line, each with a horizontal side, a vertical side and part of the line as the slanted side. Make one with a run (horizontal side) of 4 and one with a run of 8. What is the rise (vertical side) of each? Cut them out and stack them. What do you notice about their shapes?"
The run-4 triangle has a rise of 3, and the run-8 triangle has a rise of 6. When students stack them with the matching corners together, the angles line up exactly: the large triangle is the small one with every side multiplied by 2, a scale factor of 2. Both give rise ÷ run = 3/4. Ask: "Would that still happen with a triangle somewhere else on the line, not touching the origin?" Collect guesses; the lesson answers the question.
Direct Instruction20 minutes
Build the argument one step at a time on the board, and have students copy each step next to a sketch:
Slope triangle: a right triangle whose horizontal side is the run, whose vertical side is the rise, and whose slanted side lies on the line.
Similar triangles (recall from 8.G.A.5): if two angles of one triangle equal two angles of another, the triangles are similar, and their matching sides are in the same ratio.
Why two slope triangles on a line are similar: each has a right angle. Their horizontal sides are parallel, and the line crosses both of them, so the angles where the line meets the horizontal sides are equal (corresponding angles). Two pairs of equal angles make the triangles similar.
So the slope is the same everywhere: similar triangles have sides in the same ratio, so rise ÷ run is equal for both triangles. Call this number m, the slope of the line.
Deriving y = mx: for a line through the origin, compare the triangle with corners (0, 0), (1, 0) and (1, m) with the triangle with corners (0, 0), (x, 0) and (x, y) for any point (x, y) on the line. They are similar, so y ÷ x = m ÷ 1, and y = mx.
Deriving y = mx + b: if the line crosses the y-axis at (0, b), use the triangle with corners (0, b), (x, b) and (x, y). Its run is x and its rise is y - b, so (y - b) ÷ x = m. Multiply by x and add b: y = mx + b.
Vertical lines are left out: any two points on a vertical line have a run of 0, so no slope triangle can be drawn and rise ÷ run would mean dividing by 0.
Two slope triangles on one line (Diagram 1)
A line passes through (0, 1), (3, 3) and (9, 7). Draw a slope triangle from (0, 1) to (3, 3) and another from (3, 3) to (9, 7). Show that they give the same slope.
Equation: Triangle 1: run 3, rise 2. Triangle 2: run 6, rise 4. Every side of Triangle 2 is 2 times the matching side of Triangle 1, and the angles match, so the triangles are similar. Slope: 2 ÷ 3 = 4 ÷ 6 = 2/3.
Deriving y = mx (Diagram 2, left)
A line passes through the origin and (2, 3). Let (x, y) be any other point on the line. Use similar triangles to write its equation.
Equation: The triangle with corners (0, 0), (2, 0), (2, 3) and the triangle with corners (0, 0), (x, 0), (x, y) share the angle at the origin and both have a right angle, so they are similar. So y ÷ x = 3 ÷ 2, and y = (3/2)x. Check with (6, 9): (3/2)(6) = 9.
Deriving y = mx + b (Diagram 2, right)
A line crosses the y-axis at (0, 3) and passes through (4, 5). Let (x, y) be any point on it. Write its equation.
Equation: The slope triangle from (0, 3) to (4, 5) has run 4 and rise 2, so m = 2/4 = 1/2. The triangle from (0, 3) to (x, y) has run x and rise y - 3. They are similar, so (y - 3) ÷ x = 1/2, which gives y - 3 = (1/2)x and y = (1/2)x + 3. Check with (8, 7): 4 + 3 = 7.
A line with a negative slope
A line crosses the y-axis at (0, 6) and passes through (3, 4). Find its slope and its equation.
Equation: From (0, 6) to (3, 4), the run is 3 and y goes down by 2, so the rise is -2 and m = -2/3. The same similar-triangle argument works, with the triangles below the line: y = -(2/3)x + 6. Check with (9, 0): -(2/3)(9) + 6 = 0.
Use Diagram 1 to point out the two marked angles: they are equal because the two horizontal sides are parallel. Then use Diagram 2 to show that the general point (x, y) works the same way as a numbered point. Stress that b is the y-coordinate where the line crosses the y-axis, and that y = mx is the case b = 0.
Guided Practice15 minutes
Pairs solve each problem on grid paper. One partner draws the triangles and the other writes the reasoning; they switch for the next problem.
Guided practice problems with answers
Problem
Answer
A line passes through (0, 1), (2, 2) and (6, 4). Draw slope triangles from (0, 1) to (2, 2) and from (2, 2) to (6, 4). Are they similar? What is the slope?
Runs 2 and 4, rises 1 and 2: scale factor 2, so they are similar. Slope 1/2; the equation is y = (1/2)x + 1
A line passes through the origin and (5, 8). Use a general point (x, y) to derive its equation, then test (10, 16).
y ÷ x = 8 ÷ 5, so y = (8/5)x. (8/5)(10) = 16, so (10, 16) is on the line
A line crosses the y-axis at (0, 5) and passes through (2, 8). Derive its equation.
Slope 3/2. (y - 5) ÷ x = 3/2, so y = (3/2)x + 5
Can a slope triangle with run 4 and rise 3 and one with run 10 and rise 8 lie on the same line?
No. 3/4 is not equal to 8/10, so the triangles are not similar. On the same line, a run of 10 would need a rise of 7.5
Listen for students who write the run as y - b, or who forget to subtract b from y. Ask them to point to each side of the triangle on the grid before they write the ratio.
Independent Practice15 minutes
Students work alone, then compare answers with a partner.
Independent practice problems with answers
Problem
Answer
A line passes through (2, 1), (4, 6) and (6, 11). Draw two slope triangles and show that they give the same slope.
Both have run 2 and rise 5, so the slope is 5/2 each time (the triangles are the same size, which is a special case of similar)
Derive the equation of the line through the origin and (2, 7).
y ÷ x = 7 ÷ 2, so y = (7/2)x
A line crosses the y-axis at (0, -4) and passes through (3, 0). Derive its equation.
Slope 4/3; (y + 4) ÷ x = 4/3, so y = (4/3)x - 4
A line crosses the y-axis at (0, 8) and passes through (4, 6). Find the slope and the equation.
Rise -2, run 4, so m = -1/2 and y = -(1/2)x + 8
In one sentence, explain why you cannot draw a slope triangle for the line x = 3.
Every point on it has x = 3, so the run between any two points is 0 and there is no triangle
Closure5-10 minutes
Exit ticket: (1) Two slope triangles lie on the same line. The first has run 2 and rise 3. The second has run 8. What is its rise? (12, since the scale factor is 4.) (2) A line crosses the y-axis at (0, 2) and passes through (4, 5). Write its equation. (Slope 3/4, so y = (3/4)x + 2.) (3) In one sentence, explain why any two slope triangles on the same line are similar.
Differentiation Strategies
For Struggling Students
Give students pre-drawn lines with lattice points (points where grid lines cross) marked, so every run and rise is a whole number
Have students cut out two slope triangles and physically stack them to see the equal angles before writing any ratio
Use a fill-in frame for the derivation: "rise = ____, run = ____, so ____ ÷ ____ = m, so y = ____"
For Advanced Students
Ask: does the argument still work when the general point (x, y) is to the left of the y-axis, where x is negative? Have students try it with a specific point
Have students explain why a horizontal line has slope 0 and why its equation y = b fits the form y = mx + b
Extension (beyond this standard): ask students to use similar triangles to show that two lines with the same slope never meet, which previews HSG.GPE.B.5 in high school geometry
Assessment Guidance
What to Look For
A complete explanation names both pairs of equal angles (the right angles, and the angles where the line meets the horizontal sides) before it says the triangles are similar, and only then compares rise ÷ run. In derivations, check that students label the sides of the triangle with x and y - b, not with the coordinates of a single point. Watch for students who write y = bx + m, who drop the negative sign on a falling line, and who think a larger slope triangle has a larger slope.
02
Classroom Activities
3 Activities
1
Slope Triangle Hunt
20 minPairs
Pairs draw the line through (0, 2) and (5, 5) on grid paper, draw three different slope triangles under it, and test with a ruler and a protractor whether the triangles are similar.
The Three Triangles
Triangle A: from (0, 2) to (5, 5), with run 5 and rise 3
Triangle B: from (0, 2) to (10, 8), with run 10 and rise 6
Triangle C: from (5, 5) to (7.5, 6.5), with run 2.5 and rise 1.5
Procedure
Check that (10, 8) and (7.5, 6.5) really are on the line before drawing triangles B and C
Measure the angle between the line and the horizontal side of each triangle with a protractor. On a grid with square cells, each angle is about 31 degrees
Record rise ÷ run for each triangle as a fraction and as a decimal (3/5 = 0.6 each time)
Find the scale factor from Triangle A to Triangle B (2) and from Triangle A to Triangle C (0.5)
Discussion Questions
All three triangles have the same angles and give the same ratio. Which fact did you check first, and does one fact explain the other?
Can you draw a slope triangle on this line with run 4 and rise 3? Why or why not?
Where does the line cross the y-axis, and what is its equation?
Modification for Distance Learning
Students use an online graphing tool to draw the line and the three triangles, and read the angles with the tool's angle measure instead of a protractor.
2
Put the Proof in Order
15 minPairs
Pairs get 8 cards, each with one step of the derivation of y = mx + b, in scrambled order. They arrange the cards into a correct argument, then adapt it to derive y = mx.
The 8 Cards (scrambled)
Card A: Similar triangles have matching sides in the same ratio, so (y - b) ÷ x = m ÷ 1.
Card B: Draw the slope triangle from (0, b) to (1, b) to (1, b + m). Its run is 1 and its rise is m.
Card C: Add b to both sides: y = mx + b.
Card D: Let (0, b) be the point where the line crosses the y-axis, and let (x, y) be any other point on the line.
Card E: Both triangles have a right angle, and they share the angle at (0, b), so they are similar.
Card F: Multiply both sides by x: y - b = mx.
Card G: Draw a second triangle from (0, b) to (x, b) to (x, y). Its run is x and its rise is y - b.
Card H: Since (x, y) was any point on the line, every point on the line fits y = mx + b.
Answer Key
D, B, G, E, A, F, C, H. To derive y = mx, replace b with 0 on every card: the first triangle becomes (0, 0), (1, 0), (1, m), and Cards F and C combine into y = mx.
Discussion Questions
Which card uses the fact that the line is straight?
Card E says the triangles share an angle. In Diagram 1 the two triangles do not share a corner. What replaces "share an angle" there?
3
Line Detectives
20 minGroups of 3
Each group gets four line cards, each a line drawn on a grid through two marked points. For each line, groups find b, draw a slope triangle to find m, write y = mx + b, and test a third point.
The Line Cards
Line 1: through (0, 0) and (4, 5); test point (8, 10)
Line 2: through (0, 1) and (2, 6); test point (4, 11)
Line 3: through (0, 7) and (3, 5); test point (6, 3)
Line 4: through (0, -1) and (3, 5); test point (2, 3)
Answer Key
Line 1: y = (5/4)x
Line 2: y = (5/2)x + 1
Line 3: y = -(2/3)x + 7
Line 4: y = 2x - 1
Every test point is on its line
Discussion Questions
Only Line 1 passes through the origin. Which form of equation did you use for it, and why?
Line 3 is the only line with a negative slope. How did your slope triangle look different?
Line 2 is the steepest of the four. How can you tell from the equations alone?
Challenge Variation
Each group draws its own line through two lattice points, writes its equation on the back, and trades cards with another group to check.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Two Similar Slope Triangles on One Line
The line through (0, 1), (3, 3) and (9, 7), drawn to scale on a square grid. Triangle 1 has run 3 and rise 2; Triangle 2 has run 6 and rise 4. Both have a right angle, and the marked angles are equal because the horizontal sides are parallel, so the triangles are similar and both give a slope of 2/3.
Diagram 2: Deriving y = mx and y = mx + b
Left: a line through the origin and (2, 3). The small triangle and the triangle to a general point (x, y) are similar, so y ÷ x = 3 ÷ 2 and y = (3/2)x. Right: a line crossing the y-axis at (0, 3) and passing through (4, 5). The triangle from (0, 3) to (x, y) has run x and rise y - 3, so (y - 3) ÷ x = 2 ÷ 4 and y = (1/2)x + 3.
04
Homework Assignment
~30 min
8.EE.B.6 Homework: Similar Triangles and Equations of Lines
Directions: Draw every line and slope triangle on grid paper. Label the run and the rise of each triangle, and explain each step of your reasoning in words.
Part 1: Similar Slope Triangles (Problems 1-3)
A line passes through (1, 1), (3, 4) and (7, 10). (a) Draw a slope triangle from (1, 1) to (3, 4) and another from (3, 4) to (7, 10). (b) Find the run and rise of each and the scale factor between them. (c) Show that both give the same slope.
A line passes through (-2, 5), (2, 3) and (6, 1). (a) Draw two slope triangles and find the slope from each. (b) Explain, using angles, why the two triangles are similar.
Jess draws two slope triangles on what she says is one line. The first has run 4 and rise 6. The second has run 6 and rise 8. (a) Explain why the two triangles cannot both lie on one line. (b) If the first triangle is right, what should the rise of the second be?
Part 2: Derive the Equation (Problems 4-6)
A line passes through the origin and (3, 7). (a) Let (x, y) be any point on the line and use similar triangles to derive its equation. (b) Decide whether (9, 21) and (6, 15) are on the line.
A line crosses the y-axis at (0, -2) and passes through (5, 2). (a) Find the slope with a slope triangle. (b) Use a triangle from (0, -2) to a general point (x, y) to derive the equation. (c) Check that (10, 6) is on the line.
A line passes through (2, 9) and (6, 17). (a) Find the slope. (b) Use a slope triangle to find where the line crosses the y-axis. (c) Write the equation in the form y = mx + b and explain what m and b tell you about the graph.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Slope Triangles
Drawn to scale with run and rise labeled
Drawn, but some labels missing or wrong
Missing or incorrect
Similarity Reasoning
Names both pairs of equal angles and concludes the triangles are similar
Says "similar" with an incomplete reason
No reason given
Derivations
Writes the ratio from the triangle with (x, y) and solves it for y
Correct equation without the derivation, or one algebra slip
Equation missing or wrong
Checks and Accuracy
Every slope and test point correct
One or two computing errors
Many errors
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
Why are any two slope triangles on the same line similar?
Answer: D
Each slope triangle has a right angle, and the angles where the line meets the parallel horizontal sides are equal, so two pairs of angles match and the triangles are similar. Choice C is not enough: many right triangles have different shapes. Choices A and B are false, since slope triangles can have different sizes.
Question 2 of 20 · Multiple Choice
A slope triangle on a line has run 5 and rise 2. Another slope triangle on the same line has run 15. What is its rise?
Answer: C
The triangles are similar with scale factor 15 ÷ 5 = 3, so the rise is 2 × 3 = 6, and the slope is 2/5 both times. Choice A multiplies the old rise by the new run. Choice B adds 10 to the rise because the run grew by 10. Choice D multiplies 15 by 5/2, which is run ÷ rise.
Question 3 of 20 · Multiple Choice
What is the slope of the line through (2, 1) and (8, 8)?
Answer: B
The run is 8 - 2 = 6 and the rise is 8 - 1 = 7, so the slope is 7/6. Choice A divides run by rise. Choice C divides the coordinates of the single point (8, 8). Choice D gives the rise alone.
Question 4 of 20 · Multiple Choice
A line passes through the origin and (4, 9). Which equation does it have?
Answer: C
For any point (x, y) on the line, similar triangles give y ÷ x = 9 ÷ 4, so y = (9/4)x. Choice A divides run by rise. Choice B uses only the y-coordinate. Choice D adds the difference 9 - 4, but a line through the origin has the form y = mx.
Question 5 of 20 · Multiple Choice
A line crosses the y-axis at (0, 4) and has a slope of 3. Which is its equation?
Answer: A
In y = mx + b, m is the slope and b is where the line crosses the y-axis, so y = 3x + 4. Choice B swaps m and b. Choice C leaves out b, which describes a line through the origin. Choice D multiplies 3 by 4 to get b = 12.
Question 6 of 20 · Multiple Choice
In the derivation of y = mx + b, what are the run and the rise of the triangle with corners (0, b), (x, b) and (x, y)?
Answer: A
The triangle starts at (0, b), so it goes x units across and y - b units up. Choice B forgets that the triangle starts at height b, not at 0. Choice C subtracts b from the wrong coordinate.
Question 7 of 20 · Multiple Choice
Why does the similar-triangle argument not work for the vertical line x = 5?
Answer: B
Every point on x = 5 has the same x-coordinate, so the run between any two points is 0. There is no triangle, and rise ÷ run would divide by 0. Choice D confuses vertical and horizontal lines: a horizontal line has slope 0. Choice C reads the 5 as a slope, but it is where the line crosses the x-axis.
Question 8 of 20 · Multiple Choice
Which point is on the line y = -2x + 7?
Answer: B
Substitute x = 3: -2(3) + 7 = 1, so (3, 1) is on the line. Choice A drops the negative sign: 2(1) + 7 = 9, but the correct value at x = 1 is 5. Choice C writes the slope and the y-intercept as a point. Choice D uses the slope as the y-intercept; the line crosses the y-axis at (0, 7).
Question 9 of 20 · Multiple Choice
A line crosses the y-axis at (0, -3) and passes through (2, 1). Which is its equation?
Answer: D
The slope triangle from (0, -3) to (2, 1) has run 2 and rise 4, so m = 2 and b = -3: y = 2x - 3. Choice A swaps the slope and the intercept. Choice B divides run by rise. Choice C uses the y-coordinate of (2, 1) as b.
Question 10 of 20 · Multiple Choice
A line passes through the origin and (1, 4). To derive its equation, you compare the triangle with corners (0, 0), (1, 0) and (1, 4) with the triangle with corners (0, 0), (x, 0) and (x, y). Which equation comes from their similarity?
Answer: A
Similar triangles have matching sides in the same ratio: rise ÷ run is y ÷ x in the large triangle and 4 ÷ 1 in the small one. So y ÷ x = 4, and y = 4x. Choice B puts run over rise on one side and rise over run on the other. Choices C and D subtract or add sides, but similarity is about ratios.
Question 11 of 20 · Multiple Choice
A line crosses the y-axis at (0, 5) and passes through (4, 2). Which is its equation?
Answer: C
From (0, 5) to (4, 2), the run is 4 and y goes down 3, so the slope is -3/4 and y = -(3/4)x + 5. Choice A loses the negative sign of a falling line. Choice B divides run by rise. Choice D uses the y-coordinate of (4, 2) as b.
Question 12 of 20 · Multiple Choice
A slope triangle on a line has run 3 and rise 2. A larger slope triangle on the same line has rise 10. What is its run?
Answer: D
The scale factor is 10 ÷ 2 = 5, so the run is 3 × 5 = 15, and the slope is 2/3 both times. Choice A adds 8 to the run because the rise grew by 8. Choice B multiplies 10 by 2/3 instead of by 3/2. Choice C is the scale factor, not the run.
Question 13 of 20 · Multiple Choice
Which three points lie on one line?
Answer: A
In choice A, both slope triangles have run 2 and rise 3, so the slope is 3/2 each time and the points are on one line. Choice B gives slopes 3/2 and 2. Choice C gives 2 and 3/2. Choice D gives 2/3 and 1. Points are on one line only when every pair gives the same slope.
Question 14 of 20 · Multiple Choice
A line has a slope of 3 and passes through (2, 15). Where does it cross the y-axis?
Answer: B
Go from (2, 15) back 2 units to x = 0. With a slope of 3, y goes down 3 × 2 = 6, so the line crosses at (0, 9) and y = 3x + 9. Choice C subtracts the slope only once. Choice D adds 6 instead of subtracting it. Choice A uses the y-coordinate of the given point.
Question 15 of 20 · Short Answer
A line passes through (0, 1), (3, 6) and (9, 16). Draw a slope triangle from (0, 1) to (3, 6) and another from (3, 6) to (9, 16). Show that the triangles are similar and that they give the same slope.
Triangle 1: run 3, rise 5. Triangle 2: run 6, rise 10. Each side of Triangle 2 is 2 times the matching side of Triangle 1, and both have a right angle and equal angles where the line meets the horizontal sides, so they are similar. Slope: 5 ÷ 3 = 10 ÷ 6 = 5/3.
Question 16 of 20 · Short Answer
A line passes through the origin and (4, 7). Let (x, y) be any point on the line. Use similar triangles to derive its equation.
The triangle with corners (0, 0), (4, 0), (4, 7) and the triangle with corners (0, 0), (x, 0), (x, y) both have a right angle and share the angle at the origin, so they are similar. So y ÷ x = 7 ÷ 4, and y = (7/4)x.
Question 17 of 20 · Short Answer
A line crosses the y-axis at (0, 6) and crosses the x-axis at (4, 0). Find its slope and derive its equation.
From (0, 6) to (4, 0), the run is 4 and the rise is -6, so m = -6/4 = -3/2. For any point (x, y), the triangle from (0, 6) has run x and rise y - 6, so (y - 6) ÷ x = -3/2 and y = -(3/2)x + 6.
Question 18 of 20 · Short Answer
A line passes through (4, 3) and (8, 9). Find the slope, use a slope triangle to find where the line crosses the y-axis, and write its equation.
Run 4, rise 6, so m = 6/4 = 3/2. Going back 4 units from (4, 3) to x = 0, y goes down 6, to -3. So b = -3 and y = (3/2)x - 3.
Question 19 of 20 · Short Answer
A line crosses the y-axis at (0, 2) and has a slope of 2/3. Is the point (12, 10) on the line? Explain with a slope triangle.
Yes. The triangle from (0, 2) to (12, 10) has run 12 and rise 10 - 2 = 8, and 8 ÷ 12 = 2/3, the slope of the line. Using the equation: (2/3)(12) + 2 = 10.
Question 20 of 20 · Short Answer
A water tank holds 40 liters, and a pump then adds water at a steady rate. On a graph of liters y against minutes x, the line crosses the y-axis at (0, 40) and passes through (5, 115). Derive the equation of the line and explain what m and b mean.
Slope: (115 - 40) ÷ 5 = 15. For any point (x, y), (y - 40) ÷ x = 15, so y = 15x + 40. The slope m = 15 means the pump adds 15 liters each minute, and b = 40 is the 40 liters already in the tank at the start.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 8.EE.B.6 mean?
8.EE.B.6 means students explain, with similar triangles, why a line has the same slope between any two of its points. They then use that reasoning to derive y = mx for a line through the origin and y = mx + b for a line that crosses the y-axis at b. It is a reasoning standard: students explain where the equations come from, not only use them.
Is 8.EE.B.6 a geometry standard or an algebra standard?
It is in the Expressions and Equations domain, but it joins the two. The reasoning is geometric (similar triangles from 8.G.A.4 and 8.G.A.5), and the result is algebraic (the equation of a line). Many schools teach it right after the similarity unit, or together with it.
Why are slope triangles on the same line similar?
Because they have two pairs of equal angles. Each has a right angle, and the line crosses their parallel horizontal sides at equal angles (corresponding angles). By the angle-angle test, the triangles are similar, so their sides are in the same ratio and rise ÷ run is the same.
What does "derive" mean in 8.EE.B.6?
It means building the equation step by step from facts students already know, instead of being told it. Students take any point (x, y) on the line, draw the slope triangle to it, write the ratio that similarity gives, and solve for y. The result is an equation that every point on the line satisfies.
What do m and b stand for in y = mx + b?
m is the slope of the line, the rise ÷ run between any two of its points. b is the y-intercept, the y-coordinate of the point (0, b) where the line crosses the y-axis. When b = 0, the line passes through the origin and the equation is y = mx.
Does the argument work for lines that go down from left to right?
Yes. The standard covers every line that is not vertical. For a falling line, the slope triangles sit below the line and the rise is negative, but the triangles are still similar, so the slope is still the same between any two points. The equation still has the form y = mx + b, with a negative m.
Why does 8.EE.B.6 leave out vertical lines?
Any two points on a vertical line have the same x-coordinate, so the run is 0. No slope triangle can be drawn, and rise ÷ run would mean dividing by 0. A vertical line has an equation of the form x = a number, which is not of the form y = mx + b.
How is 8.EE.B.6 different from 8.EE.B.5?
8.EE.B.5 uses slope as the unit rate of a proportional relationship, whose graph passes through the origin. 8.EE.B.6 explains why slope is constant along any line and extends the equation to lines that do not pass through the origin. Together they prepare students for linear functions in 8.F.A.3 and 8.F.B.4.
What mistakes do students make with 8.EE.B.6?
A common mistake is thinking that a bigger slope triangle means a bigger slope; the triangle grows, but the ratio stays the same. Others write the rise of the triangle to (x, y) as y instead of y - b, or swap m and b in the equation. Some students say two triangles are similar because both are right triangles, which is not enough on its own.
How can parents help with 8.EE.B.6 at home?
Draw a straight line on graph paper and ask your child to draw two "staircase steps" of different sizes under it, then compare rise ÷ run for each. Ask them to explain in their own words why the answers match. Hearing the explanation, not just the number, is the goal of this standard.
07
Related Standards
6 standards
These standards connect to 8.EE.B.6: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.EE.B.5Prerequisite
Graph proportional relationships, reading the unit rate as the slope
