CCGPS Frameworks Student Edition. Mathematics. 8 th Grade Unit 4: Functions

Transcription

1 CCGPS Frameworks Student Edition Mathematics 8 th Grade Unit 4: Functions These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement.

2 TABLE OF CONTENTS Unit 4 Functions Overview... Key Standards & Related Standards...4 Enduring Understandings...6 Concepts & Skills to Maintain...6 Selected Terms and Symbols...6 Tasks Secret Codes and Number Rules...8 Vending Machines...9 Order Matters... Which is which?...5 May 0 Page of 6

3 OVERVIEW In this unit students will: recognize a relationship as a function when each input is assigned to exactly one output; reason from a context, a graph, or a table, after first being clear which quantity is considered the input and which is the output; produce a counterexample: an input value with at least two output values when a relationship is not a function; explain how to verify that for each input there is exactly one output; and translate functions numerically, graphically, verbally, and algebraically. The vertical line test should be avoided because () it is too easy to apply without thinking, () students do not need an efficient strategy at this point, and () it creates misconceptions for later mathematics, when it is useful to think of functions more broadly, such as whether x might be a function of y. Function machine representations are useful for helping students imagine input and output values, with a rule inside the machine by which the output value is determined from the input. Notice that the standards explicitly call for exploring functions numerically, graphically, verbally, and algebraically (symbolically, with letters). This is sometimes called the rule of four. For fluency and flexibility in thinking, students need experiences translating among these. Although the units in this instructional framework emphasize key standards and big ideas at specific times of the year, routine topics such as estimation, mental computation, and basic computation facts should be addressed on an ongoing basis. Ideas related to the eight practice standards should be addressed constantly as well. To assure that this unit is taught with the appropriate emphasis, depth, and rigor, it is important that the tasks listed under Evidence of Learning be reviewed early in the planning process. A variety of resources should be utilized to supplement this unit. This unit provides much needed content information, but excellent learning activities as well. The tasks in this unit illustrate the types of learning activities that should be utilized from a variety of sources. May 0 Page of 6

4 STANDARDS ADDRESSED IN THIS UNIT Mathematical standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics. KEY STANDARDS Define, evaluate, and compare functions. MCC8.F. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. MCC8.F. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. STANDARDS FOR MATHEMATICAL PRACTICE The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important processes and proficiencies with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one s own efficacy). Make sense of problems and persevere in solving them. In grade 8, students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, What is the most efficient way to solve the problem?, Does this make sense?, and Can I solve the problem in a different way? Reason abstractly and quantitatively. In grade 8, students represent a wide variety of real world contexts through the use of real numbers and variables in mathematical expressions, equations, and inequalities. They examine patterns in data and assess the degree of linearity of functions. Students contextualize to May 0 Page 4 of 6

5 understand the meaning of the number or variable as related to the problem and decontextualize to manipulate symbolic representations by applying properties of operations. Construct viable arguments and critique the reasoning of others. In grade 8, students construct arguments using verbal or written explanations accompanied by expressions, equations, inequalities, models, and graphs, tables, and other data displays (i.e. box plots, dot plots, histograms, etc.). They further refine their mathematical communication skills through mathematical discussions in which they critically evaluate their own thinking and the thinking of other students. They pose questions like How did you get that?, Why is that true? Does that always work? They explain their thinking to others and respond to others thinking. 4 Model with mathematics. In grade 8, students model problem situations symbolically, graphically, tabularly, and contextually. Students form expressions, equations, or inequalities from real world contexts and connect symbolic and graphical representations. Students solve systems of linear equations and compare properties of functions provided in different forms. Students use scatter plots to represent data and describe associations between variables. Students need many opportunities to connect and explain the connections between the different representations. They should be able to use all of these representations as appropriate to a problem context. 5 Use appropriate tools strategically. Students consider available tools (including estimation and technology) when solving a mathematical problem and decide when certain tools might be helpful. For instance, students in grade 8 may translate a set of data given in tabular form to a graphical representation to compare it to another data set. Students might draw pictures, use applets, or write equations to show the relationships between the angles created by a transversal. 6 Attend to precision. In grade 8, students continue to refine their mathematical communication skills by using clear and precise language in their discussions with others and in their own reasoning. Students use appropriate terminology when referring to the number system, functions, geometric figures, and data displays. 7 Look for and make use of structure. Students routinely seek patterns or structures to model and solve problems. In grade 8, students apply properties to generate equivalent expressions and solve equations. Students examine patterns in tables and graphs to generate equations and describe relationships. Additionally, students experimentally verify the effects of transformations and describe them in terms of congruence and similarity. 8 Look for and express regularity in repeated reasoning. In grade 8, students use repeated reasoning to understand algorithms and make generalizations about patterns. Students use iterative processes to determine more precise rational approximations for irrational numbers. During multiple opportunities to solve and model May 0 Page 5 of 6

6 problems, they notice that the slope of a line and rate of change are the same value. Students flexibly make connections between covariance, rates, and representations showing the relationships between quantities. ENDURING UNDERSTANDINGS A function is a specific type of relationship in which each input has a unique output. A function can be represented in an input-output table. A function can be represented graphically using ordered pairs that consist of the input and the output of the function in the form (input, output). A function can be represented with an algebraic rule. CONCEPTS/SKILLS TO MAINTAIN It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas. computation with whole numbers and decimals, including application of order of operations plotting points in a four quadrant coordinate plan understanding of independent and dependent variables characteristics of a proportional relationship SELECTED TERMS AND SYMBOLS The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them. The definitions below are for teacher reference only and are not to be memorized by the students. Students should explore these concepts using models and real life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers. The websites below are interactive and include a math glossary suitable for middle school students. Note Different sources use different definitions. Please preview any website for alignment to the definitions given in the frameworks. Visit or to see additional definitions and specific examples of many terms and symbols used in grade 8 mathematics. May 0 Page 6 of 6

7 Domain: Domain of a relation is the set of all x-coordinates of the ordered pairs of that relation. Function: A rule of matching elements of two sets of numbers in which an input value from the first set has only one output value in the second set. Graph of a Function: The set of all the points on a coordinate plane whose coordinates makes the rule of function true. Range of a Function: The set of all output values or the y-values of a function or a relation is called the range of the function or the relation. May 0 Page 7 of 6

8 Secret Codes and Number Rules Part Encryption is used by spies, secret societies, and other organizations to transfer information without other people reading their messages. Secret codes can be created by a simple scheme of replacing letters with numbers. More complex codes replace letters with other letters. The code scheme is called a cipher and can be written as a list or table. A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Use the cipher above to decode these messages. Question: ? Answer: Question: ? Answer: Your turn! Write, find or share a joke or riddle! Write it in code and share with your classmates. May 0 Page 8 of 6

9 Ciphers can be written in which letters are replaced with other letters, not numbers. A scheme is determined for replacement. Often letters are shifted. For example, a letter could be replaced with the letter three down the alphabet. A B C D E F G H I J K L M N O P Q R S T U V W X Y Z D E F G H I J K L M N O P Q R S T U V W X Y Z A B C Therefore TWINKLE TWINKLE LITTLE STAR becomes WZLQOH WZLQOH OLWWOH VWDU The next line is HOW I WONDER WHAT YOU ARE. Use the cipher to code it! May 0 Page 9 of 6

10 A letter to letter code can also be represented in a graph. Determine the cipher represented in the graph below. CODED LETTER Z Y X W V U T S R Q P O N M L K J I H G F E D C B A A B C D E F G H I J K L M N O P Q R S T U V W X Y Z ORIGINAL LETTER A B C D E F G H I J K L M N O P Q R S T U V W X Y Z L M N O P Q R S T U V W X Y Z A B C D E F G H I J K May 0 Page 0 of 6

11 . Use the code cipher from the preceding page to code the message Math is fun.. Use the cipher to decode: T SLGP L DPNCPE May 0 Page of 6

12 CODED LETTER Z Y X W V U T S R Q P O N M L K J I H G F E D C B A A B C D E F G H I J K L M N O P Q R S T U V W X Y Z ORIGINAL LETTER. Use the code cipher above to code the message: The homework is on page fifty. 4. Use the cipher above to decode: SLJ JCP FDSKU DW GOJEOPRW May 0 Page of 6

13 Create a cipher code similar to the ones on the preceding pages so that each letter is replaced by a letter that comes four positions later in the alphabet. CODED LETTER Z Y X W V U T S R Q P O N M L K J I H G F E D C B A A B C D E F G H I J K L M N O P Q R S T U V W X Y Z ORIGINAL LETTER 5. Use your cipher to code: April showers bring May flowers. May 0 Page of 6

14 May 0 Page 4 of 6 Determine the rule for the codes represented below. A B C D E F G H I J K L M N O P Q R S T U V W X Y Z A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

15 Secret Codes and Number Rules Part As rules can be created to code a message in letters, you can write a rule that changes numbers into other numbers. These rules can be represented by words, equations, tables and graphs. The rule in words add one to each number can be represented in the other forms. Form Representation Words verbal expression add one to each number Equation y = x + Table x 0 y 4 Graph May 0 Page 5 of 6

16 Consider these tables of values. Determine the rule. Express each function verbally and algebraically. Input Output Input Output May 0 Page 6 of 6

17 Create a table and graph for y = x + Input Output May 0 Page 7 of 6

18 Use the equation y = x + and the table and graph which you just created, to help you as you reason the following questions. Is it possible for two different inputs to give you the same output? Justify your answer. Is it possible to derive two different outputs from the same input? Justify your answer. May 0 Page 8 of 6

19 Vending Machines Marinathe, Hezza, and Samuel had been studying math all day. In the afternoon they took a break and went to the vending machine for a snack. After studying functions all afternoon, function vocabulary was on their minds. With money in hand, they stood in front of the machine. Samuel said the machine was like a function machine with input and output. Marinathe said while there was input and output, she thought the machine was more like a relation machine instead of a function machine. Hezza said she thought it could be either, depending on how one thought about it.. Under what circumstance would the machine be a function machine?. Under what circumstance would the machine be a relation machine, but not a function? May 0 Page 9 of 6

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21 Order Matters Directions: Use the picture below to Questions -4.. Is (name, shirt color) a function? Why or why not?. Prove your reasoning by showing the relation as ordered pairs and as a mapping. May 0 Page of 6

22 . Is (shirt color, name) a function? Why or why not? 4. Prove your reasoning by showing the relation as ordered pairs and as a mapping. May 0 Page of 6

23 Directions: Use the picture below to answer Questions Is (name, shirt color) a function? Why or why not? 6. Prove your reasoning by showing the relation as ordered pairs and as a mapping. May 0 Page of 6

24 7. Is (shirt color, name) a function? Why or why not? 8. Prove your reasoning by showing the relation as ordered pairs and as a mapping. May 0 Page 4 of 6

25 Which is Which? Directions: Using the vocabulary of domain, range, input, output, relation and function, explain how each real life experience is like a function.. When you pick up a phone and dial (404) , you will get Zoo Atlanta. When you pick up a phone and dial (404) , you will get the Georgia State Capital Museum. When you pick up a phone and dial (478) 6-670, you will get the Bibb County Tax Assessor s Office. When you pick up a phone and dial a specific number, you will get only one party. Problems continue on the next page. May 0 Page 5 of 6

26 . Malachi was going to make a surprise for his mother on her birthday. On the kitchen table, he had placed flour, sugar, vanilla, chocolate chips, butter, and eggs. Vanessa wanted to surprise their mother, too. On the kitchen counter, Vanessa assembled spaghetti sauce, noodles, ricotta cheese, oregano, and mozzarella cheese. While each of them was making a surprise for their mother, they each were creating a different surprise. Malachi could not use Vanessa s ingredients to make his surprise, nor could Vanessa use Malachi s ingredients to make her surprise.. Suki babysat three nights last week. One night she babysat hours and earned $45. The next night she babysat hours and earned $0. On the third night she babysat five hours and earned $75. She knows if she can babysit for 0 hours she will earn $50 May 0 Page 6 of 6