Fractions with the same denominator are just counting; fractions with different denominators have to be renamed first, and this room shows why 1/2 + 1/3 is not 2/5. Equivalent fractions and simplest form, common denominators, mixed numbers with borrowing, and sums that pass 1. Then multiplying: a whole number times a fraction, a fraction of a fraction on a bar model, a fraction of a whole number, and why multiplying by a fraction less than 1 makes the answer smaller. Word problems with recipes, ribbons and miles. Study cards, hints, a practice quiz at three levels, and four workshop activities that check themselves.
Half of a bar, and a third of the same bar. The pieces are different sizes.
You can only add pieces that are the same size. That is the entire reason for a common denominator — it is not a rule somebody made up, it is what makes the counting possible.
Two things people get wrong
⚠️People often think…
1/2 + 1/3 = 2/5.
You never add the bottom numbers — they name the piece size, not a count. It is 3/6 + 2/6 = 5/6. And notice: 2/5 is SMALLER than the 1/2 you started with, which is impossible when you add something to it.
Adding should make it bigger. If it didn't, you added wrong.
⚠️People often think…
“Half of 2/3 cup” means 1/2 + 2/3.
“Of” means times. Half OF 2/3 is 1/2 × 2/3 = 1/3 cup — and it has to come out smaller than 2/3, because you took half of it. Adding would have made it bigger, which is the opposite of what happened.
Half OF something is always less than it.
Make the pieces match first
1Watch one
1/2 + 1/3.
Halves and thirds are different sizes.
Cut both into sixths.
1/2 is 3/6. 1/3 is 2/6.
3 + 2 = 5 sixths. 5/6.
2Do one with me
1/4 + 1/2.
What size piece will both fit into?
1/2 rewritten in those pieces is
So the sum is
💬One sentence, then you move on
Why can't you just add the bottom numbers?
3Try one
A recipe calls for half of 2/3 of a cup. How much is that?
I want a hint first
“Of” means times. And half of anything has to come out smaller than what you started with.
Where this goes
Where this lives
Halving a recipe, measuring a board, reading a tape measure, splitting anything that does not come out even.
What this feeds
Next unit is decimals — the same idea, written a different way.
Name one time you have had to cut a recipe or a measurement in half.
One card at a time — tap “Show me” to check yourself, then Next. Start at Foundation; when those feel easy, climb.
Helpful Hints
🧭 Adding unlike fractions — the four moves
Look at the denominators (2/3 + 1/4: thirds and fourths — different sizes) → Find a common denominator: list multiples until one is on both lists (3, 6, 9, 12 and 4, 8, 12 → 12) → Rename each fraction, multiplying top AND bottom by the same number (2/3 = 8/12, 1/4 = 3/12) → Add the numerators and keep the denominator (8/12 + 3/12 = 11/12), then simplify if you can and check against a benchmark (a bit under 1 — yes).
Same size first. Then just count.
💡 The whole unit in one idea
The denominator is the SIZE of the piece and the numerator is the COUNT. You can only add or subtract pieces of the same size — so unlike denominators get renamed first, and the denominator never gets added. Multiplying is different: a fraction OF something cuts the pieces smaller, so the bottoms multiply, and taking a fraction less than 1 of anything makes it smaller. Every rule in this unit comes from those two sentences.
⚠️ Traps the test loves
Adding the denominators: 1/2 + 1/3 as 2/5. It is 3/6 + 2/6 = 5/6. Notice 2/5 is even smaller than 1/2 — impossible for 1/2 plus something.
Changing the denominator but not the numerator: 2/3 as 2/12 instead of 8/12. Whatever you do to the bottom, do to the top.
Multiplying the bottom when a whole number times a fraction: 3 × 2/5 as 6/15. It is 6/5 — three copies of 2/5.
Flipping the subtraction in a mixed number: 3 1/4 − 1 3/4 as 2 2/4 because “3/4 − 1/4 is easier.” Borrow a whole: 2 5/4 − 1 3/4 = 1 1/2.
Adding when the problem says OF: “half of 2/3 cup” as 1/2 + 2/3. Of means times: 1/2 × 2/3 = 1/3 cup.
Expecting multiplication to always make things bigger. 3/4 × 20 = 15, less than 20. Only a multiplier over 1 grows the number.
🗂️ Which operation, which rule — the chart
Problem
The rule
Worked
Add or subtract, same denominator
add/subtract the tops; keep the bottom
2/5 + 1/5 = 3/5 · 7/8 − 3/8 = 4/8 = 1/2
Add or subtract, unlike denominators
rename to a common denominator first
2/3 + 1/4 = 8/12 + 3/12 = 11/12
Mixed numbers
wholes with wholes, fractions with fractions; borrow a whole if you must
3 1/4 − 1 3/4 = 2 5/4 − 1 3/4 = 1 1/2
Whole number × fraction
multiply the top only
3 × 2/5 = 6/5 = 1 1/5
Fraction × fraction (“of”)
tops times tops, bottoms times bottoms
1/2 × 3/4 = 3/8
Fraction of a whole number
divide by the bottom, multiply by the top
3/4 × 20 = 3 × (20 ÷ 4) = 15
Mixed number × anything
make it improper first
2 1/2 × 4 = 5/2 × 4 = 20/2 = 10
🎯 How the test will ask
A bare sum or difference: 2/3 + 1/4, 5/6 − 1/4. Common denominator, rename, then count.
“Which fraction is equivalent to…?” Both parts multiplied (or divided) by the same number.
Mixed numbers to add or subtract — watch for the one where you must borrow a whole.
A recipe or a ribbon: “half of 3/4 cup,” “2/3 of 12 feet.” Of means multiply.
A shaded bar or an area picture — “what multiplication does this show?” Count the small pieces for the denominator.
“Without multiplying, is the product bigger or smaller than 8?” Look only at whether the fraction is under or over 1.
Someone’s wrong work: 1/2 + 1/3 = 2/5. Say what went wrong and fix it.
✅ Before the test, can you…
Add and subtract fractions with the same denominator, and simplify the answer?
Find a common denominator for thirds and fourths, and for sixths and fourths?
Add 2/3 + 1/4 and subtract 5/6 − 1/4, showing the renaming?
Add 1 1/2 + 2 1/4 and subtract 3 1/4 − 1 3/4?
Find 3 × 2/5, 1/2 × 3/4 and 3/4 × 20?
Draw a bar model for 1/2 of 3/4 and read 3/8 off it?
Explain why 1/2 + 1/3 is not 2/5, and why 3/4 × 20 is less than 20?
Pick your level
Look back at anything you missed — the hint that appeared is exactly what to reread tonight.
How sure did you feel?
Workshop
Work like a mathematician: put the common-denominator steps for 2/3 + 1/4 in order, read a fraction of a fraction off a bar model, sort products by whether they shrank or grew, then settle the oldest fraction argument there is — why 1/2 + 1/3 is not 2/5. Every activity checks itself, and hints are free.
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Your practice record — saved on this device
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