Perform Operations for RIT under 215: fraction and mixed-number word problems with like and unlike denominators, adding three or more fractions, and recipe problems.
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
Turning stories about parts of a whole into additions and subtractions
Objectives
Five MAP skills sit here, and all of them are the arithmetic you already know wrapped in a story.
The skill being tested is not the arithmetic. It is deciding what the story is asking for.
Section
Part 1
Concept
Before anything else, find what the fractions are fractions of. That is the whole, and it never changes during the problem.
Figure (svg): A bar model of a story where three eighths and two eighths of a pizza are eaten
If two fractions are parts of different wholes, they cannot be combined at all.
Concept
Fraction stories come in two shapes, and naming the shape picks the operation for you.
Figure (svg): Two story shapes: parts combining into a total, and a part being taken from a total
Pattern
Ask these in order and the arithmetic becomes routine.
Question four is the one students skip, and it is where most lost marks live.
Warm-up
Retrieve before being taught.
Discussion prompt
Write down three different questions a fraction word problem might ask at the end.
Hint: Think about the last sentence of a word problem, not the first.
Answer:
Common endings: how much altogether, how much is left, how much more one is than the other.
The first is an addition; the second and third are subtractions. Recognising the ending is often faster than reading the whole story.
Discrimination
Decide the operation only. Do not compute anything.
Sort into buckets
Sort each story by its operation.
Socratic
This is the assumption every fraction story quietly makes.
Discussion prompt
Why can you not add 1/2 of a small pizza to 1/3 of a large pizza and get 5/6 of a pizza?
Hint: Ask what each fraction is a fraction of.
Answer:
Because the two fractions are parts of different wholes. A half of a small pizza is less food than a half of a large one.
Fractions only combine when they measure parts of the same whole — which is the same requirement as matching denominators, one level up.
Trap
A story says Ana ate 1/2 of her sandwich and Ben ate 1/3 of his. A student answers that they ate 5/6 of a sandwich.
They ate 5/6 of a sandwich only if both sandwiches were identical. Otherwise the two fractions measure different things and cannot be added.
MAP does include items like this. If the wholes are not stated to be equal, the honest answer is that it cannot be determined.
Translation
The last sentence usually gives the game away.
Match the pairs
Why: Two of these add and two subtract, and the difference is visible in the final sentence alone. Reading the question before the story is a legitimate and fast strategy.
Step zero
One habit prevents most fraction word-problem errors.
Discussion prompt
You are given a fraction word problem. What should you write down before doing any arithmetic at all?
Hint: Neither of the two things is a number.
Answer:
Write down what the whole is, and what the question is actually asking for.
Those two lines take ten seconds and stop you from computing a correct total when the question wanted the remainder.
Explain it to yourself
Explaining it is what makes it usable under pressure.
Discussion prompt
In your own words, what are the two decisions you make before touching any numbers in a fraction story?
Answer:
First, what is the whole that all the fractions refer to.
Second, whether parts are being combined or removed. Everything after that is the arithmetic from the previous deck.
Section
Part 2
Concept
If both fractions have the same denominator, the story needs no preparation at all.
Figure (svg): A bar model of a story where three eighths and two eighths of a pizza are eaten
Combine or remove the numerators, and keep the denominator exactly as it is.
Worked example
A pizza is cut into 8 slices. Ana eats 3 slices and Ben eats 2. What fraction of the pizza did they eat altogether?
Find the whole
Why: The pizza, cut into 8 equal slices, so the whole is 8/8.
Write the parts
Why: Ana ate 3/8 and Ben ate 2/8.
Combining or removing
Why: Altogether means combining, so add.
\[ \frac{3}{8} + \frac{2}{8} = \frac{5}{8} \]
Verify: against the picture
Why: Five of the eight slices are shaded on the bar, which is a little over half the pizza — exactly what three slices plus two out of eight should be.
Intuition
With like denominators the bar model gives the answer directly, with no calculation.
Figure (svg): A bar model of a story where three eighths and two eighths of a pizza are eaten
This is worth doing even when you can do the arithmetic, because it checks the answer for free.
Worked example
A tank is 5/5 full. 3/5 is used. How much is left?
Find the whole
Why: The full tank, which is 5/5.
Write the part removed
Why: 3/5 was used.
Combining or removing
Why: How much is left means removing, so subtract from the whole.
\[ \frac{5}{5} - \frac{3}{5} = \frac{2}{5} \]
Verify: the answer is sensible
Why: Two fifths is less than half a tank, which fits a story where more than half was used.
Intuition
To subtract from a whole, rewrite the whole with the same denominator as the part.
Figure (svg): A whole minus a shaded part, showing what is left over
One whole is 5/5, or 8/8, or 12/12 — whatever the story is cut into.
Prediction
Naming the whole correctly is the whole battle.
Predict first
A garden is divided into 6 equal beds. 4 are planted. What fraction is unplanted?
Correct: 2/6
Writing the whole as 6/6 is what makes the subtraction possible.
Why: The whole is the garden, written as 6/6. Four beds are planted, so 6/6 minus 4/6 is 2/6, which simplifies to 1/3 of the garden left unplanted.
Faded example
A ribbon of 7/10 metre has 3/10 metre cut off.
Fill in the blanks
\frac4___ - \frac______ = \frac___}___
Why: The denominators already match, so only the numerators are subtracted. Seven tenths minus three tenths is four tenths, which simplifies to 2/5 of a metre.
Check
Solve it on paper before you click.
Check your understanding
A jug holds 9/10 of a litre. Sam pours out 4/10 of a litre. How much is left?
Answer: A
Why: The denominators match, so subtract the numerators only. Nine tenths minus four tenths is five tenths, which is half a litre.
Error analysis
A student answered: a cake in 12 slices, 5 eaten, what fraction is left?
Annotate
On: \( \frac{5}{12} \text{ is left} \)
This is the single most common fraction word-problem error, and it costs the mark despite correct arithmetic.
Real world
Anything already divided into equal parts gives like denominators for free.
Discussion prompt
Why do stories about pizza slices, tank gauges and rulers so often have like denominators, while stories about two different recipes usually do not?
Answer:
Because a single object is cut once, into one set of equal pieces, so every fraction of it shares that denominator.
Two different recipes were each divided separately, so their denominators have no reason to match — which is why Part 3 exists.
Section
Part 3
Concept
When the denominators differ, everything in Part 2 still applies — you simply re-cut first.
Figure (svg): Two story shapes: parts combining into a total, and a part being taken from a total
Find a common denominator, rewrite both, then combine the numerators as before.
Worked example
Ana walks 1/2 of a kilometre and then 1/3 of a kilometre. How far did she walk altogether?
Find the whole
Why: A kilometre, and both fractions are parts of it.
Combining or removing
Why: Altogether means combining, so add.
Common denominator
Why: Sixths: 1/2 becomes 3/6 and 1/3 becomes 2/6.
\[ \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \]
Verify: with a benchmark
Why: Half a kilometre plus a bit under half again should be just under a whole kilometre, and 5/6 is just under 1.
Intuition
A benchmark estimate tells you roughly what to expect, which catches a wrong operation instantly.
Figure (svg): A benchmark line with one half and one third marked, summing to just under one
If your answer lands above 1 or below either part, something has gone wrong.
Worked example
A recipe needs 3/4 cup of milk. There is 2/3 of a cup in the jug. How much more is needed?
Find the whole
Why: A cup, and both fractions are parts of it.
Combining or removing
Why: How much more means comparing, so subtract.
Figure (svg): Two bars over twelfths showing nine twelfths needed against eight twelfths available
Common denominator
Why: Twelfths: 3/4 becomes 9/12 and 2/3 becomes 8/12.
\[ \frac{3}{4} - \frac{2}{3} = \frac{9}{12} - \frac{8}{12} = \frac{1}{12} \]
Verify: the size is sensible
Why: Both amounts are close to each other, so the shortfall should be tiny — and 1/12 of a cup is indeed very small.
Prediction
Comparison stories need the same re-cutting as addition ones.
Predict first
Who ran further: Ana with 5/8 of a kilometre, or Ben with 2/3?
Correct: Ben, by 1/24
Both are close to a half plus a bit, so a small difference was expected.
Why: Over 24ths, 5/8 is 15/24 and 2/3 is 16/24. Ben ran further by one twenty-fourth of a kilometre, which is a very small margin.
Fill the middle
A plank of 5/6 metre has 1/4 metre sawn off.
Fill in the blanks
\frac3___ - \frac______ = \frac______ - \frac___}___ = \frac______
Why: Twelve is the common denominator. Multiplying 1/4 top and bottom by 3 gives 3/12, and 10/12 minus 3/12 is 7/12 of a metre.
Check
Solve it on paper before you click.
Check your understanding
Sam reads 1/3 of a book on Monday and 1/4 on Tuesday. What fraction has he read?
Answer: A
Why: Twelve is the common denominator, so 1/3 becomes 4/12 and 1/4 becomes 3/12. Adding gives 7/12 of the book, a little over half.
Elimination
Some options can be discarded before any arithmetic.
Eliminate the wrong options
Ana used 2/3 of a metre of ribbon from a 3/4 metre length. Which could be what is left?
Survives elimination: e1
Why: Over twelfths, 3/4 is 9/12 and 2/3 is 8/12, leaving 1/12 of a metre. Three of the four options could be eliminated on size alone.
Explain it
A classmate adds unlike denominators straight across.
Discussion prompt
How would you show them their answer is impossible, without quoting a rule?
Answer:
Ask them to check their answer against the larger of the two fractions. Adding 1/2 and 1/3 as 2/5 gives something smaller than the 1/2 they started with.
A sum of two positive amounts can never be smaller than either one, so the method must be wrong.
Section
Part 4
Concept
Mixed numbers appear whenever a story involves more than one whole unit — cups, metres, hours.
Figure (svg): A recipe card listing three ingredient amounts as mixed numbers
Handle the wholes and the fraction parts separately, exactly as in the previous deck.
Worked example
A plank is 2 and 1/2 metres and another is 1 and 3/4 metres. What is their total length?
Find the whole
Why: A metre, and both lengths are measured in metres.
Add the wholes
Why: 2 plus 1 is 3.
Common denominator for the parts
Why: Quarters: 1/2 becomes 2/4, and 3/4 stays.
Add the parts and carry
Why: 2/4 plus 3/4 is 5/4, which is 1 and 1/4, so carry the whole.
\[ 2\tfrac{1}{2} + 1\tfrac{3}{4} = 3 + \frac{5}{4} = 4\tfrac{1}{4} \text{ m} \]
Verify: with an estimate
Why: Roughly 2.5 plus 1.75 is about 4.25 metres, and 4 and 1/4 is exactly 4.25.
Intuition
Two part-metres can add to more than a metre, and when they do a whole metre appears.
Figure (svg): A recipe card listing three ingredient amounts as mixed numbers
Nothing is being invented — five quarters really is one whole metre and a quarter.
Worked example
A jug holds 3 and 1/4 litres. 1 and 3/4 litres are poured out. How much is left?
Check the fraction parts
Why: 1/4 is smaller than 3/4, so a whole must be borrowed.
Figure (svg): A whole litre being broken into quarters so the subtraction can proceed
Borrow
Why: 3 and 1/4 becomes 2 and 5/4.
Subtract the wholes
Why: 2 minus 1 is 1.
Subtract the parts
Why: 5/4 minus 3/4 is 2/4, which simplifies to 1/2.
\[ 3\tfrac{1}{4} - 1\tfrac{3}{4} = 2\tfrac{5}{4} - 1\tfrac{3}{4} = 1\tfrac{1}{2} \text{ L} \]
Verify: with an estimate
Why: Roughly 3.25 minus 1.75 is about 1.5 litres, and 1 and 1/2 is exactly 1.5.
Prediction
Decide before you start, from the fraction parts alone.
Predict first
A rope of 5 and 1/3 metres has 2 and 2/3 metres cut off. Is a borrow needed?
Correct: yes, because 1/3 is less than 2/3
The whole numbers never decide whether you borrow.
Why: Only the fraction parts decide. Since 1/3 is smaller than 2/3, borrow a whole: 5 and 1/3 becomes 4 and 4/3, and the answer is 2 and 2/3 metres.
Faded example
A recipe needs 4 and 1/4 cups; 1 and 1/2 cups are already in the bowl.
Fill in the blanks
4\tfrac2\tfrac{3}{4}___ - 1\tfrac______ = 4\tfrac______ - 1\tfrac______ = ___
Why: Since 1/4 is less than 2/4, borrow: 4 and 1/4 becomes 3 and 5/4. Then 3 minus 1 is 2, and 5/4 minus 2/4 is 3/4, so 2 and 3/4 cups are still needed.
Check
Solve it on paper before you click.
Check your understanding
A bag holds 6 1/2 kg of rice. 2 3/4 kg are used. How much is left?
Answer: A
Why: Rewrite 1/2 as 2/4. Since 2/4 is less than 3/4, borrow: 6 and 2/4 becomes 5 and 6/4. Then 5 minus 2 is 3, and 6/4 minus 3/4 is 3/4, giving 3 and 3/4 kilograms.
Error analysis
A student answered: a 4 and 1/8 metre rope with 1 and 5/8 metres cut off.
Annotate
On: \( 4\tfrac{1}{8} - 1\tfrac{5}{8} = 3\tfrac{4}{8} \)
An estimate of 4.1 minus 1.6, about 2.5, would have exposed the answer of 3.5 immediately.
Real world
Cutting materials is the classic setting.
Discussion prompt
A carpenter has a board 8 feet long and needs three pieces of 2 and 1/4 feet. Is there enough, and how much is left?
Answer:
Three pieces of 2 and 1/4 is 6 and 3/4 feet, which is less than 8, so there is enough.
8 minus 6 and 3/4 is 1 and 1/4 feet left over. Note that this needed a borrow, since 8 is 7 and 4/4.
Ranking
Put the work in the order that avoids rework.
Put in order
Why: Naming the whole and the question first is what stops a correct calculation answering the wrong thing. Carrying or borrowing can only be decided after the parts share a denominator.
Section
Part 5
Concept
With three or more fractions, find a single denominator that all of them divide into — not one pair at a time.
Figure (svg): Three fractions of a day combining on one bar
Working in pairs is legal but slower, and it invites arithmetic slips at every step.
Worked example
Ana spends 1/4 of her day at school, 1/3 asleep and 1/3 at home. What fraction of the day is accounted for?
Find the whole
Why: One day, and all three fractions are parts of it.
One common denominator for all three
Why: Twelfths works, since 4 and 3 both divide into 12.
Rewrite all three
Why: 1/4 becomes 3/12, and each 1/3 becomes 4/12.
Add the numerators
Why: 3 plus 4 plus 4 is 11.
\[ \frac{1}{4} + \frac{1}{3} + \frac{1}{3} = \frac{11}{12} \]
Verify: the answer is sensible
Why: Eleven twelfths is just under a whole day, leaving 1/12 unaccounted for — which is plausible for a day that is nearly but not fully described.
Intuition
Drawing all three parts on a single bar makes it obvious that one denominator serves them all.
Figure (svg): Three fractions of a day combining on one bar
It also shows at a glance how much of the whole is left over.
Concept
A recipe lists several fractions of the same unit, which is exactly the three-or-more situation.
Figure (svg): A recipe card listing three ingredient amounts as mixed numbers
Because everything is measured in cups, the amounts can be added directly once the denominators match.
Worked example
A recipe uses 2 and 1/4 cups of flour, 3/4 cup of sugar and 1 and 1/2 cups of oats. How many cups of dry ingredients altogether?
Find the whole
Why: A cup, and every amount is measured in cups.
Add the wholes
Why: 2 plus 0 plus 1 is 3.
Figure (svg): Three fraction parts of a cup adding to six quarters
One denominator for all the parts
Why: Quarters: 1/4 stays, 3/4 stays, and 1/2 becomes 2/4.
Add the parts and carry
Why: 1/4 plus 3/4 plus 2/4 is 6/4, which is 1 and 2/4, so carry a whole.
\[ 2\tfrac{1}{4} + \tfrac{3}{4} + 1\tfrac{1}{2} = 3 + \frac{6}{4} = 4\tfrac{1}{2} \text{ cups} \]
Verify: with an estimate
Why: Roughly 2.25 plus 0.75 plus 1.5 is about 4.5 cups, and 4 and 1/2 is exactly that.
Sorting
Pick one denominator that every fraction in the set divides into.
Sort into buckets
Sort each set by the smallest denominator that works for all of it.
Finding one denominator for the whole set is faster and safer than combining two at a time.
Estimation
An estimate tells you whether the answer is plausible before you compute it.
Predict first
Roughly how many cups is 1 and 3/4 plus 2 and 1/4 plus 1/2?
Correct: about 4 1/2
Rounding each mixed number to the nearest half is usually accurate enough for a recipe.
Why: Roughly 1.75 plus 2.25 plus 0.5 is about 4.5 cups. The exact total is 4 and 1/2 cups, so the estimate is exact here.
Check
Solve it on paper before you click.
Check your understanding
A recipe uses 1/2 cup of oil, 1/4 cup of milk and 1/8 cup of honey. How much liquid in total?
Answer: A
Why: Eighths serve all three, so 1/2 becomes 4/8, 1/4 becomes 2/8 and 1/8 stays. Adding gives 7/8 of a cup, just under a full cup.
Two truths and a lie
Two of these are sound; one is not.
Eliminate the wrong options
Which statement about fraction word problems is false?
Survives elimination: t2
Why: You may find a single denominator that serves all three at once, which is both faster and less error-prone than pairing them up. Combining in pairs is allowed but never required.
Missing information
Not every recipe question can be answered.
Discussion prompt
A recipe says: add 1/2 cup of milk and 1/3 of the flour. How much is that altogether? What is wrong?
Answer:
The two fractions are of different things — one is a fraction of a cup, the other a fraction of the flour.
Without knowing how much flour there is, and that it is measured in cups, the amounts cannot be added. Same units and same whole is the requirement.
Commit first
Decide your answer and your confidence before revealing.
Predict first
A tin holds 2 cups. A recipe uses 3/4 cup and then 1/2 cup. How much is left in the tin?
Correct: 3/4 cup
This is a two-step story: combine the parts first, then subtract from the whole.
Why: The two amounts used are 3/4 plus 2/4, which is 5/4, or 1 and 1/4 cups. Subtracting from 2 cups leaves 3/4 of a cup.
Exit ticket
One item that tells you whether the deck landed.
Predict first
Ana walks 3/8 km and then 1/4 km. How far altogether?
Correct: 5/8 km
Why: Eighths serve both, so 1/4 becomes 2/8. Adding gives 3/8 plus 2/8, which is 5/8 of a kilometre — a little over half a kilometre.
Connect it up
Your handwriting, one page, from memory.
Draw it
Draw a bar model for a fraction story where two parts combine, and another where a part is removed from the whole. Beside them, write the four questions to ask of any fraction story, and the phrases that signal add versus subtract.
If you can draw this from memory, the arithmetic is the only thing left to do.
Recap
Five skills, one reading habit.
The most expensive error on this topic is not arithmetic — it is answering how much was used when the question asked how much was left.
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