Decimal Product Reasonableness

This lesson can be used as a pre-lesson for the Multiply DecimalsLesson plan.

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This lesson can be used as a pre-lesson for the Multiply DecimalsLesson plan.

Students will be able to multiply decimals using the standard algorithm and estimating decimal point placement.

Language

Students will be able to explain how they placed the decimal point in decimal products using a place value chart and peer conversations.

(3 minutes)
• Display the expression 33 x 53And provide a story problem for students to contextualize the expression. For example, "The grocery store received 33 boxes of cereal. Each box had 53 mini boxes of cereal. How many boxes of mini cereal boxes did the grocery store receive?" Ask students to solve the problem on their whiteboards.
• Tell students to share their answers with partners while you listen to their language use. (Tip: take note of their own language and their ability to solve the multiplication problem to inform the amount of time you spend on review throughout the lesson.)
• Have a student model solving the problem and show the steps they used to find the product (i.e., 1,749).
• Explain that students will use their understanding of the standard algorithm with the multiplication of whole numbers to talk about the product of decimals.
(10 minutes)
• Solicit student help to conduct a review of decimals in context, discussing when decimals are necessary. For example, decimals are used frequently with money (to the hundredths place) and measurements (especially metric). Emphasize the size of numbers that have four digits and discuss which is bigger using a real-world example (e.g., 30.40 inch rope versus a 3.040 inch rope).
• Tie in a place value review by adding decimal place values to the number to discuss to what degree the number increases (i.e., with 3.40 and 3.409, the number as a whole barely increased when talking about inches because the nine is in the thousandths place).
• Tell students that today they will multiply decimals but they will use the standard algorithm with whole numbers and then place the decimal in the right location by thinking about the value of the digits in the ones place and what would make sense for the decimal product.
• Complete an example of whole number multiplication (e.g., 82 x 19 = 1,558) and discuss the corresponding decimal problem (8.2 x 1.9 =15.58) and a reasonable placement for the decimal given the whole number multiplication problem. Create a list of steps below as you complete the process without the hints in parenthesis:
1. Review the decimal multiplication problem and convert the decimals to whole numbers (i.e., change 8.2 x 1.9To 82 x 19).
2. Multiply 82 x 19.
3. Consider the product (1,558) and think about a reasonable decimal point placement for the number using the same digits. (For example, think aloud rounding 1.9 to two and 8.2 to eight and multiplying those numbers to get 16. Then, show how placing the decimal to create 15.58 is logical because it's close to 16. Use an empty number line to model decimal values if necessary.)
4. Tell why you think your decimal placement is correct. ("I think my decimal placement is correct because the rounded product of the decimal is 16 and the whole number answer is 1,558. I placed the decimal between the two fives to create the number 15.58 since it is close to 16.")
(8 minutes)
• Ask students to restate in partners how you were able to complete the standard algorithm multiplication. Listen to their discussions and write some of their common phrases down on the board to convert to sentence frames. Rephrase the students' explanations after the presentation using some of the following transition words to support further explanations:
• "First, I ____."
• "After I ____, I ____."
• "Then, I ____."
• "My next step was ____."
• Critique their explanations in a positive manner and write some of the sentence phrases you use on the board to model appropriate feedback for students (e.g., "How about adding/taking out...? What do you think about...? Why would you...?").
• Distribute the Standard Algorithm and Decimal Placement worksheet and review the instructions. Ask students to complete the first problem in partners. Then, have them switch partners with someone else and read their written answer.
• Call on volunteers to share their products and explanations with the class. Correct misconceptions as necessary.
• Assign students to complete the next two problems in partners (i.e., #2 and #3).
• Tell students to refer to the modeled example on the anchor chart as they complete their group assignment. For the assignment, they'll create explanations about how to multiply the whole numbers and justify the decimal placement (see worksheet answers for additional examples).
(9 minutes)
• Have students work in groups of four where each member is a number from 1–4 to share their explanation for problems 2–3 with each other. Tell group members to start with student #1 and end with student #4 for their oral presentations within the group. Have members share their explanations and allow time for listeners to share their thoughts (e.g., "I have a different answer because...").
• Challenge students to make sure they can all explain their process for placing the decimal point in the whole number product. Tell students you will choose a number from 1–4 and the student who has that number in a specific group will explain the process aloud. The student presenters can use any notes the group created to solve the problem in their explanations. (Note: allow students to adjust their answers given other students' presentations.)
• Choose a student from each group to present. (Tip: select groups with advanced ELs first so they can model the explanation process for the class first and the other students can refine their answers.)

Beginning

• Allow students to use their home language (L1) or their new language (L2) in all discussions. Provide bilingual reference materials to assist in their vocabulary word acquisition.
• Encourage them to use the vocabulary cards and terms in their conversations and writing. Allow them to draw pictures to support their understanding of the terms.

• Pair students with mixed-ability groups so they can offer explanations and provide feedback to beginning ELs when appropriate.
• Challenge them with thought-provoking questions if they are proficient mathematicians:
• Is the decimal product more or less than the ones place product when you round numbers up or down?
• When would it be easier to do the standard algorithm with the decimal numbers rather than doing the whole number multiplication?
(6 minutes)
• Assign problem 4 for students to complete on their own and then share their answers with their elbow partners.
• Monitor their explanations and write notes on the worksheet Formative Assessment: Peer Explanations Checklist to guide your assessment of their understanding and their language use.
(4 minutes)
• Ask students to compete a 3-2-1Activity. Distribute a blank sheet of paper and label it with the numbers 1–3. Have students answer:
• ThreeSteps they took to complete decimal multiplication
• TwoNew ideas they learned from the lesson
• OneOne real-world idea related to decimals
• Give students a chance to volunteer their ideas and discuss them with partners.

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