Algebra I Part 2
Algebra I Part 2 Table of Contents Unit : Exponents, Expressions and Equations... - Unit 2: Linear Equations and Graphs... 2- Unit 3: Systems of Equations and Inequalities... 3- Unit 4: Data Analysis... 4- Unit 5: Probability... 5- Unit 6: Relations, Functions, and Sequences... 6- Unit 7: Polynomials and Factoring... 7- Unit 8: Quadratics... 8- Most of the math symbols in this document were made with Math Type software. Specific fonts must be installed on the user s computer for the symbols to be read. Users can download and install the Math Type for Windows Font from http://www.dessci.com/en/dl/fonts/default.asp on each computer on which the document will be used.
202 Louisiana Transitional Comprehensive Curriculum Course Introduction The Louisiana Department of Education issued the first version of the Comprehensive Curriculum in 2005. The 202 Louisiana Transitional Comprehensive Curriculum is aligned with Grade-Level Expectations (GLEs) and Common Core State Standards (CCSS) as outlined in the 202-3 and 203-4 Curriculum and Assessment Summaries posted at http://www.louisianaschools.net/topics/gle.html. The Louisiana Transitional Comprehensive Curriculum is designed to assist with the transition from using GLEs to full implementation of the CCSS beginning the school year 204-5. Organizational Structure The curriculum is organized into coherent, time-bound units with sample activities and classroom assessments to guide teaching and learning. Unless otherwise indicated, activities in the curriculum are to be taught in 202-3 and continued through 203-4. Activities labeled as 203-4 align with new CCSS content that are to be implemented in 203-4 and may be skipped in 202-3 without interrupting the flow or sequence of the activities within a unit. New CCSS to be implemented in 204-5 are not included in activities in this document. Implementation of Activities in the Classroom Incorporation of activities into lesson plans is critical to the successful implementation of the Louisiana Transitional Comprehensive Curriculum. Lesson plans should be designed to introduce students to one or more of the activities, to provide background information and follow-up, and to prepare students for success in mastering the CCSS associated with the activities. Lesson plans should address individual needs of students and should include processes for re-teaching concepts or skills for students who need additional instruction. Appropriate accommodations must be made for students with disabilities. Features Content Area Literacy Strategies are an integral part of approximately one-third of the activities. Strategy names are italicized. The link (view literacy strategy descriptions) opens a document containing detailed descriptions and examples of the literacy strategies. This document can also be accessed directly at http://www.louisianaschools.net/lde/uploads/056.doc. Underlined standard numbers on the title line of an activity indicate that the content of the standards is a focus in the activity. Other standards listed are included, but not the primary content emphasis. A Materials List is provided for each activity and Blackline Masters (BLMs) are provided to assist in the delivery of activities or to assess student learning. A separate Blackline Master document is provided for the course. The Access Guide to the Comprehensive Curriculum is an online database of suggested strategies, accommodations, assistive technology, and assessment options that may provide greater access to the curriculum activities. This guide is currently being updated to align with the CCSS. Click on the Access Guide icon found on the first page of each unit or access the guide directly at http://sda.doe.louisiana.gov/accessguide.
Algebra I Part 2 Unit : Exponents, Expressions and Equations Time Frame: 3 weeks Unit Description This introductory unit consists of a review and extension of topics taught in Algebra I Part. Topics include work with exponents, order of operations, writing and evaluating expressions, and solving equations. Student Understandings Students solve equations in one variable and are able to rearrange a formula for a given quantity. Students write, evaluate, and simplify algebraic expressions. In addition, students understand and use the properties of exponents (including work with negative and rational exponents). Guiding Questions. Can students use the properties of exponents with both positive and negative exponents (including rational exponents)? 2. Can students use order of operations when evaluating numeric and algebraic expressions? 3. Can students write and evaluate numeric and algebraic expressions given a verbal description? 4. Can students solve equations in one variable for an unknown quantity (including rearranging formulas to isolate a quantity)? Unit Grade-Level Expectations (GLEs) and Common Core State Standards (CCSS) Grade-Level Expectations GLE# GLE Text and Benchmarks Number and Number Relations 2. Evaluate and write numerical expressions involving integer exponents (N-2-H) Algebra 8. Use order of operations to simplify or rewrite variable expressions (A--H) (A- 2-H) 9. Model real-life situations using linear expressions, equations, and inequalities (A--H) (D-2-H) (P-5-H) Algebra I Part 2 Unit Exponents, Expressions and Equations -
Grade-Level Expectations GLE# GLE Text and Benchmarks. Use equivalent forms of equations and inequalities to solve real-life problems (A--H) CCSS for Mathematical Content CCSS # CCSS Text The Real Number System N-RN. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. Reasoning with Equations and Inequalities A-REI. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Seeing Structure in Expressions A-SSE. Interpret expressions that represent a quantity in terms of its context. Creating Equations A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R. ELA CCSS CCSS # CCSS Text Reading Standards for Literacy in Science and Technical Subjects 6-2 RST.9-0. Cite specific textual evidence to support analysis of science and technical texts, attending to the precise details of explanations or descriptions. RST.9-0.4 Determine the meaning of symbols, key terms, and other domainspecific words and phrases as they are used in a specific scientific or technical context relevant to grades 9-0 texts and topics. Writing Standards for Literacy in History/Social Studies, Science and Technical Subjects 6-2 WHST.9-0.b Write arguments focused on discipline-specific content. Develop claim(s) and counterclaims fairly, supplying data and evidence for each while pointing out the strengths and limitations of both claim(s) and counterclaims in a discipline-appropriate form and in a manner that anticipates the audience s knowledge level and concerns. WHST.9-0.0 Write routinely over extended time frames (time for reflection and revision) and shorter time frames (a single sitting or a day or two) for a range of discipline-specific tasks, purposes, and audiences. Algebra I Part 2 Unit Exponents, Expressions and Equations -2
Sample Activities Activity : Exponents (GLE: 2) Materials List: paper, pencil, calculators, textbook In Algebra I Part, students worked with exponents with both whole number and integer exponents. This activity is a chance to review what was learned in that course and extend their understanding of exponents in math to see the relationship between negative exponents and 3 rewriting a number into a fraction using positive exponents (i.e., 4 = ). 3 4 Begin the lesson by asking students to consider the following questions and have them write down everything they can remember about what they know about exponents using the questions as a thought provoker to promote discussion. Question : What is an exponent and when are they used in math? Question 2: Are exponents always positive? Explain. After students have had time to think and write individually, have them get into small groups to share what they thought about each of the questions. Afterwards, find out from the groups what their thoughts were in reference to each of the questions presented. Use the questions and student responses to guide students to understand all of the following concepts: Exponents are used in math to represent repeated multiplication situations which may come up in math such as 4 x 4 x 4 which would be written using exponents as 4³ where 4 is the base or the number being multiplied repeatedly and 3 is the exponent or the power. The exponent refers to the number of times that the base is being multiplied. Some of the math topics that include the use of exponents are scientific notation, prime factorization, and geometry (area and volume formulas). o Teacher Note: Use this time as an opportunity to review using a calculator to x determine an exponent value (i.e. using the ^ or y keys). Make sure that students fully understand how to use exponents with whole numbers. Provide students extra practice on this skill using a math textbook or some other resource. Exponents are not always positive. They can be negative as well. This is evident in their use for very small numbers written in scientific notation. For example, the number 0.000034 would be written as 3.4 x 0 5. o Teacher Note: Scientific notation is also something that students should have been taught in previous courses. Use this time to make sure students fully understand how to write numbers in scientific notation using both positive and negative exponents. Provide students extra practice on this skill as needed using a math textbook or some other resource. Algebra I Part 2 Unit Exponents, Expressions and Equations -3
The fact that exponents are not always positive will be new to students, but their understanding needs to be extended beyond just the use of negative exponents to writing numbers in scientific notation. Present the following question to students and ask them to get in their groups to determine what they think the solution is. Discuss as a class afterwards. Problem: What is the value of 2? 5 o Teacher Note: Students should see that this is the same as /25 or.04. After students discuss their answers, ask students to use their calculators to find the value to 5 2 x using the y key. Students should see that both 5 2 and 2 have 5 the same value (0.04). Lead students to understand that 5 2 is actually equivalent to 2 and that anytime there is a number written with a negative 5 exponent that it can be rewritten as a positive exponent but in fractional form. Do more examples with students and make sure students understand how to evaluate negative exponents (with and without calculators) and how to rewrite these numbers as fractions with positive exponents. Use the textbook for practice problems or some other resource. Activity 2: Rational Exponents (GLE: 2; CCSS: N-RN.; WHST.9-0.b, WHST.9-0.0) Materials: paper, pencil, calculators In this lesson, continue to extend student understanding of exponents to include rational exponents. Write the following statement/prompt on the board: Exponents are always integers and will never be rational. Utilize SQPL (view literacy strategy descriptions) to guide the lesson. SQPL (student questions for purposeful learning) is a strategy designed to focus students on the content that is about to be taught by first presenting students with a thought-provoking prompt and then allowing students to ask and answer their own questions about the content being presented. In this particular case, after presenting the statement/prompt, have students individually decide if the statement is true or false and why they agree or disagree with the statement. Next, have students pair up and come up with questions based on the prompt, including reasons why they think the prompt is true or false and why. Next, elicit from students the questions they came up with based on the prompt and write them on the board. Prepare students for the presentation of the information by telling students to answer as many of the SQPL questions as they can in their pair-groups as information is presented during class discussion and teacher-led instruction. Go down the list of student SQPL questions asked by students, then use these to guide the lesson. Some sample questions from students may be: How can an exponent be a rational number? How would an exponent be dealt with if it were not an integer (i.e., if you raise 2 to a power of 3 its value is 2 x 2 x 2 or 8, but what would 2 raised to a power of 2/3 look like?) Algebra I Part 2 Unit Exponents, Expressions and Equations -4
How do you simplify numbers that have been raised to an exponent that is rational? Next, take a poll of the students to see who agrees and who disagrees with the prompt that was presented. Form one group of students who agree and another group of those who disagree. Have representatives of each group debate the issue. Use the classroom discussion and student questions/answers to drive the lesson. Act as a mediator for the discussion/debate and lead them toward the ultimate goal which is how to deal with exponents that are rational. After finding out student thinking, determine if there are any students who may have come up with legitimate arguments to support their answers. Ultimately, guide students to understand the following ideas concerning rational exponents. Exponents can be rational (they are not always integers). Rational exponents are actually related to roots of numbers (square roots, cube roots, etc.). For example, if the value of 25 2 is found using a calculator, the resulting value is 5. Why? Because when a power is written in fractional (a/b) form, in this case, ½, the numerator represents the power (which is what is typically thought of when exponents are used) while the denominator represents the root. In this example, the power is and the root is 2. Students should be familiar with square roots which is what a root of 2 is, a square root. Therefore, 25 2 can be thought of as the square root of 25 to the first power. This can be written using a traditional square root as: 25 or since the power is we can simply write 25 2 as thus the value is 5. 25 and Present more examples with various powers and roots (including with negative rational exponents). Make sure that students understand that the root and power can be dealt with first or 2 last depending on the situation. For example, 8 3 can be thought of as the cube root of 8 3 squared which can be written as ( 8) 2 or as 3 ( 8 2 ). It is important that students understand that the order in which the number is simplified at this point has some flexibility. In this case, students can take the cube root of 8 first (which is 2), then square the result (which would be 4). Another person may have found the same result by first squaring the 8 (which is 64) and then determining the cube root of the result (which is 4). In either case, the result is the same. After demonstrating examples, have students try to simplify some on their own and then provide additional practice using a textbook or other resource. In closing, ask students to look over their original SQPL questions from the beginning of the lesson and go through the answers. Check to be sure students have answered their SQPL questions accurately. Algebra I Part 2 Unit Exponents, Expressions and Equations -5
Activity 3: Properties of Exponents ((GLE: 2; CCSS: N-RN.; RST.9-0.) Materials List: paper, pencil, math textbook, calculators, Properties of Exponents BLM Provide each student a copy of the Properties of Exponents BLM. In this activity, use questioning the content (view literacy strategy descriptions). Questioning the content is a strategy designed to encourage students to use questions to help guide their understanding of the material as it is being read and learned. In this particular use of the strategy, students will read about some of the properties of exponents and how to utilize these properties when computing with exponents. Students are then to apply what they have read in order to solve problems based on the informational text. Give students the opportunity to read the text found on the BLM alone first, and then have students work in pairs on solving the problems and answering the questions presented on the BLM. Monitor the class as they go through this process. DO NOT answer the questions for them students must learn the skill of reading to learn, and actually build success in doing so. After everyone has answered the questions based on what they read, go over student answers and provide feedback as necessary to clarify any misconceptions they may have, but make sure always to refer to the text when doing so. Show students how the text can be used to justify answers in this activity. Ultimately, students should be able to apply the properties associated with exponents in numerical situations involving operations with exponents. Specifically, students need to understand the following: Whenever you multiply numbers with exponents that have the same base you can simply add the exponents. In general terms: ( x m ) ( x n ) = x ( m + n ) Whenever you have an exponent expression that is raised to a power, you can multiply the exponent and power. In general terms: ( x m ) n = x m n Whenever you divide two numbers with exponents that have the same base, you can simply subtract the denominator exponent from the numerator exponent. In general terms: m x m n n = x. x Provide additional examples using the textbook as a resource as needed to ensure student mastery of this concept. Be sure to include problems that have rational exponents (positive and negative) to build on what was learned in Activity 2. Activity 4: Order of Operations with Numerical Expressions (GLEs: 2, 8) Materials List: paper, pencil, math textbook, scientific calculators Review with students how to use the order of operations to evaluate and simplify numerical expressions with exponents and parentheses. Be sure to include expressions with rational numbers. Some examples of the types of problems that students should be able to do are shown below: Algebra I Part 2 Unit Exponents, Expressions and Equations -6
2 2 Ex: 80 4 2 2 2 Ex: 4 2 + [7 (3 5)] Provide additional practice problems as necessary (either through the use of textbook or teachermade worksheets). Include in the discussion of this topic the proper use of calculators and how the expressions must be entered differently, depending on the calculator used, to arrive at the correct answer. Make sure that students use logic, estimation, and mental math strategies throughout the course when using a calculator to catch errors and make sure their answers make sense. Activity 5: Terms, Factors, and Coefficients (CCSS: A-SSE.; RST.9-0.4) Materials List: paper, pencil, math textbook, Vocabulary Self-Awareness BLM In this lesson, the goal is for students to be able to identify and interpret the parts of an expression (i.e., terms, factors, and coefficients). Have students maintain a vocabulary selfawareness (view literacy strategy descriptions) chart utilizing the BLM for this activity. Vocabulary self-awareness is valuable because it highlights students understanding of what they know, as well as what they still need to learn, in order to fully comprehend the concept. Students indicate their understanding of a term/concept (using + for complete understanding of concept; a? for understanding of a concept to some degree but not completely; and to indicate no understanding of a concept), but then adjust or change the marking as the lesson is taught to reflect their change in understanding. The objective is to have all terms marked with a + at the end of the unit. Be sure to allow students the opportunity to revisit their vocabulary selfawareness charts often (throughout this activity) to monitor their developing knowledge about important concepts. Begin the lesson by providing students copies of the Vocabulary Self-Awareness BLM and then have students fill out the chart, as well as answer the questions found on the BLM. After students fill out their chart, discuss with students what they wrote and their answers to the questions presented. After finding out what students might know about terms, factors, and coefficients, build on their understanding and clear up any misconceptions they might have using the expression presented on the BLM (3x² + 4x + 2). Continue to have students revisit their vocabulary self-awareness chart throughout this activity (the goal is to eventually have all students comfortable with all terms being presented). After fully discussing the BLM, write the following algebraic expression on the board and have students get in pairs to determine what are the terms, factors, and coefficients of the expression. Expression: 4(x 3) + 2 After allowing students the opportunity to discuss this in pairs, elicit student responses to find out student thinking and provide feedback as needed to guide student understanding and clear up any misconceptions students may have. Students should understand that there are actually 2 terms in the expression: st Term: 4(x 3) 3 rd Term: 2 Algebra I Part 2 Unit Exponents, Expressions and Equations -7
If the original expression in words is read, it would read as follows: 4 times (x 3) plus 2. The fact that the 4 and the (x 3) are multiplied together, connect them as a single term as they are both factors of that single term. Terms are joined together by subtraction or addition, thus the plus 2 creates another term which in this case is the term, 2. In the first term, where there are two factors, 4 and (x-3), it is important that students see this as a single term that has two factors. The (x 3) should be thought of as single entity, which in this case is a factor of the term it is a part of. The coefficient of this term is 4, while the coefficient of the second term is 2. (Note: In many books, this is called a constant term (which it is) and write that it does not have a coefficient, but technically speaking, this is the coefficient of the x 0 term. Thus 2 is the constant coefficient for this term.) Provide additional examples of this type for students to become proficient at identifying terms, coefficients, and factors. Use the textbook as a resource for additional practice. Activity 6: Evaluating Expressions (GLEs: 2, 8) Materials List: paper, pencil, math textbook, scientific calculators Review evaluating and simplifying algebraic expressions that require students to replace a variable with a known quantity and simplify. An example of the type of problems that students should be able to do is provided below: 3 2 Ex: Given a = and b = 8 5 determine the value of 2 2 a b. Provide additional practice problems as necessary (either through the use of the textbook or teacher-made worksheets). Activity 7: Creating Algebraic Expressions (Modeling) (GLEs: 9) Materials List: paper, pencil, math textbook Review with students how to write algebraic expression from a verbal description from many different contexts. Some examples are shown below: Ex: Write an algebraic model for the following verbal model: four more than three times a number. Answer: 3x + 4 Ex: Write the verbal phrase as an algebraic expression: five less than the square of a number. Answer: x² - 5 Ex: Write an algebraic model to represent the area a rectangle whose length is 4 units more than its width. Answer: x (x + 4) Algebra I Part 2 Unit Exponents, Expressions and Equations -8
Students should also be able to take an algebraic model and create a verbal description of the model (going backwards so to speak). For example, for the algebraic model 4 (x 3) students might write four times three less than a number as a verbal description. Also, students should be able to see the difference between two expressions such as 4(x 3) and 4x 3. In the last expression, the verbal description would be different than the first, such as three less than four times a number. Provide additional practice as needed on these types of problems using the math textbook as a resource. Activity 8: Solving Linear Equations (GLEs: 8, ) Materials List: paper, pencil, math textbook Review how to solve algebraic equations for an unknown value. Use this opportunity to review basic operations with adding, subtracting, multiplying, and dividing integers in the context of solving more complex equations. Discuss the order of operations as a means of solving such equations. Provide students with additional practice problems as necessary using the math textbook. 203-4 Activity 9: Rearranging Formulas (CCSS: A-CED.4; A-REI.) Note: This activity addresses some new content based on CCSS and is to be taught in 203-4. Students need to learn to apply the skills associated with solving linear equations in rearranging a formula into an equivalent form for a specified variable. For example, given the equation D = RT (which relates distance, time, and rate), students need to understand that in its current form the formula is in terms of the distance, D. This same formula could be rearranged so that it was in terms or R or T (i.e., that particular variable is isolated on one side of the equal sign and the rest of the terms are on the other side). Students should connect solving equations in one variable to solving formulas for a given variable. They must understand that they are not getting a numerical answer but an equivalent formula, just rearranged in terms of a different variable. For example, if D = RT were solved for T, in order to solve for T, undo R from T. Since it s attached to T by multiplication, to undo it, use the inverse of multiplication division. Therefore, dividing both sides of the equation gives: D = R T (dividing both sides by R to isolate T) R R D = T R Algebra I Part 2 Unit Exponents, Expressions and Equations -9
An equivalent formula is found, except the formula is now in terms of T rather than D. This skill is especially important in math and science classes. If students understand how to rearrange a given formula, it simplifies having to remember all of the many different arrangements. Provide students with additional examples of this skill and provide additional practice on the skill using the math textbook as a resource. Have students justify the steps in solving the problems they are working on. This may help some students who may get confused by all the variables by making sense of the steps they are taking to solve for a given variable. This will also provide insight to the teacher on where students may be having difficulty. Sample Assessments General Guidelines Performance and other types of assessments can be used to ascertain student achievement. Following are some examples: General Assessments The student will use sample work from the activities to place in a portfolio that would showcase knowledge of inequalities. The student will describe the difference between a term and a factor in an expression. The student will write a letter to a classmate explaining how to simplify a number raised to a rational exponent. Activity-Specific Assessments Activity : The student will determine the value of 8 3 he/she got the result. and explain in words how 2 Activity 2: The student will determine the value of 26 3 explain in words how he/she got the result. Activity 4: The student will simplify the following numerical expression using order 2 2 of operations without a calculator: 5 2 + [8 4(3 5)] Activity 5: The student will identify the terms, factors and coefficients for the expression 3(4x 3) + 8 Activity 8: The student will solve the following equation for c: 5 c 2 c 4 3 = 7 0 Algebra I Part 2 Unit Exponents, Expressions and Equations -0