BIG IDEA 1
Taking Apart Prisms and Polygons

The mathematical concepts at the heart of this big idea are area and volume. Although these ideas call for students to learn through objects, holding them in their hands and exploring with them, many students are asked only to memorize formulas and so do not develop an understanding of area, volume, or the differences between them. In our Youcubed summer camp, we gave the students an activity with sugar cubes; they were invited to build different sized larger cubes with the sugar cubes. When we interviewed the students a year after they attended the camp, one of the boys told us that he now thinks about the sugar cubes every time he learns about volume, as they gave him a physical representation of a 1?×?1?×?1 cube. His experience holding the cubes and building with them contributed to a deep understanding of volume that was powerful and enduring for him. In our Investigate activity, we invite the students to build with very similar cubes-snap cubes. In all of the activities, students are asked to build with two- and three-dimensional shapes.

In the Visualize activity, we ask students to find different ways to take apart two-dimensional complex shapes as they work to find area. We have used shapes that require students to reason about how to determine the area when its boundary doesn't fit exactly on a square grid. As students reason through determining the area, they also need to break the shape into other shapes they are familiar with, such as triangles and rectangles. This type of thinking is foundational for later work in geometry and calculus.

In the Play activity, we ask students to determine the area of a complex piece of artwork that is made from different polygons. We like to connect mathematics and art in our books, as it is important for students to see that mathematics can be beautiful, creative, and applied to all sorts of different real-world situations. Because of the uneven border of the shape, students will need to come up with different creative ways to find the area. This lesson also provides students opportunities to discuss estimation. In studies of mathematics in the world, estimation has been found to be one of the most used concepts and one that is undertaught in schools. We are sure your students will enjoy making their own piece of mathematical art.

In the Investigate activity, students build off the Visualize activity as they imagine complex two dimensional shapes as the bases of buildings. Students are asked to use multilink cubes to construct the buildings, giving them an important opportunity to understand volume. In this activity, we also provide an opportunity for students to work with rational numbers. Students in sixth grade are learning to expand their number system, yet questions in traditional textbooks often ask the students only to work with whole numbers. We have provided problems that use rational numbers, fractions, to support students' growth in understanding of both volume and rational numbers. Students are asked to visualize fractions of multilink cubes as they work to determine volume and connect the idea of volume to the idea of area for the shapes that they worked with in the Visualize activity. Students often have trouble understanding the difference between area and volume because they have not had enough experience spending time connecting their numbers with visual two- and three-dimensional models. We hope that this set of activities will provide time for fun and challenge together, and that students will get an enjoyable opportunity to struggle and to use their creativity in finding different ways to see and solve problems.

Jo Boaler

How Big Is the Footprint?

Snapshot

Students develop methods for finding the area of irregular polygons by exploring ways to decompose two-dimensional figures and reason about partial square units.

Connection to CCSS

6.G.1

Agenda

• Quadrilateral in Question sheets, multiple copies per group
• Optional: colors

• Polygon 1-4 sheets, for groups to choose from
• Optional: colors

To the Teacher

Two ideas are central to this lesson, one conceptual and one mindset. At the heart of the conceptual work students are doing in this activity is making sense out of partial square units. A colleague of ours conducted a study in a sixth-grade classroom in which students engaged in an area task similar to this one (Ruef, 2016). Students developed many ways of addressing the partial units created by the angled side. Some students ignored them, believing that only whole units counted. In this method, students focus on stacks of square units, as shown in the figure here. While this does not lead to a fully accurate count of the area of the figure, it anticipates the way calculus approximates the area under a curve. If students in your class invent this way of thinking about area, it is worth naming that they have an idea that they will use in calculus to deal with the challenge of curves.

Some students count only whole squares when finding the area of an irregular shape.

Other students in Ruef's study developed various ways to create whole units out of the partial units, including ways that use the space not covered by the shape. For instance, some students visualized the partial units as half of larger rectangles, as in the methods shown in the next images. Both of these methods are accurate and have connections to thinking about slope, fractions, and decomposition of two-dimensional figures.

Some students make a rectangle and halve the area.

Some students make a large rectangle to cover the top of the larger shape that is a triangle and then halve the area.

The second central idea in this lesson is students' authority over the mathematics. This activity will challenge students to develop methods that they are uncertain about, or even, as in Ruef's study, to attempt incomplete or conflicting methods. It is crucial that students be the ones to determine whether a method makes sense, accounts for the full area, and is accurate. Ruef found that when placing the authority with students to make sense and come to consensus in this seemingly simple task, students took as long as three days to explore, debate, gather evidence, discuss, and come to agreement, and even then, they wanted their teacher to confirm that they were correct. The teacher resisted being positioned as the mathematical authority in the room, which made students responsible for deciding when they were convinced. We encourage you to take from this example the fortitude to resist students' requests that you decide who is correct and what makes sense. This is a prime activity in which to establish your students, at the beginning of the year, as the only ones who can decide whether and how a mathematical argument makes sense.

Activity

Launch

Launch the activity by showing the class the Quadrilateral in Question sheet in the document camera. Ask, How might you find the area of this shape? What do you notice that could help you? Give students a chance to turn and talk to a partner about these questions. Allow students to share some of their observations with the class. Pose the task for the day.

Quadrilateral in Question

Explore

Students work in small groups using the Quadrilateral in Question sheet to find its area. Ask students to find as many different ways as they can to make sense out of the area. You may want to provide groups with multiple copies of the Quadrilateral in Question sheet to represent each of their methods. Ask students to make a proof of their solutions on the sheet to share with the class to convince others that their solutions make sense. A proof can...