The purpose of this activity is for students to build on what they know about the language of “twice” and “twice as many” to represent comparison situations.

In this activity, students are encouraged to represent the situation in a way that makes sense to them, though discrete representations (cubes or drawings) are the focus of the Activity Synthesis. The representations used in this activity serve as a foundation for the more abstract tape diagrams that will be used later in the section.

• "Explain what you did to represent Han's cubes."
• "How could you show the difference between '2 more' and 'twice as many,' with connecting cubes?"

• Select 3–4 different student examples (a mix of cubes and drawings).
• “How does each representation show that Han has twice as many as Andre?” (In each set, Andre’s number of cubes is repeated 2 times.)
• “Another way of saying 'twice as many' is '2 times as many.'”
• “How do you know that Han's number of cubes is ‘2 times’ Andre's amount?” (Andre’s number of cubes is repeated twice. Sometimes they are stuck together in one stick and other times they are separate.)

The purpose of this activity is to extend the intuitive idea of representing “twice as many” to represent “4, 6, and 8 times as many.” Although students are prompted to draw to represent each situation, keep the connecting cubes accessible for students to use as needed.

• “How does your expression match the problem?”
• “What multiplication expression would match the problem? How do you know?”

• Invite selected students to share their representations and their reasoning.
• For each question, ask, “Where do you see _____ times as many in _____’s representation?”
• “What does it mean to have ‘_____ times as many’?” (It means we have that many groups of the original number of cubes.)
• For students who reason using multiplication expressions or equations, ask: “How does this equation show ‘times as many’? What does each number represent?” (In the example of Priya and Jada, the equation represents 6 times as many as 3. The 3 is the number of cubes Priya has, 18 represents the number of cubes that Jada has, which is 6 times as many as Priya.)

The purpose of this optional activity is for students to practice representing “ times as many,” with cubes and diagrams. Students may not have enough cubes to build each comparison. This gives students an opportunity to make sense of each quantity and how it relates to their corresponding cubes or diagram (MP2). Encourage them to draw a diagram to represent the cubes.

• “What was your method for representing ' times as many'?” (I built that many groups of the original group size. I multiplied the original number by the number rolled and counted out that many cubes.)