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Day 1 Identifying Proportional Relationships in Tables. Instructional Resources Materials Pencil paper handout. Prior Knowledge Simplifying ratios proportional relationships. Activity Lesson 20 25 minutes, Students will work on completing tables for Example 1 independently 5 minutes Students will. then collaborate with a partner if seated in pairs or their group to discuss the results of their. tables and work on each of the follow up questions together 10 minutes Group norm no. student is allowed to proceed to the next question until everyone has written down an answer. to the current question Lastly students will share their solutions with the whole class 5 10. minutes There are often times multiple ways that a problem may have been completed or. explained so the teacher should circulate during the group work time to select students who. have differing approaches to present to the class,EXAMPLE 1 Running the Race. You have decided to join a long distance running team There are two teams that you can join. Team A runs at a constant rate of 3 miles per hour Team B runs 4 miles the first hour and then. 2 miles per hour after that, Your task Complete the table for each team showing the distances that would be run for times. of 1 2 3 4 5 and 6 hours Using your tables answer the questions that follow. Team A Team B,Time hrs Distance miles Time hrs Distance miles. 1 For which team is distance proportional to time Explain your reasoning. Team A since all the ratios comparing distance to time are equivalent The equivalent. ratio is 3 Every measure of time can be multiplied by 3 to give the corresponding. measures of distance,Page 2 of 29 MCC WCCUSD WCCUSD 11 10 15.

2 Explain how you know the distance for the other team is not proportional to time. The ratios are not equivalent for Team B The ratios are 4 3 8 3 2 5 12 5 and 14 6. 7 3 Every measure of time cannot be multiplied by a constant to give each. corresponding measure of distance, 3 If the members on the team ran for 10 hours how far would each member run on each. Team A 30 miles,Team B 22 miles, 4 If the race were 3 miles long which team would win Explain. Team B would win because more distance was covered in less time. 5 If the race were 7 miles long which team would win Explain. Team A would win because more distance was covered in less time. Example 1 Takeaway Two quantities are proportional if they have a constant ratio or unit rate. For relationships in which the ratio is not constant the two quantities are nonproportional. EXAMPLE 2 5 minutes Andrew earns 18 per hour mowing lawns Is the amount of money. he earns proportional to the number of hours he spends mowing Complete the table and. provide an explanation of your reasoning,Earnings 18 36 54 72. Time h 1 2 3 4,Prompting questions from teacher, What do you need to find Whether the amount of money he earns is proportional to the. number of hours he spends mowing, How would you represent the amount of money Andrew earns per hour as a unit ratio.

How would you determine how much he makes for each number of hours worked. Multiply the number of hours worked by 18 Or add an additional 18 for each hour. Page 3 of 29 MCC WCCUSD WCCUSD 11 10 15, What will you need to do to determine if the relationship is proportional. Write the relationship between the amount he earned and hours worked as a ratio in. simplest form for each number of hours worked,amount earned 18 36 54 72. or 18 or 18 or 18 or 18,number of hours 1 2 3 4, Are the two quantities proportional Justify your thinking. All of the ratios between the two quantities can be simplified to 18 The. amount of money he earns is proportional to the number of hours he spends mowing. EXAMPLE 3 5 minutes Ticket Servant charges 7 per baseball game ticket plus a 3. processing fee per order Is the cost of an order proportional to the number of tickets ordered. Complete the table and provide an explanation for your reasoning. Cost 10 17 24 31,Tickets Ordered 1 2 3 4,Prompting questions from teacher. What do you need to find If the cost of an order is proportional to the number of tickets. How would you represent the cost of ordering one ticket as a unit ratio. How would you determine the cost of ordering two tickets. Figure out the cost of 2 tickets 14 and then add the 3 processing fee Or add 7 to. the total cost of ordering one ticket, What will you need to do to determine if the relationship is proportional.

Write the relationship between the cost and number of tickets as a ratio in simplest form. for each number of tickets,cost of order 10 17 24 31. or 10 or 8 5 or 8 or 7 75,tickets ordered 1 2 3 4, Are the two quantities proportional Justify your thinking. Since the ratios of the two quantities are not the same the cost of an order is not. proportional to the number of tickets ordered,Page 4 of 29 MCC WCCUSD WCCUSD 11 10 15. YOU TRY 1 7 minutes The tables shown below represent the number of pages Miguel and. Karlee read over time Write down at least three observations you can make about the two. tables hint proportional and or nonproportional relationships should be included in at least. two of the observations Provide justification for your observations. Pages Miguel Read 2 4 6 Pages Karlee Read 3 4 7,Time min 5 10 15 Time min 5 10 15. All of the ratios between Miguel s quantities are so Miguel s reading rate represents a. proportional relationship,pages 2 4 2 6 2,minutes 5 10 5 15 5.

The ratios between Karlee s quantities are not the same so Karlee s reading rate does not. represent a proportional relationship,pages 3 4 2 7. minutes 5 10 5 15, Miguel and Karlee read the same amount of pages after 10 minutes. Karlee read more pages than Miguel after 15 minutes. Extension questions for You Try 1, a How many pages do you think Miguel will have read after 20 minutes How do you know. Miguel will likely read 8 pages after 20 minutes based on his reading rate of 2 pages. every 5 minutes, b How many pages do you think Karlee will have read after 20 minutes How do you know. It is impossible to determine how many pages she will have read based on the. information given, c Why is it more difficult to determine how many pages Karlee will read.

Because her reading rate is not constant,Page 5 of 29 MCC WCCUSD WCCUSD 11 10 15. YOU TRY 2 7 minutes The tables shown below represent the amount of money earned per. hour for John and Jackie, John s Earnings 12 20 31 Jackie s Earnings 12 24 36. Time h 1 2 3 Time h 1 2 3, Question 1 Which situation represents a proportional relationship between the hours worked. and the amount earned Provide justification for each of the tables. The ratios between John s quantities are not the same so John s earning rate does not. represent a proportional relationship,earnings 12 20 31. or 12 or 10,hours h 1 2 3, All of the ratios between Jackie s quantities are 12 so Jackie s earning rate represents a.

proportional relationship,earnings 12 24 36,or 12 or 12 or 12. hours h 1 2 3, Question 2 How much would each earn if they worked a fourth hour. Because John s earning rate is not constant we cannot determine how much he would make. after four hours of work Jackie would make 48, Closing question How does knowing whether two quantities are proportional help answer. questions about the quantities For example if we know 1 cup 8 oz what does that allow. Understanding the relationship can allow you to find missing quantities For the above. example one cup can be substituted for 8 ounces or vice versa. Page 6 of 29 MCC WCCUSD WCCUSD 11 10 15, EXIT TICKET 5 minutes Sara ran laps around a gym Her times are shown in the table below. Sara thinks the number of laps is proportional to the time because the number of laps always. increases by 2 Is she correct Provide justification for your answer. Time min 2 3 4 5,Laps 3 5 7 9, It is not proportional because the ratio of laps to time is not consistent.

number of laps 3 5 7 9,or 1 5 or 1 75 or 1 8,time min 2 3 4 5. Page 7 of 29 MCC WCCUSD WCCUSD 11 10 15, Day 2 Identifying Proportional Relationships in Graphs. Instructional Resources Materials Pencil paper ruler handout. Prior Knowledge The coordinate plane plotting points quadrants ordered pairs x axis y. axis origin determining proportional relationships in tables. Activity Lesson,Opening activity 10 minutes, A group or pair of students will receive all four half sheets of tables and graphs For each. problem students will determine whether the two quantities in the table represent a. proportional relationship When they are finished they are to set aside the problems for the. time being STUDENTS DO NOT GRAPH THE INFORMATION FROM THE TABLES YET. The Coordinate Plane Review 5 7 minutes, Teacher should project a coordinate plane doc cam or computer and have students help label. the various parts of the grid the four quadrants the axes. Some questions to discuss,What is an ordered pair, A pair of numbers such as 1 3 used to locate or graph points on a coordinate grid.

What is the origin and where is it located, The point where the x and y axis intersect at ordered pair 0 0. When we are plotting a point where do we count from. The origin 0 0,Why are we only focusing on quadrant 1. All of the values are positive Since we are measuring or counting quantities number of. video games rented and cost the numbers in our ratios will positive. Show students how to plot the point 1 3 and label the x and y coordinate. How would we plot the point 2 5, From the origin we go 2 to the right on the x axis and then up 5 on the y axis. After graphing the points 1 3 and 2 5 connect the two points on the coordinate plane. How would you describe the graph,The points form a straight line. Page 8 of 29 MCC WCCUSD WCCUSD 11 10 15, Students should locate their piles of tables and refer to Example 1 video game example.

From a Table to Graph Examples 1 and 2 5 10 minutes. EXAMPLE 1 A swimming pool is currently filled with one gallon of water A garden hose is used. to fill the pool at a constant rate From the information from the table determine whether the. number of gallons is proportional to the number of minutes. Filling the Swimming Pool, Time Number of Is the relationship proportional or nonproportional. Minutes Gallons Provide justification for your reasoning. It is not proportional because the ratio of cost to games is not. consistent,4 9 number of gallons 3 5 7 9,or 3 or 2 5 or 2 25. time min 1 2 3 4,Page 9 of 29 MCC WCCUSD WCCUSD 11 10 15. Teacher should model how to graph the information shown in the table. Before you can graph what do you need to know,The ordered pairs. How can you find the ordered pairs,Use the table,What will the x values represent.

The time in minutes,What will the y values represent. The number of gallons in the pool,How do we plot the first ratio pair. If the relationship is 1 3 where 1 represents 1 minute and 3 is the number of gallons. then from the first point we go 1 to the right on the x axis and then up 3 on the y axis. How do we plot the second ratio pair, If the relationship is 2 5 where 2 represents 2 minutes and 5 is the number of gallons. then from the first point we go 2 to the right on the x axis and then up 5 on the y axis. Students should graph the rest of the points 3 7 and 4 9. Page 10 of 29 MCC WCCUSD WCCUSD 11 10 15, EXAMPLE 2 The slowest mammal on Earth is the tree sloth It moves at a speed of 6 feet per. minute Complete the table determine whether the number of feet the sloth moves is. proportional to the number of minutes it moves and explain your reasoning. Time min 1 2 3 4 Is the relationship proportional or nonproportional. Provide justification for your reasoning,Distance ft 6 12 18 24.

It is proportional because the ratio between,the distance and time is always 6. At this point if students are comfortable graphing in pairs or in groups they should graph. Example 2 If the majority of the students are not comfortable yet the teacher should graph. the points for Example 2 with them,Page 11 of 29 MCC WCCUSD WCCUSD 11 10 15. Comparing the Graphs Examples 1 and 2 7 minutes, Have students compare Examples 1 and 2 side by side. Have the students make observations about the two graphs Example 1 nonproportional. relationship and Example 2 proportional relationship and record the statements side by side. Using a doc cam white board or poster, Nonproportional Relationship Ex 1 Proportional Relationship Ex 2. points all fall on a straight line points all fall on a straight line. line doesn t go through the origin line goes through the origin. What are some similarities between the two examples. Both have points that lie on a straight line, What are some differences between the two examples.

The line for Example 1 does not go through the origin but the line for Example 2 does. Why are the lines straight, For every increase of 1 on the x axis there is an increase of the same amount. on the y axis, In example 1 are we able to infer how much a fifth game would cost. Yes the cost of each rental goes up by 2 for each rental after the first rental so 5. games would cost 11, In example 2 can we infer how far the sloth would walk after five minutes. Yes the sloth walks 6 ft per minute according to the table so after five minutes the. Lesson Unit Plan Name Identifying Proportional and Nonproportional Relationships in Tables and in Graphs Rationale Lesson Abstract This lesson focuses on students examining the characteristics of proportional relationships and determining when and why two quantities are actually in a proportional relationship

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