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Learning to Add and Subtract |
Addition
What is Addition? We have two piles of coins and we want to know how many coins we have all together. You can count the coins in each pile and use addition to find out the total number of coins. The symbol used for addition is "+". Here is an example:
(1 + 1 + 1)
(1 + 1)3 + 2 = 5 total coins
(1 + 1 + 1 + 1 + 1)Purpose - In the previous page, while building the trains, you actually practiced doing addition. Here we will do more exercises for you to really understand addition.
Activity 1 - In this activity we are going to do an example of addition and then give you several exercises for you to practice. Here is an example using both the bars and the equation.
We added the size 7 bar to show that the answer of 7, which is the total, is correct because it is the same size as the train made by the size 2 and size 5 bars above. Make sure you understand the problem and the solution. Here are 8 exercises for you to practice addition.
Exercises - Build up the trains for these problems and make sure you get the right answer.
<- Click on this image to start the applet
3 + 2 = 3 + 6 = 5 + 3 = 7 + 2 = 4 + 3 + 2 = 5 + 4 + 6 = 6 + 2 + 4 = 2 + 4 + 1 + 3 = To check your answers click here answers.
Commutative and Associative Rules
Purpose - To explain the meaning of two words or terms: commutative and associative rules.
Commutative Rule - This means that "5 + 3" is the same as "3 + 5". The order of the numbers that you are adding is not important. You can add them in any order you want. Here is an example using bars:
Associative Rule - This means that when you are adding more than two numbers you can combine them in any way you want. For example, "4 + 3 + 2" can be added as
(4 + 3) + 2
or
4 + (3 + 2)The parentheses, which are the symbols ( and ) mean that you add the two numbers inside the parentheses first. Here are two examples, one on the left, and one on the right::
(4 + 3) + 2 You first add 4 and 3 4 + (3 + 2) You first add 3 and 2 7 + 2 and then you add the 2 4 + 5 and then you add the 4 9 The total is 9 9 The total is 9 Combining both commutative and associative rules - Both rules can be combined when doing addition. Here is an example:
5 + (3 + 1)
(3 + 1) + 5
3 + (1 + 5)In all three equations above the total is 9.
Subtraction
What is subtraction? We'll explain subtraction in a similar way as we explained addition. You have a pile of 5 coins and you decide to remove or subtract 2 coins. You need to find out how may coins are left in the pile. The symbol used for addition is "-". Here is the example:
(1 + 1 + 1 + 1 + 1)
(1 + 1)5 - 2 = 3total coins
(1 + 1 + 1)Purpose - To learn how to do subtraction by using integer bars.
Activity 1 - Again we will start with an example and then give you some exercises for you to practice. The example is 7 - 3. Here are the black and light green bars example:
We start with a 7 bar Then we align it with the 3 bar that we want to subtract The next step is to fill in the space to the right of the green all the way to the end of the black. This is also called the difference. The bar that fills the space is the purple bar. So the answer is 4 To write this as an equation it would be:
7 - 3 = 4
Again, make sure that the example makes sense and that you understand it. Here you have 8 exercises for you to practice subtraction.
Exercises - Use the integer bars to work on these problems and make sure you get the right answer.
3 - 2 = 6 - 3 = 8 - 4 = 9 - 2 = 10 - 4 = 12 - 2 = 15 - 8 = 18 - 9 = Make sure to answer all the problems and then you can check the answers by clicking here: answers.
Purpose - To check if you can use the commutative and associative rules with subtraction.
Activity 2- We will only explain if they worl with subtraction.
Commutative Rule - Would "5 - 3" be the same as "3 - 5"? What do you think? The answer is obviously no. Removing or subtracting 3 from 5 would give you an answer of 2, but removing 5 from 3, you would remove the 3 and you would still have 2 more to remove, so the answers are completely different.
Associative Rule - Here is an example to show if this rule works when subtracting:
(6 - 3) - 2 You first subtract 3 from 6 and you get a 3 6 - (3 - 2) You first subtract the 2 from the 3 and you get a 1 3 - 2 you then subtract the 2 from the 3 6 - 1 you then subtract the 1 from the 6 1 the answer is 1 5 the answer is 5 Again, the answers are completely different.
You can not use the commutative and associative rules with subtraction.
Units
Purpose - In the exercises we have done so far we have used the white bar as a unit of 1. Now we are going to show that any size bar can be defined to be a unit of one.
Different units, dark green = 1, then red ?, which is half? How big the blue? Bar that is 3 times size of red, green, white.
Activity - As an example, let's assume that the red bar (size 2) is defined as a unit of 1. In that case, how many units would the purple bar (size 4) be? Think about it. How many red bars can fit into the space of one purple bar? Here is a picture that shows how many fit.
This shows that the purple bar would be 2 units. Even though its size is 4, since the unit of one is the red bar (with a size of 2), then the purple bar is 2 units. Now how many units would the dark green bar be and how many units would the brown bar be? Again, think about it and look at the following pictures to help you figure it out.
The dark green bar is 3 units and the brown bar is 4 units.
Exercises - Now let's assume that the light green bar is one unit. How many units would the dark green bar be? How many units would the blue bar be?
Now let's assume that the purple bar is one unit. Which bar would be 2 units? Which bar would be 3 units?
See if you got the correct answers.
Playing and Learning to Use Integer Bars
Table of Contents
Learning to Multiply Using Integer Bars
Last Updated: Saturday, 19-Jul-2003 00:45:54 GMT
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