Climbing Stairs. Goals. Launch

Transcription

1 4.1 Climbing Stairs Goals Introduce students to the concept of slope as the ratio of vertical change to between two points on a line or ratio of rise over run Use slope to sketch a graph of a line with this slope In this problem, students find the steepness of a set of stairs using carpenters guidelines for building stairs. This ratio of rise to run informally introduces the concept of slope. It provides a strong visual representation for the ratio of vertical distance to horizontal distance between two points on a line. The rise and run of a set of stairs are then compared to the vertical and between any two points on a line. Launch 4.1 Suggested Questions Discuss why stair climbing is a popular aerobic exercise. Ask: Does the steepness of a set of stairs affect the exercise? For homework, examine the stairs in your house, apartment, or school. Do all stairs have the same steepness? How can we determine the steepness? Suggested Questions Pose the questions of the Getting Ready. It asks students to think about how to describe steepness. How can you describe the steepness of the stairs? (The steepness of stairs depends on how tall each step is (the height) and on the distance from the edge of the step to the next step (the width). So really tall steps will make for a really steep set of stairs and a set of stairs where the flat part you step on isn t very wide will make for a really steep set of stairs.) Is the steepness the same between any two consecutive steps? (Yes, it seems like they would be because you want each step to be the same as all the other steps so the steepness between the consecutive steps should be the same.) For the experiment in Question A: Let students go out in groups to measure the rise and the run and then find the ratio of the rise to the run. Tell the groups that they need to measure more than one step in each set of stairs. Compare the ratios. Use different sets of stairs for each group if possible. Note: There are builders codes that limit the variability of the rise and run, so new buildings may have stairs for which the rise-to-run ratio is consistent. Sometimes steps outside or in stadium bleachers will give a different steepness than stair steps inside houses. Alternate Launch Pose this problem a day or so before you intend to do class work on it. This will allow students to do some physical experimentation. They should find several sets of stairs that have different-sized steps. Physically climbing several different sets of steps allows students to feel steepness. Challenge them to think about what makes one set of steps feel steeper than another. Suggested Question Ask: How could you use a mathematical measure to give an indication of steepness? Have them make measurements on at least a couple of different sets of steps that they think might help in indicating steepness. When you are ready to launch the in-school part of this problem, ask for suggestions on what factors seem to influence the steepness. What measures can give us a mathematical way to compare steepness? If no one mentions the rise and the run, make this suggestion as a possible measure. Draw a picture of a set of stairs and demonstrate the rise and run. You can either give the Carpenters Guidelines for the ratio now or wait until after they have collected their data and then have them compare their ratios to the official guidelines. Another Possible Launch You can have the students measure the steepness of a set of stairs and then discuss it. Then you can assign Question B. Be sure to discuss this question since it helps students make the connection between steepness of stairs and slope of a line. Let students work in small groups of three to four people. 96 Moving Straight Ahead

2 Explore 4.1 Question A When the groups have recorded their measures of a staircase in the building, have them organize what they have found out about the ratio of rise to run between several of the steps in the staircase. Also, have them look at the ratios of the rise to the run on the staircases they measured. Have the groups organize their information about steps and steepness. Question B Some students may need help in drawing the line that matches the ratio of changes. Grid paper may help. Suggest that the student draw a line that goes through the origin. Then suggest that the student draw a couple of stairs with the ratio given. Students then can connect the top of the stairs to form the desired line. Summarize 4.1 Question A Let each group report on stairs they have investigated. Make a class record of the stairs and measures on the stairs that are reported. Compare the ratios for various sets of stairs. Suggested Questions What is the steepest set of stairs in our list? The least steep set of stairs? How do you know? Are some stairs steeper than others? If so, how can you tell? Can you order the entire list of stairs from least to greatest in terms of steepness? These questions focus attention on uses of the ratio as a way of characterizing steepness in a mathematical sense. Steer the discussion to measures of rise and run. Make the connection to the Comparing and Scaling unit in which the students learned to form ratios as a way to compare situations. If you have not done so already, talk about the Carpenters Guidelines with your students. Suggested Questions Then ask questions such as the following: How do the ratios of the stairs we measured compare to the carpenters guidelines? Which ones meet the standards and which ones do not? What do you think influences a builder s decision on the run of a set of steps? What do you think influences a builder s decision on the rise of a set of steps? (In these questions students should be making comparisons using the ratio of the rise to the run.) Now let s think about another common object that you have climbed a ladder. How can you use what you have learned so far to help make sense of steepness as it applies to ladders? What would make a ladder feel steep and what would make the same ladder feel less steep? (The angle of the ladder against the house will affect the rise, hence affect how the ladder feels to climb.) Question B Collect several equations. You may want to draw several of these on a grid on the overhead projector. Suggested Questions Ask: What do you notice about these lines? (They are parallel. Many students will draw the line through the origin, but some will have y-intercepts other than (0, 0). Parallel lines will be explored further in Problem 4.3.) Discuss the strategies that students used to answer the questions in Question B. Finding a line that does not meet the carpenters guidelines is an opportunity for students to test their understanding of the ratio of rise to run. Use this summary to define slope and to launch the next problem. Use an illustration of stairs and steps when you define the slope. Help students to make visual connections between these things that they have physically experienced and the lines on a graph representing linear relationships. I N V E S T I G AT I O N 4 Investigation 4 Exploring Slope 97

3 Slope is the ratio of the change in the vertical distance to the change in the horizontal distance between two points on a line or slope = vertical change rise vertical change run 98 Moving Straight Ahead

4 4.1 Climbing Stairs Mathematical Goals At a Glance 1 PACING 1 days 2 Introduce students to the concept of slope as the ratio of vertical change to between two points on a line or ratio of rise over run Use slope to sketch a graph of a line with this slope Launch Discuss why stair climbing is a popular aerobic exercise. Ask: Does the steepness of a set of stairs affect the exercise? For homework, examine the stairs in your house, apartment, or school. Do all stairs have the same steepness? How can we find the steepness? Pose the questions in the Getting Ready. Let students go out in groups to measure the rise and the run and then find the ratio of the rise to the run. Tell the groups to measure more than one step in each set of stairs. Compare the ratios. Use different sets of stairs for each group if possible. (See Explore for Alternate Launch.) Let students work in small groups of three to four people. Transparency 4.1 Vocabulary slope Explore Question A: When the group has recorded its measures of a staircase in the building, have them organize what they have found out about the ratio of rise to run between several of the steps in the staircase. Question B: Some students may need help in drawing the line that matches the ratio of changes. Grid paper may help. Suggest that the student draw a line that goes through the origin. Then suggest that the student draw a couple of stairs with the ratio given, and connect the top of the stairs to form the desired line. Measuring tape in inches Summarize Question A: Let each group report on stairs they have investigated. Make a class record of the stairs and measures on the stairs, and compare the ratios for various sets of stairs. What is the steepest set of stairs in our list? The least steep set of stairs? Are some stairs steeper than others? If so, how can you tell? Can you order the entire list of stairs from least to greatest in terms of steepness? Talk about the Carpenters Guidelines with your students. Ask: How do the ratios of the stairs we measured compare to the carpenters guidelines? Which ones meet the standards and which ones do not? Student notebooks continued on next page Investigation 4 Exploring Slope 99

5 Summarize continued What do you think influences a builder s decision on the run/rise of stairs? Question B: Collect several equations and draw them on a grid on the projector. What do you notice about these lines? Discuss strategies students used to answer Question B. Use this summary and an illustration of stairs and steps to define slope and launch the next problem. Slope is the ratio of the change in the vertical distance to the change in the horizontal distance between two points on vertical change a line or slope = ACE Assignment Guide for Problem 4.1 Core ACE 1, 36 Other ACE Connections 37, 38, 42 Adapted For suggestions about adapting ACE exercises, see the CMP Special Needs Handbook. Connecting to Prior Units 37: Bits and Pieces II Answers to Problem 4.1 A. 1, 2. Answers will vary, but the ratio of rise to run should be approximately 0.75 and the rise plus the run should be approximately The ratio of rise to run is not the carpenter s guidelines, but it becomes important as we interpret from rise to run as slope. B =0.6 is just in the range of the carpenters guidelines. Some students may change this to a unit rate of 1 to y=(0.6)x y=0.6x y a. The coefficient of x is 0.6. b. The coefficient tells you the line s steepness. c. The coefficient tells you the stair s steepness, which is the ratio of the rise of the stairs compared to the run of the stairs. 5 x Moving Straight Ahead