5.1 The Language of Mathematics

Transcription

1 5. The Language of Mathematics Prescribed Learning Outcomes (PLO s): Use mathematical terminology (variables, degree, number of terms, coefficients, constant terms) to describe polynomials. Identify different types of polynomials (monomial, binomial, trinomial). Create a model for a given polynomial epression. Write the epression for a given model of a polynomial. Write a polynomial epression for a given situation. Warm-up: What does the eponent in 3 represent? What does the coefficient in 3 represent? Write 4 ( y ) as repeated multiplication Write 4y as repeated addition Terminology Algebra: A branch of mathematics that uses symbols to represent unknown numbers or quantities that can vary. The symbols are often letters and are called variables. Polynomial: An algebraic epression made up of terms connected by the operations of addition or subtraction. Term: An epression formed from the product of numbers and or variables. Coefficient: The number that multiples the variable in a term. Making connections: What are some words that have the following prefies? Find other eamples. MONO BI TRI Special types of polynomials: Monomial: is a polynomial with only term. Eamples: Binomial: is a polynomial with terms. Eamples: Trinomial: is a polynomial with terms. Eamples: Degree of a term: the sum of the eponents on the variables in a single term (monomial) Degree of a polynomial: the degree of the highest-degree term in a polynomial Riverside Secondary (created by Mrs. Wong) Page

2 Eample : Complete the chart. TERM DEGREE 3 yz 4 3y y ab a b c 4 3 y.5yz 4 6 Eample : Complete the chart. Epression 4y + 3 # of Terms Name Coefficients Constant Degree 7a b ab + ½ b y + z a + b + y 5 z Models using algebra tiles: Legend: Eample 4: What polynomial epression is represented by each set of algebra tiles? b) d) e) f) g) h) i) j) Riverside Secondary (created by Mrs. Wong) Page

3 Writing epressions for a given situation. Making connections: Brainstorm as many English words or phrases to indicate each of the following mathematical symbols as you can. + Eample 5: Write an epression for each statement. a. Seven and a number are added together. b. Four more than three times a number. c. The product of a number and eight is decreased by 3. d. The sum of double the number and 3. e. The length of a banner is 3cm longer than triple the width. f. 5 times a number, increased by 8 g. 5 times the sum of a number and 8 h. 5 more than 8 times a number i. 8 times the sum of a number and 5 j. Twice the sum of 5 times a number and 8 k. 9 meters shorter than twice the length l. 9 years older than half her age m. $9 cheaper than quadruple the price Eample 6: Write a polynomial for each of the given situations. a. A triangle s base and height are equal. Find an epression for the area. b. In January, David used 4 more cellphone minutes than 3 of the number he used in December. Find an epression 4 for the number of minutes used in January. c. Tickets for the Basketball game are $5 for adults and $3 for students. Write an epression that shows the total income for the game. Riverside Secondary (created by Mrs. Wong) Page 3

4 5. Equivalent Epressions Prescribed Learning Outcomes (PLO s): Use algebra tiles to show whether epressions are equivalent Identify equivalent epressions that are polynomials. ( is equivalent to Combine like terms in algebraic epressions. Write a polynomial epression for a given situation. Simplify polynomial epressions ) Eample : Write a polynomial for each of the epressions modeled by the algebra tiles. Then identify the equivalent polynomials. b) d) e) f) Terminology Like Terms: Terms that differ only by their numerical coefficients (Same variables raised to the same eponents.) Eamples: Zero Pairs: algebra tiles that have a combined value of zero Eample : Write a polynomial for each of the epressions modeled by the algebra tiles then simplify by removing the zero pairs. b) d) e) f) Riverside Secondary (created by Mrs. Wong) Page 4

5 Eample : Indicate the like terms in the list of terms below (underline/circles/squares/stars) cb 5bc 00cb 8 y y 3.4 6cb 7y y 5y 3 Eample 3: Simplify each polynomial by collecting like terms, then name it, state the coefficients and the constants. Polynomial Simplify Name Coefficients Constants 3 + b) y + y y + y d) p+ 4p + 3 3p 5+ p e) 4y 3y + y y 0 f).5 y y 3 3 Eample 4: Determine an epression for the perimeter of the following shapes then simplify. b) y 3y 5 4 d) e) f) Eample 5: A private school pays a company $5 to design gym T-shirts. It costs an additional $5 to make each shirt. A) Write an epression to represent the cost that a school is charged for n shirts. B) How much do 00 shirts cost? C) What is the cost per shirt if 00 shirts are ordered? Riverside Secondary (created by Mrs. Wong) Page 5

6 5.3 Adding & Subtracting Prescribed Learning Outcomes (PLO s): Model addition & subtraction of polynomial epressions concretely, pictorially and record the process symbolically. Solve problems using the addition & subtraction of polynomials. Simplify polynomial epressions. Adding Eample : Write a polynomial epression for the following algebra tiles then simplify. + ( ) ( ) b) Eample : Add the polynomials. 5 3 ( + ) + ( + ) b)( y y ) ( y ) Subtracting To subtract a polynomial, you can add the opposite terms and then simplify. Eample 3: Write a polynomial epression for the following algebra tiles then draw an equivalent model using addition and simplify. ( ) + ( ) ( ) ( ) b) ( ) ( ) ( ) ( ) d) ( ) ( ) Riverside Secondary (created by Mrs. Wong) Page 6

7 Eample 4: Subtract the + + b) ( ) ( ) ( a a ) ( a 3a ) d) ( ) ( ) ( y y ) ( y y ) Eample 5: Simplify ( y y) ( y y) ( y) b) ( ) ( 3 4) ( ) ( 5cd + 8c 7d ) ( 3d + 6cd 4c ) d) ( y y 3) ( 4 5y 3y ) ( y 3y ) Eample 6: Determine an epression for the perimeter of the following diagrams, then simplify..5 + b) r Eample 7: Find an epression for the missing side length given the Perimeter, then simplify. P b) P Eample 8: Identify & eplain the error, then give the correct answers. ( 3 + 3) ( + 3 ) ( y ) + ( 3+ y ) b) 7 + y 4 Eample 9: The cost to make n shirts is 5n dollars. The cost to have Riverside s logo printed on n shirts is n + 00 dollars. The Wave store sells the shirts for $0 each. Write an epression to represent the profit the Wave store makes for selling n shirts then simplify. b) What is the profit if the sell 500 shirts? Riverside Secondary (created by Mrs. Wong) Page 7

8 7. Multiplying & Dividing Monomials Prescribed Learning Outcomes (PLO s): Model multiplication of a monomial by monomial and record the process algebraically. Model division of a monomial by a monomial and record the process. Identify errors in a simplification of a polynomial epression Simplify polynomial epressions Legend: y y y y Eample : In the following models, algebra tiles are used to represent the multiplication of a monomial by a monomial. Write an algebraic representation of the models. (Multiplication statement) b) Eample : Fill in the algebra tile models and then write a symbolic representation of each model. b) Eample 3: Find the product of each pair of monomials. ( y) ( y ) b) ( 3)( y) ( m)( m) 5 d) Show you Know: How is multiplying monomials similar to the product law for powers? Eample 4: Find a simplified epression for the area of each polygon. 3 b) Eample 5: Determine a simplified epression for the volume. 0 b) y Riverside Secondary (created by Mrs. Wong) Page 8

9 Eample 5: In the following models, algebra tiles are used to represent the division of a monomial by a monomial. Complete the model and write the division statement. (Division statement) b) d)???? Eample 6: Find the quotient of each pair of monomials. 5 5 b) 6y y 4y 8 d) 8 y 4y e) 36 9 f) 0.8d 0. d g) ( ) a a h) m 5 m Show you Know: How is dividing monomials similar to the quotient law for powers? Eample 7: Given the area of each polygon find the missing dimension. b) A 8 3y A y A 6y height? 4 Eample 8: Identify & eplain the error, then give the correct answers. ( )( ) b) e) f) ( )( ) g) + h) 6y 3y i) ( 6 + ) + ( 3 5) 9 3 j) ( 6 + ) ( + 5 7) Riverside Secondary (created by Mrs. Wong) Page 9

10 7. Multiplying by Monomials Prescribed Learning Outcomes (PLO s): Model multiplication of a polynomial by monomial and record the process algebraically. Simplify polynomial epressions I can identify errors in a simplification of a polynomial epression. Eample : In the following models, algebra tiles are used to represent the multiplication of a polynomial by a monomial. Write an algebraic representation of the models. (Multiplication statement) b) Eample : Fill in the algebra tile models and then write a symbolic representation of each model. b) Distributive Property: Multiplication & Division can be distributive over addition and subtraction. Eample 3: Evaluate each of the following using two different strategies if possible. (3 + 4) b) ( 3 4) (3 + 4) Eample 4: Write each of the following using repeated multiplication or repeated addition, then simplify. ( 3 ) b) ( 3) + Eample 3: Epand and Simplify + b) ( 4) ( 3 5) d) 3( + 5) 8( 5) y y f) 3( + 4y ) g) 3( 5y 3) e) (3 4) + h) 4 y(y y) 3 y+ j) 3 4 i) ( 9 6 ) h) 3gh( gh + 4g 5h + 7) Riverside Secondary (created by Mrs. Wong) Page 0

11 7.3 Dividing by Monomials Prescribed Learning Outcomes (PLO s): Model division of a polynomial by monomial and record the process algebraically. Simplify polynomial epressions I can identify errors in a simplification of a polynomial epression. Eample : Use algebra tiles to perform the following operations b) y + 8y y d) e) 4 f) y + Eample : Determine an epression for the width of rectangle with Area 0 5 and length 5. Eample 3: Determine an epression for the height of a triangle if the and the base 4. 3 Area 6 Eample 4: The Perimeter of a square is meters. Find an epression for the length of each side. Eample 5: The Area of a parallelogram is 6 its height? square meters. If its base is 3 meters, find an epression for Riverside Secondary (created by Mrs. Wong) Page

12 More practice with modeling & simplifying polynomials Eample : Find and simplify a polynomial epression that represents the Shaded Area of the following overlapping squares with each given side length: b) 4 3 y 5y ab cm 3ab cm 5 d) e) 4 f) 4 cm 3y cm Eample : Find an epression for the volume & surface area of a cube with an edge length of 6 m? 6 Eample 3: Find an epression for the volume & surface area of a cube with an edge length of m? Eample 4: How many times larger is the volume of the cube with a side length of 6 m compared to the cube with side length of m? Eample 5: Find an epression for the shaded area of the rectangles with each given side length: b) Riverside Secondary (created by Mrs. Wong) Page

13 ) A triangle has a perimeter of ( m 5n) Measurement Worksheet 3m n cm and another side measures 7 + cm. One side measures ( ) ( m + 5n) cm. Write and simplify an epression for the length of the third side of the triangle. 4 5 ) Sarah wants to put a fence around her back yard and another fence around her pool. How much fencing is required to fence the wading pool and fence the surrounding yard? Find a simplified polynomial epression b) If represents the length of 6 m and fencing costs $3/m, how much will the project cost? 3) Find an epression for the area of each figure c b) 4y 7y 5y 0y 5c 4) Find an epression for the volume and the surface area of the rectangular prism. 9y 6 y 5) Find the height of a triangle whose area is 60 cm and whose base is 0 cm 6) Find an epression for the height of a triangle with a base of 6 f gh cm and an area of f g h cm. 7) Find an epression for the area of a meter walkway surrounding a square garden of side s meters. Riverside Secondary (created by Mrs. Wong) Page 3

14 8)Find an epression for the shaded area ) The surface area of a cone is given by SA πr + πrs where r is the radius of the base of the cone and s is the slant height. Find an epression for the surface area if r 4 and s s 0) Find an epression for the area. r ) Find an epression for the surface area of a cube with ) Find the height of the square based prism if the Volume with side length y cm. 3 is 7 9 cubic centimeters. 3 y 3 3) Find an epression for the volume of the bo + 3y l m + n cm Fence 8, b) $ c, b) Answers: ) ( 3 ) 60y 4) V 3 54 y units, SA y y y units ) h cm 6) 4 H 9 f g cm 7 A s units ) A 5 units 9) SA 8π + 0π units 4 0) A units ) 4y cm )3 3 cm 3) V 6 y y + 6y Riverside Secondary (created by Mrs. Wong) Page 4