Florida ged plus college Preparation Program Curriculum and Resource Guide

Strategy – Algebra Manipulatives

Manipulatives can play an important role in the study of algebra. Algebra tiles provide a geometric look at symbols. They also support cooperative learning and help improve discussion in the algebra class by giving students objects to think with and talk about. It is in this type of context that true learning happens. Algebra tiles have a variety of other names: algetiles, math tiles, and virtual tiles. Algebra tiles help students visualize algebraic expressions and/or equations. Many use multi-colors so that students can use both positive and negative variables.

There are many different commercial versions of algebra tiles. However, you can also use an algebra tile template and print your own. Use heavy paper or laminate them so that they will be sturdy.

The Basics of Algebra Tiles
Each tile represents a specific monomial, e.g. x, y, x², y², etc. One color represents positive, the other negative. Often green symbolizes positive and red symbolizes negative.



Large Sq. = x² Rectangle = x Small Sq. = 1

To represent a 2nd degree polynomial, simply combine tiles. A red tile and a green tile of equal size will combine to give zero. Shown below is a representation of the polynomial 2x² - x + 3.

4 + 3 = 7

2 + ( - 3) = -1

The sum of - 8 + 4

Subtracting Integers

- 8 – (-3) = -5

(Remove 3 negative tiles to get the answer.)




4 – 7 = -3

(Because you are taking away 7, you start canceling out the negative and positive, ending up with a negative 3.)

Show students how to simplify expressions and model basic expressions before going to polynomial expressions. Students should also have time to “make collections” of tiles showing how sizes fit into a rectangular shape.

Model the following expression: 3x + 2 – 4x – 4

(Remind students that subtracting is the same as adding the opposite. Point out that negative tiles are red.)

3x + 2 - 4x -4

Show students that the expression can be simplified to – x - 2.
Have students make a collection of squares before going on to solving equations. This will assist students in arranging the tiles. Generally, two rules are used when “collecting” tiles.

• 
  Big squares can’t touch little squares.
  
• 
  Little squares should all be together.
  

Multiplying polynomials using algebra tiles is a lot like using the area approach to multiplying whole numbers. To multiply binomials using algebra tiles, place one term at the top of the grid and the second term on the side of the grid. You MUST maintain straight lines when you are filling in the center of the grid. The tiles needed to complete the inner grid will be your answer. Illustrated below is the multiplication fact (x-2)(x+1) = x² - x - 2.

This figure actually shows that (x-2)(x+1) = x² - 2x + x - 2.

To make sure you've got it, try some of these problems:

(x+1)(x+1) (Notice that the resulting array forms a perfect square.)

(x+3)(x-2)

(2x+1)(x-2)

Factoring is the reverse of multiplying. Using algebra tiles, build a rectangle containing the tiles specified in this problem (1 x²-tiles, 3 x-tiles and 2 1-tiles). Remember that the lines between the tiles within your pattern must be completely vertical or horizontal across the entire pattern. To factor x² - 3x + 2, arrange the appropriate tiles into a rectangular array and determine which factors would look like this. After the pattern is established, it can be seen that the top edge of the pattern (the length) is composed of tiles with dimensions x + 1. The side edge of the pattern (the width) is composed of tiles with dimensions x + 2. Consequently, x² + 3x + 2 = (x + 1)(x + 2).

When factoring, you may need to add some red and green tiles of equal size. Remember that you need to be able to make a rectangular array. If you can't make such an array, your polynomial can't be factored over the integers. Here are a few polynomials for you to factor.

x² + 2x + 1

x² + 3x + 2

x² - 1

Another Method for Multiplying Binomials – The Grid Method

To multiply binomials using a grid, place one term at the top of the grid (separating the elements into each compartment of the grid) and the second term on the side of the grid. You then multiply the rows and columns of the grid as you would in a coordinate system. Finish by combining like terms.

Retrieved form the World Wide Web on 02/25/06 at: http://www.learner.org/channel/workshops/algebra/.

Teachers will need the following:

• 
  A bag of 40 transparent chips (20 red, 20 yellow)
  
• 
  10 paper cups
  
• 
  10 equations for use at stations (the equation should appear on one side of a strip of paper, and the solution on the other side)
  

• 
  A bag of 20 chips (red on one side, yellow on the other)
  
• 
  10 paper cups
  
• 
  Individual dry-erase boards or large sheets of paper
  

• 
  x + 10 = 15
  
• 
  y - 3 = -1
  
• 
  5 - m = -2
  
• 
  w + 4 = -5
  

• 
  If the variable is positive, place the cup(s) facing up.
  
• 
  If the variable is negative, place the cup(s) facing down.
  
• 
  The coefficient of the variable indicates the number of cups to use.
  

• 
  The chips represent the numbers.
  
• 
  If a number is positive, the chip should be yellow side up.
  
• 
  If a number is negative, the chip should be red side up.
  

• 
  5m + 1 = -9
  
• 
  2x + 3 = 4
  

Explain to students that now that they have solved the same equations using cups and chips and symbolic manipulation (or algebra), it's time to try solving similar equations with symbolic manipulation (algebra) only. At 10 stations throughout the room, post various equations for the students to solve. Do not let them know that the solutions are given on the back of each piece of paper. Have students circulate in pairs through the stations, solving each equation and checking their answers. Give students 1-2 minutes at each station, as necessary. Below are some equations you might use (make sure some of the variables have negative and fractional coefficients):

• 
  3x + 2 = 14
  
• 
  -3m - 1 = -10
  
• 
  -7x + 5 = 12
  
• 
  -w + 13 = 9
  
• 
  ½d + 7 = 10
  

Clem's balloon is 200 feet off the ground and rising at a rate of 5 feet per second. Mary's balloon is 100 feet off the ground and rising at a rate of 9 feet per second. In how many seconds will the two balloons be at the same height? Show how you found your answer.

Solve the following equation for x.

Terry is going to the county fair. She has two choices for purchasing tickets, as shown in the table below.

Ticket Choices Admission Price Cost per Ride A $6.00 $0.50 B $2.00 $0.75

Write an equation for Terry's total cost (y) for ticket

How many rides would Terry have to go on for the total cost of ticket A and ticket B to be equal? Use mathematics to explain how you determined your answer. Use words, symbols, or both in your explanation.

Terry plans to go on 14 rides. To spend the least amount of money, which ticket choice should Terry choose? Use mathematics to justify your answer.

Solution:
For Choice A, the equation is y = 6 + 0.5x; for Choice B, y = 2 + 0.75x.

For the total costs to be equal, 6 + 0.5x = 2 + 0.75x, or x = 16; therefore, Terry would have to go on 16 rides.

For 14 rides, Choice A would cost 6 + 0.5(14) = $13. Choice B would cost 2 + 0.75(14) = $12.50. Terry should choose ticket B.

Four Fours

Using exactly four fours, write expressions equal to each of the numbers from 0 to 25. You may use the addition, subtraction, multiplication, division, square root, and factorial operators. You may combine fours such as .4, .4 repeating, 44, and 444. Make sure you are using the correct order of operations. You may only use parentheses when they are needed. The first few numbers are done for you as examples.

0 = 44 - 44	13 =
1 = 44 ÷ 44	14 =
2 = 4 ÷ 4 + 4 ÷ 4	15 =
3 =	16 =
4 =	17 =
5 =	18 =
6 =	19 =
7 =	20 =
8 =	21 =
9 =	22 =
10 =	23 =
11 =	24 =
12 =	25 =

Write a word problem to represent each linear equation. For the equation: 500 = 50x

- 3	7	14	- 5
4	- 9	3	9
- 4	25	- 8	- 16
- 7	8	- 23	12

Algebra Terms Concentration

Absolute Value	The distance an integer is from zero on a number line
Exponent	A number that indicates how many times another number is used as a factor
Integer	A number from the set {… -3, -2, -1, 0, 1, 2, 3 …}
Like Terms	Terms that differ only on their numerical coefficient
Unlike Terms	Terms that do not have the same variable factors
Monomial	A number, a variable or their product
Polynomial	A monomial or the sum or difference of two or more monomials
Equation	A statement that two quantities have the same value

Positive and Negative Integers: A Card Game

Objective: Students will practice addition and subtraction of positive and negative integers using an adaptation of the card game Twenty-Five.
Materials: Standard deck(s) of playing cards
Procedure:
Arrange students into groups of two or more. Have students deal out as many cards as possible from a deck of cards, so that each student has an equal number of cards. Put aside any extra cards.
Explain to students that every black card in their pile represents a positive number. Every red card represents a negative number. For example, a black seven is worth +7 (seven), and a red three is worth -3 (three). Face cards have the following values: aces have a value of 1, jacks have a value of 11, queens have a value of 12, and kings have a value of 13.
At the start of the game, have each player place his/her cards in a stack, face down. Then ask the player to the right of the dealer to turn up one card and say the number on the card. For example, if the player turns up a black eight, he or she says “8.”
Continue from one player to the next in a clockwise direction. The second player turns up a card, adds it to the first card, and says the sum of the two cards aloud. For example, if the card is a red 9, the player says: “8” + (-9) = (-1).”
The next player takes the top card from his/her pile, adds it to the first two cards, and says the sum. For example, if the card is a black 2, the player says: “(-1) + 2 = 1.”
The game continues until someone shows a card that, when added to the stack, results in a sum of exactly 25.
Extra Challenging Version
To add another dimension to the game, you might have students always use subtraction. Playing the game this way will reinforce the skill of subtracting negative integers.
For example, if player #1 plays a red 5 (-5) and player #2 plays a black 8 (+8), the sum is -13: (-5) – (+8) = -13.
If the next player plays a red 4, the sum is -9: (-13) – (-4) = -9. (Remember, subtracting a negative number from a negative number is equivalent to adding that number.)

From Arithmetic to Algebra

Complete each of the equations. Explain your answer.

1. (3^7 + 3^8) x (3^2 x 3^6) =	2. (e^7 + e^8) x (3^2 X 3^6) =
3. 8 x 1 + 6 =	4. 8i x i + 6i =
5. 4^7 x 4^4 + 4^4 =	6. J^7 x J^4 + J^4 =
7. 6^2 + 6^8 x 6^5 x 6^3 =	8. f^2 + f^8 x f^5 x f^3 =
9. 6 x (3 + 8) + (9 + 7) =	10. 6i x (3i + 8i) + (9i + 7i) =
11. 7 x 5 + 8 =	12. 7h x 5h + 8f =
13. 7 x (6 + 9) =	14. 7h x (6h + 9d) =
15. 8 x (9+5) =	16. 8g x (9g + 5g) =
17. 1 x (5 + 7) + (6 x 4) =	18. e x (5e + 7e) + (6e x 4e) =
19. 6^4 x 6^6 + 6^ 10 =	20. c^4 x c^5 + c^10 =

Algebra problems from edHelper.com. Retrieved from the World Wide Web on 03/08/06 at: http://www.edhelper.com/algebra.htm.

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