Intent

This activitysets the stage for Border Varieties, in which students will gain additional experience using algebraic language and symbols to represent geometric situations. In addition, it strengthens students’ understanding of equivalent expressions and their skill in working with the distributive property.

Mathematics

The algebra-geometry connection is again a key mathematical element of this activity and the next. In this activity, students derive general approaches for counting the tiles along the border of a square garden. The number of tiles is a linear function of the size of the garden.

Progression

Students begin this two-activity set by creating as many ways as they can think of to count border tiles for a 10-by-10 garden, without counting one tile at a time. They discuss and compare the variety of counting methods they find, setting the stage for developing a symbolic representation of each method and for recognizing the equivalence of the expressions created.

Approximate Time

25 minutes

Classroom Organization

Individuals or groups, followed by whole-class discussion

Materials

Overhead tiles (optional)

Square tiles

Doing the Activity

You may want to present this activity without referring to the student book. If students open their books to The Garden Border, they might notice the page for Border Varieties, which includes diagrams that provide possible answers for today’s activity, and you want students to come up with these ideas on their own.

Introduce the context of the problem, noting that the garden, including the tiles, is to be 10 feet by 10 feet. Draw or display a picture of the tiles, or ask groups to agree what the tile arrangement must look like and have a volunteer share a sketch.

Tell students that their challenge is to figure out how many tiles Leslie needed, without counting the tiles individually, and to write down as many ways as they can for doing this. Then ask that they draw a diagram that indicates how each method works.

Note that you have posed the entire activity verbally. If some students would benefit from written instructions, have them turn to their own books. Or, you could display the instructions for students to refer to.

Discussing and Debriefing the Activity

Ask group representatives to report on one method for calculating the number of border tiles, giving the details of the arithmetic as well as the diagram.

One method is to start with the ten tiles along each edge and subtract 4 to account for the fact that each corner tile is on two edges. For this method, the arithmetic might be 4(10) – 4. Various diagrams can be created to represent this method. The diagram below shows the four 10s along the edges and indicates the four corner tiles that have been counted twice. (A more schematic diagram and five additional approaches appear in Border Varieties.)

Once the first presenter from every group has reported, ask for any additional ways not yet mentioned. Collect all the approaches that students have come up with.

Take a few minutes to talk about how some of the strategies collected might be applied to a 5-by-5 garden. For example, applying the method described above would show that such a garden uses 4 · 5 – 4 = 16 tiles.