Getting Started |
| Misconception/Error The student does not have a correct strategy to determine the area of the figure. |
| Examples of Student Work at this Level The student:
- Calculates the perimeter of the figure (correctly or incorrectly).
- Calculates the perimeter of several of the decomposed rectangles (correctly or incorrectly).
- Multiplies some or all the values given in the diagram.
- Calculates the average of the dimensions.
- Does not provide enough work (or clarity in the work) to determine the solution strategy.
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| Questions Eliciting Thinking What does the problem ask for? What do compose and decompose mean in this context?
How does finding the perimeter help you?
What were you trying to find by multiplying those numbers?
Can you explain the steps you have taken in your calculations? What were you trying to find? |
| Instructional Implications Ensure that the student is familiar with triangles and rectangles and the terms used to describe their dimensions such as base and height. If needed, review area and perimeter. Reinforce that perimeter is a linear measurement determined by adding the lengths of all the sides while area is measured in square units and is a measure of the surface contained in the interior of a polygon.
Guide the student to decompose the garden into non-overlapping rectangles. Assist the student in determining the dimensions of the component rectangles. Ask the student to calculate the area of each component rectangle and explain that the area of the original figure is the sum of the areas of the component parts. Provide additional rectilinear figures for the student to decompose into rectangles in order to find the area. Encourage the student to show all work neatly and completely and to label the answer with the appropriate unit. |
Moving Forward |
| Misconception/Error The student makes conceptual errors in decomposing the figure and calculating the area. |
| Examples of Student Work at this Level The student demonstrates an understanding of decomposing the figure to find the area, but is not able to correctly find the area. The student:
- Incorrectly identifies the dimensions of a decomposed rectangle.
- Decomposes the figure into overlapping rectangles.
- Finds the areas of three (or more) individual rectangles, but does not find their sum.
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| Questions Eliciting Thinking How did you determine the dimensions of each rectangle?
Can you outline each individual rectangle and label its base and height?
Do your rectangles overlap?
What is the question asking you to determine? What do you need to do to find the total area of the figure? |
| Instructional Implications Model for the student how to decompose the figure into non-overlapping rectangles. Assist the student in determining the dimensions of the component rectangles. Ask the student to calculate the area of each component rectangle and explain that the area of the original figure is the sum of the areas of the component parts. Provide additional rectilinear figures for the student to decompose into rectangles in order to find area. Encourage the student to show all work neatly and completely and to label the answer with the appropriate unit. |
Almost There |
| Misconception/Error The student makes minor mathematical errors. |
| Examples of Student Work at this Level The student makes a minor error in computation or labeling units. The student:
- Calculates incorrectly in one step of the problem but all other work is correct.
- Uses one wrong number when assigning a base or height value to a rectangle.
- Transposes a number from one step to the next.
- Uses the wrong units or no units at all.
- Indicates square units by putting the exponent of two on the numerical answer, rather than on the units (e.g.,
 rather than 319  ).
- Writes a mathematically incorrect statement in some step of the work [e.g., writes multiple steps together into one long (incorrect) equation: 14 x 3 = 42 + 192 = 234 + 85 = 319].
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| Questions Eliciting Thinking Can you check your work for errors?
What units should be used in this problem? Why?
How is different from 319  ? What does each mean?
Does 14 x 3 = 319? Can you show your work in individual steps? |
| Instructional Implications Review multi-digit multiplication and division and order of operations as needed.
Provide specific feedback concerning the error(s) made and ask the student to revise the work. Provide additional opportunities for the student to compose and decompose shapes and use formulas to find the areas of polygons.
Consider implementing the MFAS task Swimming Pool Walkway (6.G.1.1) for further practice decomposing compound shapes. |
Got It |
| Misconception/Error The student provides complete and correct responses to all components of the task. |
| Examples of Student Work at this Level The student finds a total area of 319 for the garden. The student may have found this area by:
- Calculating the area of three or more smaller rectangles and adding them together to get the total area. There are several possible ways this can be done.
 
- Finding the area of the larger rectangle composing the entire shape (29 x 20), then subtracting the sum of the areas of the “extra” rectangular spaces (9 x 24, 3 x 5 and 3 x 10).
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| Questions Eliciting Thinking Can you think of a different way you could have found the area?
What is the fewest number of rectangles you could decompose the figure into?
How could you find the perimeter of the figure? Are there any missing dimensions you would need to find first? If so, how could you find those? |
| Instructional Implications Ask the student to find the area of more complex and challenging rectilinear figures (e.g., ones with many parts and the least number of dimensions given).
Consider implementing other MFAS tasks from the standard 6.G.1.1. |
