Hands-on Algebra: Solving Multi-Step Equations

Just as we used cups and counters to solve one-step equations, they can also be used to solve equations with a variable on each side.

For the demonstration problem, 2x + 2 = -4, place 2 cups and two positive counters on one side of the mat. Place 4 negative counters on the other side of the mat. Notice it is not possible to remove the same kind of counters from each side. 

Add 2 negative counters to each side.

Group the counters to form zero-pairs and remove all zero-pairs. Separate the remaining counters into 2 equal groups to correspond to the 2 cups.

Each cup is matched with 3 negative counters. Therefore, x = -3.

Next, demonstrate solving w - 3 = 2w -1 with a cup and counters. Place 1 cup and 3 negative counters on one side of the mat. Place 2 cups and 1 negative counter on the other side of the mat. 

Remove 1 negative counter from each side of the mat.

Just as you can remove the same kind of counter from each side of the mat, you can remove cups from each side of the mat. In this case, you can remove 1 cup from each side.

The cup on the right, or the unknown, is matched with 2 negative counters. Therefore, the answer to the equation is w= -2.

Hands-on Algebra: Solving One-step Equations

Just as we have used cups and beans or chips to learn earlier math concepts, we can use these materials to teach students how to solve one-step algebra equations.

For the purposes of this lesson, a cup represents the variable, white counters represents positive integers and black represents negative integers. After representing the problem with the cup and counters, the goal is to get the cup by itself on one side of the mat by using the following rules:

• A zero-pair is formed by pairing one positive identical counter with one negative counter.
• You can remove or add the same number of identical counters to each side of the equation mat.
• You can remove or add zero-pairs to either side of the equation mat without changing the equation.

For our first teaching problem, we will use the equation x + (-3) = -5.  Place 1 cup and 3 negative counters on one side of the mat. Place 5 negative counters on the other side of the mat. 

Remove 3 negative counters from each side to get the cup by itself.

The cup on the left side is matched with 2 negative counters. Therefore, x = -2.

Now, let us solve the problem, 2p = -6.

Place 2 cups on one side of the mat. Place 6 negative counters on the other side of the mat.

Separate the counters into 2 equal groups to correspond to the 2 cups.

Each cup on the left is matched with 3 negative counters. Therefore, p = -3.

Lastly, let's solve the equation r - 2 = 3.

Let's change the equation to r + (-2) = 3. Place 1 cup and 2 negative counters on one side. Place 3 positive counters on the other side.

Notice that it is not possible to remove the same kind of counters from each side. Add 2 positive counters to each side.

Group the counters to form zero-pairs. Then, remove all zero-pairs.

The cup on the left is matched with 5 positive counters. Therefore, r = 5.

Students can now use what they have learned to solve equations you give them or they can write their own. They can justify their answer with a sketch in their math journals. As a quiz, they can write a paragraph explaining why zero-pairs can be used to solve an equation such as m + 5 = -8.

Hands-on Algebra: Adding and Subtracting Integers

Just as we have used counters to help our students learn addition and subtraction, we can use counters to help them understand addition and subtraction of integers. You will need counters of two different colors, one for positive integers and one for negative integers. We chose green blocks for our positive integers counters and black blocks for our negative integers counters.

The rules for this "game" are as follows:

A zero-pair is formed by pairing one positive counter with one negative counter.

Students can remove or add zero-pair to a set because removing or adding zero does not change the value of the set.

Using these rules, show your student how to use counters to find the sum -3 + (-2).

Place 3 negative counters and 2 negative counters on the mat to symbolize the equation.

Since there are 5 negative counters on the mat, the sum is -5. Therefore, -3 + (-2) = -5. That problem is pretty easy for students to see.

Now, use counters to find the sum -2 + 3.

Place 2 negative counters and 3 positive counters on the mat. Remind your students that it is possible to remove 2 zero-pairs.

Since 1 positive counter remains, the sum is 1. Therefore, -2 + 3 = 1.

Use the counters to find the difference between -4 - (-1).

Place 4 negative counters on the mat. Remove 1 negative counter.

Since 3 negative counters remain, the difference is -3. Therefore, -4 - (-1) = -3.

Use the counters to find the difference between 3 - (-2).

Place 3 positive counters on the mat. There are no negative counters, so you can't remove 2 negatives. 

Add 2 zero-pairs to the mat. Remember, adding zero-pairs does not change the value of the set. Now you can remove 2 negative counters.

Since 5 positive counters remain, the difference is 5. Therefore, 3 - (-2) = 5.

At this point, you can give your student a variety of simple problems that involve adding and subtracting integers, or he can make up some problems of his own. He can solve them using blocks, he can illustrate them in his journal or he can write about how he solved the problems in his math journal.

Hands-on Algebra: The Distributive Property

We have used rectangle tiles to model multiplication. They can also be used to show the Distributive Property of Algebra. 

Use a tile that is about 1 square unit. I am using about a 1-inch square of cardstock. 

Next make an "x" tile by making a unit that is 1 unit wide and as long as you wish. For our purposes, we made it about three or four times as long, but remember that it is "x" units long.

We begin by using the tiles to find the product of 2 (x + 2). The rectangle has a width of 2 units and a length of x + 2 units. We can use our area tiles to mark off the dimensions on a mat, or in this case, a dry-erase board, that will show us the product. Using the marks as a guide, we make the rectangle with the algebra tiles.

The rectangle has 2 x-tiles and 4 1-tiles. The area of the rectangle is x + 1 + 1 + x + 1 + 1 or 2x + 4. Thus, 2(x + 2) = 2x + 4

Now you just need to give your student some practice problems, (or he can even make up some of his own, if he's like.) He can use the tiles and a dry erase board and write the answers in his math journal or he could also solve problems in this way by sketching similar drawings in their math journals.

As a quiz, you could have your student write a paragraph explaining how to find the proof of such problems.