Students will build a boat from tinfoil, with the goal of floating at least 15 pennies. After some open building and exploration time, they will keep track of the changing water level in the cylinder as they add pennies.

In the follow-up lesson, students will graph (as a class) the resulting water level change-vs-pennies relationship, and use it to predict water level change for a fixed number of pennies added to another boat. In an optional third modeling lesson (ideally staged several weeks after the initial lessons), students will estimate whether a boat in a video will cause the container to overflow when pennies are added.

The primary mathematical content focus is investigating patterns of association in bivariate data: scatterplots, patterns in scatterplots, straight line models of association, informal assessment of model fit, and use of a straight-line model to make predictions. Standards in the Functions domain also play a key role, when constructing a linear function to model the water level change vs pennies relationship.

Important prior standards that are reinforced here include: seventh grade standards on use of variables to represent quantities and solve problems in real-world settings (7.EE.B.4); sixth grade standards on statistical variability (6.SP.2) and quantitative relationships between dependent and independent variables (6.EE.9); and fifth grade standards about plotting on a coordinate plane (5.G.1) and representing (5.MD.2) and interpreting (5.G.2) data.

This cycle should take two or three 50-minute class periods.

### Prior Experience:

See “About the math” for previous grades’ content that this cycle builds on

A unit on using functions to model relationships between quantities (8.F.B) should occur prior to this cycle in the school year

### Lessons:

### Learning Cycle Wrapup:

**A final all-class discussion about floating objects and displacement should lead to the following points being raised:**

- As pennies are added to the boat or cup, the weight of the pennies causes the cup or boat to “displace” some of the water--that is, to push the water out of the way. This is an important scientific principle called “Archimedes’ Principle.”
- Each penny added causes the same amount of water to be displaced: About 2.5 ml
- We can tell the amount that the water will rise for a given number of pennies, once we know the amount that each penny causes the water to rise
- When we graph the pairs of (displacement in ml, # pennies), the shape of the graph will be a straight line, because each penny causes the water level to change by the same amount

### Teacher Reflection:

**Progress in Standards for Mathematical Practice assessment**

The modeling lesson at the end of the cycle is a good chance to assess the target SMPs: 1 (Make sense of problems and persevere in solving them), 2 (Reason abstractly and quantitatively), and 4 (Model with mathematics).

Modeling: Battery Charging (extrapolating a linear relationship): https://www.illustrativemathematics.org/content-standards/tasks/641

**Progress in Mathematical content assessment**

Patterns of association in bivariate data:

- Birds’ Eggs: https://www.illustrativemathematics.org/content-standards/8/SP/A/1/tasks/41
- Texting and Grades (important to include one relationship with a “downward” trend):https://www.illustrativemathematics.org/content-standards/8/SP/A/1/tasks/975
- Animal Brains (includes constructing scatterplot): https://www.illustrativemathematics.org/content-standards/8/SP/A/1/tasks/1520
- Laptop battery charge: https://www.illustrativemathematics.org/content-standards/8/SP/A/2/tasks/1558
- US Airports https://www.illustrativemathematics.org/content-standards/8/SP/A/3/tasks/1370

**Progress in Maker mindset assessment**

[Create a Maker Mindset version of the 5x8 card--like the SMPObservationTool]

### Credits and Sources:

- Did the maker challenge provide an effective entry into the mathematics exploration?
- What mathematical learning did you see evidence of? Did it match the content goals of the cycle? What other mathematical opportunities did you notice? Do you want to use those opportunities in upcoming instruction?
- What do you want to remember the next time you teach this cycle?