Topic B focuses on spatial relationships and structuring as students organize equal groups (from Topic A) into rectangular arrays. They build small arrays (up to 5 by 5) and use repeated addition of the number in each row or column (i.e., group) to find the total. In Lesson 5, students compose arrays either one row or one column at a time and count to find the total, using the scattered sets from Topic A. For example, they might arrange one row of 3 counters, then another, and another, and another, to compose a 4 by 3 array of 12 counters, and then use the same equal groups to create an array, column by column (shown below). They count to find the total, noticing that each row contains the same number of units. Thus, for 4 rows of 3, a student might observe: “There are 4 equal groups of 3.” This is foundational to the spatial structuring students will need to discern a row or column as a single entity, or unit, when working with tiled arrays without gaps and overlaps in Topic C. In Lesson 6, students decompose one array by both rows and columns. Thus, an array of 4 rows of 3 teddy bears can be pulled apart to show either 4 rows of 3 or 3 columns of 4. Also, students see that when another row or column is added or removed, so is another group, or unit. As Lesson 6 progresses, students move the objects of the arrays closer together so that the gaps are smaller, forcing them to discern the rows and columns without the visual aid of spacing. For example, when decomposing a 4 by 3 array, students see the rows as equal groups of 3. After identifying the number in each row, or group, students realize that they can write a repeated addition sentence to find the total number of objects in the array: 3 + 3 + 3 + 3 = 12. It may be noted that since there are 4 rows, the equation will have 4 addends, or 4 threes. Students add from left to right, and write the sum such that 3 plus 3 equals 6, 6 plus 3 equals 9, and 9 plus 3 equals 12. In Lesson 7, students move to the pictorial as they use math drawings to represent arrays and relate the drawings to repeated addition. For example, students are asked to draw an array with 4 rows of 3 or 3 rows of 4 on their personal white boards then use their marker to draw horizontal lines to see the rows within the array (shown below). When counting rows containing 3 or 4 objects, students apply repeated addition strategies once again, adding from left to right to find the sum (e.g., 4 + 4 + 4 = 12, such that 4 plus 4 equals 8 and 8 plus 4 equals 12). Additionally, when representing arrays with rows of 2 or 5, students may add to find the total, and naturally point out a connection to skip-counting by twos or fives (2.NBT.2); however, the focus is on establishing a strong connection between the array and repeated addition. In Lesson 8, students work with square tiles to create arrays with gaps, composing the arrays from part to whole, either one row or one column at a time. They draw the individual, separated tiles as a foundational step for Topic C where they will be working with square tiles without gaps. As usual, students relate the arrays to repeated addition. In Lesson 9, students apply this work to word problems involving repeated addition (shown at right), interpreting array situations as either rows or columns and using the RDW process, e.g., “Mrs. Levy moves desks into 3 columns of 4 desks. How many desks does she move?” In addition to drawing objects, students may also represent the situation via more abstract tape diagrams, just as they did in the final lesson of Topic A.