Perform Operations for RIT under 215: multiplying by two and three digit numbers with partial products, multiplication word problems, and multiplying three or more factors using a helpful order.
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
Multiplying by two and three digit numbers, and multiplying three factors at once
Objectives
Five MAP skills sit here, and they are all the same idea at growing size.
The idea underneath all five: a product is an area, and an area can be cut into pieces you already know.
Section
Part 1
Concept
Multiplication counts equal groups. Drawing those groups as rows of a rectangle turns a product into an area.
Figure (svg): An array of four rows of six dots enclosed in a rectangle
Factor — One of the numbers being multiplied. In 4 times 6, the factors are 4 and 6.
Product — The result of multiplying. The product of 4 and 6 is 24.
Concept
Nobody knows 23 times 14 by heart. But everybody knows 20 times 10, and that is the trick.
Figure (svg): An area model for twenty three times fourteen split into four rectangles of two hundred, thirty, eighty and twelve
Cutting each factor at its place value produces four rectangles you can do in your head.
Warm-up
Retrieve first — the whole method is built out of facts you already own.
Discussion prompt
Write down what you get for 20 times 10, 20 times 4, 3 times 10 and 3 times 4, without any working.
Hint: These are all single-digit facts with zeros attached.
Answer:
200, 80, 30 and 12.
Add those four and you get 322, which is exactly 23 times 14. You already did the hard problem without noticing.
Pattern
Whatever the size of the numbers, the method does not change.
The written algorithm is just a compressed way of recording exactly these steps.
Socratic
The splitting point is not arbitrary.
Discussion prompt
Why do we split 23 into 20 and 3, rather than into 11 and 12?
Hint: Which split leaves you with numbers you can multiply instantly?
Answer:
Because 20 is a single digit with a zero attached, so 20 times anything is a fact you already know followed by a zero.
Splitting into 11 and 12 is perfectly legal but leaves you with two hard multiplications instead of two easy ones.
Definition probe
Recognising the easy factors is what makes a good split.
Sort into buckets
Sort each number by whether multiplying by it is instant.
Translation
MAP words a multiplication in several different ways.
Match the pairs
Why: All four describe the same product of 280. Turning a rectangle on its side does not change its area, which is why the order of the two factors never matters.
Explain it to yourself
This is the one idea the whole deck rests on.
Discussion prompt
In your own words, why is it legal to split 23 into 20 and 3 and multiply each piece separately?
Hint: Think about the picture, not the arithmetic.
Answer:
Because the rectangle really can be cut. Cutting a 23 by 14 rectangle into a 20-wide piece and a 3-wide piece does not change how much area there is.
Adding the pieces back up recovers the whole area, which is why the partial products are added at the end.
Section
Part 2
Worked example
Multiply 23 by 14, writing every piece down.
Split both factors
Why: 23 becomes 20 and 3. 14 becomes 10 and 4.
Multiply every pair
Why: 20 times 10 is 200. 20 times 4 is 80. 3 times 10 is 30. 3 times 4 is 12.
Add the four pieces
Why: 200 plus 80 plus 30 plus 12 is 322.
\[ 23 \times 14 = 200 + 80 + 30 + 12 = 322 \]
Verify: with an estimate
Why: Rounding gives 20 times 15, which is 300. The exact answer of 322 sits just above that, so no digit has gone astray.
Intuition
The standard algorithm records the same four products, just grouped into two rows.
Figure (svg): A standard multiplication showing the placeholder zero on the second line
The zero on the second line is not decoration — it is what makes that row count tens instead of ones.
Prediction
Predict the damage before seeing it.
Predict first
If a student forgets the placeholder zero when multiplying 23 by 14, what answer do they get?
Correct: 115
An estimate of 300 would have exposed 115 immediately.
Why: Without the zero the second row records 23 rather than 230, so they add 92 and 23 to get 115 instead of 322. The answer comes out roughly a third of the true size.
Trap
Multiplying 23 by 14, a student writes 92 and then 23 underneath, and adds to get 115.
The second row is 23 times the ten in 14, so it is 230, not 23. Adding 92 and 230 gives 322.
Writing the zero first, before multiplying that row, makes it impossible to forget.
Intuition
A rounded estimate before you start tells you roughly where the answer must land.
Figure (svg): An estimate of twenty times fifteen giving three hundred, next to the exact answer of three hundred twenty two
It costs five seconds and catches almost every place-value slip.
Faded example
Two of the four pieces are given.
Fill in the blanks
32 \times 21 = 600 + 20 + 30 + 2 = 672
Why: Splitting gives 30 and 2, and 20 and 1. The four products are 30 times 20 which is 600, 30 times 1 which is 30, 2 times 20 which is 40, and 2 times 1 which is 2. Those sum to 672.
Check
Solve it on paper before you click.
Check your understanding
What is 34 x 26?
Answer: A
Why: Split into 30 and 4, and 20 and 6. The partial products are 600, 180, 80 and 24, which add to 884. An estimate of 30 times 26 is 780, so 884 is the right size.
Error analysis
A student multiplied 45 by 12.
Annotate
On: \( 45 \times 12 = 90 + 45 = 135 \)
An estimate of 45 times 12 being near 45 times 10, or 450, would have flagged 135 as far too small.
Comparison
Every row is one multiplication split into its four pieces.
Comparison matrix
| problem | pieces | total |
|---|---|---|
| 21 x 13 | 200 + 30 + 20 + 3 | 273 |
| 42 x 11 | 400 + 40 + 20 + 2 | 462 |
| 15 x 12 | 100 + 50 + 20 + 10 | 180 |
Four pieces every time, however large the two factors are.
Fill the middle
The written method for 36 times 12, with one row blank.
Fill in the blanks
36 \times 12: \quad 72 + 360 = 432
Why: The ones row is 36 times 2, which is 72. The tens row is 36 times 10, which is 360 — note the placeholder zero. Adding gives 432.
Section
Part 3
Concept
A word problem hides a multiplication behind a story. Two questions pull it out.
Figure (svg): A word problem being turned into a multiplication, with the groups and the size of each group labelled
Worked example
A warehouse has 18 boxes. Each box holds 24 pencils. How many pencils are there altogether?
How many groups
Why: 18 boxes, so 18 groups.
How big is one group
Why: 24 pencils in each box.
Multiply
Why: Split into 10 and 8, and 20 and 4. The pieces are 200, 40, 160 and 32.
\[ 18 \times 24 = 200 + 40 + 160 + 32 = 432 \]
Verify: the answer makes sense
Why: Roughly 20 boxes of roughly 25 pencils is about 500, and 432 sits sensibly below that, so the size is right.
Intuition
Drawing the boxes as rows makes it obvious that this is a product and not a sum.
Figure (svg): An area model for twenty three times fourteen split into four rectangles of two hundred, thirty, eighty and twelve
If you cannot see groups, the problem is probably not a multiplication at all.
Discrimination
Sort by operation before computing anything.
Sort into buckets
Which of these stories are multiplications?
Step zero
One question prevents the most common word-problem error.
Discussion prompt
A problem says: there are 14 crates and each crate holds 30 apples. Before multiplying, what should you check about the two numbers?
Hint: Ask what each number is counting.
Answer:
Check that one number counts groups and the other counts what is inside one group. Here 14 counts crates and 30 counts apples per crate, so they play different roles.
If both numbers counted the same kind of thing, adding would be the right move rather than multiplying.
Check
Solve it on paper before you click.
Check your understanding
A theatre has 26 rows with 32 seats in each row. How many seats are there in total?
Answer: B
Why: There are 26 equal groups of 32, so multiply. Splitting gives 600, 180, 40 and 12, which add to 832 seats. An estimate of 25 times 32, or 800, confirms the size.
Real world
Two-digit multiplication is the arithmetic of ordering and stock.
Discussion prompt
A cafe orders 24 crates of milk with 12 bottles per crate. Why does the manager care about the product rather than either number on its own?
Answer:
Because the fridge holds bottles, not crates. The product, 288 bottles, is the number that has to fit.
Both original numbers are useless for that question on their own — only the product answers it.
Matching
The numbers are the same in all four; only the roles change.
Match the pairs
Why: Only the first and fourth describe equal groups, which is what multiplication counts. The second combines two piles of the same thing, and the third splits one pile up.
Error analysis
A student answered: 22 crates hold 15 apples each, how many apples?
Annotate
On: \( 22 + 15 = 37 \)
Asking what each number counts is the check that separates adding from multiplying.
Section
Part 4
Concept
A three-digit factor adds a third row, with two placeholder zeros. The method is unchanged.
Figure (svg): A standard multiplication showing the placeholder zero on the second line
| row | what it counts | placeholder zeros |
|---|---|---|
| first | the ones | none |
| second | the tens | one |
| third | the hundreds | two |
Worked example
Multiply 213 by 32.
Split the second factor
Why: 32 becomes 30 and 2.
Multiply by the ones
Why: 213 times 2 is 426.
Multiply by the tens
Why: 213 times 30 is 6,390 — that is 213 times 3, then a zero attached.
Add the rows
Why: 426 plus 6,390 is 6,816.
\[ 213 \times 32 = 426 + 6390 = 6816 \]
Verify: with an estimate
Why: Rounding to 200 times 30 gives 6,000, and 6,816 is comfortably close, so no place value has slipped.
Intuition
Each row up the second factor is worth ten times the row below, so it gains one more zero.
Figure (svg): An area model for twenty three times fourteen split into four rectangles of two hundred, thirty, eighty and twelve
Counting the zeros before multiplying is the safest way to keep the rows aligned.
Pattern
Each frame adds one row of the calculation.
Step through it
How many rows would a three-digit second factor produce?
One row per digit in the second factor, each with one more zero than the row above it.
Estimation
Estimating is the only practical check on a large multiplication.
Predict first
Roughly how big is 412 times 21?
Correct: about 8,000
Rounding both factors to one significant digit is enough to catch any misplaced zero.
Why: Rounding gives 400 times 20, which is 8,000. The exact answer is 8,652, so the estimate is the right order of magnitude.
Check
Solve it on paper before you click.
Check your understanding
What is 124 x 23?
Answer: A
Why: 124 times 3 is 372, and 124 times 20 is 2,480. Adding those gives 2,852. An estimate of 120 times 20, or 2,400, confirms the size.
Notation
The compressed method hides what each row means.
Annotate
On: \( \begin{array}{r} 213 \\ \times \; 32 \\ \hline 426 \\ 6390 \\ \hline 6816 \end{array} \)
Every digit in that layout has a place-value meaning; nothing in it is arbitrary.
Faded example
The ones row is given; supply the tens row and the total.
Fill in the blanks
312 \times 24: \quad 1248 + 6240 = 7488
Why: The ones row is 312 times 4, which is 1,248. The tens row is 312 times 20, which is 6,240. Adding gives 7,488, and an estimate of 300 times 24, or 7,200, confirms the size.
Section
Part 5
Concept
With three factors you may multiply any two first. The product comes out the same, but the work can be far easier.
Figure (svg): Three factors multiplied in two different orders, both giving the same product
Associative property — Regrouping which two factors you multiply first does not change the product.
Worked example
Work out 4 times 25 times 7.
Look for a friendly pair
Why: 4 times 25 is 100, which is the easiest number to multiply by.
Multiply that pair first
Why: 4 times 25 is 100.
Finish
Why: 100 times 7 is 700.
\[ (4 \times 25) \times 7 = 100 \times 7 = 700 \]
Verify: by regrouping the other way
Why: Doing 25 times 7 first gives 175, and 4 times 175 is also 700 — the same answer by a harder route, which confirms the regrouping was legitimate.
Intuition
Before multiplying three factors, scan for a pair that makes 10, 100 or 1,000.
Figure (svg): Three factors multiplied in two different orders, both giving the same product
Pairs worth spotting: 2 and 5, 4 and 25, 8 and 125, and anything with a 10 in it.
Sorting
Look for the pair that produces a round number.
Sort into buckets
For each triple, sort by whether an easy pair exists.
Spending three seconds scanning for a friendly pair often saves a minute of arithmetic.
Ranking
Compute 5 x 17 x 2 in the easiest order.
Put in order
Why: Scanning first is what reveals the shortcut. Multiplying 5 by 17 first would give 85, and 85 times 2 is still 170 — but the mental arithmetic is much harder that way.
Trap
For 25 x 7 x 4 a student works strictly left to right: 25 times 7 is 175, then 175 times 4.
Multiplying 25 by 4 first gives 100, and 100 times 7 is 700 — the same answer with almost no effort.
Multiplication lets you regroup freely. Scanning for a friendly pair first is always worth the few seconds.
Worked example
A school has 5 year groups. Each year group has 4 classes, and each class has 25 students. How many students are there?
Name the three quantities
Why: 5 year groups, 4 classes in each, 25 students in each class.
Scan for a friendly pair
Why: 4 times 25 is 100.
Multiply in that order
Why: 4 times 25 is 100, and 100 times 5 is 500.
\[ 5 \times 4 \times 25 = 5 \times 100 = 500 \]
Verify: the answer is sensible
Why: Five hundred students across twenty classes is twenty five per class, which matches the story exactly.
Intuition
Three factors describe groups inside groups: classes inside year groups, students inside classes.
Figure (svg): A bar model showing five year groups, each split into four classes
Each level of nesting is one more factor in the product.
Elimination
Work out 2 x 50 x 6 by elimination.
Eliminate the wrong options
What is 2 times 50 times 6?
Survives elimination: p2
Why: The friendly pair is 2 and 50, which make 100. Then 100 times 6 is 600. Doing 50 times 6 first gives 300, and 300 times 2 is also 600.
Check
Solve it on paper before you click.
Check your understanding
A shop stocks 8 shelves, each with 5 boxes, and each box holds 20 items. How many items in total?
Answer: A
Why: The friendly pair is 5 and 20, which make 100. Then 8 times 100 is 800 items. Working left to right gives 8 times 5 is 40, and 40 times 20 is also 800.
Two truths and a lie
Two of these are sound; one is not.
Eliminate the wrong options
Which statement about multiplying three factors is false?
Survives elimination: x2
Why: Changing the order never changes the product of whole numbers. It only changes how hard the arithmetic is along the way, which is exactly why choosing a good order is a strategy rather than a risk.
Explain it
A classmate always multiplies left to right and finds three factors exhausting.
Discussion prompt
How would you convince them that scanning first is worth the time?
Answer:
Give them 25 times 13 times 4 and let them do it left to right, then show them 25 times 4 is 100 and the answer is 1,300.
Feeling the difference once is more persuasive than being told the rule.
Commit first
Decide your answer and your confidence before revealing.
Predict first
What is 5 x 18 x 2?
Correct: 180
Spotting the 5 and 2 turned a two-digit multiplication into attaching a zero.
Why: The friendly pair is 5 and 2, which make 10, and 10 times 18 is 180. Multiplying 5 by 18 first gives 90, and 90 times 2 is also 180.
Missing information
Not every story has enough in it.
Discussion prompt
A problem says: a school has 6 year groups and 30 students in each class. How many students are in the school? What is missing?
Answer:
The number of classes in each year group. Without it, the two given numbers do not connect.
With three levels of grouping you need all three numbers — this is why naming what each number counts matters so much.
Exit ticket
One item that tells you whether the deck landed.
Predict first
What is 4 x 25 x 13?
Correct: 1,300
Why: 4 times 25 is 100, and 100 times 13 is 1,300. Spotting the friendly pair turns this into a one-step problem.
Invariant
Regrouping changes the route but not the destination.
Step through it
Which quantity is identical on all three lines?
Every regrouping preserves the product. That is exactly what makes choosing a convenient order safe.
Analogy
You already regroup for convenience outside a maths lesson.
Match the pairs
Why: Both habits are the associative property in disguise: you are free to decide which two things to combine first, and you choose whichever makes the counting easiest.
Edge cases
The rule should not care how many factors there are.
Discussion prompt
Work out 2 x 5 x 4 x 25 in the easiest possible order, and say what strategy you used.
Hint: There are two friendly pairs hiding in there, not one.
Answer:
Pair 2 with 5 to make 10, and 4 with 25 to make 100. Then 10 times 100 is 1,000.
The strategy scales: with any number of factors, hunt for pairs that make round numbers and combine those first.
Connect it up
Your handwriting, one page, from memory.
Draw it
Draw an area model for a two-digit times two-digit multiplication, labelling all four partial products. Beside it, write the three-step method and the rule about placeholder zeros. Add one line on scanning for friendly pairs.
If you can draw this from memory, every item in this deck becomes routine.
Recap
Five skills, one method.
Almost every error on this topic is a lost zero, and an estimate catches every one of them.
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