1. Routines for Reasoning
Connecting Representations –
Area of Rectangular Figures

2. Connecting Representations Routine Outline
IRMC Routines for Reasoning Action Research Project: Connecting Representations Routine
Teacher Name: wallin
Grade: 3
Date:
Standards Addressed: 3.OA.3, 3.OA.5, 3.OA.8, 3.MD.5-7, 4.OA.3, 4.MD.3
SMP Focus: 7, 3, 4, 1, 6, 2
Learning Target/Objective
Students will connect expressions to visual patterns to better understand ways of decomposing areas of figures.
Task description with rationale
This task is developed to help third and four-grade students connect expressions with visual representations, specifically in the area of decomposing areas into smaller multiplication problems (sums of partial products).
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 of expressions
Connect expressions to visual models
What “Ask-Yourself Questions” will you give students?
Does this problem remind me of another I’ve solved?
How can I decompose this problem to help me understand it better?
Interpret and Connect Representations
How might students think about this task? What do you anticipate seeing?
I anticipate that students may focus on the colors and make decisions based on the different colors rather than looking at the expressions provided as clues to the connections.
How will you select pairs of students to share their work?
I will look for pairs of students who have communicated connections while I am observing prior to whole group discussion.
What sentence frames will you use?
“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 for students to fully annotate parts of the expressions onto each of the representations while they discuss them. We may proceed further by having students color code the expressions as they discuss them.
Create Representations
How will you address the unmatched representation; what guiding questions will you use?
I will push students to see that we have not connected all of the representations. I will provide private think time and pair time again. I will ask them to use the second ask-yourself question to think about how the expression decomposes the problem.
How will you select pairs to present?
I will look for pair who can communicate together how they have solved the problem. I will also look for pairs who have annotated their new picture.
Discuss Representations
How will you determine the focus of the final discussion/select student work?
I will look for pairs who have annotated their new picture. I may provide some questions to the group while seated so that they can provide a deeper explanation when they come to the board.
Reflection on Student Thinking
What sentence frame will you use?
“When interpreting a mathematical representation, I learned to pay attention to _______________.”
What do you hope to learn from the student reflection?
I want students to develop comfort with connecting expressions to visual models, specifically I want them to see that each part of an expression relates to a specific part of the visual model.

3. Connecting Representations
SMP Focus: Look for and make use of structure
Chunk, Change, Connect
Interpretation of Visual Models (Visualization)
Essential Strategies
Annotation
Sentence frames and sentence starters

4. Purpose
Today we are learning how to connect visual representations to mathematical expressions. We are going to visualize how to find the areas of a shape in different ways. We are going to talk to others to clarify our own thinking about different expressions.

5. Today’s Thinking Goals
Think like mathematicians
Look at and make sense of expressions
Connect expressions to visual models

6. Thinking Questions Does this problem remind me of another I’ve solved?
How can I decompose this problem to help me understand it better?
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?

7. What do you Notice?
“I noticed ____, which makes me think _______.”

8. 1) 2) 3) 4)
2× 6×2 + 10×2
12×2 + 4×2 + 6×2
6×2 + 10×2 + 6×2
14×2 + 4×2 + 4×2
14×6 −(10×4)

9. 1) 2× 6×2 + 10×2 12×2 + 4×2 + 6×2 6×2 + 10×2 + 6×2 14×2 + 4×2 + 4×21)
“I noticed ________, so I __________.”
“They noticed ________, so they __________.”
2× 6×2 + 10×2
12×2 + 4×2 + 6×2
6×2 + 10×2 + 6×2
14×2 + 4×2 + 4×2
14×6 −(10×4)

10. 1) 2) 2× 6×2 + 10×2 12×2 + 4×2 + 6×2 6×2 + 10×2 + 6×2 14×2 + 4×2 + 4×21) 2)
“I noticed ________, so I __________.”
“They noticed ________, so they __________.”
2× 6×2 + 10×2
12×2 + 4×2 + 6×2
6×2 + 10×2 + 6×2
14×2 + 4×2 + 4×2
14×6 −(10×4)

11. 1) 2) 3)
“I noticed ________, so I __________.”
“They noticed ________, so they __________.”
2× 6×2 + 10×2
12×2 + 4×2 + 6×2
6×2 + 10×2 + 6×2
14×2 + 4×2 + 4×2
14×6 −(10×4)

12. 1) 2) 3) 4)
“I noticed ________, so I __________.”
“They noticed ________, so they __________.”
2× 6×2 + 10×2
12×2 + 4×2 + 6×2
6×2 + 10×2 + 6×2
14×2 + 4×2 + 4×2
14×6 −(10×4)

13. “They noticed ________, so they __________. “
“I noticed ________,
so I __________. “
“They noticed ________,
so they __________. “ 5)
2× 6×2 + 8×2
12×2 + 4×2 + 6×2
6×2 + 8×2 + 6×2
14×2 + 4×2 + 4×2
14×6 −(10×4)

14. Reflection
“When interpreting a mathematical representation, I learned to pay attention to _______________.”