Perform Operations for RIT under 215: multiplying unit fractions and general fractions with area models, cancelling before multiplying, and justifying whether scaling by a fraction makes a number grow or shrink.
Subject: NWEA MAP Growth Math · 60 slides · symbolic lesson
Open the interactive version of this deck · Homework for this lesson
Title
MAP Growth · The Real and Complex Number Systems · RIT under 215
Multiplying fractions with area models, and judging whether a number grows or shrinks
Objectives
Three MAP skills sit here, and they build from a picture to a rule to a judgement.
The one idea: multiplying fractions means taking a part of a part, which is why the answer gets smaller.
Section
Part 1
Concept
To find a half of a third, cut the square into thirds one way and halves the other. The overlap is the answer.
Figure (svg): A square split into halves one way and thirds the other, with one small piece shaded to show a sixth
Two cuts create rows times columns small pieces, and that count is the new denominator.
Concept
Adding needs a common denominator. Multiplying needs nothing at all — you just multiply straight across.
Figure (svg): A square cut into quarters and thirds with a two by three block shaded to show two thirds of three quarters
Intuition
Taking a part of a part leaves less than either part you started with.
Figure (svg): Three bars showing a number scaled by a fraction under one, by one, and by a fraction over one
Half of a third is smaller than a third, and smaller than a half. That is not a coincidence — it is what part of a part means.
Warm-up
Retrieve before you are taught.
Discussion prompt
Without calculating, what is half of a half? And half of a quarter?
Hint: Think about folding a piece of paper in half twice.
Answer:
A quarter, and an eighth.
You already multiply fractions — you just say the word of instead of the word times.
Pattern
Three steps, and the third is the one that earns the mark on the harder items.
No common denominator is ever needed for multiplication.
Trap
For 1/2 times 1/3 a student rewrites both over 6, getting 3/6 times 2/6, and answers 6/36.
Common denominators are only needed for adding and subtracting. Multiplying goes straight across: 1 times 1 over 2 times 3, which is 1/6.
6/36 does happen to simplify to 1/6, so this error sometimes hides — but on most numbers it gives the wrong answer outright.
Socratic
This is the question that separates the two operations properly.
Discussion prompt
Adding fractions needs matching pieces. Why does multiplying not?
Hint: Think about what each operation does to the picture.
Answer:
Because adding counts pieces, and you can only count pieces that are the same size.
Multiplying does something different: it cuts one fraction by the other, creating new pieces of its own. The new denominator comes from the two cuts, not from matching anything.
Definition probe
Sorting these correctly saves a lot of wasted work.
Sort into buckets
Sort each calculation by whether a common denominator is needed.
Explain it to yourself
Being able to say this out loud is what stops the two methods being confused.
Discussion prompt
In your own words, what is the difference between what adding fractions does and what multiplying fractions does?
Answer:
Adding puts two amounts of the same whole together, so it needs the pieces to match.
Multiplying takes a part of a part, which makes the result smaller than either piece. It needs no matching at all.
Discrimination
Judge from the fraction alone, with no arithmetic.
Sort into buckets
Sort each multiplication by what it does to 100.
Section
Part 2
Concept
A unit fraction has a numerator of 1, so the product's numerator is always 1 too.
Figure (svg): A square split into halves one way and thirds the other, with one small piece shaded to show a sixth
Unit fraction — A fraction with 1 on top, such as 1/2, 1/3 or 1/8.
Worked example
Find 1/2 of 1/3 using a model.
Figure (svg): A square split into halves one way and thirds the other, with one small piece shaded to show a sixth
Cut for the first fraction
Why: Split the square into 3 columns and shade 1 of them for the third.
Cut for the second fraction
Why: Split it into 2 rows and shade 1 of them for the half.
Read the overlap
Why: One small piece is shaded both ways, out of 6 pieces in total.
\[ \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} \]
Verify: the size
Why: One sixth is smaller than both a half and a third, which is exactly what taking a part of a part must give.
Intuition
Two cuts across each other make rows times columns pieces. That is where the new denominator comes from.
Figure (svg): A square split into halves one way and thirds the other, with one small piece shaded to show a sixth
Three columns and two rows give six pieces, so the answer is in sixths.
Prediction
Predict the denominator before drawing anything.
Predict first
If you cut a square into 4 columns and 5 rows, how many small pieces are there?
Correct: 20
The denominator of the answer is always the number of small pieces the two cuts create.
Why: Four columns crossed with five rows makes 4 times 5, which is 20 small pieces. That is why 1/4 times 1/5 is 1/20.
Faded example
The cuts are made; read off the answer.
Fill in the blanks
\frac15___ \times \frac______ = \frac______}
Why: Three columns crossed with five rows gives 15 small pieces, and exactly one of them lies in both shaded strips, so the answer is 1/15.
Check
Solve it on paper before you click.
Check your understanding
What is 1/4 x 1/6?
Answer: A
Why: Multiply straight across: 1 times 1 is 1, and 4 times 6 is 24, giving 1/24. It is smaller than both a quarter and a sixth, as a part of a part must be.
Error analysis
A student multiplied two unit fractions.
Annotate
On: \( \frac{1}{2} \times \frac{1}{4} = \frac{2}{6} \)
The size check settles it: a product of two proper fractions must be smaller than either one.
Ranking
Predict first, then check by multiplying.
Put in order
Why: The products are 1/4, 1/6, 1/12 and 1/20. As the denominators grow the pieces get smaller, so the products shrink in the order listed.
Analogy
You have done this physically even if never on paper.
Match the pairs
Why: Each fold takes a fraction of what is already there, which is exactly what multiplying fractions does. The piece gets smaller every time, which is why the products shrink.
Step zero
One glance saves the most common wasted step.
Discussion prompt
You are given two fractions to combine. What should you check first, before doing anything else?
Hint: The wasted step is one that belongs to a different operation.
Answer:
Check the operation. If it is a multiplication, skip the common denominator entirely and go straight across.
Students routinely spend time finding a common denominator for a product, which is effort that changes nothing.
Section
Part 3
Concept
With numerators bigger than 1, more columns and more rows are shaded, and the overlap grows.
Figure (svg): A square cut into quarters and thirds with a two by three block shaded to show two thirds of three quarters
The numerators multiply to give the overlap; the denominators multiply to give the total pieces.
Worked example
Find 2/3 of 3/4 using a model.
Figure (svg): A square cut into quarters and thirds with a two by three block shaded to show two thirds of three quarters
Cut and shade for the first fraction
Why: Four columns, with 3 shaded for the three quarters.
Cut and shade for the second fraction
Why: Three rows, with 2 shaded for the two thirds.
Read the overlap
Why: The overlap is 2 by 3, which is 6 pieces, out of 12 in total.
\[ \frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2} \]
Verify: the size
Why: One half is smaller than three quarters and smaller than two thirds, so the answer behaves as a part of a part should.
Intuition
If a numerator and a denominator share a factor, cancel it first. The answer is the same and the numbers stay small.
Figure (svg): A fraction multiplication with a common factor cancelled before multiplying
This is optional but saves simplifying a large fraction afterwards.
Worked example
Work out 2/3 times 3/8.
Figure (svg): A fraction multiplication with a common factor cancelled before multiplying
Look for a shared factor
Why: The 3 on the bottom of the first and the 3 on the top of the second cancel.
Cancel
Why: Both 3s become 1, leaving 2/1 times 1/8.
Multiply what is left
Why: 2 times 1 is 2, and 1 times 8 is 8, giving 2/8.
\[ \frac{2}{3} \times \frac{3}{8} = \frac{2}{8} = \frac{1}{4} \]
Verify: without cancelling
Why: Multiplying straight across gives 6/24, which also simplifies to 1/4 — so cancelling first changed the effort, not the answer.
Prediction
Predict the shape of the answer before computing it.
Predict first
For 3/5 times 2/7, what is the denominator of the answer before simplifying?
Correct: 35
6/35 cannot be simplified, since 6 and 35 share no common factor.
Why: The denominators multiply: 5 times 7 is 35. The numerators give 3 times 2, which is 6, so the answer is 6/35.
Fill the middle
Multiply straight across, then simplify.
Fill in the blanks
\frac6___ \times \frac______ = \frac___}___ = \frac______
Why: The numerators give 3 times 2, which is 6, and the denominators give 4 times 9, which is 36. Then 6/36 simplifies to 1/6.
Comparison
Each row multiplies straight across, then simplifies.
Comparison matrix
| problem | straight across | simplified |
|---|---|---|
| 1/2 x 4/5 | 4/10 | 2/5 |
| 2/3 x 3/5 | 6/15 | 2/5 |
| 3/4 x 2/3 | 6/12 | 1/2 |
Every answer is smaller than both fractions it came from.
Check
Solve it on paper before you click.
Check your understanding
What is 3/5 x 5/6?
Answer: A
Why: Multiplying straight across gives 15/30, which simplifies to 1/2. Cancelling the 5s first gives 3/1 times 1/6, or 3/6, which is also 1/2.
Elimination
Some options can be discarded on size alone.
Eliminate the wrong options
Which could be 2/3 times 1/2?
Survives elimination: e1
Why: Multiplying straight across gives 2/6, which simplifies to 1/3. Three options were eliminated on size before any arithmetic was done.
Notation
The notation packs a lot into very little space.
Annotate
On: \( \frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12} \)
Always check whether your answer can be reduced — an unsimplified answer is often a distractor rather than the key.
Real world
Scaling a recipe is the everyday version of this skill.
Discussion prompt
A recipe needs 2/3 of a cup of milk and you want to make half the recipe. Why is this a multiplication rather than a subtraction?
Answer:
Half of two thirds is a part of a part, which is a multiplication: 1/2 times 2/3 is 2/6, or 1/3 of a cup.
Subtracting would answer a different question entirely — how much less, rather than how much of.
Section
Part 4
Concept
Scaling asks what multiplying by a fraction does to a number. The answer is decided by comparing the factor with 1.
Figure (svg): Three bars showing a number scaled by a fraction under one, by one, and by a fraction over one
Worked example
Without computing exactly, is 12 times 3/4 bigger or smaller than 12? Justify your answer.
Figure (svg): Three bars showing a number scaled by a fraction under one, by one, and by a fraction over one
Compare the factor with 1
Why: 3/4 is less than 1, because the numerator is smaller than the denominator.
Apply the rule
Why: Multiplying by a factor under 1 shrinks the number.
State the justification
Why: So 12 times 3/4 must be less than 12.
\[ 12 \times \frac{3}{4} = 9 \;<\; 12 \]
Verify: by computing
Why: 12 divided by 4 is 3, and 3 times 3 is 9, which is indeed less than 12 — confirming the judgement made before any arithmetic.
Intuition
MAP asks you to justify, not to calculate. Comparing the factor with 1 is the entire justification.
Figure (svg): Three bars showing a number scaled by a fraction under one, by one, and by a fraction over one
This is worth marks even when the exact answer would be awkward.
Sorting
Compare each factor with 1. No arithmetic.
Sort into buckets
Sort each scaling by what it does to the number.
One glance at the numerator against the denominator answers the whole question.
Prediction
The justification is the answer here, not the number.
Predict first
Is 48 times 5/6 greater or less than 48, and why?
Correct: less, because 5/6 is under 1
Note the trap in the second option: multiplying does not always make things bigger.
Why: Multiplying always makes things bigger only when the factor exceeds 1. Since 5/6 is under 1, the result shrinks. The exact value is 40, which is indeed less than 48.
Trap
A student says 20 times 3/4 must be more than 20, because multiplying makes numbers bigger.
That is only true for factors greater than 1. Multiplying by 3/4 takes three quarters of the number, giving 15, which is less than 20.
The rule from whole numbers does not survive the move to fractions, and MAP tests exactly this.
Faded example
Fill the comparison that carries the argument.
Fill in the blanks
\frac<< \;___\; 1 \quad \Rightarrow \quad 32 \times \frac______ \;___\; 32
Why: Because 7 is less than 8, the factor 7/8 is under 1, so the product is less than 32. The exact value is 28.
Check
Solve it on paper before you click.
Check your understanding
Without calculating, which product is greater than 30?
Answer: A
Why: Only 4/3 has a numerator larger than its denominator, so only it is greater than 1 and only it makes the number grow. The product is 40.
Explain it
A classmate insists multiplying always makes a number bigger.
Discussion prompt
What single example would change their mind fastest, and how would you frame it?
Answer:
Ask them what half of 10 is. They will say 5 without hesitation.
Then point out that half of 10 is 10 times 1/2 — a multiplication that made the number smaller. One familiar case does more than any rule.
Error analysis
A student judged whether a product would grow or shrink.
Annotate
On: \( 45 \times \frac{6}{5} \;<\; 45 \)
Compare numerator with denominator every time — do not assume all fractions shrink.
Invariant
The square never changes, however it is cut.
Step through it
What stays the same through all three frames, and what changes?
The whole is invariant; only the number and size of pieces change. That is why the denominators multiply.
Section
Part 5
Concept
The area model explains the rule, the rule does the arithmetic, and the comparison with 1 does the judging.
Figure (svg): A square cut into quarters and thirds with a two by three block shaded to show two thirds of three quarters
A MAP item may ask for any of the three, so it is worth being fluent in all of them.
Worked example
A recipe uses 3/4 of a cup of sugar. You make 2/3 of the recipe. How much sugar, and is it more or less than 3/4 of a cup?
Figure (svg): A square cut into quarters and thirds with a two by three block shaded to show two thirds of three quarters
Recognise the operation
Why: Two thirds of three quarters is a multiplication.
Multiply straight across
Why: 2 times 3 is 6, and 3 times 4 is 12, giving 6/12.
Simplify
Why: 6/12 is 1/2 of a cup.
Justify the comparison
Why: The factor 2/3 is under 1, so the result must be less than 3/4.
\[ \frac{2}{3} \times \frac{3}{4} = \frac{1}{2} \]
Verify: the comparison
Why: A half is indeed less than three quarters, which matches the prediction made from the factor alone.
Intuition
Round each fraction to 0, a half or 1 and multiply the benchmarks for a quick sanity check.
Figure (svg): A benchmark line with two fractions and their product marked
If your computed answer is nowhere near the benchmark product, recheck the arithmetic.
Estimation
Benchmarks give a usable answer in seconds.
Predict first
Roughly what is 5/9 times 7/8?
Correct: about one half
A benchmark estimate would immediately reject an answer like 35/17.
Why: 5/9 is about a half and 7/8 is close to 1, so the product is about a half times 1, which is about a half. The exact value is 35/72, just under a half.
Matching
Judge the size before computing.
Match the pairs
Why: Only the third has a factor greater than 1, and it is the only one where the result grows. The other three all shrink, and the last shrinks dramatically because both factors are tiny.
Two truths and a lie
Two of these are sound; one is not.
Eliminate the wrong options
Which statement about multiplying fractions is false?
Survives elimination: t2
Why: Common denominators are only needed for adding and subtracting. Multiplication creates its own denominator from the two cuts, so nothing needs to match beforehand.
Missing information
Not every scaling question can be answered.
Discussion prompt
A problem says: a number is multiplied by a fraction. Is the result bigger or smaller? What do you need to know?
Answer:
Whether the fraction is less than, equal to, or greater than 1 — which means comparing its numerator with its denominator.
Without that, the question genuinely cannot be answered, since fractions can do all three things.
Commit first
Decide your answer and your confidence before revealing.
Predict first
What is 4/5 x 5/8?
Correct: 1/2
Cancelling first turned this into a one-step calculation.
Why: Cancelling the 5s gives 4/1 times 1/8, which is 4/8, or 1/2. Multiplying straight across gives 20/40, which also reduces to 1/2.
Exit ticket
One item that tells you whether the deck landed.
Predict first
Is 60 times 5/6 greater or less than 60, and what is it?
Correct: less, and it is 50
Why: 5/6 is under 1, so the result shrinks. Computing it, 60 divided by 6 is 10, and 10 times 5 is 50 — less than 60 as predicted.
Connect it up
Your handwriting, one page, from memory.
Draw it
Draw a square cut both ways to show two fractions multiplying, with the overlap shaded. Beside it, write the straight-across rule, the note that no common denominator is needed, and the three scaling outcomes with the comparison that decides each one.
If you can draw this from memory, all three skills in this deck are covered.
Recap
Three skills, one picture.
When a question says justify, the comparison with 1 is the justification — you do not need to compute anything.
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