Activity 2: The Power Tower
Rationale
The purpose of this activity is for students to focus on the relationship between two variables. The students are given a real-world example of a linear relationship, and the goal is for students to discover this relationship after watching the videos and to explore what it means for two variables to have a linear relationship. The video provides the data students will use to accomplish this goal. By the end of the activity, students should understand that a linear relationship involves a constant increase/decrease for every unit increase in the input variable. In other words, as one variable increases, the other variable increases/decreases at a corresponding amount. This activity encourages covariational reasoning since students are to consider how moving along one variable (e.g. total distance) covaries with the other variable (e.g. height). The idea is also for the students to remain quantitative as they explore the relationships between the two given quantities (i.e. height, total distance traveled). Overall, the Power Tower activity will provide the scaffolding that students will need in order to progress towards the discovery of quadratic relationships by the end of the unit.
Standards
The activity covers Common Core Georgia Performance Standards (CCGPS) for 8th Grade. The detail of standards could be found here.
Activity Links
Video Link 1 Video Link 2
Materials Needed
Video (see links above), projector/computer access, whiteboards, markers, erasers, student handout (see link at the bottom of the page)
Formative Question
See the handout attached below.
Sequence of Lesson
Warm-up: Review the ideas from the Zogs activities. For instance, discuss how to find the relationship between two points. Ask students to define slope. Provide examples on the board and ask students to describe the situations quantitatively by being specific about the variables under consideration.
Description: The purpose of this activity is for students to graph the linear relationship between height and total distance traveled by the riders of the amusement park ride, the “Power Tower.”
1. Show the two videos of the amusement park ride to the class. Ask them to identify any quantities that they could measure.
2. After the students watch the video, start a discussion on some of the quantities the students noticed (height, speed, time, total distance, etc.).
3. Split students into groups of 4-5 students each. Each group should have a whiteboard, marker, and eraser. Half the groups are responsible for answering the questions using Video 1 and the other half using Video 2.
4. Hand each group a student handout.
5. While students are working, allow the videos to loop so that students may reference it. If there are enough laptops per group, each group can watch their video. Otherwise, loop both videos (one or two at a time). If it is only possible to project one video, a modified version of this activity can be done where all the groups focuses on the same video.
6. Go around to the tables and encourage discussion of the relationships between the quantities.
7. Example Questions/Prompts:
a) What happens to the total distance as height increases/decreases?
b) How do you know how steep the lines on your graph should be?
c) How are vertical distance and total distance related?
d) Describe what happens when the ride changes directions.
e) How do you know whether the line you draw on your graph should be straight or curved?
8. When all the students are finished working, have them put up their boards so that they are visible to the class. (Note: All the boards should look the same (besides the number of peaks), regardless of their group.)
9. Allow students to share their solutions with the class. Ask them why the graphs are the same (or why one may look different). Focus the discussion around the meaning of a linear relationship between quantities. Students can also discuss the ideas of prediction, inputs and outputs of functions, and what role time played in graphing the quantities.
Post-Activity: After the class discussion, ask each student to write a reflection of the activity. The reflection should only be a few sentences/bullet points that describe what the student learned (mathematically) from the activity. This reflection is for the students to keep in their notes. Collect the handouts for grading.
Common Misconceptions and Difficulties
1. Students may want to draw curved lines on the graph. Ask them to reason through this conjecture by looking at a specific interval and describing the situation in terms of the variables. Allow the students to notice that it is impossible for a vertical change of one unit can result in a unit size increase higher or lower than an equal unit of total distance.
2. Students may mistakenly conceive of the horizontal axis as representing time. Students are typically familiar with distance-time graphs and it may be difficult for students to consider a variable other than time as the input. Remind students of the input variable representation and encourage them to reason covariationally by prompting them with questions such as, “What happens as the total distance varies?”
3. When students are asked to graph the situation again with the height of the tower doubled, students may increase the slope of the graph instead of extended the slope of 1/-1. Encourage students to remain quantitative by describing the relationship between the horizontal and vertical axes. Let them see that a slope other than 1/-1 is impossible when considering these two variables.
Resources
The purpose of this activity is for students to focus on the relationship between two variables. The students are given a real-world example of a linear relationship, and the goal is for students to discover this relationship after watching the videos and to explore what it means for two variables to have a linear relationship. The video provides the data students will use to accomplish this goal. By the end of the activity, students should understand that a linear relationship involves a constant increase/decrease for every unit increase in the input variable. In other words, as one variable increases, the other variable increases/decreases at a corresponding amount. This activity encourages covariational reasoning since students are to consider how moving along one variable (e.g. total distance) covaries with the other variable (e.g. height). The idea is also for the students to remain quantitative as they explore the relationships between the two given quantities (i.e. height, total distance traveled). Overall, the Power Tower activity will provide the scaffolding that students will need in order to progress towards the discovery of quadratic relationships by the end of the unit.
Standards
The activity covers Common Core Georgia Performance Standards (CCGPS) for 8th Grade. The detail of standards could be found here.
Activity Links
Video Link 1 Video Link 2
Materials Needed
Video (see links above), projector/computer access, whiteboards, markers, erasers, student handout (see link at the bottom of the page)
Formative Question
See the handout attached below.
Sequence of Lesson
Warm-up: Review the ideas from the Zogs activities. For instance, discuss how to find the relationship between two points. Ask students to define slope. Provide examples on the board and ask students to describe the situations quantitatively by being specific about the variables under consideration.
Description: The purpose of this activity is for students to graph the linear relationship between height and total distance traveled by the riders of the amusement park ride, the “Power Tower.”
1. Show the two videos of the amusement park ride to the class. Ask them to identify any quantities that they could measure.
2. After the students watch the video, start a discussion on some of the quantities the students noticed (height, speed, time, total distance, etc.).
3. Split students into groups of 4-5 students each. Each group should have a whiteboard, marker, and eraser. Half the groups are responsible for answering the questions using Video 1 and the other half using Video 2.
4. Hand each group a student handout.
5. While students are working, allow the videos to loop so that students may reference it. If there are enough laptops per group, each group can watch their video. Otherwise, loop both videos (one or two at a time). If it is only possible to project one video, a modified version of this activity can be done where all the groups focuses on the same video.
6. Go around to the tables and encourage discussion of the relationships between the quantities.
7. Example Questions/Prompts:
a) What happens to the total distance as height increases/decreases?
b) How do you know how steep the lines on your graph should be?
c) How are vertical distance and total distance related?
d) Describe what happens when the ride changes directions.
e) How do you know whether the line you draw on your graph should be straight or curved?
8. When all the students are finished working, have them put up their boards so that they are visible to the class. (Note: All the boards should look the same (besides the number of peaks), regardless of their group.)
9. Allow students to share their solutions with the class. Ask them why the graphs are the same (or why one may look different). Focus the discussion around the meaning of a linear relationship between quantities. Students can also discuss the ideas of prediction, inputs and outputs of functions, and what role time played in graphing the quantities.
Post-Activity: After the class discussion, ask each student to write a reflection of the activity. The reflection should only be a few sentences/bullet points that describe what the student learned (mathematically) from the activity. This reflection is for the students to keep in their notes. Collect the handouts for grading.
Common Misconceptions and Difficulties
1. Students may want to draw curved lines on the graph. Ask them to reason through this conjecture by looking at a specific interval and describing the situation in terms of the variables. Allow the students to notice that it is impossible for a vertical change of one unit can result in a unit size increase higher or lower than an equal unit of total distance.
2. Students may mistakenly conceive of the horizontal axis as representing time. Students are typically familiar with distance-time graphs and it may be difficult for students to consider a variable other than time as the input. Remind students of the input variable representation and encourage them to reason covariationally by prompting them with questions such as, “What happens as the total distance varies?”
3. When students are asked to graph the situation again with the height of the tower doubled, students may increase the slope of the graph instead of extended the slope of 1/-1. Encourage students to remain quantitative by describing the relationship between the horizontal and vertical axes. Let them see that a slope other than 1/-1 is impossible when considering these two variables.
Resources