Grade 5 · Fractions · MAT-05-NF-007
Multiply Fractions by Fractions
Represent and solve multiply fractions by fractions problems using accurate mathematical models, notation, and reasoning.
Learning objectives
What you should be able to do
- Represent and solve multiply fractions by fractions problems using accurate mathematical models, notation, and reasoning.
- Explain why a strategy for multiply fractions by fractions works and verify the result with an independent check.
Prerequisite check
Make sure the foundation is ready.
The new lesson builds directly on this prior concept, so the prerequisite should be usable rather than merely familiar.
Learn
Build the idea from meaning, not memorization.
Meaning and structure
Multiply Fractions by Fractions depends on treating fractions as numbers built from equal-sized units. The numerator counts those units and the denominator defines their size; operations are valid only when the units being combined are compatible.
Represent the idea
The Math Box uses fraction model as the primary representation. Translate the situation into numbers or geometry first, identify what stays fixed, then change one quantity at a time and observe which relationships remain invariant.
Reason, calculate, and verify
Check multiplication with an inverse division or a second model; check division by multiplying the quotient by the divisor and accounting for any remainder.
Math Box · Interactive lesson
Touch the math. Change it. See what stays true.
Use the interactive model before and after the worked examples. Change the inputs, make a prediction, then use the model to test whether your reasoning holds.
Equal-Share Lab
Keep the whole fixed and change how many equal shares it contains.
- Build one example of multiply fractions by fractions in the Math Box.
- Change one input and predict the effect before moving the control.
- Explain which mathematical relationship stayed invariant after the change.
Worked examples
Twenty different ways to see the concept work.
Multiply 1/2 × 2/5.
1/2 × 2/5 = 1/5An area model shows the overlap of the two fractional dimensions; multiply numerators and denominators, then simplify.
Multiply 2/3 × 3/6.
2/3 × 3/6 = 1/3An area model shows the overlap of the two fractional dimensions; multiply numerators and denominators, then simplify.
Multiply 3/4 × 4/7.
3/4 × 4/7 = 3/7An area model shows the overlap of the two fractional dimensions; multiply numerators and denominators, then simplify.
Multiply 4/5 × 1/3.
4/5 × 1/3 = 4/15An area model shows the overlap of the two fractional dimensions; multiply numerators and denominators, then simplify.
Multiply 5/6 × 3/4.
5/6 × 3/4 = 5/8An area model shows the overlap of the two fractional dimensions; multiply numerators and denominators, then simplify.
Multiply 6/7 × 3/5.
6/7 × 3/5 = 18/35An area model shows the overlap of the two fractional dimensions; multiply numerators and denominators, then simplify.
Multiply 1/2 × 3/6.
1/2 × 3/6 = 1/4An area model shows the overlap of the two fractional dimensions; multiply numerators and denominators, then simplify.
Multiply 2/3 × 3/7.
2/3 × 3/7 = 2/7An area model shows the overlap of the two fractional dimensions; multiply numerators and denominators, then simplify.
Multiply 3/4 × 2/3.
3/4 × 2/3 = 1/2An area model shows the overlap of the two fractional dimensions; multiply numerators and denominators, then simplify.
Multiply 2/5 × 2/4.
2/5 × 2/4 = 1/5An area model shows the overlap of the two fractional dimensions; multiply numerators and denominators, then simplify.
Multiply 1/6 × 4/5.
1/6 × 4/5 = 2/15An area model shows the overlap of the two fractional dimensions; multiply numerators and denominators, then simplify.
Multiply 6/7 × 3/6.
6/7 × 3/6 = 3/7An area model shows the overlap of the two fractional dimensions; multiply numerators and denominators, then simplify.
Multiply 1/2 × 2/7.
1/2 × 2/7 = 1/7An area model shows the overlap of the two fractional dimensions; multiply numerators and denominators, then simplify.
Multiply 2/3 × 1/3.
2/3 × 1/3 = 2/9An area model shows the overlap of the two fractional dimensions; multiply numerators and denominators, then simplify.
Multiply 3/4 × 1/4.
3/4 × 1/4 = 3/16An area model shows the overlap of the two fractional dimensions; multiply numerators and denominators, then simplify.
Multiply 4/5 × 1/5.
4/5 × 1/5 = 4/25An area model shows the overlap of the two fractional dimensions; multiply numerators and denominators, then simplify.
Multiply 2/6 × 3/6.
2/6 × 3/6 = 1/6An area model shows the overlap of the two fractional dimensions; multiply numerators and denominators, then simplify.
Multiply 6/7 × 1/7.
6/7 × 1/7 = 6/49An area model shows the overlap of the two fractional dimensions; multiply numerators and denominators, then simplify.
Multiply 1/2 × 2/3.
1/2 × 2/3 = 1/3An area model shows the overlap of the two fractional dimensions; multiply numerators and denominators, then simplify.
Multiply 2/3 × 3/4.
2/3 × 3/4 = 1/2An area model shows the overlap of the two fractional dimensions; multiply numerators and denominators, then simplify.
Practice
Guided practice with hints
01Multiply 3/4 × 2/5.
Hint: Name the quantities and representation first; then choose the operation or comparison that preserves their meaning.
Answer: 3/4 × 2/5 = 3/10
02Multiply 2/5 × 3/6.
Hint: Name the quantities and representation first; then choose the operation or comparison that preserves their meaning.
Answer: 2/5 × 3/6 = 1/5
03Multiply 3/6 × 6/7.
Hint: Name the quantities and representation first; then choose the operation or comparison that preserves their meaning.
Answer: 3/6 × 6/7 = 3/7
04Multiply 6/7 × 1/3.
Hint: Name the quantities and representation first; then choose the operation or comparison that preserves their meaning.
Answer: 6/7 × 1/3 = 2/7
05Multiply 1/2 × 2/4.
Hint: Name the quantities and representation first; then choose the operation or comparison that preserves their meaning.
Answer: 1/2 × 2/4 = 1/4
06Multiply 2/3 × 3/5.
Hint: Name the quantities and representation first; then choose the operation or comparison that preserves their meaning.
Answer: 2/3 × 3/5 = 2/5
Practice
Independent practice
01Multiply 3/4 × 3/6.
Answer: 3/4 × 3/6 = 3/8
02Multiply 4/5 × 5/7.
Answer: 4/5 × 5/7 = 4/7
03Multiply 4/6 × 2/3.
Answer: 4/6 × 2/3 = 4/9
04Multiply 6/7 × 1/4.
Answer: 6/7 × 1/4 = 3/14
05Multiply 1/2 × 2/5.
Answer: 1/2 × 2/5 = 1/5
06Multiply 2/3 × 3/6.
Answer: 2/3 × 3/6 = 1/3
07Multiply 3/4 × 4/7.
Answer: 3/4 × 4/7 = 3/7
08Multiply 4/5 × 1/3.
Answer: 4/5 × 1/3 = 4/15
Common mistakes
Learn to catch the error, not just the answer.
Track the size of the fractional unit first; fractions can be combined only when the units are compatible.
Compare complete fraction values with a model, benchmark, common denominator, or cross-product reasoning.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Scale recipes, measurements, money, and data while preserving the meaning of the unit.
- Compare quantities that are not whole numbers in science, construction, shopping, and measurement.
Challenge problems
Multiply 5/6 × 3/4.
5/6 × 3/4 = 5/8
Multiply 6/7 × 3/5.
6/7 × 3/5 = 18/35
Multiply 1/2 × 3/6.
1/2 × 3/6 = 1/4
Multiply 2/3 × 3/7.
2/3 × 3/7 = 2/7
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 5 Math Strategy Sprint
Solve and explain multiply fractions by fractions problems accurately, then check each result with a second method.
Practice
Mastery check
01Multiply 3/4 × 2/3.
Answer: 3/4 × 2/3 = 1/2
02Multiply 2/5 × 2/4.
Answer: 2/5 × 2/4 = 1/5
03Multiply 1/6 × 4/5.
Answer: 1/6 × 4/5 = 2/15
04Multiply 6/7 × 3/6.
Answer: 6/7 × 3/6 = 3/7
05Multiply 1/2 × 2/7.
Answer: 1/2 × 2/7 = 1/7
06Multiply 2/3 × 1/3.
Answer: 2/3 × 1/3 = 2/9
07Multiply 3/4 × 1/4.
Answer: 3/4 × 1/4 = 3/16
08Multiply 4/5 × 1/5.
Answer: 4/5 × 1/5 = 4/25
Terminology
Words to know
- numerator
- The number of selected equal parts or unit fractions.
- denominator
- The number of equal parts that make one whole.
- equivalent fractions
- Different fraction names for the same numerical value.
- benchmark
- A familiar reference value such as 0, 1/2, or 1 used for comparison.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
California Mathematics Framework (2023)
Grade-level instructional guidance, mathematical practices, and coherent progression.Common Core State Standards for Mathematics — Grades 4–5
Grade-level expectations for operations, number and fraction work, measurement, data, and geometry.Principles to Actions: Ensuring Mathematical Success for All
Evidence-based emphasis on reasoning, representations, discourse, and conceptual understanding.