End-of-Unit Assessment

1.

Select all the true statements.

A.

Dilations of a circle result in another circle.

B.

Dilations of a point will lie on the same line as the point and the center of dilation.

C.

Dilations of a polygon increase the measure of corresponding angles.

D.

Dilations of a polygon result in congruent corresponding angles.

E.

Dilations of a triangle are similar to the original triangle.

F.

Dilation of a triangle by scale factor

\frac13

results in a triangle that is congruent to the original triangle.

Answer: A, B, D, E

Teaching Notes

Students who select C have forgotten that angle measures before and after a dilation always remain the same. Students who select F may be confusing the term congruent with the term similar.

Students who do not select B have forgotten the definition of a dilation. Students who do not select E have forgotten that similar figures are defined as figures which can be matched by a sequence of dilations and rigid transformations.

2.

Which pair of triangles must be similar?

A.

Triangles 1 and 2 each have a $55^\circ$ angle.

B.

Triangles 3 and 4 are both right triangles. Triangle 3 has a $40^\circ$ angle and Triangle 4 has a $50^\circ$ angle.

C.

Triangle 5 has a $20^\circ$ angle and an $80^\circ$ angle. Triangle 6 has a $15^\circ$ angle and an $85^\circ$ angle.

D.

Triangle 7 has a $50^\circ$ angle and a $35^\circ$ angle. Triangle 8 has a $50^\circ$ angle and a $115^\circ$ angle.

Answer:

Triangles 3 and 4 are both right triangles. Triangle 3 has a $40^\circ$ angle and Triangle 4 has a $50^\circ$ angle.

Teaching Notes

Students who select A may believe that triangles that share only one pair of congruent angles must be similar. Students who select C may believe that having the same sum for 2 of the angles makes the triangle similar. Students who select D may have made a computation error, or think that the sum of the interior angles of a triangle is 200º instead of 180º.

3.

Select all the lines that have a slope of $\frac34$ .

A.

A

B.

B

C.

C

D.

D

E.

E

Answer: A, C, D

Teaching Notes

Students who select B or E are dividing horizontal length by vertical length rather than the other way around. Students who do not select A may not realize that

\frac68

is equivalent to

\frac34

or may not think that slope triangles with lengths other than 3 and 4 can be used.

4.

Mai’s teacher asked her to draw a polygon similar to Polygon A. Here is her work. Did Mai correctly draw a similar polygon? Explain how you know.

Answer:

No, Mai did not correctly draw a similar polygon. Sample reasoning: The vertical side of Polygon A was multiplied by a scale factor of 2 to get the corresponding side in Polygon C, but the other corresponding sides were not multiplied by a factor of 2, so the two polygons can’t be similar.

Minimal Tier 1 response:

Work is complete and correct.
Acceptable errors: students do not need to mention exact scale factors.
Sample: No. Each side of Polygon A was multiplied by a different scale factor to get the corresponding side in Polygon C.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Written explanation may be incomplete, saying that the figures are different without specifying how they are different.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Asserts that the polygons are similar.

5.

Triangles $ABC$ and $DEF$ are similar.

Find the length of segment $DE$ .
Find the length of segment $BC$ .

Answer:

$\frac65$ (or equivalent)
4 (or equivalent)

Teaching Notes

When two shapes are similar, a scale factor relates lengths in one figure to the corresponding lengths in the other. At the same time, ratios of lengths in one figure (for example, length to width) are the same as in the other figure.

6.

Write an equation for the line.
Is the point $(26,76)$ on this line? Explain or show your reasoning.

Answer:

$\frac{y-4}{x-2}=3$ or $\frac{13-y}{5-x}=3$ (or equivalent)
Yes. Sample reasoning: $\frac{76-4}{26-2}=\frac{72}{24}=3$ , so the point $(76,26)$ is on this line.

Minimal Tier 1 response:

Work is complete and correct.
Acceptable equations: In addition to the equations listed above, any equation that relates the quotients of the horizontal and vertical lengths of a slope triangle for this line is valid. Note that equations may come in different formats and may use the coordinates of any point on the line, not just the ones shown.
Sample reasoning: The point $(26,76)$ makes my equation for the line true, so this point is on the line.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Finding that the slope is $\frac13$ or the equation for the line is $\frac{x-2}{y-4}=3$ (or similar); minor computational errors when determining if the point is on the line.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Equation does not involve slope or similar triangles; response indicates point is on the line or uses incorrect reasoning.

Teaching Notes

This problem has students use slope triangles to write an equation for a line. Students can reason about slope, or use this equation to determine whether a point is on the given line.

7.

Here is triangle

ABC

and point

P

:

Draw the dilation of $ABC$ using center $P$ and scale factor $\frac13$ . Label the dilation $DEF$ .
Draw the dilation of $ABC$ with center $P$ and scale factor 2. Label the dilation $GHI$ .
Show that $DEF$ and $GHI$ are similar.

Answer:

See image
See image
Sample response: If $DEF$ is dilated by a scale factor of 3 with center at point $P$ , the result is $ABC$ . If $ABC$ is dilated by a scale factor of 2 with center at point $P$ , the result is $GHI$ .

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Sample: $DEF$ and $GHI$ are both dilations of the same triangle.

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: Scale is correct in the dilations, but the incorrect center is used; work involves a minor mistake dilating one point; explanation of similarity is something like “ $DEF$ and $GHI$ are dilations of each other” without a justification, such as referencing $ABC$ .

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: Work shows general understanding of dilations but a few points are placed incorrectly; dilations are performed using scale factors of $\frac12$ or 3; correctly drawn dilations but very weak or missing explanation of similarity.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Drawings do not resemble dilations.

Teaching Notes

Students apply dilations to a triangle off of a grid and a point not on the triangle. They then reason about similarity. Because both dilations use the same center, students can reason the triangles are similar as a result of successive dilations. They could also describe a translation and a dilation that would show the two triangles are similar.