You have learned to count, so you have already learned one way to do addition. Here are some pennies [5], and we will count them.
Here is 1 penny, and I will add 1 penny, which makes the total or sum of 2 pennies.
I will add another penny, which makes the sum of 3 pennies.
I will add another penny, which makes the sum of how many pennies?
I will add another penny, which makes the sum of how many pennies?
When you count by ones, do you understand that you keep adding one to the old sum and getting a new sum?

So, learning to count means learning a lot of addition facts such as 1 and 1 makes 2 or, in math talk, 1 plus 1 equals 2 or 1 + 1 = 2. To continue, 2 + 1 = 3, 3 + 1 = 4, 4 + 1 = 5, etc. By learning other addition facts we can add more than one penny at a time.

EXAMPLE: Jack had 3 pennies and found 2 more pennies. Then how many pennies did he have?

THINK: Do you feel that you need to combine or add 3 pennies and 2 more pennies to find the total or sum of the pennies that Jack now has? Ask yourself this question: 3 and 2 is what or 3 plus 2 equals what? Write this number question this way: 3 + 2 = ?
COMPUTE: See if you can figure out how you might answer the number question by counting pennies. Here are some pennies [5] for you to use. Take your time.
Here is one way of adding by counting pennies:
1) Count off 3 pennies and put them in a row.
2) Count off 2 more pennies and combine them with the 3 pennies at the end of the row.
3) Count the combined pennies.
You now have the answer to the number question that you wrote. Erase the question mark and write the sum, which is 5. Now you have an addition fact: 3 + 2 = 5. What is a fact?
Is 3 + 2 = 6 an addition fact? No, because it is not true.
If you remember the 3 + 2 = 5 fact, you will not need to count up the sum the next time you need to add 3 + 2.
CHECK: You can check your work by checking your thinking and by recounting the pennies, add 2 + 3 pennies. Does 2 + 3 add up to the same sum as 3 + 2? When you combine amounts, it doesn't make any difference in what order they are combined.

CHALLENGE 1: Big Tony ate 2 ice cream cones. Then he ate 3 more cones. What is the sum of the cones that he ate? What is the number question, and do you remember the addition fact that you just discovered?

CHALLENGE 2: Tricky Trevor told trusting Tessy that he saw a flying saucer land, and 4 little green men got out. Later, 3 more little green men got out. How many little green men got out of the flying saucer? Think, compute with pennies, and check your answer.

CHALLENGE 3: Silly Willie said that 4 dinosaurs stopped in front of his house. Then 5 more dinosaurs stopped. How many dinosaurs stopped in front of his house?

CHALLENGE 4: Billy's mother bought a $7 cake and a $6 carton of ice cream for his birthday. How much did she spend? Let pennies stand for dollars.

CHALLENGE 5: Kelly dreamed that 5 butterflies were fluttering over her bed. They were joined by 8 more butterflies. How many butterflies were there altogether?

CHALLENGE 6: Now here is a fun thing to do. Think of an addition story problem about two sets of kittens that comes down to this number question: 4 + 2 = ? End your story with this question: How many kittens were there altogether? Make the addition by counting pennies.

CHALLENGE 7: Here's even more fun. Think of silly addition story problems using dinosaurs or ghosts or whatever for each of the following number questions, compute by counting pennies, and check your answer.

5 + 5 = ? , 6 + 3 = ? , 6 + 5 = ? , 7 + 8 = ? , 4 + 6 = ? , 5 + 9 = ? , 8 + 8 = ?

[You will need 10 pennies, a dime, and a nickel.]

When you count backwards by ones, you are doing subtraction. Try counting backwards from 5 to 0.
Here are 5 pennies, and I will subtract or take away 1 penny, which leaves a remainder of 4 pennies.
I will take away the second penny, which leaves a remainder of 3 pennies.
I will take away the third penny, which leaves a remainder of how many pennies?
I will take away the fourth penny, which leaves a remainder of how many pennies?
I will take away the fifth penny, which leaves nothing.
When you count backwards by ones, do you understand that you subtract 1 with each count?

Here is how we write these subtractions in math language:
5 - 1 = 4, 4 - 1 = 3, 3 - 1 = 2, 2 - 1 = 1, 1 - 1 = 0.

Here is an example of subtracting more than one from another amount:

EXAMPLE 1: Jack had 5 pennies. He lost 2 pennies through a hole in his pocket. How many pennies did he have left?

THINK: Instead of combining or adding more pennies to those in his pocket 2 pennies were separated and lost. Does the problem come down to this: 5 take away 2 leaves what? In math language we say the number question is 5 minus 2 equals what or 5 - 2 = ?
COMPUTE: See if you can figure out how to answer the number question by using the addition fact that you have learned.
In the addition example Jack had 3 pennies and found 2 more pennies making a total of 5. Now we learn that he lost 2 pennies. So, how many pennies are left?
You discovered the addition fact, which is 3 + 2 = 5. You can turn this addition fact into this subtraction fact: 5 - 2 = 3.
Count out 5 pennies and take away 2 and count what is left. Sure enough, 3 are left.
Suppose he had lost 3 pennies. You can also turn the addition fact into a second subtraction fact: 5 - 3 = 2.
Count out 5 pennies and take away 3 and count what is left.
CHECK: Think about the problem again and check your work. Remember that subtraction is the opposite of addition. If you take away an amount with subtraction, you can check your computation by putting it back with addition. So, take the 3 pennies left and add back 2 pennies and you should have the original 5 pennies.

CHALLENGE 1: Trevor took 4 pieces of candy from the bowl. His mother took 2 pieces away from him. How many pieces of candy did Trevor have left? Remember to think, compute by counting pennies, and check with addition.

CHALLENGE 2: Do you remember the 9 dinosaurs that stopped in front of Willie's house? Well, 6 of them went away. How many remained?

CHALLENGE 3: Todd bought some apples costing $6 and gave the clerk a $10 bill. How much change should Todd receive?

CHALLENGE 4: Jimmy had 7 marbles. He lost 3 marbles. How many marbles did he have left?

CHALLENGE 5: There were 7 monkeys in a tree. Four of them jumped out. How many were left in the tree?

CHALLENGE 6: It's fun time again. Think of silly subtraction story problems for each of the following number questions, compute with pennies, and check.

4 - 2 = ? , 6 - 4 = ? , 7 - 3 = ? , 8 - 5 = ?

How could knowing how to subtract help you if you go to a store to buy something?

EXAMPLE 2: Suppose I buy a gift for $6 and pay for it with a $10 bill. You be the clerk and count out the change.

THINK: You could say, "$10 minus $6 leaves $4 change." The number question would be 10 - 6 = ? But you want me to see that you are not cheating me even if I don't know what 6 minus 4 equals. You learned that subtraction is the opposite of addition. So, you can turn 10 - 6 = ? into a kind of addition number question: 6 + ? = 10. You find the "missing addend," which is the remainder or the amount of change.
COMPUTE: See if you can count out the change starting with the count of 6 and letting pennies stand for dollars and letting a dime or 10-cent piece stand for a $10 bill.
Here is what a clerk might do: He takes your $10 bill, lays it to one side, and counts out the change. (Why doesn't he put the bill in the cash register? If he put it in the cash register, you could later say that you gave him a $20 bill, and he couldn't prove that you didn't.) He says, "Your purchase comes to $6." He then gives you one dollar bill at a time and says "seven, eight, nine, and ten." In other words he starts counting with 6 and continues the count (giving you a dollar on each count) up to 10, which is the amount you gave him.
CHECK: Subtract by taking away 6 pennies from 10 pennies in the usual way. What is the remainder or the amount of change you should get?

CHALLENGE 7: Let's say you are selling magazines for $2 apiece. A man takes one magazine and gives you a $5 bill. Count out the change using pennies for dollars and using a nickel for the 5-dollar bill. Check by subtracting with pennies in the usual way.

CHALLENGE 8: You make up a story problem in which Bill is selling something for $7, and someone gives him a $10 bill. You count out the change with the purchase using pennies for dollars. Check by subtracting with pennies in the usual way.

CHALLENGE 9: Count out the change for each of these purchases which are stripped down to the number question and check by subtracting with pennies in the usual way:

3 + ? = 5 , 8 + ? = 10 , 4 + ? = 10 , 15 + ? = 20 , 94 + ? = 100

[This is the student's introduction to measurement and the use of rulers. He also learns to use rulers as addition "calculators." Two rulers with a metric scale and paper and pencil will be needed. By using the metric scale all of the addition facts can be modeled. Go to and print out the rulers graphic.]

Have you ever measured the length of anything with a ruler? Here is a ruler that measures length in little units of length called inches and centimeters. How could you use the ruler to make an addition?

EXAMPLE 1: A worm crawled 4 centimeters and rested. Then he crawled 2 centimeters farther. How many centimeters did he crawl altogether?

THINK: Must you combine two amounts by addition? Write the number question.
COMPUTE: How could you make the addition using a ruler and pencil and paper?
Well, you might draw a line that is 4 centimeters long. Now, add to that line by continuing another 2 centimeters. Put the ruler beside the entire line so that it starts at the beginning of the line. What is the total length of the combined lines? Did the worm crawl 6 centimeters?
CHECK: Find the sum of 4 + 2 using your fingers. Hold 4 fingers up on one hand and 2 on the other and count the total fingers held up.

CHALLENGE 1: Sandy grew 3 centimeters in height last year and 2 centimeters the year before. How many centimeters did she grow in the two year?

Now here's a fun idea: We will now use two rulers as a calculator to add lengths.

EXAMPLE 2: Sammy had a chocolate bar that was 8 centimeters long. His little brother Jimmy had a bar that was 7 centimeters long. If they put their two bars together end to end, how long would the two of them be?

THINK: Must you combine 8 and 7 with addition? What is the number question?
COMPUTE: See if you can figure out how to use the two rulers to compute the sum. Take your time.
Here is one way to do it: See the picture of the rulers. Find 8 on the top ruler. Notice that on the bottom ruler's 0 is under the 8.
Now find 7 on the bottom ruler. What is the number on the top ruler that is above the 7? That's the sum of the chocolate bar lengths.
CHECK: Think again about the problem. This time start by finding 7 on the top ruler and 8 on the bottom ruler.

CHALLENGE 2: Teddy plays football. He carried the football 3 yards on the first down and 4 yards on the second down. Using the rulers compute the sum of the yards he carried the ball? A yard is about the length of a tall man's arm, but you can use the rulers by pretending that each centimeter is a yard.

CHALLENGE 3: Mary made some party punch. She put 4 cups of cranberry juice in a bowl. Then she put 3 cups of ginger ale in the bowl. How many cups of liquid did she put in the bowl? Use the two rulers as a calculator to compute the answer. [If it is convenient, also have the learner make the addition by adding 3 cups of water to 4 cups of water in an 8-cup measure and reading the total.]

CHALLENGE 4: Freddy had $6, and he earned $9 more from his paper route. Then how much money did he have? Use the two rulers as a calculator. Check by also using the rulers.

You can also use rulers to subtract by doing the opposite of addition.

EXAMPLE 1: Sammy had a chocolate bar that was 11 centimeters long. He ate 3 centimeters off the end of the bar. How long was the remaining piece?

THINK: Do you need to separate one amount from another and see low long the remainder is? Is the number question 11 - 3 = ?
COMPUTE: Think about how you can use the ruler and paper and pencil to find the remainder.Take your time.
Here is one way to do it: Draw a line that is eleven centimeters long. Now measure 3 centimeters back from the end and make a mark. Next, measure the rest of the line to find the remainder, which is 8 centimeters.
Here is another way to do it with two rulers: Look again at the picture of the rulers, rulers graphic. Notice that the rulers show that the sum of 8 plus 3 is 11. Since subtraction is the opposite of addition, the rulers also show that 11 minus 3 leaves a remainder of 8.

CHALLENGE 1: Kyle had a licorice stick that was 20 centimeters long. His little brother ate 5 centimeters off the end. How long was the remainder? Use one ruler and pencil and paper.

CHALLENGE 2: Sammy has 15 baseball cards. He traded 7 of the cards for a baseball. How many cards did he have left? Use the two rulers as a calculator.

CHALLENGE 3: Bryan took his 13 bugs to school, and 4 escaped and ran away. How many bugs did he have left? Use your "ruler calculator" and check your answer.

CHALLENGE 4: Kathy had $7 in her piggy bank. She took $3 out and spent it. How much did she have left in her piggy bank? Use the "ruler calculator."

EXAMPLE 2: Daniel's father is 6 feet tall. Daniel is 4 feet tall. How much taller is his father?

THINK: So far we have been doing "take away" subtraction problems and finding the remainders. We took one amount of something away from another amount. In this problem we compare one amount to another and see how much larger one amount is. We call the answer the difference instead of the remainder, but our computation is the same as in finding a remainder. The number question is this: 6 - 4 = ?
COMPUTE: Think about how you can compute the difference in their heights with rulers.
Here is one way to do it with one ruler and pencil and paper: We will let centimeters stand for feet. Take a sheet of paper and draw a line 6 centimeters high starting at the bottom of the paper. Draw another line right beside the first and make it 4 centimeters high. Now measure the difference between the lengths of the two lines.
Here is a way to do it using two rulers: Find the 6-centimeter mark on the top ruler, which is the father's height.
Put the 4-centimeter mark on the bottom ruler, which is Sammy's height, under the 6.
Look at the left end of the top ruler. What is the difference in their heights?
CHECK: Check your computation on the "ruler calculator" by adding the difference back to Sammy's height to see if you get 6. You can do this without moving the rulers.

CHALLENGE 5: Kevin caught a fish that was 9 inches long. His brother Terry caught one that was 6 inches long. Kevin's fish was how much longer than Terry's? Use one ruler and paper and pencil.

CHALLENGE 6: Rob has 7 marbles and Ted has 4. How many more marbles does Rob have compared to Ted? Use your "ruler calculator."

CHALLENGE 7: Kevin ate 9 cherries. His little brother ate 4 cherries. How many more cherries did Kevin eat compared to his brother? Use the "ruler calculator."

[Go to and print out the Addition Table.]

Here is another cool tool. When we do addition, we combine one addend with another addend. You can find the two addends in an addition table and read the sum.

EXAMPLE: Three black birds were on a telephone wire. Then 2 more birds landed on the wire. How many birds were then on the wire?

THINK: Must we combine two amount by addition? What is the number question?
COMPUTE: The numbers in the top row of the table are one set of addends, and the numbers in the far left column are the other set of addends, and all of the other numbers in the table are sums of pairs of addends. Can you figure out how you can use the table to add 3 and 2?
Here is one way to do it: 1) Find one addend, the 3, in the far left column.
2) Move your finger along the line to the right until it is under the other addend, 2, at the top.
3) The number at your finger is the sum.
CHECK: Check your thinking. Check your answer by adding 2 plus 3 instead of 3 plus 2 to see if you get the same sum.

CHALLENGE 1: Ted rode his bike 8 blocks to school. Then he rode 9 blocks to the library. How many blocks did he travel in all? Use the Addition Table.

CHALLENGE 2: Think of silly addition story problems for each of these number questions and use the Addition Table to find the sums:

2 + 3 = ? , 4 + 9 = ? , 5 + 7 = ? , 6 + 8 = ? , 5 + 5 = ?

Wait a minute! How can you use an addition table to do subtraction? Well, subtraction is the opposite of addition, so you can use the addition table for subtraction by working backwards. In addition you are given the two addends, and you look for the sum in the addition table. In subtraction you are given the sum (now called the minuend), and you are given one addend (now called the subtrahend). You look for the "missing addend" (now called the remainder.)

EXAMPLE: There were 7 big bloodthirsty mosquitoes in Mark's bedroom. He swatted 4 of them. How many were left?

THINK: Is this a combining problem or a separating problem? What do we separate from what? Is this the number question: 7 - 4 = ?
COMPUTE: See if you can figure out how to use the table to find the missing addend, which is the remainder. Take your time.
Here is one way to do it:
1) Find 4 in the left column of the table.
2) Go along the line to the right until you reach 7.
3) Go up to the top line and read the remainder, which is 3.
CHECK: Check your thinking. Using the table, add your remainder to 4 and see if the sum is 7. The table shows both the addition and the subtraction at the same place.

CHALLENGE 1: Mrs. Brown bought a dozen eggs. How many eggs is that? She boiled 8 of them for Easter eggs. How many were left? Use the table.

CHALLENGE 2: Heather has 14 pieces of chocolate hidden away. Her little brother found them and ate 5 pieces. How many did she have left? Use the table.

CHALLENGE 3: Think of silly subtraction story problems for each of these number questions and use the Addition Table to find the remainders:

7 - 2 = ? , 9 - 4 = ? , 12 - 7 = ? , 15 - 9 = ? , 16 - 8 = ?

[An electronic pocket calculator will be needed.]

The pocket calculator is the greatest invention since ice cream! All you have to do is enter the number question, and, Presto! You have the answer. But remember that as you enter each thing, you must CHECK the register to see if you entered the right numbers.

EXAMPLE: Freddie had $15 in his piggy bank. He put in $8 more. Then how much money did he have in the bank?

THINK: Is this is a combining problem or a separating problem? Write the number question.
COMPUTE: Enter the number question.
(Press 1 and 5, press +, press 8, press = and read the sum [23].)
CHECK three things: 1) Check your thinking. 2) Did you check each number in the register after you entered it? 3) Does your answer sound reasonable?

CHALLENGE 1: Skip had $23 in his savings account. He sold his baseball cards for $17 and put the money in his savings account. How much money did he then have in his savings account?

CHALLENGE 2: Mikey has $1.45 (one dollar forty-five cents.) He received his allowance of $1.50. Then how much money did he have? When you enter the numbers, also enter the decimal point that looks like a period.

CHALLENGE 3: Think of silly addition story problems for each of these number questions and use the calculator to find the sums:

14 + 27 = ? , $1.50 + $3.80 = ? , 26 + 48 = ? , $55.00 + $65.00 = ?

In doing a subtraction you enter the number question with the minus sign. Include any decimal points.

EXAMPLE: Ted has $3.75 in his pocket. If he buys an ice cream cone costing $2.25, how much money will he have left?

THINK: Do you need to combine amounts with addition or separate amounts with subtraction? Write the number question.
COMPUTE: Enter the number question and read how much money Ted will have left.
(Press , the decimal point , 7, 5, - (minus sign), 2, the decimal point , 2, 5, = and read the remainder, 1.5. It left off the ending zero, so the remainder is $1.50 or one dollar fifty cents.)
CHECK: Check your thinking. Did you check each number in the register as you entered it? Does your answer sound reasonable?

CHALLENGE 1: Mikey had $1.45, and he spent $0.85 (eighty-five cents). How much did he have left?

CHALLENGE 2: Franklin had 13 white mice. Six of them escaped when he was cleaning the cage. How many did he have left?

CHALLENGE 3: Think of silly subtraction story problems for each of these number questions and use the calculator to find the remainders:

19 - 14 = ? , $1.25 - $0.70 = ? , 55 - 39 = ? , $6.00 - $3.88 = ?