Slide 1 / 152 Slide 2 / 152 Geometry Area of Figures 2015-10-27 www.njctl.org Slide 3 / 152 Slide 4 / 152 Table of Contents Throughout this unit, the Standards for Mathematical Practice are used. Click on the topic to go to that section Area of Rectangles MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. Area of Triangles MP3: Construct viable arguments and critique the reasoning of Law of Sines others. MP4: Model with mathematics. Area of Parallelograms MP5: Use appropriate tools strategically. MP6: Attend to precision. Area of Regular Polygons MP7: Look for & make use of structure. Area of Circles & Sectors MP8: Look for & express regularity in repeated reasoning. Additional questions are included on the slides using the "Math Area of Other Quadrilaterals Practice" Pull-tabs (e.g. a blank one is shown to the right on Area & Perimeter of Figures in the Coordinate Plane this slide) with a reference to the standards used. PARCC Sample Questions If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab. Slide 4 (Answer) / 152 Slide 5 / 152 Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of Math Practice others. MP4: Model with mathematics. MP5: Use appropriate tools strategically. Area of Rectangles MP6: Attend to precision. MP7: Look for & make use of structure. MP8: Look for & express regularity in repeated reasoning. Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on [This object is a pull tab] this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab. Return to Table of Contents
Slide 6 / 152 Slide 7 / 152 Area of a Rectangle Area of a Rectangle The area of a rectangle is defined to be the number of squares of In general, the area of a rectangle is equal to its base times its area "1" that can fit within it. height. This can also be referred to as its length times its width. In the below drawing, 6 unit squares fit within the below rectangle of A rectangle = length x width (lw) = base x height (bh) height 2 and based 3. 2 h 3 b Slide 8 / 152 Slide 9 / 152 Example Area of a Rectangle The diagonal of a rectangle is 34 feet and its length is 14 feet more than its width. Find the length, width, and area of the rectangle. x + 14 Sometimes, the dimensions will not be given, so you will need to We know that the width is unknown, calculate them before calculating the area. so let's call it "x". Since a rectangle is a quadrilateral with 4 right angles, 2 right triangles x Therefore, the length will be "x + 14". 34 ft can be formed when drawing one of its diagonals. Therefore, Pythagorean Theorem can be a helpful formula. Using the Pythagorean Theorem & our Algebra skills, we can solve for x. Another helpful formula is the perimeter formula for a rectangle: x 2 + (x + 14) 2 = 34 2 P = 2 l + 2 w click x 2 + x 2 + 28x + 196 = 1,156 There might also be questions asking you about the population density click 2x 2 + 28x - 980 = 0 of a town, state, or country. It is the ratio that represents the number of Since x is the length of a side, it click people living per square mile and can be found by dividing the total 2(x 2 + 14x - 480) = 0 has to be positive. Therefore, population by the total area. our final answer is click 2(x + 30)(x - 16) = 0 x = 16 ft = width click x + 30 = 0 or x - 16 = 0 length = 30 ft click Area = 480 ft 2 x = -30 or x = 16 click click Slide 9 (Answer) / 152 Slide 10 / 152 Example 1 What is the area of a rectangle that has a length of 8.4 The diagonal of a rectangle is 34 feet and its length is 14 feet more cm and a width of 3.7 cm? than its width. Find the length, width, and area of the rectangle. x + 14 Questioning to help address MP standards: We know that the width is unknown, What information do you have? (MP1) so let's call it "x". What is this problem asking? (MP1) What strategies are you going to use? x Therefore, the length will be "x + 14". 34 ft (MP1) Math Practice How can you represent the problem with numbers and symbols? (MP2) Using the Pythagorean Theorem & Create an equation to represent the our Algebra skills, we can solve for x. problem. (MP2) x 2 + (x + 14) 2 = 34 2 What labels (or units of measurement) should you use? (MP6) click x 2 + x 2 + 28x + 196 = 1,156 Does anyone have the same answer, but a click different way to explain it? (MP7) 2x 2 + 28x - 980 = 0 Since x is the length of a side, it What concepts that we have learned before click 2(x 2 + 14x - 480) = 0 were useful in solving this? (MP8) has to be positive. Therefore, [This object is a pull tab] click our final answer is 2(x + 30)(x - 16) = 0 x = 16 ft = width click x + 30 = 0 or x - 16 = 0 length = 30 ft click Area = 480 ft 2 x = -30 or x = 16 click click
Slide 10 (Answer) / 152 Slide 11 / 152 1 What is the area of a rectangle that has a length of 8.4 2 Televisions, are advertised using the length of the cm and a width of 3.7 cm? diagonal. For example, a 26" TV could have a length of 24" and a width of 10", as shown below. 24" Answer 10" 26" A = 31.08 cm 2 What is the area of an 80" TV if the length is 69.3"? [This object is a pull tab] Slide 11 (Answer) / 152 Slide 12 / 152 2 Televisions, are advertised using the length of the 3 The population density is the amount of people living per diagonal. For example, a 26" TV could have a length of square mile. If the town of Geometryville is a rectangular 24" and a width of 10", as shown below. town that has a length of 24 miles and a width of 13 miles, and its population is 2,500 people, what is the 24" population density of the town? 10" w = 39.97 in 26" Answer A = 2,769.92 in 2 What is the area of an 80" TV if the length is 69.3"? [This object is a pull tab] Slide 12 (Answer) / 152 Slide 13 / 152 3 The population density is the amount of people living per 4 The diagonal of a rectangle is 10 feet and its width is 2 square mile. If the town of Geometryville is a rectangular feet less than its length. What is the length of the town that has a length of 24 miles and a width of 13 rectangle? miles, and its population is 2,500 people, what is the A 4 feet population density of the town? Answer B 6 feet 8 people per square mile C 8 feet D 9 feet [This object is a pull tab]
Slide 13 (Answer) / 152 Slide 14 / 152 4 The diagonal of a rectangle is 10 feet and its width is 2 5 The diagonal of a rectangle is 10 feet and its width is 2 feet less than its length. What is the length of the feet less than its length. What is the width of the rectangle? rectangle? A 4 feet A 4 feet B 6 feet B 6 feet Answer C C 8 feet C 8 feet D 9 feet D 9 feet [This object is a pull tab] Slide 14 (Answer) / 152 Slide 15 / 152 5 The diagonal of a rectangle is 10 feet and its width is 2 6 The diagonal of a rectangle is 10 feet and its width is 2 feet less than its length. What is the width of the feet less than its length. What is the area of the rectangle? rectangle? A 4 feet A 80 ft 2 B 6 feet B 60 ft 2 Answer B C 8 feet C 48 ft 2 D 9 feet D 24 ft 2 [This object is a pull tab] Slide 15 (Answer) / 152 Slide 16 / 152 6 The diagonal of a rectangle is 10 feet and its width is 2 feet less than its length. What is the area of the rectangle? A 80 ft 2 B 60 ft 2 Answer C Area of Triangles C 48 ft 2 D 24 ft 2 [This object is a pull tab] Return to Table of Contents
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