Rules, Tables, and Graphs: Part 2

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1 Rules, Tables, and Graphs: Part Objective To provide experiences with interpreting tables and graphs. epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Construct line graphs that represent two sets of data. [Data and Chance Goal ] Extend patterns in graphs and tables to solve problems. [Patterns, Functions, and Algebra Goal ] Represent rates with formulas, tables, and graphs. [Patterns, Functions, and Algebra Goal ] Key Activities Students complete a table of values that displays the distance covered by each of two students at various time intervals. They interpret the data in the table, graph the data, and interpret the graph. Ongoing Assessment: Informing Instruction See page 87. Ongoing Assessment: Recognizing Student Achievement Use an Exit Slip (Math Masters, page ). [Data and Chance Goal ] Playing Mixed-Number Spin or Fraction Spin Math Journal, pp. and 6 Math Masters, pp. 7, 7, 88, and 89 per partnership: large paper clip Students practice estimating sums and differences of fractions and/or mixed numbers. Math Boxes 6 Math Journal, p. Students practice and maintain skills through Math Box problems. Study Link 6 Math Masters, p. 6 Students practice and maintain skills through Study Link activities. READINESS Interpreting Table Data for Graphs Math Masters, p. 7 straightedge Students identify how table data translates to graph elements. ENRICHMENT Graphing Race Results Math Masters, p. 9 per group: stopwatch, or different-colored pencils or markers, straightedge Students use different movements to cover the same distance, time themselves, and graph the results. EXTRA PRACTICE Analyzing Two Rules Math Masters, pp. 7A and 7B Students derive data from two rates by making a table and graphing ordered pairs. Key Vocabulary coordinates ordered number pairs Materials Math Journal, pp. and Study Link Math Masters, p. Class Data Pad (optional) slate calculator different-colored pencils or markers (optional) straightedge Advance Preparation For the optional Enrichment activity in Part, mark off a -meter course for a race. Make copy of Math Masters, page 9 for each small group. Teacher s Reference Manual, Grades 6 pp. 6 68, 6 8 Unit Using Data; Algebra Concepts and Skills

2 Getting Started Mathematical Practices SMP, SMP, SMP, SMP, SMP, SMP6, SMP7, SMP8 Content Standards.OA.,.NBT.7,.NF.,.G.,.G. Mental Math and Reflexes Have students use slates and their calculators to practice converting between units of time. Suggestions: In years (none are Leap years), there are how many: months? 6 weeks? 6 days?,8 In, minutes, there are how many: days? hours? 7 seconds?,9, In 68 hours, there are how many: weeks? days? 7 minutes?,8 Math Message Solve Problem on journal page. Study Link Follow-Up Have partners compare answers and resolve differences. Teaching the Lesson Math Message Follow-Up (Math Journal, p. ) WHOLE-CLASS Algebraic Thinking Draw the table from journal page onto the board or Class Data Pad. Ask volunteers to fill in the x and y values from the graph. (,); (,); (,); (,); (,); and (,) Point out that although this table looks different from a What s My Rule? table, it still has a rule. In this case, the rule describes the relationship between the coordinates of the ordered number pairs (x,y). Ask students to examine the table data and name a rule that fits. Sample answer: The sum of the coordinates is always. Explain that if x and y are used as variables, the rule can be written as the equation: x + y =. Write the equation on the board or Class Data Pad. Ask students to suggest how the rule could be written in a different way. y = x, or x = y Write the equations on the board or Class Data Pad. Have students select ordered number pairs from the table to verify the equations. NOTE For practice solving problems involving numeric patterns, see Date 6 Math Message Time Rules, Tables, and Graphs. Use the graph below. Find the x- and y-coordinates of each point shown. Then enter the x and y values in the table. y x y tt Student Page x =, y = x y x, y x. Eli is years old and can run an average of yards per second. His sister Lupita is 7 and can run an average of yards per second. Eli and Lupita have a 6-yard race. Because Lupita is younger, Eli gives her a -yard head start. Complete the table showing the distances Eli and Lupita are from the starting line after second, seconds, seconds, and so on. Use the table to answer the questions below. a. Who wins the race? b. What is the winning time? c. Who was in the lead during most of the race? Eli seconds Lupita Time Distance (yd) (sec) Eli Lupita start Math Journal, p. Lesson 6 8

3 in/time (sec) Eli out/distance (yd) Start in/time (sec) Lupita out/distance (yd) Start Have students use a straightedge to draw a line that passes through the points marked on the graph. They should extend the line beyond the axes in both directions. (See page 8.) Ask students to name other ordered pairs that are on the line. Then check to see that these pairs satisfy the same x + y = rule. Guide students to locate and name ordered number pairs containing fractions or negative numbers. Fraction pairs: ( _,_ ); (_, _ ); (_,_ ); (_, _ ); and so on. Negative numbers: (6,-); (7,-); (8,-); (-,8); (- _,7_ ); and so on. Solving the Footrace Problem (Math Journal, p. ) Adjusting the Activity PARTNER Algebraic Thinking Assign Problem on the journal page. Have students work with partners or in small groups. Point out that they will be completing this table to analyze the results based on two different rules, one rule for Eli and one rule for Lupita. When most students have finished, bring the class together to discuss the answers. Ask volunteers to explain how they used the table to answer the questions in Problem. Look at the distance for Eli and Lupita at each time interval. Lupita leads most of the way, but Eli wins after seconds. Ask: After the start of the race, how many yards does Eli gain on Lupita each second? Eli gains yard each second. Ask: How did you get your answer? Sample answers: For second (input), I looked at the difference between the output for Eli and the output for Lupita. Here Lupita is ahead by, or 9 yards; after seconds, she is ahead by 8, or 8 yards; after seconds, she is ahead by, or 7 yards; and so on. Eli s speed is yard per second faster than Lupita s, so he catches up by yard per second. Have students extend the table to answer additional questions such as: Who is ahead after 8 seconds and by how much? Lupita, by yards What if the race had been yards instead of 6? The race would have ended in a tie, and it would have taken seconds. A U D I T O R Y K I N E S T H E T I C T A C T I L E V I S U A L Ask: What is the rule for the time it takes Eli to cover the distance? Distance = number of seconds, or d = t Tell students that it might help to think of the table as two What s My Rule? tables. In each table, the in values are numbers of seconds (time), and the out values are numbers of yards (distance). (See margin.) Ask: What is the rule for the time it takes Lupita to cover the distance? Distance = number of seconds +, or d = t + 86 Unit Using Data; Algebra Concepts and Skills

4 Lupita s rule is less obvious, and students might not discover it on their own. Remind students that Lupita had a -yard head start. Her actual running distance is the number of yards, but her -yard head start must be added. Write the rules on the board or a transparency, and have students verify that the rules are correct for several values from the table. Ongoing Assessment: Informing Instruction Watch for students who do not understand how Lupita s -yard head start is part of the rule. Guide them to see the pattern in the computations used to complete Lupita s data in the table. Write the computations in equation form. seconds: ( ) + = yards seconds: ( ) + = yards seconds: ( ) + = 6 yards seconds: ( ) + = yards Date 6 Time Rules, Tables, and Graphs continued. Use the grid below to graph the results of the race between Eli and Lupita. Distance (yd) Lupita Eli Time (sec). How many yards apart are Eli and Lupita after 7 seconds?. Suppose Eli and Lupita race for 7 yards instead of 6 yards. a. Who would you expect to win? b. How long would the race last? Student Page c. How far ahead would the winner be at the finish line? Math Journal, p. yards Eli seconds yards Graphing the Footrace Data (Math Journal, p. ) WHOLE-CLASS Have students graph the data in their tables, using the grid on the journal page. Students should plot two graphs on the same grid one for Eli and one for Lupita. Suggestions: Students need to plot only a few time and distance results for each runner in order to make the graph. Have them write the ordered pairs for a few results and then plot them. Suggest that students select results whose times are spread out along the interval; for example, seconds (start), seconds, and seconds. Sample answers: For Eli: (,), (,), (,); For Lupita: (,), (,), and (,) Students should then use a straightedge to connect these points for each runner and label the lines Eli and Lupita. Have students compare the two lines. How and why are they different? Sample answers: Eli s line is steeper because he is running faster. Lupita s line intersects the vertical axis at yards because she had a head start. Have students verify that all of Eli s and Lupita s times and distances shown in the table are points on the graphs. The graphs show that Eli and Lupita both cross the -yard line in seconds. The lines of the graphs intersect at the point ( sec, yd). Lesson 6 87

5 Ask additional questions that can be answered by referring to the graphs: Who was ahead after seconds and by how much? Lupita, by 6 yards How far did Eli run in seconds? yards If the race had been 7 yards, who would have won and by how much? Eli would have won by second. Use a straightedge to extend the two graphs. Adjusting the Activity ELL Have students use two colors when graphing the footrace data, one color for each data set. A U D I T O R Y K I N E S T H E T I C T A C T I L E V I S U A L Have students work with partners or in small groups to complete Problems and on journal page. Circulate and assist. Encourage students to answer the questions by using their graphs. They must use a straightedge to extend the graphs before answering the questions about the 7-yard race. Ongoing Assessment: Recognizing Student Achievement Exit Slip Use an Exit Slip (Math Masters, page ) to assess students ability to read and interpret graphs. Have students explain their answers for Problem a on journal page. Sample answer: I drew Eli s and Lupita s lines on the graph so they extended to 7 yards. The lines show that Eli is ahead of Lupita, so I would expect Eli to win. Students are making adequate progress if their writing refers to being able to extend the lines to see an increase in time and distance. [Data and Chance Goal ] 88 Unit Using Data; Algebra Concepts and Skills

6 Ongoing Learning & Practice Playing Mixed-Number Spin or Fraction Spin (Math Journal, pp. and 6; Math Masters, pp. 7, 7, 88, and 89) PARTNER Students play Mixed-Number Spin or Fraction Spin to practice estimating sums and differences of fractions and/or mixed numbers. Players use benchmarks to estimate sums and differences as they record number sentences that fit the parameters given on the Mixed-Number Spin record sheet, Math Masters, page 89. These games were introduced in Lessons 8- and 8-, respectively. Date 6 Math Boxes Time. Below are the distances (in feet) that a baseball must travel to right field to be a home run in various major league baseball parks. a. Make a stem-and-leaf plot for the data. Identify the landmarks. b. What is the maximum? c. What is the mode? d. What is the median?. Solve a = b. 8. = c., = d = 7.66 e. 9.8 = f =. Student Page Stems Leaves (s and s) (s) Complete the following equivalents. a. pint = cups b. quart = pints c. quart = cups d. gallon = quarts e. gallon = 6 cups Math Boxes 6 (Math Journal, p. ) INDEPENDENT Math Journal, p Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson -8. The skill in Problem previews Unit content. Writing/Reasoning Have students write a response to the following: Explain how you found the answer to Problem b. Include the strategies and the reasoning that you used. Answers vary. Study Link 6 (Math Masters, p. 6) INDEPENDENT Home Connection Students complete a table of values that displays race distance covered in seconds. They interpret the data in the table, graph it, and then interpret the graph. Study Link Master Name Date Time STUDY LINK 6 Interpreting Tables and Graphs Natasha is years old and runs an average of 6 yards per second. Derek is 8 years old and runs about yards per second. Natasha challenged Derek to an 8-yard race and told him she would win even if he had a -yard head start.. Complete the table showing the distances Natasha and Derek are from the starting line after second, seconds, seconds, and so on. Time Distance (yd) (sec) Natasha Derek Start Distance (yd) Derek Natasha Time (sec). Use the table to write rules for the distance covered by Natasha and Derek. Natasha s Rule: Multiply time by 6; 6 t, or 6t. Derek s Rule: Multiply time by and add ; ( t) +, t +, or t +.. Graph the results of the race between Natasha and Derek on the grid above. Label each line.. a. Who wins the race? b. What is the winning time? c. At what time in the race did Natasha take the lead? Math Masters, p. 6 Natasha. _ or _ seconds After seconds Lesson 6 88A

7 6 Interpreting Table Data There are a number of choices when making a graph from table data. The type of graph is determined by the type of data represented. The title and labels for the graph are often the easiest to recognize from the table. Deciding on the scale to use for the y-axis of a line graph is more of a challenge. The intervals in the data can guide the choice of a scale.. Make a graph for each of the tables below. Table Pinto bean plants grow an average of. inches each day. Day Plant Height (in.) Sample answers: Car Colors silver yellow white black red Teaching Master Name Date Time blue Height (in.) Table Exterior colors of cars in the movie theater parking lot Exterior Color Percent Silver % Yellow % Black % Red % Blue % White % Plant Growth Time (days). On the back of this page, explain why you chose which graph to use for each table. Math Masters, p. 7 Differentiation Options READINESS Interpreting Table Data for Graphs (Math Masters, p. 7) SMALL-GROUP Min To provide experience with constructing graphs from table data, have students identify the relationship between parts of a table and parts of a graph. Refer students to the tables on Math Masters, page 7. Discuss how students might choose the type of graph to use for each table. Circle graphs show parts of a whole, so Table data could be represented by a circle graph. Line graphs often show change over time, so Table could be represented by a line graph. Discuss how the table organization suggests the title and labels, including line-graph scales, for the graphs. Have students graph the data from the tables on the Math Masters page. ENRICHMENT Graphing Race Results (Math Masters, p. 9) WHOLE-CLASS + Min Name Date Time Meters Distance/Time Graph Teaching Aid Master To apply students understanding of constructing graphs from table data, have students explore multiple graphs on a single grid. Divide students into groups of or. Each group chooses a different method of movement (for example, hopping on one foot, taking baby steps, or walking backward) for completing the -meter course. The groups use a stopwatch to time how long each group member takes to complete the course. Each group makes a distance-over-time line graph of the results for each group member on the same grid. Plot time on the x-axis and distance on the y-axis. A student plots his or her starting point at the origin (,) and his or her ending point at the number of seconds it takes to cover the distance. The student then connects these two points with a line segment. Draw each group member s line segment in a different color. Compare the graphs. Discuss similarities and differences Seconds Math Masters, p. 9 88B Unit Using Data; Algebra Concepts and Skills

8 Teaching Master Name Date Time EXTRA PRACTICE INDEPENDENT Analyzing Two Rules (Math Masters, pp. 7A and 7B) Min Students use two rules to complete a table of values. They list and graph ordered pairs for each rule. They use the table of values and graphs to analyze the data. Planning Ahead In Lessons -8 and -9, you will need a collection of round objects for students to measure, such as rolls of tape, wide markers, food and coffee cans, round clocks, wheels, circular cardboard cutouts, and graduated cylinders. There should be at least one object per student. Ask students to bring these or similar objects from home. 6 Analyzing Two Rules Ivan earns $ every days. Elise earns $7 every days.. Write a rule to describe how much money Ivan earns in a given number of days. $ Amount earned =, or $ number of days. Write a rule to describe how much money Elise earns in a given number of days. Amount earned = $7 number of days. Complete the table for the outputs in the cells that are not shaded.. a. After 6 days, who has earned more money? Ivan b. After days, who has earned more money? Ivan. How long does it take each person to make $? Ivan: days Elise: days Days c. Who earns more money per day? Explain your answer. Ivan. Sample answers: The table shows that Ivan earns more after 6 days, and even more after days. Ivan earns _ $, or $ per day. Elise earns _ $7, or a bit more than $ per day. Money Earned Ivan Elise $ $ $ $7 $ 6 $ $ 7 8 $ 9 $ $ $ $8 $ $ 6 $ Math Masters, p. 7A Teaching Master Name Date Time 6 Analyzing Two Rules continued 6. Write three ordered pairs to show the relationship between number of days and the amount earned for each person. Ivan: Sample answer: (, ), (6, ), (, ) Elise: Sample answer: (, ), (9, ), (, ) 7. Use the grid to graph your ordered pairs. Use a straightedge to connect the points for each person, and label the lines Ivan and Elise. 7 Amount Earned (dollars) 7 7 Ivan Elise Days 8. About how much more did Ivan earn after days? About $ 9. Extend the graph to find out about how much Elise earned in 6 days. About $7 Math Masters, p. 7B Lesson 6 89

9 Name Date Time 6 Analyzing Two Rules Ivan earns $ every days. Elise earns $7 every days.. Write a rule to describe how much money Ivan earns in a given number of days. Amount earned = number of days. Write a rule to describe how much money Elise earns in a given number of days.. Complete the table for the outputs in the cells that are not shaded.. a. After 6 days, who has earned more money? b. After days, who has earned more money? c. Who earns more money per day? Explain your answer. Days Money Earned Ivan Elise. How long does it take each person to make $? 6 Copyright Wright Group/McGraw-Hill Ivan: days Elise: days 7A

10 Name Date Time 6 Analyzing Two Rules continued 6. Write three ordered pairs to show the relationship between number of days and the amount earned for each person. Ivan: Elise: 7. Use the grid to graph your ordered pairs. Use a straightedge to connect the points for each person, and label the lines Ivan and Elise. 7 Amount Earned (dollars) 7 7 Copyright Wright Group/McGraw-Hill Days 8. About how much more did Ivan earn after days? 9. Extend the graph to find out about how much Elise earned in 6 days. 7B