This Math Talk focuses on multiplying fractions. It encourages students to think about multiplying fractions and whole numbers and to rely on the structure of the fraction to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students decompose fractions found in an exponent.

To multiply the values mentally, students need to look for and make use of structure (MP7) of commutativity and decomposition of rational numbers.

The goal of this discussion is to review students’ strategies for decomposing fractions such as .

To involve more students in the conversation, consider asking:

• “Who can restate ’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to ’s strategy?”
• “Do you agree or disagree? Why?”
• “What connections to previous problems do you see?”

If no students mention the strategy of decomposing the fractions, bring it to their attention.

In this activity, students break down fractional exponents into a unit fraction times a whole number and rewrite the expressions using radicals. The first problem suggests how to break down fractions into a whole number times a unit fraction, and students can apply the same reasoning in subsequent questions.

Look for groups that write equivalent radical expressions that are different from each other to compare during the discussion.

When students articulate their reasoning, they have an opportunity to attend to precision in the language they use to describe their thinking (MP6). They might first propose less formal or imprecise language, and after sharing with a partner, revise their explanation to be clearer and stronger.

This activity connects the ideas of roots, rational exponents, graphs of exponential functions, and decimal approximations. Students graph for a few integer values of , approximate a smooth curve passing through those points, and use that curve to estimate the value of for various rational inputs. Students then test these estimates using their understanding of the connection between rational exponents and roots.

In the first problem of this activity, students are making sense of values with fractional exponents as numbers by plotting more familiar numbers, so technology is not an appropriate tool.