Ratios and proportional relationships for RIT under 215: comparing and converting customary units of length, weight and volume, and reading conversion tables.
Subject: NWEA MAP Growth Math · 62 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
Comparing and converting length, weight and volume in the customary system
Objectives
Five MAP skills sit in this deck, and every one of them runs on the same single idea.
The one idea: going to a smaller unit multiplies, and going to a bigger unit divides.
Section
Part 1
Concept
Converting does not change how long, heavy or full something is. It only changes the size of the piece you are counting with.
Figure (svg): One long yard bar above three shorter foot bars, showing that a bigger unit needs fewer of them
Unit — The size of the piece you are counting. A yard is a big piece; an inch is a small one.
So the amount stays fixed and only the number moves — and it moves in a completely predictable direction.
Prediction
Before any arithmetic, just predict the direction.
Predict first
A rope is 4 feet long. Measured in inches, will the number be bigger or smaller than 4?
Correct: bigger, because inches are smaller pieces
Smaller piece, bigger count. This is the check that catches most conversion mistakes.
Why: The rope is unchanged, but an inch is a much smaller piece than a foot, so it takes many more of them to cover the same rope. The number goes up to 48.
Concept
You never need to memorise multiply or divide. You work it out from which unit is bigger.
Figure (svg): Two arrows: going from a big unit to a small unit multiplies, and going from small to big divides
Big to small multiplies. Small to big divides. That is the whole rule for all five skills in this deck.
Warm-up
You know more of these than you think, from cooking, sport and shopping.
Discussion prompt
Write down every customary conversion fact you can already state from memory. Do not look anything up.
Hint: Think about a ruler, a bathroom scale and a milk carton.
Answer:
Most people already have: 12 inches in a foot, 3 feet in a yard, 16 ounces in a pound, 4 quarts in a gallon.
Those four alone cover a large share of the items. The rest of this deck fills in the gaps and drills the direction.
Pattern
Every conversion item in this deck answers to the same three questions.
Answer those three and the arithmetic is the easy part.
Socratic
This feels backwards to a lot of students, so it is worth saying out loud.
Discussion prompt
Why does using a smaller unit make the number bigger, when the object itself has not changed at all?
Hint: Think about money rather than length.
Answer:
Because you are counting how many pieces fit. Small pieces fit more times into the same space.
Think of paying for something with dollars versus with cents: the price is the same, but the number of cents is far larger.
Discrimination
MAP mixes the families together, so the first move is always to name the family.
Sort into buckets
Put each unit under what it measures.
Section
Part 2
Concept
Length has four units and three facts joining them.
Figure (svg): A ladder of customary length units from mile down to inch with the conversion factor on each step
| fact | in words |
|---|---|
| 1 foot = 12 inches | a ruler is one foot long and has twelve inch marks |
| 1 yard = 3 feet | a yard is about the width of a door |
| 1 mile = 5,280 feet | a mile is a long walk, about twenty minutes |
Worked example
A shelf is 4 feet long. How many inches is that?
Which unit is bigger
Why: A foot is bigger than an inch.
Which direction
Why: Going from feet to inches is big to small, so multiply.
Use the fact
Why: There are 12 inches in a foot, so multiply by 12.
\[ 4 \times 12 = 48 \text{ inches} \]
Verify: the direction is right
Why: 48 is bigger than 4, which is what moving to a smaller unit should do. If the answer had come out smaller, the operation was wrong.
Intuition
After every conversion, look only at whether the number grew or shrank.
Figure (svg): A check showing that converting feet to inches gives a bigger number, because inches are smaller
This takes two seconds and catches the single most common error on the whole topic.
Faded example
The structure is given; supply the number that does the work.
Fill in the blanks
7 \text3 = 7 \times ___ = 21 \text___
Why: There are 3 feet in a yard, and going from the bigger unit to the smaller one multiplies, so 7 yards becomes 21 feet.
Prediction
Same two units, opposite direction.
Predict first
How many yards are in 36 feet?
Correct: 12 yards
The answer got smaller, which is exactly right for a move to a bigger unit.
Why: A yard is bigger than a foot, so going from feet to yards is small to big and you divide. 36 divided by 3 is 12.
Trap
A student converts 36 feet to yards by multiplying: 36 times 3 gives 108 yards.
A yard is bigger than a foot, so 36 feet must be fewer than 36 yards. The answer has to shrink, so divide: 36 divided by 3 is 12 yards.
108 yards would be longer than a football field — the size of the answer alone shows it cannot be right.
Comparison
Each row uses one of the three length facts.
Comparison matrix
| from | to | operation | answer |
|---|---|---|---|
| 5 feet | inches | multiply by 12 | 60 |
| 24 inches | feet | divide by 12 | 2 |
| 4 yards | feet | multiply by 3 | 12 |
| 15 feet | yards | divide by 3 | 5 |
Notice the operation column never needed memorising — it came from which unit was bigger.
Check
Solve it on paper before you click.
Check your understanding
A hallway is 9 yards long. How many feet is that?
Answer: C
Why: A yard is bigger than a foot, so this is big to small and you multiply. There are 3 feet in a yard, and 9 times 3 is 27 feet.
Ranking
Convert everything to one unit first, then order.
Put in order
Why: In inches these are 24, 30, 36 and 40. Converting everything to the smallest unit is what makes the comparison possible, because you cannot compare feet against inches directly.
Error analysis
A student converted 2 miles to feet.
Annotate
On: \( 2 \text{ miles} = 2 \div 5280 = 0.00038 \text{ feet} \)
The absurd size of the answer was the tell, before any rule was checked.
Explain it to yourself
Explaining it yourself is what turns a recognised rule into a usable one.
Discussion prompt
In your own words, how do you decide whether to multiply or divide when converting between feet and inches?
Hint: Start from what should happen to the size of the number, not from the operation.
Answer:
Ask which unit is bigger. A foot is bigger than an inch, so a length in inches must be a bigger number than the same length in feet.
Going feet to inches makes the number grow, so multiply. Going inches to feet makes it shrink, so divide.
Analogy
You already convert units fluently when the units are money.
Match the pairs
Why: Money and length obey exactly the same rule. Nobody hesitates over dollars and cents, and the only difference with feet and inches is that the factor is 12 rather than 100.
Section
Part 3
Concept
Weight is the easiest of the three families, because there are only two conversion facts to hold.
Figure (svg): A chain of customary weight units: one ton is two thousand pounds and one pound is sixteen ounces
| fact | a feel for it |
|---|---|
| 1 pound = 16 ounces | a loaf of bread is about a pound |
| 1 ton = 2,000 pounds | a small car is about a ton |
Worked example
A bag of flour weighs 3 pounds. How many ounces is that?
Which unit is bigger
Why: A pound is bigger than an ounce.
Which direction
Why: Pounds to ounces is big to small, so multiply.
Use the fact
Why: There are 16 ounces in a pound.
\[ 3 \times 16 = 48 \text{ ounces} \]
Verify: the size is sensible
Why: 48 ounces is a bigger number than 3 pounds, as moving to a smaller unit requires. A three pound bag really would read 48 on an ounce scale.
Intuition
The jump from pounds to tons is the biggest in the customary weight system, which is why the numbers look startling.
Figure (svg): A chain of customary weight units: one ton is two thousand pounds and one pound is sixteen ounces
If an answer in tons comes out large, or an answer in pounds comes out tiny, check the direction.
Estimation
Estimation is a fast guard against a misplaced operation.
Predict first
Roughly how many ounces is 10 pounds?
Correct: about 160
An estimate does not need to be exact — it only needs to be close enough to rule out a wrong operation.
Why: Each pound is 16 ounces, so ten pounds is about ten sixteens, which is 160. The exact answer is 160 ounces.
Fill the middle
Going the other way now, so the operation flips.
Fill in the blanks
64 \text\div = 64 ___ 16 = 4 \text___
Why: A pound is bigger than an ounce, so going from ounces to pounds is small to big and you divide. 64 divided by 16 is 4.
Two truths and a lie
Two of these are sound; one is not.
Eliminate the wrong options
Which statement about customary weight is false?
Survives elimination: w2
Why: A ton is 2,000 pounds, not 100. The 100 is being borrowed from the metric system, where a hundred does appear in several conversions. Customary weight uses 16 and 2,000, and nothing else.
Check
Solve it on paper before you click.
Check your understanding
A crate weighs 5 pounds. How many ounces does it weigh?
Answer: B
Why: A pound is bigger than an ounce, so multiply by the 16 ounces in a pound. Five times sixteen is eighty ounces.
Anomaly
Two answers to the same question, and only one of them is believable.
\[ 4 \text{ tons} = 8000 \text{ lb} \quad \text{versus} \quad 4 \text{ tons} = 0.002 \text{ lb} \]
Predict first
Which conversion is right, and what makes the other one obviously wrong?
Correct: 8,000 pounds, because a ton is much heavier than a pound
Four tons weighing two thousandths of a pound is absurd on its face — the size check alone settles it.
Why: A ton is bigger than a pound, so four tons must be a very large number of pounds. Multiplying by 2,000 gives 8,000. The other answer divided, which would only make sense going from pounds up to tons.
Definition probe
Naming the fact before using it stops the wrong factor being grabbed.
Sort into buckets
Sort each conversion by the fact it needs.
Real world
This is one of the few topics that turns up before you leave the shop.
Discussion prompt
A recipe needs 24 ounces of chicken and the packs are sold in pounds. How many pounds do you need, and why would buying by the ounce number be a mistake?
Answer:
24 divided by 16 is 1.5, so you need one and a half pounds.
Reading 24 as pounds would have you buying sixteen times too much — this is exactly the direction error the deck keeps warning about.
Section
Part 4
Concept
Volume has the most units, but they nest neatly — each one holds two or four of the next.
Figure (svg): Nested customary volume units showing a gallon holding four quarts, a quart holding two pints, and a pint holding two cups
| fact | a feel for it |
|---|---|
| 1 cup = 8 fluid ounces | a small glass |
| 1 pint = 2 cups | a large drink |
| 1 quart = 2 pints | a carton of milk |
| 1 gallon = 4 quarts | a large jug |
Worked example
A cooler holds 3 gallons. How many quarts is that?
Which unit is bigger
Why: A gallon is bigger than a quart.
Which direction
Why: Gallons to quarts is big to small, so multiply.
Use the fact
Why: There are 4 quarts in a gallon.
\[ 3 \times 4 = 12 \text{ quarts} \]
Verify: against the picture
Why: The nesting diagram shows four quart boxes inside each gallon, and three gallons gives twelve of those boxes, matching the answer.
Intuition
Nobody memorises cups per gallon. You walk the ladder instead, one remembered step at a time.
Figure (svg): A two step conversion from gallons to cups, going through quarts and pints
Two easy steps beat one fact you are unsure of.
Pattern
Each frame is one rung, and each rung is a fact you already have.
Step through it
What would change if you were going from cups back up to gallons?
Going back up the ladder uses the same three facts, but every step divides instead.
Elimination
Convert 5 quarts to pints by elimination.
Eliminate the wrong options
How many pints are in 5 quarts?
Survives elimination: v2
Why: A quart is bigger than a pint, so multiply by the 2 pints in a quart. Five times two is ten pints.
Trap
Converting 5 quarts to pints, a student multiplies by 4 and gets 20, because 4 is the number they remember from gallons.
The 4 belongs to gallons and quarts. Quarts to pints uses 2, giving 10 pints.
Before multiplying, say the fact out loud with both unit names in it: a quart holds two pints. Naming both units stops the wrong fact being grabbed.
Matching
All of these describe the same four amounts in different units.
Match the pairs
Why: Each unit holds exactly two or four of the unit directly below it, which is what makes the volume ladder so quick to walk in either direction.
Check
Solve it on paper before you click.
Check your understanding
A jug holds 6 quarts. How many cups is that?
Answer: B
Why: Quarts to pints multiplies by 2, giving 12 pints, and pints to cups multiplies by 2 again, giving 24 cups. Both steps go to a smaller unit, so both multiply.
Sorting
Predict the direction only. Do not compute anything yet.
Sort into buckets
Sort each conversion by whether the number will grow or shrink.
Getting the direction right first makes the arithmetic almost impossible to misplace.
Notation
MAP writes units in shorthand, and misreading the shorthand is a silent error.
Annotate
On: \( 3 \text{ qt} = \square \text{ pt} \)
Read the units before the numbers. Half of the difficulty in these items is notation, not arithmetic.
Section
Part 5
Concept
The harder MAP items do not tell you whether you are in length, weight or volume. Naming the family is step zero.
Figure (svg): Two arrows: going from a big unit to a small unit multiplies, and going from small to big divides
Once the family is named, the same three questions from Part 1 finish the job.
Step zero
A mixed item rewards a moment of sorting before any calculating.
Discussion prompt
You are asked to compare 3 pints with 7 cups. What are the first two things you should establish, before doing any arithmetic at all?
Hint: You cannot compare two numbers until they are counting the same sized piece.
Answer:
First: both are volume units, so they can genuinely be compared.
Second: they are in different units, so one must be converted before comparing. Converting 3 pints gives 6 cups, which is less than 7 cups.
Worked example
Which is heavier: 2 pounds, or 40 ounces?
Name the family
Why: Both are weights, so a comparison is meaningful.
Pick one unit for both
Why: Ounces is the smaller unit, so convert the pounds into ounces.
Convert
Why: 2 pounds is 2 times 16, which is 32 ounces.
\[ 2 \text{ lb} = 32 \text{ oz} \quad \text{versus} \quad 40 \text{ oz} \]
Verify: the comparison
Why: 32 ounces is less than 40 ounces, so 40 ounces is the heavier of the two. Converting to the smaller unit avoided any fractions.
Intuition
A table hands you the rule. Your job is to spot it from a row you can see.
Figure (svg): A conversion table for feet and inches with one row left blank
Take any complete row, work out what was done to the left number to get the right one, then do the same to the unknown row.
Reverse engineer
The table shows pounds in the left column and ounces in the right.
Fill in the blanks
3 \text112 \rightarrow 48 \text___, \quad 7 \text___ \rightarrow ___ \text___
Why: From the visible row, 48 divided by 3 is 16, so the rule is multiply by 16. Applying it to 7 pounds gives 112 ounces.
Comparison
Three different families in one table, exactly as MAP mixes them.
Comparison matrix
| quantity | family | converted |
|---|---|---|
| 2 yards | length | 6 feet |
| 4 pounds | weight | 64 ounces |
| 3 gallons | volume | 12 quarts |
Same method every row: name the family, name the bigger unit, then multiply or divide.
Missing information
Not every comparison can be made.
Discussion prompt
A question asks which is more: 5 pounds or 5 quarts. What is wrong with the question?
Answer:
Pounds measure weight and quarts measure volume, so they are not in the same family and cannot be converted into one another.
Without knowing what substance is in the container, there is no way to compare them at all. Naming the family first is what exposes this.
Commit first
Decide your answer and your confidence before revealing.
Predict first
Which is longer: 100 inches, or 3 yards?
Correct: 3 yards
Converting to the smallest unit, inches, kept everything whole and made the comparison direct.
Why: Three yards is 9 feet, and 9 feet is 108 inches. Since 108 is more than 100, three yards is the longer of the two.
Explain it
A classmate keeps dividing when they should multiply.
Discussion prompt
Without giving them a rule to memorise, how would you get them to work out the direction for themselves every time?
Answer:
Get them to ask which unit is bigger, and then to predict whether the answer should be a bigger or a smaller number.
Once they have predicted the size, the operation is forced: a bigger answer means multiply, a smaller answer means divide.
Counterexample
Rules are worth more once you have tried to break them.
Discussion prompt
Is there any conversion where going to a smaller unit does not make the number bigger? Try to find one, or explain why none exists.
Answer:
There is none. If the piece you count with is smaller, more of them are needed to cover the same amount, so the count always rises.
The only apparent exception is converting a unit to itself, where the factor is 1 and nothing changes.
Exit ticket
One item that tells you whether the deck landed.
Predict first
A tank holds 24 quarts. How many gallons is that?
Correct: 6 gallons
Why: A gallon is bigger than a quart, so this is small to big and you divide. There are 4 quarts in a gallon, and 24 divided by 4 is 6 gallons.
Check
Solve it on paper before you click.
Check your understanding
Which of these is the greatest length?
Answer: A
Why: Convert every option to inches, the smallest unit present. Two yards is 6 feet, which is 72 inches; five feet is 60 inches; and one yard and two feet is also 5 feet, or 60 inches. The largest of 72, 60, 70 and 60 is 72, so two yards is the greatest.
Invariant
One thing stays fixed through every step, and naming it keeps the whole topic straight.
Step through it
What is identical on all three lines, and what is different?
The amount is invariant; only the size of the counting piece changes. That is the whole topic in one sentence.
Edge cases
Testing an idea at its extreme is how you find out whether you really believe it.
Discussion prompt
If you invented a unit smaller than an inch — say a tiny unit where 1000 of them make an inch — what would happen to the number describing a 4 foot shelf?
Hint: Apply the same direction rule you have used all deck, just with a bigger factor.
Answer:
It would become enormous. Four feet is 48 inches, and 48 times 1000 is 48,000 of the tiny units.
The rule does not break at the extremes: smaller piece, bigger count, no matter how small the piece gets.
Connect it up
One page, in your handwriting, is worth more than re-reading this deck.
Draw it
Draw three ladders side by side — length, weight and volume — with the conversion factor written on every rung. Mark one arrow going down labelled multiply, and one going up labelled divide.
If you can draw this from memory, every item in this deck becomes a two-step exercise.
Recap
Five skills, one rule, three families.
When there is no direct fact, walk the ladder one rung at a time rather than guessing a factor.
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