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The Algebra 1 Teacher's guide to the Common Core State Standards for Mathematics.

# Connecting Patterns and Functions

## Common Core Math says...

This feels like such a big unit for 20 days (on semesters). But in reality we are going to hit these concepts in much more detail as we go through the course. As is states in the Mathematics Common Core Toolbox:

The standards listed here will be revisited multiple times throughout the course, as students encounter new function families. In this unit, connections should be made from students' work in prior courses describing relationships to the general concept of a function and its attributes.

This idea helped me to get my head around everything listed in the standards below. We began to brainstorm about functions and graphing and what students understood when they came to us, as well as any common misconceptions. I began throwing together resources. Some of them look absolutely wonderful as a starting point. Be sure to look at each week to see the breakdown.

## Standards for Mathematical Practice

• 2. Reason abstractly and quantitatively
• 4. Model with mathematics
• 6. Attend to precision

## Big Picture Lesson Planning for the Common Core

Big Picture Lesson Planning forces us as teachers to answer the question, "What do I want my students to be able to do in life with the skills they obtain from my class?" Or, "Why am I teaching this?"

The student will understand patterns in business, economics, environment and behaviors to predict future outcomes to make knowledgeable decisions for success. This includes researching and formulating the patterns and understanding the target audience and how that affects the results.

• Can my students decide which job to take? Or should they begin their own business?
• Do my students know how to predict future maitenence costs for their aging car?
• Can my students use past trends to make good decisions about people, goals, and money?

What skills will they need to achieve this goal?

## Week #1 - Review

• Order of operations (A-CED.1)
• Solve (two-step) one variable equations (A-CED.1)
• Basic graphing review (A-REI.10)

During this week the goal is connecting the learning from middle school to what we will be learning in Algebra 1. The lessons are a mix of learning/connecting and formative assessments. By the end of this first week I want a good feel for where my kids are with their math understanding.

I also want them to have a good feel for the active learning that will take place in the classroom. Click here to see the week #1 plan.

## Week #2 - What is a Function?

• Define function (F-IF.1)
• Vertical line test (F-IF.1)
• Domain and range (F-IF.1)
• Evaluate a function (F-IF.2)

Week #2 gives a more formal definition for a function. Although the concept of functions is introduced in middle school, this will be the first time the students will see function notation. Click here to see the week #2 plan.

## Week #3 - Graphing equations to answer questions

• Graphing equations (F-IF.4, A-CED.2, A-REI.11)
• Understanding systems of equations (A-CED.3)
• Create equations from real world situations (A-CED.2)

The goal in week #3 is to connect real world situations with different graphs. Students should be able to create graphs of situations, tables, and simple equations. Click here to see the Week #3 plan.

## Week #4 - Can I change the answers by changing the equations?

• Comparing graphs, tables, and equations (F-IF.9)
• Deeper modeling of patterns and equations (F-IF.5, F-BF.1.a)

Week #4 is a great place to help students see the big picture of translating graphs. My experience this year gave me the realization that although the kids can graph both the image and the preimage, they often did not see the connection until it was brought forward. The notation was also very difficult for kids to grasp. Click here to see the Week #4 Plan.

## Lesson Ideas

All of these resources and more can be found in the livebinder below.

## Common Core Standards

N.Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.*

N.Q.2 Define appropriate quantities for the purpose of descriptive modeling.*

A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.*

A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.*

A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.*

A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.* (Emphasize linear, absolute value, and exponential functions)

F.IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

F.IF.2 Understand the concept of a function and use function notation. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.* (Emphasize quadratic, linear, and exponential functions and comparisons among them)

F.IF.5 Interpret functions that arise in applications in terms of the context. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*

F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, give a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

F.BF.1.a Write a function that describes a relationship between two quantities. (Emphasize linear, quadratic, and exponential functions). Determine an explicit expression, a recursive process, or steps for calculation from a context. Video explaination

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