Unit 1 Area And Surface Area — Unit Plan

Title	Assessment
Lesson 1 Tiling the Plane	What Is Area? Think about your work today, and write your best definition of “area.” Show Solution Sample responses: the amount of space inside a two-dimensional shape the measurement of the inside of a shape the number of square units inside a shape the amount of space a shape covers the amount of the plane a shape covers
Lesson 2 Finding Area by Decomposing and Rearranging	Tangram Rectangle The square in the middle has an area of 1 square unit. What is the area of the entire rectangle in square units? Explain your reasoning. Show Solution 4 square units. Sample reasoning: Put together the two small triangles to make a square. Its area is 1 square unit. Decompose each medium triangle into two small triangles that can be arranged as a square. Each of these squares has an area of 1 square unit. Together with the square in the middle, the sum of the areas of these pieces is 4 square units. A small triangle has an area of $\frac12$ square unit, and a medium triangle has an area of 1 square unit. $1 + 1 + 1 + \frac12 +\frac12=4$
Lesson 3 Reasoning to Find Area	Maritime Flag A maritime flag is shown. What is the area of the shaded part of the flag? Explain or show your reasoning. An image of a maritime flag, with a shaded portion and a triangular portion removed from the right side. Measurements are indicated around the shaded area indicate 8 inches across the top, two equivalent measurements of 6 inches on the side, and two equivalent measurements of 4 inches on the bottom. Show Solution 72 square inches. Sample reasoning: If we draw a line down the middle of the shaded area, we would have a 4 inch-by-12 inch rectangle on the left and two right triangles. The 4-by-12 rectangle has an area of 48 square inches. The two triangles on the right can be composed into a 4 inch-by-6 inch rectangle, so their combined area is 24 square inches. $48 + 24 = 72$
Section A Check Section A Checkpoint	Problem 1 Find the area of each shaded region. Explain or show your reasoning. Show Solution The areas is 16 square units. Sample reasoning: The rectangle can be decomposed into 1 square of 4 square units and 6 identical right triangles. Two right triangles make a square of 4 square units, so 6 right triangles make 3 squares with a combined area of 12 square units. $4 + 12 = 16$ The rectangle can be enclosed by a 6-by-6 square (36 square units). The square creates 2 larger right triangles and 2 smaller ones. The larger triangles can be rearranged to make a 4-by-4 square (16 square units). The smaller triangles make a 2-by-2 square (4 square units). Subtracting the areas of the triangles from the 6-by-6 square gives 16 square units. $36 - 16 - 4 = 16$ 30 sq cm. Sample reasoning: The two small triangles can be rearranged into a 3-by-3 square (9 sq cm). The two large triangles can be rearranged into a 3-by-7 rectangle (21 sq cm). The combined area is $9 + 21$ , or 30 sq cm.

Lesson 4 Parallelograms	How Would You Find the Area? How would you find the area of this parallelogram? Describe your strategy. Show Solution Sample responses: Decompose a triangle from one side of the parallelogram and move it to the other side to make a rectangle. Multiply the base and side (height) lengths of the rectangle. Draw a rectangle that just fits around the parallelogram, multiply the bottom length of that rectangle by its side length to find the area of the rectangle, and then subtract the combined area of the triangles that do not belong to the parallelogram. Count how many squares are across the bottom of the parallelogram and how many squares tall it is and multiply them.
Lesson 5 Bases and Heights of Parallelograms	Parallelograms S and T Parallelograms S and T are each labeled with a base and a corresponding height. What are the values of $b$ and $h$ for each parallelogram? Parallelogram S: $b$ = _________, $h$ = _________ Parallelogram T: $b$ = _________, $h$ = _________ Use the values of $b$ and $h$ to find the area of each parallelogram. Area of Parallelogram S: Area of Parallelogram T: Show Solution Parallelogram S: $b = 7$ , $h = 6$ Parallelogram T: $b = 3$ , $h = 6$ Area of Parallelogram S: 42 square units. $7 \boldcdot 6 = 42$ Area of Parallelogram T: 18 square units. $3 \boldcdot 6 = 18$
Lesson 6 Area of Parallelograms	One More Parallelogram Find the area of the parallelogram. Explain or show your reasoning. Was there a length measurement you did not use to find the area? If so, explain why it was not used. Show Solution 54 sq cm. Sample reasoning: A base is 9 cm and its corresponding height is 6 cm. $9 \boldcdot 6 = 54$ . The 7.5 cm length was not used. Sample reasoning: If the side that is 7.5 cm was used to find area, we would need the length of a perpendicular segment between that side and the opposite side as its corresponding height. We don't have that information. The parallelogram can be decomposed and rearranged into a rectangle by cutting it along the horizontal line and moving the right triangle to the bottom side. Doing this means the side that is 7.5 cm is no longer relevant. The rectangle is 6 cm by 9 cm; we can use those side lengths to find area.
Section B Check Section B Checkpoint	Problem 1 Select all parallelograms that show a base and its corresponding height (as a dashed segment). A. B. C. D. E. Show Solution A, C, D Problem 2 Find the area of the parallelogram. Explain or show your reasoning. Show Solution 32 sq cm. Sample reasoning: If the side that is 4 cm is the base, then its corresponding height is 8 cm. $4 \boldcdot 8 = 32$ If the side that is 10 cm is the base, then its corresponding height is 3.2 cm. $10 \boldcdot (3.2) = 32$ The parallelogram can be decomposed along the dashed line that is 3.2 cm long. Rearranging the pieces makes a rectangle that is 10 cm by 3.2 cm. $10 \boldcdot (3.2) = 32$ Problem 3 A parallelogram has an area of 60 square inches and a base that is 5 inches long. How long is the corresponding height? Show Solution 12 inches

Lesson 8 Area of Triangles	An Area of 14 Elena, Lin, and Noah all found the area of Triangle Q to be 14 square units but reasoned about it differently, as shown in the diagrams. Explain at least one student’s way of thinking and why his or her answer is correct. Three images of triangle Q labeled Elena, Lin, and Noah. Elena’s triangle has two additional triangles next to it to compose a rectangle, Lin’s triangle has a copy of the same triangle composed into a parallelogram, and Noah’s triangle shows the top portion of the triangle cut off and moved next to the bottom portion to create a parallelogram. Show Solution Sample responses: Elena drew two rectangles that decomposed the triangle into two right triangles. She found the area of each right triangle to be half of the area of its enclosing rectangle. This means that the area of the original triangle is the sum of half of the area of the rectangle on the left and half of the rectangle on the right. Half of $(4 \boldcdot 5)$ plus half of $(4 \boldcdot 2)$ is $10+4$ , so the area is 14 square units. Lin saw it as half of a parallelogram with the base of 7 units and height of 4 units (and thus an area of 28 square units). Half of 28 is 14. Noah decomposed the triangle by cutting it at half of the triangle’s height, turning the top triangle around, and joining it with the bottom trapezoid to make a parallelogram. He then calculated the area of that parallelogram, which has the same base length but half the height of the triangle. $7 \boldcdot 2 = 14$ , so the area is 14 square units.
Lesson 9 Formula for the Area of a Triangle	Two More Triangles For each triangle, identify a base and a corresponding height. Use them to find the area. Show your reasoning. A A triangle labeled A. Triangle A has sides of length 7.2, 3, and unknown. The perpendicular length from the side of length 3 to the opposite vertex is 6. The perpendicular length from the side of length 7.2 to the opposite vertex is 2.5. All lengths are in inches. B A triangle labeled B. Triangle B has sides of length 5, 6, and 5. The perpendicular length from the side of length 5 to the opposite vertex is 4.8. The perpendicular length from the side of length 6 to the opposite vertex is 4. All lengths are in centimeters. Show Solution Triangle A: 9 sq in. Sample reasoning: $b = 3$ , $h=6$ , area: 9 sq in, $\frac 12 \boldcdot 3 \boldcdot 6 = 9$ $b = 7.2$ , $h=2.5$ , area: 9 sq in, $\frac 12 \boldcdot (7.2) \boldcdot (2.5) = 9$ Triangle B: 12 sq in. Sample reasoning: $b = 6$ , $h=4$ , area: 12 sq cm, $\frac 12 \boldcdot 6 \boldcdot 4 = 12$ $b = 5$ , $h=4.8$ , area: 12 sq cm, $\frac 12 \boldcdot 5 \boldcdot (4.8) = 12$
Lesson 10 Bases and Heights of Triangles	Stretched Sideways For each triangle, draw a height segment that corresponds to the given base, and label it $h$ . Use an index card if needed. Which triangle has the greatest area? The least area? Explain your reasoning. Show Solution There are many possible locations for a height segment. The segments shown are the most straightforward. All of the triangles have the same area: 4 square units. Sample reasoning: They all have a base of 2 units and a height of 4 units.
Section C Check Section C Checkpoint	Problem 1 Identify a base and a height that you can use to find the area of each triangle. (You don’t have to actually find the areas.) Label each base with “b.” Draw a segment for each height and label it with “h.” Show Solution Sample response: Problem 2 Find the area of the triangle. Explain or show your reasoning. Show Solution 84 square inches. Sample reasoning: If the side that is 14 inches long is the base, its corresponding height is 12 inches. $\frac{1}{2} \boldcdot 14 \boldcdot 12 = \frac{1}{2} \boldcdot 168 = 84$ Problem 3 Find the area of the shaded polygon in square units. Show your reasoning. Show Solution 15 square units. Sample reasoning: The polygon can be decomposed into two right triangles. The area of the small triangle is half of a 3-by-2 rectangle, which is 3 square units. The area of the large triangle is half of a 6-by-4 rectangle, which is 12 square units. $3 + 12 = 15$

Lesson 12 What Is Surface Area?	A Snap Cube Prism A rectangular prism is 3 units high, 2 units wide, and 5 units long. What is its surface area in square units? Explain or show your reasoning. Show Solution 62 square units. Sample reasoning: $2 \boldcdot [(3 \boldcdot 5)+(2 \boldcdot 5)+(2 \boldcdot 3)]=62$
Lesson 13 Polyhedra	Three-Dimensional Shapes Write your best definition or description of a polyhedron. If possible, use the terms you learned in this lesson. Which of these five polyhedra are prisms? Which are pyramids? A B C D E Show Solution Answers might include one or more of these elements: A polyhedron is a three-dimensional figure made from faces that are filled-in polygons. Each face meets one and only one other face along a complete edge. The points where edges meet are called vertices. A, C, and D are prisms. B and E are pyramids.
Lesson 14 Nets and Surface Area	Unfolded net on a grid. 4 adjacent rectangles, from left to right, 4 by 3, 4 by 2, 4 by 3, 4 by 2. Above second rectangle 3 by 2 rectangle. Below fourth rectangle 3 by 2 rectangle. What kind of polyhedron can be assembled from this net? Find the surface area (in square units) of the polyhedron. Show your reasoning. Show Solution A rectangular prism 52 square units. Sample reasoning: $2(3 \boldcdot 4)+2(2 \boldcdot 4)+2(2 \boldcdot 3)=52$
Lesson 15 More Nets, More Surface Area	Surface Area of a Triangular Prism In this net, the two triangles are right triangles. All quadrilaterals are rectangles. What is its surface area in square units? Show your reasoning. A net of five shapes. Three rectangles in a row with a right triangle above and below the middle rectangle. The left rectangle has sides 5 and 10, the middle rectangle has sides 5 and 8, the right rectangle has sides 5 and 6. Each triangle has sides 8, 10, and 6. If the net is assembled, which of the following polyhedra would it make? A B C D Show Solution 168 square units. Sample reasoning: There are two triangular faces with area of 24 square units each. $\frac12 \boldcdot 6 \boldcdot 8 = 24$ . There is a rectangular face with area of 50 square units. $10\boldcdot 5 = 50$ . There is one rectangular face with area of 40 square units. $5\boldcdot 8=40$ . There is one rectangular face with area $5\boldcdot 6=30$ square units. $2\boldcdot 24 + 50 + 40 + 30 = 168$ Prism C
Lesson 16 Distinguishing Between Surface Area and Volume	Same Surface Area, Different Volumes Choose two figures that have the same surface area but different volumes. Show your reasoning. Show Solution Figures D and E both have a surface area of 26 square units, but D has a volume of 6 cubic units, and E has a volume of 7 cubic units.
Section D Check Section D Checkpoint	Problem 1 Select all nets that can be assembled into this triangular prism. A. B. C. D. E. Show Solution B, D, E Problem 2 Sketch a net for this square pyramid and label the known lengths. Find the surface area of the pyramid in square units. Show your reasoning. Show Solution Sample response: 72 square units. Sample reasoning: The square base is $4 \boldcdot 4$ or 16 square units. The area of each triangle is $\frac{1}{2} \boldcdot 4 \boldcdot 7$ , which is 14 square units. There are 4 triangles, so the surface area is: $16 + (4 \boldcdot 14)$ , which is $16 + 56$ or 72.

Lesson 17 Squares and Cubes	Exponent Expressions Which is larger, 5² or 3³? A cube has an edge length of 21 cm. Use an exponent to express its volume. Show Solution $3^3=27$ and $5^2=25$ , so 3³ is larger than 5². 21³ cm³ or 21³ cubic centimeters
Lesson 18 Surface Area of a Cube	From Volume to Surface Area A cube has an edge length of 11 inches. Write an expression for its volume and an expression for its surface area. A cube has a volume of 7³ cubic centimeters. What is its surface area? Show Solution Volume: 11³ or $11 \boldcdot 11 \boldcdot 11$ . Surface area: $6 \boldcdot (11 \boldcdot 11)$ (or equivalent). 294 square centimeters. $6 \boldcdot 7^2 = 294$
Section E Check Section E Checkpoint	Problem 1 A square has a side length of 15 centimeters. Select all expressions that represent the area of the square in square centimeters: A. $15^3$ B. $15 \boldcdot 15$ C. $15 + 15$ D. $15 \boldcdot 2$ E. $15^2$ Show Solution B, E Problem 2 A cube has a volume of $2^3$ cubic inches. Select all statements that are true about the cube: A.The volume of the cube is 8 cubic inches. B.The edge length of the cube is 3 inches. C.The expression $3 \boldcdot 2$ also represents its volume in cubic inches. D.The edge length of the cube is 2 inches. E. The volume of the cube is 6 cubic inches. Show Solution A, D Problem 3 A cube has an edge length of 47 units. Write an expression to represent its surface area in square units. Show Solution Sample responses: $6 \boldcdot 47^2$ $6 \boldcdot (47 \boldcdot 47)$ $47^2 + 47^2 + 47^2 + 47^2 + 47^2 + 47^2$ $(47 \boldcdot 47) + (47 \boldcdot 47) + (47 \boldcdot 47) + (47 \boldcdot 47) + (47 \boldcdot 47) + (47 \boldcdot 47)$

Lesson 19 All about Tents	No cool-down
Unit 1 Assessment End-of-Unit Assessment	Problem 1 Polyhedron P is a cube with a corner removed and relocated to the top of the cube. Polyhedron Q is a cube with the same size base as Polyhedron P. How do their surface areas compare? P Q A. Polyhedron P’s surface area is less than Polyhedron Q’s surface area. B. Polyhedron P’s surface area is equal to Polyhedron Q’s surface area. C. Polyhedron P’s surface area is greater than Polyhedron Q’s surface area. D. There is not enough information given to compare their surface areas. Show Solution Polyhedron P’s surface area is greater than Polyhedron Q’s surface area. Problem 2 Select all of the nets that can be folded and assembled into a triangular prism like this one. A. B. C. D. E. Show Solution A, D Problem 3 A cube has a side length of 8 inches. Select all the values that represent the cube’s volume in cubic inches. A. $8^2$ B. $8^3$ C. $6 \boldcdot 8^2$ D. $6 \boldcdot 8$ E. $8 \boldcdot 8 \boldcdot 8$ Show Solution B, E Problem 4 A square has a side length of 9 cm. What is its area? A square has an area of 9 cm². What is its side length? Show Solution 81 cm² 3 cm Problem 5 For each pair of expressions, circle the expression with the greater value. $13^2$ or $15^2$ $7 \boldcdot 6^2$ or $6^3$ $10^3$ or $30^2$ Show Solution $15^2$ $7 \boldcdot 6^2$ $10^3$ Problem 6 A rectangular prism measures 2 cm by 2 cm by 5 cm. What is its surface area? Explain or show your reasoning. A rectangular prism with three faces showing. Two faces are rectangles, each with a height of 2 centimeters and a length of 5 centimeters. The third face is a square, with side length of 2 centimeters. Show Solution 48 cm². Sample reasoning: The left and right faces each have area of 4 cm²( $2 \boldcdot 2 = 4$ ). The top, bottom, front, and back faces each have area of 10 cm² ( $2 \boldcdot 5 = 10$ ). So the surface area is $(2 \boldcdot 4) + (4 \boldcdot 10) = 48$ , or 48 cm². Minimal Tier 1 response: Work is complete and correct. Sample: $2 \boldcdot 2 + 2 \boldcdot 2 + 5 \boldcdot 2 + 5 \boldcdot 2 + 5 \boldcdot 2 + 5 \boldcdot 2 = 48$ , so 48 cm². Tier 2 response: Work shows general conceptual understanding and mastery, with some errors. Sample errors: Work shows the correct surface area but does not include reasoning or units. Work contains arithmetic mistakes but still indicates an intent to add up the areas of the six faces. Response (with work shown) is the sum of the areas of the three visible faces only. Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Sample errors: Student calculates a different quantity, such as volume or the area of only one face. Incorrect answer with no work is shown. Problem 7 Here is a net made of right triangles and rectangles. All measurements are given in centimeters. If the net were folded and assembled, what type of polyhedron would it make? Find the surface area of the polyhedron in square centimeters. Explain or show your reasoning. Show Solution Triangular prism 84 cm². Sample reasoning: The net is made of two triangles with a base of 4 cm and a height of 3 cm, and one big rectangle with a base of 12 cm and a height of 6 cm. The two triangles can be rearranged and put together to make a rectangle that is 4 cm by 3 cm, which results in an area of 12 cm². The area of the big rectangle is $6 \boldcdot 12$ or 72 cm². So the total (surface) area is 84 cm². Minimal Tier 1 response: Work is complete and correct, with complete explanation or justification. Sample: Triangular prism $\frac12 \boldcdot 4 \boldcdot 3 + \frac12 \boldcdot 4 \boldcdot 3 + 4 \boldcdot 6 + 5 \boldcdot 6 + 3 \boldcdot 6 = 84$ , so 84 cm². Tier 2 response: Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification. Sample errors: Arithmetic errors accompany otherwise correct work. The area of one face is missing or incorrect. Units are omitted. Surface area is correct and well-justified but the polyhedron is incorrect yet somewhat reasonable, such as a triangular pyramid or rectangular prism. Tier 3 response: Work shows a developing but incomplete conceptual understanding, with significant errors. Sample errors: The shape is correctly identified as a triangular prism but work shows little progress on finding the surface area. No reasonable attempt is made at identifying the polyhedron (or an answer like “trapezoid”). Surface area calculations involve serious mistakes, such as incorrect processes for calculating the area of a right triangle. Work includes a significant error in one part but a correct answer in the other part. Tier 4 response: Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery. Sample errors: Work shows significant omissions or Tier 3 errors across both problem parts.