Presentation on theme: "Alge-Tiles Making the Connection between the Concrete ↔ Symbolic"— Presentation transcript:
1 Alge-Tiles Making the Connection between the Concrete ↔ Symbolic (Alge-tiles) ↔ (Algebraic)
2 What are Alge-Tiles?Alge-Tiles are rectangular and square shapes (tiles) used to represent integers and polynomials.Examples: 1→1x →1x2 →
3 Objectives for this lesson Using Alge-Tiles for the following:- Combining like terms- Multiplying polynomials- Factoring- Solving equationsAllow students to work in small groups when doing this lesson.
4 Construction of Alge-Tiles 1 (let the side = one unit)For one unit tile:(it is a square tile)1Area = (1)(1) = 1x(unknown length therefore let it = x)For a 1x tile(it is a rectangular tile)1Side of unit tile = side of x tileArea = (1)(x) = 1xxSide of x2 tile = side of x tileFor x2 tile:(It is a square tile)Area = (x)(x) = x2xOther side of x2 tile = side of x tile
5 Part I: Combining Like Terms Prerequisites: prior to this lesson students would have been taught the Zero PropertyOutcomes: Grade 7 - B11, B12, B13 Grade 8 – B14, B Grade 9 – B8 Grade 10 – B1, B3Use the Alge Tiles to represent the following:3x32x2
6 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 →
7 Part I: Combining Like Terms Represent “2x” with tilesRepresent “3” with tilesCan 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)
8 Part I: Combining Like Terms Combine like terms (use the tiles):+= 4x2x + 2x →= 1x+3 (ctt)1 +1x +2 →++-2x + 3x +1→= 1x +1(ctt)++Using the zero property
9 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.
10 Part II: Multiplying Polynomials Prerequisites: Students were taught the distributive property and finding the area of a rectangle.Area(rectangle) = length x widthWhen 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
11 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 tilesThe width = 2 tilesThe tiles form a rectangle, use other tiles to fill in the rectangleOnce 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
12 Part II: Multiplying Polynomials Try: (2x)(3x)→Side: 3xSide: 2xTherefore: (2x)(3x) = 6x2Remove the sides
15 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.
16 Part III: FactoringOutcomes: Grade 9 – B9, B10, Grade 10 – B1, B3, C16Take 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 rectangleConstruct a rectangle using 2 ‘x’ tiles and 4 unit one tiles. This can be tricky until you get the hang of it.
17 Part III: FactoringNow 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
18 Part III: Factoring Try factoring 3x + 6 with your tiles. First make a rectangleMake the sidesRemove the rectangle3The sides are the factorsFactors → (1x + 2)(3)3x + 6 = (3)(1x + 2)
19 Part III: Factoring x2 + 5x + 6 = (1x + 3) (1x + 2) 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)32Then add the x tiles where needed to complete the rectangleWhen the rectangle is finished examine it to see if the tiles combine to give you the original expression → x2 + 5x + 6x2 + 5x + 6 = (1x + 3) (1x + 2)Next make the sides for the rectangleRemove the rectangle and you have the factors. (1x + 3) (1x + 2)
20 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.1When the rectangle is finished examine it to see if the tiles combine to give you the original expression → x2 + 5x + 66When 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.
21 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?”
22 Part III: Factoring (negatives) Try factoring: x2 - 1x – 6Start 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 - 31x + 2-3Next fill in the x tiles to make the rectangle.2Now the rectangle is complete check to see if the tiles combine to equalx2 - 1x – 6.Therefore x2 - 1x – 6 = (x – 3) (x + 2)Fill in the sides and remove the rectangle to give you the factors.
23 Part IV: Solving for XOutcomes: Grade 7 check, Grade 8 - C6, Grade 9 – C6, Grade 10-C 27Solve 2x + 1 = 5 using alge-tilesSet up 2x + 1= 5 using tiles=1x = 2Using the zero property to remove the 1 tile you add a -1 tile to both sidesOn the left side -1 tile and +1 tile give us zero and you are left with 2 ‘x’ tilesOn the right side adding -1 tile gives you +4 tilesNow 2 ‘x’ tiles = 4 unit tiles, (how many groups of 2 are in 4)Therefore 1 ‘x’ tile = 2 unit tiles
24 Part IV: Solving for X Solve 3x + 1 = 7 = 1x = 2 Add a -1 tile to both sidesZero Property takes placeWhat’s left? 3 ‘x’ tiles = 6 unit tiles (how many groups of 3 are in 6)Therefore 1x tile = 2 unit tiles
25 Part IV: Solving for X Solve for x: 2x – 1 = 1x + 3 Now add +1 tile to both sides… zero propertyYou are left with 2x = 1x + 4Add -1x tile to each side… zero propertyLeaving 1x = 4
26 Alge-Tile ConclusionAssessment: 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 TermsPart II: Multiplying PolynomialsPart III: FactoringPart IV: Solving for an unknown(P.S. the answers are at the end)
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