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.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Why does a straight line have just one slope, no matter which two points you pick? Draw right triangles under the line, each with a horizontal leg and a vertical leg. Because the triangles share the same angles, they are similar, so the ratio of vertical leg to horizontal leg is the same for all of them. That ratio is the slope m. A small triangle with legs 2 and 3 and a large one with legs 6 and 9 both give 3/2.
With that fact secured, students derive equations instead of memorizing them. For a line through the origin, any point (x, y) forms a slope triangle with the origin, so y/x = m, which rearranges to y = mx. If the line instead crosses the vertical axis at (0, b), the vertical leg from that point is y - b, giving (y - b)/x = m and therefore y = mx + b. The intercept b simply shifts the proportional line up or down.
A triangle with legs 4 and 8 looks larger than one with legs 1 and 2, but both ratios are 2. The size changes, the ratio does not.
For a line falling from left to right the vertical change is negative, so the slope is negative. Students who only measure lengths report a positive slope.
In y = 3x + 5 some students say the slope is 5. Connect m to steepness (the slope triangle) and b to the point where the line crosses the y-axis.
b is where the line meets the vertical axis, at x = 0. The point where it crosses the x-axis is a different value.
A line crosses the y-axis at (0, -2) and passes through (4, 6). Use a slope triangle to find m, then write the equation.
Answer: m = 2 and the equation is y = 2x - 2.
On grid paper, have students draw a line and then three different slope triangles of their own choice, recording the leg lengths in a table. Sharing results across the room shows every ratio is equal, and asking why leads naturally to similar triangles from 8.G.A.5. This builds a reason for the formula before the formula appears.
When deriving y = mx + b, keep the general point labelled (x, y) on the diagram so the algebra stays tied to the picture. Assessment items may ask students to explain why two slope triangles give the same slope, so practice writing that argument in words: the triangles have equal angles, so they are similar and their side ratios match.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 2
Vertical change 11 - 3 = 8, horizontal change 6 - 2 = 4. Slope = 8 ÷ 4 = 2.
Answer: B) 15
The triangles are similar, so 5/3 = h/9. Then h = 5 × 3 = 15.
Answer: A) Because they are similar triangles, so their leg ratios are equal
Each slope triangle has a right angle and shares the angle the line makes with the horizontal, so the triangles are similar and their side ratios are equal.
Answer: -2
The equation is y = -3x + 4. When x = 2, y = -6 + 4 = -2.
Answer: C) 7
In y = mx + b, b is the vertical intercept. Here b = 7, so the line crosses the y-axis at (0, 7).
Answer: 0.4 (also accepted: 2/5)
For a line through the origin the slope is y ÷ x = 2 ÷ 5 = 0.4, and the equation is y = 0.4x.
A full lesson with slides, activities and an exit ticket on slope with similar triangles and y = mx + b, pitched to grade 8 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 8.EE.B.6 with an answer key, ready in about a minute.
Make a worksheet →Turn slope with similar triangles and y = mx + b into a quiz students answer online that marks itself, with a class summary for you.
Build a test →They need an informal explanation using similar triangles, not a formal two-column proof. Saying the triangles have the same angles, so their leg ratios are equal, is the expected reasoning.
It produces the equation y = mx + b, which 8.F.A.3 then interprets as a linear function and 8.F.B.4 uses to model real situations.