Learning target: Convert mixed numbers to improper fractions and estimate a product before computing.
Begin with the exact prerequisite checkpoint below. Afterward, explain why your strategy works, not merely which procedure to use.
Opening task
Explore · mixed-number product
Calculate 3 2/3 × 1 2/3. Estimate before multiplying.
Show answer and worked explanation
Answer: 6 1/9
How it works: Convert exactly: 3 2/3=11/3 and 1 2/3=5/3. Multiply numerators and denominators: (11×5)/(3×3) = 55/9 = 6 1/9. Because both numbers exceed 1, their product must exceed each factor.
Lesson-specific prerequisite checkpoint
Do this without a worked example. If you cannot explain it, review the earlier skill before continuing.
Rename 2 3/4 as an improper fraction.
Show answer and worked explanation
Answer: 11/4
How it works: 2 wholes are 8/4 and 8/4 + 3/4 = 11/4.
Understand the idea
Convert mixed numbers to improper fractions and estimate a product before computing.
Convert each mixed number into one improper fraction before multiplying. Estimate using nearby whole numbers first, multiply the fractions, then convert the result back to a simplified mixed number. A rectangle can be decomposed into whole-by-whole and whole-by-fraction regions.
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: Multiplying whole parts and fractional parts separately.
Model to study: Mixed-number area decomposition.
Worked examples
Worked examples 1 · mixed-number product
Calculate 2 1/4 × 1 3/4. Estimate before multiplying.
Solution: 3 15/16
Why: Convert exactly: 2 1/4=9/4 and 1 3/4=7/4. Multiply numerators and denominators: (9×7)/(4×4) = 63/16 = 3 15/16. Because both numbers exceed 1, their product must exceed each factor.
Worked examples 2 · mixed-number product
Calculate 1 1/4 × 2 3/4. Estimate before multiplying.
Solution: 3 7/16
Why: Convert exactly: 1 1/4=5/4 and 2 3/4=11/4. Multiply numerators and denominators: (5×11)/(4×4) = 55/16 = 3 7/16. Because both numbers exceed 1, their product must exceed each factor.
Worked examples 3 · mixed-number product
Calculate 2 1/5 × 2 1/5. Estimate before multiplying.
Solution: 4 21/25
Why: Convert exactly: 2 1/5=11/5 and 2 1/5=11/5. Multiply numerators and denominators: (11×11)/(5×5) = 121/25 = 4 21/25. Because both numbers exceed 1, their product must exceed each factor.
Check understanding
Check understanding 1 · mixed-number product
Calculate 1 1/2 × 2 1/2. Estimate before multiplying.
Show answer and worked explanation
Answer: 3 3/4
How it works: Convert exactly: 1 1/2=3/2 and 2 1/2=5/2. Multiply numerators and denominators: (3×5)/(2×2) = 15/4 = 3 3/4. Because both numbers exceed 1, their product must exceed each factor.
Check understanding 2 · mixed-number product
Calculate 3 1/2 × 1 1/4. Estimate before multiplying.
Show answer and worked explanation
Answer: 4 3/8
How it works: Convert exactly: 3 1/2=7/2 and 1 1/4=5/4. Multiply numerators and denominators: (7×5)/(2×4) = 35/8 = 4 3/8. Because both numbers exceed 1, their product must exceed each factor.
Guided practice
Guided practice 1 · mixed-number product
Calculate 2 1/2 × 2 1/2. Estimate before multiplying.
Show answer and worked explanation
Answer: 6 1/4
How it works: Convert exactly: 2 1/2=5/2 and 2 1/2=5/2. Multiply numerators and denominators: (5×5)/(2×2) = 25/4 = 6 1/4. Because both numbers exceed 1, their product must exceed each factor.
Guided practice 2 · mixed-number product
Calculate 2 1/2 × 1 1/2. Estimate before multiplying.
Show answer and worked explanation
Answer: 3 3/4
How it works: Convert exactly: 2 1/2=5/2 and 1 1/2=3/2. Multiply numerators and denominators: (5×3)/(2×2) = 15/4 = 3 3/4. Because both numbers exceed 1, their product must exceed each factor.
Guided practice 3 · mixed-number product
Calculate 2 1/3 × 2 2/3. Estimate before multiplying.
Show answer and worked explanation
Answer: 6 2/9
How it works: Convert exactly: 2 1/3=7/3 and 2 2/3=8/3. Multiply numerators and denominators: (7×8)/(3×3) = 56/9 = 6 2/9. Because both numbers exceed 1, their product must exceed each factor.
Independent practice
Independent practice 1 · mixed-number product
Calculate 1 4/5 × 2 3/5. Estimate before multiplying.
Show answer and worked explanation
Answer: 4 17/25
How it works: Convert exactly: 1 4/5=9/5 and 2 3/5=13/5. Multiply numerators and denominators: (9×13)/(5×5) = 117/25 = 4 17/25. Because both numbers exceed 1, their product must exceed each factor.
Independent practice 2 · mixed-number product
Calculate 1 1/3 × 1 1/3. Estimate before multiplying.
Show answer and worked explanation
Answer: 1 7/9
How it works: Convert exactly: 1 1/3=4/3 and 1 1/3=4/3. Multiply numerators and denominators: (4×4)/(3×3) = 16/9 = 1 7/9. Because both numbers exceed 1, their product must exceed each factor.
Independent practice 3 · mixed-number product
Calculate 3 1/2 × 1 1/2. Estimate before multiplying.
Show answer and worked explanation
Answer: 5 1/4
How it works: Convert exactly: 3 1/2=7/2 and 1 1/2=3/2. Multiply numerators and denominators: (7×3)/(2×2) = 21/4 = 5 1/4. Because both numbers exceed 1, their product must exceed each factor.
Independent practice 4 · mixed-number product
Calculate 1 2/3 × 2 2/3. Estimate before multiplying.
Show answer and worked explanation
Answer: 4 4/9
How it works: Convert exactly: 1 2/3=5/3 and 2 2/3=8/3. Multiply numerators and denominators: (5×8)/(3×3) = 40/9 = 4 4/9. Because both numbers exceed 1, their product must exceed each factor.
Independent practice 5 · mixed-number product
Calculate 1 1/2 × 1 3/4. Estimate before multiplying.
Show answer and worked explanation
Answer: 2 5/8
How it works: Convert exactly: 1 1/2=3/2 and 1 3/4=7/4. Multiply numerators and denominators: (3×7)/(2×4) = 21/8 = 2 5/8. Because both numbers exceed 1, their product must exceed each factor.
Challenge and error analysis
Challenge and error analysis 1 · worked-error diagnosis
Error-analysis notebook: Learner notebook (LESSON): “7/4 × 7/4 = 7/2. I added instead of multiplying equal fractional parts.” Identify exactly which written step or assumption fails, correct it, and independently solve this new task: Calculate 1 3/4 × 1 3/4. Estimate before multiplying.
Show answer and worked explanation
Answer: 3 1/16
How it works: Wrong step/assumption: Learner notebook (LESSON): “7/4 × 7/4 = 7/2. 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: Convert exactly: 1 3/4=7/4 and 1 3/4=7/4. Multiply numerators and denominators: (7×7)/(4×4) = 49/16 = 3 1/16. Because both numbers exceed 1, their product must exceed each factor.
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). Calculate 1 2/3 × 2 1/3. Estimate before multiplying.
Show answer and worked explanation
Answer: 3 8/9
How it works: Convert exactly: 1 2/3=5/3 and 2 1/3=7/3. Multiply numerators and denominators: (5×7)/(3×3) = 35/9 = 3 8/9. Because both numbers exceed 1, their product must exceed each factor. Verification: show why the answer has a reasonable magnitude and matches the operation, referent and unit.
Exit check
Exit check 1 · mixed-number product
Calculate 1 4/5 × 2 4/5. Estimate before multiplying.
Show answer and worked explanation
Answer: 5 1/25
How it works: Convert exactly: 1 4/5=9/5 and 2 4/5=14/5. Multiply numerators and denominators: (9×14)/(5×5) = 126/25 = 5 1/25. Because both numbers exceed 1, their product must exceed each factor.
Exit check 2 · mixed-number product
Calculate 2 1/2 × 2 3/4. Estimate before multiplying.
Show answer and worked explanation
Answer: 6 7/8
How it works: Convert exactly: 2 1/2=5/2 and 2 3/4=11/4. Multiply numerators and denominators: (5×11)/(2×4) = 55/8 = 6 7/8. Because both numbers exceed 1, their product must exceed each factor.
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.