NYS COMMON CORE MATHEMATICS CURRICULUM

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Lesson 16: Subtract from multiples of 100 and from numbers with zero in the tens place.

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Lesson 16NYS COMMON CORE MATHEMATICS CURRICULUM 2•5

Crafting Lessons Through Analysis of Teaching Practice

(K‐2)

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Welcome to Crafting Lessons Through Analysis of Teaching Practice

Today’s professional development experience is intended to support the

growth of a culture of continuous learning, one characterized by studious and

collaborative preparation, dynamic teaching, on‐going and varied assessment,

and multi‐faceted analysis of data. It is a cycle of “Prepare. Teach. Assess.

Analyze. P.T.A.A.” Collaboration might be involved in each part of the

cycle, though its objective is also independent efficacy since so much of

preparation, teaching, assessment, and analysis does take place privately.

This cycle describes in a general way the daily routine of a strong teacher who prepares a lesson (based on

analysis of student work), teaches, assesses, analyzes student work, and begins to prepare once more.

This process might also take place within a one‐minute segment of instruction during, for example, a Rapid

White Board Exchange, e.g. the teacher sees that seven students have an incorrect response, analyzes the

root of the error on‐the‐spot, makes a decision to address it instantly, mentally prepares the next question,

and delivers it to the students. The cycle may occur within a two‐day time frame, e.g. the teacher sees the

same response as above but during analysis instead decides to prepare an instructional sequence and teach it

the following day. It may occur over a period of months, e.g. a teacher sees in a formal state test that 35% of

students are weak in a particular standard, and in collaborative analysis with grade level peers, prepares a

revised sequence of instruction to address the standard.

There are clearly many ways to enhance the craft of

teaching mathematics. Power tools include analysis

of student work, study and preparation using a

strong curriculum, deliberate practice (i.e. practice

which includes cycles of giving and using feedback),

and a love of student learning. Perhaps the most

important ingredient, however, is a willingness to

participate in candid, on‐going professional

conversations, to give and respond to critique, to

experience teaching is a deeply engaging profession

within which there is always more to learn.

Today’s agenda will use video of pre‐and post‐

conferences and the corresponding lessons to allow

us to peek into the experience of two thoughtful,

well‐prepared educators. The intent of the day is to

further engender a culture of continuous learning

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throughout all classrooms, schools, and districts by sharing two different processes for professional growth

with two distinct purposes. These should be seen as springboards for discerning what is best for a particular

educational setting rather than as prescriptive.

1. The teacher is the primary learner. A strong body of research suggests that an observation is not

always necessary for coaching to significantly improve the craft of teaching. What is important is

effective reflection and practice. Observations are logistically challenging and time consuming where

pre and post conferences can take place online, in professional development sessions, or before and

after school. Furthermore, when coaching mathematics, a peer may be the best partner in that he or

she has great interest in the same body of mathematical knowledge and may be more apt to

penetrate incisively into the content and how to teach it.

2. The observer is the primary learner: The teaching profession is full of mature educators whose

practice benefits the entire learning community. Being able to observe a colleague or “expert”

teacher is an honor. The process suggested today allows the observer to analyze effective practices

and generalize those to his or her own classroom immediately. The goal is immediate empowerment

through the observation.

The following four‐step process will be repeated twice throughout the day. Step 1: Prepare to watch the videos by studying the lesson being discussed in the conferences and taught in

the classroom. The first half of the day will focus on Grade 2 Module 5 Lesson 16. The second half of the day will focus on Kindergarten Module 4 Lesson 13.

Step 2: The teacher is the primary learner; the “coach” is most likely a peer or instructional leader. Watch and

analyze a pre‐and post conference without an observation including deliberate practice.

Step 3: The observer is the primary learner; the “observed” is most likely a well‐prepared teacher. Watch and

analyze a lesson in two parts, analyze the exit tickets. Uncover specific strengths and potential

weaknesses. Deliberately practice a component addressing a student need.

Step 4: Generalize best practices to a next lesson. In the afternoon session, there is the option to spend

planning time with school colleagues to consider how to return to sites with these tools.

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Crafting Lessons Through Analysis of Teaching Practice: Detailed Directions for Parts 1 ‐ 4 Part 1 Prepare by studying. 20 minutes

A: Remembering the preparation process as detailed in Appendix D: Preparing to Teach a Lesson from A Story of Units, complete the Problem Set to be done by students.

B: Study and write notes on the complexities of each problem and read through the Student Debrief questions.

C: Anticipate where students might struggle.

D: Briefly study the entire lesson, how does it lead to students’ independent success. What customizations might you consider?

Part 2 The teacher is the primary learner.

Silently read through Appendix A’s Part 1: Pre‐Instruction Preparation with Deliberate Practice and Part 2: Post‐Instruction Reflection with Deliberate Practice. Share thoughts within a partner or triad.

Prepare through collaboration and practice. 20 minutes

A: Watch a pre‐conference while making notes on Appendix A Part 1: Pre‐Instruction Preparation with Deliberate Practice.

B: After viewing, use your notes to review what took place in the conference with a partner.

C: Consider with a partner the various ways in which the conference may have advanced the teacher’s preparation and practice. Refer back to the curriculum to support your analysis.

Reflect back and deliberately practice. 28 minutes

A: Watch a post‐conference and make notes on Appendix A Part 2 Post‐Instruction Reflection with Deliberate Practice.

B: After viewing, reconstruct what happened in the conference with a partner.

C: The teacher’s peer made an effort to get into the specifics of the mathematics and how precisely it was taught. What evidence is there in the post conference of that effort? How did the dialogue’s move to precision and specificity about the mathematics change the conference?

D: Read through Appendix F: Deliberate Practice.

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E: Use the deliberate practice protocol to practice the same component as the teacher in pairs or triads.

Reflect on pre and post conferencing without an observation within a triad. How would this process advance your own teaching practice or that of your colleagues?

Part 3

The observer is the primary learner.

Silently read through Appendix B: Part 1: Interview focused on the Teacher’s Preparation and Part 2: Documentation of the Observer’s Learning.

Discuss the ways in which Appendix B is distinct from Appendix A within a partner or triad. Watch a short video of the teacher modeling from Part 1(c): Interview focused on the Teacher’s Preparation.

Observe part 1 of a Eureka lesson. (30 minutes)

A: Take on the role of a sports reporter using Appendix C: Observation Recording Sheet.

B: Watch the video while taking notes on the Fluency and Application Problem components.

C: Review your observation notes from beginning to end with a colleague recreating the action of the lesson from the notes. (Request replays if desired.) Also reflect on the experience of taking the notes.

D: Refer back to the Fluency and Application Problem of the curriculum and make notes directly on the curriculum about the customizations made by the teacher. Compare the actual classroom experience to what is in the curriculum and to what the teacher proposed in the pre‐conference.

E: Silently write about student learning that took place and the specific teacher actions and words that made it successful. Write down what you would like to incorporate into your own practice, with a focus on the mathematics rather than management. Share out writing in small groups.

Optional: Deliberately practice something the teacher wants to incorporate into his or her own practice from the fluency or application problem.

Observe part 2 of a Eureka lesson. (22 minutes)

A: Watch and take notes on the Concept Development, Problem Set, and Debrief.

B: Review your observation notes from beginning to end with a colleague recreating the action of the lesson from the notes. (Request replays if desired.) Also reflect on the experience of taking the notes.

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C: Refer back to the Concept Development, Problem Set, and Debrief of the curriculum and make notes directly on the curriculum about the customizations. Compare the actual classroom experience to what is in the curriculum and to what the teacher proposed in the pre‐conference.

Analyze the Exit Tickets: Use Appendix B: Part 2: Documentation of the Observer’s Learning. (30 minutes)

A: Distribute the Exit Tickets equally. In order to get an overview of the work, pass the set forward every 30 seconds until the set has moved around the table. Upon completion of the rotation, keep those you end up with.

B: Analyze your student work within a triad. Record your response to Appendix B: Documentation of the Observer’s Learning Part 2a. Think back to the lesson using your notes and reference the curriculum to detail what moments of instruction may have contributed to the strength exhibited in the exit tickets?

C: What specific errors are evidenced in each sample? Record your response to Appendix B: Documentation of the Observer’s Learning Part 2b. Again, think back to the lesson using both your notes and reference the curriculum to detail what specific moments of instruction may have contributed to the weaknesses exhibited in the exit tickets?

D: Record your response to Appendix B: Documentation of the Observer’s Learning Part 2c.

E: Culminate Part 3 by deliberate practice as described in Appendix B: Documentation of the Observer’s Learning Part 2: Closing Action.

Part 4

Prepare a next lesson..

A: Remembering the preparation process as detailed in Appendix D: Preparing to Teach a Lesson from A Story of Units, complete and analyze the Problem Set to be done by students.

B: What do you know from the prior lesson that may mean customizations and short‐term interventions? Tweak or replace a fluency or any lesson component to meet the needs of the students in the observed teacher’s classroom or in your own classroom.

C: Deliberately practice a component that reflects a customization.

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Lesson 16

Objective: Subtract from multiples of 100 and from numbers with zero in the tens place.

Suggested Lesson Structure

Application Problem (8 minutes)Fluency Practice (12 minutes)

Concept Development (30 minutes)

Student Debrief (10 minutes)

Total Time (60 minutes)

Application Problem (8 minutes)

Will read 15 more pages than Marcy. Marcy read 38 pages. The book is 82 pages long.

a. How many pages did Will read?

b. How many more pages does Will need to read to finish the book?

Note: This two‐step problem is intended for guided practice as students gain familiarity with the compare bigger unknown problem type. Tape diagrams enable students to make sense of the relationships between the numbers and effectively choose an operation to both represent the situation and solve.

Fluency Practice (12 minutes)

Sprint: Subtraction from Teens 2.OA.2 (8 minutes)

Coin Drop 2.OA.2 (2 minutes)

More and Less 2.NBT.5 (2 minutes)

Sprint:SubtractionfromTeens(8minutes)

Materials: (S) Subtraction from Teens Sprint

Note: Students practice subtraction from teens to gain mastery of the sums and differences within 20.

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Coin Drop (2 minutes)

Materials: (T) 10 dimes, 10 pennies, can

Note:Inthisactivity,studentspracticeaddingandsubtractingonesandtensusingcoinsinpreparationforModule7.

T: (Hold up a penny.) Name my coin.

S: A penny.

T: How much is it worth?

S: 1 cent.

T: Listen carefully as I drop coins in my can. Count along in your minds.

Drop in some pennies and ask how much money is in the can. Take out some pennies and show them. Ask how much money is still in the can. Continue adding and subtracting pennies for a minute or so. Then, repeat the activity with dimes, and then with dimes and pennies.

More and Less (2 minutes)

Materials: (T) 10 dimes, 10 pennies

Note: In this activity, students practice adding and subtracting ones and tens using coins.

T: Let’s count by tens. (Move dimes to the side while counting.)

S: 10, 20, 30, 40, 50, 60.

T: How many dimes are shown?

S: 6 dimes.

T: What is the value of 6 dimes?

S: 60 cents.

T: What is 5 cents more? (Move 5 pennies.)

S: 65 cents.

T: Give the number sentence.

S: 60 cents + 5 cents = 65 cents.

T: What is 10 cents less? (Move 1 dime.)

S: 55 cents.

T: Give the number sentence.

S: 65 cents – 10 cents = 55 cents.

Continue to repeat this line of questioning with a similar sequence of numbers.

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NOTESONMULTIPLEMEANSOFENGAGEMENT:

It is easy to lose students when

subtraction involves zeros in the

minuend. Check frequently for

understanding by establishing a quiet,

non‐verbal signal (e.g., thumbs‐up) that

students can use to indicate whether

they are following.


Materials: (S) Personal white board, math journal or paper

Note: This Concept Development extends student learning from Module 4’s Lessons 27 and 28.

Problem 1: 402 ‐ 231

T: (Write 402 – 231 horizontally.) Let’s solve this problem using a math drawing and the algorithm.

T: Rewrite the problem with me. (Write the problem vertically as students do the same.)

T: Which number is the whole?

S: 402.

T: Let’s make a chip model to show the whole. I’ll draw it on the board while you draw yours. Whisper‐count as you draw your chips.

S: (Whisper‐count and draw.) 100, 200, 300, 400, 401, 402.

T: Let’s draw our magnifying glass and get ready to subtract! (Draw a circle around 402 as students do the same.)

T: Look at your chip model. Are we ready to subtract the ones?

S: Yes!

T: Moving on, let’s look at the tens place. I don’t see any tens in the tens place on the model. Point to the digit that represents this in vertical form.

S: (Point to the 0.)

T: The zero holds the tens place open and tells us the number is 402.

T: Without that 0, what number would we read? (Write 42.)

S: 42.

T: (Erase 42.) Yes, so we must be precise when writing and representing numbers.

T: Where can we get some tens so we can subtract 3 tens?

S: The hundreds place. Decompose a hundred. Rename 1 hundred as 10 tens.

T: Let’s show that on our chip models. Count with me as we rename 1 hundred as 10 tens. (Cross off 1 hundred, draw an arrow to the tens place, and draw 10 dots, or tens.)

S: (Draw and count.) 10, 20, 30, 40, 50, 60, 70, 80, 90, 100.

T: Show that in vertical form. As I do the same, check your work with mine. (Cross off 4 and write 3 above the hundreds place, and then cross off 0 and write 10 above the tens place. Students do the same.)

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NOTESONMULTIPLEMEANSOFREPRESENTATION:

While some students prefer to rename

800 in one step (e.g. 7 hundreds, 9

tens, 10 ones), others may need the

intermediate step of renaming a

hundred as 10 tens before renaming a

ten as 10 ones. Allow students to use

place value disks or chips to model the

decomposition in two steps.

T: Are we ready to subtract now in the tens place?

S: Yes!

T: Let’s look at the hundreds place. Are we ready to subtract in the hundreds?

S: Yes!

T: Then, we’re ready to subtract! (Allow students time to complete the subtraction.)

T: Talk with your partner. Take turns sharing how you showed the subtraction on your chip model and using the algorithm. (Allow students time to share.)

T: Read the complete number sentence.

S: 402 – 231 = 171.

T: How can we prove that our answer is correct?

S: Add the parts to see if they equal the whole.

T: Yes! Please check your answer by drawing a chip model to add the two parts. If you are correct, write the number bond for this problem.

Circulate to check for understanding and support students who struggle. Project student work or call students to the board to show the chip model, vertical form, and number bond. Encourage students to use place value language to explain their work.

Problem 2: 800 – 463

Follow the above procedure to guide students as they write 800 – 463 vertically and model it.

T: Talk with your partner. What do you notice about the whole, and what will we need to do?

S: This time, there are no tens and no ones. We need to unbundle a hundred to make tens and ones. We need to rename 1 hundred as 9 tens 10 ones.

T: Let’s do that. Count aloud as you rename 1 hundred as 9 tens 10 ones. (Cross off 1 hundred, draw an arrow to the tens place, and draw 9 tens as students do the same.)

S: 10, 20, 30, 40, 50, 60, 70, 80, 90.

T: Stop! Now, count on as you draw the ones. (Draw 10 ones as students do the same.)

S: 91, 92, 93, 94, 95, 96, 97, 98, 99, 100.

T: So, 1 hundred is the same as 9 tens 10 ones. True?

S: True!

Continue using the procedure from Problem 1 to guide students as they complete the subtraction on both the model and in vertical form, share their work, and verify their solution to Problem 2 using addition.

Repeat the process for 908 – 120, 705 – 36, 600 – 316, and 500 – 327. Continue to support students who struggle, but as they demonstrate proficiency, instruct them to work on the Problem Set independently.

MP.8

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Problem Set (10 minutes)

Students should do their personal best to complete the Problem Set within the allotted 10 minutes. For some classes, it may be appropriate to modify the assignment by specifying which problems they work on first. Some problems do not specify a method for solving. Students should solve these problems using the RDW approach used for Application Problems.


Lesson Objective: Subtract from multiples of 100 and from numbers with zero in the tens place.

The Student Debrief is intended to invite reflection and active processing of the total lesson experience.

Invite students to review their solutions for the Problem Set. They should check work by comparing answers with a partner before going over answers as a class. Look for misconceptions or misunderstandings that can be addressed in the Debrief. Guide students in a conversation to debrief the Problem Set and process the lesson.

Any combination of the questions below may be used to lead the discussion.

For Problem 1(a), 304 – 53, explain how you solved this problem. How could you have solved it mentally?

For Problem 1(b), 406 – 187, what did you draw on your place value chart? How did you unbundle 400? Did you do it in one or two steps?

For Problem 1(c), 501 – 316, explain to your partner how you changed a larger unit to make more ones when there were no tens?

For Problem 1(d), what are two different ways you can unbundle 700? How can you do it in one step? How could you have solved this problem mentally?

Think like a detective: When you are subtracting three‐digit numbers, when do you choose to unbundle a hundred? When do you choose to solve mentally? What clues in the numbers help you choose a solution strategy?

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Exit Ticket (3 minutes)

After the Student Debrief, instruct students to complete the Exit Ticket. A review of their work will help with assessing students’ understanding of the concepts that were presented in today’s lesson and planning more effectively for future lessons. The questions may be read aloud to the students.

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Lesson 16 SprintNYS COMMON CORE MATHEMATICS CURRICULUM 2•5

Subtraction from Teens

1. 11–10= 23. 19 – 9 =

2. 12–10= 24. 15–6=

3. 13–10= 25. 15–7=

4. 19–10= 26. 15–9=

5. 11–1= 27. 20–10=

6. 12–2= 28. 14–5=

7. 13–3= 29. 14–6=

8. 17–7= 30. 14–7=

9. 11–2= 31. 14–9=

10. 11–3= 32. 15–5=

11. 11–4= 33. 17–8=

12. 11–8= 34. 17–9=

13. 18–8= 35. 18–8=

14. 13–4= 36. 16–7=

15. 13–5= 37. 16–8=

16. 13–6= 38. 16–9=

17. 13–8= 39. 17–10=

18. 16–6= 40. 12–8=

19. 12–3= 41. 18–9=

20. 12–4= 42. 11–9=

21. 12–5= 43. 15–8=

22. 12–9= 44. 13–7=

A NumberCorrect:_______

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Subtraction from Teens

1. 11–1= 23. 16 – 6 =

2. 12–2= 24. 14–5=

3. 13–3= 25. 14–6=

4. 18–8= 26. 14–7=

5. 11–10= 27. 14–9=

6. 12–10= 28. 20–10=

7. 13–10= 29. 15–6=

8. 18–10= 30. 15–7=

9. 11–2= 31. 15–9=

10. 11–3= 32. 14–4=

11. 11–4= 33. 16–7=

12. 11–7= 34. 16–8=

13. 19–9= 35. 16–9=

14. 12–3= 36. 20–10=

15. 12–4= 37. 17–8=

16. 12–5= 38. 17–9=

17. 12–8= 39. 16–10=

18. 17–7= 40. 18–9=

19. 13–4= 41. 12–9=

20. 13–5= 42. 13–7=

21. 13–6= 43. 11–8=

22. 13–9= 44. 15–8=

B NumberCorrect:_______

Improvement:_______

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Lesson 16 Problem SetNYS COMMON CORE MATHEMATICS CURRICULUM 2•5

Name Date

1. Solve vertically or using mental math. Draw chips on the place value chart and unbundle, if needed. a. 304 53 = __________

b. 406 187 = __________

c. 501 316 = __________

hundreds tens ones

hundreds tens ones

hundreds tens ones

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d. 700 509 = __________

e. 900 626 = __________

2. Emily said that 400 247 is the same as 399 246. Write an explanation using pictures, numbers, or words to prove Emily is correct.

hundreds tens ones

hundreds tens ones

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Lesson 16 Exit TicketNYS COMMON CORE MATHEMATICS CURRICULUM 2•5

Name Date

Solve vertically or using mental math. Draw chips on the place value chart and unbundle, if needed. 1. 604 143 = __________

2. 700 568 = __________

hundreds tens ones

hundreds tens ones

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Lesson 17

Objective: Subtract from multiples of 100 and from numbers with zero in the tens place.


Application Problem (8 minutes)Fluency Practice (12 minutes)





Colleen put 27 fewer beads on her necklace than Jenny did. Colleen put on 46 beads. How many beads did Jenny put on her necklace?

If 16 beads fell off of Jenny’s necklace, how many beads are still on it?

Note: This compare bigger unknown problem is intended for guided practice. It is one of the four most difficult subtypes of word problems in that the word fewer suggests subtraction, which would be an incorrect operation.

This type of problem highlights the importance of drawing as a way to understand the relationship between numbers in the problem. The question mark indicates the unknown because students recognize that they are looking for the whole or a missing part.


Sprint:SubtractCrossingtheTen2.OA.2,2.NBT.5 (8minutes)

UsingtheNearestTentoSubtract2.NBT.5 (2minutes)

SubtractCommonUnits2.NBT.5,2.NBT.7 (2minutes)

Sprint: SubtractCrossingtheTen(8minutes)

Materials: (S) Subtract Crossing the Ten Sprint

Note: Students practice subtracting crossing the ten to prepare for the lesson and gain mastery of the sums and differences within 20.

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16–9=____/\106 10–9=1 1+6=____

Using the Nearest Ten to Subtract (2 minutes)

Note: Reviewing the Grade 1 skill of counting up and down to 10 to subtract gives students a mental strategy to subtract fluently with larger numbers.

T: (Write 16 – 9 on the board.) Raise your hand when you know the answer to 16 – 9.

S: 7.

T: (Write in the bond.) 10 – 9 is…?

S: 1.

T: 1 + 6 is...?

S: 7.

Continue with the following possible sequence: 13 – 8, 14 – 9, 15 – 7, 16 – 7, 13 – 9, 12 – 7, 22 – 7, 25 – 7, 25 – 8, 26 – 8, 27 – 8, 27 – 18, 37 – 8, 37 – 18, 35 – 18, 45 – 18, 47 – 19, and 48 – 29.

Subtract Common Units (2 minutes)

Note: Reviewing this mental math fluency activity prepares students for understanding the importance of the subtraction algorithm and place value.

T: (Project 55.) Say the number in unit form.

S: 5 tens 5 ones.

T: (Write 55 – 22 = ____.) Say the subtraction sentence, and answer in unit form.

S: 5 tens 5 ones – 2 tens 2 ones = 3 tens 3 ones.

T: Write the subtraction sentence on your boards.

Continue with the following possible sequence: 66 – 33, 77 – 44, 177 – 44, 88 – 33, and 188 – 33.


Materials: (S) Personal white board, math journal or paper

Note: This lesson is a continuation of Lesson 16. It extends the student learning from Module 4’s Lessons 27 and 28.

Problem 1: 300 – 195

T: (Write 300 – 195 horizontally.) Talk with your partner: What do you notice about these numbers?

S: 195 is close to 200, so it’s going to be easy to solve mentally. If you add 5 to 195, it will be 200, which makes it easy to subtract from 300. Yeah, making a hundred is just like making a ten.

T: Excellent reasoning! It’s a good math habit to think about the numbers and how they relate to each other before you decide on a strategy.

T: Take a moment to solve this problem using the strategy you like best. Be prepared to explain why it works.

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Although students work extensively

with the algorithm in these modules,

they will not be assessed on the

algorithm until Grade 4. When

students build a solid conceptual

understanding of subtraction at this

level, fluency and accuracy with vertical

form should increase greatly.

T: Who would like to explain their solution?

S: I thought about this as an addition problem starting with 195. I used arrow notation and wrote 195 + 5 is 200, and 200 + 100 is 300, so 100 + 5 is 105. I broke 300 into 100 and 200, and I subtracted 195 from 200. That left 100 and 5, which is 105. 305 – 200 is 105. I added 5 to 195 and 5 to 300. The difference stays the same, 105.

T: So, we could solve this mentally, use a simplifying strategy, or use vertical form. True?

S: True!

T: Keep that in mind, even as we use models and the algorithm to solve some more problems.

Problem 2: 500 – 213

T: (Write 500 – 213 horizontally.) Let’s set this problem up to solve using the chip model and the algorithm.

T: Rewrite the problem with me. (Write the problem vertically as students do the same.)

T: Let’s make a chip model to show the whole. I’ll draw it on the board while you draw yours. Whisper‐count as you draw your chips.

S: (Whisper‐count and draw.) 100, 200, 300, 400, 500.

T: Now, draw the magnifying glass. (Draw a circle around 500 as students do the same.)

T: Are we ready to subtract?

S: No.

T: Tell your partner what we need to do.

S: There are no tens and no ones, so we have to open up a hundred. We need to rename 1 hundred as 10 tens. Then, cross out 1 ten and rename it as 10 ones. We need to change 1 hundred for 9 tens 10 ones.

T: Let’s do that. Count aloud as you rename 1 hundred as 9 tens 10 ones. (Cross off 1 hundred, draw an arrow to the tens place, and draw 9 tens as students do the same.)

S: (Draw and count.) 10, 20, 30, 40, 50, 60, 70, 80, 90.

T: Count on as you draw the ones. (Draw 10 ones as students do the same.)

S: (Draw and count.) 91, 92, 93, 94, 95, 96, 97, 98, 99, 100.

T: Look at your model. What number is 9 tens?

S: 90.

T: And, 10 ones?

S: 10!

T: 90 + 10 =…?

S: 100!

T: So, we can rename 1 hundred as 9 tens 10 ones. True?

MP.8

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NOTESONMULTIPLEMEANSOFEXPRESSION:

For students who may be impatient to

use the algorithm alone, encourage

them to use the models to show their

thinking. Explain that, while the

student may know what she is doing,

others who look at her work are helped

by seeing the work written out.

Models are also helpful for checking

work.

S: True!

T: Now, we need to show these changes using the algorithm. As I record the changes on the numbers, check your work with mine. (Cross off 5 and write 4 above the hundreds place. Cross off 0 tens and write 9 above the tens place, then cross off 0 ones and write 10 above the ones place.)

T: Complete the subtraction, and then take turns sharing how your work on the chip model matches the steps in vertical form. (Allow students time to share.)

T: Read the complete number sentence.

S: 500 – 213 = 287.

T: How can we prove that our answer is correct?

S: Add the parts to see if they equal the whole.

T: Correct! Please check your answer by drawing a chip model to add the two parts. If you are correct, write the number bond for this problem.

Circulate to check for understanding and to support students who struggle. Project student work or call students to the board to show the chip model, vertical form, and number bond. Encourage students to use place value language to explain their work.

Problem 3: 603 – 487

Follow the above procedure to guide students as they write 603 – 487 vertically, model it, and solve. Remind them to be precise in lining up the digits and drawing their chips in neat 5‐groups. Encourage students to use place value language to explain each action that they take on their model and how it is represented using vertical form. Instruct students to check their work with addition and explain why this method works.

Repeat the process for 801 – 634 and 900 – 576 or move on to the Problem Set. Continue to support struggling students, but as they demonstrate proficiency, instruct them to work on the Problem Set independently.


Students should do their personal best to complete the Problem Set within the allotted 10 minutes. For some classes, it may be appropriate to modify the assignment by specifying which problems they work on first. Some problems do not specify a method for solving. Students should solve these problems using the RDW approach used for Application Problems.

MP.8

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Lesson Objective: Subtract from multiples of 100 and from numbers with zero in the tens place.




For Problem 1, which problems did you choose to solve mentally? Why? What made some easier than others?

For Problem 1(b), how did you rename 400 using your chips? Show me what 400 looks like after you have renamed the units.

Explain to your partner the steps you took to set up Problem 1(c). How was this similar to 1(b)?

Trey solved Problem 1(d), 800 – 606, by using place value. He started with 800 – 600 = 200. Then, he added 6 more, since one part was 606, so the answer was 206. What was his error?

For Problem 1(d), explain the meaning of the 9 in the tens place.

Exit Ticket (3 minutes)

After the Student Debrief, instruct students to complete the Exit Ticket. A review of their work will help with assessing students’ understanding of the concepts that were presented in today’s lesson and planning more effectively for future lessons. The questions may be read aloud to the students.

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Subtract Crossing the Ten

23. 10–1= 45. 21 – 6 =

24. 10–2= 46. 91–6=

25. 20–2= 47. 10–7=

26. 40–2= 48. 11–7=

27. 10–2= 49. 31–7=

28. 11–2= 50. 10–8=

29. 21–2= 51. 11–8=

30. 51–2= 52. 41–8=

31. 10–3= 53. 10–9=

32. 11–3= 54. 11–9=

33. 21–3= 55. 51–9=

34. 61–3= 56. 12–3=

35. 10–4= 57. 82–3=

36. 11–4= 58. 13–5=

37. 21–4= 59. 73–5=

38. 71–4= 60. 14–6=

39. 10–5= 61. 84–6=

40. 11–5= 62. 15–8=

41. 21–5= 63. 95–8=

42. 81–5= 64. 16–7=

43. 10–6= 65. 46–7=

44. 11–6= 66. 68–9=

A NumberCorrect:_______

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Subtract Crossing the Ten

23. 10–2= 45. 21 – 6 =

24. 20–2= 46. 41–6=

25. 30–2= 47. 10–7=

26. 50–2= 48. 11–7=

27. 10–2= 49. 51–7=

28. 11–2= 50. 10–8=

29. 21–2= 51. 11–8=

30. 61–2= 52. 61–8=

31. 10–3= 53. 10–9=

32. 11–3= 54. 11–9=

33. 21–3= 55. 31–9=

34. 71–3= 56. 12–3=

35. 10–4= 57. 92–3=

36. 11–4= 58. 13–5=

37. 21–4= 59. 43–5=

38. 81–4= 60. 14–6=

39. 10–5= 61. 64–6=

40. 11–5= 62. 15–8=

41. 21–5= 63. 85–8=

42. 91–5= 64. 16–7=

43. 10–6= 65. 76–7=

44. 11–6= 66. 58–9=

B NumberCorrect:_______

Improvement:_______

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hundreds tens ones

hundreds tens ones

hundreds tens ones

Name Date

3. Solve vertically or using mental math. Draw chips on the place value chart and unbundle, if needed. a. 200 113 = __________

b. 400 247 = __________

c. 700 428 = __________

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hundreds tens ones d. 800 606 = __________

e. 901 404 = __________

4. Solve 600 367. Then, check your work using addition.

Solution: Check:

hundreds tens ones

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K NYS COMMON CORE MATHEMATICS CURRICULUM K


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Name Date

Solve vertically or using mental math. Draw chips on the place value chart and unbundle, if needed. 1. 600 432 = __________

303 254 = __________

hundreds tens ones

hundreds tens ones

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Lesson 13

Objective: Represent decomposition and composition addition stories to 6 with drawings and equations with no unknown.








CountingtheSayTenWaywiththeRekenrekK.NBT.1 (3minutes)

DotCardsof6K.OA.3 (3minutes)

DrawMoretoMake6K.CC.2 (6minutes)

Counting the Say Ten Way with the Rekenrek (3 minutes)

Materials: (T) 20‐bead Rekenrek

Note: This activity is an extension of students’ previous work with the Rekenrek in anticipation of working with teen numbers.

Conduct activity as outlined in Lesson 5.

Dot Cards of 6 (3 minutes)

Materials: (T/S) Varied dot cards of 6 (Fluency Template 1)

Note: This activity deepens students’ knowledge of embedded numbers and develops part–whole thinking.

T: (Show card.) How many do you see?

S: 6.

T: How did you see them in two parts?

S: 5 on this side and 1 on that side. 2 down and 4 up. 3 up and 3 down.

Continue with other cards of 6. Distribute the cards to the students for partner sharing time.

Have them pass on the card at a signal and repeat with a new card.

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Scaffold the Application Problem for

English language learners by providing

sentence starters such as:

“ seals and seals is seals”

and “ + = .” This will support

their oral response to the problem’s

prompt.

Draw More to Make 6 (6 minutes)

Materials: (S) Make 6 (Fluency Template 2)

Note: This activity further develops students’ understanding of the decompositions of 6.

After giving clear instructions and completing the first few problems together, allow students time to work independently. Encourage them to do as many problems as they can within a given time frame. Go over the answers, and direct students to energetically shout, “Yes!” for each correct answer.


Materials: (S) Personal white board, 6 linking cubes

4 silly seals were splashing in the water. Show the silly seals with your linking cubes. 2 more silly seals came to splash. Show the new seals. How many silly seals are splashing in the water now?

Use your cubes and talk to your partner about the seals. Can you write about the silly seals in a number bond?

Note: Composition of the number 6 serves as an anticipatory set for today’s lesson, providing students an opportunity to work with concrete materials before moving into the pictorial stage.


Materials: (T) Magnetic shapes (optional) (S) Personal white board

Draw 2 squares and 3 circles on the board.

T: Noah loves to play with magnets on his refrigerator. He has these magnets. (Show the shapes on the board.) What does Noah have on his refrigerator?

S: He has some shape magnets. There are some circles and squares. There are 5 magnets in all.

T: Copy these shape magnets onto your personal white board. Hmmm. Remember when we practiced making number bonds from shape pictures? Can anyone help me make a number bond about our picture? (Allow students to guide

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NOTESONMULTIPLEMEANSOFACTIONANDEXPRESSION:

Ask students working above grade level

to explain why they think that

5 = 2 + 3 can also be written as

2 + 3 = 5 without changing the picture

of the 3 squares and 2 circles on the

board. Encourage them to use their

math words.

you in the creation of a number bond on the board, starting with the total and then designating the parts.)

T: I want to write about this in the special Math Way in a number sentence. (Write 5 = 3 + 2 under the number bond.)

T: What does this 5 tell us about?

S: The 5 tells how many shape magnets Noah has in all.

T: I showed the parts of our picture like this: 3 + 2. Where does the 3 come from?

S: The circles!

T: Where does the 2 come from?

S: The squares!

T: Yes, there are 5 shape magnets on Noah’s refrigerator. 2 are squares, and 3 are circles. 5 equals 2 and 3 together! Write the number sentence on your board. (Circulate to ensure understanding.)

T: Erase your boards. Noah’s friend gave him another circle magnet. I’ll draw it on the board. Copy all of the shapes onto your own board. What do you notice?

S: Now, we have 4 circles! We have 2 squares and 4 circles!

T: Count the sets of shapes, and write the numbers underneath your pictures. Let’s make a number bond about our new picture. (Allow students to guide you in creation of the number bond.) Who can tell me about the number sentence?

S: 4 shapes and 2 more shapes make 6 in all.

T: Yes! We can write it like this: 4 + 2 = 6.

Talk students through the referents once again as was done with 5 = 3 + 2 (a referent is the item that the numerals represent). Switch the order of the addends, and lead them through the referents again. Have them notice which number bond might better show the total amount made into parts and which might better show the parts put together to make the total.

Lead students in more examples with totals of 6 as time allows. Release them to work in pairs as they show independence.


Students should do their personal best to complete the Problem Set within the allotted time.

Note: The final question asks students to write a number sentence. This is not a mastered skill, but this question provides an opportunity for students who are ready to write number sentences independently.

MP.4

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Lesson Objective: Represent decomposition and composition addition stories to 6 with drawings and equations with no unknown.




Look at the corn stalk problem. Did your number bond match your neighbors’?

How did your drawings help you to make your number sentences?

How did the number bond help you to make your number sentence? How are number bonds and number sentences alike or similar?

Does it matter if you put the parts first or the whole first in a number sentence?

Did you notice anything special about the parts in the gecko problem? (The parts were the same.) How do you know which part shows the spotted geckos? Does it matter?

Think back to our silly seals. Can you think of a number sentence to describe them?

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Name Date Fill in the number bond and number sentences.

How many?

There are 6 corn stalks. 5 corn stalks are in the first row. 1 corn stalk is in the second.

There are 6 cars on the road. 2 cars are big, and 4 are small.

+ =

and

6

6

is

= +

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There are 6 monkeys. 4 monkeys are swinging on the tree, and 2 monkeys are taking a nap. Draw a picture to go with the story.

Create your own story and tell your partner. Have your partner draw a picture of your story and create a number sentence to go with the picture.

+ =

+ =

3 geckos have black spots, and 3 geckos have no spots. There are 6 geckos.

+ =

+ =

3

3

3

3

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Name Date

dot cards of 6

There are 6 animals. 4 are tigers, and 2 are lions.Fill in the number sentences and the number bond.

There 3 are monkeys and 3 elephants. All 6 animals are going into the circus tent. Fill in the number sentence and the number bond.

On the back of your paper, draw some animals. Make a number bond to match your picture.

is

+ =

and

and

+ =

is

is

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Lesson 16 HomeworkNYS COMMON CORE MATHEMATICS CURRICULUM 2•5


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Draw more to make 6.

make 6

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Appendix A: Part 1: Pre‐Instruction Preparation with Deliberate Practice Part 2: Post‐Instruction Reflection with Deliberate Practice

(No observation) Educator: _________________________________________ Date and time of pre‐conference: Date and time of post‐conference: Details of Lesson Grade level_____ Module _____ Topic _____

Lesson Objective: ______________________________

Part 1: Pre‐Instruction Preparation with Deliberate Practice

a) What observations did you make about the new complexities when you completed and then analyzed the Problem Set?

b) From prior student work including formal assessments, exit tickets, and day‐to‐day observation, what customizations did you make to the lesson when preparing?

c) Let’s deliberately practice __________________________. (Either the teacher or the peer, coach, or instructional leader with mathematical interest and knowledge might generate the lesson fragment to be practiced.)

Sample Menu of Areas for Professional Growth of the Teacher

Logic of a sequence of problem

Pacing of the delivery of instruction.

Student engagement with the mathematics.

MP.1 and/or MP.3

MP.4: Modeling with mathematics

MP. 5 Strategic use of tools

MP.6 Attend to precision

MP.7 and/or 8

Efficacy of use of student work or data in

preparation.

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Part 2: Post‐Instruction Reflection with Deliberate Practice

a) Based on evidence in the lesson and student work what worked well? Why? (Question to probe for specificity, e.g. ask for references to student work and responses, ask for specific details from the lesson and curriculum, inquire into different components of the lesson (e.g. the fluency, application problem, concept development, debrief) find out what happened with the component that was deliberately practiced in the pre conference.)

b) Based on evidence, what did not work as well? Why? (Probe for specificity, e.g. refer to student work, use specific details from the lesson and curriculum, verbal explanations the students offered during the lesson.)

c) What do we know your students need more work with as you go forward?

Closing Action: Deliberately practice a component of a lesson relevant to that learning gap. Suggestions:

To start a teaching sequence for deliberate practice, consider the simplest possible problem

and build up quickly from there to close the learning gap or reach the objective.

Collaborate as desired.

Release the teacher to work independently for 2 or 3 minutes to prepare.

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Begin the deliberate practice (See appendix F for a deliberate practice protocol.)

Appendix B: Part 1: Interview focused on the Teacher’s Preparation Part 2: Documentation of the Observer’s Learning

Observer: _________________________________________ Teacher sharing practice: _________________________________________ Date and time of pre‐conference: Date and time of observation: Lesson Grade level_____ Module _____ Topic _____

Lesson Objective: ______________________________________________________________________ _

Part 1: Interview focused on the Teacher’s Preparation

a) What observations did you make about the new complexities when you completed and then analyzed the Problem Set?

b) From prior student work including formal assessments, exit tickets, and day‐to‐day observation, what customizations did you make to the lesson when preparing?

c) (The teacher sharing practice models a lesson fragment for the observers.) What was your rationale for certain choices such as _______________________? (Be precise.)

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Observer’s Learning

a) Based on evidence in the lesson and student work what worked well? Why?

b) Based on evidence, what did not work as well? Why?

c) Based on the specific learning, what practices do you, the observer, want to generalize to your own classroom? (If the observer is not a teacher, then what practices do you want generalized to a grade level, school, or district)

Closing Action: Deliberately practice one of the practices called out above that you want to generalize to your classroom, grade level, school, or district.

Appendix C: Observation Recording Sheet

Possible details to reflect on:

Precise minimalized teacher talk

Precise mathematical language used by

both teachers and students

Teaching sequences

Questioning strategies (including

differentiation)

Models and their connection to the

abstract (number)

Use of student dialogue and critique

Success of students independently

Student understanding of definitions,

notations, and descriptions

Debrief in which students articulate the

lesson’s goal.

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Time Teacher Actions, Words Student Actions, Words

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Time Teacher Actions, Words Student Actions, Words

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Appendix D: Preparing to Teach a Lesson from A Story of Units

A three‐step process is suggested to prepare a lesson. It is understood that at times teachers may need to make adjustments (customizations) to lessons to fit the time constraints and unique needs of their students. The recommended planning process is outlined below. Note: The ladder of Step 2 is a metaphor for the teaching sequence. The sequence can be seen not only at the macro level in the role that this lesson plays in the overall story, but also at the lesson level, where each rung in the ladder represents the next step in understanding or the next skill needed to reach the objective. To reach the objective, or the top of the ladder, all students must be able to access the first rung and each successive rung.

Step 1: Discern the plot.

A: Briefly review the Table of Contents for the module, recalling the overall story of the module and analyzing the role of this lesson in the module.

B: Read the Topic Overview of the lesson, and then review the Problem Set and Exit Ticket of each lesson of the topic.

C: Review the assessment following the topic, keeping in mind that assessments can be found midway through the module and at the end of the module.

Step 2: Find the ladder.

A: Complete the lesson’s Problem Set.

B: Analyze and write notes on the new complexities of each problem as well as the sequences and progressions throughout problems (e.g., pictorial to abstract, smaller to larger numbers, single‐ to multi‐step problems). The new complexities are the rungs of the ladder.

C: Anticipate where students might struggle, and write a note about the potential cause of the struggle.

D: Answer the Student Debrief questions, always anticipating how students will respond.

Step 3: Hone the lesson.

At times, the lesson and Problem Set are appropriate for all students and the day’s schedule. At others, they may need customizing. If the decision is to customize based on either the needs of students or scheduling constraints, a suggestion is to decide upon and designate “Must Do” and “Could Do” problems.

A: Select “Must Do” problems from the Problem Set that meet the objective and provide a coherent experience for students; reference the ladder. The expectation is that the majority of the class will complete the “Must Do” problems within the allocated time. While choosing the “Must Do” problems, keep in mind the need for a balance of calculations, various word problem types1, and

1 See the Progression Documents “K, Counting and Cardinality” and “K−5, Opera ons and Algebraic Thinking” pp. 9 and 23,

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work at both the pictorial and abstract levels.

B: “Must Do” problems might also include remedial work as necessary for the whole class, a small group, or individual students. Depending on anticipated difficulties, those problems might take different forms as shown in the chart below.

Anticipated Difficulty “Must Do” Remedial Problem Suggestion

The first problem of the Problem Set is too challenging.

Write a short sequence of problems on the board that provides a ladder to Problem 1. Direct the class or small group to complete those first problems to empower them to begin the Problem Set. Consider labeling these problems “Zero Problems” since they are done prior to Problem 1.

There is too big of a jump in complexity between two problems.

Provide a problem or set of problems that creates a bridge between the two problems. Label them with the number of the problem they follow. For example, if the challenging jump is between Problems 2 and 3, consider labeling these problems “Extra 2s.”

Students lack fluency or foundational skills necessary for the lesson.

Before beginning the Problem Set, do a quick, engaging fluency exercise, such as a Rapid White Board Exchange, “Thrilling Drill,” or Sprint. Before beginning any fluency activity for the first time, assess that students are poised for success with the easiest problem in the set.

More work is needed at the concrete or pictorial level.

Provide manipulatives or the opportunity to draw solution strategies. Especially in Kindergarten, at times the Problem Set or pencil and paper aspect might be completely excluded, allowing students to simply work with materials.

More work is needed at the abstract level.

Hone the Problem Set to reduce the amount of drawing as appropriate for certain students or the whole class.

C: “Could Do” problems are for students who work with greater fluency and understanding and can, therefore, complete more work within a given time frame. Adjust the Exit Ticket and Homework to reflect the “Must Do” problems or to address scheduling constraints.

D: At times, a particularly tricky problem might be designated as a “Challenge!” problem. This can be motivating, especially for advanced students. Consider creating the opportunity for students to share their “Challenge!” solutions with the class at a weekly session or on video.

E: Consider how to best use the vignettes of the Concept Development section of the lesson. Read through the vignettes, and highlight selected parts to be included in the delivery of instruction so that students can be independently successful on the assigned task.

F: Pay close attention to the questions chosen for the Student Debrief. Regularly ask students, “What was the lesson’s learning goal today?” Hone the goal with them.

respectively.

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G: Adjust the balance of the lesson’s components as necessary to support the work students are expected to do in the Problem Set or task, e.g., the Fluency, Exit Ticket, Homework, Application Problem.

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Appendix E: Protocols for Use during Professional Development Sessions Protocol A for Professional Reading: “Save the Last Word for Me”

1. Participants silently read the selection and highlight at least one statement or phrase that

stands out for whatever reason.

2. After the reading period is completed, as signaled by the facilitator, a volunteer reads one of

their highlighted statements and the other 2‐3 members of the group briefly share their

thoughts on the statement.

3. The person who read the statement shares their rationale to close the discussion of the

statement.

4. The process is repeated until all have had a chance to share.

5. After all statements are shared, participants share freely outside a protocol for the balance of

the time.

Protocol B for Analysis of New Complexities: “Problem Set Analysis”

1. Do the Problem Set independently.

2. Independently analyze the new complexities of each problem and write the analysis directly on the

Problem Set.

3. After the analysis period is complete, as signaled by the facilitator, participant A shares a complexity

of the first step in the sequence.

4. Participant B contributes more thoughts, if possible.

5. The process is repeated until no new observations are made on the first step. The pair then moves on

to the next problem and analyzes its new complexities.

Protocol C for SHARE IT: Whole Group Participation

1. Write “Share It” on four brightly colored 3 x 5 or 4 x 6 cards.

2. As you, the teacher/facilitator, circulate during small group talk, listen for an interesting insight,

thought, discussion that you think would be helpful for the whole group to hear.

3. Ask the students/participant(s) if they would be willing to share their thoughts with the whole group.

If willing, hand over a “Share It” card.

4. Once the small group sharing has concluded, call on those to whom you passed the “Share It” cards.

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Appendix F: Deliberate Practice

As adult learners, we communicate to each other and to our students that anyone’s teaching practice or learning can be improved with effective effort. We don’t give up on colleagues, on ourselves or on students!

Possible Protocol: “Teach and Repeat Using Feedback” (Time frames may be adjusted.)

1. (2 minutes) Plan the teaching segment either collaboratively or individually.

2. (1 minute) One colleague stands and delivers instruction to other team members.

3. (2 minutes) The “students” give feedback.

4. (1 minute) Partner A reteaches precisely the same segment of instruction, using the feedback.

5. (1 minute) Partners give feedback again.

6. (1 minute) Partner A uses the feedback again, etc.

Stems for Feedback during Deliberate Practice:

The connection between the pictorial and the abstract was clarified when you _____.

It engaged the students when you got them right into the math at the beginning of the lesson.

You might remain silent and demonstrate (gesture) as you_____.

Your use of the ____was incorrect. I think you meant to say ____.

You might say _____ instead of _____.

You might shorten/lengthen the wait time after _____.

It might be more efficient to tell the students ______.

It might be better to elicit _____ rather than tell.

When you check for understanding you might _____.

After you asked ________, your cue for a response could be crisper.

When you check our personal boards, you might _____.

Do it again. Do it better by_____.

Having us partner share was effective for seeing the pattern of _____.

You mention a jump between Problem ___ and Problem ___. What is a bridge problem you

could inset between the two?

You might clarify the relationship between the tape diagram and the number line by ____.

Reflection on Practice: “How are we doing at practicing deliberately?” The distractions from the deliberate practice were _____.

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NYS COMMON CORE MATHEMATICS CURRICULUM

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