Great Inheritance Mathematics Term A · Independent learning

Multiplying a Fraction by a Fraction — Part 2

Learning target: Simplify fraction products, including cross-cancellation, and explain scaling.

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 3/5 × 1/4. Explain the overlapping partitions of one rectangle.

Show answer and worked explanation

Answer: 3/20

How it works: Divide a whole into 5 columns and 4 rows. Select 3 columns and 1 rows. The overlap is 3 of 20 equal cells = 3/20.

Lesson-specific prerequisite checkpoint

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

Simplify 6/15 before multiplying by another fraction.

Show answer and worked explanation

Answer: 2/5

How it works: Divide numerator and denominator by 3; cancellation is only for factors.

Understand the idea

Simplify fraction products, including cross-cancellation, and explain scaling.

Multiply numerators and denominators; common factors can be canceled across products before computing. Cancel only factors of the multiplied numerators and denominators, not terms linked by addition. A positive proper-fraction factor scales a positive amount down.

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: Canceling terms across addition, not multiplication.

Model to study: Area model plus factor-cancellation chart.

Worked examples

Worked examples 1 · fraction product

Find 1/2 × 5/6. Explain the overlapping partitions of one rectangle.

Solution: 5/12

Why: Divide a whole into 2 columns and 6 rows. Select 1 columns and 5 rows. The overlap is 5 of 12 equal cells = 5/12.

Worked examples 2 · fraction product

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

Solution: 1/8

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

Worked examples 3 · fraction product

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

Solution: 2/15

Why: Divide a whole into 5 columns and 3 rows. Select 2 columns and 1 rows. The overlap is 2 of 15 equal cells = 2/15.

Check understanding

Check understanding 1 · fraction product

Find 3/4 × 1/6. Explain the overlapping partitions of one rectangle.

Show answer and worked explanation

Answer: 1/8

How it works: Divide a whole into 4 columns and 6 rows. Select 3 columns and 1 rows. The overlap is 3 of 24 equal cells = 1/8.

Check understanding 2 · fraction product

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

Show answer and worked explanation

Answer: 1/6

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

Guided practice

Guided practice 1 · fraction product

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

Show answer and worked explanation

Answer: 1/10

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

Guided practice 2 · fraction product

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

Show answer and worked explanation

Answer: 9/16

How it works: Divide a whole into 4 columns and 4 rows. Select 3 columns and 3 rows. The overlap is 9 of 16 equal cells = 9/16.

Guided practice 3 · fraction product

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

Show answer and worked explanation

Answer: 1/6

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

Independent practice

Independent practice 1 · fraction product

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

Show answer and worked explanation

Answer: 3/8

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

Independent practice 2 · fraction product

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

Show answer and worked explanation

Answer: 4/25

How it works: Divide a whole into 5 columns and 5 rows. Select 1 columns and 4 rows. The overlap is 4 of 25 equal cells = 4/25.

Independent practice 3 · fraction product

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

Show answer and worked explanation

Answer: 1/5

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

Independent practice 4 · fraction product

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

Show answer and worked explanation

Answer: 4/9

How it works: Divide a whole into 3 columns and 3 rows. Select 2 columns and 2 rows. The overlap is 4 of 9 equal cells = 4/9.

Independent practice 5 · fraction product

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

Show answer and worked explanation

Answer: 8/15

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

Challenge and error analysis

Challenge and error analysis 1 · worked-error diagnosis

Error-analysis notebook: Learner notebook (LESSON): “2/3 × 3/5 = 19/15. 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 2/3 × 3/5. Explain the overlapping partitions of one rectangle.

Show answer and worked explanation

Answer: 2/5

How it works: Wrong step/assumption: Learner notebook (LESSON): “2/3 × 3/5 = 19/15. 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 3 columns and 5 rows. Select 2 columns and 3 rows. The overlap is 6 of 15 equal cells = 2/5.

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

Show answer and worked explanation

Answer: 3/25

How it works: Divide a whole into 5 columns and 5 rows. Select 1 columns and 3 rows. The overlap is 3 of 25 equal cells = 3/25. Verification: show why the answer has a reasonable magnitude and matches the operation, referent and unit.

Targeted mastery: model, calculate, explain

These additional original examples address specific 5A mastery requirements. Compare the visual and the reasoning before trying the independent questions.

Targeted 1 · targeted mastery: fraction

Find 6/35 × 14/15 by canceling common factors across multiplication BEFORE multiplying.

Solution: 4/25

Why: 6/15 = 2/5 (divide by 3); 14/35 = 2/5 (divide by 7); product 2/5 × 2/5 = 4/25. Only factors, not sums, cancel.

Targeted 2 · targeted mastery: fraction

Compute 9/28 × 14/15 using legitimate cross-cancellation.

Solution: 3/10

Why: 14/28=1/2 and 9/15=3/5; 3/5 × 1/2 = 3/10.

Targeted 3 · targeted mastery: fraction

A student cancels 3 in (3 + 5)/3 to claim 1 + 5. Is that valid?

Show answer and worked explanation

Answer: no; 8/3 ≠ 6

How it works: The numerator is a sum, not a product with common factor 3; evaluate the sum first.

Targeted 4 · targeted mastery: fraction

Calculate 10/21 × 7/15 by canceling factors first.

Show answer and worked explanation

Answer: 2/9

How it works: 10/15=2/3, 7/21=1/3; 2/3×1/3=2/9.

Exit check

Exit check 1 · fraction product

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

Show answer and worked explanation

Answer: 1/5

How it works: Divide a whole into 3 columns and 5 rows. Select 1 columns and 3 rows. The overlap is 3 of 15 equal cells = 1/5.

Exit check 2 · fraction product

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

Show answer and worked explanation

Answer: 8/25

How it works: Divide a whole into 5 columns and 5 rows. Select 2 columns and 4 rows. The overlap is 8 of 25 equal cells = 8/25.

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.