Grade 5 · Fractions · MAT-05-NF-012
Divide Whole Numbers by Unit Fractions
Represent and solve divide whole numbers by unit fractions problems using accurate mathematical models, notation, and reasoning.
Learning objectives
What you should be able to do
- Represent and solve divide whole numbers by unit fractions problems using accurate mathematical models, notation, and reasoning.
- Explain why a strategy for divide whole numbers by unit 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
Divide Whole Numbers by Unit 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 divide whole numbers by unit 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.
Compute 1 ÷ 1/2.
1 ÷ 1/2 = 2Each whole contains 2 pieces of size 1/2; 1 wholes contain 2 such pieces.
Compute 2 ÷ 1/3.
2 ÷ 1/3 = 6Each whole contains 3 pieces of size 1/3; 2 wholes contain 6 such pieces.
Compute 3 ÷ 1/4.
3 ÷ 1/4 = 12Each whole contains 4 pieces of size 1/4; 3 wholes contain 12 such pieces.
Compute 4 ÷ 1/5.
4 ÷ 1/5 = 20Each whole contains 5 pieces of size 1/5; 4 wholes contain 20 such pieces.
Compute 5 ÷ 1/6.
5 ÷ 1/6 = 30Each whole contains 6 pieces of size 1/6; 5 wholes contain 30 such pieces.
Compute 6 ÷ 1/7.
6 ÷ 1/7 = 42Each whole contains 7 pieces of size 1/7; 6 wholes contain 42 such pieces.
Compute 7 ÷ 1/8.
7 ÷ 1/8 = 56Each whole contains 8 pieces of size 1/8; 7 wholes contain 56 such pieces.
Compute 1 ÷ 1/9.
1 ÷ 1/9 = 9Each whole contains 9 pieces of size 1/9; 1 wholes contain 9 such pieces.
Compute 2 ÷ 1/2.
2 ÷ 1/2 = 4Each whole contains 2 pieces of size 1/2; 2 wholes contain 4 such pieces.
Compute 3 ÷ 1/3.
3 ÷ 1/3 = 9Each whole contains 3 pieces of size 1/3; 3 wholes contain 9 such pieces.
Compute 4 ÷ 1/4.
4 ÷ 1/4 = 16Each whole contains 4 pieces of size 1/4; 4 wholes contain 16 such pieces.
Compute 5 ÷ 1/5.
5 ÷ 1/5 = 25Each whole contains 5 pieces of size 1/5; 5 wholes contain 25 such pieces.
Compute 6 ÷ 1/6.
6 ÷ 1/6 = 36Each whole contains 6 pieces of size 1/6; 6 wholes contain 36 such pieces.
Compute 7 ÷ 1/7.
7 ÷ 1/7 = 49Each whole contains 7 pieces of size 1/7; 7 wholes contain 49 such pieces.
Compute 1 ÷ 1/8.
1 ÷ 1/8 = 8Each whole contains 8 pieces of size 1/8; 1 wholes contain 8 such pieces.
Compute 2 ÷ 1/9.
2 ÷ 1/9 = 18Each whole contains 9 pieces of size 1/9; 2 wholes contain 18 such pieces.
Compute 3 ÷ 1/2.
3 ÷ 1/2 = 6Each whole contains 2 pieces of size 1/2; 3 wholes contain 6 such pieces.
Compute 4 ÷ 1/3.
4 ÷ 1/3 = 12Each whole contains 3 pieces of size 1/3; 4 wholes contain 12 such pieces.
Compute 5 ÷ 1/4.
5 ÷ 1/4 = 20Each whole contains 4 pieces of size 1/4; 5 wholes contain 20 such pieces.
Compute 6 ÷ 1/5.
6 ÷ 1/5 = 30Each whole contains 5 pieces of size 1/5; 6 wholes contain 30 such pieces.
Practice
Guided practice with hints
01Compute 7 ÷ 1/6.
Hint: Name the quantities and representation first; then choose the operation or comparison that preserves their meaning.
Answer: 7 ÷ 1/6 = 42
02Compute 1 ÷ 1/7.
Hint: Name the quantities and representation first; then choose the operation or comparison that preserves their meaning.
Answer: 1 ÷ 1/7 = 7
03Compute 2 ÷ 1/8.
Hint: Name the quantities and representation first; then choose the operation or comparison that preserves their meaning.
Answer: 2 ÷ 1/8 = 16
04Compute 3 ÷ 1/9.
Hint: Name the quantities and representation first; then choose the operation or comparison that preserves their meaning.
Answer: 3 ÷ 1/9 = 27
05Compute 4 ÷ 1/2.
Hint: Name the quantities and representation first; then choose the operation or comparison that preserves their meaning.
Answer: 4 ÷ 1/2 = 8
06Compute 5 ÷ 1/3.
Hint: Name the quantities and representation first; then choose the operation or comparison that preserves their meaning.
Answer: 5 ÷ 1/3 = 15
Practice
Independent practice
01Compute 6 ÷ 1/4.
Answer: 6 ÷ 1/4 = 24
02Compute 7 ÷ 1/5.
Answer: 7 ÷ 1/5 = 35
03Compute 1 ÷ 1/6.
Answer: 1 ÷ 1/6 = 6
04Compute 2 ÷ 1/7.
Answer: 2 ÷ 1/7 = 14
05Compute 1 ÷ 1/2.
Answer: 1 ÷ 1/2 = 2
06Compute 2 ÷ 1/3.
Answer: 2 ÷ 1/3 = 6
07Compute 3 ÷ 1/4.
Answer: 3 ÷ 1/4 = 12
08Compute 4 ÷ 1/5.
Answer: 4 ÷ 1/5 = 20
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
Compute 5 ÷ 1/6.
5 ÷ 1/6 = 30
Compute 6 ÷ 1/7.
6 ÷ 1/7 = 42
Compute 7 ÷ 1/8.
7 ÷ 1/8 = 56
Compute 1 ÷ 1/9.
1 ÷ 1/9 = 9
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 5 Math Strategy Sprint
Solve and explain divide whole numbers by unit fractions problems accurately, then check each result with a second method.
Practice
Mastery check
01Compute 2 ÷ 1/2.
Answer: 2 ÷ 1/2 = 4
02Compute 3 ÷ 1/3.
Answer: 3 ÷ 1/3 = 9
03Compute 4 ÷ 1/4.
Answer: 4 ÷ 1/4 = 16
04Compute 5 ÷ 1/5.
Answer: 5 ÷ 1/5 = 25
05Compute 6 ÷ 1/6.
Answer: 6 ÷ 1/6 = 36
06Compute 7 ÷ 1/7.
Answer: 7 ÷ 1/7 = 49
07Compute 1 ÷ 1/8.
Answer: 1 ÷ 1/8 = 8
08Compute 2 ÷ 1/9.
Answer: 2 ÷ 1/9 = 18
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.