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:

- Investigation in groups
- Compilation of ideas in poster form
- Gallery tour of posters
- 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.

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 cm^{2}:
- 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
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 cm
^{3} is the same height as cubes with volumes 16 cm^{3} and 2 cm^{3} stacked on top of each other, because

### Reference

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

**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 cm^{2} is the same height as a stack consisting of squares of 20 cm^{2} and 5 cm^{2}, then the following equation would be written:
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,
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.)