Multiplication is a nifty shortcut way of combining several of the same amounts. You will now see how that works.

EXAMPLE 1: Danny lost 2 pennies today and 2 yesterday. How many pennies did he lose altogether?

THINK: To picture the problem, make 2 rows of 2 pennies one row above the other. Do you need to combine 2 sets of 2 pennies or 2 2s to find the total?
You can think of it as an addition problem and say 2 plus 2 equals what or 2 + 2 = ? However, since you are combining two sets or amounts that are the same, you can think of it as a multiplication problem. You can say 2 times 2 equals what or 2 X 2 = ?
COMPUTE: To solve 2 + 2 = ?, you can simply count all of the pennies, and the total count is 4. So 2 + 2 = 4, which is an addition fact. It so happens that 2 times 2 is also 4 or 2 X 2 = 4, which is a multiplication fact.
Don't get the idea that any number added to itself is the same as the number multiplied by itself. This is only true if the number is 2. Note that 1 + 1 = 2 but 1 x 1 = 1 (one 1 is 1). Note that 3 + 3 = 6 (2 3s is 6) but 3 x 3 = 9 (3 3s is 9).
CHECK: You can see that we can check multiplication with addition.

CHALLENGE 1: Kyle bought 2 games that sell for $2 apiece. How much did Kyle pay for the two games? Write the multiplication question.

EXAMPLE 2: Danny gave his little sister 2 pennies on each of 3 days.

THINK: You need to combine the 3 sets of 2 pennies to find the total number of pennies. To picture this, make 3 rows of 2 pennies in each row one row above another.
You could think of this as an addition problem. The number question would be 2 + 2 + 2 = ? However, this requires two additions: 2 + 2 = 4, 4 + 2 = 6. This problem is a job for multiplication. The question is 3 times 2 is what or 3 X 2 = ? Save the grid of pennies for the next problem.
COMPUTE: To answer the number question, 3 X 2 = ?, using multiplication, you need one multiplication fact, which is 3 X 2 = 6. The total number or combined amount in multiplication is called the product.
CHECK: Check your thinking: Does the question really come down to 3 x 2 = ? Yes, it does. Now find the total number of pennies with addition by counting all of the pennies.

CHALLENGE 2: Danny gave his dog 3 biscuits today and 3 yesterday. How many biscuits did he give his dog altogether? Think of this as a multiplication problem and write the number question.
Is 2 X 3 the same as 3 X 2? Lay your head on your left shoulder and look at the grid of pennies that you saved. Now you see 2 rows of 3 pennies each, which is the same as 3 rows of 2 pennies each. When you combine things, the order makes no difference.

EXAMPLE 3: In a school room there are several rows of chairs and several chairs in each row. How could you most quickly find out the total number of chairs?

THINK: Look at the way the chairs are arranged. There are the same number in each row.

chair chair chair chair chair
chair chair chair chair chair 
chair chair chair chair chair 
chair chair chair chair chair 
chair chair chair chair chair 
chair chair chair chair chair 

So, multiplication combines sets of the same number such as 2 3s or 3 4s by using multiplication facts. Another way of looking at it is that a multiplication problem is one in which we need to combine a number with itself a few or many times. The combined amount is called the product.

CHALLENGE 3: Jimmy went to the toy store and bought 4 new baseballs costing $3 each. Picture this with a grid of pennies. How many rows are there and how many pennies in each row? What is the number question? Find the multiplication fact by counting all of the pennies.

CHALLENGE 4: Ted was with Jimmy, and he bought 3 better baseballs costing $4 each. How much did they cost? What is the number question? Find the multiplication fact by counting all of the pennies. Is that fact the same as the one in Challenge 1?

CHALLENGE 5: You want to buy 3 $5 tickets to the movies. What is the number question?

CHALLENGE 6: Now think of a multiplication story about 4 sets of 2 people each (four couples for example). What is the number question?

CHALLENGE 7: Think of silly multiplication story problems that come down to each of these number questions:

3 x 5 = ? , 5 x 3 = ? , 2 x 4 = ? , 4 x 2 = ?

[Some people would say that 6 ÷ 2 means "how many times will 2 go into 6." It is more meaningful to me to say "how many 2s can be separated from 6." This shows how division is related to subtraction.]

You will remember that when we want to take away or separate one amount from another, we use subtraction. When we want to know how many sets of the same amount we can separate from a larger amount, we use division rather than many subtractions.

EXAMPLE 1: Mikey had 6 pennies. He bought one 2-cent sticker. How many pennies did he have left?

THINK: In this problem we separate one 2 from 6 to find the remainder, and we do this by subtraction. The number question is 6 - 2 = ?

EXAMPLE 2: Mikey has 6 pennies. How many 2-cent stickers can he buy?

THINK: Instead of separating just one set of 2, the problem is to find how many 2s we can separate from 6. In other words we must divide 6 by 2. This means that we divide or separate 6 into sets of 2 and find how many sets there are.
To picture this problem, count off 6 pennies. You need to divide 6 by 2, so divide the 6 pennies into sets of 2 one set above the other. We say 6 divided by 2 equals what? or 6 ÷ 2 = ? Notice that we make the division sign like a subtraction sign with a dot above it and a dot below it.
COMPUTE: How can you find out how many 2s you can separate from 6 using the pennies? Take your time.
Here is one way: Look at the grid of pennies. Notice that the pennies were divided into to rows of 2 each. Count the rows or sets of 2 pennies. Are there 3 sets of 2 in 6. Now erase the question mark in the number question and write 3, which is called the quotient. You have discovered a division fact: 6 ÷ 2 = 3.
Remember that division is the opposite of multiplication. So, you can think of the problem as a multiplication problem with the number question written this way: 2 X ? = 6. You could look up the answer in a multiplication table or recall it if you have learned it. The multiplication fact is 2 X 3 = 6. So, Mikey can buy 3 2-cent stickers with 6 pennies. Save your grid of pennies.
CHECK: You can check the division with multiplication. 2 X 3 = 6, which is the number of pennies Mikey had.

EXAMPLE 3: Mikey has 6 pennies. How many 3-cent stickers can he buy?

THINK: Look at your grid of pennies again and think of them as stickers instead of pennies. Is the problem to find out how many 3s you can separate from 6? Is the number question 6 ÷ 3 = ?
COMPUTE: This time you need to divide 6 by 3, or divide 6 into sets of 3 and find out how many sets there are. See if you can show how you can use the grid of 6 pennies (3 rows of 2 pennies) to find out how many 3s you can separate from 6.
Here's one way to do it: Put your head on your left shoulder so that you see that there are 2 rows of 3 pennies. You can separate 2 3s from 6. So the answer is that Mikey can buy 2 3-cent stickers.
CHECK: Check your division by multiplication. We can write the number question this way: 3 X ? = 6. The grid shows that 3 x 2 = 6, which is the check. With this and the previous example you can see that with one multiplication fact, 2 X 3 = 6, you can answer two division questions: 6 ÷ 3 = ? and 6 ÷ 2 = ?

You know that 2 x 3 is the same as 3 x 2. Is 6 ÷ 3 the same as 3 ÷ 6? No, because separating is different than combining. The order matters in division.

CHALLENGE 1: Mikey has $12. How many toy dinosaurs costing $3 each can he buy? Use pennies to equal dollars, and divide 12 pennies into rows (one above the other) of 3 pennies each. Is the number of rows the number of dinosaurs he can buy? Write the division number question and also write it as a multiplication question. Later on you will find the answer by looking up the multiplication fact or recalling the fact that you have memorized.

CHALLENGE 2: Kevin has a $20 bill. How many $5 movie tickets can he buy? Write the division number question and also write it as a multiplication question.

CHALLENGE 3: Kevin decided to go to the movies by himself. He bought one $5 ticket with his $20-bill. How much change should he get from his $20? This is a separating problem, but is it a division problem? Why?
It is a subtraction problem because you will separate only one 5 from 20.

CHALLENGE 4: Think of a division problem involving 8 people with sets of 2 people each. Write the division number question and also write it as a multiplication question.

CHALLENGE 5: Think of silly division story problems that come down to each of these number questions:

4 ÷ 2 = ? , 6 ÷ 2 = ? , 10 ÷ 2 = ? , 9 ÷ 3 = ?

CHALLENGE 6: Is 2 x 8 the same as 8 x 2? Is 8 ÷ 2 the same as 2 ÷ 8?

Back to the Beginning

[Go to and print out the Multiplication Table.]

You have learned the 3 x 4 means that we must combine 3 4s which is the same as combining 4 3s. You can think of either number as a factor. In multiplication we multiply one factor by another factor to get the product. Look at the multiplication table. It has one set of factors along the left side [point] and the other set of factors along the top. The products are shown in the table.

EXAMPLE: Linda gave a party for her Outer Space Girls Club. She needed to bake 4 cookies for each of the 6 members. How many cookies should she bake?

THINK: Is the problem to combine amounts by multiplication? Is the number question 4 x 6 = ?
COMPUTE: Can you figure out how to use the Multiplication Table? Take your time.
Here is one way: Go down the far-left column of factors to 4.
Go along the line to the right with your finger until you are under 6 on the top line of factors.
Your finger should be on 24, which is the product of 4 x 6.
CHECK: Check your thinking. Check your computation by finding the product of 6 x 4, starting with 6 in the left column.

CHALLENGE 1: Billy Bob bought some white mice. He learned about "multiplication" when in about a week he discovered each of his 5 mice had 6 babies. How many baby mice were there? Use the table. Check your answer.

CHALLENGE 2: Make the following multiplications using the table:

4 x 5 = ? , 5 x 4 = ? , 3 x 7 = ? , 5 x 8 = ? , 7 x 9 = ?

[Use your printout of the multiplication table.]

Since division is the opposite of multiplication, we can use the multiplication table for division by working backwards. In multiplication you are given two factors, and you find the product. In division you are given one factor, which is now called the divisor, and you are given the product, which is now called the dividend. You must find the other factor, which is now called the quotient, which is your answer.

EXAMPLE: Suzy had $6. How many $2 scary masks could she buy for her Halloween party?

THINK: Is the problem to see how many 2s you can separate from 6 by division? Is the number question 6 ÷ 2 = ? We can write this as a multiplication number question this way: 2 x ? = 6. The problem is to find the "missing factor," which is the quotient.
COMPUTE: Try to figure out how to use the multiplication table for this division problem. Think carefully about it.
One way to do it: Think of the 6 as the product of the factor 2 and the "missing factor."
Put your finger on the factor 2 in the far-left column of the multiplication fact table.
Move your finger along the line to the right until you come to 6. You will find the "missing factor," or the quotient, in the top line directly above the 6. It is 3. Suzy can buy 3 scary masks.
CHECK: Check your thinking. You can check division by multiplication. Find the factor 2 in the left column and go to the right along the line until you are under the other factor, which is the 3 in the top line. The number at your finger should be 6.

CHALLENGE 1: Sam likes 3 eggs for breakfast. How many breakfasts can he have from a dozen eggs? How many eggs is a dozen?

CHALLENGE 2: Troy has $18. How many $3 tickets to the movie can he buy?

CHALLENGE 3: Make these divisions using the table:

12 ÷ 4 = ? , 15 ÷ 3 = ? , 21 ÷ 7 = ? , 24 ÷ 8 = ? , 45 ÷ 9 = ?

Back to the Beginning

Now you are going to do multiplication the easy way on a pocket calculator, but you can still make mistakes if you don't check your work. You can check in four ways: 1) Check your thinking, 2) After each entry, check the entry in the display. 3) See if the answer sounds reasonable. 4) Repeat the operation.

EXAMPLE: Mrs. Brown bought 14 cream puffs costing 23 cents apiece. What was the total cost?

THINK: Is this a combining problem? What is the number question?
COMPUTE: You simply enter the number question into the calculator. You enter 1, 4, x, 2, 3, = and read 322 cents or three dollars and twenty-two cents or $3.22.
CHECK: Check each number after you enter it and recheck our thinking.

In these challenges think and write the number question, compute, and check.

CHALLENGE 1: Roger bought 4 oranges selling for 15 cents apiece. How much did they cost?

CHALLENGE 2: Cindy uses 4 spoons of sugar to make a cupcake. How many spoons of sugar will she need to make a dozen cupcakes?

CHALLENGE 3: Kile bought 3 CDs costing nine dollars and fifty-three cents or $9.53 each. What was the total cost?

CHALLENGE 4: Do these multiplications with the calculator and check your work:

10 x 10 = ? , 45 x 27 = ? , 67 x 255 = ? , 12 x $3.25 = ?

Division is also a snap with the use of a pocket calculator, but you must check your work.

EXAMPLE 1: Mrs. Jones spent $27 on gifts costing $3 apiece. How many gifts did she buy?

THINK: Is the problem to find out how many 3s you can separate from 27? Write the number question.
COMPUTE: Show how you can do it with the calculator.
(Enter 27, ÷, 3, =, and read 9 or 9 gifts.)
CHECK: Check your thinking. Did you check each number in the display after you entered it? Check the reasonableness of your answer.

EXAMPLE 2: Mrs. Jones has $27 to spend on gifts for her 3 children. If she spends the same amount on each child, how much can she spend for each gift?

THINK: Compare this problem with Example 1. In Example 1 the problem was to find out how many $3s you can separate from $27. You solve this problem the same way. This may sound funny because you can't really separate 3 children from $27. Think of it this way: You need to divide $27 into 3 sets--one set for each child--and find the amount in each set. So, is the number question 27 ÷ 3 = ?
CONMPUTE AND CHECK: Show how you can do it with the calculator and check your work.

CHALLENGE 1: Paul spent $24 on 4 movie tickets. How much did each ticket cost?

CHALLENGE 2: Mrs. Jones bought 100 vitamin tablets. Her family uses 5 tablets per day. (Per day means "for each day.") How many days will the tablets last?

CHALLENGE 3: Oranges were $2.55 for a 5 pound bag. How much is this per pound?

Is the answer fifty-one cents per pound?

CHALLENGE 4: Jane has $2.55 to spend on oranges that are fifty-one cents per pound. How many pounds of oranges can she buy?

CHALLENGE 5: Do these divisions with the calculator and check your work:

144 ÷ 12 = ? , 210 ÷ 15 = ? , 990 ÷ 11 = ? , 1,470 ÷ 210 = ?