Solve multiplication and division problems in context. - ISEE Lower Level Mathematics Achievement

SECTION 1

Why Do We Use Multiplication and Division?

People have been solving math problems in everyday life for thousands of years! Long ago, farmers needed to figure out how many seeds to plant in rows. Merchants needed to split coins evenly among workers.

Multiplication and division were invented to make counting faster. Instead of adding the same number over and over, people learned shortcuts. Let's look at how these ideas grew over time!

3000 BC • Ancient Egypt

Egyptian farmers used multiplication to measure fields along the Nile River. They drew pictures to keep track of equal groups.

500 BC • Ancient Greece

Greek thinkers used division to share food and supplies fairly among soldiers.

800 AD • The × Symbol Arrives

Scholars in the Middle East wrote some of the first math textbooks. They taught people how to multiply large numbers step by step.

Today • Word Problems on Tests

On the ISEE, you solve real-world story problems using multiplication and division. The math is the same — only the stories change!

The big question is: How do you read a story problem and know whether to multiply or divide? That is exactly what this lesson will teach you!

SECTION 2

Core Principles: When to Multiply and When to Divide

Before you do any math, you need to understand what the problem is asking. Story problems give you a little scene — like a mini movie. Your job is to figure out the action happening in that scene.

Equal Groups → Multiply

When you have several groups that are all the same size, multiply to find the total. Example: 4 bags with 6 apples each.

Splitting Evenly → Divide

When you share a total equally or split it into equal parts, divide. Example: 24 stickers shared among 4 friends.

Rows and Columns → Multiply

If items sit in rows and columns (like seats in a theater), multiply rows × columns to find the total.

How Many in Each? → Divide

If you know the total and how many groups, divide to find how many go in each group.

Look for Clue Words

Words like 'each,' 'every,' 'per,' and 'times' often point to multiplication. Words like 'share,' 'split,' 'divide,' and 'equally' point to division.

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KEY TAKEAWAY

Think of multiplication like a copy machine — it makes the same number appear again and again. Division is like dealing cards — you hand out one at a time until everything is shared equally. On the ISEE, read the story carefully and ask yourself: Am I combining equal groups, or am I splitting something up?

SECTION 3

See It: Multiplication vs. Division

A picture can make things much clearer. The diagram below shows the same 12 objects arranged in two different ways. On the left, you see equal groups being combined (multiplication). On the right, you see a total being split apart (division).

Left side: Three equal groups of 4 combine to make 12 (multiplication). Right side: 12 items split into 3 equal groups give 4 in each (division). Notice they are opposite operations!

See how the left side and the right side use the same numbers? That's because multiplication and division are related — like forward and reverse on a bike. Knowing one helps you check the other!

SECTION 4

The Math Behind Story Problems

Every multiplication or division story problem boils down to a simple equation. Here are the two key formulas you need.

MULTIPLICATION

Number of Groups × Size of Each Group = Total

Use this when you know how many groups and how many in each group, and you want the total.

DIVISION

Total ÷ Number of Groups = Size of Each Group

Use this when you know the total and want to find how many are in each group, or how many groups you can make.

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ISEE Test Tip

On the ISEE, always read the last sentence of a word problem first. That tells you what you need to find. Then go back and pull out the numbers. This saves time and keeps you from getting confused!

Here is a handy trick: after you solve a multiplication problem, you can check your answer with division. If 5 × 8 = 40, then 40 ÷ 8 should equal 5. If it does, you know you got it right!

SECTION 5

Clue Words That Tell You What to Do

Story problems hide clues inside their words. Learning to spot these clue words is like having a secret decoder ring! The table below shows common clue words for each operation.

Clue Word	Operation	Example Sentence
each / every	Multiply	Each box holds 8 crayons.
times	Multiply	She rode her bike 3 times as far.
per	Multiply	He earns $5 per chore.
in all / total	Multiply	How many in all?
share equally	Divide	They share 20 grapes equally.
split / divide	Divide	She split the cards into 4 piles.
how many groups	Divide	How many teams of 5 can they make?

Common clue words for multiplication and division

This flowchart shows the two-step thinking process. First, decide what the question is asking. Then, look for clue words to pick multiply or divide.

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Watch Out!

Sometimes a problem uses the word "each" but still needs division! For example: "There are 30 chairs. Each table has 5 chairs. How many tables are there?" You divide 30 ÷ 5. Always ask yourself: Do I already know the total, or am I finding the total?

SECTION 6

Worked Examples: Step by Step

Let's solve two problems together — one multiplication and one division. Follow each step carefully. You've got this!

Example 1 — Multiplication

A teacher orders 7 packs of markers. Each pack has 8 markers. How many markers does she order in all?

Solving: 7 Packs of Markers

1
Step 1 — Read the last sentence

The question says "how many markers in all." The words in all tell us we need a total.
2
Step 2 — Pull out the numbers

We see 7 (number of packs) and 8 (markers in each pack).
3
Step 3 — Choose the operation

We have equal groups and need a total. That means multiply!
4
Step 4 — Calculate

7 × 8 = 56

The teacher orders 56 markers.
5
Step 5 — Check with division

56 ÷ 8 = 7. That matches the 7 packs. ✓ Correct!

Example 2 — Division

There are 36 oranges. They are shared equally among 9 baskets. How many oranges are in each basket?

Solving: Oranges in Baskets

1
Step 1 — Read the last sentence

The question asks "how many in each basket." The word each and shared equally tell us to divide.
2
Step 2 — Pull out the numbers

We see 36 (total oranges) and 9 (baskets).
3
Step 3 — Choose the operation

We already know the total and need to find how many in each group. That means divide!
4
Step 4 — Calculate

36 ÷ 9 = 4

Each basket has 4 oranges.
5
Step 5 — Check with multiplication

9 × 4 = 36. That matches the total. ✓ Correct!

SECTION 7

Common Mistakes and How to Avoid Them

Even great students make mistakes on word problems. Knowing the most common traps helps you avoid them. Let's look at what can go wrong and how to stay on track.

Common Mistake	Why It Happens	How to Fix It
Using the wrong operation	Rushing through without reading carefully	Read the last sentence first to find what is being asked.
Mixing up the numbers	The problem gives extra info or big numbers	Underline the numbers and label them (total, groups, each).
Forgetting a multiplication fact	Some facts like 7 × 8 are tricky to remember	Break it apart: 7 × 8 = 7 × 4 × 2 = 28 × 2 = 56.
Not checking the answer	Running out of time or feeling done	Use the opposite operation to check. It takes only seconds!

Avoid these four common mistakes on the ISEE

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KEY TAKEAWAY

Think of checking your work like re-reading a text message before you send it. It only takes a few seconds, and it can save you from a silly error. On the ISEE, checking with the opposite operation is the fastest way to make sure your answer makes sense.

SECTION 8

Building Toward Bigger Problems

The skills you're learning now are the building blocks for harder math later. As you grow, word problems will have more steps — but the thinking process stays the same!

What You Know Now	What Comes Next
One-step multiplication	Two-step problems (multiply, then add)
One-step division	Division with remainders
Whole numbers only	Multiplying and dividing with money (decimals)
Small numbers (up to 12 × 12)	Multi-digit multiplication (like 34 × 26)

Your path from here

For the ISEE Lower Level, focus on one-step problems with whole numbers. Master those, and you'll be ready for anything the test throws at you. Great job learning so far!

SECTION 9

Practice Problems

Time to show what you've learned! Try each problem on your own before reading the answer. Remember: read the last sentence first, find the numbers, pick the operation, solve, and check.

Question 1: CONCEPTUAL

Sam has 5 bags of marbles. Each bag has 6 marbles. Which number sentence shows how to find the total number of marbles? (A) 5 + 6 = 11 (B) 6 − 5 = 1 (C) 5 × 6 = 30 (D) 6 ÷ 5 = 1 R1

Show answer

The correct answer is (C) 5 × 6 = 30. Sam has 5 equal groups of 6, so we multiply. Choice (A) adds instead of multiplying — addition only works if you add 6 five separate times. Choice (B) subtracts, which doesn't make sense when combining groups. Choice (D) divides, but we aren't splitting a total — we're finding one.

Question 2: BASIC CALCULATION

A farmer plants 9 rows of corn. Each row has 7 plants. How many corn plants are there in all? (A) 16 (B) 56 (C) 63 (D) 72

Show answer

The correct answer is (C) 63. The words "rows" and "each" and "in all" tell us to multiply: 9 × 7 = 63. Choice (A) is 9 + 7, which adds instead of multiplying. Choice (B) is 8 × 7, using the wrong number of rows. Choice (D) is 9 × 8, using the wrong number per row. Always double-check which numbers go with which part of the problem.

Question 3: INTERMEDIATE

A bakery made 48 cupcakes. The cupcakes are put equally into 6 boxes. How many cupcakes are in each box? (A) 6 (B) 7 (C) 8 (D) 9

Show answer

The correct answer is (C) 8. The word "equally" tells us to divide: 48 ÷ 6 = 8. You can check by multiplying: 6 × 8 = 48. ✓ Choice (A) is just the number of boxes, not the answer. Choice (B) would mean 6 × 7 = 42, not 48. Choice (D) would mean 6 × 9 = 54, not 48.

Question 4: APPLIED

Tickets to the school play cost $4 each. Mrs. Lee buys tickets for herself and 7 students. How much does she spend in all? (A) $11 (B) $28 (C) $32 (D) $36

Show answer

The correct answer is (C) $32. Be careful! Mrs. Lee buys a ticket for herself AND 7 students, so that is 1 + 7 = 8 tickets total. Then multiply: 8 × $4 = $32. Choice (A) adds 4 + 7 instead of multiplying. Choice (B) is 7 × $4 = $28, which forgets Mrs. Lee's own ticket. Choice (D) is 9 × $4, counting one too many people. Watch out for tricky wording!

Question 5: CRITICAL THINKING

A library has 72 new books. The librarian puts 8 books on each shelf. After filling as many shelves as possible, she orders 3 more shelves. How many total shelves does she now have for these books? (A) 9 (B) 11 (C) 12 (D) 15

Show answer

The correct answer is (C) 12. This is a two-part problem. First, divide to find how many shelves she fills: 72 ÷ 8 = 9 shelves. Then she orders 3 more: 9 + 3 = 12 shelves total. Choice (A) only does the division and stops. Choice (B) adds 2 instead of 3 extra shelves. Choice (D) adds 6 instead of 3. Always re-read the full problem to make sure you answer the actual question!

SUMMARY

Lesson Summary

Summary

When you see a story problem on the ISEE, always start by reading the last sentence to find out what you need. Look for clue words like "each," "every," "per," and "in all" for multiplication. Look for "share equally," "split," and "how many groups" for division.

Use the formula Groups × Size = Total for multiplication and Total ÷ Groups = Each for division. Always check your answer using the opposite operation. Read carefully for tricky wording — sometimes the problem hides an extra step. You've got this!