Perform Operations for RIT under 215: dividing two, three and four digit numbers by two digit divisors using partial quotients, and interpreting remainders correctly 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
Dividing two, three and four digit numbers by two digit divisors, in and out of context
Objectives
Four MAP skills sit here, and they are the same method applied to bigger numbers.
The idea underneath: division asks how many of the divisor fit inside the dividend, and you may take them away in convenient chunks.
Section
Part 1
Concept
Division answers two questions that sound different but give the same number.
Figure (svg): Two pictures of twelve divided by three: sharing into three groups, and grouping into fours
Both are 12 divided by 3. Recognising either story as a division is half the battle.
Concept
MAP uses these words in the question stems, so they are worth knowing.
Dividend — The number being divided — the total you start with.
Divisor — The number you are dividing by — the size of a group, or the number of groups.
Quotient — The answer to the division.
Remainder — What is left over when the division does not come out exactly.
Warm-up
Retrieve first — the method is built from facts you already have.
Discussion prompt
Without working anything out, write down 60 divided by 20, 100 divided by 25, and 180 divided by 20.
Hint: Ask yourself what you multiply the second number by to reach the first.
Answer:
3, 4 and 9.
Every one of those is a multiplication fact read backwards. Division is not a new skill so much as multiplication asked in reverse.
Pattern
Whatever the size of the numbers, the same three steps apply.
This is called partial quotients, and it is far more forgiving than guessing one digit at a time.
Socratic
There is a real reason, and naming it helps.
Discussion prompt
Why do most students find division harder than multiplication, even when the numbers are the same size?
Hint: Think about how much guessing each operation requires.
Answer:
Because multiplication tells you exactly what to do, while division asks you to find how many fit — which involves a search.
Partial quotients removes the search. You never have to find the right digit in one go; you only have to take away a chunk you are confident about.
Definition probe
Both are divisions, but naming which one helps you picture the answer.
Sort into buckets
Sort each story by which question it asks.
Translation
All four use the same two numbers.
Match the pairs
Why: Three of the four are the same division, worded three ways, and all give 6. The fourth is an addition in disguise, which is exactly the kind of distractor MAP includes.
Section
Part 2
Intuition
Rounding the divisor tells you roughly how many digits the answer should have.
Figure (svg): Estimating a quotient by rounding the divisor to twenty
If your exact answer has a different number of digits from your estimate, something has gone wrong.
Worked example
Work out 432 divided by 18.
Estimate
Why: 18 is close to 20, and 400 divided by 20 is 20, so expect an answer near 20.
Take away a chunk you are sure of
Why: 20 lots of 18 is 360. Subtracting leaves 432 minus 360, which is 72.
Take away another chunk
Why: 4 lots of 18 is 72. Subtracting leaves nothing.
Add the chunks
Why: 20 plus 4 is 24.
\[ 432 \div 18 = 24 \]
Verify: by multiplying back
Why: 24 times 18 is 432, which matches the dividend exactly, so the quotient is right and there is no remainder.
Intuition
You are allowed to take away any chunk you like, as long as you keep track of it.
Figure (svg): Partial quotients for four hundred thirty two divided by eighteen, taking away twenty lots then four lots
Bigger chunks finish faster, but small chunks are never wrong — only slower.
Prediction
There is no single right chunk, but some are much more useful.
Predict first
Dividing 432 by 18, which first chunk gets you furthest safely?
Correct: 20 lots, because 20 times 18 is 360
A chunk is usable exactly when it is less than the number you are dividing into.
Why: Twenty lots is 360, which fits inside 432 with room to spare. Forty lots would be 720, which is more than 432, so it overshoots. One lot is safe but would take twenty four steps.
Trap
Dividing 432 by 18, a student takes 40 lots, computing 40 times 18 as 720, and then cannot subtract it.
A chunk must be smaller than what is left. 720 is bigger than 432, so 40 lots is too many — back down to 20.
The estimate is what prevents this: 400 divided by 20 is about 20, so a chunk near 20 is the right scale.
Faded example
One chunk is given; supply the other and the quotient.
Fill in the blanks
391 \div 17: \quad 20 \text3 = 340, \; \text23 51 = ___ \text___, \; \text___ ___
Why: After taking 20 lots of 17, which is 340, the remaining 51 is exactly 3 lots of 17. Adding 20 and 3 gives a quotient of 23.
Concept
Often the chunks do not use up the whole dividend. What is left is the remainder.
Figure (svg): Fourteen counters shared between four boxes, three in each box and two left over
A remainder is always smaller than the divisor. If it is not, another chunk can still be taken.
Worked example
Work out 253 divided by 12.
Figure (svg): Two hundred fifty three broken into twenty lots of twelve, one more lot, and a remainder of one
Estimate
Why: 250 divided by 12 is roughly 250 divided by 10, so expect something near 20.
Take a chunk
Why: 20 lots of 12 is 240, leaving 253 minus 240, which is 13.
Take another chunk
Why: 1 lot of 12 is 12, leaving 1.
Stop
Why: 1 is smaller than 12, so no more chunks fit. The quotient is 21 and the remainder is 1.
\[ 253 \div 12 = 21 \text{ remainder } 1 \]
Verify: by multiplying back and adding
Why: 21 times 12 is 252, and adding the remainder of 1 gives 253, which is the original dividend — so the answer is correct.
Check
Solve it on paper before you click.
Check your understanding
What is 288 / 16?
Answer: A
Why: Take 10 lots of 16, which is 160, leaving 128. Then 8 lots of 16 is 128, leaving nothing. Adding 10 and 8 gives 18, and 18 times 16 is 288, which confirms it.
Error analysis
A student worked out 195 divided by 15.
Annotate
On: \( 195 \div 15 = 13 \text{ remainder } 15 \)
Checking that the remainder is smaller than the divisor catches this instantly.
Comparison
Each row is one division with its check.
Comparison matrix
| division | quotient | check by multiplying |
|---|---|---|
| 144 / 12 | 12 | 12 x 12 = 144 |
| 210 / 14 | 15 | 15 x 14 = 210 |
| 224 / 16 | 14 | 14 x 16 = 224 |
Multiplying back is the only check that proves a quotient rather than merely suggesting it.
Explain it to yourself
This check is the reason you never have to wonder whether a quotient is right.
Discussion prompt
In your own words, why does multiplying the quotient by the divisor and adding the remainder recover the dividend?
Hint: Think of it as reassembling what you took apart.
Answer:
Because the quotient counts how many whole lots you took away, and each lot was the size of the divisor. Multiplying puts all those lots back.
The remainder is the bit you never took away, so adding it back completes the original total.
Ranking
Put the partial-quotients method in order.
Put in order
Why: Estimating first is what stops you choosing a chunk that overshoots, and checking last is what turns a plausible answer into a confirmed one.
Section
Part 3
Concept
A story is a division when a total is being broken up, rather than built up.
Figure (svg): Two pictures of twelve divided by three: sharing into three groups, and grouping into fours
Worked example
A school orders 468 exercise books and packs them into boxes of 18. How many boxes are filled?
Identify the total
Why: 468 books, so that is the dividend.
Identify the divisor
Why: 18 books per box, so that is the divisor.
Estimate
Why: 470 divided by 20 is about 23, so expect an answer in the twenties.
Divide with chunks
Why: 20 lots of 18 is 360, leaving 108. Then 6 lots of 18 is 108, leaving nothing.
\[ 468 \div 18 = 26 \text{ boxes} \]
Verify: by multiplying back
Why: 26 times 18 is 468, matching the total ordered, so every book is accounted for and no box is part full.
Intuition
Drawing the total as a bar and cutting it into equal parts makes the division visible.
Figure (svg): A bar of four hundred sixty eight cut into equal parts of eighteen
If you can draw the bar, you can always tell whether the story divides or multiplies.
Discrimination
Sort by operation before computing anything.
Sort into buckets
Which of these stories are divisions?
Step zero
One question prevents the commonest word-problem error.
Discussion prompt
A problem gives you the numbers 350 and 25. Before dividing, how do you decide which is the dividend?
Hint: Ask which number is the whole and which is the part.
Answer:
The dividend is always the total — the thing being broken up. Here 350 is the total and 25 is the size or number of groups.
Dividing 25 by 350 would give a number less than one, which no counting story can mean.
Check
Solve it on paper before you click.
Check your understanding
A florist has 336 roses and puts 14 in each bunch. How many bunches can be made?
Answer: A
Why: Take 20 lots of 14, which is 280, leaving 56. Then 4 lots of 14 is 56, leaving nothing. The quotient is 24, and 24 times 14 is 336, which confirms it.
Real world
The arithmetic is rarely the hard part outside a classroom.
Discussion prompt
A coach has 50 students and each minibus seats 15. How many minibuses are needed, and why is the plain quotient not the answer?
Answer:
50 divided by 15 is 3 remainder 5. But three buses would leave five students behind.
The answer is 4 buses. When every item must be carried, you round the quotient up regardless of how small the remainder is.
Matching
The same two numbers, four different roles.
Match the pairs
Why: Two of these divide, one multiplies and one adds. What separates them is whether a total is being broken up, built up, or simply added to.
Analogy
You divide fluently in situations that never feel like maths.
Match the pairs
Why: Each everyday situation is one of the two division meanings, and two of them force a decision about the remainder — exactly the decision Part 5 of this deck is about.
Section
Part 4
Concept
A four-digit dividend does not need a new method. It needs bigger chunks — hundreds of lots rather than tens.
Figure (svg): Partial quotients for four hundred thirty two divided by eighteen, taking away twenty lots then four lots
Start with the biggest round chunk that still fits, and work down.
Worked example
Work out 1,632 divided by 24.
Estimate
Why: 1,600 divided by 25 is about 64, so expect an answer in the sixties.
Take a big chunk
Why: 60 lots of 24 is 1,440, leaving 1,632 minus 1,440, which is 192.
Take another chunk
Why: 8 lots of 24 is 192, leaving nothing.
Add the chunks
Why: 60 plus 8 is 68.
\[ 1632 \div 24 = 68 \]
Verify: by multiplying back
Why: 68 times 24 is 1,632, which matches the dividend exactly, so the quotient is right.
Intuition
The estimate matters more as the numbers grow, because there is more room to lose a digit.
Figure (svg): Estimating a quotient by rounding the divisor to twenty
Round the divisor to the nearest ten and the dividend to something friendly, then divide.
Pattern
Each frame removes one chunk from what is left.
Step through it
What would you do differently if 5 had been left at the end instead of 0?
Anything left at the end that is smaller than the divisor is the remainder.
Estimation
The estimate tells you how many digits to expect.
Predict first
Roughly how big is 3,150 divided by 45?
Correct: about 70
Getting the number of digits right is most of the value of an estimate.
Why: Rounding gives 3,000 divided by 45, and since 45 times 70 is 3,150, the answer is about 70. The exact answer is exactly 70.
Elimination
Work out 2,208 divided by 24 by elimination.
Eliminate the wrong options
What is 2,208 divided by 24?
Survives elimination: q1
Why: Take 90 lots of 24, which is 2,160, leaving 48. Then 2 lots of 24 is 48, leaving nothing. The quotient is 92, and 92 times 24 is 2,208.
Check
Solve it on paper before you click.
Check your understanding
What is 1,845 / 15?
Answer: A
Why: Take 100 lots of 15, which is 1,500, leaving 345. Then 20 lots is 300, leaving 45, and 3 lots is 45. Adding 100, 20 and 3 gives 123, and 123 times 15 is 1,845.
Fill the middle
One chunk is given; supply the second and the quotient.
Fill in the blanks
2464 \div 22: \quad 100 \text12 = 2200, \; \text112 264 = ___ \text___, \; \text___ ___
Why: After 100 lots of 22, which is 2,200, the remaining 264 is 12 lots of 22. Adding 100 and 12 gives 112, and 112 times 22 is 2,464.
Notation
The notation itself trips students up before the arithmetic does.
Annotate
On: \( 1638 \div 26 = \square \)
Order matters in division exactly as it does in subtraction. Read which number is the whole before you start.
Invariant
Something is conserved at every step of a partial-quotients division.
Step through it
At every line, what do the lots taken so far plus the amount left add up to?
The lots removed, multiplied by the divisor, plus what is left, always equals the original dividend. That is why the multiply-back check works.
Section
Part 5
Concept
This is the part MAP tests hardest. The same division supports three different answers.
Figure (svg): Three different answers to the same division, depending on what the remainder means in the story
Worked example
43 items are packed 8 to a box. Answer all three questions.
Do the division once
Why: 5 lots of 8 is 40, leaving 3. So 43 divided by 8 is 5 remainder 3.
How many full boxes
Why: 5 boxes are completely full, so the remainder is dropped.
How many boxes needed
Why: 6 boxes, because the leftover 3 still need one.
How many left over
Why: 3 items, which is the remainder itself.
\[ 43 \div 8 = 5 \text{ remainder } 3 \]
Verify: all three against the story
Why: 5 times 8 is 40 packed with 3 spare, which accounts for all 43 items and confirms every one of the three answers.
Intuition
The arithmetic takes ten seconds. Deciding what the question wants is where the mark is won or lost.
Figure (svg): Three different answers to the same division, depending on what the remainder means in the story
Underline the actual question before you divide, not after.
Sorting
The division is the same each time; only the question changes.
Sort into buckets
For each question, sort by what happens to the remainder.
Three questions, one calculation. Read which one is being asked before you write the answer.
Trap
50 students, 15 per bus. A student divides, gets 3 remainder 5, and answers 3 buses.
Three buses seat only 45 students, leaving 5 behind. Since every student must travel, the answer is 4 buses.
Whenever the question says needed, to carry everyone, or so that none are left out, round the quotient up.
Prediction
The calculation is done; only the interpretation is left.
Predict first
There are 100 eggs and each carton holds 12. How many cartons are needed to hold all the eggs?
Correct: 9 cartons
The word needed is the signal to round up.
Why: 100 divided by 12 is 8 remainder 4. Eight cartons hold only 96 eggs, so the remaining 4 need a ninth carton. Because every egg must be held, the quotient rounds up.
Check
Solve it on paper before you click.
Check your understanding
A club has 94 members travelling in minibuses that seat 12. How many minibuses are needed?
Answer: A
Why: 94 divided by 12 is 7 remainder 10. Seven minibuses seat only 84 people, so the remaining 10 need an eighth. Since everyone must travel, the answer rounds up to 8.
Error analysis
A student answered: 77 apples, 10 per bag, how many full bags?
Annotate
On: \( 77 \div 10 = 7 \text{ r } 7 \;\Rightarrow\; 8 \text{ full bags} \)
The word full is the signal to drop the remainder, not to round up.
Two truths and a lie
Two of these are sound; one is not.
Eliminate the wrong options
Which statement about remainders is false?
Survives elimination: t2
Why: Rounding up is right only when everything must be held or carried. If the question asks for full groups, the remainder is dropped instead, and if it asks what is left over, the remainder is the answer.
Explain it
A classmate always rounds up, whatever the question says.
Discussion prompt
How would you get them to decide correctly every time, without memorising three separate rules?
Answer:
Get them to say the answer as a full sentence about the story: seven bags are full, and seven apples are left over.
Once the sentence is said out loud, the question almost always picks its own answer out of it.
Missing information
Not every question can be answered as asked.
Discussion prompt
A problem says: 200 books are packed into boxes. How many boxes are needed? What is missing?
Answer:
How many books fit in one box. Without the divisor there is no division to do.
Naming the dividend and the divisor before calculating is what exposes a missing number rather than inventing one.
Commit first
Decide your answer and your confidence before revealing.
Predict first
How many full teams of 11 can be made from 95 players?
Correct: 8 teams
The word full told you to drop the remainder rather than round up.
Why: 95 divided by 11 is 8 remainder 7. Eight complete teams use 88 players, and the remaining 7 are not enough for a ninth team, so the remainder is dropped.
Exit ticket
One item that tells you whether the deck landed.
Predict first
1,200 chairs are stacked 16 to a trolley. How many trolleys are needed to move them all?
Correct: 75 trolleys
Why: 1,200 divided by 16 is exactly 75 with no remainder, so 75 trolleys carry every chair with none left over and none wasted.
Connect it up
Your handwriting, one page, from memory.
Draw it
Draw the partial-quotients layout for a four-digit division, showing two chunks being taken away and added at the end. Beside it, write the three remainder decisions and the word that signals each one.
If you can draw this from memory, every item in this deck becomes routine.
Recap
Four skills, one method, one decision.
The arithmetic is the easy half. Reading which of the three questions is being asked is where the marks are.
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