Solving Equations by Graphing

Transcription

1 8.1 Solving Equations by Graphing Goal Use tables and graphs to solve equations. STUDENT BOOK PAGES 5 55 Guided Activity Prerequisite Skills/Concepts Create a table of values for a pattern. Use a variable to write a pattern rule as an algebraic expression. Graph a pattern to solve a problem. Assessment for Feedback Students will model linear relationships using tables and graphs Preparation and Planning Pacing (allow 5 min for previous homework) Materials Vocabulary/ Symbols 5 min Introduction min Teaching and Learning 15 min Consolidation rulers coloured pencils 1 cm Grid Paper, Masters Booklet p. 3 or.5 cm Grid Paper, Masters Booklet p. 3 Optional: Scaffolding for Lesson 8.1 (Master) p. 1 equation, solution to an equation, table of values, graph Workbook p. 77 Recommended 5*, (Application of Learning), 8, 9 Practice (Knowledge and Understanding) Additional 7 Practice Extending: Learning Skills Use of Information, Cooperation with Others Key Question 5, Application of Learning Assessment of Learning Question Mathematical Connecting, Reflecting Processes * Key Assessment of Learning Question (See chart on p..) Expectations represent, through investigation with concrete materials, the general term of a linear pattern, using one or more algebraic expressions represent linear patterns graphically (i.e., make a table of values that shows the term number and the term, and plot the coordinates on a graph), using a variety of tools determine a term, given its term number, in a linear pattern that is represented by a graph or an algebraic equation model linear relationships using tables of values, graphs, and equations, through investigation using a variety of tools translate statements describing mathematical relationships into algebraic expressions and equations make connections between solving equations and determining the term number in a pattern, using the general term What You Will See Students Doing When students understand Students will correctly make a graph from a table of values for a given algebraic expression or pattern. Meeting Individual Needs If students misunderstand Students may not remember how to make a table of values for a pattern, or how to graph the values. Review how to create a table of values and how to obtain and graph the ordered pairs. Extra Challenge Challenge students to come up with their own tile patterns in one or two colours, for example: Students should make a table of values, write a pattern rule, write an algebraic expression for the pattern rule, and draw a graph using the table of values. Extra Support For practice with algebraic expressions, give students a pattern rule stated in algebraic terms and have them create a table of values to go along with it. Repeat this a number of times. Math Background This lesson is a direct extension of Lesson.3 and Lesson.5 in Chapter. In those lessons, students learned how to use a table of values to determine a pattern rule by looking for a relationship either between the term number and the term value, or between the term values. Students could then find the general term of a pattern without finding all the terms before it. In this lesson, students also build on their understanding that graphs are an efficient way to visually represent an algebraic relationship. By substituting numbers for the variable in an equation and creating a table of values, students can graph the relationship between the term number and the term value. Students can then use the graphs to solve equations and problems, and make predictions based on the trends of the graph. Students can then find the general term or the term number (n) for a given value from the graph. 1 Chapter 8: Equations and Relationships Copyright by Thomson Nelson

2 Dealing with Homework (Whole Class) about 5 min Place students into groups of four (or into their homework groups) to compare answers to the assigned questions from Do You Remember? on Student Book page 51. Ask groups to identify any question(s) that more than half of the group members were not able to complete correctly, and take a few minutes to discuss how to solve each question. 1. Introduction (Whole Class) about 5 min This lesson deals with visually representing an algebraic expression using graphs created from tables of values. The graphs are then used to solve equations. Review with students the terms table of values, ordered pairs, coordinates, and graph. Review how a graph can be created from the data in a table of values. Go over how ordered pairs can be written using the term number and term value in a table of values. Explain that the first number in the ordered pair refers to the term number and is plotted using the horizontal axis. The second number refers to the term value and is plotted using the vertical axis. Sample Discourse How do you create a table of values from an algebraic expression? First you substitute different numbers for the variable in the expression. These numbers become the term number in a table of values. Then you evaluate the expression for each substituted number. The answers become the term value in the table of values. How do you graph an expression using the table of values? The term number and the term value can be written as ordered pairs. The first number in the ordered pair refers to the term number and is plotted using the horizontal axis. The second number refers to the term value and is plotted using the vertical axis. Tell students that in this lesson they will be learning how to solve equations using tables of values and graphs.. Teaching and Learning (Whole Class/Pairs) about min Learn about the Math Read through the problem and central question together. Read the definitions for equation and solution to an equation. Do prompts A to C as a class, and complete the table of values. Use an overhead transparency to draw the graph in prompt B. Remind students of what was done in the Introduction to generate ordered pairs from the table of values. Discuss why the data points should be connected using a dashed line (variables are discrete, not continuous), and why you can extend that line. Finally, ask students how to determine the area of the garden enclosed by 3 border tiles. This is an important concept and may not have been covered previously. Students should be familiar with how to determine the point on the line given the variable, i.e., determining the point on the vertical axis given the point on the horizontal axis. However, this question is asking students to determine the point on the horizontal axis given the point on the vertical axis. That is, students find the point on the graphed line with a term value of 3 and drop down to determine the term number. Encourage students to use a ruler or straight paper edge to draw imaginary lines from a point on the graphed line to the horizontal and vertical axes. Do this several times, reinforcing that in this particular graph the horizontal axis represents the area of the garden and the vertical axis represents the number of border tiles. Students can then work in pairs on the Reflecting questions. Answers to Learn about the Math Learn about the Math A. Area of garden Number of border (term number) tiles (term value) n n + Copyright by Thomson Nelson Lesson 1: Solving Equations by Graphing 15

3 B. Number of border tiles 3 1 C. 13 square units Number of Border Tiles Compared to Area of Garden Area of garden Reflecting Some of the ideas in the Reflecting section were discussed in the Introduction and Teaching sections of this lesson. However, it is important for students to think about them independently and put their thoughts in writing to reinforce their own understanding. Answers to Reflecting 1. The graph represented the relationship between the term number and the term value. Because this relationship is constant, the graph is a straight line. If you kept substituting different numbers for the term number, the term value would lie somewhere along the line that has been graphed. This line could go on forever, depending on the numbers you chose to substitute.. a) I graphed the coordinates (1, 8), (, ), (3, 1), (, 1), and (5, ). Then I connected these points with a dashed line and extended the line. I drew a horizontal line from 3 on the vertical axis until it touched the extended line. I drew a vertical line from the point of intersection to the horizontal axis. It touched at 13 on the horizontal axis, so the solution to the equation n + = 3 is n = 13. b) I substituted 13 for the variable n in the expression n + and got 3. So n = 13 is the correct answer for the equation n + = Consolidation about 15 min Work with the Math Solved Example (Whole Class/Pairs) Have students read the example in pairs. Students should check that their partner understands each step of using a graph to solve an equation. Each partner should know how to create a table of values, how to create a graph from the table of values, and how to read and predict information on the graph. As a class, discuss the example before assigning the Checking and Practising questions. Discuss what each of the lines on the graph means (blue dotted, red dotted, red solid). Talk about how you can check a solution to an equation by substituting that value for the variable. A Checking (Whole Class/Pairs) Have students work in pairs to solve Questions 3 and. Make sure that each partner is able to explain how to solve the problems. Answers to Checking 3. I placed a ruler horizontally from 17 on the vertical axis until it touched the dashed line. I placed a ruler vertically It touched at 7 on the horizontal axis, so the solution to the equation n + 3 = 17 is n = 7.. a) Number of tiles b) Each time, one tile is added to the previous figure, so 1 n or n is in the pattern rule, where n represents the figure number. Each number of tiles is more than n, so the algebraic expression should be n +. c) An algebraic expression for the pattern rule is n +. So an equation to determine which figure number has tiles is n + =. Chapter 8: Equations and Relationships Copyright by Thomson Nelson

4 d) Draw a horizontal line from on the vertical axis until it touches the extended line. Draw a vertical line It touches at on the horizontal axis, so there are counters in the th figure. Number of tiles Number of Tiles Compared to Figure Number B Practising (Individual) These questions provide opportunities to use tables and graphs to solve equations. If extra support is required, provide copies of Scaffolding for Lesson 8.1, Question, p. 1. Answers to Key Assessment of Learning Question 5. (Application of Learning) a) Number of counters b) In the table, three is added to the previous number, so 3n is in the pattern rule, where n represents the figure number. The first figure has 5 counters. Since 3 is taken into account by n already and figure 1 has 5 counters, the general expression should be 3n +. c) An algebraic expression for the pattern rule is 3n +. So an equation to determine which figure has 3 counters is 3n + = 3. d) Draw a horizontal line from 3 on the vertical axis until it touches the extended line. Draw a vertical line It touches at 7 on the horizontal axis, so there are 3 counters in the 7th figure. Check the solution by substituting n = 7 into 3n +. 3(7) + = 3 Number of counters Number of Counters Compared to Figure Number Key Assessment of Learning Question (See chart on p..) 3 5 C Extending (Individual) The challenge in Question is that both coloured tiles are changing in number. So the number of each colour of tile is not constant from figure to figure. However, both types of coloured tiles change relative to the figure number. The question allows for both pattern rules to be created separately. 7 Copyright by Thomson Nelson Lesson 1: Solving Equations by Graphing 17

5 Closing (Individual) Ask some students to share their answers to Question 9, and to explain their method of solution. Ask students if they could have predicted what the graphs might look like before solving the problem. Afterwards have students answer the question, How do the equations and solutions compare? Follow-Up and Preparation for Next Class At the end of the class, ask students when they might find it useful to represent an equation or a pattern using a graph. Assessment of Learning What to Look for in Student Work Assessment Strategy: short answer Application of Learning Key Assessment Question 5 a) Make a table of values for this pattern. b) Write an algebraic expression for the pattern rule. c) Create an equation to determine the number of the figure with 3 counters. d) Draw a graph to solve your equation. 1 3 demonstrates limited ability to apply mathematical knowledge and skills in familiar contexts (e.g., has difficulty using the pattern to make a table of values, write an algebraic demonstrates some ability to apply mathematical knowledge and skills in familiar contexts (e.g., demonstrates some ability to use the pattern to make a table of values, write an algebraic expression, and create an equation) demonstrates some ability to transfer mathematical knowledge demonstrates some ability to solve the equation using a graph to determine the figure number with 3 counters) demonstrates considerable ability to apply mathematical knowledge and skills in familiar contexts (e.g., uses the pattern to make a table of values, write an algebraic demonstrates sophisticated ability to apply mathematical knowledge and skills in familiar contexts (e.g., demonstrates sophisticated ability to use the pattern to make a table of values, write an algebraic demonstrates limited ability to transfer mathematical knowledge has difficulty solving the equation using a graph to determine the figure number with 3 counters) demonstrates considerable ability to transfer mathematical knowledge solves the equation using a graph to determine the figure number with 3 counters) demonstrates sophisticated ability to transfer mathematical knowledge demonstrates sophisticated ability to solve the equation using a graph to determine the figure number with 3 counters) Chapter 8: Equations and Relationships Copyright by Thomson Nelson