Singapore Math 5A · Independent learning

Multiplying a Fraction by a Unit Fraction

Learning target: Show a fraction of a fraction as a partitioned area.

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 × 1/3. 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 3 rows. Select 3 columns and 1 rows. The overlap is 3 of 15 equal cells = 1/5.

Lesson-specific prerequisite checkpoint

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

Divide a square into three columns and four rows. How many equal cells?

Show answer and worked explanation

Answer: 12 cells

How it works: 3 × 4 = 12 equal cells.

Understand the idea

Show a fraction of a fraction as a partitioned area.

Taking a unit fraction of another fraction can be modeled as cutting an already shaded rectangle into equal crosswise sections. The overlap is a smaller portion of the original whole. The grid must have denominator×denominator cells, and the product counts overlapping 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: Assuming multiplication always increases a positive amount.

Model to study: Rectangular overlap grid.

Worked examples

Worked examples 1 · fraction product

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

Solution: 1/20

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

Worked examples 2 · fraction product

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

Solution: 1/6

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

Worked examples 3 · fraction product

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

Solution: 1/15

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

Check understanding

Check understanding 1 · fraction product

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

Show answer and worked explanation

Answer: 1/25

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

Check understanding 2 · fraction product

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

Show answer and worked explanation

Answer: 1/4

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

Guided practice

Guided practice 1 · fraction product

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

Show answer and worked explanation

Answer: 1/6

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

Guided practice 2 · fraction product

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

Show answer and worked explanation

Answer: 1/8

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

Guided practice 3 · fraction product

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

Show answer and worked explanation

Answer: 1/12

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

Independent practice

Independent practice 1 · fraction product

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

Show answer and worked explanation

Answer: 1/12

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

Independent practice 2 · fraction product

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

Show answer and worked explanation

Answer: 1/16

How it works: Divide a whole into 4 columns and 4 rows. Select 1 columns and 1 rows. The overlap is 1 of 16 equal cells = 1/16.

Independent practice 3 · fraction product

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

Show answer and worked explanation

Answer: 1/18

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

Independent practice 4 · fraction product

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

Show answer and worked explanation

Answer: 1/20

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

Independent practice 5 · fraction product

Find 2/3 × 1/6. Explain the overlapping partitions of one rectangle.

Show answer and worked explanation

Answer: 1/9

How it works: Divide a whole into 3 columns and 6 rows. Select 2 columns and 1 rows. The overlap is 2 of 18 equal cells = 1/9.

Challenge and error analysis

Challenge and error analysis 1 · worked-error diagnosis

Error-analysis notebook: Learner notebook (LESSON): “1/5 × 1/6 = 11/30. 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 1/5 × 1/6. Explain the overlapping partitions of one rectangle.

Show answer and worked explanation

Answer: 1/30

How it works: Wrong step/assumption: Learner notebook (LESSON): “1/5 × 1/6 = 11/30. 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 5 columns and 6 rows. Select 1 columns and 1 rows. The overlap is 1 of 30 equal cells = 1/30.

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

Show answer and worked explanation

Answer: 1/9

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

Exit check

Exit check 1 · fraction product

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

Show answer and worked explanation

Answer: 1/24

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

Exit check 2 · fraction product

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

Show answer and worked explanation

Answer: 2/25

How it works: Divide a whole into 5 columns and 5 rows. Select 2 columns and 1 rows. The overlap is 2 of 25 equal cells = 2/25.

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