Algebraic Thinking Part C: Qualitative Graphs (90 minutes)

Problem C1

Look at the graph below. Who is represented by each point?

Tip: Don’t worry that there are no numbers on the axes. This helps you focus on the representation and its meaning rather than just a picture of the situation.

Problem C2

What would the graph for this problem look like with the axes reversed?

Problems in Part C are taken from The Language of Functions and Graphs, by Malcolm Swan and the Shell Centre Team (Nottingham, U.K.: Shell Centre Publications, 1986, 1999, www.mathshell.com).

Problem C3

Jane, Graham, Susan, Paul, and Peter all travel to school along the same country road every morning. Peter goes in his dad’s car, Jane cycles, and Susan walks. The other two children vary how they travel from day to day. The map below shows where each person lives. _{Note 8}

The following graph describes each pupil’s journey to school last Monday.

1. Label each point on the graph with the name of the person it represents.
2. How did Paul and Graham travel to school on Monday?
3. Describe how you arrived at your answer to (b).
4. In the graph, the points which correspond to Jane, Paul, and Graham lie on a line. What does this suggest about their modes of transportation? Note 9

Tip: Ask yourself which students correspond to particular values on each axis for time and distance. 

Problem C4

Peter’s father is able to drive at 30 mph on the straight section of the road, but he has to slow down for the corners. Sketch a graph on the axes below to show how the car’s speed varies along the route.

Problems in Part C are taken from The Language of Functions and Graphs, by Malcolm Swan and the Shell Centre Team (Nottingham, U.K.: Shell Centre Publications, 1999).

Problem C5

Often we are asked to sketch graphs from words or descriptions. Choose the best graph to fit each of the situations described below. _{Note 10}

1. I really enjoy cold milk or hot milk, but I loathe lukewarm milk.
   b. Prices are now rising more slowly than at any time during the last five years.
2. The smaller the boxes are, the more boxes we can load into the van.
3. After the concert there was a stunned silence. Then one person in the audience began to clap. Gradually, those around her joined in, and soon everyone was applauding and cheering.
4. If the price for movie admission is too low, then the owners will lose money. On the other hand, if admission is too high, then few people will attend, and again the owners will lose. A movie theater must therefore charge a moderate price in order to stay profitable.
5. Make up three stories of your own to fit three of the remaining graphs.

Tip: At each important place on the graph, think about whether the values in the graph should be increasing more rapidly, increasing but slowing down, remaining steady, decreasing but leveling off, or decreasing more rapidly. Additionally, the graph should reflect important changes in the real-world situation and the starting position of the graph (where it touches the axes).

Problem C6

In the following real-world situations, decide what happens. Explain each situation carefully in words, and then choose the graph that best represents the situation, as in Problem C5.

1. How does the cost of a bag of potatoes depend on its weight?
2. How does the diameter of a balloon vary as air is slowly released from it?
3. How does the time for running a race depend upon the distance run in the race?
4. How does the speed of a child vary on a swing?
5. How does the speed of a ball vary as it bounces along?

Tip: As in Problem C5, consider carefully where the graph should begin, whether it should increase or decrease, and how quickly. 

Problem C7

Draw the graphs to illustrate the following statements. Label your axes with the variables they represent.

1. In the spring, my lawn grew very quickly, and it needed cutting every week. But since we have had this hot dry spell, it needs cutting less frequently.
2. When doing a jigsaw puzzle, I usually spend the first half hour or so just sorting out the edge pieces. When I have collected together all the ones I can find, I construct a border around the edge of the table. Then I start to fill in the border with the center pieces. At first this is very slow going, but the more pieces you put in, the less you have to sort through, and so the faster you get.

Take It Further: Problem C8

Choose at least three of the situations in Problems C5-C7, and change the conditions in a way that alters the graph of the situation. Then, draw the new graphs.

Problems in Part C are taken from The Language of Functions and Graphs, by Malcolm Swan and the Shell Centre Team (Nottingham, U.K.: Shell Centre Publications, 1999).

Problem C9

_{Note 11}

Tip: When will the graph of a bottle’s height increase most slowly? Most rapidly?

Problems in Part C are taken from The Language of Functions and Graphs, by Malcolm Swan and the Shell Centre Team (Nottingham, U.K.: Shell Centre Publications, 1999).

Note 6

One of the goals of thinking algebraically is to develop different ways of representing real-world situations. Representing mathematical ideas in pictures, tables, graphs, and words allows us to use mathematics as a way of communicating. With Eric the Sheep, it appeared that a table and a description in words were good representations of the situation.

The problems in this section involve qualitative graphs. These are graphs that concentrate on the general features of a real-world situation. As an analogy, when learning a foreign language, we are given the rules of grammar, but also opportunities to express ourselves. Similarly, qualitative graphs allow us to interpret, transform, predict, and make logical deductions from the given mathematical data. In this way, even simple graphs can communicate a great deal of information, as illustrated in Problem C1.

Note 7

Groups: Work in pairs on Problem C1. All of the graphing activities in this section lend themselves to interesting and thoughtful discourse, so it’s important to allow time for everyone to discuss their own reasoning and justify their answers.

It may seem puzzling at first that there are no numbers on any of the graphs’ axes. This may be especially true for those who are accustomed to teaching precise point-plotting techniques. Consider why the axes are not labeled with values. (In fact, if there were numbers on the axes, we would not have to think about the relative positions of the points.) The graph in Problem C1 is particularly interesting because height is not on the vertical axis. This is intentional so that, once again, we have to focus on the representation and its meaning rather than on a graph that is merely a literal picture of the situation. Consider creating another graph for Problem C2, one with the axes reversed.

Note 8

Groups: Record answers to Problem C3 on an overhead or chalkboard. It is important to describe carefully the reasoning that was used to place the specific points on the graph.

Note 9

The reasoning of Problem C4 (to determine what must be true of Jane, Paul, and Graham) is sophisticated, and it foreshadows the work we will be doing in later sessions on slope and rate. Session 4 notes

Note 10

Groups: Work on Problem C5 in small groups. For each description, compare answers. Be sure to explain the reasoning for selecting the associated graphs.

Note 11

It may be helpful to illustrate the relationship between volume and height for the bottles in Problem C9 by using a conical flask and a cylindrical bottle, and filling each with water at a steady rate. Or fill each one with cups of water and measure their heights after each cupful.

Groups: If anyone has difficulty seeing how this would translate to a graph, consider drawing a figure on an overhead or chalkboard to show the conical flask. For example, the height increases by a small amount to start with (so that bottle must be wide at the bottom) and then gradually rises by larger and larger amounts (so that bottle must be getting narrower gradually towards the top). Therefore, this is the graph of the conical flask:

Problem C1

1 = Dennis

2 = Alice

3 = Freda

4 = Brenda

5 = Errol

6 = Cathy

7 = Gavin

Problem C2

Problem C3

1. Top left: Peter; top right: Jane; center: Paul; bottom left: Graham; bottom right: Susan
   
2. A good guess would be that Paul and Graham each biked to school.
3. It cannot be a ride, because Peter, who goes in his dad’s car, arrives so quickly. It cannot be a walk, because Susan takes so long to arrive. Also, notice that Graham, Paul, and Jane’s points lie on one diagonal line through the intersection of the axes. (What would the value of distance divided by rate represent?)
4. This suggests that all three used the same mode of transportation. Because Jane biked to school, it is likely that Paul and Graham did as well.

Problem C4

The graph begins at zero, climbs to (or near) 30 mph, slows down at the first turn, then stays at 30 mph for a long time, then slows at the sharp turn by Graham’s, then returns to 30 mph, then slowly drops back to zero at the school.

Problem C5

1. Number 8 is the best graph, if the vertical axis measures happiness. The horizontal axis measures the temperature of the milk, and the vertical axis measures the enjoyment of the speaker. Other answers are possible with alternate labelings of the axes; for example, number 11 could be the solution if you measured dislike on the vertical axis.
2. Number 7. Time is on the horizontal axis, and price is on the vertical axis.
3. Numbers 9, 12, and 3 are all pretty good choices. The number of boxes is on the horizontal axis, and the size is on the vertical axis.
4. Number 5 (note the concert audience’s initial stunned silence). Time is on the horizontal axis, and volume is on the vertical axis.
5. Number 11. Price is on the horizontal axis, and profit is on the vertical axis.
6. Number 1: If I work more hours, I’ll make more money. Number 3: I’ve got $100 to spend on Slurpees or gasoline. The more I spend on Slurpees, the less I have remaining for gas. Number 15: The population of frogs at the pond really exploded last year, but now it’s leveling out.

Problem C6

1. Number 2 is the best graph. A small bag still costs more than zero dollars.
2. Number 6
3. Number 4
4. Number 13 because they are slowest at the top and bottom of the swing.
5. Number 14 because speed is zero when the ball changes direction, fastest at the bouncing point

Problem C7

Problem C9

1. Since the bottle is narrower at the top, it will fill faster near the top. This means that the top half of the bottle fills in less time than the bottom half, because it holds less liquid.
2. Ink bottle: F
   Conical flask: D
   Evaporating flask: I
   Bucket: A
   Vase: E
   Plugged funnel: B
3. Graphs C and G would require quick changes to a constant width. Graph C would go from a small width to a large width; G is the opposite. As for graph H, notice the similarity to graph E.