Learning Math: Patterns, Functions, and Algebra
Proportional Reasoning Part D: Speeds, Rates, Steepness, and Lines (45 minutes)
Session 4, Part D
In Parts B and C we examined the proportionality of mixtures and scale models. In this activity, we will take a close look at graphs to learn more about the relationships between distance and time, and between steepness and speed.
Suppose seven cars are all near an intersection. The graph below show the distances between cars and the intersection as time passes. Study this graph carefully and then try to answer the following questions.
In what direction is each car moving in relation to the intersection?
Compare the cars’ speeds. How do their speeds relate to the steepness of the lines?
Tip: How could you approximate the cars’ speeds using the labels on the axes?
Does the distance vary directly with time for any of the cars? That is, is the relationship between distance and time proportional for any of the cars? How do you know?
Do any of the cars stop during their trips? If so, which cars?
Choose one of the seven cars, and describe your trip in this car. Use your imagination! Think of your observations as a passenger or driver in this car, and give the highlights of your trip for these 15 seconds. Include where and when you started the trip and what you saw going on around you — in front of the car, to the sides, and through the rearview mirror.
Car problem taken from IMPACT Mathematics Course 2, developed by Education Development Center, Inc. (New York: Glencoe/McGraw-Hill, 2000), p. 328-329. www.glencoe.com/sec/math
The graph in this section is designed to uncover a variety of common misunderstandings. First of all, some may interpret the graph as a picture of the trips the cars took, a flawed interpretation. If the graph has a segment that is horizontal, there may be more than one interpretation of the car’s trip. The car could be stopped, or it could be going in a circle around the intersection. This is often an interesting insight for people.
Some may assume two cars that are the same distance from the intersection are side by side. Others may think that the two cars in this situation are on opposite sides of the intersection, facing each other, or that they could be approaching the intersection along intersecting streets.
A negative slope indicates that a car is moving toward the intersection. A positive slope indicates that the car is moving away from the intersection, since the distance is increasing over time. Some may wonder where the intersection is. In fact, the graph is not a picture of the car’s path, but a representation of the relationship between distance and time.
Keep in mind that speed is always a positive quantity, while velocity can be positive or negative, corresponding to positive and negative slopes.
Groups: End the session by discussing Problem D5. Each group can share the “story” of their car’s trip.
First, the graph only allows us to decide whether a car is moving toward or away from the intersection, and does not tell us any specific direction. In relation to the intersection, the movement of each car is as follows:
- The black car moves to the intersection, then stops there.
- The red car does not move at all.
- The orange car is moving away from the intersection at all times.
- The green car starts at the intersection and moves away from it at all times.
- The blue car moves away from the intersection for 12 seconds, then stops.
- The yellow car stays at the intersection for 5 seconds, then moves away from it for 7 seconds, then stops.
- The purple car moves toward the intersection, then stops a distance away from it.
The steepness of the line gives the speed of each car, in meters per second. For example, the green car starts at the intersection, then is 120 meters away after 4 seconds. Its rate of speed is then 120 / 4 = 30 meters per second. Using the same technique, and any two points on the line, we can find the speed for each car.
- The black car moves at 20 meters per second, then stops.
- The red car does not move at all.
- The orange car moves at 20 meters per second.
- The green car moves at 30 meters per second.
- The blue car moves at approximately 12 meters per second, then stops.
- The yellow car starts at rest, then moves at approximately 23 meters per second, then stops.
- The purple car moves at approximately 13 meters per second, then stops.
Yes, the green car’s distance varies directly with time. Its graph is a straight line passing through the origin (0, 0).
Yes, any horizontal line represents a stopped car (because the distance from the intersection is not changing). The black, red, purple, blue, and yellow cars are stopped at some time.
You could argue that the red car, or any car that stays the same positive distance away from the intersection, is moving in a circle around the intersection! It may be true that the blue, yellow, and purple cars all entered a traffic circle at the same time; as mentioned in Problem D1, the graph does not give us enough information to tell us in what direction the cars are moving, only their distance.
Answers will vary. Here’s one scenario: The yellow car started out smoothly from a red light, and continued moving away from the intersection at 20 meters per second. Then, suddenly, the yellow car was stopped short by a collision just ahead between the purple and blue cars.
Session 1 Algebraic Thinking
In this initial session, we will explore algebraic thinking first by developing a definition of what it means to think algebraically, then by using algebraic thinking skills to make sense of different situations.
Session 2 Patterns in Context
Explore the processes of finding, describing, explaining, and predicting using patterns. Topics covered include how to determine if patterns in tables are uniquely described and how to distinguish between closed and recursive descriptions. This session also introduces the idea that there are many different conceptions of what algebra is.
Session 3 Functions and Algorithms
In Session 1, we looked at patterns in pictures, charts, and graphs to determine how different quantities are related. In Session 2, we used patterns and variables to describe relationships in tables and in situations like toothpicks and triangles. This session extends the exploration of relationships to include the concepts of algorithm and function. Note1
Session 4 Proportional Reasoning
Look at direct variation and proportional reasoning. This investigation will help you to differentiate between relative and absolute meanings of "more" and to compare ratios without using common denominator algorithms. Topics include differentiating between additive and multiplicative processes and their effects on scale and proportionality, and interpreting graphs that represent proportional relationships or direct variation.
Session 5 Linear Functions and Slope
Explore linear relationships by looking at lines and slopes. Using computer spreadsheets, examine dynamic dependence and linear relationships and learn to recognize linear relationships expressed in tables, equations, and graphs. Also, explore the role of slope and dependent and independent variables in graphs of linear relationships, and the relationship of rates to slopes and equations.
Session 6 Solving Equations
Look at different strategies for solving equations. Topics include the different meanings attributed to the equal sign and the strengths and limitations of different models for solving equations. Explore the connection between equality and balance, and practice solving equations by balancing, working backwards, and inverting operations.
Session 7 Nonlinear Functions
Continue exploring functions and relationships with two types of non-linear functions: exponential and quadratic functions. This session reveals that exponential functions are expressed in constant ratios between successive outputs and that quadratic functions have constant second differences. Work with graphs of exponential and quadratic functions and explore exponential and quadratic functions in real-life situations.
Session 8 More Nonlinear Functions
Investigate more non-linear functions, focusing on cyclic and reciprocal functions. Become familiar with inverse proportions and cyclic functions, develop an understanding of cyclic functions as repeating outputs, work with graphs, and explore contexts where inverse proportions and cyclic functions arise. Explore situations in which more than one function may fit a particular set of data.
Session 9 Algebraic Structure
Take a closer look at "algebraic structure" by examining the properties and processes of functions. Explore important concepts in the study of algebraic structure, discover new algebraic structures, and solve equations in these new structures.
Session 10 Classroom Case Studies, Grades K-2
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the K-2 grade band.
Session 11 Classroom Case Studies, Grades 3-5
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the 3-5 grade band.
Session 12 Classroom Case Studies, Grades 6-8
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the 6-8 grade band.