Students will use paper to model the partitioning of rectangles into two, three, and four equal shares. Students will transfer this concrete model to represent equal shares on a drawing, labeling the shares as halves, thirds, or fourths. Throughout the lesson, the teacher will use an interactive geoboard app to guide instruction and represent student ideas.
This activity was created as a result of the ALEX Resource Development Summit.
Topic C naturally follows Topic B, where students composed and manipulated the rows and columns of an array. This topic is designed to deepen students’ understanding of spatial structuring as they build and partition rectangles with rows and columns of same-size squares. In Lessons 10 and 11, students compose a rectangle by making tile arrays with no gaps or overlaps. They use their prior knowledge of making equal groups and the spatial relationship between rows and columns to construct rectangular arrays. In Lesson 10, given a number of tiles (up to 25) students are asked to create rectangular arrays that show equal rows or columns (up to 5 by 5). In Lesson 11, students build upon this understanding, manipulating a set of 12 square tiles to compose various rectangles (e.g., 1 column of 12, 2 rows of 6, and 3 rows of 4). As students share their rectangles, they are encouraged to ask themselves, “How can I construct this differently?” They use repeated addition to find the total number of squares, alternating flexibly between the number in each row and the number in each column as the unit. Lesson 12 introduces the added complexity of composing a rectangle by using math drawings. Once students have arranged square tiles into a specified rectangular array without gaps or overlaps, they trace to construct the same rectangle by iterating the square unit much as they iterated a length unit in Module 2 to create a centimeter ruler. Next, students use the spatial reasoning developed so far in the module to draw the same rectangle without tracing, using their understanding of equal columns and equal rows. After students compose rectangles, they decompose, or partition, them using tiles in Lesson 13. For example, when working with an array of 5 rows of 3 (and a total of 15), they see that if they remove a row of 3, they have 4 rows of 3 (and a total of 12). Alternately, they see that instead of 3 columns of 5, they have 3 columns of 4. In Lesson 14, students are encouraged to think flexibly as they use paper models to further develop their ability to visualize arrays. Students fold two congruent rectangular pieces of paper to create two 2-by-4 rectangular arrays composed of same-size squares. Next, they use scissors to cut the rectangle into 2 rows of 4 squares (first paper) and 4 columns of 2 squares (second paper). Then, students cut each row or column into individual square units. As a result, they see that just as a rectangle is composed of equal rows or columns, each row or column is composed of squares, or iterated units. Students now have 16 same-size squares and can create different rectangular arrays with them (e.g., 1 by 16, 2 by 8, and 4 by 4). Lesson 15 moves toward more abstract reasoning as students use math drawings to partition rectangles. With colored pencils and grid paper, students shade in rows or columns and relate them to the repeated addition number sentence (e.g., 5 rows of 3 squares = 3 + 3 + 3 + 3 + 3, or 5 threes). Then, given a rectangle with one row or one column missing, students draw in the remaining squares to complete the array (shown on right) and find the total by relating their completed array to repeated addition. In Lesson 16, students practice spatial structuring skills by working with grids and diagrams. They copy designs using same-size squares and triangles (half of the squares) as manipulatives. Students create their copies on paper with grid squares of the same size as the manipulative square. In order to successfully create these, they must pay careful attention to which grid square to color and how many spaces to leave. Students share designs with a partner, who then tries to copy the designs exactly. Finally, students use grid paper to design a tessellation using a core square composed of a 3 by 3 array of same-size squares. They create designs by coloring the 9 squares and then iterating that core unit. This provides students with the opportunity to sharpen their spatial structuring skills, as they must count rows and columns to successfully create a quilt of their designs.