The purpose of this Warm-up is to prepare students for the Information Gap activity that follows. First, students are given a problem with incomplete information. They are prompted to brainstorm what they need to know to solve a problem that involves scale factor. Next, they practice asking for information, explaining the rationale for their request, and persevering if their initial questions are unproductive (MP1). Once students have enough information, they solve the problem.

Tell students that the problem is a part of an Information Gap routine. In the routine, one person has a problem with incomplete information, and another person has data that can help with solving it. Explain that it is the job of the person with the problem to think about what is needed to answer the question, and then request it from the person with information.

Tell students they will try to solve the problem this way as a class to learn the routine. In this round, the students have the problem, and the teacher has the information needed to solve the problem.

• Ask students, “What specific information do you need to find out the height and radius of the original cylinder?” 

• Select students to ask their questions. Encourage students to use the format of “Can you tell me ?” Respond to each question with, “Why do you need to know ?”

• Once students justify their question, only answer questions if they can be answered using these data:

  • The dilated cylinder’s volume is  in³.

  • The original cylinder’s volume is  in³.

  • The radius of the original prism is 2 in. 

  • The height of the dilated prism is 15 in. 

  • The original prism’s surface area is  in².

  • The dilated prism’s surface area is  in².

• If students ask for information that is not on the data card, respond with, “I don’t have that information.”

When students think they have enough information, give them 2 minutes to solve the problem. (a cylinder with labeled dimensions: height 5 cm and radius 2 cm)

Tell students they will work in small groups and use the routine to solve problems in the next activity.

Math Community

After the Warm-up, display the Math Community Chart with the “Doing Math” actions added to the teacher section for all to see. Give students 1 minute to review. Then share 2–3 key points from the teacher section and your reasoning for adding them. For example, 

• If “questioning vs. telling,” a shared reason could focus on your belief that students are capable mathematical thinkers and your desire to understand how students are making meaning of the mathematics.

• If “listening,” a shared reason could be that sometimes you want to sit quietly with a group just to listen and hear student thinking and not because you think the group needs help or is off-track.

After sharing, tell students that they will have the opportunity to suggest additions to the teacher section during the Cool-down.

This activity gives students an opportunity to determine and request the information needed to find measurements of original and dilated solids, using their knowledge of one-, two-, and three-dimensional scale factors.

This is the first Information Gap activity in the course. In this activity, students represent sequences in different ways but do not initially have enough information to do so. To bridge the gap, they need to exchange questions and ideas.

The Information Gap routine facilitates meaningful interactions by positioning some students as holders of information that is needed by other students, thereby creating a need for communication. This routine supports language development by providing students with opportunities to ask for and share information, and to justify their reasoning within conversation. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Monitor for pairs that complete Problem Card 2 by using the radius and the height of the dilated cylinder in the volume formula, and for other pairs who instead apply the cube of the scale factor to the original cylinder’s volume.

In this activity, students are building skills that will help them in mathematical modeling (MP4). They recognize that a geometric solid can be a mathematical model of a real-life object, and have an opportunity to consider the accuracy of that model. They’re prompted to connect surface area and volume to the real-life context of metal to create the container and liquid to fill it.