1. U SING A LGE -T ILES IN M ATHEMATICS Adapted from: SOURCE: maths.slss.ie/resources/Algebra Tiles Full Show.ppt Presented by : Kenneth Baisden John Roopchan

2. Alge-Tiles For all Alge-Tile work it is essential to remember that RED means minus and Any other colour means plus. E.g. +1 -1 = 0

3. What are AlgeTiles? Coloured Tiles that can be (are) used as resources for developing students’ understanding of Algebra

4. 1 Defining the Variables x2x2 -x 2 x -x N.B.: The width of the x-tile is assumed to be 1 which for our purposes do not visually connect equitably to the variable x

5. Example Represent the following trinomials using alge-tiles: 1. 2x 2 +3x+5 2. x 2 -2x-3

6. Alge-Tile Uses Algebra tiles can be used for (among other things):  Section 1. Identifying ‘like’ and ‘unlike’ terms  Section 2. Adding and Subtracting Integers  Section 3. Simplifying Expressions  Section 4. Multiplying in algebra  Section 5. Factorising trinomials  Section 6. Doing linear equations

7. Section 1. Like Terms Example 1. 4x+5 Can any of these be added ? Explain your answer Example 2. 4x+5x Can any of these be added ? Explain your answer

8. Section 2. Adding and Subtracting Integers Example 1. 4-7 Example 2. –3-6

9. Section 3. Adding and Subtracting Trinomials Example 1. 2x 2 +3x+5 + x 2 -5x-1

10. Section 3. Adding and Subtracting Trinomials Example 1. 2x 2 +3x+5 + x 2 -5x-1 Answer 3x 2 -2x+4

11. Section 3. Adding and Subtracting Trinomials Example 2. 2x 2 +3x+2 - ( x 2 -2x+1 ) - A CONCRETE IDEA FOR CHANGING SIGNS.

12. Section 3. Adding and Subtracting Trinomials Example 2. 2x 2 +3x+2 - ( x 2 -2x+1 ) A CONCRETE IDEA FOR CHANGING SIGNS.

13. Section 3. Adding and Subtracting Trinomials Example 2. 2x 2 +3x+2 - ( x 2 -2x+1 ) A CONCRETE IDEA FOR CHANGING SIGNS. Answer x 2 +5x+1

14. Practice Simplify the following: 1)6-7 2)3-2-4-1 3)5x 2 +2x 4)2x 2 +4x+2x 2 -x 5)3x 2 -2x+4+x 2 -x-2 6)x 2 -3x-2-x 2 -2x+4 7)2x 2 -2x-1-3x 2 -2x-2 8)x 2 +2x+1- 3x 2 -x 9)x 2 -x+3-2x 2 +2x+x 2 -2x-5 Simplify the following: 10)2-8-1 11)-5-1-4+1 12)x 2 +2 13)x 2 +5x+x 2 -2x 14)2x 2 -x+1 - (2x 2 -2x-5) 15)x 2 - 2x 2 -2x+4 - (x 2 +2x+3) 16)3x 2 -4x+2 - (x 2 +2) 17)x 2 +x-2 - 2(x 2 +2x-3) 18)-4x-3 - (2x 2 -2x-4)

15. Multiplying & Factorising General Aim Whether multiplying or factorising, the general aim is to generate a rectangle and have no pieces left over. Also the small squares always go in the bottom right hand corner

16. Section 4. Multiplying in algebra Example 2. Multiply (x-1)(x-3) Answer: x 2 -4x+3

17. Practice Multiply the following: 1)x(x+3) 2)2(x-5) 3)3x(x-1) 4)(x+4)(x+3) 5)(x-1)(x+2) 6)(x-4)(x-2) 7)(3x-1)(x-3) 8)(x-1)(x-1) 9)(2x+1) 2 10) (x-2) 2

18. Factors and Area - a geometrical approach Review Multiplication Again Section 5. Factorising Quadratic Trinomials

19. Show (x+1)(x+3) by arranging the tiles in a rectangle. x x 3 1 + + Rearrange the tiles to show the expansion: x 2 + 4x + 3 How it works Now Arrange them into a Rectangle Remember the little guys go in the bottom right corner

20. x 2 + 6x + 8 To factorise this expression form a rectangle with the pieces. x +4 x + 2 ( x + 4 )( x + 2 )The factors are Factorise x 2 + 6x + 8

21. x+3 x - 1 NOTE: REDS ARE NEGATIVE NOW COMPLETE THE RECTANGLE WITH NEGATIVE SQUARES Show (x+3)(x-1) by arranging the tiles in a rectangle. Rearrange the tiles to show the expansion: x2x2 + 3x-1x - 3 = x 2 + 2x - 3 (x+3)(x-1)

22. Factorise x 2 - 4x + 3 x2x2 - 4x + 3 x - 3 x - 1 The factors are( x - 3 )( x - 1 ) Factorise x 2 -4x+3

23. Factorise x 2 - x - 12 x2x2 - x -12 Factorise x 2 -x-12 Clearly there is no way to accommodate the 12 small guys in the bottom right hand corner. What do you do? ? You add in Zero in the form of +x and –x. And Keep doing it to complete the rectangle.

24. Factorise x 2 - x - 12 The factors are ?( x + 3 )( x - 4 ) x - 4 x + 3

25. Section 6. Doing linear equations Solve 2x + 2 = -8 =

26. Section 6. Doing linear equations Solve 2x + 2 = -8 =

27. Section 6. Doing linear equations Solve 2x + 2 = -8 = = =

28. Section 6. Doing linear equations Solve 2x + 2 = -8 = = Solution x = -5

29. Section 6. Doing linear equations = Solve 4x – 3 = 9 + x You can take away the same thing from both sides

30. Section 6. Doing linear equations = Solve 4x – 3 = 9 + x You can add the same quantity to both sides

31. Section 6. Doing linear equations = Solve 4x – 3 = 9 + x

32. Section 6. Doing linear equations = Solve 4x – 3 = 9 + x = =

33. Section 6. Doing linear equations = Solve 4x – 3 = 9 + x = = Solution x = 4

34. Practice Solve the following: 1)x+4 = 7 2)x-2 = 4 3)3x-1 =11 4)4x-2 = x-8 5)5x+1 = 13-x 6)2(x+3) = x-1 7)2x-4 = 5x+8