Unit 7 Angles Triangles And Prisms — Unit Plan

Title	Assessment
Lesson 1 Relationships of Angles	Identical Octagons This pattern is composed of a square and some regular octagons. In this pattern, all of the angles inside the octagons have the same measure. The shape in the center is a square. Find the measure of one of the angles inside one of the octagons. Show Solution $135^\circ$ . Sample reasoning: The angles in the square are $90^\circ$ . Since the angles around a point add up to $360^\circ$ , then 2 octagon angles must be $360-90$ , or $270^\circ$ . Since all of the octagon angles are the same, each angle is $270\div2$ or $135^\circ$ .
Lesson 2 Adjacent Angles	Finding Measurements Point $F$ is on line $CD$ . Find the measure of angle $CFE$ . Angle $SPR$ and angle $RPQ$ are complementary. Find the measure of angle $RPQ$ . Show Solution $28^\circ$ $53^\circ$
Lesson 3 Nonadjacent Angles	Finding Angle Pairs Name a pair of complementary angles in the diagram. Name a pair of supplementary angles in the diagram. Name a pair of vertical angles in the diagram. Show Solution $ABC$ with one of $BAC$ , $FAD$ , or $ADG$ One of these pairs: $CAD$ with one of $ABC$ , $DAC$ , or $DAG$ $BAF$ with one of $ABC$ , $DAC$ , or $DAG$ Any 2 of $BCA$ , $ACG$ , and $CGA$ One of these pairs: $DAF$ and $BAC$ $BAF$ and $CAD$
Lesson 4 Solving for Unknown Angles	Missing Circle Angles $AD$ , $BE$ , and $CF$ are all diameters of the circle. The measure of angle $AOB$ is 40 degrees. The measure of angle $DOF$ is 120 degrees. Find the measures of the angles: $BOC$ $COD$ Show Solution Angle $BOC=80^\circ$ . Sample reasoning: Given angle $DOF=120^\circ$ , angle $AOC=120^\circ$ because they are congruent vertical angles. Consequently, angles $AOB+BOC=120^\circ$ because they are adjacent. Angle $COD=60^\circ$ . Sample reasoning: Angle $COD$ and angle $DOF$ are supplementary angles, so the sum of their measurements has to be $180^\circ$ .
Section A Check Section A Checkpoint	Problem 1 Write an equation to represent the relationship between 2 or more angles in this diagram. Find the values of $a$ , $b$ , and $c$ . Show Solution Sample responses: $a+27=90$ $b+c=180$ $a+b+c+27=360-90$ $a=63$ $b=27$ $c=153$

Lesson 6 Building Polygons (Part 1)	An Equilateral Quadrilateral When asked to draw a quadrilateral with all four sides measuring 5 cm, Jada drew a square. Does Jada’s shape meet the requirements? Is there a different shape that would also meet the requirements? Explain your reasoning. Show Solution Yes, Jada’s shape has 4 sides, all measuring 5 cm. A rhombus could be made with all four sides the same length, but without right angles.
Lesson 7 Building Polygons (Part 2)	Finishing Elena’s Triangles Elena is trying to draw a triangle with side lengths of 4 inches, 3 inches, and 5 inches. She uses her ruler to draw a 4-inch line segment, $AB$ . She uses her compass to draw a circle around point $B$ with a radius of 3 inches She draws another circle, around point $A$ with a radius of 5 inches. What should Elena do next? Explain and show how she can finish drawing the triangle. Now Elena is trying to draw a triangle with side lengths 4 inches, 3 inches, and 8 inches. Explain what Elena’s drawing means. Show Solution Elena should put a point where the two circles intersect and draw line segments connecting that point to points $A$ and $B$ to finish her triangle. Elena’s drawing means that there is no way to draw a triangle with these side lengths. The circles do not intersect, because the side lengths of 3 inches and 4 inches are too short to make a triangle with the third side of 8 inches.
Lesson 8 Triangles with 3 Common Measures	Comparing Andre's and Noah’s Triangles Andre and Noah each drew a triangle with side lengths of 5 cm and 3 cm and an angle that measures $60^\circ$ , and then they showed each other their drawings. Did Andre and Noah draw different triangles? Explain your reasoning. Explain what Andre and Noah would have to do to draw another triangle that is different from what either of them has already drawn. Show Solution These are both the same triangle. In both cases, the $60^\circ$ angle is between the 3-cm and 5-cm sides. If you trace one triangle, flip it and turn it, it can line up exactly with the other triangle. To draw a different triangle, they should try putting the $60^\circ$ angle next to the side of unknown length, instead of between the two known sides.
Lesson 10 Drawing Triangles (Part 2)	Finishing Noah’s Triangle Noah is trying to draw a triangle with a $30^\circ$ angle and side lengths of 4 cm and 6 cm. He uses his ruler to draw a 4 cm line segment. He uses his protractor to draw a $30^\circ$ angle on one end of the line segment. What should Noah do next? Explain and show how he can finish drawing the triangle. Is there a different triangle Noah could draw that would answer the question? Explain or show your reasoning. Show Solution Noah should use a compass to draw a circle with radius 6 cm and center at one end of the 4-cm side. He should then draw segments connecting both ends of the 4-cm side to the point where the circle and ray cross, and that will complete the triangle. Yes. Noah could try beginning with the same setup he has already drawn again, but this time center the circle on the other end of the 4-cm side. He could also start with the 6-cm side drawn instead of the 4-cm side and follow the same process.
Section B Check Section B Checkpoint	Problem 1 Draw a triangle with side lengths 3 in, 4 in, and 6 in. Can you draw a different triangle with these same lengths? Explain how you know. Show Solution Sample response: No. Sample reasoning: These 3 side lengths make a unique triangle. Problem 2 Priya and Han each draw a triangle that has side lengths of 2 inches and 5 inches and an angle of $30^\circ$ . Could they have drawn different triangles? Explain how you know. Show Solution Yes. Sample reasoning: If one of them drew the $30^\circ$ angle between the 2-inch and 5-inch sides, that would be a different triangle than if the other had put the $30^\circ$ angle adjacent to one of the sides but not between them.

Lesson 11 Slicing Solids	Pentagonal Pyramid Here is a pyramid with a base that is a pentagon with all sides the same length. Describe the cross-section that will result if the pyramid is sliced: horizontally (parallel to the base). vertically through the top vertex (perpendicular to the base). Describe another way you could slice the pyramid that would result in a different cross-section. Show Solution Cross-sections: A pentagon with all sides the same length, but smaller than the base of the pyramid A triangle Sample responses: You could slice the pyramid diagonally. You could slice the pyramid vertically but not through the top vertex.
Lesson 12 Volume of Right Prisms	Octagonal Box A box is shaped like an octagonal prism. Here is what the base of the prism looks like. For each question, make sure to include the unit with your answer and explain or show your reasoning. If the height of the box is 7 inches, what is the volume of the box? If the volume of the box is 123 in³, what is the height of the box? Show Solution 287 in³, because the base has an area of 41 in^2, and $41\boldcdot 7=287$ . 3 in, because $41 \boldcdot 3 = 123$ .
Lesson 13 Decomposing Bases for Area	Volume of a Pentagonal Prism Here is a prism with a pentagonal base. The height is 8 cm. What is the volume of the prism? Show your thinking. Organize it so it can be followed by others. Show Solution The volume is 232 cm³. The area of the base is 29 cm² and can be found in multiple ways, but one way is to consider a 5 by 7 rectangle with a right triangle cut off, then $5 \boldcdot 7 - \frac{1}{2} \boldcdot 4 \boldcdot 3 = 29$ . Since the height is 8 cm, the volume is calculated by $29 \boldcdot 8 = 232$ .
Lesson 14 Surface Area of Right Prisms	Surface Area of a Hexagonal Prism Find the surface area of this prism. Show your reasoning. Organize your explanation so it can be followed by others. Show Solution The surface area is 270 cm². Possible strategy: The area of the base is 27 cm². The perimeter of the base is 24 cm, so the combined area of the sides is 216 cm², because $24 \boldcdot 9=216$ . Therefore the total surface area is 270 cm², because $27 \boldcdot 2 + 216=270$ .
Lesson 16 Applying Volume and Surface Area	Preparing for the Play Andre is preparing for the school play. He needs to paint a cardboard box to look like a dresser. The box is a rectangular prism that measures 5 feet tall, 4 feet long, and $2\frac12$ feet wide. Andre does not need to paint the bottom of the box. How much cardboard does Andre need to paint? If one bottle of paint covers an area of 40 square feet, how many bottles of paint does Andre need to buy for this project? Show Solution 75 square feet. $(2.5 \boldcdot4)+2(5 \boldcdot4)+2(2.5 \boldcdot5)=75$ 2 bottles of paint. $\frac {75}{40}=1.875$
Section C Check Section C Checkpoint	Problem 1 For each situation, decide whether surface area or volume is the quantity needed. How much wrapping paper is needed to wrap a present? How much water can fill up a tank with a trapezoid-shaped base? Bees need 38 cubic inches of hive space per 1,000 bees. What is the largest number of bees that can fit in a beehive box? Cardboard costs $1.20 per square yard. How much will it cost for the cardboard needed to construct a play house? Show Solution surface area volume volume surface area Problem 2 Find the volume and surface area of this prism. Here are the dimensions of its base: Show Solution Volume: 51,975 cm³ Surface area: 9,060 cm²

Lesson 17 Building Prisms	No cool-down
Unit 7 Assessment End-of-Unit Assessment	Problem 1 Select all the true statements about the angles in this diagram. A. Angles $a$ and $b$ are supplementary angles. B. Angles $a$ and $c$ are complementary angles. C. Angles $a$ and $d$ are vertical angles. D. Angles $a$ and $e$ are supplementary angles. E. Angles $b$ and $e$ are complementary angles. Show Solution C, D Problem 2 A square pyramid is sliced parallel to the base and halfway up the pyramid. Which of these describes the cross-section? A. A square smaller than the base B. A quadrilateral that is not a square C. A square the same size as the base D. A triangle with a height the same as the pyramid Show Solution A square smaller than the base Problem 3 Which of these describes a unique polygon? A. A triangle with angles $30^\circ$ , $50^\circ$ , and $100^\circ$ B. A quadrilateral with each side length 5 cm C. A triangle with side lengths 6 cm, 7 cm, and 8 cm D. A triangle with side lengths 4 cm and 5 cm and a $50^\circ$ angle Show Solution A triangle with side lengths 6 cm, 7 cm, and 8 cm Problem 4 Priya is trying to draw a triangle with side lengths 2 cm, 5 cm, and 1 cm. Explain why Priya’s drawing is not creating a triangle. Show Solution Sample response: Because the side lengths of 2 cm and 1 cm add up to 3 cm, and 3 cm is not longer than 5 cm, the circles will never cross. This means that there is no intersection point to use to create a triangle. Problem 5 Draw as many different triangles as possible that have two sides of length 4 cm and a $\,45^\circ$ angle. Clearly mark the side lengths and angles given. Show Solution There are 2 different triangles. Minimal Tier 1 response: Work is complete and correct. Sample: Exactly 2 triangles are drawn; lengths and angles marked and reasonably accurate; no other lengths or angles are marked. Acceptable errors: Other lengths and angles, besides the ones given, may be marked with reasonable approximations of their measures; sum of marked measures of three angles close to, but not equal to, 180 degrees. Tier 2 response: Work shows general conceptual understanding and mastery, with some errors. Sample errors: 3 triangles instead of 2; 45 degree angle drawn with significant inaccuracy; sides of length 4 cm drawn with significant inaccuracy, notably if they are significantly different in length; other lengths and angles, besides the ones given, measured or marked with significant inaccuracy. Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Sample errors: 0 triangles, 1 triangle, or more than 3 triangles drawn; two or more error types from Tier 2 response; an explanation of why the triangle cannot be drawn. Problem 6 What are the values of $x$ and $y$ ? Show Solution $x = 35, y = 72.5$ (The angle marked $x$ and the angle marked as $35^\circ$ are vertical angles, so they have the same measure. The two angles marked $y$ , along with the angle marked as $35^\circ$ , form a straight line. Therefore, $2y + 35 = 180$ , and $y = 72.5$ .) Problem 7 Noah is building a planter box for the community garden near his school. The bottom of the box is wood, but the top of the planter box is left open for soil and plants. He plans to make the box 2 feet tall, with the base in this shape: How much wood will Noah need to make the planter box? If he fills the garden bed 1.5 feet deep with soil, how much soil will he need? Show Solution 43.5 ft² 17.25 ft³ Minimal Tier 1 response: Work is complete and correct, with complete explanation or justification. Sample: The area of the base is 11.5 ft². This may also be indicated by a decomposition into a rectangle, triangle, and trapezoid with areas 6 ft², 2 ft², and 3.5 ft², respectively, or other decompositions. The perimeter of the base is 16 ft. The area of the sides is $16\boldcdot2$ , or 32 ft². The total area of wood needed is $11.5+32$ , or 43 ft². The area of the base is 11.5 ft². The soil is 1.5 ft deep. The volume of soil needed is $11\boldcdot 1.5$ , or 17.25 ft². Tier 2 response: Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification. Accept a volume for Part 2 based on an incorrect calculation of the area of the base in Part 1. Sample errors: Incorrectly decomposing the base to calculate its area; including both top and bottom bases in the calculation of wood needed, or including neither base; incorrectly calculating perimeter of the base of the box; correctly calculating the quantities needed but using incorrect units to report, such as ft3 for area and surface area and ft2 for volume. Tier 3 response: Work shows a developing but incomplete conceptual understanding, with significant errors. Sample errors: Serious error(s) caused by a misunderstanding of one of the terms "volume," "area," or "base"; omission of one of the two parts. Tier 4 response: Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery. Sample errors: Serious errors caused by misunderstanding of more than one of the terms; omission of the two parts.