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 8: Perplexing Puzzle

activity under revision

Lesson Overview

Investigate the properties and characteristics of a proportional relationship.

Mathematics Overview

Students investigate proportional relationships and their characteristics. Given a puzzle, students explore how they can build a larger puzzle by using a variety of strategies. The following characteristics of proportional relationships should be discussed:

  • Comparing ratios vs. comparing differences;
  • Using a table to look for patterns that form equivalent ratios;
  • Using equivalent ratios to find the constant of proportionality (y/x = k) in both fraction and decimal form;
  • Using the constant of proportionality, k, to write an equation of the form y = kx; and
  • Graphing the ordered pairs in the table to see that they form a straight line through the origin with the equation of the form y = kx.

Throughout this activity, the teacher suggests multiple representations (verbal, concrete, pictorial, tabular, graphical, and algebraic) to provide depth of understanding to this "big idea" of proportional relationships.

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

  1. Students work in groups. Each member of the group selects one or more of the pieces of the puzzle. The group works together to arrange the pieces to make a rectangle. (There are many ways to put the puzzle together into a rectangle.)
  2. Once a rectangle is made with all eight pieces, each group member retrieves his or her pieces of the puzzle and carefully measures each side in centimeters.
  3. The task of each group is to make a puzzle larger than the one in the envelope using this clue: A side measuring 4 cm in the original puzzle must measure 6 cm in the larger puzzle.
  4. Each student of the group works independently to enlarge his or her puzzle piece using one of the four colored sheets of construction paper. (Students are likely to add 2 to 4 to get 6 and likewise add 2 to each measurement to get the larger measurement. However, the resulting puzzle pieces will not fit together because this activity calls for a proportional "multiplicative process" rather than an "additive process.")
  5. When everyone in the group has completed making his or her enlargement, the group works together again to arrange the larger pieces to complete the puzzle. Students glue the puzzle on the light-colored piece of construction paper. If any pieces of the enlarged puzzle do not fit, students should discuss the strategies used to enlarge each piece.
  6. After each group has had time to assemble the puzzle using the enlarged pieces, the teacher begins the debriefing of this activity. This is the most important part of the activity, as strategies for solving are discussed.
  7. The teacher asks each group to explain their strategy for enlarging the puzzle.
  8. Students then complete the table in Activity 1 for the new lengths and answer the questions following the table.
  9. Students look for patterns in the table and note the ratio of the new length to the original length for each measurement in the table.
  10. Students rewrite the ratio for the constant k so that it is not in fractional form. (y = kx)
  11. Next, students use 1" graph paper, markers, and peel-and-stick dots to graph the data in the table. 12. The teacher records the data on Transparency 2 and asks the students discuss the questions in Activity 2.

Teacher Notes (to personalize the lesson for your classroom)

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

  • How did you create your enlarged shape?
  • Can you describe the new shape you created?
  • How do the two shapes compare?
  • What do you notice about the angles in the original shape and the angles in the enlarged shape?
  • How do the corresponding angles of these figures compare?
  • Are the two shapes similar?
  • What has to be true for the shapes to be mathematically similar?
  • Did you use equivalent ratios to find the dimensions of the larger shape? How?
  • What do you notice about the ratio?
  • How can you change the ratio to decimal form?
  • By examining your process of enlarging the figure, can you develop an equation that can be used to find dimensions of any puzzle piece?
  • What do your variables in the equation represent?
  • How could you use your graphing calculator and your equation to find the missing dimensions?
  • What patterns do you notice in the table in Activity 1?
  • What proportional relationship do you notice?
  • What do you notice about your graph in Activity 2?
  • What does the constant ratio in the graph represent?
  • If you were to continue the graph, what might you observe?

Teacher Notes (to personalize the lesson for your classroom)

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

  • What is the relationship between each original piece and it's enlarged piece?
  • How can you tell they are similar?
  • What are the different ways to use the enlargement rule or equation for enlarging the puzzle pieces?
  • How do the ratios of the side lengths compare?
  • How do you prove that the enlarged shape is proportional to the original shape?
  • What is the constant of proportionality?
  • From the table in Activity 1, how can you generate an equation to find other dimensions of the shapes in the puzzle?
  • How can you tell from the graph in Activity 2 if the relationship of the original puzzle to the new puzzle is proportional?
  • How can the constant of proportionality be used to write an equation relating the two variables (x and y)?
  • How can the constant of proportionality be used to make a table and a graph of the original length vs. the new length of the puzzle pieces?

Teacher Notes (to personalize the lesson for your classroom)

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

  • The teacher provides students with a set of tangrams and asks them to enlarge the set so that a side measuring 5 cm will be 9 cm in the larger puzzle. The teacher asks students to use the characteristics of a proportional relationship to help complete the task. Students then graph the ordered pairs (original length, new length) and compare them to the graph they completed in Activity 2.
  • Students write a journal entry for the assessment task above, explaining the strategies they used to enlarge the puzzle pieces and describing the characteristics of proportional relationships.

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 make a puzzle which represents a reduction of the larger puzzle created in the assessment task according to the following condition: A side of the enlarged puzzle piece which measures 9 cm must measure 5.4 cm in the smaller puzzle piece.

Teacher Notes (to personalize the lesson for your classroom)