## The Crow and the Pitcher: Investigating Linear Functions Using a Literature-Based Model

- Lesson

The lesson is based upon Aesop’s fable,
“The Crow and the Pitcher,” and involves students making predictions and conducting
experiments to determine how many pebbles the crow would need to add to the
pitcher in order to bring the water to drinking height. In the course of the
investigation, students gain a real-world understanding of linear functions and
such concepts as slope, *y*-intercept,
domain, and range.

Introduce the activity by asking
what the students know about *Aesop’s
Fables*. Aesop’s fables are short, fantastical tales written by Aesop, an
ancient Greek storyteller. The tales are often characterized by a moral. Read
students the fable ”The Crow and Pitcher” from the overhead. As an option, you may wish to show
them an online video of this fable (search “Crow and Pitcher” on the web to find multiple
videos).

Crow and Pitcher Activity Sheet

Crow and Pitcher Story Overhead

Crow and Pitcher Table Overhead

After reading the story, tell students that they will conduct an experiment, collect and record data, and use that data to predict the number of pebbles needed to bring the water to drinking level.

Organize students into groups of three or four and give each group a graduated cylinder, a container of water, and a bag of marbles (about 10 marbles per group). Distribute the Crow and Pitcher Activity Sheet to students. Instruct students to do the following:

- Fill the cylinder with water to a level of 80 mL.
- Add marbles, one at a time, to the cylinder.
- Record the results on the table of values.

Tell the students to read the water level by looking at it directly from the side, not slightly above or below. If they notice a curve to the profile of the top of the water, the correct reading is indicated by the bottom of the curve.

When all groups have completed the table (and added 6 marbles, one at a time), post the Crow and Pitcher Table of Values Overhead and discuss findings so far. Instruct students to complete the activity sheet.

When students have completed the activity sheet, discuss any surprising discoveries. [The water will only rise to a certain level before the height of the marbles will exceed the height of the water.]

During the second half of class, or on the following day, each group should produce a poster presenting their results. These results should include the following:

- A table of values from their activity sheet.
- A graph.
- An equation.
- A picture that relates the equation to the physical model.
- A written explanation.

You may wish to post these requirements on the board. Hand out The Crow and Pitcher Rubric so students can refer to the expectations of the activity.The following PowerPoint shows sample student work.

Sample Student Work of Nancy Wolf's students

As a conclusion to the activity, have students speculate how results would be different if larger, or “shooter,” marbles were used. Ask students to explain their responses, describing the changes to the table of values as well as the slope of the graph.

- Standard-sized marbles
- 100 mL graduated cylinders (1 per group)
- Graph paper
- Markers
__Crow and Pitcher Story Overhead____Crow and Pitcher Activity Sheet____Crow and Pitcher Table of Values Overhead____Crow and Pitcher Rubric__- Graphing Calculator
- Computers or tablets with internet access

**Assessment Options **

- Group posters may be assessed
using the Crow and Pitcher rubric.Students may be asked to display their posters
and present their results to the class. Each student in the group may be
responsible for explaining one aspect of the multiple representations.
Different students might be asked to:

Determine how many marbles the crow needed to add to drink the water [refer to Table of Values];indicate and explain what key points of the graph represent, such as the*y*-intercept, slope, and labels; explain how you determined the equation of the line? explain how your sketch relates to the equation of the line. - Repeat the activity with a starting point of 60 mL. Students might again find that the water only rises to a certain level before the height of the marbles exceeds the water. Students may then graph the results and find the equation.

**Extensions **

- Replicate the activity using real pebbles (or varying sizes of gravel — available at home improvement stores).
- Find the line of best fit and the equation of the best-fit line using a graphing calculator or the Line of Best Fit tool.

**Questions for
Students **

1. How would the appearance of the graph change if larger marbles were used?

[The slope, or steepness of the line, would be greater as the larger marbles displace more water, therefore increasing the rate of change. It would not change the y-intercept.

2. What limitations or restrictions affect the appearance of the graph?

[The line will not increase infinitely as it is limited by the height of the cylinder and the fact that the height of the accumulated marbles will likely exceed the water level at some point.]

3. What effect would marbles of varying sizes have on the graph?

[The plots would not lie on a line but the plots would suggest a linear relationship.]

**Teacher Reflection **

- What was the general level of student engagement?
- How were the achievers challenged?
- What parts of the lesson did some of the students seem to have difficulty understanding?
- Were the struggling learners/English Language Learners able to comfortably contribute to the group?
- How did students demonstrate comprehension of the stated learning objectives by the conclusion of the lesson?
- What classroom management issues occurred during the lesson and how did you deal with them?
- Did students have prior knowledge necessary for tasks or should time be spent prior to the lesson reviewing slope,
y-intercept, domain, and range?- How did the lesson help students make connections between real-life situations and algebraic concepts in this lesson?
- Were student tasks distributed equitably?

### Learning Objectives

Students will:

- Collect and record data in a real-life linear function scenario.
- Determine multiple representations of data including graphs, tables of values, equations, sketches, and written explanations.
- Recognize slope as rate of change, and y-intercept as the value at time = zero.
- Determine the equation of a line from data, a table of values, or a graph.

### NCTM Standards and Expectations

- Explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope.

- Use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships.

- Use graphs to analyze the nature of changes in quantities in linear relationships.

### Common Core State Standards – Mathematics

Grade 6, Expression/Equation

- CCSS.Math.Content.6.EE.B.6

Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

Grade 6, Expression/Equation

- CCSS.Math.Content.6.EE.C.9

Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

Grade 8, Expression/Equation

- CCSS.Math.Content.8.EE.B.5

Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

Grade 8, Expression/Equation

- CCSS.Math.Content.8.EE.B.6

Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

Grade 8, Functions

- CCSS.Math.Content.8.F.B.4

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Grade 8, Stats & Probability

- CCSS.Math.Content.8.SP.A.2

Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

Grade 8, Stats & Probability

- CCSS.Math.Content.8.SP.A.3

Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP4

Model with mathematics.

- CCSS.Math.Practice.MP7

Look for and make use of structure.