8th Math Unit 4 - Georgia Standards

[Pages:76]Georgia Standards of Excellence Curriculum Frameworks

Mathematics

GSE Grade 8 Unit 4: Functions

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Georgia Department of Education

Georgia Standards of Excellence Framework

GSE Grade 8 Mathematics ? Unit 4

Unit 4

Functions

TABLE OF CONTENTS

OVERVIEW ................................................................................................................................... 3 STANDARDS ADDRESSED IN THIS UNIT .............................................................................. 3 STANDARDS FOR MATHEMATICAL PRACTICE .................................................................. 3 STANDARDS FOR MATHEMATICAL CONTENT................................................................... 5 BIG IDEAS ..................................................................................................................................... 5 ESSENTIAL QUESTIONS ............................................................................................................ 5 CONCEPTS/SKILLS TO MAINTAIN .......................................................................................... 6 SELECTED TERMS AND SYMBOLS......................................................................................... 7 FORMATIVE ASSESSMENT LESSONS (FAL) ......................................................................... 8 SPOTLIGHT TASKS ..................................................................................................................... 8 TASKS ............................................................................................................................................ 9

Secret Codes and Number Rules............................................................................................... 11 Foxes And Rabbits (Spotlight Task)......................................................................................... 38 Vending Machines .................................................................................................................... 43 Order Matters ............................................................................................................................ 49 Battery Charging (Spotlight Task)............................................................................................ 59 Which is Which?....................................................................................................................... 65 Modeling Situations with Linear Equations ? (FAL) ............................................................... 71 FAL: Create Matching Function Cards..................................................................................... 73 Culminating Task: Sorting Functions ....................................................................................... 74 Appendix.......... ......................................................................................................................... 76

Mathematics GSE Grade 8 Unit 4: Functions July 2020 Page 2 of 76 All Rights Reserved

Georgia Department of Education

Georgia Standards of Excellence Framework

GSE Grade 8 Mathematics ? Unit 4

OVERVIEW

In this unit students will:

? recognize a relationship as a function when each input is assigned to exactly one unique 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 (1) it is too easy to apply without thinking, (2) students do not need an efficient strategy at this point, and (3) 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.

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.

Mathematics GSE Grade 8 Unit 4: Functions July 2020 Page 3 of 76 All Rights Reserved

Georgia Department of Education

Georgia Standards of Excellence Framework

GSE Grade 8 Mathematics ? Unit 4

STANDARDS FOR MATHEMATICAL PRACTICE

Refer to the Comprehensive Course Overview for more detailed information about the Standards for Mathematical Practice.

1. Make sense of problems and persevere in solving them. Students solve real world problems through the application of algebraic and geometric concepts. They 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?

2. Reason abstractly and quantitatively. 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 understand the meaning of the number or variable as related to the problem and decontextualize to manipulate symbolic representations by applying properties of operations.

3. Construct viable arguments and critique the reasoning of others. Students construct arguments using verbal or written explanations accompanied by expressions, equations, inequalities, models, and graphs, tables, and other data displays. 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. Students model problem situations symbolically, graphically, tabularly, and contextually. They 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 scatterplots 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. 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.

Mathematics GSE Grade 8 Unit 4: Functions July 2020 Page 4 of 76 All Rights Reserved

Georgia Department of Education

Georgia Standards of Excellence Framework

GSE Grade 8 Mathematics ? Unit 4

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. 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 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.

STANDARDS FOR MATHEMATICAL CONTENT

Define, evaluate, and compare functions.

MGSE8.F.1 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.

MGSE8.F.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, give 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.

BIG IDEAS

? 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, graphically (using ordered pairs that consist of the input and the output of the function in the form (input, output), and with an algebraic rule

ESSENTIAL QUESTIONS

? What is a function? ? What are the characteristics of a function? ? How do you determine if relations are functions? ? How is a function different from a relation? ? Why is it important to know which variable is the independent variable? ? How can a function be recognized in any form? ? What is the best way to represent a function?

Mathematics GSE Grade 8 Unit 4: Functions July 2020 Page 5 of 76 All Rights Reserved

Georgia Department of Education

Georgia Standards of Excellence Framework

GSE Grade 8 Mathematics ? Unit 4

? How do you represent relations and functions using tables, graphs, words, and algebraic equations?

? What strategies can I use to identify patterns? ? How does looking at patterns relate to functions? ? How are sets of numbers related to each other? ? How can you use functions to model real-world situations? ? How can graphs and equations of functions help us to interpret real-world problems?

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 ? writing algebraic equations

FLUENCY

It is expected that students will continue to develop and practice strategies to build their capacity to become fluent in mathematics and mathematics computation. The eventual goal is automaticity with math facts. This automaticity is built within each student through strategy development and practice. The following section is presented in order to develop a common understanding of the ideas and terminology regarding fluency and automaticity in mathematics:

Fluency: Procedural fluency is defined as skill in carrying out procedures flexibly, accurately, efficiently, and appropriately. Fluent problem solving does not necessarily mean solving problems within a certain time limit, though there are reasonable limits on how long computation should take. Fluency is based on a deep understanding of quantity and number.

Deep Understanding: Teachers teach more than simply "how to get the answer" and instead support students' ability to access concepts from a number of perspectives. Therefore students are able to see math as more than a set of mnemonics or discrete procedures. Students demonstrate deep conceptual understanding of foundational mathematics concepts by applying them to new situations, as well as writing and speaking about their understanding.

Mathematics GSE Grade 8 Unit 4: Functions July 2020 Page 6 of 76 All Rights Reserved

Georgia Department of Education

Georgia Standards of Excellence Framework

GSE Grade 8 Mathematics ? Unit 4

Memorization: The rapid recall of arithmetic facts or mathematical procedures. Memorization is often confused with fluency. Fluency implies a much richer kind of mathematical knowledge and experience.

Number Sense: Students consider the context of a problem, look at the numbers in a problem, make a decision about which strategy would be most efficient in each particular problem. Number sense is not a deep understanding of a single strategy, but rather the ability to think flexibly between a variety of strategies in context.

Fluent students: flexibly use a combination of deep understanding, number sense, and memorization. are fluent in the necessary baseline functions in mathematics so that they are able to spend their thinking and processing time unpacking problems and making meaning from them. are able to articulate their reasoning. find solutions through a number of different paths.

For more about fluency, see: and:

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. The definitions below are from the Common Core State Standards Mathematics Glossary and/or the Common Core GPS Mathematics Glossary when available.

Visit or to see additional definitions and specific examples of many terms and symbols used in grade 8 mathematics.

Mathematics GSE Grade 8 Unit 4: Functions July 2020 Page 7 of 76 All Rights Reserved

Georgia Department of Education

Georgia Standards of Excellence Framework

GSE Grade 8 Mathematics ? Unit 4

? Domain: ? Function: ? Graph of a Function: ? Range of a Function:

FORMATIVE ASSESSMENT LESSONS (FAL)

Formative Assessment Lessons are intended to support teachers in formative assessment. They reveal and develop students' understanding of key mathematical ideas and applications. These lessons enable teachers and students to monitor in more detail their progress towards the targets of the standards. They assess students' understanding of important concepts and problem solving performance, and help teachers and their students to work effectively together to move each student's mathematical reasoning forward. More information on Formative Assessment Lessons may be found in the Comprehensive Course Guide.

SPOTLIGHT TASKS

A Spotlight Task has been added to each CCGPS mathematics unit in the Georgia resources for middle and high school. The Spotlight Tasks serve as exemplars for the use of the Standards for Mathematical Practice, appropriate unit-level Common Core Georgia Performance Standards, and research-based pedagogical strategies for instruction and engagement. Each task includes teacher commentary and support for classroom implementation. Some of the Spotlight Tasks are revisions of existing Georgia tasks and some are newly created. Additionally, some of the Spotlight Tasks are 3-Act Tasks based on 3-Act Problems from Dan Meyer and Problem-Based Learning from Robert Kaplinsky.

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