In this activity, students are given a diagram that shows two quantities, one of which is 10 times as much as the other. They identify possible values and possible equations that the diagram could represent. 

Students see that a single unmarked diagram could represent many possible pairs of values that have the same relationship (in this case, one is 10 times the other) and be expressed with many equations.

The activity also reinforces what students previously learned about the product of a number and 10—namely, that it ends in zero and each digit in the original number is shifted one place to the left because its value is 10 times as much (MP7).

• “How does the value of B change if A represents 6? How do you know?”
• “What expression could we write to represent this?”

Consider repeating, with another number, to reinforce the idea.

Display possible values for A, and corresponding values for B, in a table such as this:

• “What do you notice about the values of each set? How do the values compare?” (The values for B are all multiples of 10. Each value for B is 10 times the corresponding value for A.)
• Select students to share their responses and their reasoning. 
• For students who used the diagram to reason about the equations, consider asking, “How might you label the diagram to show that it represents the equations you selected?” (For the equation , I would label A “3” and B “30.” I know that 30 is 10 times as much as 3, so the diagram represents the equation.)
• For students who used their understanding of numerical patterns to support their reasoning, consider asking, “How did you know that the equation could be represented as a comparison involving ten times as many?” (I know that when we multiply a number by 10, the product will be 10 times the value. I also know that division is the inverse of multiplication, so I looked for equations that were multiplying or dividing by 10 or had 10 as a quotient.)

In this activity, students analyze situations in which one quantity is 10 times as much as another quantity. Students may use different strategies to determine the unknown quantity. For example, they may rely on counting as a strategy, or use place-value understanding to explain regularity in the products of numbers with 10 (MP8). The reasoning in this lesson prepares students to consider, in the next section, quantities that are 100 times and 1,000 times as many.

• “How did you find the value of B when A is 1,000?”
• “What do you notice about the pairs of lesser numbers you found that can help you find the value of B?”

• “What were some things you noticed about the values of A and B?” (A was less and B was always greater. The digits of A are all in B and are in the same order, but they are not in the same places in B. There's an extra 0 at the end of the value for B.)
• Consider asking:
  • “How many times as much as A was B?” (10 times.)
  • “Why do you think the digits of A are also in B, in the same order, but not in the same places?” (Because they are multiplying 10 by A.)
  • “Could we represent both and , using the same diagram of A and B? Why or why not?” (Yes, because A could represent 4 or 10, and B represents 10 times that value.)