3 that follow, the perimeter is the distance around the exterior of a figure. The area of a figure is the sum of the areas of the individual tiles.

Transcription

1 COMBINING LIKE TERMS A.. and A..2 Algebra tiles are an important part of th course and will be used throughout. The tiles provide students with the opportunit to see abstract algebraic epressions and equations with two variables. Regular use of algebra tiles will help students access abstract concepts through the use of concrete phsical representations. In the figures at left, the dimensions of each tile are shown along its sides, and the area shown on the tile itself. Algebra tiles are named b their area. For eample, the 2 -tile in the upper left corner; it has an area of 2. In diagrams with combinations of tiles, as in Eamples to 3 that follow, the perimeter the dtance around the eterior of a figure. The area of a figure the sum of the areas of the individual tiles. In algebraic epressions, as in Eamples 4 to 6 below, combining terms that have the same area to write a simpler epression called combining like terms. For additional information, refer to the Math Notes boes in Lessons A.., A..2, and A..5. Additional Math Notes boes in Lesson A..3 and A..4 can help with evaluating epressions. Students evaluate epressions in homework and Checkpoint 2. A resource page for creating algebra tiles for home use available at the CPM website, Go to the student section, and download the Lesson A.. Resource Page. 2 2 Eample 2 Perimeter = = (going clockwe from upper left) 2 Area = Core Connections Algebra

2 Appendi A Eample Perimeter = 7 (! 2) 3 = 6 8 (going clockwe from upper left) Area = 2 2 = Eample Perimeter = 8 6 Area = Eample 4 The epression can be rewritten as and simplified to b combining like terms. Eample 5 The epression can be rewritten as and simplified to b combining like terms. Eample 6 3 2! 4 3! 2 3! 7 = 3 2! 2! 4 3 3! 7 = 2 2!! 4 Parent Guide with Etra Practice 37

3 Problems Determine the area and the perimeter of each figure Simplif each epression b combining like terms ! ! ! 3! 4 2! ! ! (4 2 4! ) ( 2! 7) 7. ( ) (2 2 4! 2 ) 8. (7 2! 6! 9)! (9 2 3! 4) 9. (3 2! 8! 4)! (5 2 ) Answers. P = 4 4 A = P = 4 4 A = P = 8 6 A = P = 4 6 A = P = 4 4 A = P = 6 A = 2 3. P = 4 8 A = P = A = P = 6 4 A = ! 5 3.! !2 2! 9! 5 9.!2 2! 9! 5 38 Core Connections Algebra

4 Appendi A SIMPLIFYING ALGEBRAIC EXPRESSIONS A..3 through A..5 An epression mat a phsical representation of an algebraic epression. The upper half of an epression mat the positive (addition) region and the lower half the negative (subtraction) region. Positive algebra tiles are shaded and negative tiles are blank. (The illustration to the right reminds ou that shaded tiles are positive.) A matching pair of tiles with one tile shaded and the other tile blank represents two opposites with a value of zero. We refer to them as zero pairs. On an epression mat, tiles ma be removed or moved in one of two legal was: () Flip tiles and move them from the negative region to the positive region. That, change subtraction to adding the opposite. (2) Remove an equal number of opposite tiles (one shaded and one not shaded) that are within the same region. These pairs of opposite tiles have a value of zero. (3) Group tiles that are alike together. That, combine like terms. For additional information, see the Math Notes bo in Lesson A..6. = = Eample Simplif 3 2! (2! 3). Create epression mat: = = Flip tiles in subtraction region to addition region: Remove zero pairs: Therefore, 3 2! (2! 3) simplifies to 5. Parent Guide with Etra Practice 39

5 Eample 2 Simplif! (2! 3)! 2. Create epression mat: = = Flip tiles in subtraction region to addition region: Remove zero pairs: Therefore,! (2! 3)! 2 simplifies to 2 (!) or 2!. Problems Write the algebraic epression, then simplif Core Connections Algebra

6 Appendi A Use algebra tiles to simplif each epression ! 4! 7 8.!! 4! 7 9.!(! 3) 0. 4! ( 2). 5! (!3 2) 2.! 5! (2! ) 3.! 2! 2 4.!3 5 5! 5. 3! ( 5) 6.!( ) ! 7! (3! 7) 8.!( 2 3)! 3 Answers. 3 (!2)! 4! (!3) = 0 4.!4!! (!2) =! !! (!) = !(!2)! or!(!2 ) = 2! 3. 5! (!2)! (3) or 5! (!2 3) = (!2)! (!)! (!2) =! 5 7.!2! 8.!5! 7 9.! ! 2. 8! ! 7 3.! !! !4!! 3 Parent Guide with Etra Practice 4

7 COMPARING ALGEBRAIC EXPRESSIONS A..6 and A..7 Epressions on two side-b-side Epression Mats can be compared to determine which epression greater. To compare two epressions, represent each epression with tiles on its own separate epression mat. Work on each Epression Mat separatel to simplif the epression b moving or removing tiles using legal moves: Flip tiles and move them from the negative region to the positive region. That, change subtraction to adding the opposite. Remove an equal number of opposite tiles (one shaded and one not shaded) that are within the same region. That, remove the zero pairs. Place or remove the same tiles on or from both Epression Mats. That, add or subtract the same value from both sides. Group tiles that are alike within the same region together. That, combine like terms. Continue to make legal moves, in an order, until the epressions cannot be simplified an more. Compare the epression on the left with the one on the right to determine which epression greater. If there are variable tiles remaining after simplifing, ou do not have enough information to tell which side greater depending on what number the variable tile represents, either epression mat could be larger than the other. 42 Core Connections Algebra

8 Appendi A Eample The Epression Comparon Mat at right represents the epressions!2! 3! (! 3) and 2! 3! (! 2). Use legal moves to simplif and determine which side greater. Solution: Flip tiles and move them from the negative region to the positive region. Remove an equal number of opposite tiles (one shaded and one not shaded) that are within the same region. Also remove the same tiles from both Epression Mats. 5 < ; The right side greater. Parent Guide with Etra Practice 43

9 Students are also asked to record their steps. Different teachers have different epectations but here are two possible was to record the steps. The steps ma also be done in a different order. Recording the steps smbolicall: Epression flip!2! 3! (! 3)!2! 3! 3!! 5!(!)!5? =? =? =? = Epression flip 2! 3! (! 2) 2! 3! 2!!(!) Recording the steps with a written eplanation. Epression Epression Eplanation Starting epressions Flip 3 from to Combine like terms Flip 2 from to Remove from both sides Combine like terms 44 Core Connections Algebra

10 Appendi A Eample 2 Create the epressions! (!! 2) and 3!! (! 4) and then use legal moves to simplif and determine which side greater. () Flip tiles and move them from the negative region to the positive region. (2) Remove an equal number of opposite tiles that are within the same region. (3) Remove the same tiles from both Epression Mats. (4) Compare the Epression Mats. 3? 4 Since we do not know the value of, it not possible to determine the greater side. Parent Guide with Etra Practice 45

11 Problems Write an epression for each side of the epression comparon mat. Use legal moves to simplif and determine which side greater. Carefull record our steps as ou go. = = In problems 5 through 0, record our steps as ou use legal moves to simplif each epression and determine which side has the greater value. 5. greater: 6! (2! 4)! 3 or!! ( ) 4? 6. greater: 3! (2! ) or! 5 4 3? 7. greater:! 6! 2 4! 2 or 5! (!2 ) 3! 2? 8. greater: 2! 2 6! (!3) or! (3! 2 ) greater: 2! (2! 2) or 4! 2! (! 4)? 0. greater: 2 4!! (!2) 2 or 3 2 4! (!3 3)? Answers (epressions and eplanations will var). left side greater 2. right side greater 3. not enough information 4. left side greater 5. left side greater 6. left side greater 7. right side greater 8. not enough information 9. not enough information 0. both sides equal 46 Core Connections Algebra

12 Appendi A SOLVING EQUATIONS A.8 and A.9 An Equation Mat can be used to represent the process of solving an equation. An Equation Mat created b putting two Epression Mats side b side one for each side of the equal sign. See the Math Notes boes in Lessons A..8 and 3.2. for a lt of all the legal moves and their corresponding algebraic language. Also see the Math Notes bo in Lesson A..9 for checking a solution. When the process of solving an equation ends with different numbers on each side of the equal sign (for eample, 2 = 4), there are no real-number solutions to the problem. When the result the same epression or number on each side of the equation (for eample, 2 = 2 ) it means that all real numbers are solutions For additional eamples and practice, see the Checkpoint materials. Parent Guide with Etra Practice 47

13 Eample Solve 2! (!2) = 5! (! 3). First, build the equation on an Equation Mat. See the Math Notes bo in Lesson A..8. Second, flip the tiles in the subtraction region to the addition region (change subtraction to adding the opposite). Continue to simplif using the legal moves, in an order, from the Math Notes bo in Lesson For eample, remove zero pairs. Isolate -terms on one side and non--terms on the other b placing or removing matching tiles from both sides of the equation mat. Remove zero pairs again if needed. Finall, arrange tiles into equal-sized groups on both sides. Since both sides of the equation are equal, determine the value of. In th case, the tiles can be arranged into three groups, resulting in = Core Connections Algebra

14 Appendi A Eample 2 Solve 3 3! = (!) = (!) = = = 0 = 5 Flip all tiles from subtraction region to addition region. Combine like terms. Add to each side, remove zero pairs. Remove 4 from each side. Arrange into two groups. Eample 3 Solve!2! (!3 3) =!4 (!! 2)!2 3 (!3) =!4 (!) (!2) Flip all tiles from subtraction region to addition region. (!2) =! (!6) =! (!4) 2 =!4 =!2 Combine like terms. Add 2 to each side, remove zero pairs. Add to both sides, remove zero pairs. Arrange into two groups. Parent Guide with Etra Practice 49

15 Problems Solve each equation.. 2! 3 =! ! =! ! 3 = 2! ! (! 2) = !( 3) = 2! 6 6.!4 3! = ! 3 = ! 3 2 = ! 8! 2 = ! (! 3) = 4! (3! ). 2! 7 =!! 2.!2! 3 =! 2! 4 3.!3 7 =! 4. 2! 4 =!3! (!) 5. 2!! =! 3! (!5 ) 6.!4! 3 =!! = !(! 2) =! 5! !! 3 = 4! ! (! 3) =! 5 Answers. = 2 2. = 5 3. = 2 4. = 5. = 6. = 6 7. = 7 8. no solution 9. = 6 0. = 7. = 2 2. all numbers 3. = 2 4. = 0 5. = 2 6. no solution 7. = 3 8. = 7 9. = = 4 50 Core Connections Algebra