Perform Operations for RIT under 215: understanding why area is a product, finding areas of rectangles with fractional sides, and handling mixed-number side lengths.
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
Why area is a product, and finding areas when the sides are fractions or mixed numbers
Objectives
Three MAP skills sit here, and together they explain why the fraction multiplication rule works.
The one idea: area counts unit squares, and fractional sides simply cut those squares into smaller pieces.
Section
Part 1
Concept
The area of a rectangle is the number of unit squares that fit inside it.
Figure (svg): A rectangle three units by two units filled with six unit squares
Unit square — A square one unit long on each side. Its area is one square unit.
Because the squares sit in neat rows, counting them is a multiplication rather than a tally.
Concept
When a side is a fraction, the unit square gets cut up — and the pieces are what you count.
Figure (svg): A unit square cut into twelfths with a three by two block outlined to show three quarters by two thirds
This is the same area model as the multiplying-fractions deck, now with the sides named as lengths.
Intuition
Cutting one way into fifths and the other into quarters makes twentieths.
Figure (svg): A unit square being cut into fifths one way and quarters the other, making twentieths
So the denominator of the area is always the product of the two side denominators.
Warm-up
Retrieve before being taught.
Discussion prompt
Write down the area of a rectangle 5 by 3, and say in words why you multiplied.
Hint: Say it in terms of rows, not in terms of a formula.
Answer:
15 square units, because there are 3 rows of 5 unit squares.
That sentence — rows of squares — is the whole justification, and it survives unchanged when the sides become fractions.
Pattern
Three steps, whatever the sides look like.
Forgetting the words square units is a surprisingly common way to lose a mark.
Socratic
The name is not decoration; it describes what is being counted.
Discussion prompt
Why do we say square centimetres rather than just centimetres for an area?
Hint: Think about what physically fits inside the shape.
Answer:
Because what is being counted is squares — little tiles one centimetre on each side.
A length counts centimetres along a line; an area counts centimetre-by-centimetre tiles covering a surface. They are different kinds of quantity, so they need different units.
Definition probe
Naming the quantity keeps the units straight.
Sort into buckets
Sort each measurement by what it describes.
Discrimination
MAP mixes these deliberately, so the words matter.
Sort into buckets
Sort each question by what it is asking for.
Explain it to yourself
This is the insight the first MAP skill in this deck is testing.
Discussion prompt
In your own words, why does length times width still give the area when the sides are fractions rather than whole numbers?
Answer:
Because the rectangle is still made of neat rows of identical pieces. The pieces are just smaller than a whole unit square.
Multiplying still counts rows times pieces per row; only the size of a piece has changed.
Analogy
The formula is a shortcut for something physical.
Match the pairs
Why: Cutting the tiles smaller changes the count but not the floor. That is exactly what happens when fractional sides cut the unit square into smaller pieces.
Section
Part 2
Concept
When both sides are proper fractions, the rectangle sits inside a single unit square.
Figure (svg): A unit square cut into twelfths with a three by two block outlined to show three quarters by two thirds
So the area must be less than one square unit — a useful check.
Worked example
A rectangle is 3/4 of a unit wide and 2/3 of a unit tall. Find its area.
Figure (svg): A unit square cut into twelfths with a three by two block outlined to show three quarters by two thirds
Multiply the sides
Why: Area is width times height, so 3/4 times 2/3.
Multiply straight across
Why: 3 times 2 is 6, and 4 times 3 is 12, giving 6/12.
Simplify
Why: 6/12 reduces to 1/2.
\[ \frac{3}{4} \times \frac{2}{3} = \frac{6}{12} = \frac{1}{2} \text{ square units} \]
Verify: against the picture
Why: The outlined block covers 6 of the 12 small pieces, which is exactly half the unit square — matching the calculation.
Intuition
Both sides are less than a full unit, so the rectangle cannot fill the unit square.
Figure (svg): A unit square being cut into fifths one way and quarters the other, making twentieths
An area above 1 from two proper fractions is always an error.
Prediction
Judge before computing.
Predict first
A rectangle has sides 5/6 and 4/5 of a unit. Is its area more or less than one square unit?
Correct: less, because both sides are under 1
This is the scaling judgement from the previous deck, applied to a geometric setting.
Why: Both sides are shorter than a unit, so the rectangle fits inside the unit square with room to spare. The exact area is 20/30, which is 2/3 of a square unit.
Worked example
A strip is 1/2 of a unit wide and 3/5 of a unit long. Find its area.
Figure (svg): A unit square being cut into fifths one way and quarters the other, making twentieths
Multiply the sides
Why: 1/2 times 3/5.
Multiply straight across
Why: 1 times 3 is 3, and 2 times 5 is 10, giving 3/10.
Check it cannot simplify
Why: 3 and 10 share no factor, so 3/10 is final.
\[ \frac{1}{2} \times \frac{3}{5} = \frac{3}{10} \text{ square units} \]
Verify: the size
Why: 3/10 is under a half, which is right because taking half of 3/5 must give less than 3/5.
Faded example
Both sides are given; find the area.
Fill in the blanks
\frac6___ \times \frac______ = \frac___}___ = \frac______
Why: The numerators give 2 times 3, which is 6, and the denominators give 5 times 4, which is 20. Then 6/20 simplifies to 3/10 square units.
Check
Solve it on paper before you click.
Check your understanding
A rectangle is 2/3 unit by 3/5 unit. What is its area?
Answer: A
Why: Multiplying straight across gives 6/15, which simplifies to 2/5 square units. It is under one, as it must be when both sides are under a unit.
Error analysis
A student found the area of a 1/2 by 1/4 rectangle.
Annotate
On: \( \frac{1}{2} \times \frac{1}{4} = \frac{2}{6} \)
Comparing the area with the shorter side is a quick and reliable check here.
Comparison
Each row multiplies two fractional sides.
Comparison matrix
| width | height | area |
|---|---|---|
| 1/2 | 1/3 | 1/6 |
| 3/4 | 1/2 | 3/8 |
| 2/5 | 1/2 | 1/5 |
Every area here is smaller than both of its sides, which is the signature of two proper fractions.
Ranking
Predict first, then check.
Put in order
Why: The areas are 1/4, 1/6, 1/9 and 1/12. As the sides shrink the areas shrink faster, because both dimensions are shrinking at once.
Trap
For a 3/4 by 2/3 rectangle a student adds the sides, getting 17/12, and calls it the area.
Adding the sides gives a perimeter-style quantity, not an area. Area multiplies: 3/4 times 2/3 is 1/2 of a square unit.
The check is the units. An area must come out in square units, and only a multiplication produces those.
Elimination
Two options can be discarded on size alone.
Eliminate the wrong options
A rectangle is 1/2 unit by 2/5 unit. Which could be its area?
Survives elimination: x1
Why: Multiplying gives 2/10, which simplifies to 1/5 of a square unit. Three options were ruled out by the size check without any arithmetic.
Section
Part 3
Concept
A side longer than one unit splits into a whole part and a fraction part, and the two areas add.
Figure (svg): A rectangle with sides one and a half by two, split into four parts
This is exactly the partial-products idea from whole-number multiplication, applied to fractions.
Worked example
A rectangle is 1 and 1/2 units wide and 2 units tall. Find its area.
Figure (svg): A rectangle with sides one and a half by two, split into four parts
Split the mixed side
Why: 1 and 1/2 becomes 1 plus 1/2.
Find each piece of area
Why: The whole part gives 1 times 2, which is 2. The fraction part gives 1/2 times 2, which is 1.
Add the pieces
Why: 2 plus 1 is 3.
\[ 1\tfrac{1}{2} \times 2 = 2 + 1 = 3 \text{ square units} \]
Verify: the size is sensible
Why: A rectangle a bit wider than 1 and 2 tall should have an area a bit over 2, and 3 fits that expectation.
Intuition
Instead of splitting, you can turn the mixed number into an improper fraction and multiply straight across.
Figure (svg): A rectangle with sides one and a half by two, split into four parts
1 and 1/2 is 3/2, so the area is 3/2 times 2, which is 6/2, or 3 — the same answer.
Worked example
A rectangle is 2 and 1/2 units by 1 and 1/2 units. Find its area.
Figure (svg): A rectangle with sides one and a half by two, split into four parts
Convert both to improper fractions
Why: 2 and 1/2 is 5/2, and 1 and 1/2 is 3/2.
Multiply straight across
Why: 5 times 3 is 15, and 2 times 2 is 4, giving 15/4.
Convert back to a mixed number
Why: 15 divided by 4 is 3 remainder 3, so 3 and 3/4.
\[ 2\tfrac{1}{2} \times 1\tfrac{1}{2} = \frac{5}{2} \times \frac{3}{2} = \frac{15}{4} = 3\tfrac{3}{4} \]
Verify: with an estimate
Why: Roughly 2.5 times 1.5 is 3.75, and 3 and 3/4 is exactly 3.75.
Prediction
Both routes are correct; one is usually quicker.
Predict first
For 3 and 1/3 times 3, which route is quicker?
Correct: split into 3 times 3 plus 1/3 times 3
Splitting is usually faster when one side is a whole number.
Why: Splitting gives 9 plus 1, which is 10, all in your head. Converting gives 10/3 times 3, which is 30/3, and then needs simplifying back to 10 — the same answer with an extra step.
Fill the middle
Convert to improper fractions and multiply.
Fill in the blanks
1\tfrac7___ \times 2\tfrac______ = \frac______ \times \frac___}___ = \frac______ = 3\tfrac______
Why: 2 and 1/3 is 7/3, since 2 times 3 is 6 and adding 1 gives 7 thirds. Multiplying gives 21/6, which simplifies to 3 and 1/2 square units.
Check
Solve it on paper before you click.
Check your understanding
A rug is 2 1/2 m by 2 m. What is its area?
Answer: A
Why: Splitting gives 2 times 2, which is 4, plus 1/2 times 2, which is 1, and 4 plus 1 is 5 square metres. Converting instead gives 5/2 times 2, which is also 5.
Error analysis
A student found the area of a 1 and 1/2 by 1 and 1/2 square.
Annotate
On: \( 1\tfrac{1}{2} \times 1\tfrac{1}{2} = 1\tfrac{1}{4} \)
Mixed-number multiplication has four partial areas, not two — which is exactly why converting to improper fractions is safer.
Step zero
One decision prevents the four-partial-areas error entirely.
Discussion prompt
When both sides are mixed numbers, what should you do before multiplying anything?
Hint: Think about how many pieces the splitting method creates.
Answer:
Convert both to improper fractions. Then it is a single straight-across multiplication with no partial areas to lose.
Splitting is fine when only one side is mixed, but with two mixed sides the improper-fraction route is far safer.
Trap
For a 1 and 1/2 by 2 and 1/2 rectangle a student multiplies 1 by 2, then 1/2 by 1/2, and answers 2 and 1/4.
That misses the two cross rectangles. Converting to improper fractions gives 3/2 times 5/2, which is 15/4, or 3 and 3/4.
An estimate of 1.5 times 2.5, about 3.75, exposes the error immediately.
Notation
The conversion looks like a trick until you see what it counts.
Annotate
On: \( 2\tfrac{1}{3} = \frac{7}{3} \)
Converting is safe precisely because it changes the notation and not the quantity.
Edge cases
Test the size rule where it changes behaviour.
Discussion prompt
If one side is 1/2 and the other is 4, is the area bigger or smaller than each side? What does that tell you about the size rule?
Hint: Check the area against each side separately.
Answer:
The area is 2 square units, which is bigger than the 1/2 side but smaller than the 4 side.
So the rule that the area is smaller than both sides only holds when both sides are under one unit. State the condition, not just the conclusion.
Section
Part 4
Concept
Rooms, gardens and tiles almost never have whole-number sides, which is why this skill matters.
Figure (svg): A rectangle with sides one and a half by two, split into four parts
The method is unchanged; only the units become metres, feet or centimetres.
Worked example
A garden bed is 3/4 of a metre wide and 2 and 1/2 metres long. What is its area?
Figure (svg): A rectangle with sides one and a half by two, split into four parts
Convert the mixed number
Why: 2 and 1/2 is 5/2.
Multiply
Why: 3/4 times 5/2 gives 15/8.
Convert back
Why: 15 divided by 8 is 1 remainder 7, so 1 and 7/8.
\[ \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1\tfrac{7}{8} \text{ square metres} \]
Verify: with an estimate
Why: Roughly 0.75 times 2.5 is about 1.9 square metres, and 1 and 7/8 is 1.875 — a close match.
Intuition
Round each side to the nearest half and multiply, to know roughly what to expect.
Figure (svg): A rectangle three units by two units filled with six unit squares
An area estimate catches a misplaced denominator faster than rechecking the arithmetic.
Estimation
Rounding the sides gives a usable check in seconds.
Predict first
Roughly what is the area of a rectangle 2 and 7/8 m by 3 m?
Correct: about 9 square metres
Rounding to the nearest whole is usually accurate enough to rule out every wrong option.
Why: 2 and 7/8 is nearly 3, so the area is nearly 3 times 3, which is 9. The exact answer is 8 and 5/8 square metres.
Real world
Anyone buying flooring or fabric does this arithmetic.
Discussion prompt
A room is 3 and 1/2 m by 4 m and carpet costs by the square metre. Why is the area, rather than either side, the number that determines the cost?
Answer:
Because carpet covers a surface, and the surface is 14 square metres.
Neither the 3 and 1/2 nor the 4 tells you how much carpet is needed on its own — only the product does.
Matching
Some of these are areas and some are not.
Match the pairs
Why: Covering a surface multiplies and gives square units; going around an edge adds and gives plain units. The fractional cases behave exactly like the whole-number ones.
Check
Solve it on paper before you click.
Check your understanding
A tile is 1/2 m by 1/3 m. What area does one tile cover?
Answer: A
Why: Multiplying the sides gives 1 times 1 over 2 times 3, which is 1/6 of a square metre. Both sides are under a metre, so the area is well under a square metre.
Faded example
A path is 2/3 metre wide and 4 and 1/2 metres long.
Fill in the blanks
\frac6___ \times \frac______ = \frac______} = 3
Why: 4 and 1/2 is 9/2. Multiplying gives 2 times 9 over 3 times 2, which is 18/6, and that is exactly 3 square metres.
Comparison
Each row is a real rectangle with at least one fractional side.
Comparison matrix
| width | length | area |
|---|---|---|
| 1/2 m | 4 m | 2 sq m |
| 3/4 m | 2 m | 1 1/2 sq m |
| 2/5 m | 5 m | 2 sq m |
With one whole side and one fractional side the answer can be bigger than the fractional side but smaller than the whole one.
Section
Part 5
Concept
Whole sides, fractional sides and mixed sides all obey length times width.
Figure (svg): A rectangle three units by two units filled with six unit squares
| sides | method | check |
|---|---|---|
| both whole | multiply directly | answer bigger than both sides |
| both proper fractions | multiply straight across | answer smaller than both sides |
| one or both mixed | convert to improper, then multiply | estimate by rounding |
Sorting
Choosing the route before starting saves rework.
Sort into buckets
Sort each pair of sides by the best method.
Two mixed sides always deserve the conversion route.
Two truths and a lie
Two of these are sound; one is not.
Eliminate the wrong options
Which statement about area with fractions is false?
Survives elimination: t2
Why: With both sides under one unit the rectangle fits inside a single unit square, so its area is smaller than either side, not larger. Half by half is a quarter, which is less than a half.
Explain it
A classmate expects the area to be bigger than the sides, as it is for whole numbers.
Discussion prompt
How would you convince them with a picture rather than a rule?
Answer:
Draw a unit square and shade a half by a half block in the corner. It is visibly a quarter of the square.
Then point out that a quarter is smaller than a half, so the area came out smaller than both sides. The picture makes the argument for you.
Missing information
Not every area question is answerable.
Discussion prompt
A problem says: a rectangle has one side of 3/4 metre. What is its area? What is missing?
Answer:
The other side. Area needs two dimensions, and one length alone determines nothing.
This is worth noticing because MAP does include items where a needed measurement is deliberately absent.
Commit first
Decide your answer and your confidence before revealing.
Predict first
What is the area of a rectangle 1 1/2 units by 2/3 unit?
Correct: 1 square unit
This is a case where cancelling first makes the answer immediate.
Why: 1 and 1/2 is 3/2, and 3/2 times 2/3 gives 6/6, which is exactly 1 square unit. The 3s and the 2s cancel neatly.
Exit ticket
One item that tells you whether the deck landed.
Predict first
A poster is 2/3 m wide and 3/4 m tall. What is its area?
Correct: 1/2 square metre
Why: Multiplying the sides gives 6/12, which simplifies to 1/2 a square metre. It is smaller than both sides, as two proper fractions must give.
Invariant
However finely it is cut, one thing does not move.
Step through it
What is the same on every line, and what has changed?
The total area is invariant; only the size and number of pieces change. That is why fractional sides give fractional areas of the same square.
Counterexample
A rule is worth more once you have tried to break it.
Discussion prompt
Is it always true that a rectangle with a bigger perimeter has a bigger area? Find a counterexample using fractional sides.
Answer:
No. A 1/10 by 4 rectangle has perimeter 8 and 1/5, and area 2/5 of a square unit.
A 1 by 1 square has a smaller perimeter of 4 but a larger area of 1. Long thin shapes have big perimeters and small areas, so the two quantities do not track each other.
Connect it up
Your handwriting, one page, from memory.
Draw it
Draw a unit square cut by two fractional sides with the rectangle outlined, and label the number of small pieces. Beside it, draw a mixed-number side split into its whole and fraction parts, and write the rule for choosing between splitting and converting.
If you can draw this from memory, all three skills in this deck are covered.
Recap
Three skills, one rule.
Area multiplies and perimeter adds — MAP mixes the two deliberately, so read which one is wanted.
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