Reflect and Consolidation Prompts
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Length of Unit: 5 Days
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Students will explore subtraction in a context by decomposing up the subtrahend. A connection can be made to the combinations that make ten.
In this task, students will engage in a subtraction context and lead towards a strategy of decomposing the subtrahend.
Some of the big ideas that may emerge through this task include:
- Understanding hierarchical inclusion allows for flexible composing and decomposing of numbers
- Numbers can be decomposed by separating a whole into two or more parts
- Subtraction names the missing part in terms of the whole
- Different subtraction situations will elicit different strategies
- Number relationships provide the foundation for strategies to help students remember basic facts
- Addition and subtraction are related in that addition names the parts in the whole and subtraction uses the whole to find a missing part.
- Subtraction can be used in either take away, comparison, or missing addend situations.
- Models can be used to connect concrete to abstract
What Do You Notice? What Do You Wonder?
Show students the following video:
Then, ask students:
What do you notice?
What do you wonder?
Give students about 30-60 seconds to do a rapid write on a piece of paper or silent individual think time.
Replaying the video can be helpful here if appropriate. Ensure that students do not long enough to 1:1 count each fish. This will encourage some subitizing to occur and for students to visual what they saw.
Then, ask students to pair share with their neighbors for another 60 seconds.
Finally, allow students to individually share with the entire group. Be sure to write down these noticings and wonderings on the blackboard/whiteboard, chart paper, or some other way that is visible to all. This helps students to see the thinking of their classmates and ensures each student that their voice is acknowledged and appreciated. Adding student names or initials next to their notice/wonder is one way to acknowledge their participation and can motivate others to join in.
Some of the noticing and wondering may include:
- I notice that they are underwater
- I notice there are fish
- I notice that there are plants
- I wonder what kind of fish they are
- I wonder how many fish there are
After we have heard students and demonstrated that we value their voice, we can ask the estimation question. The students may have already made some guesses of the amount of fish in the previous section. The students will feel valued as you now ask them to make a true estimation.
How many fish are there?
Follow up that question with:
How could you convince someone that your estimation is correct?
We can now ask students to make an estimate (not a guess) as we want them to be as strategic as they can possibly be. This will force them to use spatial reasoning such as the amount of fish in a small group to help justify how many there are overall. Before collecting student estimates, students can share their estimates with neighboring students along with the reasonings.
Consider asking students to think about a number that would be “too low” and a number that would be “too high” before asking for their best estimate in order to help them come up with a more reasonable estimate.
While Students Are Estimating…
Monitor student thinking by circulating around the room and listening to the mathematical discourse. You may identify some students whose thinking would be valuable to share when the group’s estimates are collected.
Encourage students to make estimations rather than 1:1 counting each fish. The video may be paused for longer before it goes blank but we want students to make estimations based on their mathematical understanding and spatial sense.
Similar to collecting their noticings and wonderings, collect students’ range of estimates and/or best estimates along with initials or names. Having some students share justifications is an opportunity for rich, mathematical discourse.
Share the following estimation reveal video:
Celebrate students who estimated closest to 23 fish.
Crafting A Productive Struggle: Prompt
Since you have already taken some time to set the context for this problem and student curiosity is already sparked, we have them in a perfect spot to help push their thinking further and fuel sense making.
Share the following video with the prompt.
There were 23 fish swimming around but 8 of them swam off to explore. How many fish are there now? How do you know?
While Students Are Productively Struggling…
Monitor student thinking by circulating around the room and listening to the mathematical discourse. Educators are looking for students that are making their thinking visible so it can be displayed during consolidation. Select and sequence some of the student solution strategies and ask a student from the selected groups to share with the class from:
- most accessible to least accessible solution strategies and representations;
- most common misconceptions;
- most common/frequent to least common/frequent representations; or,
- choose another approach to selecting and sequencing student work.
The strategies you might see students use include:
- Direct Modeling and counting all
- Counting back
- Known addition fact
- Breaking up the 8 into 3 and 5 (Subtracting 3 from 23 to make 20 then subtracting 5 from 20 to make 15).
This checklist can be used for tracking formative assessment as students are working. The information collected can be used to form whole group, small group or one-to-one support models.
Direct modeling and counting all
Students will count the initial value, count the amount added then count all of the amounts.
Count 1: 23 blocks to start
Count 2: 8 of the blocks moved to the side
Count 3: Count the remaining 15 blocks.
Count backwards from various points
Watch for students that need to count forwards in order to start counting backwards. You may hear a student that is counting back from 8, begin their count “1, 2, 3, 4, 5, 6, 7, 8” then start counting back.
Count Back strategy
Students are holding one number in their head and continuing to count backwards and track their count
Students are finding the difference/space between in the numbers by adding up from the lower number to the higher number
Subtract to the decade
This skill demonstrates student understanding of place value. For example, 23 – 3, does the student understand that when subtracting the 3, we do not need to count back because we are just removing all of the singles/ones from 23. This skill needs to be done in one action, rather than counting back 22, 21, 20.
Subtract from the decade
This skill requires students to transfer their understanding of the “facts of ten” to other decade numbers. For example, the fact 10 – 6 helps us with 30 – 6, or 40 – 6.
Strategic and efficient strategy: Breaking Apart the Subtrahend
Students may also use benchmark numbers like decade numbers (10, 20, 30, etc) when subtracting. Breaking apart the subtrahend is an efficient strategy that involves strategically subtracting parts of the subtrahend.
Student Approach 1: Counting All with a Tool
I counted out 23 blocks for the 23 fish.
Then I counted 8 of the blocks and moved them to the side. I counted the rest of the blocks and there were 13. So there are 13 fish left.
Student Approach 2: Counting All with a Drawing
I drew 23 circles for the 23 fish. I crossed out 8 of the circles and counted the remaining amount of circles. There were 13 circles not crossed so there are 13 fish left.
Student Approach 3: Counting All with a Tool
I subtracted 3 from the 23 to get back to a friendlier number/decade number (20). Then I only had to subtract 5 more to get 15.
Show students the following reveal video:
Consolidate learning by facilitating a student discussion.
The goal of the consolation is to demonstrate the flexible way that 8 could have been decomposed in order to make subtraction easier. Students that were counting back often “get lost” as they have to travel over a decade number. As the numbers get higher or the count back becomes larger, counting is not always as reliable. There are too many pieces for a student to keep track of.
We want students to start feeling more comfortable with subtracting numbers in a way that makes sense to them rather than just counting.
During the discussion, encourage students to start by showing their work without an explanation. Classmates will use this time to understand the visual and make their own assumptions about the work in front of them. It is also an option to ask students “What do you think this group did to solve this question?”. This will engage students in the work. The group can clarify any misunderstandings.
After the reveal, check back to your estimations. The purpose here is to reflect on student reasoning, not on celebrating who was closest.
Were there some estimates in the right neighborhood?
What helped those students to get close?
Consider the strategies that the students used. Perhaps a review of counting back over the decade is necessary. The strategy of breaking up the subtrahend depends on students’ flexibility with decomposing numbers. It also relies on students applying the facts of ten to other decades, in this case subtracting from 20. Further review through number talks or games that work with the facts of ten and their subtraction pair may be helpful. For example, 6 + 4 helps with 10 – 6, etc.
Consolidation Prompt #1:
Why is it helpful to break apart/decompose 8 into 3 and 5 in this question, rather than 4 and 4?
Consolidation Prompt #2:
If you were trying to subtract 8 from 22, how should 8 be decomposed? What about 24 – 8?
We suggest collecting this reflection as an additional opportunity to engage in the formative assessment process to inform next steps for individual students as well as how the whole class will proceed.
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