Illuminations: Stacking Squares

# Stacking Squares

 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.

### 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.

### Materials

 Playing With Squares Activity Sheet Poster board, Markers, Rulers, Scissors Grid paper

### Questions for Students

 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: Similar equations would be written for other stacks.] 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.] 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, For this particular example, other equivalences are possible. The process for finding any equivalence, however, is the same.]

### Assessment Options

 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. 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. 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

 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.) 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). 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.

### 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.)

### NCTM Standards and Expectations

 Number & Operations 9-12Compare and contrast the properties of numbers and number systems, including the rational and real numbers, and understand complex numbers as solutions to quadratic equations that do not have real solutions. Judge the effects of such operations as multiplication, division, and computing powers and roots on the magnitudes of quantities.

### References

 Wagner, David. "We Have a Problem Here: 5 + 20 = 45?" Mathematics Teacher 96 (December 2003): 612–616.
 This lesson prepared by David Wagner.

2 periods

### NCTM Resources

 More and Better Mathematics for All Students
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