Perform Operations for RIT under 215: finding a fraction of a whole number with models and by calculation, choosing the easier route, and applying it in word problems.
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
Finding a fraction of a whole number, with models and in word problems
Objectives
Four MAP skills sit here, and they are the same calculation approached from four directions.
The one idea: the word of means multiply, and a fraction of a number is smaller than the number.
Section
Part 1
Concept
When a question says a fraction of a number, the word of is a multiplication sign wearing a disguise.
Figure (svg): Twelve counters split into three groups of four, with one group circled to show one third of twelve
Fraction of a number — Splitting the number into as many equal parts as the denominator says, then taking as many as the numerator says.
Concept
The denominator cuts, and the numerator collects. That order is the whole method.
Figure (svg): A bar of twenty divided into five equal parts with three of them shaded
Intuition
Taking a fraction of something means taking part of it, so the result is less than what you started with.
Figure (svg): A comparison showing that a fraction less than one makes a number smaller
If your answer is bigger than the original number, the operation went the wrong way.
Warm-up
Retrieve first — you do this arithmetic constantly outside school.
Discussion prompt
Without writing anything down, what is half of 30, a quarter of 20, and a third of 18?
Hint: Think about sharing money or splitting a bill.
Answer:
15, 5 and 6.
Every one of those was a division by the denominator. Unit fractions of a number are just divisions, which is why they feel automatic.
Pattern
Three steps, and they never change.
Step three costs one second and catches almost every error on this topic.
Socratic
Both orders are legal, so why prefer one?
Discussion prompt
To find 3/4 of 20, you could divide by 4 then multiply by 3, or multiply by 3 then divide by 4. Why is the first usually better?
Hint: Compare the size of the intermediate numbers.
Answer:
Because dividing first keeps the numbers small: 20 divided by 4 is 5, then 5 times 3 is 15.
Multiplying first gives 60, which is a bigger number to then divide. Both give 15, but the first route is easier to do mentally.
Definition probe
Unit fractions are the easy case, because they are pure division.
Sort into buckets
Sort each fraction by whether its numerator is 1.
Translation
MAP words this in several ways, and they all mean the same thing.
Match the pairs
Why: A unit fraction of a number is exactly a division, and multiplying by a fraction is exactly taking that fraction of the number. The four phrasings describe only two distinct calculations.
Trap
For 3/4 of 20, a student computes 20 times 3 and answers 60, ignoring the denominator.
60 is bigger than 20, which is impossible when taking part of something. The denominator must be used: 20 divided by 4 is 5, and 5 times 3 is 15.
The size check — the answer must be smaller than the number — catches this instantly.
Explain it to yourself
Explaining it is what makes it stick.
Discussion prompt
In your own words, what job does the denominator do and what job does the numerator do?
Hint: One of them divides and one of them multiplies.
Answer:
The denominator cuts the number into equal parts and tells you how many parts there are.
The numerator collects — it says how many of those parts you keep.
Analogy
You already take fractions of numbers whenever money is split.
Match the pairs
Why: Splitting and then collecting shares is exactly the divide-then-multiply method. The fraction notation is just a shorter way of describing something you already do.
Section
Part 2
Worked example
Find one third of 12 using a model.
Figure (svg): Twelve counters split into three groups of four, with one group circled to show one third of twelve
Cut by the denominator
Why: Split 12 counters into 3 equal groups, giving 4 in each group.
Collect by the numerator
Why: The numerator is 1, so take one group.
\[ \frac{1}{3} \times 12 = 4 \]
Verify: the size
Why: 4 is smaller than 12, as taking a part of something must be, and three groups of 4 rebuild the original 12.
Intuition
A bar is faster to draw than counters and works for any size of number.
Figure (svg): A bar of twenty divided into five equal parts with three of them shaded
Cut the bar into denominator pieces, label one piece, then shade numerator pieces.
Worked example
Find three fifths of 20 using a bar.
Figure (svg): A bar of twenty divided into five equal parts with three of them shaded
Cut by the denominator
Why: Split the bar of 20 into 5 equal parts. Each part is 20 divided by 5, which is 4.
Collect by the numerator
Why: Shade 3 of those parts, giving 3 times 4.
\[ \frac{3}{5} \times 20 = 3 \times 4 = 12 \]
Verify: the size
Why: 12 is less than 20 and more than half of it, which fits three fifths being a bit over a half.
Prediction
Finding one part is always the first move.
Predict first
To find 2/7 of 42, what is one seventh of 42?
Correct: 6
Once one part is known, every other fraction of that number is a quick multiplication.
Why: One seventh means dividing 42 into 7 equal parts, and 42 divided by 7 is 6. Two sevenths is then 2 times 6, which is 12.
Faded example
The bar has been cut; collect the parts.
Fill in the blanks
\frac39 \text___ 24: \quad 24 \div 8 = ___, \quad ___ \times 3 = ___
Why: Dividing 24 into 8 parts gives 3 in each part. Collecting 3 of those parts gives 9, which is less than 24 as expected.
Check
Solve it on paper before you click.
Check your understanding
A bar showing 30 is cut into 6 equal parts. What is 5/6 of 30?
Answer: A
Why: Each of the 6 parts is 30 divided by 6, which is 5. Collecting 5 of those parts gives 25, which is less than 30 as taking a part must be.
Error analysis
A student modelled 2/3 of 15.
Annotate
On: \( 15 \div 2 = 7.5, \quad 7.5 \times 3 = 22.5 \)
Swapping the two roles is the commonest error here, and the size check exposes it every time.
Ranking
Put the method in the order that keeps the numbers small.
Put in order
Why: Dividing before multiplying keeps every intermediate value small, and the final check costs a second while catching the swapped-roles error.
Sorting
Deciding this first tells you which route to take.
Sort into buckets
Sort each by whether the number divides cleanly by the denominator.
Checking divisibility takes a second and picks the easier route for you.
Section
Part 3
Concept
You may divide first or multiply first. Both are correct; one is usually easier.
Figure (svg): Two routes to three quarters of twenty: divide then multiply, or multiply then divide
Divide first whenever the number divides cleanly by the denominator, which it usually does in MAP items.
Worked example
Work out three quarters of 20.
Figure (svg): Two routes to three quarters of twenty: divide then multiply, or multiply then divide
Divide by the denominator
Why: 20 divided by 4 is 5.
Multiply by the numerator
Why: 5 times 3 is 15.
\[ \frac{3}{4} \times 20 = 15 \]
Verify: by the other route
Why: Multiplying first gives 20 times 3 which is 60, and 60 divided by 4 is also 15 — so both routes agree.
Intuition
If the number does not divide evenly by the denominator, multiply first instead.
Figure (svg): Two routes to three quarters of twenty: divide then multiply, or multiply then divide
For 2/3 of 10, dividing gives an awkward 3.33, but multiplying first gives 20, and 20 divided by 3 is 6 and 2/3.
Worked example
Work out two thirds of 10.
Figure (svg): A comparison showing that a fraction less than one makes a number smaller
Try dividing first
Why: 10 divided by 3 is not a whole number, so this route gets messy.
Multiply first instead
Why: 10 times 2 is 20.
Then divide
Why: 20 divided by 3 is 6 with remainder 2, which is 6 and 2/3.
\[ \frac{2}{3} \times 10 = \frac{20}{3} = 6\tfrac{2}{3} \]
Verify: the size
Why: 6 and 2/3 is less than 10 and more than half of it, which fits two thirds being a bit over a half.
Prediction
Choosing the route is a real decision, not a formality.
Predict first
For 5/6 of 42, which route keeps the numbers smallest?
Correct: divide by 6 first
Check whether the number divides cleanly by the denominator — if it does, always divide first.
Why: 42 divides cleanly by 6 to give 7, and 7 times 5 is 35. Multiplying first gives 210, which is a much bigger number to then divide by 6.
Fill the middle
Two sevenths of 56.
Fill in the blanks
56 \div 7 = 8, \quad 16 \times 2 = ___
Why: 56 divides cleanly by 7 to give 8, so one seventh is 8. Two sevenths is 2 times 8, which is 16 — comfortably less than 56.
Comparison
Each row is one fraction of one number.
Comparison matrix
| fraction of | one part | answer |
|---|---|---|
| 3/5 of 25 | 5 | 15 |
| 2/9 of 45 | 5 | 10 |
| 4/7 of 21 | 3 | 12 |
| 5/8 of 40 | 5 | 25 |
The middle column is the only real work; the last column is one multiplication.
Check
Solve it on paper before you click.
Check your understanding
What is 3/8 of 40?
Answer: A
Why: 40 divided by 8 is 5, so one eighth is 5. Three eighths is 3 times 5, which is 15 — less than 40 as expected.
Elimination
Some options can be discarded before any arithmetic.
Eliminate the wrong options
Which could be 2/5 of 35?
Survives elimination: e1
Why: 35 divided by 5 is 7, and 7 times 2 is 14. Three of the four options were eliminated on size alone, without computing anything.
Notation
The same calculation appears in three different notations on MAP.
Annotate
On: \( \frac{3}{4} \times 20 = \frac{3 \times 20}{4} = \frac{60}{4} = 15 \)
Recognising all three notations as the same question is worth more than any single method.
Section
Part 4
Concept
In a story, the whole number is the size of the group, and the fraction is the part being asked about.
Figure (svg): A bar model for a story where two fifths of thirty students walk to school
Drawing the whole group first is what stops the fraction being applied to the wrong number.
Worked example
A class has 30 students. Two fifths of them walk to school. How many walk?
Figure (svg): A bar model for a story where two fifths of thirty students walk to school
Find the whole group
Why: 30 students is the number the fraction applies to.
Divide by the denominator
Why: 30 divided by 5 is 6, so one fifth is 6 students.
Multiply by the numerator
Why: 2 times 6 is 12.
\[ \frac{2}{5} \times 30 = 12 \text{ students} \]
Verify: the answer is sensible
Why: 12 is less than 30 and less than half of it, which fits two fifths being under a half. The other 18 students do not walk.
Intuition
A story often contains two or three numbers, and the fraction only applies to one of them.
Figure (svg): A bar model for a story where two fifths of thirty students walk to school
The fraction always applies to the total group, not to a part that has already been separated out.
Step zero
One question prevents the most common story error.
Discussion prompt
A story says: a shop has 48 apples and 12 oranges. One quarter of the apples are sold. What number does the fraction apply to?
Hint: Read the noun immediately after the fraction.
Answer:
The 48 apples, not the 60 pieces of fruit and not the 12 oranges.
Naming which number the fraction belongs to, before calculating, is what stops the wrong total being used.
Discrimination
Identify the target number only. Do not calculate.
Sort into buckets
Sort by whether the fraction applies to the first or second number given.
Check
Solve it on paper before you click.
Check your understanding
A library has 84 books. Three quarters of them are fiction. How many are fiction?
Answer: A
Why: 84 divided by 4 is 21, so one quarter is 21 books. Three quarters is 3 times 21, which is 63 books — less than 84 as it must be.
Real world
Sales, tips and surveys all run on fractions of a number.
Discussion prompt
A shop offers one third off a 60 pound coat. Why is it a mistake to answer 20 pounds when asked what the coat now costs?
Answer:
20 pounds is the discount, not the price. One third of 60 is 20, so the saving is 20 pounds.
The new price is 60 minus 20, which is 40 pounds. The question asked for what remains, not for the part removed.
Error analysis
A student answered: a team of 24 players, two thirds are forwards, how many are not forwards?
Annotate
On: \( 24 \div 3 = 8, \quad 8 \times 2 = 16 \text{ are not forwards} \)
Correct arithmetic answering the wrong question is the most expensive mistake on this topic.
Matching
Same numbers, same fraction, different question.
Match the pairs
Why: Three of these give 8 but by different routes, and only the first asks for the fraction itself. Reading the final question is what separates them.
Section
Part 5
Concept
A three-digit number and an awkward denominator use exactly the same two steps.
Figure (svg): Two routes to three quarters of twenty: divide then multiply, or multiply then divide
The only new decision is which route keeps the arithmetic manageable.
Worked example
Work out 5/8 of 240.
Figure (svg): A bar of twenty divided into five equal parts with three of them shaded
Check the division
Why: 240 divides cleanly by 8, so divide first.
Divide by the denominator
Why: 240 divided by 8 is 30, so one eighth is 30.
Multiply by the numerator
Why: 5 times 30 is 150.
\[ \frac{5}{8} \times 240 = 150 \]
Verify: the size
Why: 150 is less than 240 and more than half of it, which fits five eighths being just over a half.
Intuition
Before computing, decide roughly what fraction of the number you expect.
Figure (svg): A comparison showing that a fraction less than one makes a number smaller
Five eighths is a bit over a half, so the answer should be a bit over half the number.
Estimation
A benchmark estimate rules out three of four options instantly.
Predict first
Roughly what is 7/8 of 320?
Correct: a bit under 320
Placing the fraction against the benchmarks of 0, a half and 1 is usually enough to pick the answer.
Why: Seven eighths is close to a whole, so the answer should be a little under 320. The exact value is 280, which is indeed just under.
Faded example
Four ninths of 360.
Fill in the blanks
360 \div 9 = 40, \quad 160 \times 4 = ___
Why: 360 divides cleanly by 9 to give 40, so one ninth is 40. Four ninths is 4 times 40, which is 160 — under half of 360, as four ninths should be.
Check
Solve it on paper before you click.
Check your understanding
What is 3/4 of 176?
Answer: A
Why: 176 divided by 4 is 44, so one quarter is 44. Three quarters is 3 times 44, which is 132 — less than 176 and clearly over half of it.
Two truths and a lie
Two of these are sound; one is not.
Eliminate the wrong options
Which statement about a fraction of a number is false?
Survives elimination: t2
Why: A fraction less than one takes only part of the number, so the result is smaller. It would only be larger if the fraction were an improper one greater than 1, which this deck does not use.
Explain it
A classmate keeps dividing by the numerator.
Discussion prompt
How would you fix this for them without just repeating the rule?
Answer:
Ask them to do a case they already know by heart: half of 10. The denominator 2 is what they divide by, and they will do that automatically.
Then extend it to 3/4 of 20 and ask which number played the role that the 2 played. The pattern names itself.
Missing information
Not every question can be answered.
Discussion prompt
A problem says: three fifths of the students passed. How many passed? What is missing?
Answer:
The total number of students. A fraction on its own is not a quantity — it needs a number to be a fraction of.
This is exactly why the first step of the method is to name the number the fraction applies to.
Commit first
Decide your answer and your confidence before revealing.
Predict first
What is 2/3 of 96?
Correct: 64
Two thirds is a bit over a half, so an answer a bit over 48 was expected.
Why: 96 divided by 3 is 32, so one third is 32. Two thirds is 2 times 32, which is 64 — less than 96 and clearly over half of it.
Exit ticket
One item that tells you whether the deck landed.
Predict first
A farm has 45 animals and 4/9 of them are sheep. How many are sheep?
Correct: 20
Why: 45 divided by 9 is 5, so one ninth is 5 animals. Four ninths is 4 times 5, which is 20 sheep — under half the farm, as four ninths should be.
Invariant
Two different orders of working, and something identical at the end of both.
Step through it
What is identical about the two routes, and what differs?
The answer is invariant; only the size of the intermediate numbers changes. That is why you may always pick the easier route.
Edge cases
Testing an idea at its extreme shows whether you really believe it.
Discussion prompt
What happens if the fraction is 5/4 rather than 3/4? Is the answer still smaller than the number?
Hint: Compare the numerator with the denominator first.
Answer:
No. 5/4 of 20 is 25, which is bigger than 20, because 5/4 is more than one whole.
The size check needs stating precisely: a fraction less than one shrinks the number, and a fraction greater than one grows it. Everything in this deck uses proper fractions, so the check holds throughout.
Connect it up
Your handwriting, one page, from memory.
Draw it
Draw a bar being cut into denominator parts with numerator parts shaded. Beside it, write the two steps, the note about which route to choose, and the size check that a fraction of a number must be smaller than the number.
If you can draw this from memory, every item in this deck becomes a two-step calculation.
Recap
Four skills, one method.
The two commonest errors are swapping the roles of numerator and denominator, and answering the part when the question wanted the rest.
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