The purpose of this activity is for students to practice solving problems that involve volumes of pyramids and cones. Monitor for various problem-solving strategies—in particular, for the problem in which students find the radius or height of a cone given a particular volume. Some students may substitute values into a formula and solve the resulting equation, and others may follow a process of deductive reasoning.

For the pyramid with a triangular base, students may forget that the triangle area formula involves dividing by 2, resulting in a base area of 6 square units. Prompt them to draw and label the base prior to calculating its area.

The purpose of this discussion is to allow students to see multiple approaches to solving these multi-step problems. Ask students to share their strategies for volume calculations. For example, for the cone, did they calculate the area of the base first, and substitute it into the formula ? Or, perhaps they wrote out and performed the calculations all at once.

Select previously identified students to share their thought process on the “working backward” cone problem. If possible, find one student who solved an equation algebraically and another who used backward reasoning rather than formally solving an equation.

In this activity, students are building skills that will help them in mathematical modeling (MP4). They apply their understanding of pyramid volumes to find possible dimensions of a pyramid given its volume. They are asked to create multiple designs of an ice sculpture that meets the given criteria. 

In the digital version of the activity, students use an applet to create three-dimensional shapes that match given information about the volume. The applet allows students to first create a two-dimensional base before extruding it to a three-dimensional solid and checking its volume. The digital version may be preferable to help students visualize their pyramids accurately.

Monitor for students who start by using trial and error to find values, and for those who begin by setting one value and calculating the others based on their assumed value.

Many students may use squares and rectangles for their bases. Listen for students who consider other shapes.

The purpose of this discussion is to highlight aspects of modeling that students used during the activity. Ask previously selected students to share their strategies. Here are some questions for discussion:

• "What assumptions did you make while designing your ice sculpture?"
• "Once you found one set of dimensions, how did you go about working on the second set?"
• "Could the base be a circle? If so, how could we find the dimensions of a circle that would work?"

Math Community

Conclude the discussion by inviting 2–3 students to share a norm they identified in action. Provide this sentence frame to help students organize their thoughts in a clear, precise way:

• “I noticed our norm ‘’ in action today, and it really helped me/my group because .”

Students may not know how to get started because this problem has few constraints. Ask students to choose a shape for the base first, then to consider what information from the problem could help them choose a size for that base.