##
* *Reflection: Developing a Conceptual Understanding
Naming Arrays - Section 1: Vocabulary Rule Building

I have used this strategy of giving the students the answer to a problem and then having them define what it is or why it works for several years now, knowing that it helps build individual understanding. However, now that I am paying more attention to the Common Core and the 8 Mathematical Practices, I recognize this teaching strategy also points students to practice attending to precision and making sense of a problem and persevering to solve them. A lot of bang for your buck

*More Bang for Your Buck*

*Developing a Conceptual Understanding: More Bang for Your Buck*

# Naming Arrays

Lesson 2 of 15

## Objective: Students will be able to correctly name an array and define whether a configuration is an array or not.

#### Vocabulary Rule Building

*15 min*

Now that the students have worked a bit with learning to share their thinking in journals and in the Turn and Talk practice, we are ready to work a little more with the actual math at hand! The students have been working with a problem found in the book “One Hundred Angry Ants” by Elinor J. Pinczes over the past several days. They have been working with arrays in their journals, but have not had any formal instruction of what a true array is or is not.

There are times in my instruction when I like to give the answer to the students, instead of a question, and see what sense they can make of the math. If you choose, this is a really good time to use this strategy. Again, at the beginning of the year, you and I are getting to know our learners…how they see things, make sense of information, and communicate their thinking and their confusion. Try to keep in mind that the math spirals around all year and it is just as important to teach our young students to **think and make meaning**, not just answer and move to the next problem on the page.

*Students, will you please gather? Thank you for walking over to the community area so safely and quietly. That really helps us get going quickly with our learning. *

*As you remember, we have been working with these hungry ants that were trying to get to the picnic. Yesterday you pretended that there were 20 ants going to find food and you and your partner worked on configurations that we called arrays for those 20 ants. You did a wonderful job and we all saw each other’s work. *

*Now I would like you to look up at the board to the array from the book. I am going to tell you that this array has a special name. It is a 10 by 10. Can you figure out why? Look at it silently and decide what you think. Ok, now turn and talk.*

*Great, I heard a lot of interesting theories. Next, look at these two configurations. This one is an array and this one is not. Gather your thoughts silently. Ok, turn and talk. *

After allowing the students time to think independently, talk with a partner, and then as a group, you will have worked them through the first steps in Mathematical Practice 3, which is to construct viable arguments and critique the reasoning of others. I spend a lot of time on this practice, as I believe it is a cornerstone of math understanding.

*Boys and girls, let’s use what you have all noticed and your theories and come up with some rules we can all use to define what an array is.*

Step the students through this thinking. It helps make the vocabulary their own and they will be more likely to use the word and really understand the concept. You will see on the video provided in the next section that I have their thoughts written on the white board.

#### Resources

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#### Making Vocabulary Meaningful

*25 min*

After we come up with the rules for what an array is and is not, I will have the student partnerships return to working with each other to define an array in drawing and in words. This is a fantastic way to have them take what we talked about and make it meaningful to themselves. After their time working, I will spend a majority of the math time allowing students to share, field questions and comments, and revise their own thinking.

*Ok mathematicians, we have a few really good ideas listed here on what arrays are and are not. I would like you to go off and work with your partner to draw and write in your journals something to show or “defend” our thinking here. You might even try something that doesn’t work. That is ok, because often our errors lead us to better understanding. When we have worked for about 15 or 20 minutes, we will re-gather and share. Okay - off you go.*

If you have some students that are not able to draw all of the arrays for various reasons, you may want to think of giving them the total number of cubes and let them arrange the cubes in the different configurations. Then a partner could draw for them, or the student could copy one of his/her arrays into the journal. Note the photo in the the attached resources.

You may want to listen to the student’s conversations. I always hope to hear some debate among partnerships, as it reveals thinking. Again, all of this can seem loose, but it is the beginning of a true math community, in which struggling in a problem leads to understanding, not just an answer on the paper.

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#### Sharing and Critiquing

*15 min*

Bring students back together. If you have created a gathering norm and a talking moves chart, direct their attention to those as you begin allowing the students to share. Don’t worry if you spend extra minutes modeling and practicing the norms, as it will create an amazing sharing community later on.

I have the students put their work up on the document camera and use a microphone. If you have these sorts of tools, this may be a great time to use them. If not, think of a way for students to display their work while they focus on talking through it. As they share, you may want to consider helping prompt them to speak more in depth. In my video, you will hear me pipe up every now and then.

#### Resources

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Today, I am introducing a 12 Ants homework assignment to allow the children more practice using their math journals. This assignment is beneficial in three ways.

First, it allows the children to work alone on journaling applying what we've learned during prior lessons on how to use the Math Journal. Journaling is not an easy task for a beginning third grader, so the more practice, the better.

Secondly, it is a wonderful way to allow parents to see what math looks like in our room. Parents are always surprised to see that there may not be a “right” answer, but that their son or daughter must think and expand their answers into understanding. This sets the homework tone for the rest of the year.

Last, I am assigning a prompt that is front loading a skill that will come up in about a week - division. We will go back and make connections to it when we begin the division unit.

#### Resources

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- UNIT 1: Developing Mathematical Practices
- UNIT 2: Understanding Multiplication
- UNIT 3: Using Multiplication to Find Area
- UNIT 4: Understanding Division
- UNIT 5: Introduction To Fractions
- UNIT 6: Unit Fractions
- UNIT 7: Fractions: More Than A Whole
- UNIT 8: Comparing Fractions
- UNIT 9: Place Value
- UNIT 10: Fluency to Automoticity
- UNIT 11: Going Batty Over Measurement and Geometry
- UNIT 12: Review Activities

- LESSON 1: It's As Basic As That
- LESSON 2: Naming Arrays
- LESSON 3: Variables
- LESSON 4: X Represents Groups Of
- LESSON 5: Each Orange Had 8 Slices
- LESSON 6: Creating a Word Problem Book
- LESSON 7: Equal Groups on a Number Line
- LESSON 8: Collections of Equal Groups
- LESSON 9: Explaining Multiplication
- LESSON 10: The Story Of Multiplication Day 1
- LESSON 11: The Story of Multiplication Day 2
- LESSON 12: Making Equal Groups Part 1
- LESSON 13: Making Equal Groups Part 2
- LESSON 14: Associative Property
- LESSON 15: Associative Property With Manipulatives