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CED-A: Create equations that describe numbers or relationships

**Standards**:

- CED-A.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*. - CED-A.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
- CED-A.3. Represent constraints by equations or
inequalities, and by systems of equations and/or inequalities, and
interpret solutions as viable or nonviable options in a modeling
context.
*For example, represent inequalities describing nutritional and cost constraints on combinations of different foods*. - CED-A.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*.

**Technology-enhanced investigations:**

Mathwords.com: Math Dictionary: Key vocabulary for this domain. Use with CED-A:

Concord Consortium STEM resources: Mathematics: Each of the following includes an applet, student activity, teacher lesson plan, and student assessment.

- Linear Equations: Ski Slope: Learn about the numeric representation of the slope of a line, including using rise and run to graph lines with a given slope. Aligns with standard CED-A.2.
- y= mx + b: Learn to graph a line using its slope and y-intercept, or to identify the slope and y-intercept from a linear equation written in slope-intercept form. Aligns with standard CED-A.2, and REI-D.10.
- Linear Equations: Points, Intercepts, and Slopes, Oh My!: Students learn to graph lines using the x- and y-intercepts of equations, as well as how to graph a line when given an equation in point-slope form. Aligns with standard CED-A.2, and REI-D.10.
- Linear Equations: Word Problems: Learn to apply linear equations and their graphs to real-world problems. Write an equation based on a word problem, and graph the line described by your equation. Relate the equation and the line to the situation described in words. Aligns with CED-A.2 and REI-D.10.
- Solving Systems of Equations: Two linear equations, with two variables, are presented and students use graphs to solve the system of equations. Students graph the lines and solve the problem using the graphs. Aligns with CED-A.2 and REI-C.6.
- Systems of Equations: Word Problems 1: Students explore two real-life scenarios that involve solving systems of equations. The first situation involves mixing chemicals to create a new solution while the second situation involves cost and revenue for a bake sale. Aligns with CED-A.2 and REI-C.6.
- Systems of Equations: Word Problems 2: Students explore two new scenarios that involve solving systems of equations. The first situation involves the travel time for a round trip on an airplane, while the second situation involves calculating the numbers of students and adults attending a high school musical. Aligns with CED-A.2 and REI-C.6.

MathsIsFun.com: Explore multiple types of graphs with Make Your Own Graphs. Select a Function Grapher and Calculator (also graphs polar forms), an Equation Grapher (e.g., great for conics), and more. You can begin by exploring linear and quadratic graphs. Aligns with CED-A.2.

MIT BLOSSOMS: Video lessons with additional teacher and learner resources. Descriptions are from the video summaries.

- Free Fall: "This video lesson is an example of "teaching for understanding" in lieu of providing students with formulas for determining the height of a dropped (or projected) object at any time during its fall. The concept presented here of creating a chart to organize and analyze data collected in a simple experiment is broadly useful." Aligns with Algebra standard CED-A.2 and Function standards IF-B.6, IF-C.7.a, and LE-A.1.a.
- The Quadratic Equation: It’s Hip to Be Squared: "The lesson begins with three
quadratics and their graphs (three parabolas): y = x
^{2}- 2x + (0 or 1 or 2). The prerequisite or co-requisite is some working experience with algebra, like factoring x^{2}-2x into x(x-2). The objective is to connect four things: the formula for y, the graph of y (a parabola), the roots of y and the minimum or maximum of y." Aligns with Algebra standards SSE-B.3.a, CED-A.2, and REI-B.4.b and Function standards IF-B.4, IF-C.7.a, IF-C.8.a, BF-B.3. - The Power of Exponentials, Big and Small: "Exponential growth is keenly applicable to a variety of different fields ranging from cell growth in biology, nuclear chain reactions in physics to computational complexity in computer science. In this lesson, through various examples and activities, [the presenters] have tried to compare exponential growth to polynomial growth and to develop an insight about how quickly the number can grow or decay in exponentials." Aligns with Algebra standards SSE-B.3.c and CED-A.2 and Function standards IF-A.3, IF-C.7.(a, c, e), IF-C.8.b, and LE-A.1.(a, c).
- Selfish Drivers: The Braess Paradox and Traffic Planning: "The idea of this lesson is to introduce, in a simplified manner, the so-called Braess Paradox by providing simple examples to clarify that the addition of some new roads to a network does not always lead to an improvement in the liquidity of the traffic; in some cases it might even increase the time required to get from one point to another if all drivers are selfish. The educational objectives for this lesson can be summarized as follows: providing a practical application of mathematics; indicating the importance of scientific research; training the students on methods of counting and surveying all the possibilities in a given problem; and introducing Nash Equilibrium and Braess Paradox in a simplified manner by visual examples." Aligns with CED-A.3 and REI-C.6.

Ohio Resource Center on YouTube: Tutorials for High School Mathematics. Select the following:

- Solving One-Step Literal Equations. Using inverse operations to solve simple equations.
- Solving Multi-Step Literal Equations. Using inverse operations to solve complex equations based on real world examples.
- Mixture and Concentration Problems for setting up and solving mixture and concentration problems with science examples.
- Using Rates and Proportions for working with unit rates, ratios, and proportions.

LearnZillion:

- Lesson set: Solve problems with equations and inequalities in one variable: This is a series of six video lessons that also include guided practice on this topic. Videos demonstrate how to create and solve: 1. linear inequalities, 2. quadratic equations, 3. quadratic inequalities, 4. rational equations by expressing relationships as unit rates, 5. simple exponential functions by examining exponential growth and decay problems, and 6. exponential equations by using a table of values. Aligns with standard CED-A.1.
- Lesson set: Create equations in two or more variables to represent relationships between quantities (non-linear functions): This is a series of five video lessons that also include guided practice on this topic. Videos demonstrate how to create an equation for a geometric sequence involving exponential growth or decay by making a table and drawing a graph; how to represent solutions and constraints to systems of linear inequalities by graphing; how to solve problems involving quadratic functions by using a table of values; and how to create an equation for a simple rational function by using a graph and a table of values. Aligns with standard CED-A.2.
- Lesson set: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities: This is a series of four video lessons that also include guided practice on this topic. Aligns with standard CED-A.3.

YouTube video lessons. Use the following with CED-A.4:

- Rearrange formulas to isolate specific variables
- 3b Rearrange Formulas
- Solving for a variable in literal equations

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives:

- US Common Core State Standard Math HSA-CED.A.1: 20 manipulatives. Among those: Maximizing area of rectangle with fixed perimeter (uses a quadratic), carbon dating (uses an exponential), and more.
- US Common Core State Standard Math HSA-CED.A.2: 26 manipulatives. Among those: Finding the linear function rule, finding a pattern (linear equations), linear function game, slope and equations of lines through points, and more.
- US Common Core State Standard Math HSA-CED.A.3: 10 manipulatives. Among those: Graphing systems of inequalities, graph of inequalities, the greatest fenced area along a barn, a word problem about boats, and more.
- US Common Core State Standard Math HSA-CED.A.4: 2 manipulatives: The Wire problem and Maximizing the area of some geometric figures of fixed perimeter.

Thinking Mathematics:

- Fitting Quadratics to Data: A YouTube video on the easy way to write down the quadratic that fits three given data points.
- Fitting Exponentials to Data: A YouTube video on the easy way to find the exponential function that fits two given data points.
- The Equation of a Circle: A YouTube video.

McDougal Littell ClassZone: Select resources for interactive practice: Animations (Virtual manipulatives), Games/Vocabulary Flashcards, and PowerPoint lesson examples, and interactive problems within the eWorkbook. See multiple choice below for sections to use within the eWorkbook for each of the following. Use with CED-A:

- Algebra 2, 2011: Select resources in Ch. 1 Equations and Inequalities, Ch. 2 Linear Equations and Functions, Ch. 4 Quadratic Functions and Factoring, and Ch. 7 Exponential and Logarithmic Functions
- Geometry, 2011: Select resources in Ch. 3 Parallel and Perpendicular Lines

NCTM Illuminations:

- Compound Interest Simulator: This applet will allow learners to investigate savings account earnings, credit card debt, and a stock market simulation.
- Escape from the Tomb: Learners solve a system of linear equations.
- Pedal Power: Learners investigate slope as a rate of change. Learners compare, contrast, and make conjectures based on distance-time graphs for three bicyclists climbing to the top of a mountain.
- Supply and Demand: Learners explain factors that affect supply and demand, and create and solve a system of linear equations in a real-world setting. By solving the system, Learners find the equilibrium point for supply and demand. They also translate between table, graph and equation representations for supply and demand data.

Purplemath: Translating Word Problems: Keywords and Worked Examples Following this are sections with explanations and examples of typical word problems that learners encounter, including the following:

- Age problems
- Area/volume/perimeter problems involving basic geometric formulas
- Coin problems
- Distance problems involving the formula
*d = rt* - Investment problems involving the formula
*I = Prt* - Mixture problems
- Number problems
- "Percent of" problems
- Quadratic word problems, such as projectile motion and max/min
- Work problems involving two or more people or things working together to complete a task

Khan Academy video lessons:

- Age Word Problems
- Mixture Problems 1 (Coffee mixture) posted at YouTube
- Mixture Problems 2 (Salt solution)
- Mixture Problems 3 (Sugar solution)
- Area and Perimeter word problem: using a table
- Sums of consecutive integers

Additional YouTube video lessons:

- Distance = Rate x Time (Airplanes traveling in opposite directions)
- Coins (coins consisting of dimes and nickels)
- Rate of Work Problems (How much time does it take to complete the job if two people or things work together?)
- Investments (Use Interest =Principal x Rate: Money was invested, part at one rate and part at another rate. The total interest in one year is provided. How much was invested at each rate?)

**Multiple Choice:**

CK-12: Algebra: Video examples, practice problems online, and real world applications. The following align with CED-A.2:

Glencoe Online Learning Center, Algebra 1 (2010) self-check quizzes:

- Ch. 0, Lesson 1: Plan for Problem Solving
- Ch. 0, Lesson 6: The Percent Proportion involves "percent of" problems
- Ch. 2, Lesson 1: Writing Equations. Aligns with CED-A.1.
- Ch. 2, Lesson 8: Literal Equations and Dimensional Analysis. Aligns with CED-A.4.

MathsIsFun.com: Tutorials on equations. Examples are shown. Learners then complete 10 multiple choice problems in each section. Aligns with CED-A.2:

McDougal Littell ClassZone, Algebra 2, 2011:

- Ch. 2.4 Quiz: Write Equations of Lines: Aligns with CED-A.2.
- Ch. 4.10 Quiz: Write Quadratic Functions and Models: Aligns with CED-A.2.
- Ch. 7.7 Quiz: Write and Apply Exponential and Power Functions: Aligns with CED-A.2.
- Ch. 1.4 Quiz: Rewrite Formulas and Equations: Aligns with CED-A.4.

McDougal Littell ClassZone, Geometry, 2011:

- Ch. 3.4 Quiz: Find and Use Slopes of Lines: Aligns with CED-A.2.
- Ch. 3.5 Quiz: Write and Graph Equations of Lines: Aligns with CED-A.2.
- Ch. 3.5B Quiz: More Questions on Write and Graph Equations of Lines: Aligns with CED-A.2.

**Constructed-response:**

Achieve the Core: Quadratic Equations Mini-Assessment (pdf or Word docx) can be used to evaluate students' understanding of algebra standards REI-B.4 and CED-A.1 in order to prepare to teach this material or to check for student ability to demonstrate understanding and apply these concepts.

West Texas A & M University: Virtual Math Lab: Intermediate Algebra: Tutorial 16: Equations of Lines. Use with CED-A.1.

Khan Academy: Also see College Algebra Brush Up: Quadratic Equations: Practice questions with videos. Use the following with CED-A.2:

- Write the equation of a line in any form
- Constructing linear functions word problems: Write the linear equation to model a situation.
- Graphing solutions to two-variable linear equations.
- Graphing parabolas in all forms: vertex and standard forms

Algebra Lab:

- Word Problem Exercises: Inequalities
- Writing a Linear Equation: Real World Situations: Learners write a linear equation from a word problem and use it to answer various questions.
- Solving Literal Equations: Lesson with practice problems. This aligns with standard CED-A.4.

Consortium for Mathematics and Its Applications (COMAP): Mathematical Modeling Handbook (Sample). This sample mathematical modeling module aligns to Common Core Standards. It contains teacher guidance for implementation and student questions to complete during their investigation:

- The module
*For the Birds*aligns to number and quantity standards Q-A.1 and Q-A.2, and algebra standard CED-A.4.

OpenEd:

- Create equations and inequalities in one variable and use them to solve problems: 4 questions combine free response and multiple choice. Includes related review resources. Aligns with CED-A.1.
- Create equations in two or more variables: 5 free response questions. Includes related review resources. Aligns with CED-A.2.
- Represent constraints by equations or inequalities: 5 questions combine free response and multiple choice. Includes related review resources. Aligns with CED-A.3.
- Rewrite formulas: 5 free response questions. Includes related review resources. Aligns with CED-A.4.

**Performance tasks:**

Illustrative Mathematics: Algebra:

- Cash Box: Aligns with CED-A and REI.C.6.
- Sum of Angles in a Polygon: Aligns with CED-A.1.
- Paying the Rent: Aligns with CED-A.1.
- Buying a Car: Aligns with CED-A.1.
- Planes and Wheat: Aligns with CED-A.1.
- How Much Folate?: Aligns with CED-A.2 and CED-A.3.
- Fishing Adventures 3: Students are given a real world problem in which they write equations and graph inequalities to solve it. Aligns with REI-D.12 and CED-A.3.
- Writing Constraints: Aligns with CED-A.3.

Inside Mathematics: MARS Tasks:

- Number Towers: Learners form and solve equations obtained from a number pattern. This task aligns with algebra standards CED-A.2 and CED-A.3.
- Expressions: Learners work with algebraic expressions for areas and perimeters of parallelograms and trapezoids. This task aligns with standard CED-A.4.
- Sorting Functions (pdf) or see all MARS Tasks: Functions & Relations: Course 1 Algebra: Scroll down the page to select Sorting Functions: Learners match graphs of four functions with their equations, tables, and rules. This task aligns with function standard IF-C.7 and algebra standard CED-A.2.

Mathematics Assessment Project: Standards: High School: Algebra: http://map.mathshell.org/stds.php?standardid=1190 Choose from the following:

- Task 403: Inscribing and Circumscribing Right Triangles
- Task 406: Equations of Circles 1
- Task 209: Solving Linear Equations in Two Variables
- Task 425: Equations of Circles 2
- Task 207: Optimization Problems: Boomerangs

Mathematics Vision Project, Secondary 1 Student Edition:

- Getting Ready Module: This module contains 10 classroom tasks. Task 5: Cafeteria Actions and Reactions aligns to algebra standard CED-A.1; Task 6: Elvira’s Equations and Task 7: Solving Equations, Literally align with CED-A.4.
- Module 1: Systems of Equations and Inequalities: This module contains 10 classroom tasks. Standards addressed include REI-C.5, REI-C.6, REI-D.12, and CED-A.2, CED-A.3, and CED-A.4.
- Task 2: Too Big or Not Too Big, That is the Question; Task 3: Some of One, None of the Other; Task 4: Pampering and Feeding Time align with CED-A.2;
- Task 1: Pet Sitters; Task 4: Pampering and Feeding Time; Task 5: All for One, One for All, parts 1 & 2; and Task 9: Food for Fido and Fluffy align with CED-A.3.
- Module 4: Features of Functions: This module contains nine classroom tasks. Task 4: Water Park; Task 5: Pooling it Together; and Task 6: Interpreting Functions align with CED-A.4.

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