Lesson 15

The purpose of this Number Talk is to elicit strategies and understandings students have for addition, subtraction, multiplication, and division. These understandings help students develop fluency and will be helpful later in this lesson when students compute with numbers in scientific notation. While four problems are given, it may not be possible to share every strategy. Consider gathering only two or three different strategies per problem, saving most of the time for the final question.

Display one problem at a time. Give students 1 minute of quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Students may write \(3 \times 10^9 - 2 \times 10^7 = 1 \times 10^2\), or something similar. Ask these students to evaluate each product first before subtracting.

Ask the class how the first two questions are different than the second two questions. Note that we have to pay more attention to place value to answer the second two than the first two because when we add or subtract, we can only add or subtract digits that correspond to the same powers of 10. Consider asking:

• “Which problem was easier? Why?”
• “Of the four operations, which operations are easier to do with scientific notation? Which are harder?” (Multiplying and dividing is easier with scientific notation, because the exponent rules involve multiplication and division. Addition and subtraction are harder, because the exponent rules don’t involve those operations.)

Students attend to precision when adding numbers in scientific notation, taking care that the numbers are first written as a decimal or with powers of 10 with the same exponent (MP6). Students critique the reasoning of Diego, Clare, and Kiran as they make sense of adding numbers in scientific notation (MP3).

Arrange students in groups of 2. Give 10–12 minutes to work followed by a whole-class discussion. Encourage students to share their thinking with a partner as they work.

Students may make various mistakes that show a misunderstanding of exponent rules, for example multiplying the exponents like \(4.7 \times 10^4 + 1.2 \times 10^5 = 5.9 \times 10^{20}\) or adding exponents like \(4.7 \times 10^4 + 1.2 \times 10^5 = 1.67 \times 10^9\). For these students, emphasize that the terms are being added, not multiplied, so the factors that are 10 are not being grouped in the same way.

Select students to share their reasoning about how to add \(4.7 \times 10^4\) km and \(1.2 \times 10^5\) km. It is important that students understand that \(4.7 \times 10^4\) and \(1.2 \times 10^5\) are off by roughly a factor of 10. In order to compare them, they either have to be written as decimal numbers or with the same power of 10.

Discuss some of the following questions:

• “How are Clare’s and Kiran's approaches alike?” (They both reached the same sum by attending to place value.) 
• “How are their approaches different?” (Kiran wrote the numbers as decimals and added them. Clare wrote the numbers with the same power of 10 and added them. Kiran’s method might not work very well if the numbers are very large or very small. The decimal form of those number would be unwieldy. ) 
• “Why must the terms have the same power of 10 to be added?” (We can only add digits that are of the same place value. If the powers of 10 are different, the place values of the digits in the first factors of the two expressions would be different. For example, the 4 in \(4.7 \times 10^4\) means 4 ten-thousands and the 1 in \(1.2 \times 10^5\) means 1 hundred-thousand, so we cannot add 4.7 and 1.2.)
• “How might Clare have reasoned that \(1.2 \times 10^5\) can be written as \(12 \times 10^4\)?” (One way is to see that changing 1.2 into 12 requires multiplying by 10. To keep the value of the expression the same, we must divide it by 10, which decreases the exponent by 1, from \(10^5\) to \(10^4\).)
• “How did you compare Clare and Kiran’s results to the width of Jupiter?” (Converting the widths to scientific notation makes it easy to compare.)

In this activity, students add quantities written in scientific notation in order to answer questions in context. To add numbers in scientific notation, students must attend to precision by aligning place value (MP6).

As students work, notice the different strategies used to align place value. One strategy would be to convert all the distances to decimal, align the place values vertically, and then add in the usual way. Another example would be to rewrite all the addends to use the same power of 10 before adding.

Arrange students in groups of 2. Tell students to discuss their thinking with a partner and work to reach agreement. Give students 12 minutes to work, followed by a brief whole-class discussion.

The main point to highlight is that values given in scientific notation can be added by carefully aligning the place values of all of the addends. Select students who show different ways of aligning the place values. Record their strategies and display them for all to see. Ask students to explain how they decided to scale the objects in their drawing.

Consider taking the time to engage with this activity if students need more experience with negative exponents and additional practice with adding quantities expressed in scientific notation. Students work with positive and negative exponents simultaneously.

Arrange students in groups of 2. Tell students to explain their thinking to their partner and work to reach agreement. Give students 10-12 minutes to work followed by a whole-class discussion.

In a whole-class discussion, talk about what the difference between multiplying or dividing numbers in scientific notation and adding or subtracting numbers. Consider asking: “Which is easier? What do you have to be careful about?”