Math 6 Unit 9 Notes: Measurement and Geometry, Area/Volume

Transcription

1 Math 6 Unit 9 Notes: Measurement and Geometry, rea/volume erimeter Objectives: (5.5) The student will model formulas to find the perimeter, circumference and area of plane figures. (5.6) The student will apply formulas to find the perimeter, circumference and area of plane figures. reviously in Unit 8 notes, we worked on the conceptual understanding, modeling of and deriving the formulas for perimeter, circumference and area of plane figures. Now we are moving into the application of these formulas. The perimeter of a polygon is the sum of the lengths of the segments that make up the sides of the polygon. m Example: Find the perimeter of the regular pentagon. Since the pentagon is regular, we know all five sides have a measurement of meters. So we simply multiply 5 for an answer of 0 meters for the perimeter. Example: Find the perimeter for a rectangle with length 7 feet and width feet. Since opposite sides of a rectangle are equal in length, 7 or 8 feet. Example: The perimeter of a square is 64 inches. Find the length of each side. = 4s 64=4s 6=s Each side of the square is 6 inches. Example: Find the length of the missing side if you know the perimeter is 67 meters. erimeter is equal to the sum of the sides k 4 6 k k The length of the missing side is 6 meters. k 3 m 3 m 5 m Math 6 Notes Unit 09: Measurement and Geometry age of 7 Revised 0 CCSS

2 Example: Find the perimeter for the polygon.. Students must determine the dimensions of the missing side lengths. 8 mm 6 mm 4 mm 4 mm Find the missing dimensions. 4 mm Since this is a rectangle (opposite sides congruent) and the bottom length is 0 mm and the top piece is 6 mm, then 0 6=4. The left side of the rectangle is 8 mm and on the right we have 4 mm, so the far right section must be 8 4=4. 0 mm. Then compute the perimeter by finding the sum of the edge lengths. or since bottom = 0, top pieces =0 = since left =8, right pieces=8 = 36 SO, = = 36 The perimeter is 36 mm. Example: Find the perimeter for the polygon. = = = The perimeter is 36 units. Circles: Circumference circle is defined as all points in a plane that are equal distance (called the radius) from a fixed point (called the center of the circle). The distance across the circle, through the center, is called the diameter. Therefore, a diameter is twice the length of the radius, or d r. centerpoint diameter radius Math 6 Notes Unit 09: Measurement and Geometry age of 7 Revised 0 CCSS

3 We called the distance around a polygon the perimeter. The distance around a circle is called the circumference. There is a special relationship between the circumference and the diameter of a circle. Let s get a visual to approximate that relationship. Take a can with 3 tennis balls in it. Wrap a string around the can to approximate the circumference of a tennis ball. Then compare that measurement with the height of the can (which represents three diameters). You will discover that the circumference of the can is greater than the three diameters (height of the can). There are many labs that will help students understand the concept of pi,. Most involve measuring the distance around the outside of a variety of different size cans, measuring the diameter of each can and dividing to show that the relationship is 3 and something. You may want to use a table like the following to organize student work: Can # Circumference Diameter Circumference Diameter = Once students record the measurements for a variety of cans, students should find the quotient of the circumference and the diameter is close to 3.4. Some quotients are not exact, or are off, due to measurement errors or lack of precision. So initially get students to comprehend the circumference diameter x 3. This should help convince students that this ratio will be the same for every circle. C We can then introduce that π or C πd. Then, since d r, we can also write C πr. d lease note that π is an irrational number (never ends or repeats). Mathematicians use to represent the exact value of the circumference/diameter ratio. The formulas for the circumference of a circle are C d and C r. Example: circle has a radius of 4 m. Estimate the circumference of the circle. (Use 3 to approximate pi.) C r Using the formula: C C 34 4 The circumference is about 4 meters. Note: Many standardized tests (including the CRT and the district common exams) ask students to leave their answers in terms of π. Be sure to practice this! Math 6 Notes Unit 09: Measurement and Geometry age 3 of 7 Revised 0 CCSS

4 Example: If a circle has a radius of 5 feet, find its circumference. Do not use an approximation for π. Using the formula: C C πr π 5 The circumference is 0π feet. C 0π Example: If a circle has a diameter of 35 inches, find its circumference. Leave your answer in terms of π. C d Using the formula: C 35 C 35 Example: circle has a diameter of 4 m. Using π answer to the nearest whole number. The circumference is 35π inches. 3.4, find the circumference. Round your Using the formula: C πd C (3.4) 4 C The circumference is about 75 meters. Example: If a circle has a diameter of 4 m, what is the circumference? Use 3.4 to approximate π. State your answer to the nearest 0. meter. Using the formula: C C πd (3.4)(4) The circumference is about.6 meters. C.56 CRT Example: Solution: The circumference of the original young tree would be: C d The circumference of the older tree would be: C d The difference between the two circumferences is: nswer: B 9.4 cm Math 6 Notes Unit 09: Measurement and Geometry age 4 of 7 Revised 0 CCSS

5 rea of lane Figures rea of Rectangles, Squares and Rhombi One way to describe the size of a room is by naming its dimensions. room that measures ft. by 0 ft. would be described by saying it s a by 0 foot room. That s easy enough. There is nothing wrong with that description. In geometry, rather than talking about a room, we might talk about the size of a rectangular region. For instance, let s say I have a closet with dimensions feet by 6 feet. That s the size of the closet. ft. 6 ft. Someone else might choose to describe the closet by determining how many one foot by one foot tiles it would take to cover the floor. To demonstrate, let me divide that closet into one foot squares. ft. 6 ft. By simply counting the number of squares that fit inside that region, we find there are squares. If I continue making rectangles of different dimensions, I would be able to describe their size by those dimensions, or I could mark off units and determine how many equally sized squares can be made. Rather than describing the rectangle by its dimensions or counting the number of squares to determine its size, we could multiply its dimensions together. utting this into perspective, we see the number of squares that fits inside a rectangular region is referred to as the area. shortcut to determine that number of squares is to multiply the base by the height. More formally, area is defined as the space inside a figure or the amount of surface a figure covers. The rea of a rectangle is equal to the product of the length of the base and the length of a height to that base. That is bh. Most books refer to the longer side of a rectangle as the length (l), the shorter side as the width (w). That results in the formula lw. So now we have formulas for the areas of a rectangle that can be used interchangeably. The answer in an area problem is always given in square units because we are determining how many squares fit inside the region. Of course you will show a variety of rectangles to your students and practice identifying the base and height of those various rectangles. Math 6 Notes Unit 09: Measurement and Geometry age 5 of 7 Revised 0 CCSS

6 Example: Find the area of a rectangle with the dimensions 3 m by m. lw 3 6 The area of the rectangle is 6 m. Example: The area of a rectangle is 6 square inches. If the height is 8 inches, find the base. Solution: = bh 6 = b 8 = b The base is inches. Example: The area of the rectangle is 4 square centimeters. Find all possible whole number dimensions for the length and width. Solution: rea Length (in cm) Width (in cm) Example: Find the area of the rectangle. 9 ft. yd. Be careful! rea of a rectangle is easy to find, and students may quickly multiply to get an answer of 8. This is wrong because the measurements are in different units. We must first convert feet into yards, or yards into feet. Since yard = 3 feet we start with yards feet x We now have a rectangle with dimensions 3 yd. by yd. 3 x x Math 6 Notes Unit 09: Measurement and Geometry age 6 of 7 Revised 0 CCSS

7 height height height lw 3 6 The area of our rectangle is 6 square yards. If we were to have a square whose sides measure 5 inches, we could find the area of the square by putting it on a grid and counting the squares as shown below. gain we could count the boxes to find the area of the square ( 5units ), but more easily we could multiply the base height or the length the width ( 5x5 5units ). So again, =bh or lw. base Students may also see that since the base and height or length and width of a square are congruent, they may choose to use s ( 5x5 5units ). Counting the squares is a viable method but eventually students will begin to see the area can be computed more efficiently. If they multiply the base times the height, then again bhor some will see the area as the square of one side, so s. In this example, s units. bh 5 5 5units or Next, if we begin with a 6 x 6 square and cut from one corner to the other side (as show in light blue) and translate that triangle to the right, it forms a new quadrilateral called a rhombus (plural they are called rhombi). Since the area of the original square was 6 x 6 or 36 square units, then the rhombus has the same area, since the parts were just rearranged. So we know the rhombus has an area of 36 square units. gain we find that the rea of the rhombus = bh. base base Be sure to practice with a number of different rhombi so your students are comfortable with identifying the height versus the width of the figures. Math 6 Notes Unit 09: Measurement and Geometry age 7 of 7 Revised 0 CCSS

8 height height rea of arallelograms, Triangles and Trapezoids If I were to cut one corner of a rectangle and place it on the other side, I would have the following: base base We now have a parallelogram. Notice, to form a parallelogram, we cut a piece of a rectangle from one side and placed it on the other side. Do you think we changed the area? The answer is no. ll we did was rearrange it; the area of the new figure, the parallelogram, is the same as the original rectangle. So we have the rea of a parallelogram = bh. h Example: The height of a parallelogram is twice the base. If the base of the parallelogram is 3 meters, what is its area? First, find the height. Since the base is 3 meters, the height would be twice that or (3) or 6 m. To find the area, bh The area of the parallelogram is 8 m. b NOTE: If students are having difficulty with correctly labeling areas with square units considering having them write the units within the problem. For example, instead of bh square meters bh try 3meters 6meters 8 square meters Say aloud 3 6 = 8 and meters meters = square meters Math 6 Notes Unit 09: Measurement and Geometry age 8 of 7 Revised 0 CCSS

9 We have established that the area of a parallelogram is understand the area formula for a triangle and trapezoid. bh. Let s see how that helps us to h base h base For this parallelogram, its base is 4 units and its height is 3 units. Therefore, the area is 4 3 units. If we draw a diagonal, it cuts the parallelogram into triangles. That means one triangle would have one-half of the area or 6 units. Note the base and height stay the same. So for a triangle, bh, or units h base h b b b b base For this parallelogram, its base is 8 units and its height is units. Therefore, the area is 8 6 units. If we draw a line strategically, we can cut the parallelogram into congruent trapezoids. One trapezoid would have an area of one-half of the parallelogram s area (8 units ). Height remains the same. The base would be written as the sum of b and b. For a trapezoid: ( b ( b ) h b ) h Triangles s demonstrated above, one way to introduce the area formula of a triangle is to begin with a rectangle, square, rhombus and/or a parallelogram and show that cutting the figure using a diagonal produces congruent triangles. This aids students in remembering the rea formula of bh a Triangle is bh or. Math 6 Notes Unit 09: Measurement and Geometry age 9 of 7 Revised 0 CCSS

10 Example: Find the area of each of the following triangles. a. b c. cm 6 cm 3 cm 8 in 0 in 8 mm 6 mm 5 mm 7 cm 6 in mm Solutions: a. bh (7)(6) (7)(3) 5 b. bh (6)(8) 3(8) 4 c. bh ()(8) 6(8) 48 5 square cm 4 squares inches 48 square mm Important things here: In each example, students must determine which dimensions given are the base and height. In example a, the Commutative roperty of multiplication was employed to take half of the even number to make the arithmetic easy. In example c, the height must be given outside the triangle. Why? Example: triangular piece of fabric has an area of 54 square inches. The height of the triangle is 6 inches. What is the length of the triangle s base? Beginning with the formula and substituting in the values we know, we get: 54 bh ( b)(6) The triangle s base is 8 inches. 54 3( b) 8 b Reflection: When can you use two side lengths to find the area of a triangle? In this situation, does it matter which side is the base and which side is the height? Solution: When you have a right triangle you use two side lengths to find the area of a triangle. It doesn t matter which side is the base or height. Math 6 Notes Unit 09: Measurement and Geometry age 0 of 7 Revised 0 CCSS

11 CRT Example: Solution: bh 8 Trapezoids s demonstrated on page 9, the formula used to find the rea of a trapeziod is b b h. We must be sure students see this formula in several ways. Be sure to teach the other forms b b rea of a trapeziod is ( ) h a nd ( verage of the bases) height. Example: a. 4 ft 0 mm b. 4 ft 8 mm 6 mm 6 ft 4 mm Solutions: a. b b b. ( ) h ( verage of the bases) height (6) 0 44 (6) c. (6) 3 c. cm ( b b ) h (5 )(8) (7)(8) 4(7) 8 8 cm 5 cm 0 ft 3 mm 8 cm Math 6 Notes Unit 09: Measurement and Geometry age of 7 Revised 0 CCSS

12 Notice: In example a, it was easy to mentally find the average of 6 and 4 (the bases) thus the use of the formula = (verage of the bases) height. In example c, students will need to correctly identify the bases to solve this problem. Be sure to give students exposure to this orientation of the trapezoid. Example: trapezoid has an area of 00 cm and bases of 5 cm and 5 cm. Find the height. Solution: ( b b ) h 00 (5 5) h 00 (40) h 00 0h 0 h The height is 0 cm. rea of Circles You can demonstrate the formula for finding the area of a circle. First, draw a circle; cut it out. Fold it in half; fold in half again. Fold in half two more times, creating 6 wedges when you unfold the circle. Cut along these folds. Rearrange the wedges, alternating the pieces tip up and down (as shown), to look like a parallelogram. radius (r) This is ½ of the distance around the circle or ½ of C. We know that C π r, so C π r C πr Math 6 Notes Unit 09: Measurement and Geometry age of 7 Revised 0 CCSS

13 The more wedges we cut, the closer it would approach the shape of a parallelogram. No area has been lost (or gained). Our parallelogram has a base of πr and a height of r. We know from our previous discussion that the area of a parallelogram is bh. So we now have the area of a circle: bh (π r)( r) πr πr radius (r) Example: circle has a radius of 4 m. Estimate the area of the circle. (Use 3 to approximate pi.) r Using the formula: The area of the circle is approximately 48 m 48 Example: student estimated the area of the circle to the right as shown. Is the solution correct? Explain your thinking. r Solution: 8 mm mm Example: Find the area of the circle in terms of pi, if the diameter is 0 feet. Students must note: If diameter is 0 feet, then the radius is 0 feet. r Using the formula: The area of the circle is 00 feet. Example: Find the area of the circle if the diameter is 0 inches. Leave your answer in terms of π. Using the formula: πr π 5 5π The area of the circle is 5π square inches. Remember if the diameter is 0 inches, the radius must be 5 inches. Math 6 Notes Unit 09: Measurement and Geometry age 3 of 7 Revised 0 CCSS

14 Many standardized tests (including the CRT and the district common exams) ask students to leave their answers in terms of π. Be sure to practice this! Example: Find the area of the circle to the nearest square meter if the radius of the circle is m. Use 3.4. Using the formula: πr (3.4)() (3.4) The area of the circle is about 45 square meters or 45 m. Example: Find the area of the circle if the diameter of the circle is 00 m. Use 3.4. Students must note: If diameter is 00 m, then the radius is 00 m. Using the formula: r ,000 3, 400 The area of the circle is 3,400 square meters or 3,400 m. Example: Find the area of the circle if the radius of the circle is 7 m. Use r 7. Using the formula: The area of the circle is 54 m. 54 Math 6 Notes Unit 09: Measurement and Geometry age 4 of 7 Revised 0 CCSS

15 Example: Find the area of the circle if the diameter is inches. Use Using the formula: r The area of the circle is 346 in Reflection - Check for Understanding For which plane figure(s) does the formula bh work? For which plane figure(s) does the formula lw work? For which plane figure(s) does the formula bh work? For which plane figure(s) does the formula s work? For which plane figure(s) does the formula r work? For which plane figure(s) does the formula ( verage of thebases) height work? For which plane figure(s) does the formula ( b b ) h work? Example: When a tree was planted, its diameter measured 6 cm. Five years later, the diameter of the tree measured 0 cm. Find the difference between the two areas. (Use =3.4) Solution: The area of the original young tree would be: r The area of the older tree would be: r Math 6 Notes Unit 09: Measurement and Geometry age 5 of 7 Revised 0 CCSS

16 The difference between the two areas is: nswer: The difference in the areas is 50.4 cm. rea of Composite Figures Students should also practice finding the area of irregular figures by composing and decomposing into shapes they know. 6 cm Example: Find the area of the polygon. 8 cm 5 cm Solutions: Method (Whole to art) Composing 6 cm 8 cm 5 cm 0 cm. Create a rectangle using the outside borders of the original figure. (Composing). Find the area of the large new rectangle.(whole) 3. Subtract the negative (white) space. (art) 0 cm = rea = 68cm Math 6 Notes Unit 09: Measurement and Geometry age 6 of 7 Revised 0 CCSS

17 Method (art to Whole) Decomposing 6 cm 8 cm 5 cm. Break the polygon into smaller figures (rectangles, squares, triangles, etc. that you know). (Decomposing). Find the area of each piece.(arts) 3. dd the areas of the pieces. (Whole) 0 cm = rea = 68 cm Example: Find the area of the polygon. Solutions: Method (Whole to art) 5. Create a rectangle using the outside borders of the original figure. (Composing). Find the area of the large new rectangle.(whole) 3. Subtract the negative (white) spaces. (arts) 6 NOTE: Students will need to compute outer side length and width (shown in red). Math 6 Notes Unit 09: Measurement and Geometry age 7 of 7 Revised 0 CCSS

18 5 6 = 4 ( ) rea = 6 square units Method (art to Whole) Method 3 (art to Whole). Break the polygon into smaller figures (rectangles, squares, triangles, etc. that you know). (Decomposing). Find the area of each piece.(arts) 3. dd the areas of the pieces. (Whole) 5 6 = = = = = = = 6 = 6 6 units Math 6 Notes Unit 09: Measurement and Geometry age 8 of 7 Revised 0 CCSS

19 5 m Example: Find the area of the polygon. 0 m 4 m 0 m Solution: Method Method = = m m NOTE: Students will need to compute the outer side length and width (shown in red). Example: The dimensions of a church window are shown below. Find the area of the window to the nearest square foot. = + 8 feet bh r 0 8 (3.4)(5) 80.57(5) feet We are given the diameter, so the radius would be half of the 0 feet or 5 feet. Math 6 Notes Unit 09: Measurement and Geometry age 9 of 7 Revised 0 CCSS

20 The area of the church window is about 9 square feet to the nearest foot. Comparing erimeter and rea In this section students are given polygons that are either enlarged (dimensions doubled or tripled) or reduced (dimensions halved) to create similar figures, and students are asked to compare the perimeters and/or areas. Many students will need to build or draw these figures to visually and mentally determine what happens when. the dimensions are When dimensions are doubled: Original Figure Dimensions Doubled Example : cm cm cm erimeter ( l w) ( ) cm ( l w) ( ) () (4) 4 8 rea 4 cm l w 8 cm l w 4 cm 4 cm Example : Original Dimensions Figure Doubled 4 units 8 units units Math 6 Notes Unit 09: Measurement and Geometry age 0 of 7 Revised 0 CCSS 4 units

21 erimeter ( l w) ( 4) (6) units ( l w) (4 8) () 4 4 units rea l w l w units 3 units Example 3: Original Figure Dimensions Doubled units 4 units erimeter units ( l w) ( ) (4) 8 8 units 4 units ( l w) (4 4) (8) 6 6 units rea l w l w units 6 units Math 6 Notes Unit 09: Measurement and Geometry age of 7 Revised 0 CCSS

22 Charting the perimeters from Examples -3 when the dimensions of a figure are DOUBLED we find the following: erimeter of original erimeter of figure doubled Example 4 cm 8 cm Example cm 4 cm Example 3 8 cm 6 cm Looking at these example results, the pattern that occurs is that when the dimensions of a figure are doubled, the perimeter is doubled. Charting the area from Examples -3 when the dimensions of a figure are DOUBLED we find the following: rea of original rea of figure doubled Example cm 4 cm Example 8 cm 3 cm Example 3 4 cm 6 cm Looking at these example results, the pattern that occurs is that when the dimensions of a figure are doubled, the area is four times or. When dimensions are tripled: Dimensions Example 4: Original Tripled Figure erimeter cm cm ( l w) ( ) () 4 4 units 3 cm 3 cm ( l w) (3 3) (6) units Math 6 Notes Unit 09: Measurement and Geometry age of 7 Revised 0 CCSS

23 rea l w units l w units Example 5: Original Figure Dimensions Tripled units 4 units units erimeter rea ( l w) ( 4) (6) units l w units 6 units ( l w) (6 ) (8) units l w units Example 6: 5 in Original Figure in Dimensions Tripled 5 in Math 6 Notes Unit 09: Measurement and Geometry age 3 of 7 Revised 0 CCSS 3 in

24 erimeter ( l w) (5 ) (6) units ( l w) (5 3) (8) units rea l w l w units 45 units Charting the perimeters from Examples 4-6 when the dimensions of a figure are TRILED we find the following: erimeter of original erimeter of figure tripled Example 4 4 units units Example 5 units 36 units Example 6 units 36 units Looking at these example results, the pattern that occurs is that when the dimensions of a figure are tripled, the perimeter is tripled. Charting the area from Examples 4-6 when the dimensions of a figure are TRILED we find the following: rea of original rea of figure tripled Example 4 units 9 units Example 5 8 units 7 units Example 6 5 units 45 units Looking at these example results, the pattern that occurs is that when the dimensions of a figure are cut in tripled, the area is 9 times or 3. When dimensions are halved: Original Figure Halved Figure Example 7: cm cm cm cm Math 6 Notes Unit 09: Measurement and Geometry age 4 of 7 Revised 0 CCSS

25 erimeter ( l w) ( ) (4) 4 8 cm ( l w) ( ) () 4 4 cm rea l w 4 4 cm l w cm Example 8: Original Figure Halved Figure units 4 units unit units erimeter rea ( l w) ( 4) (6) cm l w cm ( l w) ( ) (3) 6 6 cm l w cm Math 6 Notes Unit 09: Measurement and Geometry age 5 of 7 Revised 0 CCSS

26 Original Figure Halved Figure Example 9: 3 units 6 units units erimeter rea 4 units ( l w) (4 6) (0) 0 0 cm l w cm ( l w) ( 3) (5) 0 0cm l w cm Charting the perimeters from Examples 7-9 when a figure is HLVED we find the following: erimeter of original erimeter of figure halved Example 7 8 units 4 units Example 8 units 6 units Example 9 0 units 0 units Looking at these example results, the pattern that occurs is that when the dimensions of a figure are cut in half, the perimeter is cut in half. Charting the area from Examples 7-9 when a figure is HLVED we find the following: rea of original rea of figure halved Example 7 4 units units Example 8 8 units units Example 9 4 units 6 units Math 6 Notes Unit 09: Measurement and Geometry age 6 of 7 Revised 0 CCSS

27 Looking at these example results, the pattern that occurs is that when the dimensions of a figure are cut in half, the area is cut to one- fourth or. In summary, when the dimensions of a rectangle are: doubled, the perimeter is doubled. tripled, the perimeter is tripled. halved, the perimeter is halved. In summary, when the dimensions of a figure are: doubled, the area is or four times as great. tripled, the area is 3 or nine times as great. halved, the area is or one fourth as great. Math 6 Notes Unit 09: Measurement and Geometry age 7 of 7 Revised 0 CCSS