The Common Core State Standards for Mathematics references the use of arrays at every elementary grade level. Arrays are used for counting as well as for organization, measurement, multiplication, and representation of fractions. As adults we have a well-developed understanding of both discrete (separate objects) and continuous (area model) arrays. It is sometimes difficult for us to see when children don’t observe features of an array that are obvious to us. .
What should we consider?
Arranging discrete items into rows of equal numbers of elements to assist in counting is probably one of the earliest experiences young students have with arrays. Helping children to understand the counting and repeated addition possibilities of these arrangements is important to address early in the learning process.
The multiplicative features of an array will probably not be perceived by students until later, so teachers should be aware of differences between their own understanding of “obvious features” and children’s developing awareness of those same features. Transitioning from discrete arrays to continuous area models is another step that requires some assistance from the teacher. Using individual units to “cover” the area of a rectangle may not lead to an understanding of the multiplicative aspect of such a measurement. Students need time and assistance to discover for themselves the relationship of equal numbers of units in each row and a specific number of rows.
What should we do?
Providing many activities and spending time talking with students to determine their level of understanding is important at the early stages because arrays will be used later for both measurement activities and fraction activities. Some of the resources included in this collection will provide examples of the types of activities students need to develop the deep understanding of the features of both discrete and continuous arrays. Teachers will want to help young students see and use the characteristics of arrays to organize and count objects. Allow students to discover how arrays provide information about factors, how they can be used to identify prime numbers, even vs. odd numbers, or square numbers. Work with older students to be sure that they see the relationship between arrays of discrete objects and continuous area models. Explore the resources in this collection for ideas and strategies that ensure your students develop expertise in recognizing and using the characteristics of arrays.
In this Focus on Math blog entry, Carollee Norris describes how she uses a 100-dot array to develop number sense and mental math skills. As she covers some rows (and later rows and parts of rows) on a projected grid, students learn to identify quickly the number of rows and dots that are visible, as well as how many are hidden (to complete 100). Through practice, children are able to visualize the structure of the array and apply it to mental math and problem solving tasks.
This one-page article describes and illustrates how arrays can be used to represent many number concepts, including building multiplication facts, commutativity, parity (odd/even), and exploring factors, prime numbers, and square numbers.
This brief article advocates the use of arrays to model the commutative and distributive properties as well as the inverse relationship between multiplication and division. The author explains how arrays also help children form mental pictures that support their memory and reasoning.
This interactive Flash applet helps students develop the concept of equal groups as a foundation for multiplication and division. The applet displays an array of dots, some of which are covered by a card. Student use the visible number of rows and columns to determine the total number of dots. Clicking on the card reveals the full array, and a voice announces the total.
In this lesson, students first create factor posters for a variety of different numbers that will be displayed in the classroom to be utilized as a resource throughout the school year. They make discoveries about factors using color tiles, represent their discoveries using graph paper, and display their information on poster board as find factors of an assigned number. The plan includes a list of materials, questions, assessment options, and extensions.
This interactive applet allows a student to visually explore the concept of factors by creating different rectangular arrays for a number. The user constructs the array by clicking and dragging on a grid. The length and width of the array are factors of the number. A student can elect an option of a randomly selected number or the student selects his own number between 2 and 50. Exploration questions are included to promote student discovery of mathematical concepts with factors.
This interactive Java applet helps students explore the relationship between area and multiplication. Users are asked to draw all possible factorizations of a number as rectangular arrays by clicking and dragging across a grid. Options include the use of the commutative property (e.g., user must enter both 2x4 and 4x2 for factors of 8 represented by different orientations of the array), entering a number of the user's own choice, and an optional scoring feature allowing users to keep track of the number correct.
This lesson helps students develop conceptual understanding of 2-digit multiplication. Students decompose 2-digit numbers, model area representations using the distributive property and partial product arrays, and carry out paper-and-pencil calculations from the arrays. Included are student activity sheets, a link to a supporting online applet, and extension suggestions.
This interactive Wolfram Demonstration helps students visualize the relationship between multiplication and the area of a rectangle. On a 10x10 multiplication table, with 1 in the lower left and 100 in the upper right, users adjust the size of a rectangle by dragging a vertex or sliders, or by choosing a product from a pulldown list. The resulting rectangle highlights the width, height and area (factors and product) as well as other equivalent areas (products) on the board, illustrating commutativity and multiple factor pairs. Wolfram CDF Player, a free download, is required to view this resource.
This interactive Java applet use arrays and three different computational models (Grouping, Common, or Lattice) to help students visualize and understand the process of multiplication of whole numbers. In each method the user drags sliders to adjust the two factors and then observes the resulting changes in the rectangular array and in the respective algorithm and product.
This article from the NCTM journal Teaching Children Mathematics, explains how to help students see the structure of two-dimensional arrays and suggests activities that connect spatial structuring to multiplicative thinking.
In this article from the NCTM journal Teaching Children Mathematics, students explore different characteristics of the numbers 1–25 and have opportunities to represent the commutative property of multiplication, construct an understanding of the difference between prime and composite numbers, and explore what makes a number “square.” The exploration is a way to introduce students to the array model of multiplication, a tricky concept for them to construct. They interchange rows and columns until they have worked with arrays many times and in many different contexts.