Reflect and Consolidation Prompts
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Length of Unit: 5 Days
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Students will explore subtraction in a context encouraging students to use the “think addition” strategy
In this task, students will engage in a subtraction context and lead towards the “think addition” strategy.
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
- Subtraction can be used in either take away, comparison, or missing addend situations. Missing addend is explored in this task.
- Models can be used to connect concrete to abstract
Before starting this unit, students should be familiar with:
- Facts of 10 ( e.g., 6 + 4 = 10, 10 – 4 = 6)
- Flexibility when decomposing numbers (e.g., 13 can be decomposed into 10 and 3, but also 9 and 4, 8 and 5, etc)
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 there is a book shelf
- I notice there books on the shelf
- I notice different colours of books on the shelf
- I notice that the books are piled in different ways
- I wonder what kind of books they are
- I wonder how many books are on the shelf
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 books are on the shelf?
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 books in one of the groupings to help justify how many there are overall. Before collecting student estimates, students can share their estimates with neighbouring students along with the reasoning.
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 book. 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.
Brian has read 19 books since the beginning of school. He would like to read 27 books. How many more books does Brian need to read? 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 model and counting all
- Counting back
- Adding up with a tool
- Adding up with a drawing
- Adding up on an open number line
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 forwards from various points
Watch for students that need to count forwards starting from 1. You may hear a student that is counting on from 8, whisper their count “1, 2, 3, 4, 5, 6, 7, 8” then start counting on while tracking out loud.
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
Add to the decade
This skill requires students to transfer their understanding of the “facts of ten” to other decade numbers. For example, the fact 6 + 4 helps us with 26 + 4 or 36 + 4.
Add on from a decade number
This skill demonstrates student understanding of place value. For example, 20 + 3, does the student understand that when adding the 3, we do not need to count on by ones. Instead, this skill needs to be done in one action, rather than counting on 21, 22, 23.
Strategic and efficient strategy: Think Addition
The Think Addition strategy demonstrates the understanding of difference. Students are using addition to find the difference between two numbers. This strategy is efficient when the minuend and subtrahend are close together. For example, 34 – 28. We can add on 2 to get 30 then add on 4 more. Therefore, 34 – 28 = 6.
Student Approach 1: Counting All with a Tool
I counted out 27 blocks for the 27 books that he wants to read.
Then I counted 19 blocks and put them to the side.
I counted that there was 8 blocks left so he needs to read 8 more books
Student Approach 2: Counting All with a Drawing
I drew 27 circles for the 27 books. I crossed out 19 of the circles and counted the remaining amount of circles. There were 8 circles not crossed so he needs to read 8 more books.
Student Approach 3: Adding up with a tool to track
I started with 19 and then I counted on my fingers. Each time I said a number, I put up a finger. 20, 21, 22, 23, 24, 25, 26, 27. I have 8 fingers up so he needs to read 8 more books.
Student Approach 4: Adding up with a drawing to track
I started with 19 and then I counted on with a tally. Each time I said a number, I put up a tally. 20, 21, 22, 23, 24, 25, 26, 27. I have 8 tallies so he needs to read 8 more books.
Student Approach 5: Adding up with a numberline to track
I drew a numberline. I put 19 on the left side. I added 1 to get to 20 then I added 7 more to get to 27. Since I added 1 and 7, that makes 8. So he needs to read 8 more books.
RevealShow students the following reveal video: Facilitator Note: The open number line (or empty number line) is an incredible tool for students to use to demonstrate their thinking. It allows flexibility from the traditional number line because students do not have to count the “ticks’ or “spaces”, instead they may jot their thinking anywhere on the line. Notice that the arrows are going to the right or up the number line which demonstrates an increase in value.
Consolidate learning by facilitating a student discussion.
The goal of the consolation is to demonstrate the flexible way that we can add up to find the difference between two values. The “join with the change unknown” problem type will naturally elicit the Think Addition strategy. This problem type can be illustrated through the equation:
19 + ? = 27
Although it is not a subtraction question, this problem type is a good introduction to the Think Addition strategy to subtract. Students are often more familiar with addition so opening the gateway that we can use an addition technique when subtracting will benefit students.
This problem may become more obvious by modeling the thinking on an open number line. It will allow students the opportunity to see that we are trying to figure out the space between 19 and 27 or the difference between them. It is very important for students to understand where the answer is. Often when students use a numberline, the answer is one of the numbers on the number line. In this case, the answer lies in the number of “jumps” or the
Since the student added 1 then added 7 more, the difference between the two values is 1 + 7= 8.
Also during the consolidation, consider highlighting how we can add up in an efficient way. 19 was purposefully chosen due to its proximity to 20. The idea is that students can add 1 to get to a more friendly decade number. Then they would just need to add 7. Ideally, the later step can be done without direct counting.
Students that were counting back often “get lost” as they have to travel over a decade. 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.
Think addition is an effective way to subtract. Students may use “think addition” for their more basic facts and also solve subtraction problems as numbers get bigger. Specific problem types such as join with the change unknown or missing part problems may encourage the use of think addition strategies for subtraction. It may also be helpful to ask the difference between two numbers to practice this strategy.
Consider the strategies that the students used. Perhaps a review of adding up over the decade is necessary. To efficiently use this strategy, we want students to be flexible 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 decade partners may be helpful. For example, knowing 7 + 3 helps with 17 + 3, 27 + 3, etc.
Consolidation Prompt #1:
How does adding help us to subtract?
Consolidation Prompt #2:
Brian has read 26 books so far this year, how many more books will he need to read if he wants to read 38 books?
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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