Perform Operations for RIT under 215: dividing whole numbers by unit fractions and unit fractions by whole numbers, using area models and number lines, and knowing when the answer grows or shrinks.
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
How many fit, and how to share — with area models and number lines
Objectives
Three MAP skills sit here, and the whole topic turns on which number is being divided by which.
The one surprise: dividing by a fraction smaller than one makes the answer bigger.
Section
Part 1
Concept
The most useful reading of division here is: how many of the second number fit inside the first?
Figure (svg): Three whole bars each cut into quarters, showing twelve quarters fit into three wholes
Twelve quarters fit into three wholes, so 3 divided by 1/4 is 12.
Concept
The other reading is sharing: split the first number into as many equal parts as the second says.
Figure (svg): One quarter of a bar being shared between three people, each getting a twelfth
A quarter shared between 3 people gives each a twelfth.
Intuition
Which number sits on which side of the division sign changes the answer completely.
Figure (svg): Two divisions compared: a whole divided by a fraction grows, a fraction divided by a whole shrinks
3 divided by 1/4 is 12. But 1/4 divided by 3 is 1/12. Same numbers, wildly different answers.
Warm-up
Retrieve before being taught.
Discussion prompt
How many halves are there in 4 whole pizzas? Answer without any written work.
Hint: Think about cutting each pizza in half and counting the pieces.
Answer:
Eight. Each pizza gives 2 halves, and 4 pizzas give 8.
You just divided 4 by 1/2 and got 8 — an answer bigger than what you started with. That is the whole surprise of this deck, and you already knew it.
Pattern
Before any model is drawn, settle these two.
Getting this right is worth more than any drawing technique.
Trap
A student says 3 divided by 1/4 must be less than 3, because dividing makes numbers smaller.
Dividing by a number less than one makes the answer bigger. Three wholes contain twelve quarters, so the answer is 12.
The rule from whole numbers does not survive the move to fractions, and MAP tests exactly this.
Socratic
This is the conceptual heart of the deck.
Discussion prompt
Why does dividing by a smaller piece give a bigger number?
Hint: Think about measuring the same room in metres and then in centimetres.
Answer:
Because you are counting how many pieces fit, and smaller pieces fit more times.
It is the same logic as unit conversion: measuring a length in centimetres rather than metres gives a bigger number, because the unit is smaller.
Definition probe
Judge from the divisor alone, with no arithmetic.
Sort into buckets
Sort each division by what happens to the starting number.
Discrimination
Naming the reading picks the model to draw.
Sort into buckets
Sort each story by which reading of division it uses.
Explain it to yourself
Being able to say this stops the two cases being confused.
Discussion prompt
In your own words, what is the difference between 6 divided by 1/2 and 1/2 divided by 6?
Answer:
6 divided by 1/2 asks how many halves fit into 6, which is 12 — a bigger number.
1/2 divided by 6 asks what each of 6 people gets from half a thing, which is 1/12 — a much smaller number.
Analogy
You have already met answers that grow when the unit shrinks.
Match the pairs
Why: Measuring in a smaller unit is literally a division by that unit, and the count always rises. Metric conversion and fraction division are the same operation wearing different clothes.
Ranking
Predict the order before computing anything.
Put in order
Why: The answers are 32, 16, 8 and 2. The smaller the divisor, the more of them fit, so the count is largest when the piece is smallest.
Section
Part 2
Concept
This is the case where the answer grows, and the model is a straight count.
Figure (svg): Three whole bars each cut into quarters, showing twelve quarters fit into three wholes
Each whole contains as many pieces as the denominator says, so multiply the whole by the denominator.
Worked example
Work out 3 divided by 1/4 using bars.
Figure (svg): Three whole bars each cut into quarters, showing twelve quarters fit into three wholes
Draw the amount you start with
Why: Three whole bars.
Cut each into the size of the divisor
Why: Each whole cuts into 4 quarters.
Count the pieces
Why: 3 bars times 4 quarters is 12 pieces.
\[ 3 \div \frac{1}{4} = 12 \]
Verify: by multiplying back
Why: 12 quarters is 12 times 1/4, which is 12/4, or 3 — the amount we started with, so the answer is right.
Intuition
Each whole gives denominator pieces, so the answer is the whole number times the denominator.
Figure (svg): Three whole bars each cut into quarters, showing twelve quarters fit into three wholes
6 divided by 1/3 is 6 times 3, which is 18. No drawing needed once the pattern is seen.
Prediction
Predict the answer from the pattern, not from a drawing.
Predict first
What is 5 divided by 1/6?
Correct: 30
Multiplying the whole number by the denominator is the shortcut this whole section is building to.
Why: Each whole contains 6 sixths, so 5 wholes contain 5 times 6, which is 30 sixths. The answer is much bigger than 5, as dividing by a small piece must give.
Faded example
Four wholes cut into fifths.
Fill in the blanks
4 \div \frac520 = 4 \times ___ = ___
Why: Each whole holds 5 fifths, so 4 wholes hold 4 times 5, which is 20 fifths. The answer grows because the pieces are small.
Check
Solve it on paper before you click.
Check your understanding
How many 1/3 cup scoops are needed to fill 4 cups?
Answer: A
Why: Each cup holds 3 thirds, so 4 cups hold 4 times 3, which is 12 scoops. Dividing by a piece under one gives an answer bigger than 4.
Error analysis
A student worked out 6 divided by 1/2.
Annotate
On: \( 6 \div \frac{1}{2} = 3 \)
Dividing by 1/2 and multiplying by 1/2 do opposite things, and confusing them is the commonest error here.
Comparison
Each row counts how many pieces fit.
Comparison matrix
| division | pieces in one whole | answer |
|---|---|---|
| 2 div 1/4 | 4 | 8 |
| 3 div 1/5 | 5 | 15 |
| 6 div 1/2 | 2 | 12 |
Every answer is larger than the number being divided, which is the signature of a fractional divisor.
Real world
Cooking and building both ask this question constantly.
Discussion prompt
A recipe needs 3 cups of flour and your only scoop holds 1/4 cup. Why is this a division rather than a multiplication?
Answer:
You are asking how many scoops fit into 3 cups, which is 3 divided by 1/4, or 12 scoops.
Multiplying 3 by 1/4 would tell you how much is in three quarters of a cup, which answers a completely different question.
Notation
Which number goes where is the entire question.
Annotate
On: \( 4 \div \frac{1}{3} = 12 \qquad \text{versus} \qquad \frac{1}{3} \div 4 = \frac{1}{12} \)
Reading the order correctly matters more here than any calculation technique.
Edge cases
Testing the extreme confirms the pattern.
Discussion prompt
What happens to 4 divided by 1/n as n gets larger and larger — say 1/100, then 1/1000?
Hint: Apply the multiply-by-the-denominator shortcut with bigger denominators.
Answer:
The answers are 400 and 4,000. The smaller the piece, the more of them fit, without limit.
This is why dividing by something close to zero gives an enormous answer, and why dividing by zero itself has no answer at all.
Section
Part 3
Concept
Here the answer shrinks, because one small amount is being split further.
Figure (svg): One quarter of a bar being shared between three people, each getting a twelfth
A quarter shared between 3 people cuts each quarter into 3, giving twelfths.
Worked example
Work out 1/4 divided by 3 using bars.
Figure (svg): One quarter of a bar being shared between three people, each getting a twelfth
Draw the amount you start with
Why: One bar with a single quarter shaded.
Split that quarter into 3
Why: Cutting every quarter into 3 turns the bar into 12 pieces.
Read one share
Why: The shaded quarter becomes 3 of those twelfths, and one share is 1 twelfth.
\[ \frac{1}{4} \div 3 = \frac{1}{12} \]
Verify: by multiplying back
Why: Three shares of 1/12 is 3/12, which is 1/4 — the amount we started with, so the answer is right.
Intuition
Splitting each piece into 3 multiplies the denominator by 3.
Figure (svg): One quarter of a bar being shared between three people, each getting a twelfth
So 1/5 divided by 4 is 1/20, because the fifths become twentieths.
Prediction
Predict from the pattern.
Predict first
What is 1/3 divided by 5?
Correct: 1/15
Multiplying the denominator by the whole number is the shortcut for this case.
Why: Splitting each third into 5 pieces makes fifteenths, so one share is 1/15. The answer is smaller than 1/3, as sharing must give.
Faded example
One sixth shared between two.
Fill in the blanks
\frac12___ \div 2 = \frac______}
Why: Splitting each sixth in two makes twelfths, so each share is 1/12 — half of one sixth, as expected.
Trap
For 1/4 divided by 3, a student writes 3/4, multiplying the top by 3.
Sharing makes each piece smaller, so the answer must be less than 1/4. Multiplying the denominator gives 1/12.
3/4 is three times bigger than what we started with, which sharing between three people can never produce.
Check
Solve it on paper before you click.
Check your understanding
1/2 of a pizza is shared equally between 4 friends. What fraction of a whole pizza does each get?
Answer: A
Why: Splitting each half into 4 makes eighths, so each friend gets 1/8 of a whole pizza. Four eighths is one half, which checks the answer.
Elimination
Two options can be discarded on size alone.
Eliminate the wrong options
1/3 of a cake is shared between 2 people. What does each get?
Survives elimination: e1
Why: Splitting each third in two makes sixths, so each person gets 1/6. Two sixths is a third, which confirms it.
Step zero
One reading decision prevents the entire family of errors in this deck.
Discussion prompt
You see the calculation 1/5 divided by 4. What should you establish before doing anything?
Hint: Look at which side of the division sign each number is on.
Answer:
That the fraction is the amount being divided and the whole number is the divisor — so this is a sharing question.
That tells you the answer must be smaller than 1/5, which immediately rules out any answer bigger than a fifth.
Comparison
Each row shares one unit fraction between a whole number of people.
Comparison matrix
| shared amount | between | each share |
|---|---|---|
| 1/2 | 3 | 1/6 |
| 1/4 | 2 | 1/8 |
| 1/5 | 3 | 1/15 |
In every row the denominator is multiplied by the number of people, and every share is smaller than the amount shared.
Real world
This case appears whenever a leftover is divided up.
Discussion prompt
Half a cake is left and three people want to share it equally. Why is the answer a sixth rather than a half of three?
Answer:
Each person gets a third of a half, which is 1/6 of the whole cake.
Answering half of three would treat the cake as the divisor rather than the amount, which reverses the question entirely.
Section
Part 4
Concept
A number line turns how-many-fit into a count of equal jumps.
Figure (svg): A number line from zero to three marked in quarters with jumps of one quarter counted
Mark the line in pieces the size of the divisor, then count the jumps to reach the target.
Worked example
Work out 3 divided by 1/4 using a number line.
Figure (svg): A number line from zero to three marked in quarters with jumps of one quarter counted
Draw the line to the target
Why: Mark 0 to 3.
Cut it into divisor-sized pieces
Why: Mark every quarter, giving 4 marks between each whole.
Count the jumps
Why: From 0 to 3 there are 12 jumps of one quarter.
\[ 3 \div \frac{1}{4} = 12 \]
Verify: against the area model
Why: The bars in Part 2 also gave 12, so the two models agree.
Intuition
For the sharing case, you split a single jump rather than counting many.
Figure (svg): A number line from zero to one half showing the half split into three equal jumps
Splitting the jump from 0 to 1/2 into three equal steps makes each step 1/6.
Worked example
Work out 1/2 divided by 3 using a number line.
Figure (svg): A number line from zero to one half showing the half split into three equal jumps
Mark the amount being shared
Why: Draw from 0 to 1 and mark 1/2.
Split that distance into 3
Why: Three equal steps from 0 to 1/2 needs the line cut into sixths.
Read one step
Why: Each step is 1/6.
\[ \frac{1}{2} \div 3 = \frac{1}{6} \]
Verify: by multiplying back
Why: Three steps of 1/6 is 3/6, which is 1/2 — the amount shared, so the answer is right.
Pattern
Each frame adds one more jump of a quarter.
Step through it
How many jumps would it take to reach 5?
The jumps per whole is always the denominator, so the total is the whole number times the denominator.
Matching
Each model answers exactly one of these.
Match the pairs
Why: Counting many pieces answers a how-many-fit question and gives a large answer; splitting one piece answers a sharing question and gives a small one.
Check
Solve it on paper before you click.
Check your understanding
On a number line, how many jumps of 1/5 are needed to reach 2?
Answer: A
Why: Each whole takes 5 jumps of a fifth, so 2 wholes take 2 times 5, which is 10 jumps. This is 2 divided by 1/5.
Notation
The notation and the picture say the same thing in different languages.
Annotate
On: \( 3 \div \frac{1}{4} = 12 \)
Recognising that the answer counts pieces rather than measuring an amount explains why it grows.
Section
Part 5
Concept
Once the reading is settled, each case has a one-step shortcut.
Figure (svg): Two divisions compared: a whole divided by a fraction grows, a fraction divided by a whole shrinks
| calculation | reading | shortcut | answer |
|---|---|---|---|
| whole div unit fraction | how many fit | multiply by the denominator | grows |
| unit fraction div whole | share out | multiply the denominator | shrinks |
Sorting
Sort before calculating.
Sort into buckets
Sort each division by which shortcut it needs.
Which number sits after the division sign decides the whole approach.
Estimation
A rough judgement rules out most wrong options.
Predict first
Roughly how big is 7 divided by 1/8?
Correct: a bit under 60
Just knowing the answer must be much bigger than 7 eliminates three of the four options.
Why: Each whole contains 8 eighths, so 7 wholes contain 7 times 8, which is 56 — a bit under 60. Dividing by a small piece always gives a large count.
Two truths and a lie
Two of these are sound; one is not.
Eliminate the wrong options
Which statement about dividing with fractions is false?
Survives elimination: t2
Why: That rule holds only when dividing by a number greater than one. Dividing by a fraction under one makes the answer bigger, which is the central idea of this deck.
Explain it
A classmate refuses to believe 3 divided by 1/4 is 12.
Discussion prompt
What concrete example would convince them fastest?
Answer:
Ask how many quarter-pizzas they could serve from 3 whole pizzas. They will count 12 without hesitation.
Then show them that the question they just answered is exactly 3 divided by 1/4. A familiar situation beats any rule.
Missing information
Not every question is answerable as written.
Discussion prompt
A problem says: a ribbon is cut into pieces. How many pieces are there? What do you need?
Answer:
The total length of the ribbon and the length of each piece.
Without both, there is no division to do. Naming what is being divided and by what is always the first step.
Invariant
The total never changes, however it is cut.
Step through it
What is the same on every line?
The total amount is invariant. That is why multiplying the answer by the divisor always checks a division.
Commit first
Decide your answer and your confidence before revealing.
Predict first
What is 1/3 divided by 4?
Correct: 1/12
The answer had to be smaller than 1/3, which rules out three of the options immediately.
Why: A third shared between 4 people splits each third into 4, making twelfths. Each share is 1/12, and four twelfths is one third, which confirms it.
Exit ticket
One item that tells you whether the deck landed.
Predict first
How many 1/2 metre pieces can be cut from a 6 metre rope?
Correct: 12 pieces
Why: Each metre gives 2 half-metre pieces, so 6 metres gives 6 times 2, which is 12 pieces. This is 6 divided by 1/2, and the answer is bigger than 6.
Connect it up
Your handwriting, one page, from memory.
Draw it
Draw three whole bars cut into quarters with the twelve pieces counted, and beside it one quarter split between three people. Label which is how-many-fit and which is sharing, and write what happens to the size of the answer in each case.
If you can draw both pictures from memory, you will never confuse the two directions again.
Recap
Three skills, two readings.
The single most valuable habit here is deciding, before any arithmetic, whether the answer should be bigger or smaller than the number you started with.
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