## Explaining and figuring out the meaning of rows vs. columns - Section 4: Application Using Manipulatives

*Explaining and figuring out the meaning of rows vs. columns*

# Introduction: Discovering an array

Lesson 1 of 12

## Objective: SWBAT define, identify and use an array to show multiplication facts.

## Big Idea: Students find arrays within the school building, use counters to create arrays to describe how they find the product of given factors and assess themselves using a rubric.

*65 minutes*

#### Warm Up: Tic Tac Toe Review

*10 min*

Tic Tac Toe Review Place Value

I started todays's lesson with a review game of tic tac toe. Students chose three problems to do that were either diagonal, vertical, or horizontal with the rule that the center problem had to be one of their three. I wanted them to show me if they could multiply using two digits by one digit. I was checking to see if they had covered it in third grade.

They worked quickly and quietly. I told them that the first one done with all three could go to the board and write out one of the answers. One girl raised her hand and chose to write out the expanded form problem.

We continued on as students gradually volunteered to write the answers on the Smart Board. I discovered that they had all picked two of the same sets of problems. They either went diagonally or up and down. So, some were left undone. I purposely didn't solve those because students picked the ones they chose for a reason and wanted to be sure they understood what they were doing, since this activity acts as a spiral review. We will return to this tic tac toe again in the near future to revisit concepts that they may still struggle with.

#### Resources

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I introduced the word "array" on the Smart Board. We defined the word using our text book glossary and iPad Dictionary ap for two sources.

I returned to the tic tac toe board to and asked students if they could see rows and columns. They all quickly said that they could see it in the tic tac toe board and that they could see 9 squares. One boy said..."Yeah, three times three makes nine and it makes a square." *I left it be for now, but strongly suspect that he will understand square numbers without a hitch. *

I told them that we were going on an array hunt in the building. We took our iPads and explored the building and I had them photograph an array that they saw. After they took their photo, they were to close their iPad. Soon, all 21 students were ready to come back to the classroom with a photo. (If you don't have iPads, you can use this website for example photos). Using Apple TV I asked everyone to share photos on the Smart Board and discuss the idea of an array. One student had taken a photo of the bricks outside our door. The class said out loud almost all at once " That's not an array!"That's Not an Array! I was afraid he was embarrassed. One student explained to him that it couldn't be an array because even though the bricks are lined up, they are not in equal rows and columns. We shared other photos and talked about how we could multiply to find all of the squares or objects in the array.

These questions proved important ones we considered.

1. How can you use multiplication to find out how many items are in the photo?

2. How would an array be useful to solve for an unknown factor?

They were completely stumped on the second question and answered the first without any trouble.

Below are just some samples of their photographs.

This little camera activity got them thinking about rows and columns, and it helped those who didn't understand what an array is just through the photographs.

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**Supplies: **LOTS of counters. Rubric Sheet

I started this lesson by using the Smart Board to help them compare drawings of groupings that they were accustomed to in third grade when they first learned to multiply basic facts.Smart Board Lesson

The first two pages of the Smart board lesson defined that grouping is not an array. The second page shows another example of what an array isn't. Students sketched these in their notebooks. Next they drew the concept of 2 rows of 10. I drew a fresh page of 10 rows of 2. We talked about the Commuative Property of Multiplication as I referred to the white board. The whiteboard shows the vocabulary we were learning and the drawings of the concepts of Commutative Property. I wanted these visuals permanently there so I didn't use a SmartBoard Notebook Page. They copied the whiteboard notes down in their journals.

I referred back again to the importance of rows and columns as we talked about being an architect who is designing a movie theater. The owner wanted 2 rows of ten seats across the front of his screen. I flipped to the next page we had sketched of 10 rows of two and asked; Why is it important to keep the idea of row and column as 2 rows of 10. One boy raised his hand and said..." You would have weird seating at the theatre if you had it like that." as he referred to the 10 rows of 2. " Yeah, somebody won't see very good," was the next comment.

Why is it important to follow the language and idea in our facts?

"Because you would get fired if you were that architect!" the first boy blurted out.

Well, at least they are starting to see the value of their math education!

We practiced some more facts that I had written on the board. I used simple ones so that they could draw the arrays easily. 3x4 ( as we see on the last page) 3x2, 4x5 etc. They all had a good chance to practice and I thought it was solid!

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I told them it was time to try it using counters. I showed them on the carpet using 5x2 how to arrange the 5 rows of two. I showed them a grouping example of 5 groups of 2 and explained once again that this was not an array. I used this fact because it would transition nicely into their activity/ rubric/ assessment work.

I passed out their counters and a rubric assessment. I told them they needed to read the directions and create an array as directed on the sheet. They also needed to create the Commutative pair and array. I told them that after they were finished, to raise their hand and I would come visit them to see if they could explain their model. After I visited with them, they would fill out their rubric, grading themselves on their understanding and evaluate what needed to be done to improve understanding.

These video clips are what I found. I was astounded to see that figuring out what a row was with the counters was extremely difficult for most of my students. Even though we had several examples of practice together, when it came down to explaining it, they couldn't! Wow.Explaining and figuring out the meaning of rows vs. columns & Rows go across.

It is so strange to see this happen. But, with CCSS, I think that the conceptual accountability is what makes the difference. Explaining our understanding and not just doing something really is a challenge for almost all of my students. This is all the more reason that this lesson is so important.

Finally, toward the end, one student who had struggled at the beginning, showed improvement in Explaining what we have learned. He still stumbles, but eventually, he got it!

When students were done, they filled out their self assessment rubrics to monitor their progress.

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#### Homework: Array

*10 min*

Homework: Using Graph paper, students will create arrays representing the 15 most difficult facts to remember.I listed on the board:

4x3,6x3,7x3,8x3,9x3,6x4,7x4,8x4,9x4,7x6,8x6,96,8x7,9x7,9x8 These are noted as the 15 facts most difficult to memorize.

I passed out graph paper to each student. I stopped at three of my low students and drew 3x2 arrays in their graph paper by just drawing dots. I explained that this was their example of how I expected the homework to look.

I went back to the board and drew an example for everyone to follow as I explained that I needed them to practice these 15 facts by using graph paper to make arrays for each of the facts.

I closed the lesson by explaining that this skill will build them a foundation for many other concepts in multiplication.

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- UNIT 1: Place Value and Multi-Digit Addition & Subtraction
- UNIT 2: Metric Measurement
- UNIT 3: Graphing and Data
- UNIT 4: Concepts of Multiplication
- UNIT 5: Geometry
- UNIT 6: Fractions 1: Understanding Equivalence in Fractions and Decimals
- UNIT 7: Fractions 2: Addition and Subtraction Concepts/ Mini unit
- UNIT 8: Fractions 3 Mini Unit: Multiplying Fractions by Whole Numbers
- UNIT 9: Division Unit
- UNIT 10: Addition and Subtraction: Algorithms to One Million
- UNIT 11: Place Value
- UNIT 12: Addition and Subtraction Word Problems
- UNIT 13: Multiplication Unit

- LESSON 1: Introduction: Discovering an array
- LESSON 2: How To Easily Find All Those Factor Pairs
- LESSON 3: Factor Pairs, Arrays, Patterns and Fun! Building factor pairs to 50
- LESSON 4: Using a Factor Pair Calculator to Check Factor Pairs from 51-100
- LESSON 5: Circle and Stars: Grouping Compared to Arrays
- LESSON 6: What's My Rule? : An introduction to function tables.
- LESSON 7: Prime, Composite and Square Numbers
- LESSON 8: Assessment: Quiz 1 Groupings, Arrays, & Factor Pairs Quiz
- LESSON 9: A Quick Quiz: Writing about an Array
- LESSON 10: Order of Operations: Solving Simple Equations Using PEMDAS
- LESSON 11: A Student Driven Review for the Test on Multiplication Concepts
- LESSON 12: Testing Multiplication Concepts