End-of-Unit Assessment

1.

Quadrilateral $EFGH$ is a scaled copy of quadrilateral $ABCD$ . Select all of the true statements.

Quadrilateral EFGH is a scaled copy of quadrilateral ABCD.<br>

A.

Segment $EF$ is twice as long as segment $AB$ .

B.

Segment $CD$ is twice as long as segment $FG$ .

C.

The measure of angle $HEF$ is twice the measure of angle $DAB$ .

D.

The length of segment $EH$ is 16 units.

E.

The area of $EFGH$ is twice the area of $ABCD$ .

Answer: A, D

Teaching Notes

Students who select choice B are missing the fact that the two segments are not corresponding, possibly because they have not read the question carefully. Choice C may be tempting for some students because the side lengths of $ABCD$ are doubled, but the angle measures are not. Students who select choice E are making a common mistake: When a figure is scaled up by a factor of two, its area quadruples rather than doubles.

2.

Rectangle A measures 9 inches by 3 inches. Rectangle B is a scaled copy of Rectangle A. Select all of the measurement pairs that could be the dimensions of Rectangle B.

A.

4.5 inches by 1.5 inches

B.

8 inches by 2 inches

C.

10 inches by 4 inches

D.

13.5 inches by 4.5 inches

E.

90 inches by 30 inches

Answer: A, D, E

Teaching Notes

The correct answer choices all involve sides lengths that are in a 9:3 ratio. Students who select choice B or choice C may believe that adding or subtracting the value "1" from both lengths will maintain their ratio. The work in this problem helps students build toward developing proportional reasoning in the next unit.

3.

A scale drawing of a rectangular park is 5 inches wide and 7 inches long. The actual park is 280 yards long. What is the area of the actual park, in square yards?

<p>A scale drawing of a rectangular park. The width is 5 inches and the length is 7 inches.</p>

A.

35

B.

200

C.

1,400

D.

56,000

Answer:

56,000

Teaching Notes

Students who select choice A have calculated the area of the scale drawing rather than of the park. Students who select choice B have found the correct width of the park, but not the area. Students who select choice C have multiplied the area of the drawing by the scale factor, neglecting the fact that both the length and the width of the park are scaled by a factor of 40, so the area of the park will be 1,600 times greater than the area of the drawing.

4.

Here is a polygon. Draw a scaled copy of the polygon using a scale factor of $\frac 1 2$ .

Answer:

<p>Two polygons drawn on a grid. One polygon is a scaled copy of the other polygon.</p>

Minimal Tier 1 response:

Work is complete and correct.
Sample: See above. Acceptable errors: Figure is somehow in a different orientation; figure overlaps original.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Correct drawing with scale factor 2 instead of scale factor $\frac 1 2$ ; minor error in determining dimensions of figure, such as a pair of segments 1 unit longer than they should be.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Drawing shows lack of use of any scale factor; multiple errors in determining dimensions; incorrect attempt at drawing with scale factor 2.

Teaching Notes

Students draw a scaled copy of a figure on a grid.

5.

The scale of a map says that 4 cm represents 5 km.

What distance on the map represents an actual distance of 10 km?
What actual distance is represented by 10 cm on the map?

Answer:

8 cm (or equivalent)
12.5 km (or equivalent)

Teaching Notes

Students are given the scale of a map and asked to find lengths and distances using that scale. Common errors for this problem may include reversing the numbers (5 cm represents 4 km) and other errors typical of calculating and using unit rates, such as multiplying rather than dividing.

6.

Tyler has two different maps of Ohio.

On the first map, the scale is 1 cm to 10 km. The distance from Cleveland to Cincinnati is 40 cm.
On the second map, the scale is 1 cm to 50 km.

What is the distance from Cleveland to Cincinnati on the second map? Explain your reasoning.

Answer:

The distance is 8 cm. Sample explanations:

On the 1 cm : 50 km scale map, each centimeter represents 5 times as much actual distance as on the 1 cm: 10 km map. That means that on the 1 cm : 50 km map the distance from Cleveland to Cincinnati will be one fifth as much, 8 cm.
The actual distance from Cleveland to Cincinnati is 400 km, because $40 \boldcdot 10 = 400$ . The distance on the second map that represents 400 km is 8 cm, because $400 \div 50 = 8$ .

Minimal Tier 1 response:

Work is complete and correct.
Sample: Lengths on the second map are five times smaller because 1 cm represents 50 km instead of 10 km. Divide 40 cm by 5 to get 8 cm.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Multiplication or division errors in otherwise correct work; work involves a correct substantive intermediate step (such as the actual distance from Cleveland to Cincinnati) but goes wrong after that; one mistake involving an “upside down” scale factor (or multiplying when division is called for); a correct answer without explanation.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Work does not involve proportional reasoning; an incorrect answer without explanation, even if close; multiple mistakes that involve inversion of scale factors.

Teaching Notes

Students reason about maps drawn at two different scales. This problem can be solved using a wide variety of approaches, including ratio tables and double number lines.

7.

Elena's bedroom is a rectangle with length 5 meters and width 3 meters.

Make a scale drawing of Elena’s bedroom, using a scale of 1 to 50. Be sure to label the dimensions of your drawing.
Elena’s bedroom door is 0.8 meters wide. How wide should the door be on the scale drawing? Explain how you know.
Elena’s bed measures 4 centimeters by 3 centimeters on the scale drawing. What are the actual measurements of her bed?

Answer:

A rectangle labeled with length 10 cm and width 6 cm (or equivalent units, such as 0.1 m and 0.06 m)
1.6 cm (or equivalent units). Sample reasoning: 0.8 m is 80 cm. At 1 to 50, the width of the door is 1.6 cm, because $80 \div 50 = 1.6$ .
200 cm by 150 cm (or equivalent units). Actual measurements are 50 times as long as the corresponding measurements on the drawing.

Minimal Tier 1 response:

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

A rectangle, labeled 10 cm and 6 cm, with the larger side labeled 10 cm.
1.6 cm. Because the scale is 1 to 50, the door’s 80 cm width becomes $\frac{80}{50}$ cm in the scale drawing.
2 m by 1.5 m

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: One calculation or conversion error; clear error in relative shape of rectangle; incomplete explanation of 1.6 cm calculation; describing instead of building and labeling the scale drawing.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: Two or more error types from Tier 2 response; multiple calculation and conversion errors; scaling in wrong direction (multiplying or dividing when inappropriate); using incorrect scale factor.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Two or more error types from Tier 3 response; adding or subtracting when working with scale factor; misunderstanding of the meaning and use of “1 to 50.”

Teaching Notes

Students make a scale drawing of a bedroom using a given scale. They then use that scale to calculate lengths of various objects.