End-of-Unit Assessment

1.

Select all the true statements.

A.

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

B.

Dilations of a circle result in another circle.

C.

Dilations of a polygon result in congruent corresponding angles.

D.

Dilations of a polygon increase the measure of corresponding angles.

E.

Dilation of a triangle by scale factor

\frac12

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

F.

Dilations of a triangle are similar to the original triangle.

Answer: A, B, C, F

Teaching Notes

Students selecting D have forgotten that angle measures before and after a dilation always remain the same. Students selecting E may be confusing the term congruent with the term similar.

Students who do not select A have forgotten the definition of a dilation. Students who do not select F 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 $35^\circ$ angle.

B.

Triangles 3 and 4 are both right triangles. Triangle 3 has a $40^\circ$ angle and Triangle 4 has a $60^\circ$ angle.

C.

Triangle 5 has a $30^\circ$ angle and a $100^\circ$ angle. Triangle 6 has a $60^\circ$ angle and a $70^\circ$ angle.

D.

Triangle 7 has a $50^\circ$ angle and a $25^\circ$ angle. Triangle 8 has a $50^\circ$ angle and a $105^\circ$ angle.

Answer:

Triangle 7 has a $50^\circ$ angle and a $25^\circ$ angle. Triangle 8 has a $50^\circ$ angle and a $105^\circ$ 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 B may have made a computation error, or think that the sum of the interior angles of a triangle is $190^\circ$ instead of $180^\circ$ . Students who select C may believe that having the same sum for 2 of the angles makes the triangle similar.

3.

Select all the lines that have a slope of $\frac52$ .

A.

A

B.

B

C.

C

D.

D

E.

E

Answer: A, E

Teaching Notes

Students selecting B have miscounted the vertical length of the slope triangle. Students selecting C or D are dividing horizontal length by vertical length rather than the other way around. Students not selecting A may not realize that $\frac{10}4$ is equivalent to $\frac52$ or may not think that slope triangles with lengths other than 5 and 2 can be used.

4.

Han’s teacher asked him to draw a polygon similar to polygon

A

. Here is his work. Did Han correctly draw a similar polygon? Explain how you know.

Answer:

Yes, Han correctly drew a similar polygon. Sample reasoning: Polygon A can be translated 7 units to the right and then dilated with center of dilation at the upper-left vertex and a scale factor of 3 to get to Polygon C.

Minimal Tier 1 response:

Work is complete and correct. Acceptable for sequence of transformations to take Polygon C to Polygon A.
Acceptable errors: Use of language like “move” or “shift” instead of “translate.”
Sample: Yes. Dilate A by a factor of 3 using the bottom left corner as the center, and then translate it down and right until it matches the location of C.

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: Incorrect scale factor; not describing which polygon is being transformed; incorrectly counting the distance for a polygon to be translated.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Polygons are not identified as similar; the sequence of rigid motions and dilations does not take one polygon to the other (and is not close).

5.

Triangles $ABC$ and $DEF$ are similar.

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

Answer:

5
$\frac{14}3$ (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 $(40, 74)$ on this line? Explain or show your reasoning.

Answer:

$\frac{y-2}{x-2}=2$ or $\frac{10-y}{6-x}=2$ (or equivalent)
No. Sample reasoning: $\frac{74-2}{40-2}=\frac{72}{38}$ which is not equal to 2, so the point (40, 74) is not 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 $(40,74)$ does not make the equation for the line true, so this point can't be on the line.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Finding that the slope is $\frac12$ or the equation for the line is $\frac{x-2}{y-2} = 2$ or $\frac{x-2}{y-2} = \frac12$ ; 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 a polygon:

Draw the dilation of $ABCD$ using center $B$ and scale factor 2. Label the dilation as Figure F.
Draw the dilation of $ABCD$ with center $B$ and scale factor $\frac{1}{3}$ . Label the dilation as Figure G.
Show that Figure F and Figure G are similar.

Answer:

See image
See image
Sample response: If Figure G is dilated by a scale factor of 3 with center at point $B$ , the result is $ABCD$ . If $ABCD$ is dilated by a scale factor of 2 with center at point $B$ , the result is Figure F.

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Sample: Figure F and Figure G are both dilations of polygon $ABCD$ .

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 “Figure F and Figure G are dilations of each other” without a justification, such as referencing $ABCD$ .

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 polygon without a grid, and then reason about similarity. Because of the way the two polygons are constructed, there is a natural sequence of dilations that takes one polygon to the other, making them similar.