Use Similar Triangles to Derive y = mx + b

Use Slope Triangles on One Line Are Similar first so students see the equal ratios. Then work through the derivation sheet together, part by part.

Use similar slope triangles to derive y = mx through the origin and y = mx + b through (0, b). The directions on the page read: Slope triangles on one line are similar, so they have equal rise over run. In Parts 1 and 2, count the small triangle and use the letters on the large one. In Part 3, write each equation. An answer key is included.

Yes, both. We made it for 8th grade linear equations, and it works the same at home or at school. At home, a student can work through it alone and then go over it with you, which shows quickly what is solid and what needs another look. In class it works as independent practice, homework or a warm-up, and you can print one copy or a full set.

Each slope triangle has a right angle, and the line meets each triangle's run at the same angle because they lie on one line. Triangles with two equal angles are similar. Slope Triangles on One Line Are Similar asks students to explain this in one sentence.

Similar triangles have proportional sides, so rise over run is the same for a small triangle and a large one on the same line. That shared ratio is the slope m. Students check this by counting both triangles on four lines.

For a line through (0, b), a triangle from (0, b) to any point (x, y) has rise y minus b and run x. Setting that ratio equal to m and multiplying by x, then adding b, gives y = mx + b. Use Similar Triangles to Derive y = mx + b guides this in parts.