## HOW MANY COOKIES? HOW MANY CRACKERS? [DAY 3]

### ADDING AND SUBTRACTING WITHIN 100

Introduction to using place value strategies to add and subtract within 100

## Intentionality & Unit Overview

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Students will determine how many cookies remain in the box of 54 crackers after a handful are removed.

## Intentionality…

In this task, students will determine how many cookies remain in the box of 54 crackers after a handful are removed.

Some of the big ideas that may emerge through this task include:

• The action of removal is represented by subtraction.
• Removal problems can be solved by addition or subtraction.
• Unitizing 10 single cookies into one group of 10 cookies
• Apply knowledge of decomposition within 10 to solve problems involving addition and subtraction of two digit numbers.
• Addition and subtraction problems within 100 can be solved using strategies along a developmental continuum of counting all, counting on/back, and using place value strategies.
• When students are using place value strategies to solve addition and subtraction problems within 100, they are using their understanding of tens and ones and decomposition to accurately, flexibly, and  efficiently solve problems. See the Student Approaches section in this guide for examples.

Before starting this unit, students should be familiar with:

• Fluency within 10.
• Solving Result Unknown Take From story problems (eg: 27 – 5 = ?)
• Difference Unknown Comparison story problems. (eg: 25 – ? = 7)
• Understanding subtraction as an unknown-addend problem (eg: 7 + ? = 25)
• Understanding that the two digits of a two-digit number represent amounts of tens and ones. (in 25, the 2 has a value of 20 and the 5 has a value of 5)
• Composing and decomposing numbers within 10.
• Composing and decomposing two digit numbers. (25 can be decomposed into 20 and 5 or into 15 and 10)
• Subtract multiples of 10 in the range 10 – 90 from multiples of 10 in the range 10 – 90

## What Do You Notice? What Do You Wonder?

Show students this video:

Then, ask students:

What do you notice?

What do you wonder?

Give students 60 seconds (or more) to do a rapid write on a piece of paper or silent individual think time.

Replaying the video can be helpful here if appropriate.

Then, ask students to pair share with their neighbours for another 60 seconds.

Finally, allow students to individually share with the entire group. Be sure to record 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 noticings and wonderings that may come up:

• I noticed a box of crackers. They’re called Triscuits I think.
• This box is original flavor. There are also other flavors of Triscuits, like black pepper!
• I noticed the box was unopened at the beginning of the video.
• I noticed the person took a lot of crackers.
• I noticed that there were less cookies in the box after the person took some out because the level of crackers was lower.
• I wonder how many crackers were taken out of the box.
• I wonder how many crackers were left in the box.
• I wonder how many crackers were in the box before it was opened.
• I wonder if we will get to eat some Triscuit crackers.
• And many more.

At this point, you can answer any noticings and wonderings that you can cross off the list right away. Things like “I wonder whether we will get any Triscuit crackers?” can be addressed right away to show students that we are indeed listening to their contributions and that we value student voice.

## Estimation: Prompt

After we have heard students and demonstrated that we value their voice, we can land on today’s question. Acknowledging that the question came from students underscores their identities as thinking mathematicians and Math Practice 1, Making Sense of Problems.

How many crackers are left in the box?

Follow up that question with:

How could you convince someone that the quantity you come up with 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 comparing the amount of crackers in the box at the beginning and end of the video or possibly some estimation based on the crackers in the top layer. Before collecting student estimates,  students can share their estimates in small groups along with a justification for the reasonableness of their prediction.

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 use precise mathematical language (including greater than, less than, more, less, half, higher than, lower than, layer) to articulate their defense of their estimate.

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.

At the end of this task, during The Reveal, revisit the estimates to see how accurate students were and to reflect on the justifications they made.

## Crafting A Productive Struggle: Prompt

Share the following image and prompt with students:

One handful of 27 cookies was taken out of the box.

How many cookies remain in the box?

Since we’ve already invested 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.

Ask the students what information they will need and how they will use it in order to find out how many cookies remain in the box. Here we are asking students to envision a solution path. By considering what information will be needed and how it can be used to solve the problem, students are being asked to make sense of the problem and reason abstractly and quantitatively (Math Practices 1 and 2). By not simply providing this information, we are inducing students to engage more deeply with the problem and think about all of its parts, known and unknown, before they begin crunching the numbers.

## While Students Are Productively Struggling…

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## Student Approach #1: Counting All with a Tool

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## Student Approach #2: Place Value Thinking with a Tool

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## Student Approach #3: An Open Number Line Diagram for Counting Back/Up by Ones

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## Student Approach #4: An Open Number Line Diagram to Show Place Value Strategies for Taking Away

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## Student Approach #5: Base 10 Block Drawing to Represent Place Value Thinking

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## Student Approach #6: A Bar Model Diagram With An Open Number Line Diagram Using Place Value Strategies to Find the Difference

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## Reveal

Show students different variations of the reveal videos below.

Here is a screenshot of the final frame of reveal video:

## Consolidation: Making Connections During Classroom Discourse

Consolidate learning by facilitating a student discussion.

Students present their work one by one or group by group according to the sequence that the teacher determined during the Sense Making portion of the lesson. During student presentations, the teacher can promote discussion by facilitating commenting and question asking between classmates. The twin goals of discussion are to get students to engage one another and to generalize that place value strategies are an effective for adding and subtracting within 100 when starting from a number that is not a multiple of ten.

To promote student engagement and facilitate discussion, teachers can:

• Create a psychologically safe space to promote intellectual risk taking, new ideas, and broader participation during discussions,
• Provide guided practice in advance regarding how to present work, appropriate voice volume, how to listen, how to agree/disagree respectfully, and other behaviors  and practices to promote discussion.
• Make student work easily visible by using a document camera or other tool so that the image is large enough or all students to clearly see. so that the image is large.
• When an important point is made by a student, prompt other students to repeat/rephrase it. This encourages listening to one another and allows for an important point to be stated 2 – 3 times so that more people can hear it.
• Instead of answering questions asked by students, redirecting the question back to the class. This promotes student to student dialogue.
• While it is important to keep the consolidation purpose in mind, also being open to other important thoughts and ideas that arise. Honoring these rich insights builds excitement and students’ vision of themselves as mathematicians making discoveries.

To promote greater understanding of place value strategies, teachers can familiarize themselves with the following Place Value Strategies. As students explain their thinking during the class discussion, listen for and highlight their explanations and representations that demonstrate place value strategies.

Look for and highlight examples of students using their understanding of tens and ones to help them flexibly solve the problem in a way that is efficient and accurate. This will resemble the work samples in Student Approaches  4, 5, or 6. In these approaches, students are decomposing the amount being taken away into tens and ones. Students can show this understanding using Base 10 block drawings, open number lines, or just equations.

It is not necessary for students in your class to use each and every one of these approaches. Rather, utilize the examples that your students come up with as the examples of place value thinking that can be discussed by the whole group. By guiding the discussion in this direction with classroom based examples, teachers will help their class to construct their own understandings of adding and subtracting using place value strategies.

• Adding or Subtracting One Number in Parts (see Student Approach #4 for diagram)
• Students who use this strategy for 54 – 26 will break the 26 into two parts by place, 20 and 6. Then they will subtract each part separately. Some students will first subtract the tens place number (54 – 20 = 34). Often, they will then decompose the 6 ones into 4 ones and 2 ones. They will first subtract 4 ones to get to the benchmark number (34 – 4 = 30). Then they subtract the remaining 2 ones to get the final answer (30 – 2 = 28)

Other students will decompose the 26 in the same way, but subtract in a different order. These students will first subtract 4 ones to get to the benchmark number (54 – 5 = 50). Then they subtract the 2 tens (50 – 20 = 30). Finally they subtract the remaining 2 ones to get the final answer (30 – 2 = 28).

• Adding Up (see Student Approach #6 for diagram)
• Some students may be aware of the inverse relationship of addition and subtraction. These students know that if 10 – 3 = ?, then 3 = ? = 10. These students may find it easier to add up than subtract back. To solve 54 – 26 = ?, then rewrite it as 26 + ? = 54. They will then use their understanding of place value to add on tens and ones separately. Here are two ways students might add up.

• Making an Easier Problem
• Some students like to take a problem such as 54 – 26 = ? by adjusting each number by subtracting 4 to make the easier 50 – 22 = ? that they can solve mentally. These students know that it is easier to subtract from multiples of tens. They also know that when each number in a subtraction problem is adjusted in the same way, the difference remains constant.
• Ask questions such as,
• How did this student take away 26 from 54?
• How is taking away 26 ones different from taking away 20 and then taking away 6?
• Why did this student first take away 20, then take away 4, and then take away 2. Why not just take away 20 and 6?
• Why not just take it all away at once? Why not just take away 26 ones?
• How did the student make the problem easier by showing 54 as 5 sticks of ten and 4 singles?
• Why did the student add 4 to 26 first? How did that help them to solve the problem?
• How did this student use their understanding of place value to solve the problem?
• How did the bar model diagram help some students to understand what was happening in the situation?

Have students present their strategies and reasoning for how to represent the number of cookies remaining in the box. Ask them to convince you and their classmates that their answer is correct by sharing their mathematical model. It can be very effective to project the student work with a document camera or project a photo of the student work from your computer so that everyone can easily see what the presenting student is referring to. When the other students can easily see the work sample, they remain more engaged and understand more deeply.

Discuss student strategies and elicit the questions and comments of other students during your consolidation. The goal is to help them to  build off of their current prior knowledge and understanding. We want students to be socially constructing their understanding in a way that makes sense for them.

Here you are listening to this discourse with an ear for what is important for your students’ emerging understanding and application of place value strategies for addition and subtraction problems.

## Reflect and Consolidation Prompts

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? How did they think about it?

Without a lot of information, mathematicians have to make assumptions all of the time. What assumptions made some estimates way too low or way too high?

Now that we have figured out how many cookies were left in the box, what new questions does this raise in your mind? You may choose to explore one or more of these student generated questions on another day or as homework.

## Consolidation Prompt #1:

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## Consolidation Prompt #2:

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## Consolidation Prompt #3:

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## Consolidation Prompt #4:

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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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