# Unit Overview: The Tortoise and the Hare

Aim: How can we interpret the meaning of slope and y-intercept as they relate to linear functions?

Objectives: Students will accomplish the following:

• Experiment with different walking rates, starting points, and directions of travel to walk various linear and piece-wise linear graphs with the aid of a CBR.

- • Complete a table, graph the data, and find a linear equation given a speed and starting point.
- • Discover the relationship between slope, speed, and the coefficient of x in a linear equation.
- • Discover the relationship between different speeds and directions, and positive, negative, and zero slopes.
- • Discover the relationship between the starting point, y-intercept, and the value that is added to the x term in the equation of a linear function.
- • Translate between representations of table, graph, equation, and scenario.
- • Correctly interpret and write stories for graphs that contain both positively and negatively sloped linear segments.

Grade Levels: Sixth-ninth

Number of 60-Minute Periods: 5 math periods

Source: Original from Char and Kara with adaptations suggested by Tara and through several pilots with students.

NCTM Principles

• Teaching and Learning, Access and Equity, Curriculum, Tools and Technology, Assessment, Professionalism

NCTM Mathematics Standards

• Algebra: Understand patterns, relations, and functions.

CCSSM Standards

- 6.EE.B.5-6.EE.B.8. Reason about and solve one-variable equations and inequalities.
- 6. EE.C.9. Represent and analyze quantitative relationships between dependent and independent variables.
- 7. RP.A.2, A.2. Analyze proportional relationships and use them to solve real-world and mathematical problems.
- 7. EE.B.4. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
- 8. EE.B.5. Understand the connections between proportional relationships, lines, and linear

equations.

8.F.B.4, B.5. Use functions to model relationships between quantities.

HSA.REI.D.10. Represent and solve equations and inequalities graphically.

HSF.IF.B.4. HSF.IF.B.6. Interpret functions that arise in applications in terms of the context. HSF.LE.B.5. Interpret expressions for functions in terms of the situation they model. HSF.IF.C.7. Analyze functions using different representations.

NCTM Processes/CCSSM Mathematical Practices

• Problem Solving

SMP1: Make sense of problems and persevere in solving them.

• Reasoning and Proof

SMP2: Reason abstractly and quantitatively.

SMP3: Construct viable arguments and critique the reasoning of others. SMP8: Look for and express regularity in repeated reasoning.

• Communication

SMP3: Construct viable arguments and critique the reasoning of others. SMP6: Attend to precision.

• Connections

SMP5: Use appropriate tools strategically.

• Representation

SMP4: Model with mathematics.

SMP5: Use appropriate tools strategically.

SMP7: Look for and make use of structure.

Prerequisites

- • Basic understanding of and ability to work with the Cartesian coordinate plane
- • Ability to compile tables or t-charts
- • Ability to solve an equation that is in the form of ax + b = c
- • A limited amount of prior experience graphing motion (especially through the use of motion detectors)

Mathematical Concepts: Students use a CBR to explore and explain the properties for linear and piece-wise linear graphs of distance versus time. They discover properties of slope and y-intercept of a line.

Materials and Tools

- • Per student: handouts, scientific calculators
- • Per class: masking tape and meter sticks on floor, stopwatch, centimeter measuring tape, CBR (motion detector) and graphing calculator, whiteboard transparency of coordinate axes or board marked with grid

Management Procedures

- • Prepare Figures 11.1-11.6
- • Think-Pair-Share cooperative group work (all students positively interdependent)

Assessment

- • Observation and questioning of students' work (made while circulating the room during class period)
- • Homework
- • Final stories using vocabulary learned through the series of activities, which should reflect correct use of vocabulary and deep understanding of the concepts of linear function, constant speed, slope, and y-intercept

# Resources

Aesop. (1954). The Hare and the Tortoise. In *Childcraft, Vol. 3, Folk and Fairy Tales.* Chicago: Field Enterprises: 239.

Anderson, R. (2007). Being a mathematics learner: Four faces of identity. *The Mathematics Educator, *17(1): 7-14.

Beckmann, C. (1989, May). Interpreting graphs. *Mathematics Teacher,* 82: 353-360.

Beckmann, C., and K. Rozanski. (1999). Graphs in real time. *Mathematics Teaching in the Middle School,* 5(2): 92-99.

Hastings, E. and D. Yates. (1993). Microcomputer unit: Graphing straight lines. In *Activities for Active Learning and Teaching,* edited by C. Hirsch and R. Laing. Reston, VA: National Council of Teachers of Mathematics: 105-109.

Lappan, G., W Fitzgerald, S. Friel, J. Fey, and E. Phillips. (1997). Variables and patterns. In *Connected Mathematics Grade* 7. White Plains, NY: Dale Seymour.

Town, R. J. and A. Espinosa. (2015). Racing towards algebra and slope. *Mathematics Teaching in the Middle School,* 21(3): 169-175.