Lesson 5

In this warm-up, students use the structure (MP7) of a set of multiplication equations to see the relationship between two numbers that differ by a factor of a power of 10. Students evaluate the expression \(x\boldcdot 10\) by considering the effect of multiplication by 10. 

Ask students to answer the following questions without writing anything and to be prepared to explain their reasoning. Follow with whole-class discussion.

Ask students to share what they noticed about the first four equations. Record student explanations that connect multiplying a number by 10 with moving the decimal place. 

Discuss how students could use their observations from the first question to multiply 0.81 by a number to get 810. 

In grade 5, students recognize that multiplying a number by \(\frac{1}{10}\) is the same as dividing the number by 10, and multiplying by \(\frac{1}{100}\) is the same as dividing by 100. In this lesson, students will recognize and use the fact that multiplying by 0.1, 0.01, and 0.001 is equivalent to multiplying by \(\frac{1}{10}\), \(\frac{1}{100}\), and \(\frac{1}{1,000}\), respectively. In all cases, the essential point to understand is that in the base-ten system, the value of each place is \(\frac{1}{10}\) the value of the place immediately to its left. Writing the decimals 0.1, 0.01, and 0.001 in fraction form will help students recognize how the position of the decimal point in the product is affected by the positions of the decimal points in the factors.

The structure of the base-ten system can serve as a guide for calculating products of decimals, and the goal of this lesson is to begin to uncover that structure (MP7).

Students might need to see decimal and fraction names. If so, display a place-value chart for reference.

Arrange students in groups of 2. Ask one student in each group to complete the questions for Partner A, and have the other take the questions for Partner B. Then ask them to discuss their responses, answer the second question together, and pause for a brief class discussion.

Students may readily see that \(36 \boldcdot (0.1)\) but be unsure about what to do when the factor being multiplied by the decimal 0.1 and 0.01 is also a decimal (e.g., \((24.5) \boldcdot (0.1)\)). Encourage them to try writing both decimals (e.g., \(24.5\) and \(0.1\)) as fractions, multiply, and convert the resulting fraction back to a decimal.

The purpose of this discussion is to highlight the placement of the decimal point in a product. Consider asking some of the following questions:

• “How does the size of a product compare to the size of the factor when the factor is multiplied by 0.1? How does the placement of the decimal point change?” (Multiplying by 0.1 makes the product ten times smaller than the factor. The decimal point moves to the left one place.)
• “How does the size of a product compare to the size of the factor when the factor is multiplied by 0.01?” (Multiplying by 0.01 makes the factor one hundred times smaller. The decimal point moves to the left two places.)
• “Can you predict the outcome of multiplying 750 by 0.1 or 0.01 without calculating? If so, how?” (Multiplying 750 by 0.1 would produce 75. Multiplying 750 by 0.01 would produce 7.5.)

In this activity, students continue to think about products of decimals using fractions. They use what they know about \(\frac{1}{10}\) and \(\frac{1}{100}\), as well as the commutative and associative properties, to identify and write multiplication expressions that could help them find the product of two decimals.

While students may be able to start by calculating the value of each decimal product, the goal is for them to look for and use the structure of equivalent expressions (MP7), and later generalize the process to multiply any two decimals.

As students work, listen for the different ways students decide on which expressions are equivalent to \((0.6) \boldcdot (0.5)\). Identify a few students or groups with differing approaches so they can share later.

Arrange students in groups of 2. Give groups 3–4 minutes to work on the first two questions, and then pause for a whole-class discussion. Discuss:

• Why is \((0.6) \boldcdot (0.5)\) equivalent to \(6 \boldcdot \frac{1}{10} \boldcdot 5 \boldcdot \frac{1}{10}\)? (0.6 is 6 tenths, which is the same as \(6 \boldcdot \frac{1}{10}\), and 0.5 is 5 tenths, or \(5 \boldcdot \frac{1}{10}\))
• Why is the expression \(6 \boldcdot \frac{1}{10} \boldcdot 5 \boldcdot \frac{1}{10}\) equivalent to \(6 \boldcdot 5 \boldcdot \frac{1}{10} \boldcdot \frac{1}{10}\)? (We can ‘switch’ the places of 5 and \(\frac{1}{10}\) in the multiplication and not change the product. This follows the commutative property of operations.)
• How did you find the value of \(30 \boldcdot \frac{1}{100}\)? (Multiplying by \(\frac{1}{100}\) means dividing by 100, which moves the decimal point 2 places to the left, so the result is \(0.30\) or \(0.3\).

If students try to use vertical calculation to find the products, ask them to instead do so by thinking of the decimals as fractions and about any patterns they observed.

Select several students to share their responses and reasonings for the last question.

To conclude, ask students to consider how writing \(6 \boldcdot 5 \boldcdot \frac{1}{100}\) might be a favorable way to find \(0.6 \boldcdot 0.5\). Students may respond that using whole numbers and fractions makes multiplication simpler; even if there is division, it is division by a power of 10. In future lessons, students will apply this reasoning to find products of more elaborate decimals, such as \((0.24) \boldcdot (0.011)\).