Intent

This activity gives students additional experience using algebraic language and symbols to represent geometric situations. It also strengthens their understanding of equivalent expressions and skill in working with the distributive property.

Mathematics

In The Garden Border, students perceived a geometric context in many ways. The variety of approaches they developed lead to different-looking, but equivalent, methods for counting tiles. Here are three such methods for counting border tiles, along with rules that reflect these three ways of viewing the problem.

4(s – 1)

2s + 2(s – 1)

4(s – 2) + 4

These rules are equivalent because each gives the same total number of tiles for a given value of s.

Progression

In this follow-up activity to The Garden Border, students compare a variety of methods for counting border tiles for a 10-by-10 garden and then review and generalize these methods for gardens of any size. The classroom conversation emphasizes the equivalence of the resulting expressions and the occurrence of the distributive property.

Approximate Time

5 minutes for introduction

20 minutes for activity (in class or at home)

20 to 40 minutes for discussion

Classroom Organization

Individuals, followed by whole-class discussion

Doing the Activity

This activity asks students to create “formulas” and offers the example 4s – 4. In the early use of symbolic algebra in IMP, words like formula and rule are not precisely defined. In the context of this activity, students will be comparing expressions, so an equation is not necessary.

Have students read up until Question 1, and then ask a volunteer to summarize what was stated. Invite others to add points of clarification. Once the important points have been covered, tell students that they will be creating similar formulas for several more methods.

This might be a good assignment to collect to help assess and support students’ understanding of using variables to express generalizations.

Discussing and Debriefing the Activity

Ask students to share their answers. They will probably want to use the diagrams in the activity to explain how they found their formulas. The diagrams will help make the generalizations more understandable.

To help clarify the techniques, ask students to test another specific case, such as s = 100. Would this method work for a 100-by-100 square? For example, for Question 1, the diagram suggests that the border for a 100-by-100 garden would have 100 tiles along the top and along the bottom, leaving 98 for each of the other two sides. Getting students to express this as 2 · 100 + 2(100 – 2) will help elicit the expression 2s + 2(s – 2). You might use the phrase “imitating the arithmetic” to describe this technique for developing algebraic formulas to represent situations. Students might also express the method in Question 1 with the formula s + s + (s – 2) + (s – 2).

The various expressions students develop provide an excellent opportunity to review equivalent expressions, the distributive property, and the idea of combining terms.

As much as possible, have students demonstrate equivalence for all the formulas they found for the various diagrams. For example, for Question 1, you might ask a volunteer to explain how to be sure that the expressions s + s + (s – 2) + (s – 2) and 2s + 2(s – 2) are equivalent, independent of the problem setting. How can you be sure these expressions are equivalent? Students might recognize that s + s is the same as 2s and that (s – 2) + (s – 2) is the same as 2(s – 2). They might also see why these expressions are equivalent to 4(s – 2) + 4.

At any time during this discussion, you might use numeric examples to check equivalence or to simply help students see that the operations being carried out in the different orders do yield equal values. Do communicate that such examples do not prove equivalence, just help to confirm it.

To wrap up the conversation, remind students of the distributive property and ask them to identify instances in which they determined that two expressions were equivalent and in which they can see the distributive property in action, such as 2(2s – 2) = 4s – 4. You might ask groups to work on this question for a short period of time and then to share with the class any other instances they found. Record their observations as equalities; for example, 4(s – 1) = 4s – 4 and 2(s – 1) = 2s – 2. Students will likely see that the distributive property is just a subset of the bigger idea of equivalent expressions.

A general statement of the distributive property might look like this.

The expressions N(a + b) and Na + Nb are equivalent.

You might want to post this statement, or another that students develop, for students to refer to.

Key Questions

Would your method work for a 100-by-100 square?

How can you be sure these expressions are equivalent?

Supplemental Activities

More About Borders (extension) contains variations on the Border Varieties activity.

Programming Borders (extension) asks students to write a program that answers some or all of the questions posed in More About Borders.