End-of-Unit Assessment

1.

Polyhedron P is a cube with a corner removed and relocated to the top of the cube. Polyhedron Q is a cube with the same size base as Polyhedron P. How do their surface areas compare?

cube. top front corner of cube is removed and placed on top of cube. — P

A.

Polyhedron P’s surface area is less than Polyhedron Q’s surface area.

B.

Polyhedron P’s surface area is equal to Polyhedron Q’s surface area.

C.

Polyhedron P’s surface area is greater than Polyhedron Q’s surface area.

D.

There is not enough information given to compare their surface areas.

Answer:

Polyhedron P’s surface area is greater than Polyhedron Q’s surface area.

Teaching Notes

Students who select Choice A may think (or may just be guessing) that removing a piece of a solid always decreases its surface area. Students who select Choice B are correct that the large cube has the same surface area before and after the small cube is removed, but they are forgetting that putting the small cube back on top also contributes to the surface area. Students who select Choice D may not believe that the problem can be solved without specific measurements, such as a side length.

2.

Select all of the nets that can be folded and assembled into a triangular prism like this one.

A triangular prism with faces of equilateral triangles and squares.

A.

Net A is a square bordered by a square, an equilateral triangle, a square, and an equilateral triangle.

B.

Net B is a rectangle bordered by 4 triangles.

C.

Net C is an equilateral triangle bordered by 3 squares.

D.

Net D is a row of 3 squares with an equilateral triangle bordering the top of the middle square and an equilateral triangle bordering the bottom of the right square

E.

Answer: A, D

Teaching Notes

Students who fail to select Choice A or Choice D may be having trouble visualizing how the nets can be folded to form the prism. Students who select Choice B or choice C may not understand that each face of the net corresponds to a face of the prism: There are 2 triangular faces and 3 rectangular faces in the prism, but that is not true of the nets in Choice B or Choice C.

3.

A cube has a side length of 8 inches.

Select all the values that represent the cube’s volume in cubic inches.

A.

$8^2$

B.

$8^3$

C.

$6 \boldcdot 8^2$

D.

$6 \boldcdot 8$

E.

$8 \boldcdot 8 \boldcdot 8$

Answer: B, E

Teaching Notes

Students who select Choice A may be thinking of the area of a square. Students who fail to select Choice B may not understand the meaning of the exponent. Students who select Choice C may be thinking of the surface area of a cube. Students who select Choice D may also be thinking of surface area, treating each face of the cube as if it has an area of 8 square inches. Students who select Choice B, but fail to select Choice E, may have memorized that the volume of a cube is the cube of its side length, without thinking about what it means to cube a number.

4.

A square has a side length of 9 cm. What is its area?
A square has an area of 9 cm². What is its side length?

Answer:

81 cm²
3 cm

Teaching Notes

This problem has students find the area of a square given a side length, and then find the side length of a square given the area. Watch for students who treat 9 as a side length in the second part of the problem.

5.

For each pair of expressions, circle the expression with the greater value.

$13^2$ or $15^2$
$7 \boldcdot 6^2$ or $6^3$
$10^3$ or $30^2$

Answer:

$15^2$
$7 \boldcdot 6^2$
$10^3$

Teaching Notes

The first two parts of this problem can be solved without direct computation, by comparing factors. For the third part, students are very likely to compute each value and then directly compare the results. Some students will say that the value of 30² is larger because 30 is larger than 10.

6.

A rectangular prism measures 2 cm by 2 cm by 5 cm. What is its surface area? Explain or show your reasoning.

A rectangular prism with three faces showing.

Answer:

48 cm². Sample reasoning: The left and right faces each have area of 4 cm²( $2 \boldcdot 2 = 4$ ). The top, bottom, front, and back faces each have area of 10 cm² ( $2 \boldcdot 5 = 10$ ). So the surface area is $(2 \boldcdot 4) + (4 \boldcdot 10) = 48$ , or 48 cm².

Minimal Tier 1 response:

Work is complete and correct.
Sample: $2 \boldcdot 2 + 2 \boldcdot 2 + 5 \boldcdot 2 + 5 \boldcdot 2 + 5 \boldcdot 2 + 5 \boldcdot 2 = 48$ , so 48 cm².

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Work shows the correct surface area but does not include reasoning or units. Work contains arithmetic mistakes but still indicates an intent to add up the areas of the six faces. Response (with work shown) is the sum of the areas of the three visible faces only.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Student calculates a different quantity, such as volume or the area of only one face. Incorrect answer with no work is shown.

Teaching Notes

Students use an understanding of area in rectangles to find the surface area of a rectangular prism.

7.

Here is a net made of right triangles and rectangles. All measurements are given in centimeters.

If the net were folded and assembled, what type of polyhedron would it make?
Find the surface area of the polyhedron in square centimeters. Explain or show your reasoning.

Answer:

Triangular prism
84 cm². Sample reasoning: The net is made of two triangles with a base of 4 cm and a height of 3 cm, and one big rectangle with a base of 12 cm and a height of 6 cm. The two triangles can be rearranged and put together to make a rectangle that is 4 cm by 3 cm, which results in an area of 12 cm². The area of the big rectangle is $6 \boldcdot 12$ or 72 cm². So the total (surface) area is 84 cm².

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Sample:

Triangular prism
$\frac12 \boldcdot 4 \boldcdot 3 + \frac12 \boldcdot 4 \boldcdot 3 + 4 \boldcdot 6 + 5 \boldcdot 6 + 3 \boldcdot 6 = 84$ , so 84 cm².

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: Arithmetic errors accompany otherwise correct work. The area of one face is missing or incorrect. Units are omitted. Surface area is correct and well-justified but the polyhedron is incorrect yet somewhat reasonable, such as a triangular pyramid or rectangular prism.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: The shape is correctly identified as a triangular prism but work shows little progress on finding the surface area. No reasonable attempt is made at identifying the polyhedron (or an answer like “trapezoid”). Surface area calculations involve serious mistakes, such as incorrect processes for calculating the area of a right triangle. Work includes a significant error in one part but a correct answer in the other part.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Work shows significant omissions or Tier 3 errors across both problem parts.

Teaching Notes

Students identify the polyhedron associated with a net, then calculate its surface area.