Unit 3: Algebra. Date Topic Page (s) Algebra Terminology 2. Variables and Algebra Tiles 3 5. Like Terms 6 8. Adding/Subtracting Polynomials 9 12

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1 Unit 3: Algebra Date Topic Page (s) Algebra Terminology Variables and Algebra Tiles 3 5 Like Terms 6 8 Adding/Subtracting Polynomials 9 1 Expanding Polynomials Introduction to Equations One Step Equations Two Step Equations 0 1 Multi Step Equations 4 Algebra Applications 5 6 Review 1

2 Algebra Terminology Algebra: The branch of math that deals with general statements of relations and uses letters or other symbols to represent specific numbers. Variable: A letter used to represent a value that can change or vary. Ex. In the expression z + 4, z is the variable Constant: A term that contains no variables. Its value does not change. Ex. In x + 5, the constant term is 5. Coefficient: The number by which a variable is multiplied. Ex. In the term 8y, 8 is the coefficient. Like Term: Terms that have the same variable(s) raised to the same exponent(s). Ex. 3xy and xy are like terms. Monomial: A polynomial with one term. Ex. 7y Binomial: A polynomial with two terms. Ex. 3x + 3 Polynomial: An algebraic expression formed by adding and/or subtracting terms. Ex. 3x + y 4 is a polynomial. Expression: A mathematical phrase made up of numbers and variables, connected by operators. Ex. 3x + is an expression. Simplify: Get an expression in the simplest form by combining like terms. Ex. 3x + x + 7 simplifies to x + 9 Equation: A mathematical statement made of two expressions that equal each other. Ex. 3x + = 8 Solve: Find all possible values of the variable that make the equation true. Ex. If 3x + = 8, then x = is the solution

3 Variables 1. Consider the following situation. In your class, your teacher has decided to give each student three pencils at the start of the year. a) How many pencils would he or she need if there were i) 0 students? ii) 5 students? iii) 8 students? b) Create an expression that would determine the number of pencils if you didn t know how many students there were.. In the previous question expression should have included a variable. Explain what a variable is when it is in an expression. 3. Sometimes variables are used in mathematical expressions or equations. Explain what the expression 5w means mathematically? 3

4 Introduction to Algebra Tiles Algebra Tiles can be used to represent different variables. The different sizes represent different variables, whereas the different colours represent positive or negative variables Positive Tiles : (BLUE) Negative Tiles : (RED) We can use Algebra tiles to help us model algebraic expressions. Red Tiles Blue Tiles Zero Principle Represents 1 Represents = 0 ( 1 ) + ( +1 ) = 0 Represents x Represents + x + = 0 ( x ) + (+ x) = 0 Represents x Represents + x + = 0 ( x ) + (+ x ) = 0 We can use algebra tiles to help is write algebraic expressions Writing Algebraic Expressions Example: Write the expression represented by each group of tiles a) b) c) d) Example: Write the expression represented by each group of tiles, remember the zero principal! a) b) 4

5 Practice: Write the expression represented by each group of tiles. a) b) c) Modeling Algebraic Expressions Example: Use algebra tiles and diagrams to model each of the following expressions a) x + x b) 3 + 4x x c) x + x 3 Practice: Use algebra tiles and diagrams to model each of the following expressions a) x 4x + 5 b) 4x + x 3 c) x + 7 5

6 Simplifying Algebraic Expressions Like Terms To simplify a collection of algebra tiles, we need to collect like tiles or like terms Like terms have the variables raised to the exponent. Unlike terms have variables or the same variables raised to a exponent. To simplify an algebraic expression, we need to collect like terms An expression is simplified when all the like terms are combined and any zero pairs are removed. Examples of Like Tiles Examples of Like Terms 1) Like Terms 1) Not Like Terms ) Like Terms ) Not Like Terms 10x + 3x 4x 5x + 6x + 1 CAN Simplify Cannot Simplify CAN Simplify Cannot Simplify = = = = Example: Simplify a) b) = = 6

7 We can also simplify algebraic expressions without using the algebra tiles by using paper and pencil. Example: Simplify without using algebra tiles 5 + x + 3x x + 4 x + 3 = = STEPS Step # 1: Identify like terms Step # : Group like terms Step # 3: Add coefficients of like terms = Practice: 1. Write the algebraic expression modeled by the tiles below. a) b) c) d) e) f). Use algebra tiles to model each algebraic expression below. Sketch the tiles you used. a) x + 6 b) x 3x + 5 c) 5 + 3x + 7x 3. Circle the terms that are like terms to the first term given in brackets. 5 b) ( x ): x, 3y, x, - 4y, 3x a) ( y) : 3y, y, xy, 4y, 9y 7

8 4. Simplify the algebraic expression by combining like terms and removing any zero pairs. a) b) c) d) e) f) 5. Simplify each expression using algebra tiles a) x 3x + 3x + 5x b) x + + x 1 c) 3x + x + x + x Simplify each expression without using algebra tiles. Show all steps. a) + 4x x + 3 b) x + x + 3x + 1 c) 15x x x 8 9x Algebra Tile Assignment Using the algebra tiles, create a tile pattern of your choice. Transfer the picture to a white piece of paper and colour it appropriately. Create a mathematical expression to represent your pattern and then simplify it. 8

9 Adding and Subtracting Polynomials Adding Polynomials Using Algebra Tiles To add polynomials is not much different than simplify a polynomial. To simplify the polynomial (5x 7) + ( 3x + 4) Algebra Tile Representation Algebraic Representation 1. Represent each polynomial with algebra tiles:. Collect all similar sized tiles: 3. Use the zero principle to reduce the number of tiles. Final answer: Practice: Use Algebra tiles to simplify the following expressions a) (6x + 7) + (3x + 5) b) (x 8) + ( 5x + 7) c) (3x + x + ) + (x + 5x + 3) d) (x 3x + 1) + (x 4x 3) 9

10 Adding Polynomials without Algebra Tiles To add polynomials without algebra tiles, we simply remove the bracket then simplify by collecting and combining like terms. Examples: a) (5x + 4) + ( 7x 1) b) ( 4x x + 9) + (x x 7) Steps: 1. Remove brackets (and double signs if present). Collect like terms 3. Simplify (combine like terms) Practice: 1. Complete the following question in the space provided. Be sure to show ALL your work a) (3x + 1) + (4x ) b) (3x + x + 1) + (x x + 1) c) ( x + 7x 5) + (x x 1). a) Determine a simplified expression for the perimeter of the following triangle. x + 1 5x + 3x 4 b) Using your expression from part a), determine the perimeter if x = 3m. 3. Two polynomials are added. The sum (answer) is 3x 5x. One polynomial is x x + 3. What is the other polynomial? Explain how you found it. 10

11 Subtracting Polynomials with Algebra Tiles To subtract polynomials requires one more step than adding polynomials. To simplify the polynomial (5x 7) ( 3x + 4) Algebra Tile Representation Algebraic Representation 1. Represent each polynomial with algebra tiles:. Reverse each of the tiles in the nd polynomial. 3. Collect all similar sized tiles: 4. Use the zero principle to reduce the number of tiles. Final answer: Practice: 1. Simplify the following polynomials using algebra tiles. a) (3x + 5) (x +1) b) (x + x) ( 3x + x) c) (x + 3x ) ( x x + 1) d) (4x 6x 10) (3x + x 1) 11

12 Subtracting Polynomials without Algebra Tiles To Subtract Polynomials we need to remove the brackets and change the signs of ALL the terms in the SECOND bracket then simplify by collecting and combining like terms Examples a) (5x + 4) ( 7x 1) b) (3x x + 5) (x x + 4) Steps: 1. Change all signs in second bracket and make it an addition question. Remove brackets (and double signs if present). Collect like terms. 3. Simplify Practice: Complete the following question in the space provided. Be sure to show ALL your work 1. Subtract the following polynomials. a) (3x + 1) (4x ) b) (3x + x + 1) (x x + 1) c) (x + 3x 5) ( 5x x + 4). John subtracted these polynomials ( x 4x + 6) ( 3x + x 4) a) Explain why his solution is incorrect b) What is the correct solution? ( x 4x + 6) ( 3x + x 4) = x = x = x 4x + 6 3x + x 4 3x 4x + x x + 1

13 Expanding Polynomials Using Algebra Tiles Expanding Polynomials We can use an area model to multiply monomials and polynomials. Multiplying polynomials is like calculating the area of a rectangle. Steps: 1. Determine the polynomials being multiplied.. Trace the algebra tile representation of each term in the multiplication template; one on the top and one on the left. 3. Fill the rectangle in with algebra tiles so the side lengths match the top and side. 4. Write your answer Ex. 3(x + ) What are the two polynomials being multiplied? Represent these polynomials with algebra tiles Place the tiles for the first polynomial on the left hand side, and the tiles that represent the second polynomial on the top. Fill in the rectangle with the product. *Remember your sign rules for multiplying integers Ex. x( 3x + 1) What are the two polynomials being multiplied? Represent these polynomials with algebra tiles Place the tiles for the first polynomial on the left hand side, and the tiles that represent the second polynomial on the top. Fill in the rectangle with the product. *Remember your sign rules for multiplying integers Would you get the same answer if you switched the location of the tiles? 13

14 Practice: 1. Use algebra tiles and the area model to expand the following: a) x(3x +4) b) ( x + 1) c) x(5x ) d) 3x( x 3). Use algebra tiles and the area model to expand the following: a) 4(x) = b) 3x(5) = c) x(3x) = d) 3x(4x) = 3. Look at your solutions for question. Is there a pattern that might help determine the answer without using algebra tiles? Explain your pattern. 4. Use your pattern to simplify the following: a) 6(3x) = b) 8(5x) = c) 9x( 3x) d) 7x( 6x) = Expanding polynomials without algebra tiles To multiply polynomials, we use the distributive property. This tells us to multiply the term in front of the brackets by EVERY term inside the brackets. Example: Expanding using the distributive property: a) (x + 5) b) ( x + 7) Some expressions may involve multiplying more than one polynomial and then collecting like terms. If so, multiply one set at a time. Be sure to watch the signs of the number in front of the brackets when you are multiplying. Example: Expand and simplify (collect and combine like terms) (x + 1) 3(x ) 14

15 Practice: 1. Expand and simplify when possible: a) 3(x + ) b) (5x 8) c) 6( 4x + 7) d) 5x(x 3) e) 4( 9x + 11) 3( 10x 1) f) 3 (9x + 4) + 4( x + 5). a) Determine a simplified expression for the area of the following rectangle. [HINT: A = lw] x + 7 x b) Use your simplified equation from part a) and determine the area of the rectangle if x = 5 m. [HINT: use substitution] 15

16 Introduction to Equations 1. Draw an X through the example that does not belong. Justify your answer. a) b) x + 4 = 8 x 4 = 3 3x 3x = 9 + x = 8 x + 4 x = 8 -x = 4 c) d) 3x 3 = 3 4 x y = 3x + 1 C = 10t + 1 x 1 = 5 -x = 4 y + 3x P = l + w. Answer True (T) or False (F). Be prepared to justify your answer. a) Every equation has exactly two sides. b) Every equation has one equal sign. c) Every equation has one variable. 16

17 What does the Answer Mean? *Match the solution that is most likely the answer to each of the equations by placing the appropriate number in the space provided. Equations 5x 3 = = -x x = 0 3x 9 = -9 x + 1 = -6 9x 7 = -x = = 5x - 3 3x 6 = 0 Possible Solutions #1 x = 1 #5 5 = x #9 4 = x # x = #6 4 = x #10 x = - #3 x = 4 #7 #11 x = 0 x = 4 #4-5 = x #8 x = -3 ½ #1 10 = x 17

18 One Step Equations To solve an equation, we are trying to find the number that makes the statement true. *We must isolate the variable (get the variable by itself) Solve each equation using algebra tiles. Check your answer. x + 5 = 4 x - 1 = 6 3x = 6 x = 8 Practice: x + 1 = -3 x + 6 = 9 4x = 1 5x = 0 18

19 One Step Equations To solve an equation, we are trying to find the number that makes the statement true. We must isolate the variable (get the variable by itself) REMEMBER : What we do to one side of an equation, we need to do to the other side too! To solve an equation we need to isolate the variable To do this, we need to perform the opposite operations: The opposite of adding is The opposite of subtracting is The opposite of multiplying is Examples a) 4 x = 16 To get x by itself, we need to do the opposite of multiplying by 4 b) x + 5 = 8 To get x by itself, we need to do the opposite of adding 5 c) x 4 = 18 To get x by itself, we need to do the opposite of subtracting 4 Practice: 1. Solve each equation. Show your work. a) 3 w = 1 b) a + = 5 c) x 4 = d) p 7 = 3 e) 6 z = 4 f) 1 = x + 7 g) 5 x = 10 h) 36 + e = 84 19

20 Solve each equation using algebra tiles. Two Step Equations x + 3 = 5 3x 1 = - 7 Practice: Solve each equation using algebra tiles + x = x = - 4x + 1 = -3 3x + 6 = 9 0

21 Two Step Equations To solve equations with a coefficient in front of the variable, we still want to get the variable by itself using opposite operations. Examples: Solve each of the following and check your answer. a) 3x + 4 = 6 Formal check: 3x Steps: (backwards BEDMAS) 1.. b) 3 6x = 15 Formal check: - 3-6x 15 Practice: Solve the following equations. Show all your work. Check your answer for part e. a) 4x + 4 = 44 b) 3 + 4x = 11 c) 1 + 6x = 41 d) 1 + 5x = 54 e) 1 + 7x = 43 Formal check 1 + 7x 43 1

22 Multi Step Equations Solve each equation using algebra tiles. Check your answer. 3x + = x 4 x + 7 = x 3 Practice: Solve each equation using algebra tiles. Check your answer. 5x + 6 = x x 5 = - 4x x 4 = 0 x -x = 1 x

23 Multi Step Equations Multi step equations are equations that have brackets and/or variables on both sides of the equal sign. Equations with brackets: Example: 4( 4x + 10) = 7 Steps: Equations with variables on both sides of the equal sign: Example: 5x + 6 = x + 18 Steps: Formal Check: 5x + 6 x

24 Practice: 1. Solve the following equations. Proper form is expected. Show ALL steps. a) 3(x + 7) = 30 b) 5(x + 4) = 0 c) 8x 10 = 4x + 14 d) 4(x + 1) = 1 e) x = x 13 f) 4(x 3) + 9x = 38 g) 3(x 4) = 3 h) 10x 5 = 1 3x i) 3(x 5) = (5x 11) 3 ( x 4) 3 10x 5 1 3x 3(x 5) (5x 11) 4

25 Algebra Applications 1. Ivan shows his steps in solving the following equation for x: In which step has Ivan made an error? Complete the problem correctly. Pauline builds a fence around her garden, which is shaped like a parallelogram, as shown below. Pauline uses 100 metres of fencing along the perimeter of the garden. Find the dimensions of her garden. Show your work. 5

26 3. A field in the shape of a trapezoid has a perimeter of 460 m. A fence is being built along the field s perimeter. Determine the length of fencing needed for each side of the field. Show your work. 4. Peter has two part-time jobs. His earnings for one week are represented by the equation below: E = 7.50r + 8.5v E is his total earnings in one week; r is the number of hours he works at the restaurant and v is the number of hours he works at the video store. Peter earns a total of $ in one week. If he works 8 hours at the restaurant, how many hours does he work at the video store? 6