Great Inheritance Mathematics Term A · Independent learning

Multiplying a Fraction by a Fraction — Part 1

Learning target: Compute products of proper fractions from overlapping equal subdivisions.

Begin with the exact prerequisite checkpoint below. Afterward, explain why your strategy works, not merely which procedure to use.

Opening task

Explore · fraction product

Find 4/5 × 1/2. Explain the overlapping partitions of one rectangle.

Show answer and worked explanation

Answer: 2/5

How it works: Divide a whole into 5 columns and 2 rows. Select 4 columns and 1 rows. The overlap is 4 of 10 equal cells = 2/5.

Lesson-specific prerequisite checkpoint

Do this without a worked example. If you cannot explain it, review the earlier skill before continuing.

What fraction is 2 of 8 equal parts of one whole?

Show answer and worked explanation

Answer: 1/4

How it works: 2/8 reduces to 1/4.

Understand the idea

Compute products of proper fractions from overlapping equal subdivisions.

For fraction×fraction, subdivide one whole in two perpendicular directions. Rows represent one fractional factor, columns the other. The doubly shaded overlap counts numerator×numerator cells out of denominator×denominator equal cells.

Vocabulary focus: fraction of a quantity, overlapping partitions, reciprocal. Make a small labeled model, work through the arithmetic, and explain the result in words.

Watch for: Treating 2/3 of 3/4 as 2/3 + 3/4.

Model to study: Exact grid with shaded intersection.

Worked examples

Worked examples 1 · fraction product

Find 1/2 × 1/2. Explain the overlapping partitions of one rectangle.

Solution: 1/4

Why: Divide a whole into 2 columns and 2 rows. Select 1 columns and 1 rows. The overlap is 1 of 4 equal cells = 1/4.

Worked examples 2 · fraction product

Find 2/5 × 2/3. Explain the overlapping partitions of one rectangle.

Solution: 4/15

Why: Divide a whole into 5 columns and 3 rows. Select 2 columns and 2 rows. The overlap is 4 of 15 equal cells = 4/15.

Worked examples 3 · fraction product

Find 1/3 × 2/3. Explain the overlapping partitions of one rectangle.

Solution: 2/9

Why: Divide a whole into 3 columns and 3 rows. Select 1 columns and 2 rows. The overlap is 2 of 9 equal cells = 2/9.

Check understanding

Check understanding 1 · fraction product

Find 2/5 × 1/4. Explain the overlapping partitions of one rectangle.

Show answer and worked explanation

Answer: 1/10

How it works: Divide a whole into 5 columns and 4 rows. Select 2 columns and 1 rows. The overlap is 2 of 20 equal cells = 1/10.

Check understanding 2 · fraction product

Find 3/4 × 1/2. Explain the overlapping partitions of one rectangle.

Show answer and worked explanation

Answer: 3/8

How it works: Divide a whole into 4 columns and 2 rows. Select 3 columns and 1 rows. The overlap is 3 of 8 equal cells = 3/8.

Guided practice

Guided practice 1 · fraction product

Find 1/3 × 5/6. Explain the overlapping partitions of one rectangle.

Show answer and worked explanation

Answer: 5/18

How it works: Divide a whole into 3 columns and 6 rows. Select 1 columns and 5 rows. The overlap is 5 of 18 equal cells = 5/18.

Guided practice 2 · fraction product

Find 1/4 × 2/5. Explain the overlapping partitions of one rectangle.

Show answer and worked explanation

Answer: 1/10

How it works: Divide a whole into 4 columns and 5 rows. Select 1 columns and 2 rows. The overlap is 2 of 20 equal cells = 1/10.

Guided practice 3 · fraction product

Find 2/3 × 3/4. Explain the overlapping partitions of one rectangle.

Show answer and worked explanation

Answer: 1/2

How it works: Divide a whole into 3 columns and 4 rows. Select 2 columns and 3 rows. The overlap is 6 of 12 equal cells = 1/2.

Independent practice

Independent practice 1 · fraction product

Find 1/2 × 3/5. Explain the overlapping partitions of one rectangle.

Show answer and worked explanation

Answer: 3/10

How it works: Divide a whole into 2 columns and 5 rows. Select 1 columns and 3 rows. The overlap is 3 of 10 equal cells = 3/10.

Independent practice 2 · fraction product

Find 3/5 × 2/3. Explain the overlapping partitions of one rectangle.

Show answer and worked explanation

Answer: 2/5

How it works: Divide a whole into 5 columns and 3 rows. Select 3 columns and 2 rows. The overlap is 6 of 15 equal cells = 2/5.

Independent practice 3 · fraction product

Find 2/3 × 5/6. Explain the overlapping partitions of one rectangle.

Show answer and worked explanation

Answer: 5/9

How it works: Divide a whole into 3 columns and 6 rows. Select 2 columns and 5 rows. The overlap is 10 of 18 equal cells = 5/9.

Independent practice 4 · fraction product

Find 2/3 × 1/2. Explain the overlapping partitions of one rectangle.

Show answer and worked explanation

Answer: 1/3

How it works: Divide a whole into 3 columns and 2 rows. Select 2 columns and 1 rows. The overlap is 2 of 6 equal cells = 1/3.

Independent practice 5 · fraction product

Find 1/3 × 3/4. Explain the overlapping partitions of one rectangle.

Show answer and worked explanation

Answer: 1/4

How it works: Divide a whole into 3 columns and 4 rows. Select 1 columns and 3 rows. The overlap is 3 of 12 equal cells = 1/4.

Challenge and error analysis

Challenge and error analysis 1 · worked-error diagnosis

Error-analysis notebook: Learner notebook (LESSON): “1/2 × 2/3 = 7/6. I added instead of multiplying equal fractional parts.” Identify exactly which written step or assumption fails, correct it, and independently solve this new task: Find 1/2 × 2/3. Explain the overlapping partitions of one rectangle.

Show answer and worked explanation

Answer: 1/3

How it works: Wrong step/assumption: Learner notebook (LESSON): “1/2 × 2/3 = 7/6. I added instead of multiplying equal fractional parts.” Diagnosis: Multiply fraction factors and overlap same-whole partitions; cancel only multiplicative factors, never terms joined by plus. Correct computation: Divide a whole into 2 columns and 3 rows. Select 1 columns and 2 rows. The overlap is 2 of 6 equal cells = 1/3.

Challenge and error analysis 2 · reasoning transfer

Reasoning and transfer: Solve, then explain a different way to check your answer (inverse operation, estimate, model, or units as appropriate). Find 2/3 × 1/3. Explain the overlapping partitions of one rectangle.

Show answer and worked explanation

Answer: 2/9

How it works: Divide a whole into 3 columns and 3 rows. Select 2 columns and 1 rows. The overlap is 2 of 9 equal cells = 2/9. Verification: show why the answer has a reasonable magnitude and matches the operation, referent and unit.

Exit check

Exit check 1 · fraction product

Find 2/5 × 5/6. Explain the overlapping partitions of one rectangle.

Show answer and worked explanation

Answer: 1/3

How it works: Divide a whole into 5 columns and 6 rows. Select 2 columns and 5 rows. The overlap is 10 of 30 equal cells = 1/3.

Exit check 2 · fraction product

Find 2/3 × 4/5. Explain the overlapping partitions of one rectangle.

Show answer and worked explanation

Answer: 8/15

How it works: Divide a whole into 3 columns and 5 rows. Select 2 columns and 4 rows. The overlap is 8 of 15 equal cells = 8/15.

Target for independent mastery: all exit-check problems correct plus a justified explanation. A instructor may reteach the concept if calculations are correct without meaningful reasoning.