10 minute read
Organize, count, and represent a collection of objects.
Lesson at a Glance
1. What unit did you use to count your collection? Explain why you chose that unit.
Sample:
I chose fives. Counting by fives is quick. I found the total using fewer counts than I would have if I had counted by ones.
This student-driven lesson provides an opportunity to gather formative assessment data as students work with counting collections. Students decide how to organize, count, and represent the items. They analyze the work of others and discuss efficient strategies as a class.
There is no Problem Set for this lesson. Instead, use classroom observations and student work to analyze student thinking after the lesson. The Exit Ticket for this lesson serves as an opportunity for students to reflect on their counting strategies.
Key Question
• How can we use groups to help us organize?
2. If you counted your collection again, would you choose the same unit? Explain.
Sample:
I would use a different unit. I would use tens because I could find the total using fewer counts than I used when I counted by fives.
Achievement Descriptor
This lesson is foundational to the work of grade 3 and builds from 2.NBT.A.2. Its content is intended to serve as a formative assessment and is therefore not included on summative assessments in grade 3.
Agenda Materials
Fluency 5 min
Launch 10 min
Learn 35 min
• Organize, Count, and Record
• Share, Compare, and Connect
Land 10 min
Teacher
• 100-bead rekenrek
• Color tiles, plastic, 1 inch (60–150)
• Interlocking cubes, 1 cm (100)
• Computer or device*
• Projection device*
• Teach book*
Students
• Color tiles, plastic, 1 inch (60–150 per student pair)
• Interlocking cubes, 1 cm (60–150 per student pair)
• Organizational tools
• Dry-erase marker*
• Eraser*
• Learn book*
• Pencil*
• Personal whiteboard*
• Personal whiteboard eraser*
*These materials are only listed in lesson 1. Ready these materials for every lesson in this module.
Lesson Preparation
• Create collections of interlocking cubes or color tiles that have between 60 and 150 objects (per student pair). Organize each collection in a bag or small box. Although this lesson uses interlocking cubes and color tiles, you can incorporate other items of high interest.
• Display tools for students to choose from to help organize their counts. Tools may include envelopes, cups, bags, rubber bands, or graph paper.
Fluency
Counting on the Rekenrek by Tens
Materials—T: Rekenrek
Students count by tens in unit and standard form to develop an understanding of multiplication.
Show students the rekenrek. Start with all the beads to the right side.
Say how many beads there are as I slide them over.
Slide the top row of beads all at once to the left side.
10
The unit is 10. In unit form, we say 1 ten. Say 10 in unit form.
1 ten
Slide the second row of beads all at once to the left side.
How many beads are there now? Say it in unit form.
2 tens
Continue sliding over each row of beads all at once as students count.
3 tens, 4 tens, 5 tens, 6 tens, 7 tens, 8 tens, 9 tens, 10 tens
Slide all the beads back to the right side.
Now let’s practice counting by tens in standard form. Say how many beads there are as I slide them over.
Let’s start at 0. Ready?
Slide over each row of beads all at once as students count.
0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100
Counting the Math Way by Tens
Students relate counting on the rekenrek to counting the math way to develop a strategy for multiplying beginning in lesson 2.
Let’s count the math way. Each finger represents 10, just like a row on the rekenrek. Face the students and direct them to mirror you. Show a fist with your right hand, palm facing out.
Show me your left hand. Make a fist like me. That’s 0. Now, raise your right pinkie.
Show me your left pinkie. That’s 10. 01020304050
Let’s put up the very next finger. Raise your right ring finger. Students raise their left ring finger. That’s 20. Put up the next finger. 30.
Student View of Your Hands
Differentiation: Support
Students with fine motor delays may find it easier to use their fingers when they lay their hands flat on the desk or floor. The flat surface helps them hold out the fingers they want up and keep the others tucked in.
Student View of Student's Hands
Now that students understand the routine, switch to having them say the count as they show fingers. Guide students to continue counting the math way by tens to 100 and then back down to 0.
Student View of Your Hands Student View of Student’s Hands
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Launch
Materials—T: Counting collection; S: Counting collection, organizational tools
Students estimate the total of a class collection and prepare to count one of their own.
Gather students and display a counting collection.
Language Support
Consider using strategic, flexible grouping throughout the module.
• Pair students who have different levels of mathematical proficiency.
• Pair students who have different levels of English language proficiency.
• Join pairs to form small groups of four.
As applicable, complement any of these groupings by pairing students who speak the same native language.
Invite open-ended discussion by asking students what they notice or wonder about the collection.
Then ask students to estimate the total number of objects in the collection. Ask questions such as the following:
• What prediction, or estimate, would be too big? Why?
• What prediction, or estimate, would be too small? Why?
Briefly orient students to the materials and procedure for the counting collection activity:
• Partners will collaborate to count a collection.
• Partners will make their own recordings to show how they counted.
• Partners may use organizational tools. Organizational tools may include readily available classroom items such as cups, rubber bands, personal whiteboards, etc.
Partner students and distribute a different counting collection to each pair.
Before they begin to count, invite partners to work together to predict how many objects are in their collection. Have them write down their estimates. Then encourage them to talk about how they will organize their collections to count.
Invite students to select organizational tools they would like to use, with the understanding that tools may be exchanged as plans are refined.
Transition to the next segment by framing the work.
Today, we will count our collections and record the ways we organize and count.
Teacher Note
Plan for what students should do when they finish counting their collection and recording how they counted:
• Try another way to organize and count.
• Switch collections with another student pair, and count to confirm the total.
• Explain their recording to another pair.
• Clean up and get another collection.
Learn
Organize, Count, and Record
Materials—S: Counting collection, organizational tools
Partners organize and count a collection and record their process.
Ask partners to begin counting their collections. Circulate and notice how students engage in the following behaviors:
Organizing: Strategies may include counting a scattered configuration, separating counted and uncounted objects, lining up objects as they are counted, making equal groups, creating 5-groups, forming arrays, and writing expressions or equations. Students may also organize their collections using attributes that do not support counting efficiently, such as by color or size.
Counting: Students may count by ones, twos, fives, or tens. Some may count subgroups and then add to find the total.
Recording: Recordings may include drawings, numbers, expressions, equations, and written explanations.
Circulate and use questions and prompts such as the following to assess and advance student thinking:
• Show and tell me what you did.
• How can you organize your collection to make it easier to count?
• How does the way you organized your collection make it easier to count?
• How did you keep track of what you already counted and what you still needed to count?
• How close was your estimate to your actual count?
Promoting the Standards for Mathematical Practice
Students look for and make use of structure (MP7) as they decide how to organize their counting collections to make them easier to count.
Ask the following questions to promote MP7:
• What is another way to organize your collection that would help you count?
• How does what you know about counting by tens help you to count your collection?
Differentiation: Support
If students sort by size or color or in other ways not related to equal groups, help them transition to more efficient ways of organizing and counting. Use prompts such as the following:
• How can you organize your collection to help you count?
• What organizational tools can help you count?
• Let’s visit another group to see what helps them count their collection.
Select two or three pairs of students to share their counting work in the next segment. If possible, take pictures to show the class in the next segment. The samples show possible strategies. They demonstrate
• making equal groups and counting on by that unit (e.g., grouping by fives and skip-counting),
• counting by a basic unit (e.g., sorting by an attribute and then adding the totals of the groups), and
• organizing groups as an array and counting on by that unit (e.g., making an array with ten in each row and knowing the total without having to skip-count).
Teacher Note
Students may progress at different paces with counting and recording, so they may count and record with different levels of sophistication.
Organize in an Array and Recognize the Total
For this counting collection, I am partners with .
We are counting .
We estimate there are about of them.
This is how we organized and counted the collection:
We counted altogether.
An equation that describes how we found the total is
Self-Reflection
Write one thing that worked well for you and your partner. Explain why it worked well.
It worked well for me and my partner to connect the cubes into groups of 10 to organize our collection. We both agreed that 10 was a number we could count by.
Write one challenge you had. How did you work through the challenge?
At first we thought we should make groups of 20, but the sticks kept breaking, and we lost track of our count. We decided to make groups of 10 instead.
Share, Compare, and Connect
Students discuss strategies for organizing and compare the efficiency of each.
Gather the class to view the selected work samples and lead a discussion. Invite the selected pairs to share their counting process. The following dialogue models a sample discussion.
Teacher Note
Consider reserving time for the class to engage in discussion after partners have time to complete the self-reflection questions. Development of metacognitive strategies may support students in understanding how they learn best and help them self-monitor their progress.
Group
and Count by Fives (David and James’s Way)
This sample uses a skip-count. Invite partners to share.
How did you know the total?
We counted by fives.
Why did you decide to count that way?
We know how to count by fives and tens, so we made groups of 5 with our cubes and put 2 fives together to make tens on the sheet.
Can you count your collection a different way? How?
Yes, we can put 2 tens together and skip-count by twenties. Then we can add what’s left.
Ask students to raise their hand if they used a skip-count.
UDL: Representation
Consider creating a three-column chart and, as partners share, recording each strategy. After all pairs have shared, compare the different organizations of the count, the methods for finding the total, and the efficiency of each strategy. For example, emphasize that each pair organized in a different way (by color, by 5-groups, and in rows of 10) and found the total in different ways (counted on, skip-counted, and used a known fact).
Group by Color and Count by Ones (Ivan and Jayla’s Way)
Note that sorting by color is a common strategy but that it is not always efficient. Invite partners to share their counting strategy.
Could we use David and James’s way, a skip-count, to find the total of this collection?
Yes. We could count each color group by fives or tens.
How would the recording change if we skip-counted by fives or by tens?
We wouldn’t have to add all the numbers together. We could just count in our head.
Teacher Note
As students count, they exhibit different levels of sophistication in their counting strategies. Select students to share their work so that students with less sophisticated counting strategies have an opportunity to hear new ideas. If time allows, encourage students to count their collection a second time by using a strategy they heard from another group.
Organize in an Array and Recognize the Total (Mia and Amy’s Way)
Invite partners to share.
How did your organization help you count?
We made 6 rows of 10 tiles. We know that 6 rows of 10 is 60, so we didn’t have to count all the tiles or skip-count. We had 2 extra tiles, so we added 2 to 60 and got 62.
Invite students to turn and talk about how Mia and Amy’s strategy could be used for 8 rows of 10.
Some students might notice that they can find the total of their array by multiplying, but this is not expected of students at this point in the year. If students make the connection, invite them to share their thinking. Formal introduction of multiplication using different units begins in lesson 2.
If time permits, have students show a new partner their recording and explain their work. Use the following prompts to guide a discussion about how the organization of a collection helps in finding the total.
What were you successful with when counting?
I organized my cubes into equal groups so I could skip-count to find the total.
What did you find challenging about counting?
At first, I counted by twos. It was hard to keep track of my count. Then I switched to making and counting groups of 5 and that was much easier.
Did you see something you’d like to try next time we count collections? Why do you want to try it?
Next time, I’ll use groups of 5 again, but I’ll organize the groups better so that I know which ones I’ve already counted and which ones I haven’t.
I’d like to use cups to put my collection in the next time. We counted cubes and they were hard to line up. We could just count 10 and put 10 in each cup. Then we could skip-count by tens to find the total.
Next time, I’d like to try multiplication to find the total number.
How does organizing help you count?
It helps me to keep track of which things I’ve counted, and it makes it faster to count. If I make a mistake, I don’t have to start all over again.
