Mid-Unit Assessment

1.

Which parallelogram has an area of 60 square units?

A.

Parallelogram A has base 10, height 12, diagonal length 15.

B.

Parallelogram B has base 6, height 10, diagonal length 10.2.

C.

Parallelogram C has base 15, height 12, diagonal length 15.

D.

Parallelogram D has base 6, height 6, diagonal length 10.

Answer:

Teaching Notes

Students who select Parallelogram A are likely applying the formula for the area of a triangle, 12bh. Students who fail to select Parallelogram B may be multiplying one base of the parallelogram by the other base, rather than multiplying one of the bases by the height. They may also be confused by the fact that the height is horizontal. Students who select Parallelogram C are picking the parallelogram with a perimeter of 60 units, and may need to revisit the conceptual differences between area and perimeter. Students who select Parallelogram D may be multiplying the two bases together, or they may be multiplying the height by the incorrect base.

2.

Select all the triangles that have an area of 30 square units.

A.

Triangle with sides of length 10, 6, and unknown. The sides of length 10 and 6 form a right angle.

B.

Triangle with sides of length 10 and a height of 8.66.

C.

Triangle with sides of length 10, 10, and unknown. The height of the triangle from the side of length 10 is 6.

D.

Triangle with sides of length 10, 6, and unknown. The height of the triangle from the side of length 6 is 8.

E.

Triangle with sides of length 3, 10, and unknown. The sides of length 3 and 10 form a right angle.

Answer: A, C

Teaching Notes

Students who select Triangle B may be calculating the perimeter rather than the area. Students who select Triangle D may be treating the side of length 10 as the height. Students who select Triangle E may be multiplying the base and height but may not be multiplying that product by ${1 \over 2}$ . Students who fail to select Triangle A may have a major misconception about the area of a triangle or may be neglecting to divide by 2. Students who fail to select Triangle C may not recognize the external height.

3.

Select all the parallelograms that have an area of 16 square units.

Four parallelograms in a grid. They are labeled A, B, C, and D.

A.

Parallelogram A

B.

Parallelogram B

C.

Parallelogram C

D.

Parallelogram D

Answer: B, C, D

Teaching Notes

Students who select Parallelogram A may be decomposing and rearranging the parallelogram, but miscounting the number of unit squares; the area is 15 square units. Students who fail to select Parallelogram B may be confusing the side length with the height. Students who fail to select Parallelogram C may be using the area formula instead of decomposing and rearranging, and failing to use the vertical side length as the base. Students who fail to select D may think that a square is not a parallelogram.

4.

On each triangle, draw a segment to represent the height that corresponds to the given base. Label each height with the word “height.”

Two identical triangles: one with the shortest side labeled as the base; the other with the longest side labeled as the base.

Answer:

Sample response:

Teaching Notes

Identifying the height for a chosen base is important for calculating the area of a triangle.

5.

Draw two parallelograms, both with areas of 18 square units. The two parallelograms should not be identical copies of each other.

Answer:

Sample responses:

Minimal Tier 1 response:

Work is complete and correct.
The two parallelograms may have the same base and height as long as they are not congruent.
Sample: Two parallelograms (rectangles allowed) with base and height lengths that when multiplied equal 18.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Only one parallelogram is drawn. Only one parallelogram has the correct area. An equation such as “ $6 \boldcdot 3 = 18$ ” is written, but the base or height of the parallelogram is slightly off.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: A length or a side that is not perpendicular to the base is used as the height. Shapes drawn are not parallelograms.

Teaching Notes

This question purposely admits the possibility that students might draw rectangles if they know that a rectangle is a particular kind of parallelogram. The question could be modified to instruct students to draw parallelograms that are not rectangles.

6.

Find the area of the figure. Explain or show your reasoning.

Answer:

31 square units. Sample reasoning:

Decompose the figure into parallelograms and triangles with known bases and heights. Find the sum of the areas of the components.
Enclose the figure in a 9-by-6 rectangle. From 54 square units, subtract the areas of the right triangles that are not part of the figure.

Minimal Tier 1 response:

Work is complete and correct.
A diagram is included.
Acceptable errors: No units are included.
Sample: A box around the shape has an area of 54. The triangles on the outside have areas of 8, 5, 8, and 2. The area of the box minus the triangles is 31.

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: The area of a small number of the partitioned shapes are calculated incorrectly. Addition or subtraction to find the area of the polygon is done incorrectly. The area of one of the partitioned shapes cannot be correctly calculated because the base or height does not lie on a vertical or horizontal line. The partitioned shapes are not shown on the diagram.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: The partitioned shapes are not shapes for which students can find the area. Incorrect area formulas for triangles or rectangles are used. Calculations show many errors.

Teaching Notes

Watch for students who break the shape into parts in unusual ways. These students may not be recalling well the methods developed in the lessons, and are likely to create parts for which they cannot determine the area.

7.

The figure is a diagram of a wall. Lengths are given in feet.

A room has two walls that are this shape and size. A painter has a bucket of paint that can cover 160 square feet of wall surface. Is that enough paint to cover both walls? Explain or show your reasoning.

Answer:

No, that is not enough paint. Sample reasoning: Each wall is made up of a rectangle that is 11 feet by 7 feet and a triangle with a base of 11 feet and a height of 2 feet.

The area of one wall is $(11 \boldcdot 7) + (\frac{1}{2} \boldcdot 11 \boldcdot 2)$ , or $77 + 11$ , which is 88 square feet.
The area of two walls is $2 \boldcdot 88$ , which is 176 square feet.
The total area is more than 160 square feet.

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Sample: Not enough, because the combined area of both walls is more than 160 square feet. The area of one wall is the area of a 9-by-11 rectangle reduced by the area of 2 triangles, each with a base of 5.5 feet and a height of 2 feet.
- Area of rectangle: $9 \boldcdot 11 = 99$
- Area of 2 triangles: $2 \boldcdot \frac{(5.5) \boldcdot 2}{2} = 2 \boldcdot (5.5) = 11$
- Area of one wall: $99-11$ or 88 square units
- Area of two walls: $2 \boldcdot 88$ or 176 square units

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: Units are omitted. Arithmetic errors are made when calculating area. Acceptable errors: Incorrect total area due to an error in computing a partial area.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: A plausible but flawed strategy is used for decomposing or enclosing the wall. Incorrect area formulas are used with correct decomposition strategy.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: No reasonable strategy is used for decomposing or enclosing the wall.

Teaching Notes

Students may decompose each wall into a rectangle and one or more triangles, and rearrange the triangles into rectangles. For each wall, students may enclose the given shape with a rectangle and subtract the areas of the triangles that are not part of the wall.