This Number Talk encourages students to compose or decompose multiples of 7 and to rely on properties of operations to mentally solve problems. The ability to compose and decompose numbers will be helpful when students divide multi-digit numbers. It also promotes the reasoning that is useful when finding multiples of a number, or when deciding if a number is a multiple of another number.
Launch
Display one expression.
“Give me a signal when you have an answer and can explain how you got it.”
1 minute: quiet think time
Activity
Record answers and strategy.
Keep expressions and work displayed.
Repeat with each expression.
Student Task Statement
Find the value of each expression mentally.
Student Response
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Advancing Student Thinking
Activity Synthesis
“What do the expressions have in common?” (They all involve division by 7. The dividends are all multiples of 7. The results have no remainders.)
“How did the first three expressions help us find the value of the last expression?”
Consider asking:
“Who can restate _______ 's reasoning in a different way?”
“Did anyone have the same strategy but would explain it differently?”
“Did anyone approach the expression in a different way?”
“Does anyone want to add on to____’s strategy?”
Activity 1
20 mins
Write Multiples
Standards Alignment
Building On
4.NBT.B.5
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.
Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
The purpose of this activity is to prompt students to use the relationship between multiplication and division and their understanding of factors and multiples to solve problems about an unknown factor (MP7). Students recognize such problems as division situations. In this activity, the dividends are three-digit numbers and the divisors are one-digit numbers. Students use the context of factors and multiples to interpret division that results in a remainder.
One approach for solving these problems is by decomposing the dividend into familiar multiples of 10 and then dividing the remaining number (which is now smaller) by the divisor. Another is to find increasingly greater multiples of the divisor until reaching the dividend. Both are productive and appropriate. In the Activity Synthesis, help students see the connections between the two paths.
Engagement: Develop Effort and Persistence.If students don’t recognize this as a division situation at first, they may do so when presented with a more accessible value. Consider offering this situation first: “Han starts writing multiples of a number. When he reaches 12, he has written 3 numbers.” Invite students to identify what number Han is writing multiples of (4), and how they know. Then ask how they might apply that reasoning to the task as presented here. Supports accessibility for: Conceptual Processing
Launch
“Who can remind the class of the meaning of ‘multiple?’”
Activity
4–5 minutes: independent work time for the first set of questions
Monitor for students who:
Decompose 104 into a multiple of 8 and another number.
Compose 104 from increasingly larger multiples of 8.
Use multiplication or division equations to show their reasoning.
Pause for a discussion before the second set of questions.
Select students who use different decomposition strategies to share responses. Record and display their reasoning.
“Let’s revisit each question we just answered. What equations could we write to represent them?”
Display equations and highlight their connection to the questions:
or
or
or
4–5 minutes: independent work time for the second set of questions
Activity Synthesis
Invite students to share responses for the last set of questions.
Highlight that to divide a number by a smaller number—say, divide 150 by 7, we can:
Use familiar multiples or multiplication facts to help us. For example, if we know , we know the result is 21 with a remainder of 3, or .
Think of the dividend in smaller chunks. For example: We can see the 150 as and divide each 140 and 10 by 7 separately, which gives with a remainder of 3.
Activity 2
15 mins
Jada’s Mystery Number
Standards Alignment
Building On
Addressing
4.NBT.B.6
Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
The purpose of this optional activity is for students to continue to use the relationship between multiplication and division to reason about situations that involve division. Students divide three-digit numbers by single-digit divisors and find results with and without a remainder. This activity provides more practice applying the relationship between multiplication and division.
MLR8 Discussion Supports. Before they begin, invite students to make sense of the situation. Monitor and clarify any questions about the context. Advances: Reading, Representing.
Launch
Groups of 2
Activity
5 minutes: independent work time on the first question
2 minutes: partner discussion
2–3 minutes: partner work time on the second question
Monitor for students who clearly show how they use partial products or partial quotients to answer the questions.
Student Task Statement
Jada writes multiples of a mystery number. After writing some numbers, she writes the number 126.
Mai says 6 is the mystery number.
Priya says 8 is the mystery number.
Andre says 9 could be the mystery number.
Which student do you agree with? Show how you know using equations.
Jada gives one more clue: “If I keep writing multiples, I’ll get to 153.”
What is the mystery number? Explain or show your reasoning.
Student Response
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Advancing Student Thinking
Activity Synthesis
“How could we use division to help us find the mystery number?” (If the result has a remainder, then the divisor could not be the mystery number. 153 divided by 6 has 3 as a remainder. 153 divided by 9 has no remainder, so 9 is the mystery number.)
Activity 3
15 mins
Watch Your Remainder!
Standards Alignment
Building On
Addressing
4.NBT.B.6
Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
The purpose of this activity is to introduce the center activity Watch Your Remainder, Stage 1. Students spin a spinner to get the divisor for the round. Each student picks 5 cards and chooses 3–4 of them to create a dividend. Each student finds their quotient. The score for the round is the remainder from each expression. Students pick new cards so they have 6 cards in their hand and then start the next round. The player with the lowest score after 6 rounds wins.
The Activity Synthesis provides suggested cards to display to discuss strategies for the game. Consider having a group of 2 share one of the rounds from their game instead.
Launch
Groups of 2
Give each group a set of Number Cards and 2 copies of the recording sheet.
“We are going to learn a game called Watch Your Remainder.”
Give students several minutes to read over the directions.
Answer any clarifying questions from students.
Consider playing a sample round with the class.
Activity
10 minutes: partner work time
Monitor for students who play a round where there is no remainder and one where there is a remainder.
Student Task Statement
Directions:
Spin the spinner to get your one-digit divisor.
Each partner:
Use 3–4 cards to create a dividend.
Write a multiplication equation to represent the quotient. (For example, is written as and your score is 1.)
Check your partner’s work to make sure you agree.
Your score for each round is the remainder.
Take new cards so that you have 4 cards to start the next round.
The partner who has the fewest points once the recording sheet is full wins the game.
Student Response
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Advancing Student Thinking
Activity Synthesis
Invite selected students to show the number cards for the round with no remainder or display the number cards 0, 1, 2, 3, 7 with a divisor of 5.
“How would you use these cards to get the smallest remainder?”
Invite selected students to show the number cards for the round with a remainder or display the number cards 0, 1, 2, 3, 7 with a divisor of 9.
“How would you use these cards to get the smallest remainder?”
Lesson Synthesis
“Today we tried to find out if a number is a multiple or a factor of another number. For example: Is 267 a multiple of 8?”
“Is this question a division problem?” (Yes)
“Why?” (We’re looking for how many 8s are in 267.)
“What are we dividing?” (267 by 8)
“One way to answer the question is by using familiar multiplication facts or by finding partial products. How would you start?” (One way is to start with 10 x 8 or its multiples, build the products up to 267 or close to it, and then try smaller multiples of 8.)
“Why might it be helpful to start with multiples of 10?” (They’re easy to find and easy to add.)
“Can we use division to answer the question?” (We can start with a number close to 267 that is a multiple of 8, divide it by 8, see what is left, and find a multiple of 8 that is close to that number. For example: . After taking 160 away, there’s 107 left. . After taking 80 away, there’s 27 left. . After taking 24 away, there’s 3 left, which is not enough to make 8.)
“How are the two approaches alike?” (They involve using smaller multiples of a number to see if a larger number is a multiple of that number.)
Student Section Summary
We solved different problems that involved dividing whole numbers.
We recalled two ways of thinking about division.
For example, if represents a situation where 274 markers are put into equal groups. The value of can tell us:
How many markers are in each group if there were 8 groups.
How many groups can be made if there were 8 markers in each group.
We learned that in , the is called the dividend, and the is called the divisor. We then identified many ways to find the value of a quotient —or the result of the division. For , we can:
Think about whether one number is a multiple or factor of another number. For example, “Is 274 a multiple of 8?” or “Is 8 a factor of 274?”
Divide by place value and think about putting 2 hundred, 7 tens, and 4 ones into 8 equal groups.
Divide in parts and find partial quotients. For example, we can first find (which is 20), and then (which is 10), and then (which is 4).
Think in terms of multiplication. For example, we can think of , , and so on.
Here is one way to record division using partial quotients.
Divide. 2 hundred seventy 4 divided by 8, 11 rows. First row: 34. Second row: 4. Third row: 10. Fourth row: 20. Fifth row: 8, long division symbol with 2 hundred seventy 4 inside. Sixth row: minus 1 hundred 60. Side note, 8 times 20. Horizontal line. Seventh row: one hundred 14. Eighth row: minus 80, side notes 8 times 10. Horizontal line. Ninth row: 34. Tenth row: minus 32, side note 8 times 4. Horizontal line. Eleventh row: 2.
Sometimes a division results in a leftover that can’t be put into equal groups or is not enough to make a new group. We call the leftover a remainder. Dividing 274 by 8 gives 34 and a remainder of 2.
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Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
