Lesson 4.2.2 – Teacher Notes Standard: 7.RP.A.2a, d Recognize and represent proportional relationships between quantities. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. Fully mastery can be expected with the exception of problems relating to distance, rate, and time. Lesson Focus: The focus for this lesson is for students to create tables and graphs of proportional relationships. Students should be able to explain the meaning of any given point on a graph, in terms of its situation. (4-35) I can explain what the point (1, r) means in context, where r is the unit rate. Calculator: No Literacy/Teaching Strategy: Reciprocal Teaching (Intro and 4-36); Hot Potato (4-34)

Skill Review Decide whether you think the relationship described is proportional or non-proportional and justify your reasoning. Be prepared to share your decisions and justifications with the class. Make a table for each situation to prove your answer.        c. Vu is four years older than his sister.  Is the relationship between Vu and his sister’s age proportional?   d. Janna runs at a steady pace of 7 minutes per mile.  Is the relationship between the number of miles she ran and the distance she covered proportional?   e. Carl just bought a music player and plans to load 50 songs each week. Is the relationship between the number of weeks after Carl bought the music player and the number of songs on his player proportional?  

In Lesson 4.2.1, you learned that you could identify proportional relationships by looking for a constant multiplier. In fact, you have already seen a relationship with a constant multiplier in this course. Today you will revisit the earlier situation that contains a proportional relationship.

4-34. GRAPHING THE PENNY TOWER DATA In Chapter 1, you found a multiplicative proportional relationship between the height of a stack of pennies and the number of pennies in the stack.  You could always find one piece of information by multiplying the other by a constant number. Fill in the missing values.  What strategies did you use to determine the missing numbers? How many pennies are in a tower with a height of 0 cm?  Add a row to your table withthis value.    

Graph this data.  Be sure that all of the points in your table are visible on your graph.  What do you notice about the graph of height and the number of pennies?  How does this graph compare to Sonja’s graph of birdseed weight and cost?   What do the graphs have in common?  How are they different?

4-35. Kaci loves cheese and buys it whenever she can 4-35. Kaci loves cheese and buys it whenever she can. Recently, she bought 5 pounds of mozzarella cheese for $15.00 and 3 pounds of havarti for $7.50. a. Work together to record, plot, and label Kaci’s two cheese purchases.  b. Find another point (x,y) that you could plot on the graph for each kind of cheese.  Record these points in the tables.  That is, find another combination of pounds of cheese and the associated cost for the mozzarella and then another combination of pounds and cost for the havarti.  

4-35. Kaci loves cheese and buys it whenever she can 4-35. Kaci loves cheese and buys it whenever she can. Recently, she bought 5 pounds of mozzarella cheese for $15.00 and 3 pounds of havarti for $7.50. c. Think about and answer the following questions. Then decide how best to complete the two tables and graphs that you started in parts (a) and (b). Can you find any other points that should be in the mozzarella table and graph? Add them.  Can you find any other points that should be in the havarti table and graph? Add them.  Should the points on each graph be connected?  If so, why does that make sense? If not, why not? 

4-35. Kaci loves cheese and buys it whenever she can 4-35. Kaci loves cheese and buys it whenever she can. Recently, she bought 5 pounds of mozzarella cheese for $15.00 and 3 pounds of havarti for $7.50. d. How do the graphs for each type of cheese compare? What is the same and what is different?  e. Which cheese is more expensive (costs more per pound)? How can you tell by looking at the graph? How can you tell by looking at the table?  f. What is significant about the point (1, y) for each line on the graph or in your table? What do we call it?

4-36. Look back at the tables and graphs you created for proportional relationships in the previous problems. How can you use a table to decide if a relationship is proportional?  How can you use a graph to decide if a relationship is proportional?

4-37. Which of the tables below shows a proportional relationship between  x  and  y?  How can you tell?

4-38. The following graphs show examples of relationships that are not proportional.  For each graph, explain what makes the relationship different from the proportional relationships you have studied.

4-38. The following graphs show examples of relationships that are not proportional.  For each graph, explain what makes the relationship different from the proportional relationships you have studied.

Practice William made cookies over consecutive hours. If a proportional relationship exists between time and the number of cookies made, create a graph to prove it. Mason made omelets. If there is a proportional relationship between the number of eggs used and the number of omelets made, create a graph to prove it.

Practice Isabella made necklaces with beads. Create a chart and a graph to show a proportional relationship of 2 necklaces for every 24 beads. Marcus can run 1 mile in 7 minutes, create a chart and a graph to prove this is a proportional relationship.