ࡱ> _ J0bjbj >2bbY>X jjj~~~8<~H,,B$V:&d&4$cjm($"$m(m(,Bˤ+++m(8,jB+m(++b~$RB*6(ZV0U*&j&L'6+T','&&&C+&&&m(m(m(m(&&&&&&&&& : Rigorous Curriculum Design Unit Planning Organizer Subject(s)MathematicsGrade/Course7thUnit of StudyUnit 6: GeometryUnit Type(s)Q'Topical X Skills-based Q' Thematic Pacing25 days Unit Abstract  In this unit, students will recognize two- and three-dimensional figures and their construction. They will use nets of figures to determine surface area; area and circumference of circles; and draw, construct, and describe the relationship between geometric figures.  Common Core Essential State Standards Domain: Geometry (7.G) Clusters: Solve real-world and mathematical problems involving area, surface area, and volume. Draw, construct, and describe geometrical figures and describe the relationship between them. Standards: 7.G.2 DRAW (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. FOCUS on CONSTRUCTING triangles from three measures of angles or sides, NOTICING when the conditions DETERMINE a unique triangle, more than one triangle, or no triangle. 7.G.3 DESCRIBE the three-dimensional figures that RESULT from SLICING three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. 7.G.4 KNOW the formulas for the area and circumference of a circle and USE them to SOLVE problems; GIVE an informal derivation of the relationship between the circumference and area of a circle. 7.G.5 USE facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to WRITE and SOLVE simple equations for an unknown angle in a figure. 7.G.6 SOLVE real-world and mathematical problems INVOLVING area, volume and surface area of two- and three-dimensional objects COMPOSED of triangles, quadrilaterals, polygons, cubes, and right prisms.  Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.  UNPACKED STANDARDS7.G.2 Students draw geometric shapes with given parameters. Parameters could include parallel lines, angles, perpendicular lines, line segments, etc. Example 1: Draw a quadrilateral with one set of parallel sides and no right angles. Students understand the characteristics of angles and side lengths that create a unique triangle, more than one triangle or no triangle. Example 2: Can a triangle have more than one obtuse angle? Explain your reasoning. Example 3: Will three sides of any length create a triangle? Explain how you know which will work. Possibilities to examine area: a. 13 cm, 5 cm, and 6 cm b. 3 cm, 3 cm, and 3 cm c. 2 cm, 7 cm, 6 cm Solution: A above will not work; B and C will work. Students recognize that the sum of the two smaller sides must be larger than the third side. Example 4: Is it possible to draw a triangle with a 90 angle and one leg that is 4 inches long and one leg that is 3 inches long? If so, draw one. Is there more than one such triangle? (NOTE: Pythagorean Theorem is NOT expected this is an exploration activity only.) Example 5: Draw a triangle with angles that are 60 degrees. Is this a unique triangle? Why or why not: Example 6: Draw an isosceles triangle with only one 80 angle. Is this the only possibility or can another triangle be drawn that will meet these conditions?  Through exploration, students recognize that the sum of the angles of any triangle will be 180 and the angles of any quadrilateral will sum to 360 Other explorations would include: Base angles of an equilateral triangle are equal Angle and side length relationships between scalene, isosceles, and equilateral triangle Angle and side length relationships between obtuse, acute and right triangles 7.G.3 Students need to describe the resulting face shape from cuts made parallel and perpendicular to the bases of right rectangular prisms and pyramids. Cuts made parallel will take the shape of the base; cuts made perpendicular will take the shape of the lateral (side) face. Cuts made at an angle through the right rectangular prisms will produce a parallelogram.  If the pyramid is cut with a plane (green) parallel to the  base, the intersection of the pyramid and the plane is a square cross section (red). If the pyramid is cut with a plane (green) passing through the top vertex and perpendicular to the base, the intersection of the pyramid and the plane is a triangular cross section (red).  If the pyramid is cut with a plane (green) perpendicular to the base, but not through the vertex, the intersection of the pyramid and the plane is a trapezoidal cross section (red).  HYPERLINK "http://intermath.coe.uga.edu/dictnary/descript.asp?termID=95" http://intermath.coe.uga.edu/dictnary/descript.asp?termID=95  7.G.4 Students understand the relationship between radius and diameter. Students also understand the ratio of circumference to diameter can be expressed as pi. Building on these understandings, students generate the formula for circumference and area. The illustration shows the relationship between the circumference and area. If a circle is cut into wedges and laid out as shown, a parallelogram results. Half of an end wedge can be moved to the other end and a rectangle results. The height of the rectangle is the same as the radius of the circle. The base length is EMBED Equation.3  the circumference (2r). The area of the rectangle (and therefore the circle) is found by the following calculations:    HYPERLINK "http://mathworld.wolfram.com/Circle.html" http://mathworld.wolfram.com/Circle.html Students solve problems (mathematical and read-world) involving circles or semi-circles. Note: Because pi is an irrational number that neither repeats nor terminates, the measurements of area approximate when 3.14 is used in place of . Example 1: The seventh grade class is building a mini-golf game for the school carnival. The end of the putting green will be a circle. If the circle is 10 feet in diameter, how many square feet of grass carpet will they need to buy to cover the circle? How might someone communicate this information to the salesperson to make sure he receives a piece of carpet that is the correct size? Use 3.14 for pi. Solution: Area = r2 Area = 3.14 (5)2 Area = 78.5 ft2 To communicate this information, ask for a 9 ft by 9 ft square of carpet. Example 2: The center of a circle is at (2, -3). What is the area of the circle?  Solution: The radius of the circle is 3 units. Using the formula, Area = r2, the area of the circle is approximately 28.26 units2. Students build on their understanding of area from 6th grade to find the area of left-over materials when circles are cut from squares and triangles or when squares and triangles are cut from circles. Example 3: If a circle is cut from a square piece of plywood, how much plywood would be left over?  Solution: The area of the square is 28 x 28 or 784 in2. The diameter of the circle is equal to the length of the side of the square, or 28, so the radius would be 14. The area of the circle would be approximately 615.44 in2. The difference in the amounts (plywood left over) would be 168.56 in2 (784 615.44). Example 4: What is the perimeter of the inside of the track?  Solution: The ends of the track are two semicircles, which would form one circle with a diameter of 62 m. The circumference of this part would be 194.68 m. Add this to the two lengths of the rectangle and the perimeter is 2194.68 m. Know the formula does not mean memorization of the formula. To know means to have an understanding of why the formula works and how the formula relates to the measure (area and circumference) and the figure. This understanding should be for all students. 7.G.5 Students use understandings of angles and deductive reasoning to write and solve equations. Example 1: Write and solve an equation to find the measure of angle x.  Solution: Find the measure of the missing angle inside the triangle (180 90 40) or 50. The measure of angle x is supplementary to 50, so subtract 50 from 180 to get a measure of 130 for x. Example 2: Find the measure of angle x.  Solution: First, find the missing angle measure of the bottom triangle (180 30 30 = 120). Since the 120 is a vertical angle to x, the measure of x is also 120. Example 3: Find the measure of angle b.  Note: Not drawn to scale. Solution: Because, the 45, 50 angles and b form are supplementary angles, the measure of angle b would be 85. The measures of the angles of a triangle equal 180 so 75 + 85 + a = 180. The measure of angle a would be 20. 7.G.6 Students continue work from 5th and 6th grade to work with area, volume and surface area of two-dimensional and three-dimensional objects. (composite shapes) Students will not work with cylinders, as circles are not polygons. At this level, students determine the dimensions of the figures given the area or volume. Know the formula does not mean memorization of the formula. To know means to have an understanding of why the formula works and how the formula relates to the measure (area and volume) and the figure. This understanding should be for all students. Surface area formulas are not the expectation with this standard. Building on work with nets in the 6th grade, students should recognize that finding the area of each face of a three-dimensional figure and adding the areas will give the surface area. No nets will be given at this level; however, students could create nets to aid in surface area calculations. Students understanding of volume can be supported by focusing on the area of base times the height to calculate volume. Students solve for missing dimensions, given the area or volume. Students determine the surface area and volume of pyramids. Volume of Pyramids Students recognize the volume relationship between pyramids and prisms with the same base area and height. Since it takes 3 pyramids to fill 1 prism, the volume of a pyramid is  EMBED Equation.3  the volume of a prism (see figure below).  To find the volume of a pyramid, find the area of the base, multiply by the height and then divide by three. V = Bh B = Area of the Base 3 h = height of the pyramid Example 1: A triangle has an area of 6 square feet. The height is four feet. What is the length of the base? Solution: One possible solution is to use the formula for the area of a triangle and substitute in the known values, then solve for the missing dimension. The length of the base would be 3 feet. Example 2: The surface area of a cube is 96 in2. What is the volume of the cube? Solution: The area of each face of the cube is equal. Dividing 96 by 6 gives an area of 16 in2 for each face. Because each face is a square, the length of the edge would be 4 in. The volume could then be found by multiplying 4 x 4 x 4 or 64 in3. Example 3: Huong covered the box to the right with sticky-backed decorating paper. The paper costs 3 per square inch. How much money will Huong need to spend on paper?  Solution: The surface area can be found by using the dimensions of each face to find the area and multiplying by 2: Front: 7 in. x 9 in. = 63 in2 x 2 = 126 in2 Top: 3 in. x 7 in. = 21 in2 x 2 = 42 in2 Side: 3 in. x 9 in. = 27 in2 x 2 = 54 in2 The surface area is the sum of these areas, or 222 in2. If each square inch of paper cost $0.03, the cost would be $6.66. Example 4: Jennie purchased a box of crackers from the deli. The box is in the shape of a triangular prism (see diagram below). If the volume of the box is 3,240 cubic centimeters, what is the height of the triangular face of the box? How much packaging material was used to construct the cracker box? Explain how you got your answer.  Solution: Volume can be calculated by multiplying the area of the base (triangle) by the height of the prism. Substitute given values and solve for the area of the triangle. V = Bh 3,240 cm3 = B (30 cm) 3,240 cm3 = B (30 cm) 30 cm 30 cm 108 cm2 = B (area of the triangle) To find the height of the triangle, use the area formula for the triangle, substituting the known values in the formula and solving for height. The height of the triangle is 12 cm. The problem also asks for the surface area of the package. Find the area of each face and add: 2 triangular bases: (18 cm)(12 cm) = 108 cm2 x 2 = 216 cm2 2 rectangular faces: 15 cm x 30 cm = 450 cm2 x 2 = 900 cm2 1 rectangular face: 18 cm x 30 cm = 540 cm2 Adding 216 cm2 + 900 cm2 + 540 cm2 gives a total surface area of 1656 cm2.  Unpacked Concepts (students need to know)Unwrapped Skills (students need to be able to do)COGNITION DOK7.G.2 Drawing geometric figures (freehand, ruler, protractor or technology)  I can draw geometric figures freehand, ruler and protractor or with technology with given conditions. 2 7.G.3 Description of cross sections  I can describe the shape of the cross section resulting from cutting through a three-dimensional figure. 27.G.4 Formulas for area and circumference of a circle  I can use formulas to find the area and circumference of circles. I can use a formula to find the diameter and radius of a circle when the circumference is given. I can explain how the circumference and area of a circle are related to each other. 2 2 37.G.5 Angle pairs  I can identify supplementary, complementary, vertical and adjacent angles and find the measure of one angle when the measure of another angle is known.  27.G.6 Area, volume and surface area of two- and three-dimensional figures  I can solve real-world and mathematical problems that involve area of shapes that can be decomposed into smaller shapes (squares, rectangles, triangles, trapezoids) whose areas can be found by applying formulas. I can solve real-world and mathematical problems involving surface area and volume of three-dimensional figures that are made up of smaller figures such as cubes and right prisms whose surface areas and volumes can be found by applying formulas. 3 3  Essential QuestionsCorresponding Big Ideas7.G.2 How can I draw geometric figures (freehand, ruler, protraction or technology) with given conditions?  Students will draw geometric figures freehand, by ruler and protractor or with technology with given conditions.7.G.3 How can I describe cross sections that result from slicing three-dimensional figures as in plane sections of right rectangular prisms and right rectangular pyramids? Students can describe the shape of the cross section when cutting through a three-dimensional figure.7.G.4 How can I use formulas to find area and circumference of a circle? How can I use the formulas to find the area of a circle when the circumference is given? How can I describe the informal derivation of the relationship between area and circumference of a circle? Students can use formulas to find the area and circumference of circles. Students can use a formula to find the diameter and radius of a circle when the circumference is given. Students can use a formula to find the area of a circle when the circumference is given. Students can explain how the circumference and area of a circle are related to each other.7.G.5 How can I use facts about angle pairs to write and solve simple equations for an unknown angle in a figure? Students can identify angle pairs and find the measure of one angle when the measure of another angle is known. 7.G.6 How can I solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional figures that are composed of triangles, quadrilaterals, polygons, cubes, and right prisms? Students can solve real-world and mathematical problems that involve area of shapes that can be decomposed into smaller shapes (squares, rectangles, triangles, trapezoids) whose areas can be found by applying formulas. Students can solve real-world and mathematical problems involving surface area and volume of three-dimensional figures that are made up of smaller figures such as cubes and right prisms whose surface areas and volumes can be found by applying formulas. Vocabulary inscribed, circumference, radius, diameter, pi, , supplementary, vertical, adjacent, complementary, pyramids, face, base, decompose, area, surface area, volume, net, vertices, height, trapezoid, isosceles, right triangle, squares, right rectangular prisms, cross section Language Objectives Key Vocabulary  7.G.2 7.G.5SWBAT define, give examples of, and use the key vocabulary specific to this standard orally and in writing (inscribed, circumference, radius, diameter, pi, , supplementary, vertical, adjacent, complementary, pyramids, face, base, decompose, area, surface area, volume, net, vertices, height, trapezoid, isosceles, right triangle, squares, right rectangular prisms, cross section) Language Function 7.G.2SWBAT write step-by-step directions to draw geometric shapes with given conditions.7.G.4SWBAT use pictures, words, and number to show the formulas for area and circumference of a circle. 7.G.6SWBAT use examples, words, and pictures of two- and three-dimensional figures that are made up of triangles, quadrilaterals, polygons, cubes, and right prisms.Language Skill 7.G.4SWBAT read a real-world story problem and decide which formula will be used to solve the problem.7.G.5SWBAT listen to a teacher describe supplementary, complementary, vertical and adjacent angles and determine which angles are each to a partner.7.G.6SWBAT listen to a teacher describe the parts of two- and three-dimensional figures and label these parts with a partner.Grammar and Language Structures7.G.2SWBAT use comparative phrases such as greater, more, less, fewer, or equal with a partner when describing the side lengths or angle measurements of triangles.Lesson Tasks 7.G.4SWBAT explain how they use models to find the area and circumference of a circle.7.G.5SWBAT explain how they use models to locate and label complementary, supplementary, vertical, and adjacent angles.7.G.6SWBAT explain how they use models to locate and label the parts of a two- or three-dimensional figure.Language Learning Strategies 7.G.4SWBAT listen to a partner describe how to find the area and circumference of a circle and write the steps.7.G.5SWBAT listen to a partner describe the types of angles in a figure and label them. Information and Technology Standards 7.SI.1.1 Evaluate resources for reliability.7.TT.1.1 Use appropriate technology tools and other resources to access information.7.TT.1.2 Use appropriate technology tools and other resources to organize information (e.g., graphic organizers, databases, spreadsheets, and desktop publishing).7.RP.1.1 Implement a collaborative research process activity that is group selected.7.RP.1.2 Implement a collaborative research process activity that is student selected. 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h3.hQph3.hQp53`````:agUC-- & Fd$If^gdB7d$If^gdd$If^gdO;kd$$Ifl0H$   t0644 lapyt~:a;aMa[ana{aaaaaaaa2HIo & F(d$Ifgdhd$If^hgd & F&d$If^gdB7d$Ifgdhd$If^hgdLessons for Learning (DPI) Changing Surface Areas Packing to Perfection Mathematics Assessment Project (MARS)  HYPERLINK "http://vsimington.pbworks.com/w/session/login?return_to=http%3A%2F%2Fvsimington.pbworks.com%2Fw%2Fpage%2F5774312%2FFrontPage"WSFCS Math Wiki HYPERLINK "http://maccss.ncdpi.wikispaces.net/Seventh+Grade"NCDPI Wikispaces Seventh Grade HYPERLINK "https://www.georgiastandards.org/Common-Core/Common%20Core%20Frameworks/CCGPS_Math_7_7thGrade_Unit5SE.pdf"Georgia Unit HYPERLINK "http://www.graniteschools.org/depart/teachinglearning/curriculuminstruction/math/secondarymathematics/Pages/Math7.aspx"Granite Schools Math7 HYPERLINK "http://illuminations.nctm.org/LessonDetail.aspx?ID=L570"Illuminations NCTM Building a Box HYPERLINK "http://illuminations.nctm.org/LessonDetail.aspx?ID=L270"Illuminations NCTM Polygon Capture HYPERLINK "http://illuminations.nctm.org/LessonDetail.aspx?ID=L616"Illuminations NCTM Cubes Everywhere HYPERLINK "http://illuminations.nctm.org/LessonDetail.aspx?ID=L763"Illuminations NCTM Planning a Playground HYPERLINK "http://katm.org/wp/wp-content/uploads/flipbooks/7th_FlipBookEdited21.pdf"KATM Flip Book7 HYPERLINK "http://www.shodor.org/interactivate/discussions/SurfaceAreaRectangular/"Shodor Interactive Discussions Surface Area Rectangular Prism HYPERLINK "http://www.shodor.org/interactivate/activities/SurfaceAreaAndVolume/"Shodor Interactive Activities/Surface Area And Volume/ HYPERLINK "http://www.shodor.org/interactivate/activities/Angles/"Shodor Interactive Activities Angles HYPERLINK "http://www.mathvillage.info/node/116"Mathvillage Surface Area Rectangular Prisms  HYPERLINK "http://www.uen.org/Lessonplan/LPview.cgi?grade=7" UEN Lesson Plans Grade 7      PAGE \* MERGEFORMAT 17 Revised 7/23/13 Anet = Base x Height Area =  EMBED Equation.3 (opqropklIKd$If^gd~d$Ifgd~d$-DIfM gd~d$If^gd & F'd$IfgdNlmnpqz{|EFGᲧyngYnjՔhcih5 U hcih5 jhcih5 U hcih~ hcihQphcihq$0J^Jjܓhcih h3.hQphNMh5 6hcihO;0J^JjќhcihO;UjhcihO;Ujhcih5 U hcihO; 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