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1 of 19 9/2/ :09 PM Answers Teacher Copy Plan Pacing: 1 class period Chunking the Lesson Example A #1 Example B Example C #2 Check Your Understanding Lesson Practice Teach BellRinger Activity Students should recall that an absolute value of a number is its distance from zero on a number line. Have students evaluate the following: [6] 6 [6] Then have students solve the following equation. 3. x = 6 [x = 6 or x = 6] Example A Marking the Text, Interactive Word Wall Point out the Math Tip to reinforce why two solutions exist. Work through the solutions to the equation algebraically. Remind students that solutions to an equation make the equation a true statement. This mathematical understanding is necessary for students to be able to check their results College Board. All rights reserved.
2 of 19 9/2/ :09 PM Developing Math Language An absolute value equation is an equation involving an absolute value of an expression containing a variable. Just like when solving algebraic equations without absolute value bars, the goal is to isolate the variable. In this case, isolate the absolute value bars because they contain the variable. It should be emphasized that when solving absolute value equations, students must think of two cases, as there are two numbers that have a specific distance from zero on a number line. 1 Identify a Subtask, Quickwrite When solving absolute value equations, students may not see the purpose in creating two equations. Reviewing the definition of the absolute value function as a piecewisedefined function with two rules may enable students to see the reason why two equations are necessary. Have students look back at Try These A, parts c and d. Have volunteers construct a graph of the two piecewisedefined functions used to write each equation and then discuss how the solution set is represented by the graph. Example B Marking the Text, Simplify the Problem, Critique Reasoning, Group Presentation Start with emphasizing the word vary in the Example, discussing what it means when something varies. You may wish to present a simpler example such as: The average cost of a pound of coffee is $8. However, the cost sometimes varies by $1. This means that the coffee could cost as little as $7 per pound or as much as $9 per pound. Now have students work in small groups to examine and solve Example B by implementing an absolute value equation. Additionally, ask them to take the problem a step further and graph its solution on a number line. Have groups present their findings to the class. ELL Support For those students for whom English is a second language, explain that the word varies in mathematics means changes. There are different ways of thinking about how values can vary. Values can vary upward or downward, less than or greater than, in a positive direction or a negative direction, and so on. However, the importance comes in realizing that there are two different directions, regardless of how you think of it. Also address the word extremes as it pertains to mathematics. An extreme value is a maximum value if it is the largest possible amount (greatest value), and an extreme value is a minimum value if it is the smallest possible amount (least value). Developing Math Language An absolute value inequality is basically the same as an absolute value equation, except that the equal sign is now an inequality symbol: <, >,,, or. It still involves an absolute value expression that contains a variable, just like before. Use graphs on a number line of the solutions of simple equations and inequalities and absolute value equations and inequalities to show how these are all related.
3 of 19 9/2/ :09 PM Example C Simplify the Problem, Debriefing Before addressing Example C, discuss the following: Inequalities with A > b, where b is a positive number, are known as disjunctions and are written as A < b or A > b. For example, x > 5 means the value of the variable x is more than 5 units away from the origin (zero) on a number line. The solution is x < 5 or x > 5. See graph A. This also holds true for A b. Inequalities with A < b, where b is a positive number, are known as conjunctions and are written as b < A < b, or as b < A and A < b. For example: x < 5; this means the value of the variable x is less than 5 units away from the origin (zero) on a number line. The solution is 5 < x < 5. See graph B. This also holds true for A b. Students can apply these generalizations to Example C. Point out that they should proceed to solve these just as they would an algebraic equation, except in two parts, as shown above. After they have some time to work through parts a and b, discuss the solutions with the whole class. Teacher to Teacher Another method for solving inequalities relies on the geometric definition of absolute value x a as the distance from x to a. Here s how you can solve the inequality in the example: Thus, the solution set is all values of x whose distance from is greater than. The solution can be represented on a number line and written as x < 4 or x > 1. 2 Quickwrite, Self Revision/Peer Revision, Debriefing Use the investigation regarding the restriction c > 0 as an opportunity to discuss the need to identify impossible situations involving inequalities. Check Your Understanding
4 of 19 9/2/ :09 PM Debrief students answers to these items to ensure that they understand concepts related to absolute value equations. Have groups of students present their solutions to Item 4. Assess Students answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination for the activity. Adapt Check students answers to the Lesson Practice to ensure that they understand basic concepts related to writing and solving absolute value equations and inequalities and graphing the solutions of absolute value equations and inequalities. If students are still having difficulty, review the process of rewriting an absolute value equation or inequality as two equations or inequalities. Activity Standards Focus In Activity 4, students identify and graph various piecewisedefined functions. They explore functions made up of parts of linear functions. They look at the absolute value and step functions. Finally, they transform various parent piecewise functions. Throughout this activity, emphasize the use of technology to graph piecewisedefined functions as well as how changes in coefficients and constants affect the graphs of functions. Plan Pacing: 1 class period Chunking the Lesson #1 2 #3 4 #5 Check Your Understanding #9 #10 Check Your Understanding Lesson Practice
5 of 19 9/2/ :09 PM Teach BellRinger Activity Ask students to graph the following and be ready to share a description of the features of each graph. 1. y = 3x + 5 [graph of line, slope of 3, yintercept of 5] 2. y = x 2 6 [graph of parabola, opening upward, with yaxis as line of symmetry, and with vertex at (0, 6)] 3. y = x + 1 [Vshaped with the point of the V at (0, 1); note nothing is below the xaxis ] Have students discuss the methods they used to graph these functions. 1 2 Activating Prior Knowledge, Create Representations, Quickwrite If students have previous experience with piecewisedefined functions, their descriptions will provide formative information about their understanding. Students are asked to graph two equations on the same grid and recognize the combined graphs as a function. Students may describe the graphs as pieces that are connected or pieces that are not connected. It is imperative that students recognize the graphs on each grid as functions, as this provides the access to defining piecewisedefined functions. Common Core State Standards for Activity 4 HSFBF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. HSFIF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. HSFIF.C.7b Graph piecewisedefined functions, including step functions and absolute value functions. 3 4 Create Representations
6 of 19 9/2/ :09 PM The graph shown in Item 3 is discontinuous at x = 0. It is important for students to note the endpoints of the intervals of the piecewise definitions. In Item 4, take special care when discussing the values of the functions at x = 0. Technology Tip A piecewisedefined function such as can be graphed on a graphing calculator by following the steps below: Press the [y=] key. For Y1, type in (x 2 2)(x < 3). Note: In order to access the < symbol, press the [2nd] key, followed by the [MATH] key to access the Test. 3. For Y2, type in (2x + 1)( 3 x)(x < 5). Note: You key in the domain of Y2 separately For Y3, type in (8)(x 5). Press the [GRAPH] key. For additional technology resources, visit SpringBoard Digital. Developing Math Language A piecewisedefined function is so called because it is a function that follows a different set of rules as the domain changes. Each set of rules provides a different piece of the function. The pieces can be straight or curved. Piecewisedefined functions exist because in the real world not every situation can be modeled using only a single function. The rules may change as the domain changes. 5 Create Representations As students complete the table, make sure that they are using the correct rule for the given domain values. Additionally, when graphing, students may stop the graph for the lefthand piece before x = 1 and not continue to show x = 1 with an open circle. If this occurs, ask them to consider the domain of the left part of the graph and then ask them which rule applies to points between 2 and 1.
7 of 19 9/2/ :09 PM MiniLesson: Recognizing and Evaluating Functions If students need additional help identifying functions; recognizing functions from graphs, equations, and tables; or evaluating functions; a minilesson is available to provide practice. See the Teacher Resources at SpringBoard Digital for a student page for this minilesson. Check Your Understanding Debrief students answers to these items to ensure that they understand concepts related to graphing a piecewisedefined function. 9 Activating Prior Knowledge Defining the domain and range for the function will require students to look not only at the function rules and their restricted domains, but also at the graph for the range values. Formative information can be gathered as a result of this exercise. Use additional practice as needed. 10 Activating Prior Knowledge Be sure that students recognize that the function is not defined for x = 1 in Item 10b. Use these items to assess understanding and to start a class discussion of domain and range. Teacher to Teacher Note that there are three different ways to express the domain and range of a function: (1) interval notation, (2) inequalities, and (3) set notation. While all three can be written to represent the same domain and range, each takes on a different look in expressing them. In Item 9, note the domain is all real numbers, and the range is all real numbers greater than zero. In the inequalities format and the set notation format, the variables x and y are used to represent the domain and range, whereas in interval notation, the ± (infinity symbol) and parentheses are used. Set notation is the only format that uses braces { } and the vertical bar to represent the words such that. The set notation also uses the symbols (element) and R (set of all real numbers). Check Your Understanding Debrief students answers to these items to ensure that they understand different notations for the domain and range of piecewise functions. Assess
8 of 19 9/2/ :09 PM Students answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination for the activity. Adapt Check students answers to the Lesson Practice to ensure that they understand how to graph and evaluate piecewisedefined functions and how to identify the domain and range. If students have difficulty with the endpoints in Item 14b, encourage them to evaluate the function at x = 1 to determine which endpoint is closed and which is open. Learning Targets p. 57 Graph piecewisedefined functions. Write the domain and range of functions using interval notation, inequalities, and set notation. Activating Prior Knowledge (Learning Strategy) Definition Recalling what is known about a concept and using that information to make a connection to a new concept Purpose Helps students establish connections between what they already know and how that knowledge is related to new learning Quickwrite (Learning Strategy) Definition Writing for a short, specific amount of time about a designated topic Purpose Helps generate ideas in a short time
9 of 19 9/2/ :09 PM Create Representations (Learning Strategy) Definition Creating pictures, tables, graphs, lists, equations, models, and /or verbal expressions to interpret text or data Purpose Helps organize information using multiple ways to present data and to answer a question or show a problem solution Interactive Word Wall (Learning Strategy) Definition Visually displaying vocabulary words to serve as a classroom reference of words and groups of words as they are introduced, used, and mastered over the course of a year Purpose Provides a visual reference for new concepts, aids understanding for reading and writing, and builds word knowledge and awareness Marking the Text (Learning Strategy) Definition Highlighting, underlining, and /or annotating text to focus on key information to help understand the text or solve the problem Purpose Helps the reader identify important information in the text and make notes about the interpretation of tasks required and concepts to apply to reach a solution ThinkPairShare (Learning Strategy) Definition Thinking Typesetting through math: 35% a problem alone, pairing with a partner to share ideas, and concluding by sharing results with the
10 0 of 19 9/2/ :09 PM class Purpose Enables the development of initial ideas that are then tested with a partner in preparation for revising ideas and sharing them with a larger group Discussion Groups (Learning Strategy) Definition Working within groups to discuss content, to create problem solutions, and to explain and justify a solution Purpose Aids understanding through the sharing of ideas, interpretation of concepts, and analysis of problem scenarios Suggested Learning Strategies Activating Prior Knowledge, Quickwrite, Create Representations, Interactive Word Wall, Marking the Text, ThinkPairShare, Discussion Groups The graphs of both y = x 2 for x < 3 and y = 2x + 7 for x 3 are shown on the same coordinate grid below. p. 58 Discussion Group Tips As you listen to your group s discussions as you work through items 1 4, you may hear math terms or other words that you do not know. Use your math notebook to record words that are frequently used. Ask for clarification of their meaning, and make notes to help you remember and use those words in your own communications. 1. Work with your group on this item and on Items 2 4. Describe the graph as completely as possible. Sample answer: The graph shows an increasing linear function with a slope of 1 for xvalues from negative infinity Typesetting to math: 3; it 35% then shows a decreasing linear function with a slope of 2 for xvalues from 3 to infinity.
11 1 of 19 9/2/ :09 PM 2. Make use of structure. Why is the graph a function? Sample answer: It is a function because for each xvalue, there is only one yvalue, and graphically, it passes the vertical line test. 3. Graph y = x 2 3 for x 0 and for x > 0 on the same coordinate grid. 4. Describe the graph in Item 3 as completely as possible. Why is the graph a function? Sample answer: The graph has a break at x = 0. To the left of x = 0, it is a decreasing quadratic function with a minimum at y = 3, which is included. To the right of x = 0, but not including x = 0, it is an increasing linear function with a slope of. The graph is a function because for each xvalue, there is only one yvalue. Math Terms A piecewisedefined function is a function that is defined using different rules for the different nonoverlapping intervals of its domain. The functions in Items 1 and 3 are piecewisedefined functions. Piecewisedefined functions are written as follows (using the function from Item 3 as an example): Math Tip A piecewisedefined function may have more than two rules. For example, consider the function below. 5. Model with mathematics. Complete the table of values. Then graph the function.
12 2 of 19 9/2/ :09 PM x g(x) Check Your Understanding p Critique the reasoning of others. Look back at Item 5. Esteban says that g( 1) = 2. Is Esteban correct? Explain. Yes. The function rule that applies when x = 1 is x + 3. Substituting 1 for x in the expression x + 3 gives = 2; g( 1) = Explain how to graph a piecewisedefined function. Graph the first function rule for the values of x given after the first if statement. Then graph the second function rule for the values of x given after the second if statement. Repeat this process if there are more than 2 function rules.
13 3 of 19 9/2/ :09 PM 8. If a piecewisedefined function has a break, how do you know whether to use an open circle or a closed circle for the endpoints of the function s graph? If the restriction on x for a function rule includes < or >, use an open circle for the endpoint. If the restriction on x for a function rule includes or, use a closed circle for the endpoint. Math Terms The domain of a function is the set of input values for which the function is defined. The range of a function is the set of all possible output values for the function. The domain of a piecewisedefined function consists of the union of all the domains of the individual pieces of the function. Likewise, the range of a piecewisedefined function consists of the union of all the ranges of the individual pieces of the function. You can represent the domain and range of a function by using inequalities. You can also use interval notation and set notation to represent the domain and range.
14 4 of 19 9/2/ :09 PM Math Tip Interval notation is a way of writing an interval as a pair of numbers, which represent the endpoints. For example, 2 < x 6 is written in interval notation as (2, 6). Use a parenthesis if an endpoint is not included; use a bracket if an endpoint is included. In interval notation, infinity,, and negative infinity,, are not included as endpoints. Set notation is a way of describing the numbers that are members, or elements, of a set. For example, 2 < x 6 is written in set notation as {x x R, 2 < x 6}, which is read the set of all numbers x such that x is an element of the real numbers and 2 < x Write the domain and range of g(x) in Item 5 by using: a. inequalities Domain: < x < ; range: y > 0 b. interval notation Domain: (, ); range: (0, ) c. set notation. Domain: [Math Processing Error]; range: 10. Graph each function, and write its domain and range using inequalities, interval notation, and set notation. Show your work. Domain: a. < x <, (, ), ; Typesetting range: math: < 35% y <, (, ), ;
15 5 of 19 9/2/ :09 PM Domain: b. x 1, (, 1) and (1, ), ; range: y > 1, ( 1, ), Check Your Understanding p. 60 Writing Math You can use these symbols when writing a domain or range in set notation. such that is an element of R the real numbers Z the integers N the natural numbers 11. The domain of a function is all positive integers. How could you represent this domain using set notation? {x x R, x > 0} 12.Explain how to use interval and set notation to represent the range y 3. The range includes the interval from 3, which is included, to, which is not included. The interval notation for the range is [3, ). In set notation, the range is {y y R, y 3}
16 6 of 19 9/2/ :09 PM 13. What can you conclude about the graph of a piecewisedefined function whose domain is {x x R, x 2}? Sample answer: The graph has a break or hole at x = 2. Lesson 41 Practice 14. Graph each piecewisedefined function. Then write its domain and range using inequalities, interval notation, and set notation. domain: a. < x <, (, ), {x x R}; range: y 0, [0, ), {y y 0} domain: b. < x <, (, ), {x x R}; range: y 3, (, 3], {y y 3} 15. The range of a function is all real numbers greater than or equal to 5 and less than or equal to 5. Write the range of the function using an inequality, interval notation, and set notation. 5 y 5, [ 5, 5], {y 5 y 5} 16.Evaluate the piecewise function for x = 2, x = 0, and x = 4. g( 2) = 4, g(0) = 2, g(4) = 8
17 7 of 19 9/2/ :09 PM 17. Model with mathematics. An electric utility charges residential customers a $6 monthly fee plus $0.04 per kilowatt hour (kwh) for the first 500 kwh and $0.08/kWh for usage over 500 kwh. a. Write a piecewise function f(x) that can be used to determine a customer s monthly bill for using x kwh of electricity. b. Graph the piecewise function. c. A customer uses 613 kwh of electricity in one month. How much should the utility charge the customer? Explain how you determined your answer. $35.04; Sample explanation: The customer uses 613 kwh, so x = 613. This value of x satisfies the domain restriction of the second rule of the piecewise function, so I evaluated the second rule for x = 613.
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Answers Teacher Copy. Systems of Linear Equations Monetary Systems Overload. Activity 3. Solving Systems of Two Equations in Two Variables
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