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Divide by Powers of Ten

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This lesson can be used as a pre-lesson for the Dividing Decimals By Powers of 10Lesson plan.

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This lesson can be used as a pre-lesson for the Dividing Decimals By Powers of 10Lesson plan.

Students will be able to discuss the powers of 10 and explain decimal placement patterns in the quotient.

Language

Students will be able to discuss decimal placement patterns using sentence stems and vocabulary instruction.

(5 minutes)
• Display the answer sheet for the Growing by Powers of Ten Chart worksheet and discuss what they see happening to the numbers as they are multiplied by 10 in the standard form column (e.g., the number gets larger by adding more zeros, but the rest of the digits stay the same). Focus on the standard form column so students can notice the pattern.
• Remind students about the inverse relationship between division and multiplication with simple calculations (e.g., 5 x 5 = 25And 25 ÷ 5 = 5). Ask students to discuss how the product is different from the factor, and how the quotient is different from the dividend (e.g, "The product is bigger than both of the factors, while the quotient is smaller than the dividend.").
• Refer back to the answer sheet of the worksheet Growing by Powers of Ten Chart. Have students turn and talk to their partners regarding the following question: "What would happen to the quotient if I changed all the multiplication problems to division problems?"
• Explain to students that the worksheet shows the Power of tenFor multiplication, but the power of ten also applies in division as well. The quotient will get smaller and smaller exponentially as the dividend is divided by ten (e.g., 234.5 ÷ 10 = 23.45And 23.45 ÷ 10 = 2.345). Tell them they will begin to notice PatternsFor decimal point movement within division by the end of this lesson.
(7 minutes)
• Define the terms Patterns, Drawing conclusions, and Powers of ten. Have students say the meanings with you and write your notes from the board on the Vocabulary Cards or Glossary. Write some examples of the words on the board with assistance from students.
• Distribute all the vocabulary cards and ask students to add a visual to the cards. For example, the visual for powers of ten could have a number being multiplied by ten however many times. Additionally, students can create a short pattern with numerals or shapes, or a number that increases by two every time, on the pattern vocabulary card.
• Write "Patterns of Ten: Division" on a piece of chart paper. Write the expression 983.4 ÷ 10On the chart paper and ask students to tell their partners what the quotient might look like without doing any calculations ("I think the answer will be..."). Listen for their rationale and include some of their thinking when you share the quotient.
• Share that the answer for the division expression is 98.34 and that you did not even have to use the standard algorithm because of the powers of ten with division. Show the calculator to prove your answer and write your observations on the chart paper (e.g., "I noticed that the decimal point moved one place to the left when the number was divided by 10, making the value smaller.").
• Ask students to consider 98.34 ÷ 10And discuss with partners what the answer may be. Ask them to share their ideas aloud and confirm or deny their guesses by using the calculator again. Mention that the decimal point moved again to the left when the dividend was multiplied by 10.
• Have them consider 983.4 ÷ 100Given the last two quotients. Have students share aloud what they think their answer will be (e.g., "Since 10 x 10 = 100, I think the decimal will move two places to the left. The quotient will be 9.834.").
• Explain that patterns can be easy to find with repeated calculations.
(6 minutes)
• Display the worksheet Multiply and Divide with Powers of 10 and model completing a box with division problems using a calculator. Discuss how the answer is different from the dividend. Students should understand that the value of the dividend gets smaller and that they could essentially eliminate a zero in the answer for every zero in the divisor.
• Distribute the worksheet Multiply and Divide with Powers of 10 and calculators. Ask students to write the answers of a set of division problems (i.e., one box of division expressions) using their calculators to solve the problems. (Note: tell students to skip all the multiplication questions and those with exponents.)
• Ask students to share their observations about their division problems and their quotients (e.g., "I noticed that for every zero in the divisor, there was one less zero in the quotient.") Allow them to complete another box of division problems and share observations with partners.
• Choose a few more students to share their observations aloud and write them on the chart paper Patterns of Ten: Division.
• Model asking a student a clarifying question about their explanation after they have shared it with the class. Use language such as:
• "What do you mean when you say...?"
• "Could you clarify how you did this part?"
• "What step did you take to solve this step?"
• "This sentence might be better if you said..."
• Have students repeat some of the clarifying questions you asked as you write them on the board.
• Tell students they will now discuss in partnerships their observations about patterns involving the powers of ten.
(10 minutes)
• Display and distribute the worksheet Clearer Stronger Activity with Notes and discuss the expectations you have for their discussions (see worksheet instructions). Have an advanced student summarize the expectations for the activity.
• Choose a student to model the discussion with you using the equations from the Explicit section (e.g., 983.4 ÷ 10 = 98.34And 98.34 ÷ 10 = 9.834). Share a conclusion you've drawn based on the example equations and have your student partner ask clarifying questions. Write a note in the teacher copy of the Partnership Notes section ("My partner said to...").
• Tell students they will now conduct a Clearer Stronger Activity where they will choose a set of boxed expressions (i.e., a set of division problems that does not include exponents) from the Multiply and Divide with Powers of 10 worksheet. They will discuss the patterns they saw with one box of expressions two different times using examples and sentence frames. The goal of the activity is to improve their explanations after each round.
• Tell students they will have two different partners throughout the activity. Each partner will share an observation they have about their chosen expressions and the effect of the power of ten on the quotient, while the other partner asks clarifying questions. The partners will switch roles of listener and presenter and follow the same sequence. Then, they will find their new, second partner and repeat the same process.
• Have students take notes at the end of each partnership on what they can change to make their own explanations stronger and clearer.
• Encourage students to ask their partners clarifying questions, such as "What happens to the dividend when you divide it by 100?"
• Monitor students and offer feedback as needed, whether individually or to the whole class if you see they need assistance.
• Have one student share their explanation with the class. Offer a chance for students to ask and answer each other's questions as a whole group.

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.
• Encourage students to draw ideas on their whiteboards and speak aloud to themselves prior to sharing their ideas with others.
• Preteach a lesson on the powers of ten within multiplication so they can draw conclusions based on some of their background knowledge.
• Connect the relationship of the powers of ten in division to that of multiplication if they have a strong understanding from a previous lesson. Ask students to restate the connection and provide an example.

• Pair students with mixed ability groups so they can offer explanations and provide feedback to beginning ELs when appropriate.
• Encourage them to add onto other students' explanations and rephrase their wording as necessary.
• Solicit example sentence frames or stems that they think would be helpful for others to use prior to each partnership activity.
• Have a student give their answer to the open-response question first to serve as a model for their peers. Have listeners agree or disagree with the presenter's opinion and give an example to support their own ideas.
• Ask students to use a tablet to complete some problems for the exercise Division and Powers of Ten. Have them write down some problems and share some generalizations with the class.
(6 minutes)
• Present the following question and ask students to answer it in partners using their whiteboards: "What will happen to the number 23.4 if I multiply it by 10 and then divide it by 10?" A potential answer is as follows: "My expression is... I think... Since this is about the power of ten, I know the decimal point will..."
• Ask students to consider the power of ten and all that they had learned already. Then tell them to discuss their ideas in partners given the information they learned from the Clearer and Stronger activity.
• Have partnerships share their ideas and write any relevant, correct conclusions that pertain to division with the power of ten to the chart paper Patterns of Ten: Division.
• Choose one student to summarize the class's stance on what would happen to the number 23.4 and prove it with a calculator. Ask students to agree or disagree with the explanation and add onto the explanation if they can ("I agree with ____. I also think ____.").
(5 minutes)
• Present the following question to students to consider and discuss in groups: "Are there any other numbers that can move the decimal to the right or left without changing the digits of the dividend or quotient?"
• Have students use their whiteboards or calculators to quickly test out their theories.
• Choose volunteers to share their ideas. Students should understand that they can divide by another decimal number with one and zeros as the digits (e.g., 0.75 ÷ 0.001 = 750.0) and move the decimal point to the right rather than the left (i.e., the value of the number gets bigger, not smaller).

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