EDITED Resources. The resources on this page have been aligned with the revised K-12 mathematics TEKS. Necessary updates to the resources are in progress and will be completed Fall 2006. These revised TEKS were adopted by the Texas State Board of Education in 2005–06, with full implementation scheduled for 2006–07.

Clarifying Lessons

Grade 7: Pipe Cleaners

activity under revision

Lesson Overview

Students build rectangles with pipe cleaners. Since the pipe cleaners are all the same length, the rectangles will all have the same perimeter. Students investigate the relationship between the base and height of each of these rectangles. Using multiple representations, students collect data in a table, graph the data, and build a symbolic rule relating base and height of rectangles of a given perimeter.

Mathematics Overview

Students gather data to determine the relationships between the base and height of rectangles with fixed perimeters and express the relationships using words and symbols.

Set-up (to set the stage and motivate the students to participate)

  1. The teacher gives students one pipe cleaner each and asks them to measure the length of the pipe cleaner to the nearest half-centimeter. Students agree on the length and record it on Activity Sheet 1, question 1a.
  2. Each student builds a rectangle from a pipe cleaner so that the ends of the pipe cleaner meet at a corner of the rectangle. (The ends should not overlap.) See the following pictures for examples:

    pipe cleaner rectangles

  3. In small groups, students share their rectangle-building strategies. A member of each group shares one of the group's strategies with the class, taking turns until all the different strategies used have been mentioned. Possible strategies include:
    • Guess-and-check bending
    • Folding the pipe cleaner in half first and then folding the remaining halves
    • Starting with an oval and forming a rectangle by pinching the oval
  4. In their groups, students compare rectangles and change them if necessary so that all the rectangles in the group are different from one another (for example, short and wide, long and skinny, square).
  5. Students mark the corners of their rectangles with a marker or piece of tape and then straighten the pipe cleaner. Students line up the pipe cleaners with others in their group, compare them, and discuss their observations. The teacher should make sure to elicit the following two important points:
    • The pipe cleaners are all the same length.
    • There is a second mark in common, in the middle.
  6. Students complete Activity 1 by measuring the length of each piece of their rectangles to the nearest half-centimeter, filling in the table, and answering the questions.
  7. Students complete Activity 2 by graphing the data from the table in Activity 1 and answering the questions. Each group of students should make a group graph on chart-size 1-inch graph paper with peel-and-stick dots, display the graph on the wall, and compare their graph to the other groups' graphs.

Teacher Notes (to personalize the lesson for your classroom)

Guiding Questions (to engage students in mathematical thinking during the lesson)

  • How did you find different rectangles?
  • Are there any other rectangles you can create?
  • Did you agree on a measurement for the length of the pipe cleaner?
  • What do you notice about the corner marks of your rectangles?
  • What information should be included in the table?
  • What patterns do you notice in the table?
  • If you add the base and the height of each rectangle, what do you notice?
  • If you know the base of the rectangle, what can you conclude about the height?
  • What do you notice about the relationships between the base and height of the rectangle and length of your pipe cleaner? How can you write this sentence symbolically?
  • Are there other relationships that you noticed?
  • If the length of your pipe cleaner changed to 30 centimeters, how would that change your rule?
  • If I give you the length of the base and the height of a rectangle, can you tell me the perimeter of the rectangle?
  • If you know the perimeter and the length of the base or the height can you find the missing measurement? What process would you use to do this?
  • How did you label your graph?
  • How did you scale your axes when you graphed the data?
  • How can you describe your graph (e.g., its appearance, shape, direction, etc.)?
  • Can your calculator be used to display this information? How?
  • As one dimension increases, what do you notice about the other?
  • How are the table, graph, and symbolic rule related?

Teacher Notes (to personalize the lesson for your classroom)

Summary Questions (to direct students' attention to the key mathematics in the lesson)

  • Can you take the information from the table and make a rule? How?
  • The sum of the base and the height has what relationship to the perimeter?
  • Can you find all the whole number bases and the heights for rectangles with a perimeter of 40 centimeters?
  • How do you know if you have found all the rectangles that can be made with a perimeter of 40 centimeters?
  • Would all these rectangles have the same area?
  • Does the rule you found in your table work for all perimeters?
  • If you create a scatterplot of the data on your table and graph the rule, what will the graph look like?

Teacher Notes (to personalize the lesson for your classroom)

Assessment Task(s) (to identify the mathematics students have learned in the lesson)

  • Given a table of data showing base and height of a fixed perimeter, can the student create a rule for the data and graph it?
  • Given a table of data showing base and height of rectangles, can the student determine if the perimeters of the rectangles are constant or changing?
  • From a graph, can the student generate a table and a rule?
  • From a rule, can the student generate a table and a graph?

Teacher Notes (to personalize the lesson for your classroom)

Extension(s) (to lead students to connect the mathematics learned to other situations, both within and outside the classroom)

Students build a table of heights, bases, and areas of the rectangles with fixed perimeter. They graph area vs. base and discuss the nonlinear plot. Which rectangle gives the most area?

Teacher Notes (to personalize the lesson for your classroom)