## Walk the Plank

• Lesson
6-8,9-12
1

When one end of a wooden board is placed on a bathroom scale and the other end is suspended on a textbook, students can "walk the plank" and record the weight measurement as their distance from the scale changes. The results are unexpected— the relationship between the weight and distance is linear, and all lines have the same x‑intercept. This investigation leads to a real world occurrence of negative slope, examples of which are often hard to find.

To prepare for this lesson, draw a line on a sturdy plank of wood that is 6 inches from one end. Then, draw additional lines at 12‑inch intervals. During the activity, students will stand with their feet straddling these lines.

Place a bathroom scale and a textbook on the floor, about six feet apart. Place the plank of wood so that one end rests firmly on the scale and the other end rests on the book. The line drawn 6 inches from one end of the plank should lie along the center of the scale. (Be sure to test the arrangement prior to class to ensure that it is safe for students.)

Provide the following explanation to students about the forthcoming math investigation. (Since the lesson is called "Walk the Plank," it may be fun to invoke a pirate accent while reading.)

Belay your talk, lads and lasses! Yo ho ho… ye all have performed handsomely as math students, but I’m afraid there are just too many of you in this here classroom. So today, some of ye are going to walk the plank!

(Point to the plank.) Aye, mateys! This here plank stretches between a scale and a textbook. Don’t ye be scared — it’s plenty sturdy. See? (Demonstrate its strength by walking across the plank.) As ye walk across it, we’ll record the weight shown on the scale. To show ye scallywags how to do it properly, I’ll go first.

Distribute the Walk the Plank Activity Sheet to students, and explain how the chart is to be filled in.

Step on the plank so that your feet straddle the line down the center of the scale. Read aloud the weight shown on the scale. (The weight shown will be significantly more than your actual weight, because it includes the weight of the plank.) On the chalkboard or overhead projector, make a note of the weight.

Ask the class, "Do you think this is my actual weight?" Students should realize that the weight shown on the scale includes the weight of the plank. (Although it may seem trivial, this is an important question to ask. When students realize that their actual weight will not be displayed, they will be more likely to participate. Still, when students walk the plank, use care with those who are particularly self conscious.) Step off the plank.

Start at the line nearest the scale. Use the chart on the activity sheet to record the weight. Step left, and move to the next line on the plank. Again, read and record the weight. Continue moving to the left and recording the weight at each line. If it becomes difficult to read the weight, invite a student to read the weight as you move across the plank. As you move and say the weight aloud, remind students to fill in their charts.

After you have moved the entire way along the plank, ask the following questions:

• Plot the points on a graph. What do you notice? [The points occur in a straight line; that is, the relationship between weight and distance is linear.]
• Where is the y intercept? [The y‑intercept is approximately equal to the weight of the teacher plus the weight of the plank.]
• Where is the x intercept? [The x‑intercept is approximately equal to the length of the plank.]
• Approximately, what is the slope? Is it positive or negative? [The slope is negative, and its absolute value is equal to the combined weight of the teacher and the plank divided by the length of the plank.]

Then, allow student(s) to walk the plank. If possible, select a student whose weight is approximately half of your own weight. When the line for this student is graphed, the slope of the line will be half of the slope for your line. Then, select several other students at random. (Because weight is a sensitive subject, choose students carefully, and do not force any student to participate. To avoid an awkward situation, you may want to ask for volunteers rather than select students.)

Allow students to discuss the questions on the activity sheet. To fill the time and extend the thinking of those groups who finish the worksheet and are waiting for others to finish, use the extension activities below.

The activity sheet can be reviewed after all groups have discussed the question, or you may have students complete it for homework.

If necessary, you can refer to the Walk the Plank Answer Key.

Assessment Options

Observe student answers during the class discussion, and check their written answers on the activity sheet.

Extensions

1. Use a longer plank to perform the same experiment. How do the results differ?
2. What happens when two students simultaneously walk the plank?
3. Place two textbooks 12 feet apart. Between them, place a scale. Arrange a 12 foot plank so that its ends rest on the textbooks and the middle of it lies on the scale. What happens when two students simultaneously walk the plank, one on either side of the scale?
4. For the linear equation y = mx + b, what is the value of the x-intercept in terms of m and b?

Questions for Students

1. Why is the slope of the graph negative?

[As the person moved away from the scale, the weight displayed on the scale decreased.]

2. Why does the weight shown on the scale not accurately reflect your weight?

[The weight shown will be significantly more than your actual weight, because it includes the weight of the plank.]

3. When a student whose weight was about half of the teacher's walked across the plank, what did you notice about the slope of that student's line on the graph?

[The slope of the line was about half of the teacher's line.]

Teacher Reflection

• Was students’ level of enthusiasm/involvement high or low? Explain why.
• What student actions allowed you to determine that students did or did not have an adequate understanding of the material? How did you use that information to adjust the lesson?
• Were you able to challenge the high achievers in your class? If so, how? If not, what could have been done to provide more challenge?
• Was this lesson appropriate for your students? If not, what could you do to make it more appropriate?

### Learning Objectives

Students will:

• Recognize that a real world situation is linear.
• Create a graph and write an equation for various linear functions.
• Determine the slope, equation, and x‑intercept of a linear function.

### NCTM Standards and Expectations

• Model and solve contextualized problems using various representations, such as graphs, tables, and equations.
• Use graphs to analyze the nature of changes in quantities in linear relationships.
• Analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior.
• Understand and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions.
• Approximate and interpret rates of change from graphical and numerical data.

### Common Core State Standards – Mathematics

• CCSS.Math.Content.6.EE.A.3
Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

• 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.

• 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.

• CCSS.Math.Content.7.EE.A.2
Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that ''increase by 5%'' is the same as ''multiply by 1.05.''

• 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.

• 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.

• 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.

• CCSS.Math.Content.8.SP.A.1
Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

• CCSS.Math.Content.8.F.A.3
Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s^2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP5
Use appropriate tools strategically.
• CCSS.Math.Practice.MP7
Look for and make use of structure.