1. Welcome to Day One

2. Welcome and Agenda 5 Practices for Mathematical Discourse
Re-engaging in division and ratio
Looking at algorithms
Lessons and Reflections

3. The Standards for Mathematical Practice Student Reasoning and Sense Making about Mathematics
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Sherry – Have teachers look at their SMP pages we passed previously and state an exemplar of evidence from a student enacting the SMP. Ex. 1 is of SMP 2 – the student is contextualizing a mathematics problem to help solve it.
Ex. 2 is of SMP 8 – where the students looked for and expressed regularity in repeated reasoning.
Sherry – Tell the teacher to be thinking about examples you can write about students; Explain to Teachers that we will reflect for ourselves about the tasks we do and the connections and evidence of SMP’s, at the end each grant day.

4. Bring your ideas…
As a group of professionals we have made a commitment to helping children attain success in life and a voice in the world.
Many times the best part of these kinds of professional development is simply the chance to share ideas, raise questions, and work with other practitioners to improve our own understandings and practice.
Please bring your stories of children’s learning with you.

5. Our Socio-mathematical Norms
Listen intently when someone else is talking avoiding distractions
Persevere in problem solving; mathematical and pedagogical
Solve the problem in more than one way
Make your connections explicit - Presentation Ready
Contribute by being active and offering ideas and making sense
Limit cell phone and technology use to the breaks and lunch unless its part of the task.
Be mindful not to steal someone else’s “ice cream”
Respect others ideas and perspectives while offering nurturing challenges to ideas that do not make sense to you or create dissonance.
Limit non-mathematical and non-pedagogical discussions
Gabriel - Make any agreed upon revisions to the norms

6. Presentation Norms
Presenters should find a way to show mathematical thinking, not just say it
Presenters should indicate the end of their explanation by stating something like “Are there any questions, discussion, or comments?”
Others should listen and make sense of presenters’ ideas.
Give feedback to presenters, extend their ideas, connect with other ideas, and ask questions to clarify understandings
Gabriel - Make any agreed upon revisions to the norms

7. Break

8. Fractions/Ratios – Division Connections
What is the meaning of multiplication?
5 x 3 = 15
Be sure to discuss Factors and Product in addition to the number of groups and the group size

9. Considering Division
In Multiplication (in the US) when we write 4 x 5 we mean
“4 groups of 5”
But when we write 5 x 4 we mean
“5 groups of 4”

10. Fractions/Ratios – Division Connections
Use the snap cubes to represent the solution to 15 3
Observe how teachers represent, pull out examples of partitioning methods.
Ask, which one is the correct way to divide and why?

11. Fractions/Ratios – Division Connections
What is the meaning of division?
Be sure to discuss the dividend, divisor, and quotient. What do each mean?

12. Essence of Division
“The essence of division is this; we consider the number of groups of a particular size OR the size of a particular number of groups that is contained within a total amount (dividend). The difficulty for many students is making sense of the fact that the divisor and the quotient can represent either the group size or the number of groups.”
-Matney (2014)

13. Fractions/Ratios – Division Connections
Group Size Unknown (Fair Share/Partitioning)
The pet kennel has five puppies and 40 treats. How many treats would each puppy get if they were to each get the same amount?
Kendra paid 42 cents for seven apples. What was the cost for each apple?
Have the teachers answer these mentally then devise a statement as to “why” these problem types might be called “Group Size Unknown”
In these question types the whole is known and must be partitioned into a known number of groups, to find the “size of each group”.
Ask teachers to write down the division problem represented by each problem

14. Fractions/Ratios – Division Connections
Number of Groups Unknown (Grouping/Measurement)
Kendrae has 42 apples. He put them into bags containing three apples each. How many bags did he use?
Blanca walked 24 miles at a rate of 3 miles per hour. How many hours did this take Blanca?
Have the teachers answer these mentally then devise a statement as to “why” these problem types might be called “Number of Groups Unknown”
In these question types the whole is known and must be measured off into groups of a known size to find the “number of groups”.
Ask teachers to write down the division problem represented by each problem

15. Fractions/Ratios – Division Connections
Group Size Unknown (Fair Share or Partitive)
Number of Groups Unknown (Grouping or Measurement)
Day 1 handout pages 4
The ideas in this introduction to division will be thought about again throughout the two weeks in various other mathematical capacities.
To the presenter: At the moment, the connections between division and fractions have yet to be ferreted out, but this will be developed more on subsequent days.

16. Fractions/Ratios – Division Connections
Hand Out
Five friends go on a hike together. One of the friends brought along 30 strawberries. How many strawberries would each person get if they shared them equally?
A class of 36 students travels together on a school fieldtrip. The teacher wishes to divide the students into tour groups of 9 students. If each group needs one student leader, how many student leaders will there be for the tour?
Pass out page 4 of handout and ask if directions 1-3 are clear?
Have teacher’s present their ideas and discuss solutions, problem types, and children’s thinking
Pass out pages 5 and work through
Pass out page 6 – read and discuss

17. Fractions/Ratios – Division Connections
Consider: “Suppose four conference speakers are giving a presentation that is 3 hours long; how much time will each person have to present if they share the presentation time equally?”
Day 1 handout page 7

18. Fractions/Ratios – Division in the Common Core1

19. LUNCH

20. Understanding Division
Anything Strange about this picture?
Mathematical Language - Quotient, Dividend, Divisor
Horizontal bar in a fraction 1/3 is called the “Vinculum”
The traditional division symbol 25 “divided by” 5 is called an “Obelus”.
Teacher Hats Off - I want you to step out of your mindset of teacher for a series of connected mathematics tasks. Let’s just make sense of these ideas together.

21. Understanding Division
Handout Page 9
Let’s take a step back and explore the standard algorithm and think about what is happening:

22. Understanding Division
Handout page 10

23. Understanding Division
Handout page 11

24. BREAK

25. Math Content for our Classrooms
Each day we will spend time with grade level teams making lesson plans.
Each of us will make one plan that is part of a unit of plans the grade level team is working on.
Each plan must have the following:
Connected mathematics content focus from Ohio’s Mathematics Learning Standards
A focus SMP
Designed to Orchestrate Productive Mathematics Discussions (The 5 Practices)
Handout Page 15

26. Math Content for our Classrooms
Three checks must be made for the completion of lesson plans:
Check 1) Consult with Sandy and/or Mary about the mathematics content of the lesson and explain to her its mathematical appropriateness. When the lesson is complete Sandy, our resident mathematician, will sign off on its content (SMC’s).
Check 2) Consult with Sherry about the design of the lesson to promote mathematical discourse (5 Practices). Sherry must sign off on the lessons discourse elements.
Check 3) Consult with Dr. Matney about the design of the lesson having a focus Standard for Mathematical Practice. Dr. Matney must sign off on the lessons mathematics proficiency elements (SMP’s)
?Questions about COMP Lesson Plans?
Handout Page 15

27. On a sticky note tell us one thing you learned today.
Time of Reflection
On a sticky note tell us one thing you learned today.
Tell us one think you liked or one thing you are still struggling with.
Handout Page 15

28. Stay Safe Please help us put the room in proper order.
Please leave your name tents for next time.