Multiplying a fraction by a whole number
Repeated addition of fractions, multiply the numerator, and mixed-number results
About three to four lessons of 45 to 60 minutes
Three friends, three slices of the same pizza
Three friends share a movie night. A pizza is cut into 5 equal slices, and each friend eats 2 of those slices, so each eats 2/5 of the pizza. How much pizza did they eat altogether? You could add it up: 2/5 and 2/5 and 2/5. But when you add the same amount over and over, there is a faster tool you already own for whole numbers: multiplication.
Three lots of 2/5 is 3 x 2/5, exactly the way three lots of 4 is 3 x 4. Today we take multiplication, which you have only ever used on whole numbers, and stretch it to cover fractions. By the end you will multiply a whole number and a fraction in one step, and turn a top-heavy answer like 6/5 back into the friendly mixed number 1 and 1/5.
- 3 friends each eat 2/5 of a pizza3 x 2/5, three copies of two fifths joined together
- 4 bottles each 3/4 full4 x 3/4, four copies of three quarters of a bottle
- 6 ribbons each 2/3 of a meter6 x 2/3, six copies of two thirds
- 2 people each eat 3/8 of a bar2 x 3/8, two copies of three eighths
What students will be able to do
Students will understand multiplying a fraction by a whole number as repeated addition of that fraction, compute the product by multiplying the whole number and the numerator while keeping the denominator, and rewrite an improper-fraction result as a mixed number, simplifying where possible.
- I can write a repeated addition of the same fraction as a whole number times that fraction.
- I can multiply a whole number by a fraction by multiplying it by the numerator and keeping the denominator.
- I can show the product on a number line as equal jumps of the fraction.
- I can rewrite an improper-fraction answer as a mixed number.
- I can simplify the answer to lowest terms when it is possible.
Standards this unit teaches
- 4.NF.B.4Common Core (US)Multiply a fraction by a whole number
Apply and extend understanding of multiplication to multiply a fraction by a whole number. Understand a fraction a/b as a multiple of the unit fraction 1/b, so that n copies of a/b is (n x a)/b, building the idea as repeated addition of unit fractions.
- 5.NF.B.4Common Core (US)Multiply fractions, including by a whole number
Apply and extend understanding of multiplication to multiply a fraction by a whole number or by another fraction, interpreting the product with area models, number lines and repeated addition, and connecting a whole number times a fraction to scaling.
- AC9M4N02Australian Curriculum v9 (ACARA)Unit fractions and their multiples (Year 4)
Recognise and represent unit fractions and their multiples in different ways, and combine same-denominator fractions to make a whole. A multiple of 1/b, such as 3/5 being three copies of 1/5, is the foundation this unit multiplies from.
- AC9M5N05Australian Curriculum v9 (ACARA)Add and subtract fractions (Year 5)
Solve problems that add and subtract fractions using knowledge of equivalent fractions. Multiplying a fraction by a whole number is repeated addition of same-denominator fractions, so it sits inside this Year 5 descriptor.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Naming and understanding fractionsa fraction is a count of unit fractions, the idea this unit multiplies
- Fractions on a number lineequal jumps of a unit fraction from 0
- Multiplication as equal groupsn x a is n copies of a, now applied to a fraction
- Equivalent and simplified fractionsreducing an answer to lowest terms
Words to teach and display
- Fraction
- a number that names equal parts of one whole, such as 2/5
- Numerator
- the top number, how many equal parts you have
- Denominator
- the bottom number, how many equal parts the whole is cut into
- Unit fraction
- a fraction with 1 on top, one single part such as 1/5
- Product
- the answer to a multiplication
- Improper fraction
- a fraction whose top is as big as or bigger than its bottom, such as 6/5
- Mixed number
- a whole number and a fraction together, such as 1 and 1/5
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Multiplying is repeated adding of the same fraction
ConcreteStart where whole-number multiplication started. Three lots of 4 is 4 + 4 + 4, and we shortened it to 3 x 4. Fractions work the same way. Three friends each eat 2/5 of a pizza, so the total is 2/5 + 2/5 + 2/5, three copies of the same fraction. That repeated addition is exactly what 3 x 2/5 means.
Draw three identical pizza bars, each with 2 of its 5 slices shaded. Line them up and the point is plain: you have three groups of 2/5, and multiplication is the tool for equal groups. So 3 x 2/5 is the same total as 2/5 + 2/5 + 2/5.
Keep the meaning front and centre before any rule: a whole number times a fraction is that many copies of the fraction joined together. The whole number counts the copies, the fraction is the size of each copy.
- Write 3/8 + 3/8 as a multiplication.
- In 4 x 2/3, how many copies are there, and what is the size of each copy?
- Why is multiplying a quicker way to add the same fraction many times?
2. Multiply the numerator, keep the denominator
PictorialNow find the total without drawing every time. The trick is to count in unit fractions. 2/5 is two of the fifth-size pieces, 1/5. Three copies of 2/5 is three lots of two fifth-pieces, which is 6 of those fifth-pieces altogether, so 6/5. The pieces never changed size, there are just more of them.
That gives the rule, and the number line proves it. To work out a whole number times a fraction, multiply the whole number by the numerator and keep the denominator the same: 3 x 2/5 = (3 x 2)/5 = 6/5. The denominator names the size of each piece, and that size does not change when you take more pieces, so it stays put.
The number line shows it as equal jumps. Start at 0 and jump 2/5 three times: you land on 2/5, then 4/5, then 6/5. Three jumps of 2/5 reach 6/5, and 6/5 is just past 1.
Work out 4 x 3/8.
- 3/8 is three eighth-size pieces. Four copies is 4 lots of 3 pieces.
- Multiply the whole number by the numerator and keep the denominator: (4 x 3)/8 = 12/8.
- The denominator stayed 8 because the piece size did not change.
Answer: 4 x 3/8 = 12/8 (which we will simplify and rewrite as a mixed number next).
- What is 2 x 3/8 as a fraction over 8?
- In 5 x 2/9, which number do you multiply the 5 by, and which number stays the same?
3. Turning a top-heavy answer into a mixed number
PictorialProducts often come out top-heavy, like 6/5, where the top is bigger than the bottom. That is a perfectly correct answer called an improper fraction, but a mixed number tells the story more clearly: how many whole ones, and how much is left over.
6/5 means six fifth-pieces. Five of them make one whole (5/5 = 1), and one fifth-piece is left over. So 6/5 is one whole and 1/5, written 1 and 1/5. The number line agrees: 6/5 sits one fifth past 1.
The routine is a division. To change an improper fraction to a mixed number, divide the numerator by the denominator. The quotient is the number of whole ones and the remainder, over the same denominator, is the leftover fraction. For 6/5, 6 divided by 5 is 1 remainder 1, giving 1 and 1/5.
Rewrite 12/8 as a mixed number in lowest terms.
- Divide the numerator by the denominator: 12 divided by 8 is 1 remainder 4, so 12/8 = 1 and 4/8.
- Simplify the fraction part: 4/8 = 1/2, dividing top and bottom by 4.
- Combine the whole and the simplified fraction.
Answer: 12/8 = 1 and 4/8 = 1 and 1/2.
- How many fifth-pieces make one whole? So how many wholes are in 6/5?
- Change 7/3 to a mixed number.
4. The one-step rule and simplifying
AbstractPull the moves into a single clean method the class can rely on. To multiply a whole number by a fraction: multiply the whole number by the numerator, keep the denominator, then simplify and rewrite as a mixed number if the top ends up as big as or bigger than the bottom.
You can also write the whole number as a fraction over 1 and multiply straight across: 4 x 3/8 = 4/1 x 3/8 = 12/8. This is the same rule and it is the bridge to multiplying two fractions in a later unit, so it is worth showing once the repeated-addition meaning is secure.
Always finish the job: simplify to lowest terms and present the answer as a mixed number when it is improper. 4 x 3/8 = 12/8 = 3/2 = 1 and 1/2. An answer left as 12/8 is correct but not complete.
Work out 6 x 2/3, giving the answer as a whole or mixed number in lowest terms.
- Multiply the whole number by the numerator, keep the denominator: (6 x 2)/3 = 12/3.
- Divide to change to a whole or mixed number: 12 divided by 3 is 4 exactly, remainder 0.
- There is no leftover fraction, so the answer is a whole number.
Answer: 6 x 2/3 = 12/3 = 4.
- What is 5 x 2/5, and why does it come out to a whole number?
- After multiplying to get 10/4, what two finishing steps still remain?
5. Solving real problems
AbstractFractions times whole numbers show up whenever equal fractional amounts are repeated: cups in a recipe, lengths of ribbon, portions of food, distances walked each day. The skill is to spot the equal groups, write the multiplication, then simplify the answer.
Read for the two numbers: how many copies (the whole number) and the size of each copy (the fraction). A recipe uses 3/4 cup of flour per batch and you make 5 batches, so the flour is 5 x 3/4 cups.
Then compute and interpret. 5 x 3/4 = 15/4 = 3 and 3/4 cups. Always send the answer back to the story so it makes sense: three and three-quarter cups of flour is a believable amount for five batches.
A recipe uses 3/4 cup of flour per batch. How much flour is needed for 5 batches?
- Five equal copies of 3/4 cup is 5 x 3/4.
- Multiply the whole number by the numerator, keep the denominator: (5 x 3)/4 = 15/4.
- Change to a mixed number: 15 divided by 4 is 3 remainder 3, so 15/4 = 3 and 3/4.
Answer: 5 batches need 15/4 cups, which is 3 and 3/4 cups of flour.
- A path is walked 4 times, and each walk is 2/3 of a kilometer. Write and solve the multiplication.
- Why does it help to turn the final answer into a mixed number for a real-world problem?
Common misconceptions and how to address them
MisconceptionTo multiply, multiply the whole number by both the top and the bottom, so 3 x 2/5 = 6/15.
Why it happens: Students apply multiply everything, or confuse this with making an equivalent fraction where both parts are multiplied.
How to address it: Return to unit-fraction counting: 2/5 is fifth-size pieces, and taking more pieces does not change their size, so the denominator stays 5. Only the number of pieces grows: (3 x 2)/5 = 6/5. Multiplying the bottom too would shrink the pieces, which taking more copies never does.
MisconceptionAn answer like 6/5 is wrong because the top cannot be bigger than the bottom.
Why it happens: Early fraction work only ever showed proper fractions inside one whole.
How to address it: Show 6/5 on the number line just past 1. It is a correct number, an improper fraction, worth more than one whole. It can be rewritten as the mixed number 1 and 1/5, but it was never wrong.
MisconceptionThe product of a whole number and a fraction is always smaller than the whole number, because multiplying by a fraction makes things smaller.
Why it happens: Students overgeneralise from multiplying by a proper fraction as scaling down.
How to address it: Here the whole number is doing the copying, so the result is bigger than one copy of the fraction, not smaller than the whole number. 3 x 2/5 = 6/5 is three times as much as 2/5. Keep the meaning as copies of the fraction.
MisconceptionLeave the answer as 12/8, since that is what the multiplication gave.
Why it happens: Students stop at the raw product and skip the finishing steps.
How to address it: Insist on two finishing moves every time: simplify to lowest terms, and rewrite as a mixed number if it is improper. 12/8 = 3/2 = 1 and 1/2. A correct answer is not a complete answer until it is simplified.
MisconceptionWhen changing 6/5 to a mixed number, the whole number is 6 divided by 1 or the top number itself.
Why it happens: Students are unsure which number to divide by when converting.
How to address it: Divide the numerator by the denominator, top by bottom: 6 divided by 5 is 1 remainder 1, giving 1 and 1/5. The denominator counts how many pieces make one whole, so it is the divisor.
Misconception3 x 2/5 and 3 + 2/5 are the same thing.
Why it happens: The words times and plus blur together, and both involve the numbers 3 and 2/5.
How to address it: Contrast the two on the line. 3 + 2/5 is one jump to 3 then a small 2/5 jump, landing near 3. 3 x 2/5 is three equal jumps of 2/5, landing on 6/5, just past 1. Multiplying means copies of the fraction, not the whole number plus the fraction.
Guided practice (with answers)
1. Write 2/3 + 2/3 + 2/3 + 2/3 as a multiplication and solve it.
Answer: 4 x 2/3 = (4 x 2)/3 = 8/3 = 2 and 2/3.
2. Work out 3 x 2/5.
Answer: (3 x 2)/5 = 6/5 = 1 and 1/5.
3. Work out 5 x 3/4 and give the answer as a mixed number.
Answer: (5 x 3)/4 = 15/4 = 3 and 3/4.
4. Work out 6 x 1/3.
Answer: (6 x 1)/3 = 6/3 = 2. It is a whole number because 3 pieces make one whole and there are six of them.
5. Rewrite 12/8 in lowest terms as a mixed number.
Answer: 12 divided by 8 is 1 remainder 4, so 1 and 4/8, and 4/8 = 1/2, giving 1 and 1/2.
6. A bottle holds 3/4 of a liter. How much do 4 full bottles hold?
Answer: 4 x 3/4 = 12/4 = 3 liters.
Independent practice worksheets
Set the matching ChalkBee fraction worksheets for independent work. The answer keys are computed in code, so they are never wrong. Start with same-denominator addition to lock in the repeated-addition meaning, then move to the multiplication and simplifying sets.
Differentiation
- Stay concrete for longer: draw or fold each copy of the fraction before writing the multiplication.
- Keep denominators small (halves, thirds, quarters, fifths) until the rule is secure.
- Give the repeated-addition line already written so the student only counts the total pieces.
- Use a fraction wall or number line on every question so the improper answer is always seen, not just computed.
- Multiply where the answer simplifies to a whole number and ask why, such as 3 x 2/6.
- Work backwards: 3 x what fraction equals 9/4?
- Compare 3 x 2/5 with 2/5 x 3 and check the answer is the same, previewing that multiplication is commutative for fractions too.
- Introduce a fraction of a set as related, such as 2/3 of 12, and discuss how it connects to whole number times fraction.
Assessment: exit ticket
A three-question exit ticket for the last five minutes, sampling the repeated-addition meaning, the one-step rule, and converting an improper answer.
1. Write 3/5 + 3/5 as a multiplication and solve it.
Answer: 2 x 3/5 = 6/5 = 1 and 1/5.
2. Work out 4 x 2/3.
Answer: (4 x 2)/3 = 8/3 = 2 and 2/3.
3. Rewrite 10/4 as a mixed number in lowest terms.
Answer: 10 divided by 4 is 2 remainder 2, so 2 and 2/4, and 2/4 = 1/2, giving 2 and 1/2.
Teacher notes and timings
- Rough timing across three to four lessons: Lesson 1 repeated addition and the meaning (section 1), Lesson 2 the rule with the number line (section 2), Lesson 3 improper to mixed and simplifying (sections 3 to 4), Lesson 4 word problems plus the exit ticket (section 5 and assessment).
- Language to keep saying: copies of the fraction, multiply the top and keep the bottom, the pieces stay the same size, finish the job by simplifying. These pre-empt most of the misconceptions.
- The number-line diagrams use fraction tick labels only (no decimals shown), so the jumps read as fifths and quarters. If your class has met decimals, you can note that 6/5 is 1.2, but the fraction labels are what the lesson needs.
- The single most common error is multiplying the denominator too (6/15 instead of 6/5). Meet it head on with unit-fraction counting: more pieces, same-size pieces.
- Curriculum note and a US and AU alignment: the US introduces multiplying a fraction by a whole number as repeated addition of unit fractions in Grade 4 (4.NF.B.4) and generalises it in Grade 5 (5.NF.B.4). ACARA has no dedicated multiply-a-fraction-by-a-whole-number descriptor at Year 5 or 6. The skill is met as multiples of unit fractions at Year 4 (AC9M4N02) and repeated addition of same-denominator fractions at Year 5 (AC9M5N05), with formal fraction multiplication arriving at Year 7 (AC9M7N06), so in an Australian setting this unit suits Year 5 or early Year 6.
- Present mode and print both work: use the Print button for a clean teacher copy or a student handout, and project the page to teach straight from the diagrams.