1. Cypher IV Mathematics Leadership Project
Teaching Student-Centered Math Book Study 3-5 Group Session 2 Number & Operation Sense

2. (Re)Introductions Kendra Haines (Grades 5-6, Ross River)
Jane MacArthur (Grades 4-6, Carcross)
Lois Moore (Grades 4-5, Whitehorse)
Pam Harry (Grades 3-4, Whitehorse)
Sherry MacInnis (Grades 3-5, Atlin)
Kalvin Beuerlein (Grades 3-9, Telegraph Creek)
Mike Emerick (Grades 7-8, Dease Lake)

3. Homework Review (Small Group)
Based on the homework assigned in the previous session, discuss the following questions in a small group:
What have you tried in your classroom as a result of the last session?
What role did you play in the teaching and learning of math?
What role did the students play in their learning?
What discoveries did you and your students make?
What misconceptions, if any, surfaced about the topic? How did you redirect the students?
What suggestions do you have for others when they try this?

4. Objectives Focus on the Big Ideas of number and operation sense
Explore various number relationships
Investigate problem structures and models for multiplication and division
Discuss important issues related to solving story problems

5. Materials (Electronic Transfer)
Evaluation Form
Copy of the curriculum (sent big poster)
Grade 3, 4, 5, and/or poster (smaller size)
Ten-frames (BLM 1)
Little ten-frame cards (BLM 3,4)
Hundreds Chart (BLM 5,6)
Graph Paper (BLM 7-13 & 16-19)
Thousand Chart Skip-by-Ten
How Bear Got A Short Tail Materials (Problem Structures)
What’s Your Number? Game Sheets
The Snowsnake Game

6. Number Sense (Partner)
Discuss the following with a partner:
What does it mean for a student to have a good sense or intuition of numbers?
What implications does teaching to encourage number sense have on how we work with students and what we emphasize in our classrooms?
Be prepared to share a few of your ideas with the large group in 10 minutes. (If the breakout room whiteboard is used to record ideas, I can copy it into the main room.)

7. Big Ideas Look at the Big Ideas for this chapter (p. 39)
Does the curriculum have a strong emphasis on the key points about number sense that are articulated in the Big Ideas?
Consider ways that you can emphasize the Big Ideas of number sesne and provide number sense strengthening activities for individuals or groups of students.

8. Anchoring Numbers to 5 & 10
In grades K-3, the numbers 5 & 10 can be used as anchors. They are especially useful when thinking about combinations of numbers. A key model to use with students to illustrate these relationships is the 10-frame.

9. Anchoring Numbers to 5 & 10 (cont’d)
Here is a ten frame. To build the 7-frame:
Always fill the top row first, starting on the left - the same way that you read.
When the top row is full, counters can be placed in the bottom row, also starting on the left.

10. Anchoring Numbers to 5 & 10 (cont’d)
Build 8
Share what you know about the number 8 from looking at the ten-frame.
How could you use ten-frames to help students develop 5 and 10 benchmark relationships?
8 is 3 more than 5.
3 away from 5.
8 is 2 less than 10.
8 is 5 and 3.
8 is 3 and 5.
Important variations of “Ten-Frame Flash Cards” include:
There are 2 empty spots.
2 more than 8 is 10.
2 less than 10 is 8.
2 away from 10.
Say the ten fact =10. Read it as “is the same as”

11. Part-Part-Whole Relationships
& Extending Number Relationships to Larger Numbers - Lauren Resnick (1983) states:
Probably the major conceptual achievement of the early school years is the interpretation of numbers in terms of part and whole relationships. With the application of a Part-Whole schema to quantity, it becomes possible for children to think about numbers as compositions of other numbers. This enrichment of number understanding permits forms of math problem solving and interpretation that are not available to younger children.
Lauren Resnick

12. Part-Part-Whole Relationships
& Extending Number Relationships to Larger Numbers (continued)
Find as many ways as possible to represent the following numbers (with the ten-frame cards): 80 35 67
As participants share there ideas about how to represent the numbers, emphasize the part-whole relationships that foster flexible thinking. Each of the forms of 67 may be useful in a variety of situations. For example, if your are adding 67 and 56, thinking of 67 as 50 and 17 allows you to add 50 and 50 (from the 56) to get 100. Now add 17 and 6 to get 23. Combine the 100 and 23 to get 123. Once students are able to think flexibly like this, they will be able to do additions such as 67 and 56 mentally much faster tha using a pencil-and-paper procedure.

13. Relative Magnitude Small Group Activity
Complete Activity 2.6, Close, Far, and In Between (p. 45) using 219, 364, & 457.
Explain the thinking behind your responses to the questions in the activity.
Consider other trios of numbers that you could use and the prompts you would use with students to get them thinking about the relative magnitude of numbers.

14. Thinking About Parts of #s
To continue to build on part-part-whole ideas together, and in particular to focus on the missing part, we will work through one or more of:
Activity and Some More (p. 54)
Acitivity 2.18, The Other Part of 100 (p. 54)
Activity 2.19, Compatible Pairs (p. 55)
Work with a partner to consider ways to use these activities in your class. Be prepared to share with the whole group.
For 2.17, be sure to use larger numbers such as 483, where the answer would be “450 and 33 more.”
For 2.18, have participants share the thought processes that they used to determine the missing part of 100. It is through this sharing theat they will begin to recognize that we all use different strategies when solving problems in our heads.
For 2.19, the compatible-pair searches are on the following four slides. Compatible numbers for addition and subtraction are numbers that go together easily to make nice numbers. Numbers that make tens or hundreds are the most common examples. Compatible sums also include numbers that end in 5, 25, 50, or 75, since these numbers are easy to work with as well. The teaching task is to get students accustomed to looking for combinations that work together and then looking for these combinations in computational situations.

15. Compatible Pairs - Make 5037 41 28 9 31 12 38 19 22 13

16. Compatible Pairs - Make 1000
815
635
435
565
760
550
240
365
280
720
450
185

17. CP - Using 5s to Make 100 25 35 5 45 95 85 15 65 75 55

18. Compatible Pairs - Make 500
240
150
415
165
375
350 85
260
125
335

19. Thinking About Parts of #s
To continue to build on part-part-whole ideas together, and in particular to focus on the missing part, we will work through one or more of:
Activity and Some More (p. 54)
Acitivity 2.18, The Other Part of 100 (p. 54)
Activity 2.19, Compatible Pairs (p. 55)
Work with a partner to consider ways to use these activities in your class. Be prepared to share with the whole group.
For 2.17, be sure to use larger numbers such as 483, where the answer would be “450 and 33 more.”
For 2.18, have participants share the thought processes that they used to determine the missing part of 100. It is through this sharing theat they will begin to recognize that we all use different strategies when solving problems in our heads.
For 2.19, the compatible-pair searches are on the following four slides. Compatible numbers for addition and subtraction are numbers that go together easily to make nice numbers. Numbers that make tens or hundreds are the most common examples. Compatible sums also include numbers that end in 5, 25, 50, or 75, since these numbers are easy to work with as well. The teaching task is to get students accustomed to looking for combinations that work together and then looking for these combinations in computational situations.

20. Problem Structures for x and ÷
Multiplication and division problems can be categorized according to the types of relationships involved. The two most common structures for multiplication and division are equal-group problems and multiplicative comparison problems.
Review Equal-Group Problems (pp ) & Multiplicative Comparison Problems (pp )

21. Problem Structures (cont’d)
In groups of 2 or 3, solve the problems in the text as directed in the Stop and Reflect box (p. 60).
Create one of each type of problem with your partners.
Share the cognitively guided instruction work that I did with the group.

22. Using Models For x and ÷
An important model for multiplication and division is the array. An array is any arrangement of things in rows and columns, such as a rectangle of square tiles or blocks.
With a partner, represent the factors of 30 using arrays.
Record your arrays on the grid paper and write the multiplication expression that the array represents beside it.
Be prepared to share with the whole group.
A grid is provided on the next slide.

24. Using Models for x and ÷ Large group discussion:
How and why do arrays support students in their understanding of multiplication.
Share your ideas about how arrays could be used to support students in their understanding of division.
Make links to how arrays support the development of the order property in multiplication and the distributive property (p. 66).

25. After Round table sharing
Impressions and/or questions that have surfaced as a result of the session/
Your responses to impressions and/or questions that have surfaced as a result of the session
What does it mean for a student to have a good sense or intuition of numbers?
What implications does teaching to encourage number sense have on how we work with students and what we emphasize in our classrooms?

26. Homework
Try the two strategies for problem solving, Think About the Answer Before Solving the Problem (p. 68) and Work a Simpler Problem (p. 69), with a small group of students in your class. Reflect on how your students solved problems and be ready to share your experiences at the next session.
Read Chapter 3, Helping Children Master the Basic Facts (pp )

27. Evaluation
Please consider completing the evaluation form and ing it to me after this session.