Unit 8 Table of Contents Unit 8: Polynomials Video Overview Learning Objectives 8.2 Media Run Times 8.3 Instructor Notes 8.4 The Mathematics of Monomials and Polynomials Teaching Tips: Conceptual Challenges and Approaches Teaching Tips: Algorithmic Challenges and Approaches Instructor Overview 8.8 Tutor Simulation: Roman Numerals and Polynomials Instructor Overview 8.9 Puzzle: Polynomial Poke Instructor Overview 8.11 Project: It's All Fun and Games Glossary 8.16 Common Core Standards 8.17 Some rights reserved. Monterey Institute for Technology and Education 2011 V1.1 "#$
Unit 8 Learning Objectives Unit 8: Polynomials Lesson 1: Operations on Monomials Learning Objectives Topic 1: Multiplying and Dividing Monomials Learning Objectives Multiply and divide monomials. Lesson 2: Operations on Polynomials Topic 1: Polynomials Learning Objectives Identify monomials, binomials and polynomials. Write polynomials to describe real world situations. Topic 2: Adding and Subtracting Polynomials Learning Objectives Add and subtract polynomials. Topic 3: Multiplying polynomials Learning Objectives Multiply polynomials and collect the like terms of the resulting sum of monomials. Topic 4: Special Products of Polynomials Learning Objectives Identify and multiply binomial products. L e ar ni n g O bj e ct iv e s "#%
Unit 8 Media Run Times Unit 8 Lesson 1 Topic 1, Presentation 4 minutes Topic 1, Worked Example 1 1.8 minutes Topic 1, Worked Example 2 4 minutes Topic 1, Worked Example 3 2.3 minutes Lesson 2 Topic 1, Presentation 3.8 minutes Topic 1, Worked Example 1 4.6 minutes Topic 1, Worked Example 2 1.3 minutes Topic 2, Presentation 4.4 minutes Topic 2, Worked Example 1 2 minutes Topic 2, Worked Example 2 1.7 minutes Topic 2, Worked Example 3 2.4 minutes Topic 3, Presentation 1 2.7 minutes Topic 3, Worked Example 1 4.5 minutes Topic 3, Worked Example 2 4.7 minutes Topic 3, Worked Example 3 4.3 minutes Topic 4, Presentation 5.3 minutes Topic 4, Worked Example 1 3.3 minutes Topic 4, Worked Example 2 5 minutes Topic 4, Worked Example 3 2.5 minutes "#&
Unit 8 Instructor Notes Unit 8: Polynomials Instructor Notes The Mathematics of Monomials and Polynomials Unit 8 introduces polynomials and teaches students how to work with them no matter how many terms they contain (in this course, monomials are included in the definition of polynomials). Students will learn how to carry out all the basic mathematical operations on polynomials. They ll also gain experience writing polynomials from verbal descriptions of real world situations. The ability to work fluently with polynomials will be critical for students who progress into higher math classes like Algebra 2 and beyond. Teaching Tips: Conceptual Challenges and Approaches Working with polynomials can present a significant challenge for many students. Most of the mathematics concerns symbolic manipulation, and if students don t build a meaningful conceptual understanding of how and why the techniques work, they will get lost as the terms become more numerous and complicated. Using a visual model can be very helpful when working with polynomials. Try beginning with terms that describe real objects, which can be sketched out and then manipulated and counted up. "#'
Example In the presentation for Lesson 2, Topic 3, the concept of multiplying two polynomials is introduced through the visual of planting a garden. Students see the result of multiplying polynomials as an understandable collection of objects instead of just symbols. They can count the various vegetables, compare them to the original terms, and by doing so, learn to appreciate what actually happens when polynomials are multiplied. Hands-On Opportunities The example above used the area model as a basis for understanding polynomial multiplication. Students can use virtual algebra tiles for further practice of this technique. One of the most useful virtual manipulative websites can be found here [MAC users will need to copy/paste url into browser]: http://courses.wccnet.edu/~rwhatcher/vat/simplifyingpolynomials/ Sketches and manipulatives are powerful tools that can help students build understanding and practice techniques. However, there are two very important ideas to keep in mind: 1. Students will need significant guidance to understand manipulatives, especially in the beginning. We suggest demonstrating any virtual tool in the classroom, either just by projecting the image and solving the problem on the computer, or better still, by using an interactive whiteboard and showing students how to solve this "#(
problem. Students could also use the board to demonstrate their ideas about how to take next steps. After seeing it done in the classroom, they ll be in a position to work with a tool like this either alone or in small groups. 2. Visual representation tools help students get started, but they are not an alternative method for ultimately doing the mathematics. Students still need to be fluent with the relevant symbolic manipulation. It is very important to discuss the connections between a visual model and its symbolic counterpart when working with polynomials. Teaching Tips: Algorithmic Challenges and Approaches It s tempting to teach students tricks for memorizing algebraic techniques. A lot of traditional teaching materials suggest using the acronym FOIL as a mnemonic device to remember how to multiply two binomials. While there is nothing inherently wrong with this memorization approach, it does have limitations. This mnemonic only works when multiplying two binomials when students who have grown comfortable with it are confronted by a more challenging situation like x + y + 2 ( )( 3 2x), they ll often struggle. As a result, it is more productive to teach a more general rule from the beginning: When multiplying two polynomials, multiply everything in the first parenthesis to everything in the second parenthesis. That approach is used exclusively in this course. If students are struggling to keep track of all of the terms when multiplying polynomials, they may find it useful to create a rectangular table (which is obviously connected to the visual area model) to diagram this operation. Example ( x + y + 2) ( 3 2x) x y 2 3 3x 3y 6 2x 2x 2 2xy 4x Once students have completed the multiplication, they can easily collect the like terms and find the answer. "#)
Summary This unit focuses on the addition, subtraction, multiplication, and division of polynomials. It uses general rules and visual models to explain the conceptual basis and the procedures involved in these operations. Students who struggle with these ideas may benefit from virtual or hands-on manipulatives, but they must learn how to carry out strictly symbolic manipulations. "#*
Unit 8 Tutor Simulation Unit 8: Polynomials Instructor Overview Tutor Simulation: Roman Numerals and Polynomials Purpose This simulation is designed to challenge a student s understanding of polynomials. Students will be asked to apply what they have learned to solve a real world problem by demonstrating understanding of the following areas: Polynomials Multiplying Polynomials The Distributive Property The Associative Property The Commutative Property Applying Properties to Polynomials Problem Students are given the following problem: You will take a look at Roman numerals and see how working with them is similar to working with polynomials. Once familiar with Roman numerals, you'll learn how to multiply them, then apply the same steps to multiply polynomials. Recommendations Tutor simulations are designed to give students a chance to assess their understanding of unit material in a personal, risk-free situation. Before directing students to the simulation, make sure they have completed all other unit material. explain the mechanics of tutor simulations o Students will be given a problem and then guided through its solution by a video tutor; o After each answer is chosen, students should wait for tutor feedback before continuing; o After the simulation is completed, students will be given an assessment of their efforts. If areas of concern are found, the students should review unit materials or seek help from their instructor. emphasize that this is an exploration, not an exam. "#"
Unit 8 Puzzle Unit 8: Polynomials Instructor Overview Puzzle: Polynomial Poke Objective Polynomial Poke challenges students' familiarity with polynomial nomenclature. To play the game successfully, they must be able to distinguish between cubic, quadratic, and linear terms, and recognize monomials, binomials, and trinomials. Figure 1. Polynomial Poke asks players to pop balloons that contain specified types of polynomials. "#+
Description There are three levels in this puzzle, which each consist of 10 groups of floating balloons containing polynomials. In the first level, learners are challenged to pop balloons in order of degree of monomials, from cubic to quadratic to linear. In the second level, players must pop balloons depending on the number of terms in their polynomials. In the third level, players are asked to pop only those balloons that contain a specified degree of polynomial. Players earn points for correct answers and lose points for popping balloons out of sequence. The puzzle is primarily designed for a single player but in a classroom it could be played in a group with learners identifying the order or the degree and calling out the balloon for one to pop. "#$,
Unit 8 Project Unit 8: Polynomials Instructor Overview Project: It's All Fun and Games Student Instructions Introduction In algebra variables are used to represent unknowns. When first beginning algebra, the symbolic representation can be difficult. By now you should be quite comfortable with x and y, as symbols for unknowns, however, the roots of mathematics are engrained in complex symbol-based number systems. Get ready to explore these ancient systems and become an expert at interpreting and using the symbols found within Task Working together with your group, you will research one of four ancient number systems. Then, based upon what you have learned, you will design a team game based on that number system. The game needs to be complete with rules, scoring guidelines, and dimensions of the field based on the ancient number system. Finally, you will calculate the perimeter of your field using the number system. Instructions Solve each problem in order. Save your work along the way, as you will create a presentation at the conclusion of the project. Your audience will be the Mayan, Egyptian, Sumerian, or Roman people. You may use multi-media, make a movie, or create a website to highlight your game and how it connects to the number system. 1 First problem: With your group, choose one of the ancient number systems below to research. You may use the following links to begin, but there are multiple websites dedicated to the number systems. Egyptian: http://www.touregypt.net/featurestories/numbers.htm Mayan: http://www-history.mcs.stand.ac.uk/histtopics/mayan_mathematics.html Roman: http://turner.faculty.swau.edu/mathematics/materialslibrary/roman/ Sumerian: http://www.storyofmathematics.com/sumerian.html "#$$
Use the following questions to guide your research: How do you write the following numbers: 1, 5, 10, 20, 50, & 100? What is the base of the number system? How has the number system impacted the base ten system that is used today? How do you perform basic addition using the number system? 2 Second problem: Once your group has a good understanding of the number system, begin thinking about games that are played today and how they might be adapted to fit with the number system. For instance, consider football. The field is based on 100 yards and advancing the ball is based on ten-yard increments. This game would fit well for the Egyptians and Romans, but not for the Sumerians and Mayan. Creativity and originality will make your presentation stand out. A good example of a made-up game is Quidditch from the Harry Potter book series. Information about the fictional game, including rules and field dimensions can be found at: http://en.wikipedia.org/wiki/quidditch First, decide the dimensions of your field. Be sure to keep in mind the foundation of your number system when making your decisions. Your dimensions should be written using the symbols from the system. You can draw a picture of your field using drafting software such as, Google Sketchup. The free download is available at http://sketchup.google.com/. If you have an artistic flair, a hand-drawn field or court is another option. 3 Third Problem: Now your team needs to work on developing rules and scoring guidelines for your game. What tools are necessary to play? How many points are various tasks worth? How many points are necessary to win? How many players are on the field at once? Make sure to consider how the answers to each of these questions would be impacted based on the number system that is studied. Hint: (Remember that you will be presenting the game at the end of the project. It may help to actually go outside and attempt to play the game in order to discover what works and what does not.) 4 Fourth problem: Your final task is to calculate the perimeter of the field, using the number system studied. Begin by learning to add small numbers within the system and then work your way to larger numbers. You will need to include the detailed process that was used to add the perimeter as part of your presentation. "#$%
Collaboration Get together with another group to discuss your game and how it is played. Discuss how the rules and field dimensions relate to what you have studied about the number system. Finally, work together to check each other s perimeter calculations. While reviewing the perimeter calculations, answer the following questions: How is adding within your number system related to adding polynomials? Do you see any other ties to algebra within your calculations? Conclusions Now you will get to present your game using modern technology to your classmates, who will be considered citizens of the ancient civilization that you researched. Some options for the presentation include using multi-media, making a movie, or creating a website to highlight your game and how it connects to the number system. Your presentation should include answers to each of the four problems above. Instructor Notes Assignment Procedures Problem 1 It is important for students to master basic addition and regrouping within the number system they have chosen before moving on. By the end, each group will calculate the perimeter of the playing field. Without a solid understanding of basic addition within their system, completing more difficult calculations will not be possible. Problem 4 If a group chooses to make their field LXV by XXV, the calculation for perimeter would be: By Addition: LXV + LXV + XXV + XXV By combining like terms: LLXXXXXXVVVV By simplifying: LL = C and VVVV = XX By substitution: CXXXXXXXX By simplifying: XXXXX=L By substitution: CLXXX By showing their work and justifying each step, the students should be able to see how addition within the number system relates very closely to adding polynomials in algebra. Recommendations: have students work in teams to encourage brainstorming and cooperative learning. assign a specific timeline for completion of the project that includes milestone dates. provide students feedback as they complete each milestone. ensure that each member of student groups has a specific job. "#$&
Technology Integration This project provides abundant opportunities for technology integration, and gives students the chance to research and collaborate using online technology. The students instructions list several websites that provide information on numbering systems, game design, and graphics. The following are other examples of free Internet resources that can be used to support this project: http://www.moodle.org An Open Source Course Management System (CMS), also known as a Learning Management System (LMS) or a Virtual Learning Environment (VLE). Moodle has become very popular among educators around the world as a tool for creating online dynamic websites for their students. http://www.wikispaces.com/site/for/teachers or http://pbworks.com/content/edu+overview Allows you create a secure online Wiki workspace in about 60 seconds. Encourage classroom participation with interactive Wiki pages that students can view and edit from any computer. Share class resources and completed student work with parents. http://www.docs.google.com Allows students to collaborate in real-time from any computer. Google Docs provides free access and storage for word processing, spreadsheets, presentations, and surveys. This is ideal for group projects. http://why.openoffice.org/ The leading open-source office software suite for word processing, spreadsheets, presentations, graphics, databases and more. It can read and write files from other common office software packages like Microsoft Word or Excel and MacWorks. It can be downloaded and used completely free of charge for any purpose. Rubric Score Content Presentation 4 Your project appropriately answers each of the problems. Background research is thorough. A detailed drawing of the game field is given, with dimensions labeled using the number system. Rules and scoring guidelines are complete and relate to the number system studied. A detailed calculation of perimeter, in the number system, is included. 3 Your project appropriately answers each of the problems. Background research is thorough. A detailed drawing of the game field is given, with dimensions labeled using the Your project contains information presented in a logical and interesting sequence that is easy to follow. Your project is professional looking with graphics and attractive use of color. Your project contains information presented in a logical sequence that is easy to follow. Your project is neat with graphics and attractive use of color. "#$'
number system. Minor errors may be noted. Rules and scoring guidelines are complete and relate to the number system studied. A detailed calculation of perimeter, in the number system, is included. Minor errors may be noted. 2 Your project attempts to answer each of the problems. Background research is present, but not complete. A drawing of the game field is given, with dimensions labeled using the number system. Major errors may be noted &/or some information is missing. Rules and scoring guidelines present and relate to the number system studied. The perimeter is given, but the detailed work used to obtain the answer is not given. Major errors may also be noted. 1 Your project attempts to answer some of the problems. Major errors are noted and information is missing. Your project has minimal information on rules and scoring guidelines. The perimeter calculation is missing. Your project is hard to follow because the material is presented in a manner that jumps around between unconnected topics. Your project contains low quality graphics and colors that do not add interest to the project. Your project is difficult to understand because there is no sequence of information. Your project is missing graphics and uses little to no color. "#$(
Unit 8 Glossary Glossary Unit 8: Algebra - Polynomials area model a graphic representation of a multiplication problem, in which the length and width of a rectangle are the factors and the area is the product binomial a sum of two monomials, such as 3x 2 + 7 coefficient like terms monomial a number that multiplies a variable two or more monomials that contain the same variables raised to the same powers, regardless of their coefficients. For example, 2x2y and -8x2y are like terms because they have the same variables raised to the same exponents. a number, a variable, or a product of a number and one or more variables with whole number exponents, such as -5, x, and 8xy3 polynomial a monomial or sum of monomials, like 4x2 + 3x 10 special product term a product resulting from binomial multiplication that has certain characteristics. For example x2 25 is called a special product because both its terms are perfect squares and it can be factored into (x + 5)(x 5). a value in a sequence--the first value in a sequence is the 1st term, the second value is the 2nd term, and so on; a term is also any of the monomials that make up a polynomial "#$)
Unit 8 Common Core NROC Algebra 1--An Open Course Unit 8 Mapped to Common Core State Standards, Mathematics Algebra 1 Polynomials Operations on Monomials Multiplying and Dividing Monomials Grade: 9-12 - Adopted 2010 CATEGORY / CLUSTER A-SSE. Seeing Structure in Expressions Interpret the structure of expressions. EXPECTATION A-SSE.2. Use the structure of an expression to identify ways to rewrite it. For example, see x^4 - y^4 as (x^2)^2 - (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 - y^2)(x^2 + y^2). Algebra 1 Polynomials Operations on Polynomials Polynomials Grade: 9-12 - Adopted 2010 CATEGORY / CLUSTER A-SSE. Seeing Structure in Expressions Interpret the structure of expressions. EXPECTATION A-SSE.1. Interpret expressions that represent a quantity in terms of its context. GRADE EXPECTATION A-SSE.1.a. Interpret parts of an expression, such as terms, factors, and coefficients. CATEGORY / CLUSTER A-CED. Creating Equations Create equations that describe numbers or relationships. EXPECTATION A-CED.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. EXPECTATION A-CED.3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling "#$*
context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. STRAND / DOMAIN CC.F. Functions CATEGORY / CLUSTER F-BF. Building Functions Build a function that models a relationship between two quantities. EXPECTATION F-BF.1. Write a function that describes a relationship between two quantities. GRADE EXPECTATION F-BF.1.a. Determine an explicit expression, a recursive process, or steps for calculation from a context. Algebra 1 Polynomials Operations on Polynomials Adding and Subtracting Polynomials Grade: 9-12 - Adopted 2010 CATEGORY / CLUSTER A-SSE. Seeing Structure in Expressions Interpret the structure of expressions. EXPECTATION A-SSE.1. Interpret expressions that represent a quantity in terms of its context. GRADE EXPECTATION A-SSE.1.a. Interpret parts of an expression, such as terms, factors, and coefficients. CATEGORY / CLUSTER A-SSE. Seeing Structure in Expressions Interpret the structure of expressions. EXPECTATION A-SSE.2. Use the structure of an expression to identify ways to rewrite it. For example, see x^4 - y^4 as (x^2)^2 - (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 - y^2)(x^2 + y^2). CATEGORY / CLUSTER A-APR. Arithmetic with Polynomials and Rational Functions Perform arithmetic operations on polynomials. EXPECTATION A-APR.1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, "#$"
subtraction, and multiplication; add, subtract, and multiply polynomials. Algebra 1 Polynomials Operations on Polynomials Multiplying Polynomials Grade: 7 - Adopted 2010 STRAND / DOMAIN CC.7.EE. Expressions and Equations CATEGORY / CLUSTER Use properties of operations to generate equivalent expressions. 7.EE.1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. Grade: 9-12 - Adopted 2010 CATEGORY / CLUSTER A-SSE. Seeing Structure in Expressions Interpret the structure of expressions. EXPECTATION A-SSE.1. Interpret expressions that represent a quantity in terms of its context. GRADE EXPECTATION A-SSE.1.a. Interpret parts of an expression, such as terms, factors, and coefficients. CATEGORY / CLUSTER A-SSE. Seeing Structure in Expressions Interpret the structure of expressions. EXPECTATION A-SSE.2. Use the structure of an expression to identify ways to rewrite it. For example, see x^4 - y^4 as (x^2)^2 - (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 - y^2)(x^2 + y^2). CATEGORY / CLUSTER A-APR. Arithmetic with Polynomials and Rational Functions Perform arithmetic operations on polynomials. EXPECTATION A-APR.1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. "#$+
Algebra 1 Polynomials Operations on Polynomials Special Products of Polynomials Grade: 7 - Adopted 2010 STRAND / DOMAIN CC.7.EE. Expressions and Equations CATEGORY / CLUSTER Use properties of operations to generate equivalent expressions. 7.EE.1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. Grade: 9-12 - Adopted 2010 CATEGORY / CLUSTER A-SSE. Seeing Structure in Expressions Interpret the structure of expressions. EXPECTATION A-SSE.1. Interpret expressions that represent a quantity in terms of its context. GRADE EXPECTATION A-SSE.1.a. Interpret parts of an expression, such as terms, factors, and coefficients. CATEGORY / CLUSTER A-APR. Arithmetic with Polynomials and Rational Functions Perform arithmetic operations on polynomials. EXPECTATION 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. 2011 EdGate Correlation Services, LLC. All Rights reserved. 2010 EdGate Correlation Services, LLC. All Rights reserved. Contact Us - Privacy - Service Agreement "#%,