## Lemon Calculus

Source: Wikimedia

**Type:** Investigation

**Theme:** Calculus

**Grades:** 5, 6, 7, 8, 9, 10

**Source:** Youcubed.org

**Learning Target:** Students will calculate the volume of a lemon using various techniques: estimation, displacement, and Riemann sums.

Instructions

### Materials

- One lemon per group of two
- Centimeter ruler
- Graduated cylinder or transparent 500 mL measuring cup with measuring lines
- Knife
- Cutting surface
- Sugar and container (if making lemonade)

### What to do

- Divide into groups.
- Discussion - Before students are given tools for measuring the volume of the lemon, students should discuss their ideas about how to find volume of a lemon. The teacher should remind younger students that volume of a cube is
*length x width x height*. Groups should capture this brainstorming session on paper or digitally as an artifact for submission. - Groups will estimate their volume using a centimeter ruler.
- Initially, the teacher will use the displacement method to find the volume of a lemon using a graduated measuring cup. Teacher will compare cubic centimeters with milliliters.
- The teacher will explain the technique of Riemann sums for finding area under a curve. Students will be using a similar method for finding the volume of a lemon.
- The teacher will demonstrate how to slice the lemon to determine volume by summing the estimated volume of each cylindrical slice (π x radius squared x height): https://www.desmos.com/calculator/rwstvakjfd
- Students should compare their three values: (a) estimation, (b) displacement, (c) Riemann sum.

### What to Submit

- One Sway or PowerPoint
- Photographic documentation of the process
- One paragraph explanation (copy/capture your brainstorming sessions)
- Volume of lemon using estimation (cubic centimeters)
- Volume of lemon using water displacement (mL)
- Volume of lemon in cubic centimeters using 'calculus'

### Resources

- https://www.desmos.com/calculator/vfqpn7xlsn
- https://www.desmos.com/calculator/noozvg9bij
- https://www.youcubed.org/resources/the-volume-of-a-lemon/

(This task was inspired by an activity used by Laura Evans and Carlos Cabana while at San Lorenzo High School, California)

Exit Ticket

CCSS Math Practice

- I can reason abstractly and quantitatively.
- I can attend to precision.

NGSS Crosscutting Concepts

- Scale, Proportion, and Quantity