Lesson 13

The purpose of this warm-up is to encourage students to apply the properties of integer exponents to reason about equivalent expressions. While students may evaluate each side of the equation to determine if it is true or false, encourage students to reason about the properties of exponents and operations in their solution.

Display one problem at a time. Tell students to give a signal when they have decided if the equation is true or false. Give students 2 minutes of quiet think time followed by a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Do you agree or disagree? Why?”
• “Who can restate ___’s reasoning in a different way?”
• “Does anyone want to add on to _____’s reasoning?”

After each false equation, ask students how the problem could be changed to make the equation true. 

In this activity, students answer questions about quantities in context. They use scientific notation as a tool for working with small and large numbers—to describe quantities, make estimates, and make comparisons (e.g., to express how many times as much one is as the other).

If possible, display an attention-grabbing image of cats and people for all to see. Ask students, “How many humans do you think there are for each cat in the world?” Ask for estimates that are too high, too low, and as reasonable as possible. 

Explain that large numbers like populations are often estimated using scientific notation. There are an estimated \(7.5 \times 10^9\) humans and \(6 \times 10^8\) cats in the world. Guide students through the example \(\frac{7.5 \times 10^9}{6 \times 10^8} = \frac{7.5}{6} \times 10^{9-8} = 1.25 \times 10^1 = 12.5\). So there are roughly 12.5 humans for each cat.

Arrange students in groups of 2 to allow partner discussions as they work. Select a student to read the first paragraph before telling students to answer the questions. Tell students that estimation will help answer these questions much more easily, so if they get stuck computing, they should try to make reasonable estimates. For example, estimating the number of humans for each cat could have looked like \(\frac{7.5 \times 10^9}{6 \times 10^8} \approx \frac{6 \times 10^9}{6 \times 10^8} = 1 \times 10^1 = 10\). In this case, the final estimate is within 20% of the original calculation (not very accurate, but within a power of 10).

Give groups 10–12 minutes to work, followed by a brief whole-class discussion.

Students may need help remembering how to estimate using a single digit times a power of 10. Remind these students that \(1.9 \times10^5\), for example, could be estimated as \(2 \times 10^5\) to make calculation easier.

Select students to explain how they used scientific notation and estimation to compare quantities. Record and display their reasoning for all to see. After each student presents, ask others if they reasoned the same way and if there are other approaches. 

Students should see that being able to multiply and divide quantities in scientific notation is particularly helpful for reasoning and estimating about very large or very small quantities, which would be challenging to work with in their decimal representations. 

In this info gap activity, students continue to use scientific notation as a tool for working with small and large numbers—to describe quantities, make estimates, and make comparisons (e.g., to express how many times as much one is as the other).

The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. 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).

As students work, identify those who use slightly different estimations to compare during whole-class discussion. While students may have slightly different results based on their estimations, the results should be relatively close in value compared to the power of 10.

Here is the text of the cards for reference and planning:

Some students may not recognize that “width” and “diameter” refer to the same measurement in this context. If needed, prompt students with Data Card 1 to give the information about the diameter when their partner asks for the width.

Select students to explain how they used scientific notation and estimation to compare the sizes of objects in the solar system. Poll the class on whether they estimated the same way. Ask previously identified students (who had slightly different ways of estimating) to compare their estimates and to show that any reasonable estimate will differ by an amount that is much smaller than the given quantities.

If time allows, discuss how scientific notation was useful in answering these questions.

This activity gives students additional practice using scientific notation to work with small and large numbers and answering questions about quantities in context. Students express numbers in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, as well as to determine how many times as much one is than the other.

Give students a minute of quiet time to observe the values in the table. Ask them to be prepared to share at least one thing they notice and one thing they wonder. Then, invite students to share their observations and questions. Consider recording their responses for all to see and using students' observations or questions about scientific notation, order of magnitude, and place values to orient students to the work in the activity.

Tell students that estimation will make it much easier to answer these questions. Give students 10–12 minutes to work, followed by a brief whole-class discussion.

Select students who use slightly different estimations and discuss how their answers are different, but very close compared to the size of the numbers they are working with. The main idea is that students can use a single digit times a power of 10 to estimate numbers and use what they know about exponent arithmetic to compare how many times larger one value is than another.