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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 For a 1x tile (it is a rectangular tile) (unknown length therefore let it = x) 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) Other side of x2 tile = side of x tile Area = (x)(x) = x2 x 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): + 2x + 2x → 1 +1x +2 → + -2x + 3x +1→ Using the zero property = 4x = 1x+3 (ctt) + + + = 1x +1(ctt) 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 Remove the sides Therefore: (2x)(3x) = 6x2 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: Simplify to get: x2 + 2x – 1x – 2 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 i. 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(1x with+ 2) the unit tiles. Then add the x tiles where needed to complete the rectangle 3 2 When the rectangle is finished examine to see if+the tiles combine to give you 2it + x 5x 6 = (1x + 3) (1x + 2 the original expression → x + 5x + 6 Next make the sides for the rectangle Remove the rectangle and you have the factors. (1x + 3) (1x + 2) 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. 1x - 3 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 1x + 2 middle term, -1. -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)