Algebra 1 Unit 3: Systems of Equations s6

Course Name: Foundations of Math I Unit # 2 Unit Title: Exponen ts and Polynomials

Enduring understanding (Big Idea): Students will understand that manipulation of polynomials will enable them to model and analyze real- world non-linear situations.

Essential Questions: 1. How are the properties of real numbers related to polynomials? 2. How can two algebraic expressions that appear to be different be the same? BY THE END OF THIS UNIT: Students will know… Students will be able to… 8.EE.1  Properties of exponents (non-negative only) I can recognize integers  Rectangle: Perimeter = 2L + 2W I can add and subtract, multiply and divide integers  Rectangle: Area = LW I can recognize and fluently read exponents  I can read and generate equivalent expressions with Rectangular Prism: Volume = LWH exponents  Area of a Triangle: A = ½ LW or ½ BH I can identify the laws of exponents including  xm + x m = 2xm multiplication,  xm • xn = xm+n division, power of a power, and zero exponents  (xm)n = xm•n I can apply the laws of exponents when multiplying and dividing  (xm)/(xn) = xm-n  0 like and unlike bases x = 1 I can convert bases with negative exponents to  -n -n x = 1/x fractions  (a + b)2 = a2 + 2ab + b2 and vice versa I can simplify algebraic expressions, involving zero  (a – b)2 = a2 – 2ab + b2 and vice versa exponents  (a – b)(a + b) = a2 – b2 and vice versa I can simplify algebraic expressions, involving negative exponents I can simplify algebraic expressions, by applying the Vocabulary: multiplication, Area, Base, Binomial, Coefficient, Constant, power, division properties of exponents and a Difference of squares, Distribute, Equivalent, combination of them. Evaluate, Exponent, Expression, Factor, Like Terms, 8.EE.2 Monomial, Perimeter, Polynomial, Product , Radical, I can read and define perfect squared and cubed Unit Resources numbers MARS Concept Development Lessons: I can define square and cube root  Interpreting Algebraic Expressions I can solve square and cube root equations  Sorting Equations and Identities I can understand that non-perfect squares and cubes are irrational  Manipulating Polynomials I can recognizing the inverse operation of squared is Projects: square rooting  Polynomial Project I can recognizing the inverse operation of cubed is cube  Area and Volume Project rooting Unit Review Game: I can define and recognize a rational number  Polynomial Jeopardy (powerpoint) I can define and recognize an irrational number I can evaluate perfect squares thru 144 fluently Released test items: #3, #5, #33, I can evaluate perfect cube roots thru 125 fluently I can use prime factorization to find the cube root of a positive number Mathematical Practices in Focus: I can recall the perfect squares and perfect cubes of 2. Reason abstractly and quantitatively numbers less 6. Attend to precision than or equal to 100 7. Look for and Make Use of Structure 7.G.6 8. Look for and express regularity in repeated I can identify types of triangles/quadrilaterals/polygons. reasoning I can solve real-world problems involving the area of triangles, StandardsCCSS-M are Included listed in alphabetical: /numerical order not suggested teaching quadrilaterals, and other polygons. order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Course Name: Foundations of Math I Unit # 2 Unit Title: Exponen ts and Polynomials

CORE CONTENT Cluster Title: Expressions and Equations: Work with radicals and integer exponents. Standard: 8.EE.1: Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example: 3² x = = =

Concepts and Skills to Master:  Know the properties of integer exponents.  Apply the properties of integer exponents to simplify and evaluate numerical expressions SUPPORTS FOR TEACHERS Critical Background Knowledge  Understand exponents as repeated multiplication. (6.EE.1)  Compute fluently with integers (add, subtract, and multiply). Academic Vocabulary • Exponent • base • power • integer

Suggested Instructional Strategies: Resources:  Use repeated multiplication and division to  CMP2 Growing, Growing, Growing Unit informally derive the exponent rules. (online codes can be found on the intranet) . Investigations 5  Have students examine equivalent o Common Core Investigation (CMP2) numerical expressions with exponents. . Investigation 1.1 Negative Exponent  Write the patterns for each rule on the board and challenge students to come up  Algebra textbook 7-1, 7-3, 7-4, 7-5 (online with the rule: codes can be found on the intranet)

 MARS AssessmentTask (HS): E06: “Ponzi” Pyramid Schemes

 Texas Instrument 8.EE.1 Lessons

 CMP2 Resources

Sample Assessment Tasks Skill-based Task Problem Task

Simplify ( ( (2)²)²)² * 4 Explain why * = and not .

Write three expressions equivalent to * .

Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Course Name: Foundations of Math I Unit # 2 Unit Title: Exponen ts and Polynomials

CORE CONTENT Cluster Title: Expressions and Equations: Work with radicals and integer exponents. Standard: 8.EE.2:Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

Concepts and Skills to Master:  Evaluate the square roots of small, perfect squares and cube roots of small perfect cubes.  Represent the solutions to equations using square root and cube root symbols.  Understand that all non-perfect square roots and cube roots are irrational.

SUPPORTS FOR TEACHERS Critical Background Knowledge  Understand and use inverse operations to solve equations.  Use and understanding of repeating multiplication  Order of operations Academic Vocabulary • Square • square root • cube • cube root • radical

Suggested Instructional Strategies: Resources:  Use the geometric representations of square  CMP2 Looking for Pythagoras Unit (codes can area and cube volumes and their relation to be found on the intranet) the side length. . Investigations 2, 3, and 4

 Use the idea of inverse operations to  Squares, Square Roots and Exponential introduce the concept of roots. Expressions

 Have students draw squares with side  MARS Formative Assessment Lesson (MS): lengths 1, 2, 3, 4, and 5 on graph paper. The Pythagorean Theorem: Square Areas Have them count the squares to find the area of each. This will help students with  CMP2 Resources conceptual understanding of perfect squares. Explain that the root number is the  Texas Instrument 8.EE.2 Lessons side length of the square.

Sample Assessment Tasks

Skill-based Task Problem Task

If a square has an area of 9/16 square inches, what Is the square root of a number always smaller than is the length of a side? the number itself? Explain. If a cube has a volume of 0.125 cubic meters, what are the dimensions of the cube?

CORE CONTENT Cluster Title: Draw, construct, and describe geometrical figures and describe the relationships between them. Standard: 7.G.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. Concepts and Skills to Master:  Determine the height and base area of three dimensional objects composed of triangles, quadrilaterals, polygons, cubes and right prisms necessary to demonstrate understanding of volume.  Find perimeters and areas of two dimensional figures using polynomial expressions  Find volumes of cubes and boxes using polynomial expressions. SUPPORTS FOR TEACHERS Critical Background Knowledge  Properties of exponents  Perimeter, area, volume formulas  Combining like terms  Properties of real numbers  Properties of squares, rectangles, cubes, and rectangular boxes. Academic Vocabulary • Cubic units • square units • volume • area • perimeter

Suggested Instructional Strategies: Resources:

 Have students draw squares and  CMP2 Filling and Wrapping Unit (online codes rectangles on a piece of graph paper. Use can be found on the intranet). the square units to find the perimeter and  http://www.mathplayground.com/area_perimet area of each. Have the students make a er.html conjecture for the formula of each. Video lessons: Repeat for volume of boxes.  http://www.mathplayground.com/howto_samea readiffperimeter.html  http://www.mathplayground.com/mv_volume_p risms.html  MARS Tasks: Fearless Frames:

http://map.mathshell.org/materials/tasks.php? taskid=277#task277

Sample Assessment Tasks Skill-based task Problem Task

1. Find the perimeter and area of a square The 7th graders at Sunview Middle School were helping with a side length of “x”. to renovate a playground for the kindergartners at a 2. Find the volume of a box with height of 2 nearby elementary school. City regulations require inches, a width of “x” inches and a length that the sand underneath the swings be at least 15 of “x2” inches. inches deep. The sand under both swing sets was only 12 inches deep when they started.

The rectangular area under the small swing set measures 9 feet by 12 feet and required 40 bags of sand to increase the depth by 3 inches. How many bags of sand will the students need to cover the rectangular area under the large swing set if it is 1.5 times as long and 1.5 times as wide as the area under the small swing set?

(illustrativemathematics.com)

CORE CONTENT Cluster Title: Perform Arithmetic Operations on Polynomials Standard: A.APR.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Concepts and Skills to Master:  Look for and make use of the structure of addition and subtraction extended to linear and quadratic polynomials  Understand the Closure Property for addition, subtraction, and multiplication of polynomials (adding/subtracting/multiplying polynomials will give you another polynomial. Division is not included because the answer is sometime rational (NCDPI)).  Add, subtract, multiply polynomials (limit to linear and quadratic functions) SUPPORTS FOR TEACHERS Critical Background Knowledge  Distributive property  Exponent properties

 Formulas for perimeter and area of two-dimensional figures  Formula for volume of a cube and rectangular prism (box) Academic Vocabulary • Polynomial • monomial • trinomial • binomially • linear terms • quadratic terms Suggested Instructional Resources: Strategies:  Algebra I textbook (online codes can be found on the intranet): 8-1, 8-2, 8-3, 8-4 Discoveryeducation.com video:  Discoveryeducation.com: Algebra: multiplying Algebra: Exponents- polynomials Have students use individual white http://app.discoveryeducation.com/player/view/assetGu boards to do problems on the video. id/0D66EF0B-5B78-4918-BDEB-93135ECFC174 (see an administrator for passcode) Use Algeblocks/tiles to help visualize adding, subtract, multiplying polynomials. Discoveryeducation.com: Tile Trials: Modeling and Adding Polynomials Sample Assessment Tasks Skill-based task Problem Task A square green rug has a blue square in You are given the task of creating a rug to fit into a square the center. The side length of the blue room. You must design the rug so that you create a square in square is x inches. The width of the the middle. In order for the rug to fit perfectly it must have an green band that surrounds the square is area of 49 – 4x2. What is the length of the sides of the outer 6in. What is the area of the green band? square and the inner square of the rug?

If the radius of a circle is 5x – 2 kilometers, what would the area of the circle be?

Explain why (4x + 3)2 does not equal 16x2 + 9

CORE CONTENT Cluster Title: Interpret the Structure of Expressions Standard: A.SSE.2: Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

Concepts and Skills to Master:  Rewrite an algebraic expression in different forms such as using different properties of real numbers, factoring or combining like terms.

 Use factoring techniques such as common factors, the difference of two squares, the sum or difference of cubes, or a combination of methods to “undo” the distributive property. SUPPORTS FOR TEACHERS Critical Background Knowledge  Properties of exponents  Adding, subtracting, multiplying polynomials  Properties of real numbers  Distributive property  Greatest common factor  Perfect squares Academic Vocabulary  Perfect square trinomial  Factors  Equivalent expressions Suggested Instructional Strategies: Resources:

 Use algeblocks/algebratiles to explore  Algebra I textbook (online codes can be found structure of expressions. on the intranet): 8-2, 8-7, CC-12  Discoveryeducation.com : Factoring: Part 1: http://app.discoveryeducation.com/player/vie w/assetGuid/1E47576E-CBE2-4C79-AA33- 9B1116D8B5BE  MARS task: Circle pattern: http://map.mathshell.org/materials/tasks.php? taskid=253#task253 Sample Assessment Tasks Skill-based task Problem Task

Factor 6x2 + 2x Expand the expression 2(x – 1)2 – 4 to show that it is a quadratic expression of the form ax2 + bx + c.

Smarter Balanced Problem task: http://www.ode.state.or.us/wma/teachlearn/commoncore/ mat.hs.cr.2.0asse.a.005_v1.pdf

Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.