Unit Plan 13 (Grade 5 Math): Multiplying Fractions—Models and Meaning
5th graders multiply fractions by whole numbers and by other fractions using models like equal groups, arrays, and number lines. Students connect visuals to the rule of multiplying numerators and denominators, reason about product size, and justify results with clear explanations.
Focus: Multiply a fraction by a whole number and a fraction by a fraction using area/array models, equal groups, and number lines; connect models to the meaning of multiplication and to the general method (multiply numerators and denominators).
Grade Level: 5
Subject Area: Mathematics (Number & Operations—Fractions)
Total Unit Duration: 5 sessions (one week), 45–60 minutes per session
I. Introduction
Students develop a grounded understanding of fraction multiplication by first modeling fraction of a whole number and then fraction of a fraction. They use equal groups, arrays, and area models to see why (a/b) × q represents taking a parts out of b equal parts of q, and why (a/b) × (c/d) can be viewed as “a/b of c/d”. They reason about size (e.g., products between 0 and 1) and transition from models to the numerator×numerator / denominator×denominator method, justifying each step with visuals.
Essential Questions
- What does it mean to take a fraction of a quantity?
- How do area/array and equal-groups models show the structure of fraction multiplication?
- When is a product less than, equal to, or greater than one of the factors—and why?
- How do we justify the general rule for multiplying fractions using models?
II. Objectives and Standards
Learning Objectives — Students will be able to:
- Interpret and compute (a/b) × q as taking a out of b equal parts of q, using equal groups or number lines.
- Model and compute (a/b) × (c/d) with area/array models, explaining why the product represents a/b of c/d.
- Explain and use the general procedure (a/b) × (c/d) = (a×c) / (b×d), grounded in area tiling (unit squares of side lengths 1/b and 1/d).
- Estimate products using benchmarks and reason about size (0–1, 1–factor comparisons).
- Communicate thinking with models, equations, units (when contextual), and clear explanations.
Standards Alignment — CCSS Grade 5
- 5.NF.4a: Interpret the product (a/b) × q as a parts of q partitioned into b equal parts; use models to represent.
- 5.NF.4b: Find the area of a rectangle with fractional side lengths by tiling with unit squares of side length 1/b and 1/d; show that the area is the product of the side lengths and represent it as a fraction.
- Mathematical Practices emphasized: MP.1 (persevere), MP.3 (justify/critique), MP.4 (model), MP.6 (precision), MP.7 (structure), MP.8 (regularity).
Success Criteria — Student Language
- I can explain (a/b) × q as taking a out of b equal parts of q.
- I can use an area/array to show (a/b) × (c/d) and label what each region means.
- I can compute products by multiplying numerators and denominators and connect that to my model.
- I can estimate whether my answer should be less than 1 or bigger than one factor and explain why.
- I can write a clear explanation matching my model and equation.