Teaching Resources for Algebra II
- Quadratic Transformer - demonstrate the effect on the graph of changes in the coefficients and the relationship between zeros and intercepts.
- Talk or Text - In this lesson, students compare different costs associated with two cell phone plans. They write equations with 2 variables and graph to find the solution of the system of equations. They then analyze the meaning of the graph and discuss other factors involved in choosing a cell phone plan.
- Egg Launch Contest- Students will represent quadratic functions as a table, with a graph, and with an equation. They will compare data and move between representations. In this activity, students encounter data that comes in different forms in the context of the description of an egg launch contest. The data for team A are shown in a table, the data for team B are expressed by an equation, and the data for team C are displayed in a graph. The data are available to students on the activity sheet.
- Finding the Domain of a Function- This applet guides the user through the process of finding the domain of a function. Hints and feedback are plentiful and useful. New problems are generated at the click of a button.
- Movement with Functions: Lesson 3- In this lesson, students use remote-controlled cars to create a system of equations. The solution of the system corresponds to the cars crashing. Multiple representations are woven together throughout the lesson, using graphs, scatter plots, equations, tables, and technological tools. Students calculate the time and place of the crash mathematically, and then test the results by crashing the cars into each other.
Title Quadratic Transformer URL http://seeingmath.concord.org/resources_files/QuadraticGeneral.html Materials needed Java-enabled web browser to run applet Learning Objectives Students understand how changes in the coefficients of a quadratic function change the graph. Students the relationship between the roots of a quadratic equation and the x-intercepts of a parabola.
Grade Levels Algebra 1 and Algebra 2 CA 97 Standards Algebra I: 21.0 Students graph quadratic functions and know that their roots are the x-intercepts. Algebra II: 9.0 Students demonstrate and explain the effect that changing a coefficient has on the graph of quadratic functions; that is, students can determine how the graph of a parabola changes as a, b, and c vary in the equation y = a(x-b)^ 2+ c. Algebra II: 10.0 Students graph quadratic functions and determine the maxima, minima, and zeros of the function. CA Common Core State Standards Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoningFunctions-IF 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. Functions-IF 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. Functions-IF 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. F-LE 6. Apply quadratic equations to physical problems, such as the motion of an object under the force of gravity. How could you use this resource? This Java applet could be shown to a class to demonstrate visually how changing the coefficients in a quadratic function changes the graph, and to show the relationship between roots and x-intercepts. Could be explored by students using a computer individually or in small groups. An included activity handout guides students to develop rules for how changes to the quadratic and constant coefficients of a quadratic function change its graph, and to briefly explore the "polynomial," "root" and "vertex" forms of a quadratic function. EL and Special Needs Primarily a visual demonstration of effects of changing coefficients; helps make the concepts accessible. Lesson Plans Teacher Comments Cost Free Copyright (c) 2005, The Concord Consortium
Title Talk or Text URL http://illuminations.nctm.org/LessonDetail.aspx?id=L780 Materials needed Computer with internet access (optional)
Information about current cell phone plans (optional)
Talk or Text? Activity Sheet
Talk or Text? Answer KeyLearning Objectives Students will:
- Compare two cell phone plans through examples of different usage
- Write equations to model allocation of money for cell phone usage
- Graph and solve a system of equations
- Analyze the solution and the meaning of the graph
Grade Levels Algebra I, Algebra II CA 97 Standards Algebra I 9.0 Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets Algebra II 2.0 Students solve systems of linear equations and inequalities by substitution, with graphs, or with matrices CA Common Core State Standards Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoningAlgebra-REI 5. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. Algebra-REI 6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. How could you use this resource? Students can use this activity in the lab to solve systems of equations or at home to find which plan would be better. Teachers can use this activity to help students discover the practicality of using systems of equations in two variables.
EL and Special Needs Group discussions can be used to help EL or special needs students to understand the concepts. Other differentiation can be used by teachers based on their own class needs.
Lesson Plans Teacher Comments Cost Free Copyright (c)2008 NCTM
Title Egg Launch Contest URL http://illuminations.nctm.org/LessonDetail.aspx?id=L738 Materials needed Egg Launch Activity Sheet
Graphing CalculatorLearning Objectives Students will:
- Move between representations of a function as a table, a graph and an equation
- Determine the maximum value of a quadratic function
- Compare quadratic functions
Grade Levels Algebra I, Algebra II
CA 97 Standards Algebra I: 21.0 Students graph quadratic functions and know that their roots are x-intercepts.
Algebra I: , 23.0 Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity. Algebra II: 8.0 Students solve and graph quadratic equations by factoring, completing the square, or using the quadratic formula. Students apply these techniques in solving word problems. They also solve quadratic equations in the complex number system. Algebra II: 10.0 Students graph quadratic functions and determine the maxima, minima, and zeros of the function. CA Common Core State Standards Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoningFunctions-IF 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. Functions-IF 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Functions-IF 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. F-LE 6. Apply quadratic equations to physical problems, such as the motion of an object under the force of gravity. How could you use this resource? Students can use this resource in a lab to apply quadratic functions to real world problems. Teachers can use this resource as a review for a unit because it has students use equations, tables, and graphs of quadratic functions.
EL and Special Needs The activity sheet has pictures and three representations of information. The students must then fill out the information for the other two representations of the information and compare it to similar data.
Lesson Plans Teacher Comments Cost Free Copyright (c)2000-2009 NCTM
Title Finding the Domain of a Function URL http://www.ltcconline.net/greenl/java/IntermedCollegeAlgebra/Domain
Equations/DomainEquations.htmlMaterials needed Java-enabled web browser to run applet Learning Objectives Students will be able to find the domain of a function and graph the domain on a number line, knowing whether to include the end point or not.
Grade Levels Algebra 1 and Algebra 2 (no standard) CA 97 Standards Algebra I: 17.0 Students determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression. CA Common Core State Standards Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
5. Use appropriate tools strategically
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoningFunctions-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). Functions-IF 2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. How could you use this resource? Students can use this resource as extra practice on finding the domain of a function and graphing the domain on a number line. Teachers can use this resource to demonstrate to students how to find the domain of a function and visually show them how to graph the domain of a function or can use the questions and other links on the page to find information to use in lesson on finding the domain of a function. EL and Special Needs There is a link on the bottom of the web page that goes to another link where a function and relation are defined using terms easily understandable with the correct math vocabulary after it. Many examples are given for students to investigate what is a function and what is not. There are examples given with solutions so students can see how the solution is derived. Lesson Plans Teacher Comments Cost Free Copyright Not available on web site
Title Movement with Functions: Lesson 3 URL http://illuminations.nctm.org/LessonDetail.aspx?ID=L770 Materials needed Stop watches
Remote-controlled cars (strongly suggested, but alternatives are described below)
Rulers
Colored Masking Tape
Collision Activity Sheet (optional pre-activity)
Road Rage Activity Sheet
What If? Activity Sheet (optional)
Road Rage Answer KeyLearning Objectives Students will:
- Collect data and graph a scatter plot to determine the speed of a remote-controlled car
- Create a line of best fit using estimation and technology
- Use tables, graphs, and algebraic calculation to determine when their cars will crash with another group's car
- Validate their calculations by crashing the cars into each other
- Analyze why their time and location estimates for the crash may not be the same as a real-life trial
Grade Levels Grade 7, Algebra I, Algebra II, Probability & Statistics, AP Probability & Statistics CA 97 Standards Grade 7: AF 3.3 Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same and know that the ratio (“rise over run”) is called the slope of a graph. Grade 7: SDAP 1.2 Represent two numerical variables on a scatterplot and informally describe how the data points are distributed and any apparent relationship that exists between the two variables. Algebra 1: 9.0 Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets. Algebra II: 2.0 Students solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices. Probability and Statistics: 8.0 Students organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots. AP Probability and Statistics: 12.0 Students find the line of best fit to a given distribution of data by using least squares regression. AP Probability and Statistics 14.0 Students organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line graphs and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots. CA Common Core State Standards Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoningGrade 6.SP.4; Display numerical data in plots on a number line, including dot plots, histograms, and box plots. Grade 8.EE.5: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Grade 8.EE.6: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. Grade 8-EE.8: Analyze and solve pairs of simultaneous linear equations.
a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because
points of intersection satisfy both equations simultaneously.
b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.
c. Solve real-world and mathematical problems leading to two linear
equations in two variables.Algebra-CED.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. Algebra-REI.7; Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Algebra-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). Algebra-REI.12; Graph the solutions to a linear inequality in two variables as a half plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. plots on a number line, including dot plots, histograms, and box plots. Statistics & Probability-ID.1; Represent data with plots on the real number line (dot plots, histograms, and box plots). Statistics & Probability-ID.5; Summarize categorical data for two categories in two-way frequency tables. Recognize possible associations and trends in the data. Statistics & Probability-ID.6; Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential model
b. Informally assess the fit of a function by plotting and analyzing residuals.
c. Fit a linear function for a scatter plot that suggests a linear association.Statistics & Probability-ID.7; Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Statistics & Probability-ID.8; Compute (using technology) and interpret the correlation coefficient of a linear fit. How could you use this resource? This activity can be used by students to create a systems of equations using manipulatives and data. Teachers can use this activity to create real life connections to systems of equations. EL and Special Needs This is a hands-on activity using manipulatives and group work to help students make connections for concepts and real life applications. Lesson Plans Teacher Comments Cost Free Copyright © 2000-2010 NCTM