1. Routines for Reasoning
Connecting Representations -
Early Multiplication

2. IRMC Routines for Reasoning Action Research Project: Connecting Representations Routine 
Teacher Name: Wallin
Grade: 3
Date: 11/13/18
Standards Addressed: 3.OA.1, 5, 7; 3.MD.7
SMP Focus: 7,
Learning Target/Objective
Students will explore strategies for decomposing arrays. This action supports students learning their multiplication facts from memory.
Task description with rationale
When students have flexible ways of thinking about multiplication problems it allows them to learn a specific fact and then apply that thinking to other facts that they may not have already learned. These strategies help move towards fluency of facts.
Connecting Representations Routine Outline
Routine Step
Planning Questions
Planning Notes
Launch
What are your thinking goals for this lesson?
Think like mathematicians
Look at and make sense number sentences
Connect arrays to number sentences
What “Ask-Yourself Questions” will you give students?
What do I see?
Look at and make sense of number sentences
Connect number sentences to visual models
Interpret and Connect Representations
How might students think about this task? What do you anticipate seeing?
I assume that students may struggle initially with the idea of knowing the fact and may rely on counting to find the answer. I am hoping that numbering the circles will help them to see the answer sooner and will help them at least consider that the problem is 8 x 6. In the task I hope that the students will see every array refers to 8 x 6
How will you select pairs of students to share their work?
I will look for pairs that are using the sentence starters during their discussions. Groups that annotate their work would be helpful when I have them come to the board.
What sentence frames will you use?
I notice __________, which makes me think __________.
“I noticed ________, so I __________.”
“They noticed ________, so they __________.”
How will you manage the discussion? What annotations will you want to focus on during discussion?
I want to focus on the numbers and get students to see that the array is an 8 by 6 and that relates to the multiplication problem 8 x 6.
Create Representations
How will you address the unmatched representation; what guiding questions will you use?
I will put space on the worksheet for students to create their own decomposition of the (8*5)+(8*1).
How will you select pairs to present?
I will look for groups that have a picture, annotate the picture, and who are using the sentence frames when discussing their picture.
Discuss Representations
How will you determine the focus of the final discussion/select student work?
The task solution is convergent in that there should be a representation where students have created a 8 by 6 array broken into sections of 8 by 5 and 8 by 1.
Reflection on Student Thinking
What sentence frame will you use?
“Breaking an array into smaller pieces helps me to _______________.”
What do you hope to learn from the student reflection?
I want to know how students will generalize the act of decomposing arrays. I want them to discuss how looking at different expressions can all represent the same general array.

3. Purpose
We are learning how to write number sentences from a picture. We are going to listen to what others say and think about what they mean. We are going to create a picture for a number sentence.

4. Today’s Thinking Goals
Think like mathematicians
Look at and make sense number sentences
Connect arrays to number sentences

5. Thinking Questions What do I see? What is the number sentence saying?
How does the picture match the number sentence?
How many thinking questions should we give kids at one time? Do we develop these as we go?
How is this situation behaving?
What kind of problem is this?
Does this problem remind me of another I’ve solved?
How can I decompose this problem to help me understand it better?

6. I notice __________, which makes me think __________.1 2 3 4 5 6 7 8

7. 1 2 3 a) (4 X 6) + (4 X 6) b) (8 X 5) + (8 X 1) c) (8 X 3) + (8 X 3)
“I noticed ________, so I __________.” 1
“They noticed ________, so they __________.”
a) (4 X 6) + (4 X 6) 2
b) (8 X 5) + (8 X 1)
c) (8 X 3) + (8 X 3) 3
d) (5 X 6) + (3 X 6)

8. What would (8 X 5) + (8 X 1) look like?
“I noticed ________, so I __________.”
“They noticed ________, so they __________.”

9. Reflection:
“Breaking an array into smaller pieces helps me to _______________.”
Use one reflection or the other depending on objectives

10. Reflection:
“When I have to remember (8 X 6), I will think about _______________.”
Use one reflection or the other depending on objectives