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The purpose of this How Many Do You See? routine is to prompt students to decompose a rectilinear figure to find its area and to recognize that there are many ways to do so. Students are also reminded that area is additive. The reasoning in this routine prepares students to reason flexibly about the area of rectilinear figures later in the lesson.
How many shaded squares do you see? How do you see them?
In this activity, students solve geometric problems by reasoning about length and area, decomposing and recomposing of rectangles, considering units of measurements, and performing operations.
Each question can be approached in a variety of ways. Consider asking students to create a visual display of their approach and to share it with the class.
The first problem offers students an opportunity to make sense of a problem and persevere in solving it (MP1). They may focus on the area of the banner and poster paper or think about cutting up the poster paper into pieces that can be used for the banner. Students will also need to convert between feet and inches at some point in their solution.
Jada’s teacher buys a poster paper that measures 36 inches by 24 inches. Her plan is to cut it into pieces, rearrange them, and tape them back together to create a welcome banner that is 8 inches tall and 8 feet long.
In this activity, students perform operations on multi-digit numbers to solve situations about perimeter and area. They use operations to convert units of measurements along the way. Converting inches to feet could be done by dividing by 12, but this is not an expectation at this point. Students could perform the conversion with multiplicative reasoning. To convert 180 inches into feet, they could reason , or and .
In IM Grade 3, students learned that area is additive, and that the area of rectilinear figures can be found by decomposing them into non-overlapping rectangles. Students apply that understanding in this activity, after converting lengths in different units into the same unit.
A classroom orders new carpet and baseboards. Tyler and a couple of friends are helping to take measurements.
Here is a sketch of the classroom and the measurements they record.
For each question, show your reasoning.
How many feet of baseboard do they need to replace in the classroom? How many inches is that?
1,200 inches of baseboard material was delivered. Is that enough?
“Today we used all kinds of operations on large numbers to solve problems about measurements.”
“Can you find examples where it was helpful or necessary to multiply, divide, add, or subtract multi-digit numbers?” (Multiplication and division were handy for converting from one unit to another. Multiplication was needed to find the area of rectangular shapes. Addition helped us find a total length or area. Subtraction was useful for finding a difference or amount left over.)
“What challenges did you come across when solving the problems? When multiplying or dividing?”