Alge-Tiles Making the Connection between the Concrete ↔ Symbolic (Alge-tiles) ↔ (Algebraic)
What are Alge-Tiles? • Alge-Tiles are rectangular and square shapes (tiles) used to represent integers and polynomials. Examples: 1→ 1x → 1x2 →
Objectives for this lesson • Using Alge-Tiles for the following: • - Combining like terms • - Multiplying polynomials • - Factoring • - Solving equations Allow students to work in small groups when doing this lesson.
Construction of Alge-Tiles 1 (let the side = one unit) For one unit tile: (it is a square tile) 1 Area = (1)(1) = 1 x (unknown length therefore let it = x) For a 1x tile (it is a rectangular tile) 1 Side of unit tile = side of x tile Area = (1)(x) = 1x x Side of x2 tile = side of x tile For x2 tile: (It is a square tile) Area = (x)(x) = x2 x Other side of x2 tile = side of x tile
Part I: Combining Like Terms • Prerequisites: prior to this lesson students would have been taught the Zero Property • Outcomes: Grade 7 - B11, B12, B13 Grade 8 – B14, B15 Grade 9 – B8 Grade 10 – B1, B3 • Use the Alge Tiles to represent the following: • 3x • 3 • 2x2
Part I: Combining Like Terms • For negative numbers use the other side of each tile (the white side) • Use the Alge Tiles to represent the following: -2x → -4 → -3x - 4 →
Part I: Combining Like Terms • Represent “2x” with tiles • Represent “3” with tiles • Can 2x tiles be combined with the tiles for 3 to make one of our three shapes? Why or why not? • Therefore: simplify 2x + 3 = • 2x + 3 can’t be simplified any further (can’t touch this)
Part I: Combining Like Terms Combine like terms (use the tiles): + = 4x 2x + 2x → = 1x+3 (ctt) 1 +1x +2 → + + -2x + 3x +1→ = 1x +1(ctt) + + Using the zero property
Part I: Combining Like Terms • After mastering several questions where students were combing terms you could then pose the question to the class working in groups: “Is there a pattern or some kind of rule you can come up with that you can use in all situations when combining polynomials.” • In conclusion, when combining like terms you can only combine terms that have the same tile shape (concrete) → Algebraic: Can combine like terms if they have the same variable and exponent.
Part II: Multiplying Polynomials • Prerequisites: Students were taught the distributive property and finding the area of a rectangle. • Area(rectangle) = length x width • When multiplying polynomials the terms in each bracket represents the width or length of a rectangle. • Find the area of a rectangle with sides 2 and 3. Two can be the width and 3 would be the length. • The area of the rectangle would = (2)•(3) = 6
Part II: Multiplying Polynomials • We will use tiles to find the answer. The same premise will be used as finding the area of a rectangle. Make the length = 3 tiles The width = 2 tiles The tiles form a rectangle, use other tiles to fill in the rectangle Once the rectangle is filled in remove the sides and what is left is your answer in this case it is 6 or 6 unit tiles
Part II: Multiplying Polynomials • Try: (2x)(3x)→ Side: 3x Side: 2x Therefore: (2x)(3x) =6x2 Remove the sides
Part II: Multiplying Polynomials • Try (1x + 2)(3) Side: 1x + 2) Side: 3 Therefore: (1x + 2)(3) = 3x + 6 (ctt) Make rectangle, fill rectangle Remove sides
Part II: Multiplying Polynomials • Try (1x +2)(1x -1) Side: 1x - 1 Side: 1x + 2 Tiles remaining: x2 + 2x – 1x – 2 Simplify to get: x2 + 1x – 2 (ctt)
Part II: Multiplying Polynomials • Pattern: After mastering several questions where students were combing terms you could then pose the question to the class working in groups: “Is there a pattern or some kind of rule you can come up with that you can use in all situations when multiplying polynomials.” This can lead to a larger discussion where students can put forth their ideas.
Part III: Factoring • Outcomes: Grade 9 – B9, B10, Grade 10 – B1, B3, C16 • Take an expression like 2x + 4 and use the rectangle to factor. • You will go in reverse when being compared to multiplying polynomials. (make the rectangle to help find the sides) • The factors will be the sides of the rectangle • Construct a rectangle using 2 ‘x’ tiles and 4 unit one tiles. This can be tricky until you get the hang of it.
Part III: Factoring • Now make the sides; width and length of the rectangle using the alge-tiles. Side 1 : (1x + 2) Side 2 : (2) 2x + 4 = (2)(1x +2) Remove the rectangle and what is left are the factors of 2x +4
Part III: Factoring • Try factoring 3x + 6 with your tiles. 1x + 2 First make a rectangle Make the sides Remove the rectangle 3 The sides are the factors Factors → (1x + 2)(3) 3x + 6 = (3)(1x + 2)
Part III: Factoring • Try factoring x2 + 5x + 6 (make rectangle) (1x + 3) **Hint: when the expression has x2, start with the x2 tile. Next, place the 6 unit tiles at the bottom right hand corner of the x2 tile. You will make a small rectangle with the unit tiles. (1x + 2) 3 2 Then add the x tiles where needed to complete the rectangle When the rectangle is finished examine it to see if the tiles combine to give you the original expression → x2 + 5x + 6 x2 + 5x + 6 = (1x + 3) (1x + 2) Next make the sides for the rectangle Remove the rectangle and you have the factors. (1x + 3) (1x + 2)
Part III: Factoring • What if someone tried the following: Factor: x2 + 5x + 6 (make rectangle) Start with the x2 tile, now make a rectangle with the 6 unit tiles. Now complete the rectangle using the x tiles. 1 When the rectangle is finished examine it to see if the tiles combine to give you the original expression → x2 + 5x + 6 6 When the tiles are combined, the result is x2 + 7x + 6, where is the mistake? The unit tiles must be arranged in a rectangle so when the x tiles are used to complete the rectangle they will combine to equal the middle term, in this case 5x.
Factoring • Have students try to factor more trinomials (refer to Alge-tile binder – Factoring section: F – 3b for additional questions) • After mastering several questions where students were factoring trinomials you could then pose the question to the class : • “Is there a pattern or some kind of rule you can come up with that you can use when factoring trinomials?”
Part III: Factoring (negatives) Try factoring: x2 - 1x – 6 Start with x2 tile, then fill in the unit tiles in this case -6 which is 6 white unit tiles. Remember to make a rectangle at the bottom corner of the x2 tiles where the sides have to add to equal the coefficient of the middle term, -1. 1x - 3 1x + 2 -3 Next fill in the x tiles to make the rectangle. 2 Now the rectangle is complete check to see if the tiles combine to equal x2 - 1x – 6. Therefore x2 - 1x – 6 = (x – 3) (x + 2) Fill in the sides and remove the rectangle to give you the factors.
Part IV: Solving for X • Outcomes: Grade 7 check, Grade 8 - C6, Grade 9 – C6, Grade 10-C 27 • Solve 2x + 1 = 5 using alge-tiles • Set up 2x + 1= 5 using tiles = 1x = 2 Using the zero property to remove the 1 tile you add a -1 tile to both sides On the left side -1 tile and +1 tile give us zero and you are left with 2 ‘x’ tiles On the right side adding -1 tile gives you +4 tiles Now 2 ‘x’ tiles = 4 unit tiles, (how many groups of 2 are in 4) Therefore 1 ‘x’ tile = 2 unit tiles
Part IV: Solving for X • Solve 3x + 1 = 7 = 1x = 2 Add a -1 tile to both sides Zero Property takes place What’s left? 3 ‘x’ tiles = 6 unit tiles (how many groups of 3 are in 6) Therefore 1x tile = 2 unit tiles
Part IV: Solving for X • Solve for x: 2x – 1 = 1x + 3 = 1x = 4 Now add +1 tile to both sides… zero property You are left with 2x = 1x + 4 Add -1x tile to each side… zero property Leaving 1x = 4
Alge-Tile Conclusion • Assessment: While students are working on question sheet handout, go around to each group and ask students to do some questions for you to demonstrate what they have learned. • For practice refer to handout of questions for all four sections: • Part I: Combining Like Terms • Part II: Multiplying Polynomials • Part III: Factoring • Part IV: Solving for an unknown • (P.S. the answers are at the end)