Unit 2 Dilations Similarity And Introducing Slope — Unit Plan

Title	Assessment
Lesson 1 Projecting and Scaling	Scaled Copies Rectangle G measures 9 inches by 12 inches. Which of these rectangles are scaled copies of Rectangle G? Show Solution Rectangles H, J, L, M
Lesson 2 Circular Grid	Dilating Points on a Circular Grid Dilate $A$ using $P$ as the center of dilation and a scale factor of 3. Label the new point $A'$ . Dilate $B$ using $P$ as the center of dilation and a scale factor of 2. Label the new point $B'$ . Show Solution
Lesson 3 Dilations with No Grid	A Single Dilation of a Triangle Lin drew a triangle and a dilation of the triangle with scale factor $\frac{1}{2}$ : What is the center of the dilation? Explain how you know. Which triangle is the original and which triangle is the dilation? Explain how you know. Show Solution The center of dilation is $A$ . Sample reasoning: The original and dilated points all lie on rays that start at $A$ . Triangle $ACD$ is the original and triangle $ABE$ is the dilation. Sample reasoning: Since the scale factor is less than 1, the dilated triangle is smaller than the original triangle.
Lesson 4 Dilations on a Square Grid	A Dilated Image Draw the image of rectangle $ABCD$ after a dilation using point $P$ as the center and scale factor $\frac12$ . Show Solution
Section A Check Section A Checkpoint	Problem 1 Triangle $L$ is dilated so that its image is triangle $M$ . Which point is the center of dilation? A.point $A$ B.point $B$ C.point $C$ D.point $D$ Show Solution point $C$ Problem 2 Draw the image of triangle $XYZ$ after a dilation with center at $(0,0)$ and scale factor of 2. What are the coordinates of the image of point $Z$ ? Show Solution $(6,\text-8)$

Lesson 6 Similarity	Showing Similarity Elena gives the following sequence of transformations to show that the 2 figures are similar by transforming $ABCD$ into $EFGD$ . Dilate using center $D$ and scale factor 2. Reflect using the horizontal line through $D$ . Is Elena’s method correct? If not, explain how you could fix it. Show Solution Elena’s method is not correct. Sample response: After dilating using $D$ as the center with a scale factor of 2, Elena can reflect over the vertical line through $D$ rather than the horizontal line through $D$ .
Lesson 7 Similar Polygons	How Do You Know? Are these 2 figures similar? Explain how you know. Show Solution The 2 figures are not similar. Sample reasoning: Sides $CD$ and $AB$ are multiplied by a scale factor of $\frac34$ to get sides $GH$ and $EF$ , but sides $AD$ and $BC$ are multiplied by a scale factor of $\frac56$ .
Lesson 8 Similar Triangles	Finding Similar Triangles Here is triangle $ABC$ . Select all triangles that are similar to triangle $ABC$ . A B C D E F Show Solution A, B, E
Lesson 9 Side Length Quotients in Similar Triangles	Similar Sides The 2 triangles shown are similar. Find the value of $\frac{a}{b}$ . Show Solution $\frac32$ or 1.5 (or equivalent)
Section B Check Section B Checkpoint	Problem 1 Explain why triangle $RST$ is similar to triangle $TYZ$ . Show Solution Sample reasoning: Triangle $RST$ is similar to triangle $TYZ$ because triangle $RST$ can be dilated by a scale factor of 2 using point $R$ as the center, and then translated so that point $R$ goes to point $T$ . Problem 2 Triangle $ABC$ and triangle $DEF$ are similar. What is the length of side $DE$? What is the length of side $EF$? Show Solution $\frac{35}{9}$ (or equivalent) $\frac52$ (or equivalent)

Lesson 10 Meet Slope	Finding Slope and Graphing Lines Lines $\ell$ and $k$ are graphed. Which line has a slope of 1, and which has a slope of 2? Use a ruler or straightedge to help you graph a line whose slope is $\frac35$ . Label this line $a$ . Show Solution Line $\ell$ has a slope of 1, and line $k$ has a slope of 2. Sample response:
Lesson 11 Writing Equations for Lines	Matching Relationships to Graphs Line $a$ is shown on the coordinate plane. Explain why the slope of line $a$ is $\frac26$ . Label the horizontal and vertical sides of the slope triangle with expressions representing their length. Use the slope triangle to write an equation for any point $(x,y)$ on line $a$ . Is the point $(95,37)$ on line $a$ ? Explain or show your reasoning. Show Solution Sample reasoning: The points $(5,7)$ and $(11,9)$ are on the line. A slope triangle drawn using these points as vertices will have a vertical length of 2 and a horizontal length of 6, giving a slope value of $\frac{2}{6}$ . The vertical side has length $y-7$ , and the horizontal side has length $x-5$ . $\frac{y-7}{x-5}=\frac{2}{6}$ (or equivalent). Equations such as $\frac{7-y}{5-x}=\frac13$ or $\frac{y-6}{x-2}=\frac26$ are also correct, being derived from different slope triangles than the one shown. Yes, point $(95,37)$ is on line $a$ . Sample reasoning: Those $x$ - and $y$ -coordinates make the line’s equation true: $\frac{37-7}{95-5}=\frac{30}{90}=\frac26$ .
Lesson 12 Using Equations for Lines	Is the Point on the Line? Is the point $(20,13)$ on this line? Explain your reasoning. Show Solution Yes, point $(20,13)$ is on the line. Sample reasoning: One possible equation for the line is $\frac{y-3}{x}=\frac12$ . Since $\frac{13-3}{20}=\frac12$ , the point $(20,13)$ is on this line.
Section C Check Section C Checkpoint	Problem 1 Of the lines on the graph, line $\ell$ has a slope of 1 and line $m$ has slope of 3 and line $n$ has a slope of $\frac12$ . Label lines $\ell$ , $m$ , and $n$ . Show Solution Problem 2 A line can be described by the equation $\frac{y-1}{x-3}=\frac13$ . Is the point $(33,12)$ on this line? Explain or show your reasoning. Show Solution No. Sample reasoning: Since $\frac{12-1}{33-3}=\frac{11}{30}$ and not $\frac13$ , the point $(33,12)$ does not make the equation true. Problem 3 Select all equations that describe the line. A.$\frac{y-6}{x-7}=\frac12$ B.$\frac{y-7}{x-6}=\frac12$ C.$\frac{x-6}{y-7}=\frac12$ D.$\frac{y-3}{x-4}=\frac12$ E.$\frac{y-4}{x-3}=\frac12$ F.$\frac{x-4}{y-3}=\frac12$ Show Solution A, E

Lesson 13 The Shadow Knows	No cool-down
Unit 2 Assessment End-of-Unit Assessment	Problem 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. Show Solution A, B, C, F Problem 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. Show Solution Triangle 7 has a $50^\circ$ angle and a $25^\circ$ angle. Triangle 8 has a $50^\circ$ angle and a $105^\circ$ angle. Problem 3 Select all the lines that have a slope of $\frac52$ . A.A B.B C.C D.D E.E Show Solution A, E Problem 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. Show Solution 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). Problem 5 Triangles $ABC$ and $DEF$ are similar. Find the length of segment $BC$ . Find the length of segment $DF$ . Show Solution 5 $\frac{14}3$ (or equivalent) Problem 6 Write an equation for the line. Is the point $(40, 74)$ on this line? Explain or show your reasoning. Show Solution $\frac{y-2}{x-2}=2$ 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. Problem 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. Show Solution 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.