1. Using Algebra Tiles to Solve Equations, Combine Like Terms, and use the Distributive Property
Objective: To understand the different parts of an equation, and use algebra tiles to help us solve problems.

2. Important Vocabulary!
Equation – An equation is a mathematical statement that uses an equal sign to show that two expressions have the same value.
To solve an equation that contains a variable, find the value of the variable that makes the equation true. This value of the variable is called the solution of the equation.
Term – the parts of an expression that are added or subtracted.
Like Term – Two or more terms that have the same variable raised to the same power.
Coefficient – The number that is multiplied by a variable in an algebraic expression.
Constant – A value that does not change.
Equivalent Expression – Equivalent expressions have the same value for all values of the variables.

3. Parts of an Equations! 5x + 4x + 5 = 50 Like Terms constant
coefficient
variable

4. Your Turn… 6y + 5x + 2y = 42 Coefficients? Variables? Like Terms?
Constant?
Coefficients = 6, 5, 2
Variables = y and x
Like Terms = 6y and 2y
Constant = 42

5. Discovery What do you think the different tiles stand for? Why?
Even though students may have had experience with the algebra tiles, most probably have not. Allow students 5 minutes to “play” with the tiles so they can get it out of their system.
Have them make a separate pile of each shape on their desks. Have them place their Tile Mat in front of them.
Ask students what they think the tiles stand for and why. (Some students may already know…)
Most students will try to stack the unit tiles end to end on top of the x tile to equate say the x tile equals 5. It is important you clarify this misconception. When the student stack them end to end there is a small section which the unit tiles do not cover. This is the “unknown” portion of the x tile and the reason it is represents x. The students may discover that the “short” side of the x tile is the same height as the unit tile. So, x times 1 equals x. This discovery of how the sides of the tiles relate to each other will be important as they finish middle school math and move into high school math.
Allow the student to discover why the blue square is x2. Allow them to take 2 x tiles and put them against the length and width of the blue square to see it stands for x • x or x2
The single unit tile stands for 1.

6. What do these stand for? Why?
Algebra Tiles
Have student flip the tiles over and see that the opposite sides are red. Ask them why they think they are red and what they stand for? If students do not discover the red tiles are negative representations of that particular tile, tell them.
What do these stand for? Why?

7. Let’s Try It
Represent the following equations on your tile mat. Compare your answer with a neighbor. Assist each other as needed.
5 + x = 2
5 – 5x = -1
2x – 5 = 9
Ask student to use the tiles to represent (NOT SOLVE) the following equations on the tile mat:
Now ask students to return to the first equation, build it again and model how to solve the equation.
Note:
Ask students “what does it mean to isolate?” Give real life examples of what “isolate” means and relate it back to what it means to isolate x.
Since we need to isolate the x to solve the equation, we will need to turn the 5 positives into zero. How can we get those 5 positives to make zero? We will use inverse operations by placing 5 negatives with the 5 positives. An equation is like a balance. If we place 5 negatives on one side of the equation, we MUST do it to the other side of the equation as well. We will be left with x on one side of the equal sign, and 2 positives and 5 negatives on the other side of the equal sign. Remove the zero pairs, and students will have 3 negatives left. This means the answer will be x = -3.

8. Build this equation 5 + x = 2 On your own: x – 2 = 3 x + 3 = 7
Allow students to build the equation THEN show them the example.
Allow them to solve for x as you remind them about inverse operation and isolating the x to solve for x.
Remind students that a negative and positive tile is considered a zero pair because a negative and a positive of the same quantity make zero.
Remind them whatever you do to one side of the equal sign, you must be do to the other side of the equal sign. Only then should you show them the example of the answer.
Allow students to solve 1 step equations in collaborative groups. Provide them with the following problems:
x – 2 = 3
x + 3 = 7
7 – x = 9
x – 5 = 1
2 = -x – 4
It is important for students to understand why they can’t leave a variable negative. Although you probably already know…if a variable is left negative, you truly haven’t isolated the variable. If a variable is negative, that means there is a value of -1 attached to it. It means there is a -1 times x. The inverse operation is division, therefore you divide both sides of the equal sign by -1 to truly isolate the variable.

9. To solve for the variable, you must do the inverse operation.
Build this equation
2x = 6
-3x = 15
-12 = -4x
3x = 12
6x = 3
5 = 5x
Ask students what the inverse operation of multiplication is? After they say division you can have the discussion about putting tiles into even groups with the x tiles and unit tiles. AND if the unit tiles do not fit into even groups, then they are probably dealing with a fraction or a decimal.
Have student work through various problems collaboratively with the tiles.
To solve for the variable, you must do the inverse operation.
With tiles, in order to divide, you must create even groups of x tiles and unit tiles.
x = 3

10. When should we NOT use tiles?
Let’s say this piece of paper represents our whole x.
How many sections are there on the paper?
How many positive tiles will go in each section?
Using the visual, what is the value of x?
Students must realize when the tiles are not a good idea to use. With division problems such as x/6 = 2, tiles can be confusing and are truly more of a nuisance than a help.
Instead, student should see the pattern of using the inverse operations when solving equations and begin to apply this practice with division problems.
Students should view a problem like x/6 = 2 as “x groups of 1/6 equals 2”
We have to think of 𝑥/6 as 1/6 times x. We can solve for x by thinking, I need 6 groups of 2 to make one whole. This
means we will need 6 parts to make one whole, and there are 2 positives in each part. How many positives are there in 1 whole x?
What the students will find, is that they are multiplying 6 by 2 (6 groups OF 2) to find what x equals.
Intervention

11. To solve for the variable, you must do the inverse operation.
Build this equation
This can stand for x/5
(5)
(5)
Provide this type of problem and discuss what portion of the equation can be solved with tiles and what cannot.
Allow students who need to use tiles to add 6 to each side of the equal sign, to do so.
However, when they are solving for x, allow them to see that it is easier to perform the inverse operation to solve for x.
Ask students what the inverse operation of division is? After they say multiplication you can have the discussion about how the “x” cannot be split or cut. Students should discover that there will be 10 tiles in each of the 5 sections that make up x. They will quickly see that they can just multiply to solve for x.
Have student work through various problems collaboratively with the tiles.
Gauge your students understanding and provide extra problems to use with algebra tiles as needed.
To solve for the variable, you must do the inverse operation.
With tiles, you must isolate x first, then you can figure out what x equals.
x = 50

12. Make even groups with each x
Activity…
Use your algebra tile mat and algebra tiles, to solve the following equations.
2x – 3 = 9
5 – 5x = -1
3x – 1 = 8
7 = 5x + 2
2x + 3 = 3x
4x – 2 = 3x + 6
Zero pairs
Work with the students on the following problems. Students might have an issue with using tiles with division. (Example is provided).
2x – 3 = 9 (example provided)
5 – 5x = -1 (you may need to have a conversation regarding the negative attached to the 5x.)
When you feel your students are ready for independent practice allow them to do so.
2x + 3 = 3x (allow students to determine they have to combine like terms here)
Gauge your students understanding and provide extra problems to use with algebra tiles as needed.
Make even groups with each x
x = 6

13. Summary!
How will algebra tiles be useful to you in solving equations and combining like terms?

14. Combining Like Terms x2
What does this tile represent? What do these tiles represent?
-x2 x -x
Students should be familiar with the algebra tiles from the activity in week 1 and know what each tile represents. This is just an overview.
When combining like terms, students need to discover that “plus negative” is the same as “minus” or subtraction (Most students will come to the conclusion on their own, but some will be confused with how we know we are subtracting if there is no minus sign. Teachers must clarify the negative sign IS the minus sign).
Have a brief discussion about zero pairs. Clarify any misconceptions.
Briefly talk about combining tiles. Provide students with examples, “Do you think I can combine the x2 tile with the unit tile? Why or why not?”
Allow students to discover (with your lead) that only like-tiles can be combined.
Gauge your students understanding and provide extra problems to use with algebra tiles as needed. 1 -1

15. These are NOT the same shape
Combining Like Terms
4x + 5 Can these be combined? Explain your reasoning. 4x + 5x Can these be added together? Explain your reasoning.
These are NOT the same shape
In the first example, the tiles are different shapes and different sizes (different colors too since they are positive).
In the second example, the tiles are all the same size and same shape. Therefore, they can be combined together.
Have a discussion about whether tiles are be combined if they are all red (or negative). Clarify that in order for tiles to be combined, the shape of the tile is important.
Gauge your students understanding and provide extra problems to use with algebra tiles as needed.

16. Let’s Try It! 3x + 4 – 2x 3x + 5 2x2 – 6x +2 x2 – 2x – 3 3x2 + 3x – 5x
Represent the following expressions on your tile mat. Compare your answer with a neighbor. Assist each other as needed.
3x + 4 – 2x
3x + 5
2x2 – 6x +2
x2 – 2x – 3
3x2 + 3x – 5x
Ask student to use the tiles to represent (NOT SIMPLIFY) the following expressions on the tile mat:
3x + 5
3x + 4 – 2x
2x2 – 6x +2
X2 – 2x – 3
3x2 + 3x – 5x
Some misconceptions include: Some students will be very literal thinkers with the tiles. When asked to display 2x some students will put down 2 unit tiles and an x tile instead of 2 x tiles.
Now ask students to return to the last expression, build it again and model how to re-group the tiles by combining “like” tiles together. If there are any zero-pairs, remove them from the mat.
Gauge your students understanding and provide extra problems to use with algebra tiles as needed.

17. Combining Like Terms: Build It!x2 x2 x x x 1 1 1
2x2 + 3x + 5 +x2 – 5x – 1
Try these:
2x2+4x+2x2 – x
3x2 – 2x – 1 – 3x2 – 2x – 2
x2+2x+1 – 3x2 – x
3x2 – 3x + x2 – 1 + 2x – 3 1 1 x2 -x -x -x -x -x -1
The teacher will display on the PPT the first polynomial. Students can display the tiles as you display the tiles on the PPT. Allow students to see the zero pairs and the resulting answer.
You may need to complete additional problems with students. Have students put the like tiles together. This will help them see the zero pairs and be able to rewrite the expression easier.
Gauge your students understanding and provide extra problems to use with algebra tiles as needed.
3x2 – 2x + 4
What’s left??

18. Summary
Write 2 – 3 sentences explaining how you use algebra tiles to combine like terms. Pretend you are teaching this concept to a 4th grader.

19. Distributive Property
Using algebra tiles, we will use Distributive Property to help us combine like terms and solve equations.
Distributive Property - The property that states that if you multiply a sum by a number, you will get the same result if you multiply each addend by that number and then add the products.

20. How does it work?
Represent the following expression using algebra tiles:
3 (x + 2)
3 groups of x plus 2
For this particular example, it should be read as “3 groups of x plus 2.” Walk students through modeling this expression x + 2 will be represented 3 times.
This will help students visualize how distributive property works.
Once they have the expression modeled, they can move their like tiles together to see what they have.
In this example, they will have 3 x’s and 6 unit tiles. Put together, it would be 3x + 6.
Gauge your students understanding and provide extra problems to use with algebra tiles as needed.
When we group our like tiles, what expression
do we have?
3 (x + 2) = 3x + 6

21. Let’s Practice! Simplify the following expressions: 2(x – 4)
On your own:
(2x + 1)4
6(-x – 2) + 3
(3 – 2x)3 + x
2 groups of x – 4
Walk students through modeling the first example.
The standard emphasizes students simplifying using the distributive “forward and backward.”
Have students try to represent the following expressions with their groups:
4(2x + 1)
6(-x – 2) + 3
3(3 – 2x) + x
Gauge your students understanding and provide extra problems to use with algebra tiles as needed.
= 2x – 8
After grouping like tiles, what do we have?

22. Distributive Property and Equations
Use distributive property to solve the following equations!
2(x + 3) = 10
You try:
(-x – 4)3 = 3
2 + 2(2x – 3) = 8
4 = 3x – (-x + 3)2
Walk students through how to model the 2 groups of x plus three on one side of the tile mat, and 10 tiles on the other side of the tile mat.
Remind students about isolating the variable to be able to solve for x. In this example, students will have to place 6 negative unit tiles on both sides of the equal sign to create zero pairs and get the x tiles isolated.
When they do this, they will have 2 x tiles on one side, and 4 positive unit tiles on the other side. We divide the unit tiles between the two x’s and find that x equals 2.
Have students plug 2 back into the original equation to check if their answer is correct. If it is not, they will go back and find where a mistake was made.
Once you model how to solve the first example, have students try a few more with a partner.
Gauge your students understanding and provide extra problems to use with algebra tiles as needed.
x = 2
After making zero pairs, we are left with 2 x’s
And 4 unit tiles. What does x equal?

23. Summary
Pair up with a partner. Each partner will make up a problem that uses the concepts learned in today’s lesson. Switch problems with your partner and solve.