# Arrays, Arrays, Arrays

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## Objective

SWBAT construct different arrays using the same quantity and describe the array using rows and columns.

#### Big Idea

Rearranging arrays to understand equivalent products.

## Introduction

5 minutes

Distribute square tiles for to students to explore, and then ask the students to follow along with you and create the same type of model you are creating.

Because the students have not used the tiles recently, I allow a few minutes for students to explore using the tiles in different ways on their own.  I find this eliminates some of the off task behavior that occurs with their natural interest in a manipulative.

Display or draw an array of 4 rows with 3 columns. Tell them they will be creating arrays to understand multiplication.  This is where I introduce the word multiplication and announce they are learning to multiply.  Some students may be familiar with this from second grade, but this is their first experience with it in my classroom.  I explain to the students they are creating a model of multiplication. I write the words array and multiplication to display using the document camera. These vocabulary words will be recorded in their math journals later in the lesson.

## Modeling With Square Tiles

15 minutes

As I build models of arrays, I create a concrete idea of rows by giving students the visual model of a movie theater. You sit in a row, with your family and friends, at the movie theater. That way they're sitting next to you. For columns, I create a concrete idea by having the students think of what holds up the roof, or the floors of a tall building.

I think it is important to spend time defining the difference between rows (as in a movie theater that are horizontal) and columns (vertical and hold up building structures).

If students are struggling with this they can also lay tiles in rows and stack the tile into a column.

Demonstrate to students how to build models of different dimensions and record the multiplication number sentence to match each model.  I create one array and number sentence at a time, emphasizing which is the row and which is the column.  During this time I repeat questions including, How many tiles are in each row? How many tiles are in each column? How can you find out how many tiles are in the whole array? I avoid asking How many tiles are there? because it can be answered by just counting.

As each model is built, rotate the model and show the new number sentence. I try to keep the modeling of factors below 5 at this time, and I also include arrays with one as a factor. 1 x 4, 1 x 3, etc. I usually start with arrays including the factor of two.  These are followed the factor of three, and then I go back to the factor with one.

Because I want them to focus on the vocabulary and understand the difference between columns and rows, I avoid doubles at this point of the lesson.  Number sentences are recorded using:  rows x columns = whole amount

Once students are showing confidence with the difference between rows and arrays, you can start showing square arrays using the same factors.  3 x 3, 4 x 4, 5 x 5, etc.

I ask students to describe the shapes of the arrays either as a square or a rectangle. Tell students, If an array is a square shape it is a square number, meaning its factors are the same.  Even though this is not part of the Common Core third grade standards, I feel it is important to identify square numbers.

Continue building arrays for students as needed for student understanding.

## Modeling With Graph Paper

10 minutes

Move from building an array to include recording the array on graph paper. Record the number of rows and columns next to each array, and then record the number sentence.   Include single row and single column arrays in your modeling.

During this section of the lesson, as each array is built I draw it on the graph paper on the document camera.  I also review the terminology of horizontal for rows, vertical for columns. Because precision is an important mathematical practice, I make sure to use opportunities such as this to emphasize how critical it is to check our work, make sure our representations are accurate to our problems, and that we are using the correct mathematical vocabulary (MP6). Drawing the arrays precisely on the lines of the graph paper is demonstrated.  I also show the students an array that is not on the lines of the graph paper for them to compare.

Rather than creating all of the arrays, and then moving to drawing them, I make a point to draw each array just after it is built and I keep these two side-by-side.  It is important to make explicit connections between the two, so we periodically step back to compare the model to the array.  I would also recommend using a larger scale graph paper during this lesson for precision in drawing.

## Let's Practice Together

15 minutes

I progress with this slowly to give each students time to complete the each step, alternating between providing examples with the square tiles, graph paper, or number sentences. When working with one approach, students practice by completing the problem using the other two approaches. I give my students needing support with fine motor skills larger scale graph paper because this can quickly become an overwhelming task if a great deal of their energy and focus is on working within a constrained area.

You may find that some students need support with how to rotate the array.  They can be helped by drawing different colored lines on the opposite edges of a piece of paper.  They can start with red on the top and bottom and blue on the sides for the first array.  The second array would have the blue on the top and bottom and the red on the sides.

Remind students about the structure of the number sentence of rows x columns = whole amount. My English language learners benefitted from my use, and theirs, of gestures to practice the rows and columns vocabulary.  The gesture for the rows is moving the arm horizontally in front, and vertically for the columns.

Once students are successfully completing each component, we move on to working independently.

## Try It Yourself

10 minutes

Students build their own arrays, record on graph paper, and write the number sentence.  Students should also rotate each array to demonstrate the related array using the same factors.

Students began with 9 tiles and built all the possible arrays.

After every few minutes, students added three tiles to create new arrays.  As the numbers get larger, they will be building more arrays.

As the students worked, I would ask the to describe their array using the vocabulary rows and columns. Some sentences I asked them to use were, I have _____ rows and _____ columns.  I have _______ tiles altogether.

I have _____ tiles in rows of _____ and columns of ________.

This next video demonstrates how the array is recorded on grid paper with the multiplication sentences.

## Wrap Up

5 minutes

Working in partners on whiteboards or paper, one student writes an array or number sentence, and the other student writes the array or number sentence to show the Commutative Property of Multiplication.  Although this property was not specifically addressed in this lesson or in this standard, my students transferred their knowledge of the Commutative Property of Addition to the idea of multiplication.  My teaching sequence this year included a review of addition and subtraction before moving to multiplication.  We recorded this term in their vocabulary journal for addition, and after this lesson we added it for multiplication.