2Modeling AlgebraMiddle school students are primarily concrete thinkers. Developmentally, most are not ready to think abstractly.Making the transition from concrete arithmetic to abstract reasoning and symbolism in algebra presents a unique challenge to teachers.One way to bridge this gap is through using manipulatives to make algebraic symbols and procedures concrete.One such manipulative is Algebra Tiles.Algebra tiles are made by various companies with many different names, each with strengths and weaknesses, but all are used in basically the same manner.
3Algebra TilesSupport cooperative learning, improve discourse in classroom by giving students objects to think with and talk about.When I listen, I hear.When I see, I remember.But when I do, I understand.
4Algebra TilesAlgebra tiles can be used to model operations involving integers.Let the small blue square represent +1 and the small red square (the flip-side) represent -1.The blue and red squares are additive inverses of each other.
5Zero PairsCalled zero pairs because they are additive inverses of each other.When put together, they cancel each other out to model zero.
6Building ZeroHow many different ways can you model ZERO using algebra tiles? Build zero in the space below.
7Addition of Integers Addition can be viewed as “combining”. Combining involves the forming and removing of all zero pairs.For each of the given examples, use algebra tiles to model the addition.Draw pictorial diagrams which show the modeling.
9Addition of Integers (+3) + (-1) = (+4) + (-4) = After students have seen many examples of addition, have them formulate rules.
10Subtraction of Integers Subtraction can be interpreted as “take-away.”ORSubtraction can also be thought of as “adding the opposite.”For each of the given examples, use algebra tiles to model the subtraction.Draw pictorial diagrams which show the modeling process.
11Subtraction of Integers (+5) – (+2) =(-4) – (-3) =
12Subtracting Integers (+3) – (-5) +3 “remove” -5, but you do not have any negatives to remove.Add zeros!
13Subtracting Integers (+3) – (-5) Try: (-4) – (+1) Zero(+3) – (-5)Now you can remove -5Try:(-4) – (+1)
14Subtracting Integers (+3) – (-3) After students have seen many examples, have them formulate rules for integer subtraction.
15Modeling Algebra Naming Tiles To begin using algebra tiles, students need to understand the name of each tile. This is done most effectively if you discuss why the tiles are named what they are, and have students figure out the name of most of the tiles based on the name of the x tile.Next step would be representing expressions, then operations with polynomials. You would need to start with naming all the tiles.x xName each of the remaining tiles based on the shape of each tile, and what you know about the x and –x tiles. Hint: Consider area.
16Modeling Algebra Naming Tiles: Answers x -x x xNotice the connection between naming the tiles with the area of the tiles. From the x tile, you can see the length is x and the width is 1. This leads to the x2 tiles, which are x by x for an area of x2. The single unit tiles are 1 by 1. Each of the tiles can be lined up to see the length of the sides are either x or 1.
17Modeling Algebra Naming Tiles: Answers This is a simple activity that won’t take long in class, but helps students make a connection between algebra and geometry (specifically with a concept with which they are already comfortable), as well as introducing them to the manipulative before beginning formal algebraic procedures.Research shows that students who work with sense-making procedures, such as the cover-up method and manipulatives, before beginning formal algebraic procedures and operations make the transition to algebra more easily and with more success.