1. Routines for Reasoning
Connecting Representations –
The Structure of Rounding

2. IRMC Routines for Reasoning Action Research Project: Connecting Representations Routine 
Teacher Name: wallin
Grade: 4th
Date:
Standards Addressed: 4.NBT.3
SMP Focus: 7, 6, 3, 8
Learning Target/Objective
Students will create and communicate a precise generalization for how to round a number based on mathematical structure.
Task description with rationale
By fourth grade students should have a deep understanding of the mathematical structures involved in rounding numbers to the nearest specified place. This task serves as a means of assessing student understanding in this regard.
Connecting Representations Routine Outline
Routine Step
Planning Questions
Planning Notes
Launch
What are your thinking goals for this lesson?
- Think like a Mathematician - Create a system for rounding any number to any place
What “Ask-Yourself Questions” will you give students?
- What do I know about rounding numbers that could help me answer this question?
- How could I explain my thinking to others and would my explanation make sense?
Interpret and Connect Representations
How might students think about this task? What do you anticipate seeing?
Initial task (15392) students will want to know which place to round the number to – I will not provide this so that they can see they use different rules based on the digits. For main task the students will struggle with how one rounded number could have two “original numbers” and how another will have zero. This will make them upset. This is important to anticipate and understand that frustration will help push towards students creating their own representation.
How will you select pairs of students to share their work?
I will look for students who annotate their worksheet in order to make their choice; I will press them in individual time to make sure that I understanding their thinking prior to bringing them to the board.
What sentence frames will you use?
The number __ could be rounded to ______. I know this because I noticed ______.
How will you manage the discussion? What annotations will you want to focus on during discussion?
I will want to have students focus on the digit that will need to be changed and the digit that they use to determine how to round the number. A system of circling or underlining numbers could work – just have to be consistent. I also want to have others re-voice any generalizations that we can create.
Create Representations
How will you address the unmatched representation; what guiding questions will you use?
I will have students to the 40,000 and create their own number that could be rounded to this value. I want to have wide range of numbers.
How will you select pairs to present?
I will look for a wide range of numbers – I want to list as many as possible, looking for those where students rounded up and where students rounded down.
Discuss Representations
How will you determine the focus of the final discussion/select student work?
I will use the list of numbers and focus on the smallest and the largest of the list (that work). I will create a number line and the class can determine whether the other numbers fit. We will use whatever generalizations we create in the main task to evaluate these values.
I know that __ could be rounded to 40,000 because _.
Reflection on Student Thinking
What sentence frame will you use?
The number ___ could be rounded to 127,000. I know this because _________.
What do you hope to learn from the student reflection?
I hope to collect data to assess the depth of student understanding with rounding. This routine will be used to determine whether I need to spend more time with this topic or if students generally understand the structure of rounding numbers. If the later, then I would move on with a discussion of the powers of ten. If not, I would spend a little more time attempting to generalize this phenomenon.

3. Purpose
We are going to use what we know about rounding numbers to make decisions. We are going to generalize rules about rounding using what we know about how rounding works.

4. Today’s Thinking Goals
Think like mathematicians
Create a system for rounding any number to any place

5. Thinking Questions
What do I know about rounding numbers that could help me answer this question?
How could I explain my thinking to others and would my explanation make sense?
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. Critical Vocabulary
Number
Digit

7. 15,392 Think about rounding this number…
“I would round this number to ______________ because __________________.”

8. 15,392 Talk to the person next to you
“I would round this number to ______________ because __________________.”

9. The Number After Being Rounded
The Original Number
The Number After Being Rounded
37,463 1)
36,923 2)
39,496 3)
37,827 4)
36,392 5)
c) 38,000
b) 37,000
a) 36,000
d) 39,000
e) 40,000
He/She said…
The number _________ could be rounded to _____________. I know this because I noticed ______________.
He/She
He/She knew…

10. What numbers could be rounded to 40,000?
I know that _________ could be rounded to 40,000 because _______.

11. Reflection
“The number _________ could be rounded to 127,000. I know this because______.”

12. 1,392 Think about rounding this number…
“I would round this number to ______________ because __________________.”

13. 1,392 Talk to the person next to you
“I would round this number to ______________ because __________________.”

14. The Number After Being Rounded
The Original Number
The Number After Being Rounded
7,463 1)
6,923 2)
9,496 3)
7,827 4)
6,392 5)
c) 8,000
b) 7,000
a) 6,000
d) 9,000
e) 10,000
He/She said…
The number _________ could be rounded to _____________. I know this because I noticed ______________.
He/She
He/She knew…

15. What numbers could be rounded to 40,000?
I know that _________ could be rounded to 10,000 because _______.