Using their boats from the last lesson, teams will gather data about the relationship of water level change to number of pennies floating. They will graph that data on their own plot, then on an all-class plot. Using the all-class plot, they will discuss trends/patterns, and try to predict water level change for a larger quantity of pennies.

The purpose of this lesson is to develop students’ understanding of relationships that can occur in bivariate (two changing quantities) data, and their ability to informally model such a relationship with a linear function.

### Activities:

**Setup**

Beakers, water, pennies available as in previous lesson. Each team must have a boat that floated at least 15 pennies available. Each team needs a blank sheet of graph paper and a data sheet.

**Narrative**

[include purposes and a discussion of what students should be doing/thinking/discussing/struggling with]

Students will choose 3 different numbers of pennies between 0 and the maximum that their boat will float, record water level for each of those, and compute water level change for each.

The main issues are precision and attention to detail.

**Launch (narrative could include comments on timing)**

- We noticed yesterday that the water level goes up when we add pennies to the boat. Does anyone have any idea why that is? Are we adding more water to the beaker? [not more water. Some students may explain that the weight of the pennies is “pushing the water out of the way” which is basically accurate. We say “the mass of the pennies displaces some of the water”]
- I’m curious what the relationship is like between the change in water level and the number of pennies. How could we investigate this?
- I’ve got a data sheet for each team to keep track of their data.
- You’ll need to pick three numbers of pennies between 0 and 20 (or the maximum number that your boat would float), not closer than 4 pennies and not including the maximum. For example, I could pick 7, 11, and 18 pennies.
- I would first check and record the water level with no pennies in the boat. How should I read the water level if it is between marks on the side of the beaker? [make an agreement in the class--you could agree to record the nearest mark, but you can probably read more precisely than that and agree to record to the nearest 5 ml or even nearest 2 ml]
- Then, using dry pennies, I would carefully put in the smallest number of pennies I chose, and carefully record the water level. Then add more pennies up to my second number of pennies, etc.

**Task statement**

- Choose three numbers of pennies between 0 and 20, always differing by at least 4 pennies, and write them in the left column on your data sheet
- For each of those numbers of pennies (including 0), record the water level in the beaker
- For each of those numbers of pennies, compute the water level change from 0 pennies

**Orchestrating & Monitoring (narrative could include comments on timing)**

- Make sure students record water level with just the boat
- Be sure students aren’t adding or removing water during the activity
- Encourage careful reading and recording of water levels, including double-checking each others’ work
- Keep an eye on the data sheets to make sure students are correctly computing water level change
- Note any tricky issues that arise: How to record a level that is between marks?
- If anyone records a water level that goes down (or up by an unreasonable amount) when adding more pennies, express scepticism (hmm, I haven’t seen that before. Can you do it again and show me?)

**Discussion (could be synthesis; narrative could include comments on timing)**

- Brief discussion: Did anyone notice anything? Did the water level always go up when adding more pennies? [should be yes]
- [choose a student and ask] [Shayna], what was the biggest gap between numbers of pennies that you had? What were the two numbers of pennies? [gap should be at least 4, probably more]
- [choose a number roughly halfway between: if student says 7 and 13 pennies, choose 10]
- If I were to put 10 pennies in your boat, how much do you think the water would go up by? Could you predict how much? Do others have ideas for how we might predict?
- Take ideas, and note that we will investigate one way to predict in the next activity

**Anticipated solutions**

Each penny should cause a water level increase of about 2.5 ml (so 4 pennies should be 10 ml increase). If numbers are too far off, express scepticism and ask to see for yourself (i.e. ask them to repeat).**Anticipated misconceptions and challenges**

- Precision, confusion on calculations
- It is fairly common to have one team or more that over-generalize from their first measurement: For example, if a team decides that with 4 pennies the water level went up by 20 ml, they might just add 20 to the water level for each 4 additional pennies, and only cursorily look at the measurement on the beaker. In other words, they believe their assumption about proportionality and their first measurement over subsequent evidence. If this happens, first encourage careful looking at the measurements (including asking them to show you), and if they stick to their story, note for discussion later in the lesson--following the evidence over what we have convinced ourselves is true may be one of the most important lessons here

**Setup**

Same as previous

**Narrative**

[include purposes and a discussion of what students should be doing/thinking/discussing/struggling with]

Teams will first graph their own data on their graph paper, and use that graph to predict water level change for another number of pennies.

**Launch (narrative could include comments on timing)**

- It is helpful to have a visual picture of data to see trends and patterns. How might we get a picture of your boat’s data? [draw a graph; maybe other ideas]
- Work with graphing on a coordinate plane to see if you can spot any trends
- We will put water level change on the vertical axis and the number of pennies on the horizontal. Any thoughts about why we would choose this and not the other way around? [it is a convention that we put the thing we control--the independent variable--on the horizontal axis, and the thing that is the effect of that--the dependent variable--on the vertical axis. Simply an agreement so that we know to do things the same way]

**Task statement**

- Choose appropriate scales for your axes, and label them (numbers and units), and graph all four of your data points from your boat (including 0 pennies)
- Choose a number of pennies that is in between two of your tested numbers, and predict the water level change for that number of pennies. Write your prediction on the “Prediction #1” table on the data sheet and put it on your graph in another color. Does it look like it will fit the pattern of the graph?
- Test! Record the water level for the boat with no pennies, then with your new number of pennies, and compute the new water level change.
- Compare your prediction with your results and add your new data point to the graph

**Orchestrating & Monitoring (narrative could include comments on timing)**

- Watch to make sure teams choose appropriate axes so all data fits but isn’t bunched in one corner or along one side of the paper
- Check for consistent scales on the axes, and encourage revision if needed
- Be sure students are graphing water level change, and not water level
- Be sure students are not fudging/changing data to fit some pattern that isn’t really there

**Discussion (could be synthesis; narrative could include comments on timing)**

- Did all your data points fit on a straight line? [most no; could have some that said yes especially if they assumed proportionality and didn’t measure as precisely as possible]
- How did you use your graphs to predict?
- Take ideas and ask people to critique. Most common will likely be connecting dots on either side or, equivalently, choosing a water level change that is halfway between the levels on either side. Some may have sketched a line or other curve to approximate the data; if so, make sure it gets mentioned but don’t highlight it as the “right” way

- How close were your actual measurements to your predictions?

**Anticipated solutions**

If care was taken in data gathering, all graphs should show increasing water level change with increasing numbers of pennies.

**Anticipated misconceptions and challenges**

Consistent and appropriate scales, fudging data to fit how they think it “should” look are pitfalls.

**Setup**

Large graph grid with axes & scales on whiteboard or grid poster paper

Each team with 4 sticky dots

2 liter graduated beaker with ~1.5 liters of water in it, and pre-made boat from 6” square of foil, approx 1” sides (so 4” square bottom). Make sure it fits in the beaker.

**Narrative**

[include purposes and a discussion of what students should be doing/thinking/discussing/struggling with]

Teams will put all of their data on a class graph,

The mathematical meat of the whole cycle is in the preparation and interpretation of the class scatterplot, so the bulk of this activity is the discussion after creating the scatterplot. The class will discuss trends they see in the data and how they might use the whole class’s data to predict. This will connect the idea of central tendency for a single varying quantity (mean, median) with the notion of a line modeling a trend in a set of bivariate data.

Finally, the class will informally fit a line to the data, and teams will use that line to predict water level change for 45 pennies.

**Launch (narrative could include comments on timing)**

**Text**

- Now I’d like to see all of our data together. I have a big graph here that I’ve already put scales on. What should I label the axes? [number of pennies, water level change]
- We all have one data point in common. What is that? [(0,0)]. So let’s go ahead and add that to our graph. Where does it go?
- Please write your initials on all 4 of your dots, then come up and carefully add your data to the class graph [each team has three original numbers of pennies, then the one additional one they used to predict--they should be graphing their measurement, not their prediction]
- Before sticking a dot down, make sure you and your partner both agree where it should go

**Task statement**

Create a class graph with data from all teams. While everyone is adding their data, look for patterns or strange things you didn’t expect.

**Orchestrating & Monitoring (narrative could include comments on timing)**

- Be sure teams graph water level change, not water level
- Be sure they graph their “actual” data for the fourth data point, not their prediction
- Encourage teams to agree before placing a dot
- By this point you should not have massive outliers or surprises

**Discussion (could be synthesis; narrative could include comments on timing)**

- First discuss trends or patterns in the class graph [it goes up--probe for more precision; it looks like a fuzzy line; it’s all over the place]
- Data should be roughly along a line through the origin. Lead class discussion about the shape of the data, including why the data points aren’t all exactly on a line (measurement imprecision, perhaps varying weights of pennies). Using a yardstick, draw in a line that you can get general agreement on. It should pass through the origin. Extend the line only as far as your data goes (perhaps 28 pennies or so).
- Next ask teams to predict how much they think the water level will change if we float 45 pennies in a boat (show them larger boat), and be ready to explain their prediction method. They should use the data on the board.
- Give 3 minutes for teams to estimate. Encourage them to come up to the board if it is helpful. Leave the yardstick with the graph and watch to see whether any teams use it.
- While teams are working, watch for
- indications that they are extending the line of best fit, either by gesture or with the yardstick
- Proportional reasoning: “22 pennies made it go up 50 ml, so 45 pennies will make it go up about 100 ml”

- After 3 minutes, invite selected groups to share their method and then prediction. Could start with a extend-the-line group, then a proportional reasoning group, and compare results and methods
- A proportional reasoning group might use one data point to extrapolate, rather than the whole class data set. Or they could extrapolate from a point on the line of best fit. If both occur, highlight the difference
- It is important that the method of extending the line be raised and discussed, so if no team uses that idea, the teacher will have to raise it to begin the discussion
- Discussion: Why might the sketched line be a good way to use all the data to estimate? [compared to methods that only take into account one or a few data points, this is a method to take all the data into account, just like using the “mean” when summarizing one repeated measurement]

- Invite each group to write down their prediction for the total number of pennies for your new boat, and their final prediction for water level change for 45 pennies:
- I have a boat here that I have made from a 6” square of foil. I’ve tested it, and it will hold 45 pennies. I’d like your team to write down on your data sheet:
- How many pennies you think my boat can hold before it sinks, and why
- How much you think the water level in my beaker will change with 45 pennies in my boat, and why

- I have a boat here that I have made from a 6” square of foil. I’ve tested it, and it will hold 45 pennies. I’d like your team to write down on your data sheet:
- Test using your boat! Record starting water level (just boat; make sure you’ve left at least 200 ml of graduations above the water level), then gently place 45 and see how people did on displacement estimates, then keep going until boat sinks to find max it will hold before sinking (it can hold close to 100 pennies)

**Anticipated solutions**

It is possible that the data will not look particularly linear. It is acceptable to use a different curve to approximate the data. However, this will make extrapolating to 45 pennies difficult

To predict for 45 pennies, students might:

- Double the displacement for 22 or 23 pennies
- Extend the sketched line of best fit up to 45 pennies, and read across to the vertical axis to read the corresponding displacement

**Anticipated misconceptions and challenges**

If the data do not have an approximately linear trend (proportional, in fact), then that indicates some problems in measurement or record-keeping along the way.

Let’s look back over the boat challenge. What are some things we learned and practiced? [Take ideas}

- Measuring volume
- Recording data
- Noticing trends and patterns
- Building scatterplots
- Reasoning about predictions
- Building boats
- Working together
- Listening to team member ideas
- Learning from others
- others?

Main discussion: the scatterplot and what it can tell us. Be sure to raise the following:

- This is different than a graph where all the points lie on a line or a nice curve (like y = 4x or y = x2)
- Even when the points aren’t all on a line or nice curve, we can still see a relationship
- We can sketch in a line or curve to model the relationship
- If there are points we know are part of the relationship, we can choose to include those points on our line or curve (in this case, (0,0)).
- Once we have that model (the line or curve), we can use it to make predictions or to talk about “typical” behavior
- Predicting displacement for different numbers of pennies--sometimes in between measured values, sometimes beyond. Which kind of prediction gives you more confidence? [typically an in between prediction--interpolation]
- Describing the relationship: How much displacement is produced by each additional penny? How can we tell?
- If the model is a line, the per-penny change is the same all along the model
- Look at difference in displacement--as shown by the model--between 0 and 20 pennies (or 5 and 15, or…), and divide by 20 (or 10…)

- That per-penny change is what we call the slope of the line

- So a scatterplot, and a sketched-in model, is a powerful way to understand the relationship between two quantities that change together

### Timeline:

**Activity 1: **Data Gathering: 20 minutes**Activity 2: **Graphing: 15 minutes**Activity 3: **Class Graph & Prediction: 30 minutes**Lesson synthesis: **10 minutes