Show that when a circle is inscribed inside a square the diameter of the circle is the same length as the side of the square.


 Elizabeth Morrison
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1 Week & Day Week 6 Day 1 Concept/Skill Perimeter of a square when given the radius of an inscribed circle Standard 7.MG:2.1 Use formulas routinely for finding the perimeter and area of basic twodimensional figures and the surface area and volume of basic threedimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders. Materials CAHSEE Math Released Item #104 Overview for Week This week will provide students with a quick review of the vocabulary, methods for finding perimeter, area, and volume, and some practice problems from the CAHSEE released items. It is important that students determine what the questions is asking so they do not calculate the wrong answer. Review vocabulary perimeter distance around the outside circumference the perimeter of a circle radius distance from the center of a circle to the side diameter distance from side to side of a circle through the center of a circle which is 2 times the radius of a circle. Show that when a circle is inscribed inside a square the diameter of the circle is the same length as the side of the square. Have students review how to find the perimeter of a square if given the radius of an inscribed circle: 1. Find the length of one side of the square. (Since the radius is the distance from the center of the circle to the side, the radius is ½ the length of the side of the square. To find the length of a side of the square multiply the radius by 2) 2. Since the perimeter is the length around the square add up the length of all 4 sides or multiply the length by 4. Have students do the guided practice on Day 1 of the student CAHSEE Math Problem of the Day  Week 6 worksheet... 2 In the figure above, the radius of the inscribed circle is 2 inches. What is the perimeter of square ABCD? Find the length of the side of the square: Since the radius is _2_ inches the length of the side of the square is 2 x _2_ in. = 4_ in. Find the perimeter of the square: 4 in x 4 = 16 in (Day 1 continued on next page) SAUSD, Educational Services Week 6 Teacher  Page 1 of 10
2 Have students do the CAHSEE released test item on Day 1 of the student CAHSEE Math Problem of the Day  Week 6 worksheet. Day 1 (CAHSEE Math Released Item #104) 104. In the figure above, the radius of the inscribed circle is 6 inches (in.) What is the perimeter of square ABCD? A 12π in. Students are told that the radius is 6 inches so the diameter is 12 inches. The side of the square is the same as the diameter of the circle so the side of the square is 12 inches. The students must find the perimeter or the distance around the square which is 4x12=48. The correct answer is D. B C D 36π in. 24 in. 48 in. SAUSD, Educational Services Week 6 Teacher  Page 2 of 10
3 Week Week 6 Day 2 Concept/Skill Finding areas of rectangles and triangles. Standard 7.MG:2.1 Use formulas routinely for finding the perimeter and area of basic twodimensional figures and the surface area and volume of basic threedimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders. Materials CAHSEE Math Released Item # 106 Review that area is the space inside a rectangle or triangle and will be measured in square inches (in 2 ), square feet (ft 2 ), square centimeters (cm 2 ) or square meters (m 2 ). Have students review how to find the area of a rectangle (this skill is used later this week): area of rectangle = length width This formula works for all rectangles, including squares. Have students do the guided practice on Day 2 of the student CAHSEE Math Problem of the Day  Week 6 worksheet. 2 ft What is the area of the rectangle shown above? First determine the length and the width. The length is 3 ft and the width and 2 ft. Then put the numbers in the formula: area of rectangle = length width area of rectangle = 3 ft_ 2ft_ area of rectangle = 6 ft 2 Have students review how to find the area of a triangle Multiply ½ the length of the base by the height (distance from the base to the top) of the triangle. area of triangle = ½ base height This formula works for all triangles. (continued on next page) 3ft SAUSD, Educational Services Week 6 Teacher  Page 3 of 10
4 Have students do the guided practice on Day 2 of the student CAHSEE Math Problem of the Day  Week 6 worksheet. 5 in 4 in 9 in What is the area of the triangle shown above? First determine the base and the height. The base is _9 in and the height is 4 in. Then put the numbers in the formula: area of triangle = ½ base height area of triangle = ½ _9 in 4 in area of triangle = 18 in 2 Note: Tell students that the base is the vertical line with the. They will not use the side of the triangle; that is just extra information. Have students do the CAHSEE released test item on Day 2 of the student CAHSEE Math Problem of the Day  Week 6 worksheet. Day 2 (CAHSEE Math Released Item #106) Students need to know the formula for finding the area of a triangle: area of triangle = ½ x base x height The base is 15 The height is 8 ½ x 15 x 8 = 60 Since the units are not indicated on the drawing they only need to know that the answer is in square units The ANSWER is B. SAUSD, Educational Services Week 6 Teacher  Page 4 of 10
5 Week Week 6 Day 3 Concept/Skill Finding the area of a rectangle that has an inside piece missing Standard 7.MG:2.1 Use formulas routinely for finding the perimeter and area of basic twodimensional figures and the surface area and volume of basic threedimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders. Materials CAHSEE Math Released Items #105, 107 Review Steps to find the shaded area: 1. Calculate the area of the whole rectangle or square: area = length x width 2. Calculate the area of the cut out part. 3. Subtract the area of the cut out part from the area of the rectangle or square. Have students review how to find the area of a circle. area of a circle = π r 2 or π x radius x radius where π is 3.14 Have students do the guided practice on Day 3 of the student CAHSEE Math Problem of the Day  Week 6 worksheet. Example: Calculate the area of the square: Area of square = 10 feet 10 feet Area of square = 100 feet 2 _ Calculate the area of the circle: Since the circle is fits exactly inside the square the length of the square is the same as the diameter of the circle. The radius of the circle is ½ the diameter so the radius is 5 feet. Area of circle = π r 2 Area of circle = feet 5 feet Area of circle = 78.5 feet 2 Since you only need to know the approximate area you can round the area to nearest 10. Subtract the area of the circle from the square: _100 feet 2  _80 feet 2 = 20 feet 2 The correct answer is _A. Have students do the CAHSEE released test item on Day 3 of the student CAHSEE Math Problem of the Day  Week 6 worksheet (see next page) SAUSD, Educational Services Week 6 Teacher  Page 5 of 10
6 Have students do the CAHSEE released test item on Day 3 of the student CAHSEE Math Problem of the Day  Week 6 worksheet Day 3 (CAHSEE Math Released Item 107) Students need to know to subtract the area of the smaller rectangle from the area of the larger rectangle. Since the question says approximately all of the numbers can be rounded to the nearest ten. Area of the larger rectangle 100 x 240 = 24,000 Area of the smaller rectangle 40 x 70 = 2, =21,200 which is closest to 21,000 The correct answer is C SAUSD, Educational Services Week 6 Teacher  Page 6 of 10
7 Week Week 6 Day 4 Concept/Skill Find the volume of a box (prism) Standard 7.MG:2.1 Use formulas routinely for finding the perimeter and area of basic twodimensional figures and the surface area and volume of basic threedimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders. Materials CAHSEE Math Released Items # 108 Review the steps to find the volume of a box (aka prism): Multiply the length by the width by the height: volume of box = length width height Have students do the guided practice test item on Day 4 of the student CAHSEE Math Problem of the Day  Week 6 worksheet. Example: 3 cm 2 cm 4 cm What is the volume of the box shown above in cubic centimeters (cm 3 )? First determine the length, width and height. The length is 4cm The width is 2cm The height is 3cm_ Then put the numbers in the formula and do the calculation: volume of box = length x width x height volume of box = 4cm_ _2cm 3cm volume of box = 24cm 3 Day 4 continued on next page. SAUSD, Educational Services Week 6 Teacher  Page 7 of 10
8 Have students do the CAHSEE released test item on Day 4 of the student CAHSEE Math Problem of the Day  Week 6 worksheet Day 4 (CAHSEE Math Released Item # 108 ) Students need to make sure they are looking for the correct answer. The key word in this question is volume. Volume of a prism = length x width x height Volume = 15 x 5 x 9 Volume = 675 The correct answer is D SAUSD, Educational Services Week 6 Teacher  Page 8 of 10
9 Week Week 6 Day 5 Concept/Skill Surface area of a prism Standard 7.MG:2.1 Use formulas routinely for finding the perimeter and area of basic twodimensional figures and the surface area and volume of basic threedimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders. Materials CAHSEE Math Released Item # 118 Review the steps to find the the surface area of a box: A box is composed of six surfaces: Top and bottom Left side and right side Front and back To find the total surface area you need to add the areas of all six sides. Have students do the guided practice test item on Day 5 of the student CAHSEE Math Problem of the Day  Week6 worksheet. Example: 3 cm 2 cm 4 cm What is the surface area of the rectangle shown above in square centimeters (cm 2 )? The area of the bottom = _4_ cm _2_ cm The area of the bottom = 8 cm 2 The top and the bottom are the same size. The total for top and bottom _8 cm 2 + _8 cm 2 = _16 cm 2 The area of the right side = _2_ cm _3_ cm The area of the right side = _6_ cm 2 The right side and the left side are the same size. The total for right and left side _6 cm cm 2 = _12 cm 2 The area of the front = _4_ cm _3_ cm The area of the front = 12_ cm 2 The front and the back are the same size. The total for front and back _12_ cm _cm 2 = 24_ cm 2 The total surface area of the box is _16 cm _ cm _ cm 2 = 52_ cm 2 SAUSD, Educational Services Week 6 Teacher  Page 9 of 10
10 Have students do the CAHSEE released test item on Day 5 of the student CAHSEE Math Problem of the Day  Week worksheet. Day 5 (CAHSEE Math Released Item 118) Students need to make sure they are looking for the correct answer. The key word in this question is surface area. Calculate the area of all six sides: Top = 24 x 10 = 240 Top + Bottom = = 480 Right side = 12 x 10 = 120 Right + Left = = 240 Front = 24 x 12 = 288 Front + Back = = 576 Total of all six sides = = 1296 The correct answer is C At the end of the week remind students to keep the student CAHSEE Math Problem of the Day  Week 6 worksheet to use as a study guide as they continue to prepare for the CAHSEE. SAUSD, Educational Services Week 6 Teacher  Page 10 of 10
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