2 A Look At Expressions and Equations A manipulative, like algebra tiles, creates a concrete foundation for the abstract, symbolic representations students begin to wrestle with in middle school.Tell participants that they will use the algebra tiles on their tables as we work with expressions and equations.2
3 What do these tiles represent? 1 unitArea = 1 square unitTile BinUnknown length, x unitsArea = x square units1 unitx unitsNote to presenters: It would be good to mention that with an interactive white board works nicely with this type of exploration. It might also be good to tell the audience that no special program was used to create the representations of the tiles.We want to start by establishing the value of the algebra tiles. (~10 minutes) Many students will relate these tile to the base ten blocks and want to name them very concretely with a number value.We start with the yellow tile which does represent one unit.We declare that it is one unit tall and one unit wide, so it has an area of one square unit.Next, we move to the green tile. It is one unit tall. However, its width can vary.The tiles shown above are about 2.5 units long. The manipulatives our participants will have are about 5.1 units long.The width is unknown, so we name it with a variable, x. The green tile is one unit by x units, so it has an area of x square units.(The Multiplicative Identity can be mentioned here.)After this, we take a look at the blue tile. It is x units tall and x units wide.(Many students will still go back to the yellow tiles as a unit of measurement.)The blue tile has an area of x2 square units.Finally, discuss that the red tiles are used to denote negative quantities.x unitsArea = x2 square unitsThe red tiles denote negative quantities.
4 Modeling expressions x + 5 5 + x Tile Bin Our next idea is to focus us modeling algebraic expressions. (~20 minutes)Now that we have established the value of each tile, have participants model each expression using tiles.a) x + 5 (one green tile and 5 yellow tiles)b) 5 + x (5 yellow tiles and 1 green tile)These two expressions represent the same group of tiles… Commutative Property of Addition4
5 Modeling expressions x - 1 Tile Bin c) x - 1 Start with x and take away one. (Put up one green tile.)We don't have one to take away. We only have an x.This is another picture of x where we can take one away. (Still have one green tile. Add one yellow tile and one red tile.)We added a zero pair. (The red tile and the yellow tile combine to make the zero pair… Additive Inverse)After we take one away, or subtract one, this is what we are left with. (Remove the yellow tile representing one, leaving one green and one red tile.)This process with the tiles is helpful in building understanding that subtracting one is the same as adding negative one.5
6 Modeling expressions x + 2 2x Tile Bin Compare the expressions x + 2 and 2x (Have participants model each expression at their tables.)Should they look the same?x is an x with 2 ones added onto the end. (one green tile and two yellow tiles)2x means you have an x two times. (two green tiles)6
7 Modeling expressions x2 + 3x + 2 Tile Bin Now, we are ready to move to simplifying algebraic expressions. (~20 minutes)(combining like terms and applying the Distributive Property)a) Build x2 + 3x + 2(one blue tile, three green tiles, and two yellow tiles)This expression has three terms and each term looks quite different.7
8 Simplifying expressions x2 + x - 2x2 + 2x - 1Tile BinBuild x2 + x - 2x2 + 2x – 1(From left to right, pull in one blue tile, one green, two red large square tiles, two green, and one small red square tile.)Some of these terms do look “like.”Let’s rearrange, using the Commutative Property of Addition.(Rearrange so like tiles are together.)We see one zero pair with our x2 and -x2 (Additive Inverse). We see three x tiles.It is easy to see that the expression simplifies to x2 + 3x - 1.zero pairSimplified expression-x2 + 3x - 18
9 Simplifying expressions Tile BinWhat would the model of this expression look like?Invite a volunteer to come up and draw what they have modeled. Ask if anyone has a different model to share (and compare).It is likely that most participants will make two sets of tiles that each contain 2 greens and 3 yellows. It is possible that someone may think about the area model. This should lead nicely into the next slide.This expression is the first one we have seen with parentheses. One would use the Distributive Property to simplify the expression.Simplified expression4x + 69
10 Two methods of illustrating the Distributive Property: Example: 2(2x + 3)One way to model the Distributive Property is to copy and paste.The copy and paste method reinforces the idea that you will be getting the whole quantity within the parentheses multiple times.Another method to model the Distributive Property is to use an area model for multiplication and build an array.The area model connects to the earlier work we did with multi-digit multiplication, and it carries over into Algebra I when multiplying and factoring polynomials is introduced.Both of these methods have value for the students.
11 ** What does this mean for Course 2 students who go to Algebra 1? Solving Equations How does this concept progress as we move through middle school?6th grade:The student will solve one-step linear equations in one variable involving wholenumber coefficients and positive rational solutions.7th grade:7.14 The student willsolve one- and two-step linear equations in one variable; andsolve practical problems requiring the solution of one- and two-step linear equations.Finally, we can work with equations.6th grade one-step equations remember that these students have not learned about integer operations.7th grade up to two-step equations we can now apply operations with integers.8th grade up to four-step equations simplifying expressions within an equation and working with variables on both sides of the equation make their debut.8th grade:The student willa) solve multistep linear equations in one variable on one and two sides of theequation;b) solve two-step linear inequalities and graph the results on a number line; andc) identify properties of operations used to solve an equation.** What does this mean for Course 2 students who go to Algebra 1?
12 Solving Equations Tile Bin We should start with a discussion of the word equation.It is a statement that two expressions are equal.This idea is mirrored in the equation mat.Each rectangle should show the model of an expression. The two expressions are equivalent.
13 Solving Equations x + 3 = 5 Tile Bin The student will solve one-step linear equations in one variable involving wholenumber coefficients and positive rational solutions.x + 3 = 5Tile BinLet’s start with some examples from 6th grade SOL.x + 3 = 5What would that look like on the equation mat?(Pull in one green and three yellow on left and five yellow on the right.)We want to isolate the x term.How can we achieve that goal?Take three yellows off each sideAdd two negatives to each side (Note this is not a 6th grade response.)
14 Solving Equations x + 3 = 5 x + 3 = 5 ̵ 3 ̵ 3 x = 2 x + 3 = 5 ̵ 3 ̵ 3 The student will solve one-step linear equations in one variable involving wholenumber coefficients and positive rational solutions.Pictorial Representation:Symbolic Representation:Condensed Symbolic Representation:x + 3 = 5x + 3 = 5̵ 3 ̵ 3x = 2x + 3 = 5̵ 3 ̵ 3Here we start with a pictorial representation of what we just built with the tiles.How can we solve for x ?You can take three away from each side. You can see a pictorial representation as well as the symbolic representation of that step.Finally, we are left with the green tile that represents x alone on the left and two yellow tiles on the right. The solution is x = 2.Note: the “condensed symbolic representation” column allows you to rework the problem in a more stand-alone symbolic version.x = 2
15 Solving Equations 2x = 8 Tile Bin The student will solve one-step linear equations in one variable involving wholenumber coefficients and positive rational solutions.2x = 8Tile BinLet’s try another 6th grade example, 2x = 8. Model this on your equation mats. What does it look like?We have two green tiles on the left side, and eight yellow tiles on the right.If we want to split the green tiles to determine what each one is worth individually, how would we separate the yellow tiles to distribute them fairly into 2 equal groups?(Pull the yellow tiles one at a time into two separate groups, distributing one at a time to show the fair sharing idea.Discuss that the picture still shows 2x = 8, it has just been reorganized.)How many x’s do we want? Can you tell me what 1x is worth? Delete all but one section by editing.
16 Solving Equations 3 = x - 1 Tile Bin 7.14 The student will solve one- and two-step linear equations in one variable;and solve practical problems requiring the solution of one- and two-step linear equations.3 = x - 1Tile Bin3 = x - 1 (This is a 6th grade problem that would be challenging with the tiles. It wouldrequire a discussion of zero pairs. Although integer operations and the AdditiveInverse Property are 7th grade curriculum, we could approach problems like thisto help create a bridge to the 7th grade content.)To model with tiles:Put three yellow tiles on the left and one green tile on the right.We don’t have any unit tiles to remove (or subtract), but we could add a zero pair (one yellow and one red unit tile that combine to equal zero). Then, we could remove the yellow unit tile (take away one) and this would leave x and negative one, or x plus -1 on the right.This example would cause sixth grade students to consider what might happen and would be an early example for seventh graders as they formulate their own rules for operating with integers.