MULTIPLE REPRESENTATIONS OF WORD PROBLEMS. Relating Word Problems, Tables and Line Graphs using the Graphing Calculator. Unit Title Linear Sentences

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1 MULTIPLE REPRESENTATIONS OF WORD PROBLEMS Relating Word Problems, Tables and Line Graphs using the Graphing Calculator Unit Title Linear Sentences University of Chicago School Mathematics Project ALGEBRA Grade 9 Time Span Five to Eight Days Graphing Calculator Overhead Calculator Screen Maria Cane Gomlak

2 Objectives of the unit 1. Recognize properties associated with linear sentences. 2. Use linear equations and inequalities of the form ax + b = cx + d or ax + b < cx + d to solve real-world problems. 3. Use tables to solve real-world problems involving linear situations. 4. Graph horizontal and vertical lines. 5. Use graphs to solve problems involving linear expressions. NCTM and New York State Standards 1. Operations 2. Modeling and Multiple Representations 3. Patterns and Functions Resources Textbook Scott Foresman Publishers The University of Chicago School Mathematics Project Algebra Copyright 1996 Authors John W. McConnell Susan Brown Zalman Usiskin Sharon Senk Ted Widerski Chapter 4 Lesson 4-6, Lesson 4-9, Chapter 5 - Lesson 5-1, Lesson 5-2 Accompanying Lesson Masters A and B

3 Materials Needed Graph paper, lined paper, rulers, colored pencils, graphing calculator, scientific calculators, overhead, overhead view screen, Scott Foresman s, Algebra. OVERVIEW Day 1 Exploring a constant sum as a linear graph. Day 2 Exploring graphing equations of the form y = ax + b, y = ax b Day 3 Exploring horizontal and vertical lines formed by equations y = a constant, x = a constant. Day 4 Using tables, spreadsheets to solve real-world problems involving linear situations. Finding information previously unavailable in the question from graphs. Day 5 Drill and Practice. Pairing to reinforce knowledge.

Day 1 Graphing Sums and Differences Objectives: to develop a line for a real life situation; to describe algebraically the real life situation; to be able to create a table of values; to graph the

4 Day 1 Graphing Sums and Differences Objectives: to develop a line for a real life situation; to describe algebraically the real life situation; to be able to create a table of values; to graph the values on a coordinate grid. To note the changes in the inclination of the line when the problem is altered; to note information that develops the concept of the y- intercept. Materials: graphing calculator, text, Lesson Masters 4-6,A&B Presentation: Exploring a constant sum. Tim was asked to buy a dozen muffins. He was told to buy only raisin or bran muffins. What would be an equation for this situation? Create a table of possible purchases. Here is the table. Using L1 and L2 as (x,y) coordinates graph these points. Go to 2 nd, Y= (stat plot)

Notice Plot 1 is ON, and is using L1 and L2. Enter. We now need to choose the type of graph: scatter plot, line, histogram, box and whisker, etc. I turned Plot 1 on, and selected a scatter plot.

5 Notice Plot 1 is ON, and is using L1 and L2. Enter. We now need to choose the type of graph: scatter plot, line, histogram, box and whisker, etc. I turned Plot 1 on, and selected a scatter plot. Scrolling down to the end and hit enter. I need to go to window and choose the limits for x and y. Go to zoom and choose 5 Zsquare, and make the axes the same size. Note a constant sum, x + y = 12, is a set of points with a decreasing slope. Imagining the line that would exist if our set of points were the real numbers. The line in the form of y = mx + b, would have the equation is y = -x + b. Notice for every increase of one on the x-coordinate the y-coordinate decreases by one. Since slope is the change in y / change in x, we see that

6 the slope is 1/1 or 1. Notice where the x-coordinate is zero, the y-value would be 12. So the y-intercept, the point where the line cuts the y-axis is 12. So m = -1 and b = 12. Seatwork: Students read through examples 2 and 3, ppp247,248. Ask any questions. Students work on Lesson Master 4-6A, questions 1&2 in class. Homework: Students work on Lesson Master 4-6B 1-4 for homework.

7 Day 2 Graphing Linear Patterns Objectives: Graph equations in the form y = ax + b or y = ax-b by making a table of values. Presentation: Give the students a practical problem to solve using a table to collect their data. Have the graph the data. Students draw a line if the domain and range are real numbers. Ask questions whose answers are not in the data but can be found by using the line graph. (This method is used extensively in the Carnegie Math computer program, which has been instituted in Buffalo. The courses are Integrated Math I & II. These courses parallel Algebra and Geometry and use a workbook for 3 days and students use a computer lab for 3 days, in a six day cycle. Questions are randomly generated so that students are working on individual questions.) A typical problem: Jon has $16 in his savings account. He just started a part-time job and will be able to save $12 per week for holiday shopping. It is September 1 st, and he will get his first paycheck on the 8 th. He plans on saving $12 every week. Make a table to show his savings for 5 weeks. Make a graph of your data. In 8 weeks will he have enough to spend $15 on each of his 4 brothers and still have enough for a present for his parents that costs $65? If not how many weeks must he work? Fill in the following table.

8 Go to 2 nd Stat plot. After adjusting the window

9 Graph To get a line of best fit, go to Stat Calc

10 4:linReg(ax + b),enter VARS

11 Y-VARS ENTER ENTER

12 ENTER, ENTER, Therefore the equation of the line is y = 12 x GRAPH

Now we can answer the question, In 8 weeks will he have enough saved to spend $15 on each of 4 brothers and $65 on his parents? $15*4 + 65 = $125.

13 Now we can answer the question, In 8 weeks will he have enough saved to spend $15 on each of 4 brothers and $65 on his parents? $15* = $125. We can use the trace on the calculator. Apparently not. He will have only $112 in 8 weeks. How many weeks must he save? He must save for ten weeks and he will have $136, enough to buy the gifts.

14 Seatwork: 269. Students will read and work through examples 1 & 2, pages268 and Homework: Lesson Master 4-9 A

Day 3 Linear Sentences- Horizontal and Vertical Lines Objective: Students are to get a sense of a line whose equation is x = a. If x = a, then x is always a, but y has an infinite number of values.

15 Day 3 Linear Sentences- Horizontal and Vertical Lines Objective: Students are to get a sense of a line whose equation is x = a. If x = a, then x is always a, but y has an infinite number of values. If y = a constant number, then y has a constant value and x has an infinite numbers of values. Students are to see that a line x = a, a constant is a vertical line perpendicular to the x-axis at the constant. A line y = a, is a horizontal line perpendicular to the y-axis at the constant. For example, from 1974 to 1987 the national speed limit was 55mph. In July, 1987 the speed limit was raised to 65 mph. We can put this information in a table and then graph the information. Notice the speed limit is constant, 55 mph, between 1974 and 1987, and then again from 1987 to the present. Go to 2 nd STAT PLOT, ENTER,

16 ENTER, scroll through the screen, ENTER., ZOOM FIT

Seatwork: Students will read pages 284, 285, 286 and study

17 ENTER,GRAPH This shows that where Y = a constant like 55 mph and 65 mph, we have a horizontal line graph. Seatwork: Students will read pages 284, 285, 286 and study the examples, asking questions. Homework: Students will do Lesson Master 5-1A

Day 4 Using Tables to Compare Linear Expressions Objectives: Use tables or spreadsheets to solve real-world problems involving linear situations.

18 Day 4 Using Tables to Compare Linear Expressions Objectives: Use tables or spreadsheets to solve real-world problems involving linear situations. Presentation: We will discuss comparing parking costs at two locations. We will make a table and a graph and then compare the two situations. At Gil s garage it costs $1.50 for the first hour to park and $0.80 for each additional hour. At Patsy s garage it costs $1.80 for the first hour and $0.75 for each additional hour. When is it cheaper to park at Gil s? When is it cheaper to park at Patsy s? We can chart these costs. 2 nd Stat Plot plot 2

Zoom Stat Zoom in TRACE At x = 6, y =.80*x + 1.

19 Zoom Stat Zoom in TRACE At x = 6, y =.80*x is 6.3, but y =.75 x is 6.3. Where the number of hours, x < 6, then Gil s is cheaper, but after 6 hours Patsy s Parking is cheaper. Window

20 Zoom Fit, Stat,Edit Calc 4 Linear regression Vars Y-vars1:Function enter enter yields ENTER

21 GRAPH Go back to stat lists 2 nd Stat Plot Plot2 Enter Scroll Through Enter Stat Calc 4: LinReg(ax + b) enter LinReg(as +b) L 1,L 3, Vars Y-Vars Enter Seatwork: Students will read through example2, pages and ask questions. They will prepare tables with the data.

22 Homework: Students will graph the data. Each equation will be in a different color. Axes must be labeled. A title must be at the top. The scale of the axes must be noted.

23 Day 5 Drill and Practice: Objective: Students will work in pairs to prepare charts and a graph to describe a problem. Students will learn to organize their work, cooperate, and prepare for a presentation. Presentation: Directions: Tables must be neat and labeled on each column. There must be a title at the top of the graph. The axes must be labeled and numbered. All lines must be drawn with a ruler. Only neat work will be accepted. Each line must be in a different color. Supplies are lined paper, graph paper, and calculators. Seatwork: Complete the chart for question 1, Lesson Master 5-2B, make an appropriate graph. Homework: Complete the chart for question 2, Lesson Master 5-2B, and make an appropriate graph.

************************************** FINDING WAYS TO USE TECHNOLOGY TO ASSIST OUR STUDENTS WITH GRAPHING A VARIETY OF LINEAR EQUATIONS!!

************************************** FINDING WAYS TO USE TECHNOLOGY TO ASSIST OUR STUDENTS WITH GRAPHING A VARIETY OF LINEAR EQUATIONS!! ************************************** PROJECT SUMMARY 1. Title