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Stacking Squares

  • Lesson
9-12
2
Number and Operations
David Wagner
Location: unknown

This lesson prompts students to explore ways of arranging squares to represent equivalences involving square- and cube-roots. Students’ explanations and representations (with their various ways of finding these roots) form the basis for further work with radicals.

The investigation in this lesson should be done by students at the beginning of their work on radicals. Before they see traditional algorithms for adding radicals, they should be given the chance to explore the meaning of square roots. This lesson gives them this opportunity. After they conduct their investigations, you can help them see connections between their various methods as well as connections between these and traditional algorithms.

To begin, give the class an overview of the structure of the lesson:

  1. Investigation in groups
  2. Compilation of ideas in poster form
  3. Gallery tour of posters
  4. Class discussion about the posters

Also, inform students of the supplies that you have available to them (poster board, markers, grid paper, scissors, rulers, etc.). Be careful not to place too much emphasis on any of the supplies, though; if you suggest the use of a certain tool, students will take this as a hint about how to do the investigation.

Organize students into groups of four, and give each group one copy of the Playing with Squares activity sheet. Distributing just one copy to each group may encourage discussion between students. (Alternatively, you could distribute a copy of the activity sheet to each student, allow them to think about it for a few minutes, and then share their thoughts with the group.) Before they begin the task, inform students that this is not an easy task and that they should not be embarrassed to struggle. Encourage them to focus on whatever aspects of the task they find most interesting or most accessible.

pdficon  Playing With Squares Activity Sheet 

Allow about fifty minutes for the groups to complete their posters. What should you do during this time? If you resist the urge to give hints to your students, they are likely to find a number of different methods for completing the task. One thing that you will have to do is keep them aware of the time. With this investigation, students tend to get quite immersed in the mathematics, to lose track of time, and to have insufficient time for effectively communicating their findings on a poster.

At the beginning of the next class, have groups hang their posters on a wall, and invite the class on a "gallery tour," a silent perusal of all the posters.

Conclude the lesson by leading a class discussion about student work. Possible questions to pose:

  • After seeing other groups’ posters, what questions do you have for your peers?
  • Identify and classify the different methods that you and your classmates found for finding equivalent stacks of squares. (Note: This is not a simple question. You are not asking students to identify things they have been told how to identify. They have to notice differences and similarities between the various groups’ approaches to the problem.)
  • For any given method, what sets it apart from the other methods used by other groups?
  • Find something that is common to all the methods. [All methods will likely involve perfect squares.]
  • After seeing each other’s work, can you think of any other approaches?

Possible Solutions to the Playing With Squares Activity Sheet 

  • The stacks of squares with the exact same height as a square with area 72 cm2:
    • 2-2-2-2-2-2
    • 8-2-2-2-2
    • 8-8-2-2
    • 8-8-8
    • 18-2-2-2
    • 18-8-2
    • 18-18
    • 32-2-2
    • 32-8
    • 50-2
    This is the complete list, unless the order of the squares is significant. Note that students are not asked to give the complete list; they are only asked to find some.
  • Squares that have no stacks that are the exact same height:
    • If the area of the square is not divisible by a perfect square (4, 9, 16, …), then it cannot be partitioned into squares that have an area that is a natural number.
     
  • How to find the stacks that would match a given square in height:
    • This is the heart of the problem. There are at least a few ways of doing this. Of course, one could find the height of the original square, find an equivalence, and make squares with these heights to form the stack. For the example on the activity sheet, this works because

      2076 root 45 

      But this method requires algorithms for working with radicals, and this lesson is supposed to precede the teaching of these algorithms. Your students will have to find other ways to address the task. The beauty of this investigation is that the methods your students find will connect nicely to the traditional algorithms that they will learn later.

     
  • For cubes instead of squares:
    • If we use cubing and cube roots instead of squaring and square roots, we can stack cubes. For example, a cube of volume 54 cm3 is the same height as cubes with volumes 16 cm3 and 2 cm3 stacked on top of each other, because

      2076 cube54 

Assessments 

  1. This investigation provides an excellent opportunity for informal assessment. As you walk around and listen to students work on the problem, you will gain insight into the way students think about mathematics. Some of the things you hear students say are likely to be shocking; some will leave you wondering, "How can they think that after all these years of mathematics?" Others will leave you amazed at their insight, when they discover methods that you did not anticipate.
  2. If you want to evaluate student work, various options are available. Consider what you want to evaluate. You might want to evaluate problem-solving ability, strength of reasoning, quality of communication, accuracy of mathematics, or level of completion. You can mark the posters (and thus the group’s result). You can mark students on the quality of their contribution to the investigation, by walking around and observing this interaction. You can evaluate individual students based on what they got out of the experience by assigning an extension as a journal prompt.
  3. It is important for students to know how they are being assessed. If you do not have a classroom culture in which students already know how they are assessed in such problem-solving situations, you should discuss a rubric with your class before distributing the task.

Extensions 

  1. Ask students to consider another extension into three dimensions. Say to students, "When stacking squares, you compared two-dimensional objects (squares) in one dimension (height). When stacking cubes, you compared three-dimensional objects (cubes) in one dimension (height). How about comparing three-dimensional objects (cubes) in two dimensions (area)?"

    Find a way of using smaller cubes to cover the face of a larger cube. For instance, the figure below shows six smaller cubes that exactly cover the face of a 54 cm3 cube (the yellow one). What is the volume of each smaller cube? Describe a method for finding other arrangements of cubes with matching faces. (As with the stacks of squares, use only natural numbers.)

    2076 face cover 

  2. Provide your students with copies of alternative methods that they did not consider. Ask them to explain how these methods relate to the methods they came up with. You might have some interesting student responses from other times you’ve used this lesson. Or you can photocopy some of the student responses to this investigation from the Mathematics Teacher article "We Have a Problem Here: 5 + 20 = 45?" (see reference below).
  3. Write numbers on slips of paper (and folded up for secrecy). Have each student pick a number out of a hat. When you say go, they open their numbers and find partners with whom they could make equations—for example, students with 5, 20, and 45 can form a group by making the equation from this lesson. The idea is that everyone should be in a group; each person should use their number exactly once.

    In preparation, you will have to make a set of numbers that will work. For example, if there are 25 students in your class, you could use the following set of numbers: 3, 5, 5, 6, 12, 12, 12, 12, 20, 20, 24, 27, 27, 45, 45, 45, 48, 54, 54, 75, 80, 96, 108, 125, 150. Perhaps this preparation could occupy you while students work on the investigation.

    This puzzle has numerous extension possibilities. You could ask students how many different solutions there are. Or, you could ask students to compile other sets of numbers that will work for a given number of participants.

 

Questions for Students 

1. Using square-roots to represent the heights of each square, construct an equation that shows that the heights of two stacks are the same. (This question should be repeated using various stack pairings.)

[The equation will depend on the stacks that students use. For instance, if students identify that a square of 45 cm2 is the same height as a stack consisting of squares of 20 cm2 and 5 cm2, then the following equation would be written:

2076 equation 45 

Similar equations would be written for other stacks.]

2. Sketch a pair of matching stacks that does not appear on any of the posters. Exercise your imagination; try to make one that no one else will think of. Then, write an equation that represents the equivalency.

[Answers will vary, but if students are encouraged to use very large numbers, they will be more likely to generalize the results.]

3. Describe a method for writing equivalencies without sketching squares. If necessary, use a specific example to describe a method that will work for any radical.

[Determine if the number under the radical contains a factor that is a perfect square. If so, the square root of that number can be pulled in front of the radical sign. Then, decompose the number in front to various quantities. For instance, the number 18 contains the factor 9, which is a perfect square. Therefore,

2076 equations 18 

For this particular example, other equivalences are possible. The process for finding any equivalence, however, is the same.]

Teacher Reflection 

  • How engaged were the students in the investigation? How could the class be organized differently (e.g., other ways of grouping students) to promote better engagement?
  • What parts of this lesson were especially good for advanced students, and which were good for the students who struggle? How might I consider such students’ needs the next time I use this lesson?
  • How did the quality of the students’ posters compare to the quality of their mathematical discussion? How can I help my students become better at explaining their mathematics in writing?
  • As I progress through this unit (and others), what concepts can be connected to the students’ work on their posters? (Make a note of these connections so you can relate them to your students at appropriate times, which will be very meaningful for your students.)

Learning Objectives

Students will:

  • Investigate ways of finding equivalencies that involve square- and cube-roots.
  • Make and test conjectures about connections between geometric and numeric representations of squares and between whole numbers and their square-roots.
  • Communicate their reasoning.