Learning Math: Patterns, Functions, and Algebra
Linear Functions and Slope Part B: Slope (40 minutes)
Session 5, Part B
In This Part
- Thinking About Slope
- Comparing Slopes
- Slopes and Architecture
Thinking About Slope
Slope is an important concept in mathematics, and in Part B we’ll explore how it is used to solve problems.
Take a minute to think about what you already know about slope. What does it mean? Where is it used?
You may be familiar with the idea of slope as a measure of steepness. The formula for slope is usually described as:
slope = (change in y) / (change in x)
The slope of a line is often described as a ratio of rise/run. Another way to think of slope is as the amount that the dependent variable changes for each increase by 1 in the independent variable. In other words, as x changes by 1, what happens to y?
Look at these four graphs. For each graph, select four pairs of points, and calculate the slope of the line between each pair of points. Remember that slope = (change in y) / (change in x). As you calculate the slopes for each of the graphs, ask yourself why the slope between pairs of points would change or why it would stay the same.
What happened when you tried to find the ratio of rise/run for the fourth example, a curved object?
The drawing below shows a cable attached to a wall.
Calculate the ratio rise/run for each pair of points:
- Points P and Q
- Points P and R
- Points Q and R
Describe the difference between the rise/run ratios for the graph in Problem B3 and the ratios for the graph of a line.
Here are graphs of nine different lines. Look at the equation and the slope of each line. What changes when the slope becomes positive or negative, and when slope is larger or smaller than 1?
Consider this line. What are the coordinates of the points R and S?
Find the slope of the line through points R and S.
Many people describe “slope” as a measure of the “steepness” of a line. Look at the two graphs below. Which line has larger slope? Which line appears to be steeper? Explain what is happening.
Tip: Is there anything about the graphs that makes comparison more difficult?
In this video segment, the onscreen participants discuss methods of comparing the slope of two lines. Watch this segment after you have completed Problem B7 and compare your methods of comparison with those of the onscreen participants.
If two graphs are drawn with the same scale, is it true that the graph that appears steeper has the larger slope?
You can find this segment on the session video, approximately 11 minutes and 32 seconds after the Annenberg Media logo.
Do the following problems on graph paper:
On one graph, create three different lines with slope 2. Describe what is the same and what is different about these graphs.
On one graph, create three different lines with negative slope. Describe what is the same and what is different about these graphs.
Problems B5 and B6 taken from IMPACT Mathematics Course 3, developed by Education Development Center, Inc. (New York: Glencoe/McGraw-Hill, 2000), p. 27.
Slopes and Architecture
Architects and carpenters use rise and run to describe and build staircases. The picture below shows the rise and run for each step and the total rise and run for the staircase.
Take It Further
An architect is designing a staircase for a house with a difference of 10 feet between floors. The staircase has 18 steps and a total run of 14 feet. What is the ratio (total rise/total run) of the staircase?
Find the rise and run for each step (what we’ll call the step rise and run) in inches. What is the ratio (step rise/step run)? Explain your answer.
Find a nearby staircase and measure the rise and run of one step. How could you use this to estimate the total rise and run of the staircase?
Design a staircase with a total rise of 14 feet, a step rise between 6 and 8 inches, and a sum of step rise and step run between 17 and 18 inches. All steps should have the same rise and the same run. Your answer should include
- the number of steps
- the height of each step
- the run of each step
- the total run
- the ratio (step rise/step run)
- the ratio (total rise/total run) for the staircase.
Tip: There are many different solutions to this problem. Remember that the total of the step rise and the step run must be between 17 and 18 inches.
In this video segment, taken from the “real world” example at the end of the Session 5 video, master carpenter Norm Abram describes the importance of slope in construction. Watch this segment after completing Part B.
In what other professions is slope used on a regular basis?
You can find this segment on the session video, approximately 21 minutes and 8 seconds after the Annenberg Media logo.
Problems B10-B13 taken from IMPACT Mathematics Course 3, developed by Education Development Center, Inc. (New York: Glencoe/McGraw-Hill, 2000), p. 41. www.glencoe.com/sec/math
Groups: Discuss what everyone already knows about slope. Go over the definition of slope in the course materials, and then work on Problems B1-B4. Graph paper is needed for this exercise.
While working with the first Interactive Activity, choose different pairs of points. For each pair, it’s important that the difference between the y coordinates and the difference between the x coordinates are computed by considering the pairs in the same order. That is, for points (x1, y1) and (x2, y2), the slope can be computed as either
(y1 – y2) / (x1 – x2) or (y2 – y1) / (x2 – x1).
Note that students sometimes confuse this by computing (y2 – y1) / (x1 – x2).
For Problem B3, consider measuring directly on the picture or tracing the picture onto graph paper to compute the ratios.
Think about the main points for this part of the session: The rise/run for a line is the same between any pair of points on the line; lines are the only kind of graph for which this is true.
Problem B7 focuses on the relationships between slope, scale, and the appearance of a line’s graph. It’s an important idea that slope only relates to “steepness” if you are comparing two lines with the same scale.
In Problem B8, participants should notice that the lines are all parallel and therefore do not intersect. You may want to talk briefly about what they know about parallel lines, as we will return to this idea in future sessions.
On a basic level, slope measures vertical change over some horizontal distance — a road can “slope up” or “slope down.” The slope of a line describes how much vertical change (change in y) there is per horizontal change (change in x).
On a line, the ratio of rise to run is always constant; on a curved object, this value is constantly changing. So, the slope of a curved object changes depending on the points selected, while the slope of a line is always constant.
From P to Q, the ratio is roughly 0.6.
From P to R, the ratio is roughly 1.3.
From Q to R, the ratio is roughly 4.
The ratios for the graph of a line would be constant throughout. For example, a line connecting P to R would have a rise/run ratio of 1.3, regardless of where a new point was located on it. This is different from the graph used in Problem B3, which has a slope that varies depending on which two points are used.
The coordinates are R(1, 6) and S(4, 12).
The slope is (change in y) / (change in x). The change in y is 12 – 6 = 6, and the change in x is 4 – 1 = 3. Therefore, the slope is 6/3 = 2. Note that the line may not appear to have slope 2, since the vertical axis is labeled by twos, while the horizontal axis is labeled by ones.
Graph A has a larger slope, even though graph B appears steeper. This appearance is caused by the different scale used in the two graphs. If it were placed on the other scale, the line in graph B would appear nearly flat!
All the lines have the same slope at all points. None of the lines intersect, and they are always the same distance from each other — they are all parallel lines.
A line with negative slope travels down and to the right (“decreasing”). Lines with negative slope may still intersect one another. Any line with negative slope will travel through the top left and bottom right quadrants of the graph paper.
The total rise / total run is 10/14, which is roughly 0.71.
To calculate step rise and run, divide both rise and run by 18. The step rise is 10/18 feet (6 2/3 inches) and the step run is 14/18 feet (9 1/3 inches). The ratio of step rise / step run is (10/18) / (14/18), which again equals 5/7 or 0.71 — the step rise and step run are in proportion to the total rise and total run.
The total rise and run can be estimated accurately by multiplying the rise and run of a single step by the total number of stairs, because each stair has roughly the same rise and run.
There are a lot of answers here!
Number of steps: between 21 and 28
Height of each step: between 6 and 8 inches
Run of each step: between 9 and 12 inches
Total run: between 189 and 336 inches
Ratio of step rise to step run: between 6/12 and 8/9
Ratio of total rise to total run: between 6/12 and 8/9
Regardless of what you did, the ratios of step rise to step run and total rise to total run should be identical.
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.