Find your own rectangles
Construct and evaluate different valid rectangular decompositions of a whole region.
Explore this lesson →6 practice questions · Optional read aloud · Thinking pad
Let’s understand the idea.
- A rectilinear region has straight sides that meet at right-angle turns. It can have an inward corner, such as the notch in an L-shaped room.
- Partition its surface into rectangles. They must cover the entire region without overlapping interiors.
- Multiply each rectangle's whole-number side lengths to find its area in square units. Add every rectangular part once.
- Multiplying the overall width and height would include empty space in a missing corner. That larger bounding rectangle is not the original region.
- Different correct decompositions can use different rectangles and still give the same whole area. Shared cut lines have no surface area and are not overlapping interiors.
Something to notice
Observe physical cut-and-cover work and independent explanations of why the pieces cover once. The structured model accepts any 2–6 positive grid-aligned rectangular parts whose exact cell union equals the region with no overlapping interiors, and checks every part area and the total. The finite part limit is a disclosed interface constraint, not a theorem about area. Counts are exact square units; drawings are not actual-size measurements. Original L, T, U, stair and cross regions include inward corners but no enclosed holes. Free scribbles are not automatically evaluated. Retention, spontaneous decomposition without a grid and independent real measurement require observation.
Try it. Change it. Explain it.
Open the lesson to use its pictures and controls. In How it works, connect the picture to the idea. In Play & discover, change the quantities and notice what stays the same. In Your turn, answer questions and use the explanation to check your reasoning.
You can ask for help, draw on the thinking pad, or listen to the question and answer options. Progress distinguishes independent work from work completed with help.
Let’s try it →Curriculum connection
Grade 3 · 3.MD · US Common Core
This lesson is mapped to these standards. A completed activity is practice evidence, not a claim of mastery. Content is checked against the linked official standards and its mathematical models are tested.