Unit Plan 13 (Grade 7 Math): Proportional Problem Solving—Rates to Equations
7th graders connect ratios to algebra by writing equations of the form y = kx. They find and label unit rates with correct units, interpret graphs and key points like (1, k), test for proportionality, and apply equations to real-world rate and scaling problems.
Focus: Blend 7.RP and 7.EE: translate word problems to equations y = kx; justify unit rate and scaling moves.
Grade Level: 7
Subject Area: Mathematics (Ratios & Proportional Relationships • Expressions & Equations)
Total Unit Duration: 5 sessions (one week), 45–60 minutes per session
I. Introduction
This unit connects ratio reasoning to algebraic modeling. Students extract a unit rate (k) from contexts, then represent the situation with the equation y = kx and (when appropriate) with tables and graphs. They practice scaling moves on ratio tables, interpret key points like (1, k) and (x, y), and decide when a situation is proportional (through the origin) versus non-proportional (fixed fee or offset). Emphasis: units on k, clear equations, and reasonableness checks.
Essential Questions
- How do I find and justify the unit rate (k), and why must it carry units?
- How does y = kx capture a proportional situation, and what do points like (1, k) and (x, y) mean in context?
- How do I know when a context is not proportional, and which equation form fits instead?
II. Objectives and Standards
Learning Objectives — Students will be able to…
- Determine and label the constant of proportionality k from descriptions, tables, or graphs (with units).
- Write y = kx for proportional contexts; use it to find missing values and to explain scaling in ratio tables.
- Interpret graph points in context, especially (1, k) and (x, y); explain why proportional graphs pass through (0, 0).
- Translate non-proportional fee/rate contexts into px + q = r or p(x + q) = r and solve (connection to 7.EE.4a).
- Use estimation and bounds (7.EE.3) to assess reasonableness and communicate conclusions with units.
Standards Alignment — CCSS Grade 7
- 7.RP.2c–d: • Represent proportional relationships by equations (y = kx). • Explain points on the graph of a proportional relationship (particularly (1, k)).
- 7.EE.3: Solve multistep real-life problems with positive/negative rational numbers; assess reasonableness with mental math and estimation.
- 7.EE.4a: Use variables to model and solve equations from contexts (px + q = r and p(x + q) = r).
- Mathematical Practices: MP.1 (make sense), MP.3 (justify), MP.4 (model), MP.6 (precision), MP.7 (structure).
Success Criteria — Student Language
- I can find k and write y = kx, with units on k.
- I can explain what (1, k) and any (x, y) mean in the situation.
- I can use a ratio table and scaling to get missing values and check my work.
- I can tell when a situation is not proportional and choose a different equation form.
- I can estimate before solving and give a clear conclusion with units.