ALGEBRA TILES Dawne Spangler In this presentation, the words “participant” and “facilitator” may also be understood as “teacher” and “student”, although this PowerPoint is not appropriate or intended for use with students. The facilitator, however will be modeling parallel practice. It will be helpful to have an overhead projector and the overhead tiles from the kit. When a problem is posed, participants will solve the problem and then the facilitator will ask for a volunteer to show a solution on the overhead. The solution on the slide is given last, as a validation of the answer rather than the source of the answer.

ALGEBRA TILES Demonstrate Reaffirm Reassure Using Algebra Tiles makes algebraic logic simpler and easier to comprehend. The transition into the abstractions of algebra is difficult for some students. A concrete model for abstract concepts assists students in connecting what they are learning to what they already know. It also assists them in in visualizing abstract outcomes. This presentation is designed to familiarize teachers with the structure and potential of the tiles. It is important to note that algebra tiles do not provide a means to solve problems, but a model. This allows students to shift into abstraction comfortably without relying on a system of rules, which become confused and distorted without internalization of the underlying structure of the problem.

OBJECTIVES Section 1 Use rectangular arrays to model numerical values Define value in terms of measurement First, we will apply the rectangular model to numbers. As we demonstrate how numbers can be “seen” as rectangles, we can then associate a unique value to each rectangle, because a rectangle has only one area. (The inverse is not true, as a number might be associated with more than one rectangle.) Having done so, we can then use the rectangles to model negative numbers and variables. The model is intended to strengthen the students intuitive ability to work with negative numbers and variables. Rules will be developed AFTER students know and understand the behavior of numbers other than whole numbers

MODELING A WHOLE NUMBER The number 9 - 7 8 9 10 11 As we look for a way in which we can show, rather than tell, we need to find a way to model numbers. Students have seen counters, which are useful in some applications. They have also worked with number lines, which express a number in terms of its position in a sequence. But these models break down when they try to address abstractions.  

RECTANGULAR ARRAYS The area model does not rely upon counting, but it is countable. Like counters and number lines, area models numbers using a physical reference. It is a move towards abstraction because it can also be measured rather than counted.. The shape of the rectangle may vary but the attribute of size is consistent and predictable. This concept will also transfer to other applications.

THE VALUE OF THE PIECE IS DETERMINED BY ITS AREA. IF THIS IS “1”UNIT, THEN THIS IS 11. THE AREA IS 1 SQUARE UNIT, SO THE VALUE OF THE PIECE IS “1”. 1 1 In early discussions, we will use the area model to define the number 1, but we will not require that the pieces be formed into rectangles to express different values, such as two or 12. At this point, the countability is useful. If tiles are not already in participant hands, they should be now. The value of any tile is its area, length times width. A unit tile is 1 unit long and one unit wide. 11=1. Two can be expressed as 2 unit tiles, or a rectangle 2 units long and one unit wide. SO, WHAT WOULD TWO LOOK LIKE?

OBJECTIVES Section 2 Introduce negative models Develop an INATE understanding of the behavior of negative numbers in addition and subtraction problems. Again, the objective is to make students familiar and comfortable with the behavior of negative numbers. They have heard the rules. They have used the rules. They do not, however, have the deep understanding of what is making those rules work. The result is that they believe that -2 + -1 = +3, because they KNOW that two negatives make a positive. Use of algebra tiles builds the understanding that puts rules into perspective.

HOW DO WE MODEL A NEGATIVE NUMBER? WE CAN USE DIFFERENT COLORS… OR WE CAN MARK PIECES WITH +/- Elicit answers from participants prior to showing the answers.. Neither solution is concrete – both color and sign are arbitrary and abstract. In this set of tiles, red is negative. The color for positive numbers varies for 1, x and x2. Students will adapt quickly to either concept and it allows them a visual and manipulative way to interpret problems. This can be a vital step in assisting a struggling student from arithmetic understandings to algebraic understandings. The concepts in signed numbers are counterintuitive, and freshmen have seen them before. They have formed abstract strategies and rules which are largely incorrect. The visual, manipulative model allows them to “unlearn” incorrect notions by replacing unsupportable abstract misconceptions with new, correct and demonstrable thinking. - +

COMBINE THE TILES Find the value of -2 + -1 At this stage, the tiles are only counters, and the only “algebra” is the use of red for negative numbers. This concept is not among those that ordinarily confuse students but it is a necessary first step in building an understanding of the model. We do not want to begin building rules, but it is a good time to ask what happens if you have red tiles and add more red tiles. Will the answer turn yellow?

SHOW ME ZERO -1 +1 = 0 Participants should on fact hold up the tiles to show zero. The facilitator will acknowledge right answers and encourage rethinking for those who do not have the correct answer. (Among teachers the opportunity may not arise, create it. Choose a fictional “student” to encourage.) Emphasize that responses should be positive and encouraging. Participants will check answers for one another as well. Zero pairs are an important element in the success of this model.

What happens to the zero pairs? FIND -4 + 2 = -2 The presence of zero pairs does not affect the answer. What happens to the zero pairs?

ADDITION OF INTEGERS MODEL THE FOLLOWING EXPRESSIONS, USING TILES 1 + (-3) -5 + 2 -3 + (-4) = -2 = -3 All of these examples are addition problems, which students may find simple. Do not encourage rules at this point. Answers on the slides are only validation of student answers. They are shown only after participants have worked the problem, found answers, shared and corrected their answers. = -7

SUBTRACTION OF INTEGERS 3 – 1 -4 – (-2) = 2 Both examples are simple problems where there is no need to resort to “new” information. In each case, the arithmetic definition of subtraction as “take away” is demonstrated. = -2

HOW WOULD YOU MODEL 1 - 3? -2 METHOD 1 There are still not enough. Add another . There are not enough positive tiles to take away 3. Add zero. There are not enough positive tiles on the board to “take away” 3. When zeros are pulled in, students will not find the process comfortable. Reassure them that a number + zero has not changed its value. Adding zeros allows students to used the take away model for subtraction. The presence of zero pairs does not affect the outcome of the problem, so pulling in zeros which will help with subtraction does not change the problem. Now it is possible to subtract.

ADDING ZEROS Determine the value of each set of tiles. Students know that a number + 0 is unchanged, but it is a concept that needs to be reinforced visually, in order to give them confidence using the tiles. This method does not translate into the eventual rule they will use, but it validates the process they will use in method 2. Find the value after each click. The first value is 2, the second is 2 + 0 = 2, the third, 2 etc. Reassure students that adding zeros does not affect the value of the set. It does, however, allow them to use the take away model. Because the zeros do not affect the answer, they can fade out. Use the take away model to find 2 – (-3). 2 – (-3) = 5

EXAMPLES -2-(-3) =1 3-6 = -3 In the problem -2-(-3), there are not enough negatives to use the take away model…so pull in some zeros. Student usually ask how many. It doesn’t matter. If you don’t have enough, pull in more. If you get too many, the final answer will remain the same. In the problem 3 – 6, we intentionally do not provide enough zeros. Get more.

HOW WOULD YOU MODEL 1 - 3? METHOD 2, “ADD THE OPPOSITE” 1 +(-3) -2 In this problem, it is not possible to “take away” without some manipulation. Method two relies heavily on rules…change both signs and so on. It is not natural for students to see how subtraction is really addition. Reinforce the validity of the process using the take away model. Will it always work?

EXAMPLES 5 – 2 5 + (-2) 3 3 -4 – (-3) -4 + (+3) -1 -1 Both examples lend themselves to a take away process so students can compare. -1 -1

TRY THESE 3 – (-5) 3 + (+5) = 8 -2 – 4 -2 + (-4) = -6 1 – (4) 1 + (-4) Have students work these independently, and model answers on the overhead. These answers are for validation only. 1 – (4) 1 + (-4) = -3

OBJECTIVES Section 3 Introduce variables as rectangles Create algebraic expressions Perform addition and subtraction on algebraic expressions The rectangle model provides a concrete example of abstractions, which allows students to understand the operations which follow. By now, students have accepted and become comfortable with the use of tiles. The only new concept is the appearance of the variable x, so they are not confused or challenged by the move into algebra.

ALGEBRAIC EXPRESSIONS THE VALUE OF EACH PIECE IS DETERMINED BY ITS AREA. A NEW PIECE IS CREATED BY ESTABLISHING A NEW DIMENSION, “x”. FOR PRACTICAL REASONS, “x” IS NOT A MULTIPLE OF THE DIMENSION REPRESENTING ONE. THE VALUE OF THIS PIECE IS x, BECAUSE 1  x = x. 1 For the variables in this set of algebra tiles, the positive pieces are green, and the negatives are red. Students want to “measure” the piece, and determine the value of x in terms of the known unit, 1. Continue to insist that we just don’t now what x is. As they work problems, they will become more comfortable with the changing values of x. x

EXAMPLES Use tiles to express the following: x + 2 3x 2x -1 Students need practice modeling algebraic expressions. Have them create the expressions as you walk around the room. Students should show and explain their answers. In an algebraic expression, the operation of subtraction is modeled by using negatives. Students understand the concept, but reinforce often.

COMBINING EXPRESSIONS 3x – 2x x + 4 + x – 3 2x + 3 – x + 1 = x = 2x + 1 Using algebra tiles allows students to strengthen their understanding of the processes involved in combining algebraic expressions. Both subtraction models (take away and adding the opposite) are used on this slide, as well as a zero pair involving variables. It would be good to model adding zeros on the overhead, to demonstrate that the outcome is not changed. = x + 4

MORE ALGEBRAIC EXPRESSIONS (5x - 6) -(3x - 2) 3x + 2 The distributive property is well modeled with the tiles. It allows students to see what the property really means, and how easily it is applied. In the second problem, the disrtibution of the subtraction sign is simply taking away the two negatives. 2x - 4 THE TILES MAKE THE CONCEPTS SIMPLE.

OBJECTIVES Section 4 Solve one variable equations Solving equations using tiles is simple, and very intuitive. Students master the skill quickly, and are more confident and accurate with the algebra in the abstract.

= SOLVING EQUATIONS MODEL 2x + 1 = 5 x = 2 TO SOLVE, SUBTRACT 1 TILE FROM EACH SIDE. NEW EQUATION 2x = 4 ISOLATE x BY DIVIDING INTO TWO GROUPS = Walk participants through the process. Each algebraic step is performed concretely. x = 2

SOLVING EQUATIONS 2x + 3 = 7 2x + 3 – 3 = 7 – 3 2x = 4 x = 2 2x = 4 Subtract 3 from each side of the equation x = 2 2x = 4 x = 2 Divide by 2 on each side of the equation Again, it is important to remember that Algebra Tiles are a model and not a method. Using the tiles to routinely solve equations is silly and impractical. As the student works through the problems here, they should always be reminded of the algebra being modeled. What we are providing is a way for students to see and know what is happening, so that when we ask “what is the next step”, they have an innate understanding of the process. This is not a set of rules. When tiles are modeled and used, students become more confident and their understandings do not fade. The operations and the result are exactly the same.

SOLVE x – 4 = 5 In each example, allow participants to attempt on their own. What might be a concern in this problem? The negatives cannot be “subtracted” using the take away model. Add the opposite to form zero pairs. The solutions should be shown on the overhead by volunteers. These answers are merely confirmation. x = 9

Solve 3x + 2 = 11 3x = 9 x = 3

Solve 2(x - 3) = 10 2x - 6 = 10 2x = 16 x = 8