Perform Operations for RIT under 215: adding and subtracting fractions and mixed numbers with unlike denominators, carrying and borrowing wholes, and estimating sums and differences with benchmarks.
Subject: NWEA MAP Growth Math · 61 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
Unlike denominators, mixed numbers, and estimating with benchmarks
Objectives
Six MAP skills sit here, and every one rests on a single requirement.
The requirement: you can only add or subtract pieces that are the same size. Everything else is bookkeeping.
Section
Part 1
Concept
A denominator says how big each piece is. Halves and thirds are different pieces, so they cannot be counted together.
Figure (svg): A bar cut into halves above a bar cut into thirds, showing the pieces are different sizes
Denominator — The bottom number — how many equal pieces the whole was cut into. A bigger denominator means smaller pieces.
Numerator — The top number — how many of those pieces you have.
Concept
You do not change the amount. You cut both bars more finely until the pieces line up.
Figure (svg): Halves and thirds both re-cut into sixths so the pieces match
Sixths work because both halves and thirds can be cut into sixths without any leftover.
Intuition
Multiplying the top and bottom by the same number cuts every piece into smaller pieces without changing the amount shaded.
Figure (svg): One half shown as two quarters and as three sixths, all the same shaded amount
This is why one half, two quarters and three sixths all shade the same portion of the bar.
Warm-up
Retrieve first — you have more fraction sense than you think.
Discussion prompt
Write down three fractions that are equal to one half, without doing any calculation.
Hint: Think about cutting a cake into more and more pieces.
Answer:
Common answers: 2/4, 3/6, 4/8, 5/10, 50/100.
Notice the pattern: the top is always exactly half the bottom. That single observation is the benchmark you will use to estimate later in this deck.
Pattern
Four steps, and they never change no matter how awkward the numbers look.
Step three is where most errors happen, and they are almost all the same error.
Trap
A student computes 1/2 plus 1/3 as 2/5, adding the tops and the bottoms.
The denominator is not a quantity being added — it is the name of the piece. Adding halves to thirds gives 3/6 plus 2/6, which is 5/6.
Sanity check: 1/2 is already bigger than 2/5, so a sum of two positive fractions could never be smaller than one of them.
Socratic
This is the question that unlocks the whole topic.
Discussion prompt
When you add 3/6 and 2/6 to get 5/6, why does the 6 not change?
Hint: Replace the word sixths with the word apples and read the sentence again.
Answer:
Because the 6 is not a number being combined; it is the name of the piece. Three sixths plus two sixths is five sixths, exactly as three apples plus two apples is five apples.
You would never add apples to get apples squared, and the denominator behaves the same way.
Definition probe
Before adding anything, check whether the pieces already match.
Sort into buckets
Sort each pair by whether it can be added immediately.
Translation
Re-cutting is the only tool you need for step two.
Match the pairs
Why: In each case the bottom was multiplied by a whole number, and the top was multiplied by exactly the same number, which keeps the value unchanged while making the pieces finer.
Explain it to yourself
Explaining it is what makes it stick.
Discussion prompt
In your own words, why must denominators match before adding, but not before multiplying?
Hint: Think about what each operation is physically doing to the bar.
Answer:
Adding counts pieces, and you can only count pieces that are the same size.
Multiplying does something different — it takes a part of a part — so the sizes do not have to line up first. That is why multiplying fractions is actually easier than adding them.
Section
Part 2
Worked example
Work out 1/2 plus 1/3.
Find a common denominator
Why: 6 works, because both 2 and 3 divide into it.
Rewrite both fractions
Why: 1/2 becomes 3/6, and 1/3 becomes 2/6.
Add the numerators only
Why: 3 plus 2 is 5, so the answer is 5/6.
\[ \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \]
Verify: against the picture
Why: The bar shows five of six pieces shaded, which is a little over a half — exactly what adding a half and a third should give.
Intuition
The product of the two denominators always works. The smallest one is neater but never required.
Figure (svg): Halves and thirds both re-cut into sixths so the pieces match
| denominators | product works | smallest that works |
|---|---|---|
| 2 and 3 | 6 | 6 |
| 4 and 6 | 24 | 12 |
| 3 and 9 | 27 | 9 |
Prediction
More than one answer is correct, but one is tidier.
Predict first
For 3/4 plus 5/6, which common denominator is smallest?
Correct: 12
If you cannot spot the smallest, use the product — it is never wrong, only untidier.
Why: Both 4 and 6 divide into 12, so 12 is the smallest that works. The product 24 also works but leaves bigger numbers and a fraction that then needs simplifying.
Worked example
Work out 3/4 minus 1/6.
Find a common denominator
Why: 12 works, since 4 and 6 both divide into it.
Rewrite both fractions
Why: 3/4 becomes 9/12, and 1/6 becomes 2/12.
Subtract the numerators only
Why: 9 minus 2 is 7, so the answer is 7/12.
\[ \frac{3}{4} - \frac{1}{6} = \frac{9}{12} - \frac{2}{12} = \frac{7}{12} \]
Verify: the size is sensible
Why: 7/12 is a little over a half, which is right — taking a small sixth away from three quarters should leave something just above a half.
Intuition
Nothing changes for subtraction except the sign between the numerators.
Figure (svg): Halves and thirds both re-cut into sixths so the pieces match
The denominators still have to match, and the denominator still stays put.
Faded example
The common denominator is chosen; supply the rewritten numerators.
Fill in the blanks
\frac85 + \frac______ = \frac___}___ + \frac___}___ = \frac______
Why: To turn fifths into twentieths, multiply top and bottom by 4, so 2/5 becomes 8/20. To turn quarters into twentieths, multiply by 5, so 1/4 becomes 5/20. Adding gives 13/20.
Error analysis
A student added two fractions with unlike denominators.
Annotate
On: \( \frac{2}{3} + \frac{1}{4} = \frac{3}{7} \)
The size check alone — a sum must be bigger than either part — catches this every time.
Check
Solve it on paper before you click.
Check your understanding
What is 1/3 + 1/4?
Answer: A
Why: Twelve is the smallest common denominator, so 1/3 becomes 4/12 and 1/4 becomes 3/12. Adding the numerators gives 7/12, which is a little over a half.
Comparison
Every row uses the same four-step method.
Comparison matrix
| problem | rewritten | answer |
|---|---|---|
| 1/2 + 1/5 | 5/10 + 2/10 | 7/10 |
| 3/4 - 1/2 | 3/4 - 2/4 | 1/4 |
| 2/3 + 1/6 | 4/6 + 1/6 | 5/6 |
The middle column is the whole job; the last column is just addition.
Elimination
You can often discard options before computing anything.
Eliminate the wrong options
Which could be the value of 5/6 minus 1/4?
Survives elimination: e1
Why: Rewriting over 12 gives 10/12 minus 3/12, which is 7/12. Two of the wrong options could be eliminated purely on size, without any calculation at all.
Section
Part 3
Concept
Mixed numbers are easier than they look, because the wholes and the parts can be handled separately.
Figure (svg): A mixed number three and one quarter shown as three whole bars and one quarter
Mixed number — A whole number and a proper fraction written together, such as 3 and 1/4.
Worked example
Work out 2 and 1/3 plus 1 and 1/2.
Add the wholes
Why: 2 plus 1 is 3.
Give the fractions a common denominator
Why: Sixths: 1/3 becomes 2/6, and 1/2 becomes 3/6.
Add the fractions
Why: 2/6 plus 3/6 is 5/6.
Recombine
Why: 3 wholes and 5/6.
\[ 2\tfrac{1}{3} + 1\tfrac{1}{2} = 3 + \frac{5}{6} = 3\tfrac{5}{6} \]
Verify: with an estimate
Why: The two numbers are roughly 2.3 and 1.5, so the answer should be near 3.8 — and 3 and 5/6 is about 3.83, which matches.
Intuition
Sometimes the fraction part spills over into another whole, and that whole must be carried.
Figure (svg): A mixed number three and one quarter shown as three whole bars and one quarter
This is exactly like carrying a ten in whole-number addition.
Worked example
Work out 1 and 3/4 plus 2 and 1/2.
Figure (svg): Three quarters and two quarters combining into one whole and one quarter
Add the wholes
Why: 1 plus 2 is 3.
Common denominator for the fractions
Why: Quarters: 3/4 stays, and 1/2 becomes 2/4.
Add the fractions
Why: 3/4 plus 2/4 is 5/4, which is more than one whole.
Carry the extra whole
Why: 5/4 is 1 and 1/4, so add that 1 to the wholes, giving 4 and 1/4.
\[ 1\tfrac{3}{4} + 2\tfrac{1}{2} = 3 + \frac{5}{4} = 4\tfrac{1}{4} \]
Verify: with an estimate
Why: The two numbers are roughly 1.75 and 2.5, giving about 4.25, and 4 and 1/4 is exactly 4.25.
Prediction
Predict before computing — it tells you whether to expect a carry.
Predict first
In 2 and 2/3 plus 1 and 2/3, will the fraction part be more than one whole?
Correct: yes, because 2/3 plus 2/3 is more than 1
Comparing each fraction to a half is the fastest way to predict a carry.
Why: Each 2/3 is more than a half, so two of them must exceed one whole. Their sum is 4/3, which is 1 and 1/3, so a whole is carried and the answer is 4 and 1/3.
Fill the middle
The wholes are done; finish the fraction part.
Fill in the blanks
3\tfrac3___ + 2\tfrac______ = 5 + \frac___}___ = 5\tfrac______
Why: Rewriting 1/2 as 2/4 gives 1/4 plus 2/4, which is 3/4. Since that is less than one whole, nothing is carried and the answer is 5 and 3/4.
Check
Solve it on paper before you click.
Check your understanding
What is 1 1/2 + 2 1/3?
Answer: A
Why: The wholes give 3. The fractions become 3/6 and 2/6, which add to 5/6, and that is less than a whole so nothing carries. The answer is 3 and 5/6.
Real world
This is the arithmetic of recipes, timber and fabric.
Discussion prompt
A recipe needs 1 and 1/2 cups of flour for the base and 3/4 of a cup for the topping. How much flour in total, and why does the answer need a mixed number?
Answer:
1 and 1/2 is 1 and 2/4, and adding 3/4 gives 1 and 5/4, which carries to 2 and 1/4 cups.
A measuring jug is marked in wholes and parts, so a mixed number is the form you can actually pour.
Ranking
Put the method in the order that avoids rework.
Put in order
Why: Carrying has to come after adding the fractions, because you cannot know whether a carry is needed until the fraction sum exists. Simplifying last avoids simplifying twice.
Analogy
You already carry fluently; only the thing being carried changes.
Match the pairs
Why: In both systems you carry once you have collected a full unit. Ten ones make a ten; four quarters make a whole. The denominator simply tells you how many pieces fill one unit.
Section
Part 4
Concept
If the fraction you are subtracting is bigger than the one you have, break a whole into pieces first.
Figure (svg): A whole being broken into fifths so a subtraction can be completed
This is borrowing, exactly as in whole-number subtraction — only the thing borrowed is a whole rather than a ten.
Worked example
Work out 4 and 3/4 minus 1 and 1/2.
Subtract the wholes
Why: 4 minus 1 is 3.
Common denominator
Why: Quarters: 3/4 stays, and 1/2 becomes 2/4.
Subtract the fractions
Why: 3/4 minus 2/4 is 1/4, and it is positive so no borrowing is needed.
\[ 4\tfrac{3}{4} - 1\tfrac{1}{2} = 3 + \frac{1}{4} = 3\tfrac{1}{4} \]
Verify: with an estimate
Why: Roughly 4.75 minus 1.5 is about 3.25, and 3 and 1/4 is exactly 3.25.
Worked example
Work out 4 and 1/5 minus 1 and 3/5.
Check the fractions
Why: 1/5 is smaller than 3/5, so there are not enough fifths to subtract from.
Borrow a whole
Why: Take 1 from the 4, leaving 3, and turn it into 5/5. Now 1/5 becomes 6/5.
Subtract the wholes
Why: 3 minus 1 is 2.
Subtract the fractions
Why: 6/5 minus 3/5 is 3/5.
\[ 4\tfrac{1}{5} - 1\tfrac{3}{5} = 3\tfrac{6}{5} - 1\tfrac{3}{5} = 2\tfrac{3}{5} \]
Verify: with an estimate
Why: Roughly 4.2 minus 1.6 is about 2.6, and 2 and 3/5 is exactly 2.6.
Intuition
One whole becomes as many pieces as the denominator says, and those pieces join the ones you already had.
Figure (svg): A whole being broken into fifths so a subtraction can be completed
A whole broken into fifths is 5/5, so 4 and 1/5 becomes 3 and 6/5.
Prediction
Deciding this before you start prevents a negative fraction appearing.
Predict first
For 5 and 1/4 minus 2 and 3/4, will borrowing be needed?
Correct: yes, because 1/4 is smaller than 3/4
The whole numbers never decide whether you borrow; only the fractions do.
Why: Borrowing is decided by the fraction parts alone. Since 1/4 is less than 3/4, a whole must be broken, giving 4 and 5/4 minus 2 and 3/4, which is 2 and 2/4, or 2 and 1/2.
Trap
For 4 and 1/5 minus 1 and 3/5, a student computes 3/5 minus 1/5 to avoid a negative, and writes 3 and 2/5.
Flipping the subtraction changes the problem. The correct move is to borrow: 4 and 1/5 becomes 3 and 6/5, and 6/5 minus 3/5 is 3/5, giving 2 and 3/5.
An estimate settles it: 4.2 minus 1.6 is about 2.6, not 3.4.
Faded example
The borrow has started; finish it.
Fill in the blanks
6\tfrac43\tfrac{2}{3} - 2\tfrac______ = 5\tfrac___}___ - 2\tfrac______ = ___
Why: Borrowing one whole turns it into 3/3, and adding the 1/3 already there gives 4/3. Then 4/3 minus 2/3 is 2/3, and 5 minus 2 is 3, so the answer is 3 and 2/3.
Check
Solve it on paper before you click.
Check your understanding
What is 5 1/4 - 2 3/4?
Answer: A
Why: Since 1/4 is less than 3/4, borrow a whole: 5 and 1/4 becomes 4 and 5/4. Then 4 minus 2 is 2, and 5/4 minus 3/4 is 2/4, which simplifies to 1/2. The answer is 2 and 1/2.
Error analysis
A student subtracted two mixed numbers.
Annotate
On: \( 7\tfrac{1}{6} - 3\tfrac{5}{6} = 4\tfrac{4}{6} \)
Estimating first — about 7.2 minus 3.8 is about 3.4 — would have flagged 4 and 2/3 immediately.
Notation
The borrowed form looks wrong until you know what it is saying.
Annotate
On: \( 4\tfrac{1}{5} = 3\tfrac{6}{5} \)
An improper fraction inside a mixed number is legal and temporary; it exists only long enough to complete the subtraction.
Step zero
One check, done first, prevents the commonest mixed-number error.
Discussion prompt
Before subtracting two mixed numbers, what single comparison should you make, and what does each outcome tell you?
Hint: The whole numbers are a distraction here.
Answer:
Compare the two fraction parts only. If the top fraction is the larger, subtract normally.
If the top fraction is smaller, borrow a whole before doing anything else. The whole numbers play no part in this decision.
Section
Part 5
Concept
Benchmark estimation asks only one question of each fraction: which of the three is it closest to?
Figure (svg): A number line marked at zero, one half and one, with several fractions placed on it
Worked example
Estimate 7/8 plus 1/12 without computing exactly.
Place the first fraction
Why: 7 is nearly 8, so 7/8 is close to 1.
Place the second fraction
Why: 1 is tiny compared with 12, so 1/12 is close to 0.
Add the benchmarks
Why: About 1 plus about 0 is about 1.
\[ \frac{7}{8} + \frac{1}{12} \approx 1 \]
Verify: against the exact value
Why: The exact sum is 23/24, which is just under 1 — so the benchmark estimate was accurate and took a fraction of the time.
Intuition
Double the numerator. If it is about the denominator, the fraction is about a half.
Figure (svg): A number line marked at zero, one half and one, with several fractions placed on it
5/9 doubles to 10, which is close to 9, so 5/9 is just over a half.
Sorting
One question per fraction: nearest 0, a half, or 1?
Sort into buckets
Sort each fraction by its nearest benchmark.
Three buckets is all the precision an estimate ever needs.
Estimation
Benchmarks are fastest when the exact answer would be ugly.
Predict first
Roughly what is 11/12 minus 5/11?
Correct: about one half
The exact calculation needs a denominator of 132 — the estimate gets you a usable answer in seconds.
Why: 11/12 is close to 1 and 5/11 is close to one half, so the difference is about one half. The exact answer is 61/132, which is just under a half.
Check
Solve it on paper before you click.
Check your understanding
Which is the best estimate for 9/10 + 2/5?
Answer: A
Why: 9/10 is close to 1 and 2/5 is close to one half, so the sum is about 1 and a half. The exact answer is 13/10, or 1 and 3/10, which rounds to that estimate.
Two truths and a lie
Two of these are sound; one is not.
Eliminate the wrong options
Which statement about adding fractions is false?
Survives elimination: t2
Why: The denominator names the piece and is never added. Adding 1/2 and 1/3 as 2/5 gives an answer smaller than the 1/2 you started with, which is impossible for a sum of positive numbers.
Explain it
A classmate flips the fractions to avoid borrowing.
Discussion prompt
How would you show them that flipping changes the answer, without simply telling them the rule?
Answer:
Ask them to estimate first: 4 and 1/5 minus 1 and 3/5 is roughly 4.2 minus 1.6, which is about 2.6.
Their flipped answer of 3 and 2/5 is 3.4, which is nowhere near. The estimate exposes the error without any appeal to authority.
Missing information
Not every fraction question is answerable as written.
Discussion prompt
A problem says: Ana ate 1/2 of a pizza and Ben ate 1/3 of a pizza, so together they ate 5/6 of a pizza. What assumption is hidden in that answer?
Answer:
That both pizzas are the same size. A half of a small pizza and a third of a large one cannot be added meaningfully.
Fractions only combine when they are fractions of the same whole — which is the same requirement as matching denominators, one level up.
Commit first
Decide your answer and your confidence before revealing.
Predict first
What is 3/4 - 2/3?
Correct: 1/12
The benchmark check agrees: both fractions are a bit over a half, so the difference should be tiny.
Why: Twelve is the common denominator, so 3/4 becomes 9/12 and 2/3 becomes 8/12. Subtracting gives 1/12, a very small amount — which makes sense because 3/4 and 2/3 are close together.
Exit ticket
One item that tells you whether the deck landed.
Predict first
What is 2 1/3 + 1 1/2?
Correct: 3 5/6
Why: The wholes give 3, and the fractions become 2/6 and 3/6, which add to 5/6. Since 5/6 is less than a whole, nothing carries, and the answer is 3 and 5/6.
Connect it up
Your handwriting, one page, from memory.
Draw it
Draw two bars with unlike denominators being re-cut so their pieces match. Beside them write the four-step method, the borrowing rule for mixed numbers, and the three benchmarks with one example fraction each.
If you can draw this from memory, every item in this deck becomes routine.
Recap
Six skills, one requirement.
Almost every error on this topic is either adding the denominators or forgetting to borrow, and an estimate catches both.
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