Unit 2 Dilations Similarity And Introducing Slope — Unit Plan

Title	Takeaways	Assessment
Lesson 1 Projecting and Scaling	—	Scaled Copies (1 problem) 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 (1 problem) 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 (1 problem) 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 (1 problem) Draw the image of rectangle $ABCD$ after a dilation using point $P$ as the center and scale factor $\frac12$ . Show Solution
Lesson 5 More Dilations	—	Identifying a Dilation (1 problem) The smaller triangle is dilated to create the larger triangle. The center of dilation is plotted, but not labeled. A triangle, it‘s image after dilation and a point on a coordinate plane, origin O. Horizontal axis scale negative 2 to 11 by 1’s. Vertical axis negative 7 to 4 by 1’s. The point has coordinates (3 comma 0). The triangle has the coordinates (0 comma 0), (6 comma negative 6) and (9 comma 3). The image of the triangle has the coordinates (2 comma 0), (negative 2 comma 4) and (5 comma 1). Describe this dilation. Be sure to include all of the information someone would need to perform the dilation. Show Solution Sample response: The triangle being dilated has vertices at $(2,0)$ , $(4, \text-2)$ , and $(5,1)$ . The center of dilation is $(3,0)$ and the scale factor is 3.
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 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 (1 problem) 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? (1 problem) 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 (1 problem) 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 (1 problem) 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 (1 problem) 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 (1 problem) 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? (1 problem) 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. Show Solution A, E

Lesson 13 The Shadow Knows	—	No cool-down
Unit 2 Assessment End-of-Unit Assessment