Covers Unit 4 (Multiply by 2-digit numbers) and Unit 5 (Division). The examples use different numbers from the diagnostic test, but the methods, models, and problem types are identical.
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Unit 4 — Multiply by 2-Digit Numbers
1. Multiplying multiples of 10 and 100 (The Zero Rule)
When you multiply numbers that end in zeros, use basic multiplication facts and place value patterns.
Step 1: Multiply the basic fact digits (non-zero numbers).
Step 2: Count the total number of ending zeros in both factors and attach them to your product.
Example: 40 × 60
• Basic fact: 4 × 6 = 24
• Count zeros: 40 has 1 zero, 60 has 1 zero (2 zeros total).
• Answer: 2,400
Watch Out for Hidden Zeros:
When multiplying numbers like 5 × 4 or 5 × 6, the basic fact already creates a zero!
Example: 50 × 40
• Basic fact: 5 × 4 = 20 (already has a zero!)
• Attach the two zeros from 50 and 40: 2,000 (3 zeros total!).
Common Mistake: Forgetting the zero from the basic fact. In 50 × 80, 5 × 8 = 40, so the answer is 4,000, NOT 400!
Check yourself
aCalculate 60 × 70
bCalculate 50 × 800
cFill in the missing number: 80 × [____] = 3,200
Show check-yourself answer
a. 4,200 (6 × 7 = 42, add 2 zeros)
b. 40,000 (5 × 8 = 40, add 3 zeros: one from 50 and two from 800)
c. 40 (since 32 ÷ 8 = 4, and 3,200 has 2 zeros while 80 has 1)
Before doing long calculations, estimating helps you check if your answer makes sense.
To estimate the product of two 2-digit numbers:
1. Round each number to the nearest ten (look at the ones digit: 5 or higher rounds UP, 4 or lower stays).
2. Multiply the rounded numbers using the zero rule.
Example: Estimate 59 × 42
• 59 rounds to 60 (9 is 5 or more)
• 42 rounds to 40 (2 is 4 or less)
• 60 × 40 = 2,400
Overestimate vs. Underestimate:
• Overestimate: If both numbers were rounded UP, the estimate will be greater than the exact product.
• Underestimate: If both numbers were rounded DOWN, the estimate will be smaller than the exact product.
• When one number rounds up and one rounds down, the estimate is very close to the true answer!
Common Mistake: Rounding to the wrong place value or multiplying the original numbers instead of the rounded numbers. Round first, then multiply!
Check yourself
A library orders 38 boxes of books. Each box contains 23 books.
aEstimate the total number of books by rounding each number to the nearest ten.
bIs 38 rounded up or down? Is 23 rounded up or down?
Show check-yourself answer
a. 38 rounds to 40. 23 rounds to 20.
Estimated books: 40 × 20 = 800 books.
b. 38 rounds UP to 40. 23 rounds DOWN to 20. (Since one went up and one went down, 800 is a very close estimate to the real total of 874).
Your understanding:
3. Area models for 2-digit multiplication (The 4-Box Rectangle)
An area model turns a big multiplication problem into 4 friendly, easy rectangular areas.
To solve 28 × 35 using an area model:
1. Break each factor into tens and ones:
• 28 = 20 + 8
• 35 = 30 + 5
2. Build a 2-by-2 box grid. Write 20 and 8 on the left, and 30 and 5 along the top.
×
30
5
20
20 × 30600
20 × 5100
8
8 × 30240
8 × 540
3. Add all 4 partial areas together:
600 + 100 + 240 + 40 = 980
So, 28 × 35 = 980!
Common Mistake: Mixing up rows and columns. In the top-right box, make sure you multiply the top header (5) by the left header (20), NOT 8!
Check yourself
Draw a 4-box area model for 42 × 16 = (40 + 2) × (10 + 6):
The standard algorithm is a fast vertical way to write down multiplication by combining partial products into two rows.
Solve 47 × 32:
47
× 32
────
94 ← Row 1: Multiply 47 by 2 (the ones digit)
+1410 ← Row 2: Put a 0 placeholder in the ones place! Then multiply 47 by 3
─────
1504 ← Row 3: Add both rows together (94 + 1,410 = 1,504)
Why do we put a 0 in the second row?
Because the 3 in 32 is in the tens place! You are really multiplying 47 × 30, not 47 × 3. The placeholder zero shifts everything into the correct tens place.
Common Mistake: Forgetting the placeholder zero in the second row! If you forget it, 1,410 becomes 141, and your final answer will be completely wrong.
Check yourself
Solve 52 × 27 vertically:
aWhat is Row 1 (52 × 7)?
bWhat is Row 2 (52 × 20, including the placeholder 0)?
cWhat is the final sum (total product)?
Show check-yourself answer
a. Row 1: 52 × 7 = 364
b. Row 2: 52 × 20 = 1,040 (with the 0 in the ones place)
c. Final sum: 364 + 1,040 = 1,404
Real-life math often needs more than one step. Break the story down into Step 1 and Step 2 before calculating.
Problem: A soccer club has 6 teams. Each team has 18 players. Each uniform jersey costs $25. What is the total cost for all players' jerseys?
Step 1: Find the total number of players.
6 teams × 18 players = 108 players
Step 2: Find the total cost.
108 players × $25 = $2,700
Total cost = $2,700
Common Mistake: Stopping after Step 1! Always re-read the question's final sentence to make sure you answered what it actually asked for.
Check yourself
A bakery bakes 16 trays of muffins every morning. Each tray holds 12 muffins. If they sell each muffin for $3:
aHow many muffins do they bake in total?
bIf they sell all muffins, how much money do they collect?
Show check-yourself answer
a. Step 1: 16 trays × 12 muffins = 192 muffins
b. Step 2: 192 muffins × $3 = $576
Division is splitting a total into equal groups. If things cannot be divided evenly, the leftovers are called the remainder.
Example: 29 ÷ 4
• How many groups of 4 fit into 29?
4 × 7 = 28
• Leftover: 29 - 28 = 1
• Quotient = 7, Remainder = 1 (written as 7 R 1)
The Golden Rule of Remainders:
The remainder MUST ALWAYS be smaller than the divisor!
If you divide by 4, the only possible remainders are 0, 1, 2, or 3.
If your remainder is 4 or bigger, you could have made another full group!
The Check Formula:
(Divisor × Quotient) + Remainder = Dividend
(4 × 7) + 1 = 28 + 1 = 29 ✓ It checks out!
Common Mistake: Leaving a remainder that is equal to or bigger than the divisor. If you get a remainder of 6 when dividing by 5, your quotient needs to be 1 higher!
Check yourself
aFind the quotient and remainder: 47 ÷ 6 = ?
bWrite the multiplication and addition equation that checks your answer for part a.
Show check-yourself answer
a. 47 ÷ 6 = 7 R 5 (since 6 × 7 = 42, and 47 - 42 = 5)
b. Check equation: (6 × 7) + 5 = 42 + 5 = 47
Use basic division facts and place value thinking to divide large round numbers quickly in your head.
Think in terms of place value bundles:
• 180 ÷ 3 → 18 tens ÷ 3 = 6 tens = 60
• 2,400 ÷ 6 → 24 hundreds ÷ 6 = 4 hundreds = 400
• 35,000 ÷ 7 → 35 thousands ÷ 7 = 5 thousands = 5,000Division Fact Family Check:
If 2,400 ÷ 6 = 400, then 6 × 400 must equal 2,400!
Common Mistake: Losing track of zeros when the basic fact has a zero. For example, in 3,000 ÷ 5: the basic fact is 30 ÷ 5 = 6. The remaining zeros are 2, so the answer is 600, NOT 6,000!
Compatible numbers are friendly numbers that divide evenly without any remainder. They are close to the real numbers and easy to compute mentally.
To estimate 263 ÷ 4:
Look at the first two digits (26) and find multiples of 4 close to 26:
• 4 × 6 = 24 → 240
• 4 × 7 = 28 → 280
263 is between 240 and 280.
• 240 ÷ 4 = 60
• 280 ÷ 4 = 70
So 263 ÷ 4 is between 60 and 70. (Since 263 is closer to 280, a great estimate is 70).
Common Mistake: Rounding using standard rules to numbers that don't divide cleanly. In 263 ÷ 4, rounding 263 to 260 doesn't help because 260 doesn't easily divide by 4. Use 240 or 280 instead!
Check yourself
aChoose a compatible number close to 478 to estimate 478 ÷ 6.
bWhat is the estimated quotient?
cIs 315 ÷ 5 between 50 and 60, or between 60 and 70?
Show check-yourself answer
a. 480 is very close to 478 and is a multiple of 6 (since 6 × 8 = 48).
b. 480 ÷ 6 = 80.
c. Between 60 and 70 (since 5 × 60 = 300 and 5 × 70 = 350; 315 is between 300 and 350).
In division, the area model works backwards: you know the total inside area (dividend) and the height (divisor). You find the missing lengths across the top (quotients).
Solve 132 ÷ 4 by breaking 132 into friendly multiples of 4:
• Break 132 into 120 + 12 (both divide easily by 4).
Common Mistake: Breaking the dividend into numbers that don't divide by the divisor. For 132 ÷ 4, breaking into 100 + 32 is also fine (25 + 8 = 33), but breaking into 130 + 2 makes it much harder!
Check yourself
Solve 185 ÷ 5 by breaking 185 into two friendly multiples of 5:
aWhat two friendly numbers can you split 185 into?
bDivide each part by 5 and add them to find the total quotient.
Show check-yourself answer
a. 150 and 35 (since 150 + 35 = 185, and both end in 0 or 5).
b. (150 ÷ 5) + (35 ÷ 5) = 30 + 7 = 37.
When solving division word problems, the math gives you a quotient and remainder. What you DO with the remainder depends on the real-world story!
There are 3 main ways to handle a remainder:
Path 1: Round Up (Add 1 to quotient)
Used when everyone or everything must fit (vans, buses, tents, tables, boxes).
• Example: 35 students ride in cars that hold 4 students each. 35 ÷ 4 = 8 R 3.
You need 9 cars so the 3 leftover students aren't left behind!
Path 2: Drop the Remainder (Use only the whole number)
Used when you can only count whole completed items (packages made, cakes sold, full teams).
• Example: You have 25 wheels. Each toy car needs 4 wheels. 25 ÷ 4 = 6 R 1.
You can only make 6 full cars. (The 1 extra wheel is not enough for another car).
Path 3: The Remainder IS the Answer
Used when the question asks about the leftovers!
• Example: You have 25 cookies shared equally among 4 friends. How many cookies are left over?
Answer: 1 cookie.
Common Mistake: Giving the raw math answer (like "8 R 3") instead of answering the real-world question (like "9 cars"). Read carefully to see which path is needed!
Check yourself
There are 86 campers. Each cabin sleeps 8 campers.
aCalculate 86 ÷ 8 (quotient and remainder).
bHow many cabins are needed so EVERY camper has a bed?
cIf all full cabins are filled, how many campers will sleep in the last cabin?
Show check-yourself answer
a. 86 ÷ 8 = 10 R 6
b. 11 cabins (Path 1: Round up, because all 86 campers need a place to sleep!)
c. 6 campers (Path 3: The remainder is the number of campers in the partially filled cabin).