Proportional Reasoning—Part 1
Count out and bring back to your desk 6 graham crackers and 16 candies (for each group). With your group, write down (on your white board) all the ways you can of describing what you have.
Proportional Reasoning—Part 2
If you want to keep this ratio of candies to graham crackers the same, how many candies should you have if you have one graham cracker?
If you have 2 graham crackers?
3 graham crackers?
On your white board, make a plot that represents this relationship.
Write an equation that describes the relationship between the number of graham crackers and the number of candies.
Proportional ReasoningPart 3:
- Suppose you wrap a string around the equator of the Earth and pull it tight. Now, add 2 meters to the length of the string so that it still forms a circle around the Earth. Would you expect to be able to fit a playing card, a baseball or a basketball under the string (between the string and the earth)?
- If you plotted the equation on a C versus R plot, what sort of a graph would you expect to see? (linear, quadratic, exponential, trigonometric, etc.)
What would be the slope of this equation? It’s intercept?
Is C proportional to R?
The equation is plotted below:
- Recalling that you know the slope of the line above, use this plot to answer the following question: Suppose you have a circle of circumference C. If you increase the circumference, C, by some amount (call it ) so that the new circumference is C’ = C + , how much does the radius change?
- Now does your answer to question #1 change? Approximately how much space is there for fitting things under the rope?
Proportional Reasoning Part 4:
Certain chemical salts, called hydrates, can absorb water into their structure. Each salt molecule has space around it for a certain number of water molecules. Different kinds of salts hold different numbers of molecules. For example, every molecule of barium chloride (BaCl2) holds 3 molecules of water (H2O), which is written as BaCl23H2O.
Here is a list of some hydrates:
BaCl23H2O
CoCl26H2O
CuSO45H2O
MgSO47H2O
CaSO4H2O
You obtain 1.00 mole of a dry unknown salt. The dry salt absorbs 90.1 g of water
(18.02 g/mole). What is the unknown salt?
Proportional Reasoning Part 5:
You are the managing engineer of a polymer production facility. In the production of silly putty, no synthesis of the polymer (polymerization) occurs until the reaction temperature reaches 192° C. The volume of silly putty that can be produced increases linearly with temperature. Make a plot showing the volume of polymer produced (units?) versus the temperature of the process.
Suppose you increase the temperature of your process from 200 ° C to 212 ° C, how much does the volume increase?
What piece of information are you missing?
Make your own assumption to solve this problem.
Proportional Reasoning
Bring to class:
-Graham Crackers
-Candies (individually wrapped-one type of candy if possible)
-Paper Towels
Teacher’s notes:
The goal is to explore the difficulties encountered in ratios in the concrete case of the graham crackers and candy first, then to try the more abstract cases with a ratio of the circumference to radius, and the ratio of water molecules to salts in a hydrate material. If there is extra time at the end, students can experimentally verify their circumference/radius prediction with the materials provided (tape measure, round things.)
Part 1:
The question in part 1 is intentionally vague, but after 5 minutes or so, direct them to consider the numbers of items and the relationship between those numbers, to keep them from paying too much attention to other details for too long.
Leading questions as you walk around the room in part 1 include:
- “How many graham crackers do I have for eachcandy?”
- “Break the graham crackers.”
- “How many candies do I have for each graham cracker?”
- “Can you write an equation which shows the relationship between the number of graham crackers and number of candies?”
You really need to push them into breaking the crackers and making piles which show how many crackers for each candy, and vice versa.
Draw together everybody as a group and have each group contribute a way of representing their situation. Main point:
-“Is the ratio 3/8: 1 the same as the ratio of 3:8 or 6:16?”
-What does it mean to have a fraction in my ratio?
Our goal is to have them write down various ratios using whole numbers and fractions, as well as equations and vocabulary words such as “ratio” and “per” to represent the relationship of graham crackers to candy. (Possibly they can express it as a percentage as well…suggest for groups who are getting ahead.) We also want them to understand that all these ways express equivalent information. If they have an equation make sure they check it to see if it’s correct.
Part 2:
While they are making their plots walk around the room. If they don’t make a line (only a few points) ask about how many candies they would have if there were 1.5 graham crackers. The point is for them to know it is continuous.
-Group discussion: Have everybody show their plots and equations and/or write them on the board.
It is likely they will write an equation in terms of y & x…ask what each variable represents. Write down what they tell you next to their equation.
Using your equation, how many candies will you have if you have 3 graham crackers? (Is that right? Show me a pile of g.c. & c. like that.)
Now we can make the points that it helps us to keep things straight if we:
-Use units in our slope (8/3 candies/graham cracker).
-Use variables whose names remind us what they represent (g,c rather than x,y)
-Test our equation to see if its reasonable.
Part 3 (string-earth):
In question 3, students have trouble. They don’t see that they can answer the question (find change in radius, given change in circumference) from the slope. They don’t have a plan yet how to do the problem. We want to give them two pieces of advice:
1)Name what you don’t know. They need to choose names (normally r and delta r) for the points on the x-axis of their graph that figure into the slope calculation.
2)Write down what you do know. They can write some sort of equation that has delta r in it, even though it won’t be in the form delta r = … Even if they don’t yet know how they’ll use it, the best way to start is by writing down true equations, then seeing if any of them help them. (Another hint: What’s the definition of slope?)
Part 4:
After they do the hydrate question, we can ask them what that question had to do with proportional reasoning, and what are some ratios that they used in the problem. One point we hope they can make is that the ratio of salt molecules to water molecules (for each salt) is the same no matter how many molecules we have. Leading question: Would your answer change if we changed the definition of a mole, so that 1 mole = 12 atoms?
Post-Class Notes:
In Part 3, students don’t really seem to get that ∆ C is proportional to ∆ R from the graph. They do not use proportional reasoning to find the relationship between a change in circumference and the change in radius. They managed, with instructor assistance, to write and equation like slope = ∆C/∆R = Pi, but were not moved to write it themselves, and weren’t sure what do with it.
We think part three should be rewritten to really get at this point. They need something intermediate between the simple graham crackers/candies proportional reasoning and the more difficult reasoning using a graph. (Aarons points out that students have a problem with this.) Or, part 3 could be taken out, and students can go straight to the hydrate and polymer questions. This year, no groups finished part 3 in time to do these parts.
Also, Part 4 is done on the following week. Students worked on it individually and most of them did well.