1. April 25, 2014 SSOS Central Center of Excellence Team Meeting
Doing Mathematics
April 25, 2014
SSOS Central Center of Excellence Team Meeting

2. Ability and Math Mindset
Video –
Jo Boaler, Professor Math Education Stanford University

3. Standard and Benchmarks
STANDARD 3.1.2 Add and subtract multi-digit whole numbers; represent multiplication and division in various ways; solve real-world and mathematical problems using arithmetic
BENCHMARK:  Represent Multiplication & Division Facts: Represent multiplication facts by using a variety of approaches, such as repeated addition, equal-sized groups, arrays, area models, equal jumps on a number line and skip counting. Represent division facts by using a variety of approaches, such as repeated subtraction, equal sharing and forming equal groups. Recognize the relationship between multiplication and division.
BENCHMARK:  Multiplication & Division in the Real-World: Solve real-world and mathematical problems involving multiplication and division, including both "how many in each group" and "how many groups" division problems.

4. Big Ideas and Essential Understandings
Standard Essential Understandings
Students build on previous work with addition and subtraction of 2-digit numbers to include multi-digit whole number addition and subtraction.  They use various strategies to solve real-world and mathematical problems involving addition and subtraction.  Third grade students begin their formal work with multiplication and division by representing basic facts in a variety of ways.  They represent multiplication facts in a variety of ways including:
repeated addition, equal groups, arrays, equal jumps on a number line (skip counting)
They represent division facts in a variety of ways including:
repeated subtraction, equal sharing, forming equal groups
Students develop an understanding of the relationship between multiplication and division which will lead to the development of a variety of strategies for multiplying and dividing.  They will solve real-world and mathematical problems involving multiplication and division. 
Problem solving involving division will include both "how many in each group" and "how many groups" problems.
Multiplication extends to multiplying a two- or three-digit number by a one-digit number. Strategies may include mental strategies, partial products, the standard algorithm, and the commutative, associative and distributive properties.  Strategic thinking when solving problems is the focus, not procedures. 
From SciMath Frameworks

5. Learning Objective
Students will be able to write a situation for a given equation and be able to represent division facts using equal sharing and forming equal groups.

6. We will be working with the following equation:
48 ÷ 3 = 16

7. 48 ÷ 3 = __ 48 ÷ __= 16 Write a situation for: Partner work
Ask them to write out a situation that describes this number sentence
Anticipate
When walking around, be sure two different type situations are evident – if not, be prepared to have one group adapt one of the following types of examples :
Example Type 1: (equal sharing – How Many in Each Group?)
Billy has 48 marbles and needs to split them evenly between his three friends. How many marbles does each friend get?
Example Type 2: (equal groups – How Many Groups?)
Mrs Smith needs to place 48 students into groups that hold 16 students each. How many groups are needed?

8. 48 ÷ 3 = __ 48 ÷ __= 16 Solve the equation:
Handout plates and jelly beans to each group
Encourage groups to draw the situation on paper
Monitor
Questions that can be asked while they are working:
How can you organize the information?
Can you make a drawing (model) to explain your thinking? What are other possibilities?
What would happen if ...?
Can you describe an approach (strategy) you can use to solve this?
What do you need to do next?
Do you see any relationships that will help you solve this? (like repeated subtraction)
How does this relate to your written situation?
Why did you...?
What assumptions are you making?
After they have the answer:
How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?
How can you convince me your answer makes sense?
What did you try that did not work? Has the question been answered?

9. Different Standard and Benchmark
Now an Algebra Standard (instead of Number and Operation)
STANDARD Use number sentences involving multiplication and division basic facts and unknowns to represent and solve real-world and mathematical problems; create real-world situations corresponding to number sentences.
  Use multiplication and division basic facts to represent a given situation using a number sentence. Use number sense and multiplication and division basic facts to find values for the unknowns that make the number sentences true.

10. Share Solutions Share out: written situation and representation
Select and sequence –
I am going to have a group share first that has the most frequent type of situation
If time can have all groups share out (not necessary)
Last – have group with different solution share out
Allow other team members to ask clarifying questions.

11. Discussion Show the two situations –
equal sharing (how many in each group?)
forming equal groups (how many groups?)
With partner - discuss
How are they different?

12. Reflect and Connect: What We Learned about the Two Types of Division Problems
Equal Sharing:
How many in a group?
Also referred to as Partitive or Distribution Division
Example: Mrs H has 48 students and needs to split into 3 equal groups. How many in each group?
Need clarification – - (this link to the equation is where I messed up the first time):
if the wording says, “split into 16 equal groups” instead of ‘split into 3 equal groups” does it matter if it is 48/3 = 16 or 48/16 = 3
Student examples from lesson today
Forming Equal Groups:
How many groups?
Also referred to as Quotative or Measurement Division
Example: Mrs H needs to form groups of three students. How many groups can she form if 48 students are in the classroom?
From Ontario:
Junior students need experiences with partitive and quotative contexts to understand that both are division problems, and they need to see the connections between them. It is not necessary for students to know the names or definitions of the terms “partitive” and “quotative”.
Equal Sharing: How many in each group? (Partitive or Distributive)
Mrs H has 48 students and needs to split into 3 equal groups. How many in each group?
To solve this problem, students may try to guess and check to get the answer, or may use a dealing out strategy until the wheels are gone. These strategies are very different from the ones used in quotative division problems and are difficult to connect to repeated subtraction.
Forming Equal Groups: How many groups? (Quotative or Measurement Division)
Mrs H needs to form groups of three students. How many groups can she form if 48 students are in the classroom?
To solve this problem, many students will start with 4, add up to the whole of 24, and then count the number of fours they used. They use repeated addition instead of repeated subtraction. While it could be argued that this solution could be explained using repeated subtraction, the term would not describe the strategy the students used or how they thought about the problem.

13. Related Benchmarks Multiplication Place value benchmarks
Equal size groups is what place value is about!!
We use a base ten system, so the most common group size that we see in math is 10

14. Other Related Benchmarks
Look at other benchmarks
in 3rd grade
in previous grades (K – 2nd)
in latter grades (4th - 5th)
Split up team into groups of twos and have them highlight benchmarks related to division

15. Solve the following division problem
The situation is:
I have 1459 apples and need to place into boxes that hold 18 each. How many boxes do I need.
Have them solve any way they would like

16. Adding on to your thinking
Consider this approach:
Tough to do using paper plates and jelly beans!!! And also tough to do using pictures!
CRA is not what really is important – what is important is the sense making and always connecting
On white board (flip paper) (partial products)
Help them with recording thinking (have them write down as I write down)