The purpose of this Number Talk is to elicit strategies and understandings students have for adding within 1,000, particularly around adjusting numbers in a sum to make them easier to add. These understandings help students develop fluency for adding within 1,000.
When students notice that the same value is being removed from one addend and added to the other and the value of the sum does not change, they look for and make use of structure (MP7).
Launch
Display one expression.
“Give me a signal when you have an answer and can explain how you got it.”
“What pattern do you see as you look at the expressions and their values? Why is that happening?” (The values are all the same. This happens because you add an amount to one addend, but subtract the same amount from the other addend. Addition and subtraction are opposites, so the value will stay the same.)
As needed, record student thinking with expressions or equations. Consider testing any student generalizations with smaller addends to make the thinking accessible to all students.
Consider asking:
“Did anyone notice a different pattern”?
“Did anyone notice the same pattern but would explain it differently?”
Activity 1
Standards Alignment
Building On
Addressing
3.OA.A.2
Interpret whole-number quotients of whole numbers, e.g., interpret as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as .
The purpose of this activity is for students to physically represent the difference between making 2 groups and making groups of 2. Ten students will put themselves into 2 groups and then groups of 2. The rest of the students observe how the groups were made to highlight the difference between “how many groups?” problems and “how many in each group?” problems.
Launch
Groups of 2
Invite 10 students to come to the front of the class.
“These students are going to put themselves into groups in different ways. If you are observing, take notes on what you notice about how they make the groups.”
Activity
Ask the 10 students to put themselves into groups of 2.
Give observers a chance to take notes.
Ask the 10 students to put themselves into 2 groups.
Give observers a chance to take notes.
Ask the students to return to their seats.
“Talk with a partner about what you noticed about how the students put themselves into groups of 2 and 2 groups.”
2–3 minutes: partner discussion
What did you notice about how the students put themselves into groups of 2?
What did you notice about how the students put themselves into 2 groups?
Student Response
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Advancing Student Thinking
Activity Synthesis
Ask students who observed to share what they noticed.
Highlight ideas that help clarify differences between “how many groups?” and “how many in each group?”
Activity 2
Standards Alignment
Building On
Addressing
3.OA.A.2
Interpret whole-number quotients of whole numbers, e.g., interpret as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as .
The purpose of this activity is for students to match a division situation to a drawing of equal groups. Students should be able to explain why the situation matches Drawing A, which shows 2 groups of 6, and why it does not match Drawing B, which shows 6 groups of 2.
This activity uses MLR1 Stronger and Clearer Each Time. Advances: reading, writing.
Launch
Groups of 2
“Today we are going to look at drawings to represent division situations. Take a minute to read this situation.”
1 minute: independent work time
Activity
“Work independently to decide which drawing matches this situation and explain your reasoning.”
2–3 minutes: independent work time
Elena has 12 colored pencils. She has 2 boxes and wants to put the same number of colored pencils in each box. How many colored pencils should go in each box?
Which drawing matches the situation? Explain your reasoning.
A
B
Student Response
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Advancing Student Thinking
Activity Synthesis
MLR1 Stronger and Clearer Each Time
“Share your response with your partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
2–3 minutes: structured partner discussion
Repeat with 2 different partners.
“Which drawing did you decide matches? How do you know?”
“How do you know the other drawing does not match this situation?” (Drawing B is 6 groups of 2 colored pencils. That would be like if she had 6 boxes, not 2 boxes.)
Activity 3
Standards Alignment
Building On
Addressing
3.OA.A.2
Interpret whole-number quotients of whole numbers, e.g., interpret as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as .
The purpose of this activity is for students to relate division situations and drawings of equal groups (MP2). Each given drawing matches two different situations. Students learn that the same drawing can represent both a “how many groups?” problem and a “how many in each group?” problem because the drawing shows the end result, not how the groups were made. When students interpret one diagram as representing two different story types, they state clearly how each part of the diagram corresponds to the story, including what corresponds to the unknown in the story (MP6).
MLR8 Discussion Supports. Students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frame for all to see: “I noticed ___ , so I matched . . . .” Encourage students to challenge each other when they disagree. Advances: Listening, Speaking, Representing
Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select at least 3 of the 6 problems to complete. Supports accessibility for: Organization, Attention, Social-emotional skills
Launch
Groups of 2
“We’re going to look at some situations that involve writing or drawing tools. What are some things we use to write or draw?”
30 seconds: quiet think time
Share and record responses.
Activity
“You are going to match six situations and drawings that could represent them. Take a few minutes to decide which drawing matches each situation.”
3-5 minutes: independent work time
“Share your ideas with your partner.”
2–3 minutes: partner discussion
Match each situation to a drawing. Be prepared to explain your reasoning.
Mai has 8 markers. She puts 4 markers in each box. How many boxes of markers are there?
Kiran has 20 pens. He wants to put 2 pens at each table. How many tables can he put pens on?
Lin has 8 colored pencils. She puts them into 2 bags. Each bag has the same number of colored pencils. How many colored pencils are in each bag?
Priya has 15 crayons. She puts 5 crayons on each desk. How many desks have crayons?
Noah has 20 pencils. He puts the same number of pencils in 10 boxes. How many pencils are in each box?
Jada has 15 markers. She puts the same number of markers on 3 tables. How many markers are on each table?
A
B
C
Student Response
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Advancing Student Thinking
If students say that the drawing can’t match both situations, consider asking:
“How could we make a drawing for each situation?”
“What might we draw first to represent the first situation with 8 objects? What could we draw for the second situation with 8 objects?”
Activity Synthesis
Invite students to share which drawing matches each situation.
Focus on one drawing and the two situations it can represent, such as:
Mai has 8 markers. She puts 4 markers in each box. How many boxes of markers are there?
Lin has 8 colored pencils. She puts them into 2 bags. Each bag has the same number of colored pencils. How many colored pencils will be in each bag?
“How can the same drawing represent both situations?” (We didn’t see how the groups were made, but in the end, the same number and size of groups were made in both situations. The drawing can represent putting 8 markers into boxes with 4 markers in each box and finding that they fit into 2 boxes. It can also represent putting 8 pencils into 2 bags with the same number of pencils in each bag and finding that you can put 4 pencils in each bag.)
Lesson Synthesis
Continue to display the drawing and situations from the last activity, such as:
Mai has 8 markers. She puts 4 markers in each box. How many boxes of markers are there?
Lin has 8 colored pencils. She puts them into 2 bags. Each bag has the same number of colored pencils. How many colored pencils will be in each bag?
“Today we matched drawings to division situations. There are two types of division situations, and we saw today that the same drawing can represent both types of situations.”
“What is the same and what is different about these division situations?” (Both situations have the numbers 8, 2, and 4 in them. Both involve putting objects into equal groups. The objects are different—one is about markers, and the other is about colored pencils. One situation tells us how many items go into each container, and the other tells us how many containers there are.)
“In the first situation, we need to figure out how many groups there are. We know there are 4 markers in each box, but we don’t know how many boxes there will be. In the second situation, we need to figure out how many in each group. We know there are 2 bags, but we don’t know how many colored pencils will be in each bag.”
“Now that we are dividing, we need a new symbol to write division expressions.”
Display:
“The symbol in the middle of this expression is the division symbol. This expression can be read as ‘8 divided into groups of 4.’” Record the meaning below the expression. Discuss how both relate to Mia’s situation.
Display:
“This expression can be read as ‘8 divided into 2 equal groups.’” Record the meaning below the expression. Discuss how both relate to Lin’s situation.
Standards Alignment
Building On
Addressing
3.NBT.A.2
Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.
