Square Metre. 1 cm 2 = 1 cm 1 cm. 10 cm. 1 m = 100 cm. 1 m 2 = 1 m 1 m = 100 cm 100 cm = cm 2

Transcription

1 Square Metre 1 cm 2 = 1 cm 1 cm 1 m = 100 cm 10 cm 1 m = 100 cm 1 m 2 = 1 m 1 m = 100 cm 100 cm = cm 2 G-70 Blackline Master Measurement Teacher s Guide for Workbook 7.1

2 Area and Perimeter Review This shape covers 4 square centimetres. Its area is 4 cm 2. This shape has 6 sides. The distance around it is 1 cm + 3 cm + 2 cm + 1 cm + 1 cm + 2 cm = 10 cm. Its perimeter is 10 cm. one square centimetre 1 cm 1 cm = 1 cm 2 1. Find the area of these figures in square centimetres. a) b) c) Area = cm 2 Area = cm 2 Area = cm 2 2. Look at the rectangle in 1 c). Write the length and width of the rectangle: width = cm length = cm Write a multiplication statement for the area: cm 2 Write an addition statement for the perimeter: 3. Using a ruler, divide each rectangle into squares 1 cm 1 cm. (The sides of the first two rectangles have already been marked in centimetres.) Write a multiplication statement for the area of each rectangle in cm 2. Write an addition statement for the perimeter of each rectangle in cm. a) b) c) cm 4. On grid paper, draw 3 different shapes that have an area of 10 cm 2 (the shapes don t have to be rectangles). 5. On grid paper, draw a rectangle with area 12 cm 2 and perimeter 14 cm. 6. If you know the length and width of a rectangle, how can you find its area? 7. Find the area of each rectangle using the clues. a) Width = 2 cm Perimeter = 10 cm Area =? b) Width = 4 cm Perimeter = 18 cm Area =? Blackline Master Measurement Teacher s Guide for Workbook 7.1 G-71

3 Dividing Decimals by Decimals 1. Estimate the quotients. a) b) c) d) e) f) Estimate the quotients. a) b) c) d) e) f) Use estimation to place the decimal point correctly in each quotient. a) b) c) d) e) f) The decimal point is in the wrong place. Estimate the quotient to correct the answer. a) b) c) Multiply both the dividend and divisor by 10, 100, or 1000 to change them to whole numbers. Then divide using a calculator. Estimate the quotient to check your answer. a) b) c) d) e) f) Calculate using a calculator. Estimate to check your answer. Round answers to two decimal places. a) b) c) d) G-72 Blackline Master Measurement Teacher s Guide for Workbook 7.1

4 Distance Between Parallel Lines A. Measure the line segments with endpoints on the two parallel lines with a ruler. Write the lengths of the line segments on the picture. B. Use a square corner to draw at least three perpendiculars from one parallel line to the other, as shown. Measure the distance between the two parallel lines along the perpendiculars. What do you notice? C. Explain why all the perpendiculars you drew in part B are parallel. D. A parallelogram is a 4-sided polygon with opposite sides parallel. You can draw parallelograms by using anything with parallel sides, like a ruler. Place a ruler across both of the parallel lines and draw a line segment along each side of the ruler. Use this method to draw at least 3 parallelograms with different angles. E. Measure the line segments you drew between the two given parallel lines in part D. What do you notice? F. To measure the distance between two parallel lines, draw a line segment perpendicular to both lines and measure it. Does the distance between parallel lines depend on where you measure it? Blackline Master Measurement Teacher s Guide for Workbook 7.1 G-73

5 Parallelograms G-74 Blackline Master Measurement Teacher s Guide for Workbook 7.1

6 Area of Parallelograms 1. Albert cuts a parallelogram into two triangles and rearranges the pieces to find the area. a) The shaded triangle in the first diagram was moved. Shade the same triangle where it was reattached in the second diagram. b) Explain why the vertical line segments in each diagram are equal. c) Why are the areas of both parallelograms the same? d) Does the formula for the area of a parallelogram (base height) still work when the height falls outside the base? Explain. 2. Regina wants to find the area of a parallelogram a different way. She cuts her parallelogram in half and rearranges the pieces as shown. a) Measure the side AB. AB = b) Does the area of the parallelogram change? A B c) Regina wants to use the horizontal side of the new parallelogram as a base. What is the length of the base of the new parallelogram? How is it related to AB? d) Draw and measure the height of the new parallelogram. Height = e) Which fraction of the height of the old parallelogram is the height of the new parallelogram? f) Verify that the formula for the area of a parallelogram gives the same answer for both the new parallelogram and the old parallelogram. 3. Find the area of EFGH in two ways. a) Use EH as the base. EH = Extend EH and draw a perpendicular to EH from vertex G. What is the height of EFGH? Height = H F Area = b) Use EF as the base. EF = Draw the height to EF. Height = Area = Did you get the same answer both ways? Explain. G Blackline Master Measurement Teacher s Guide for Workbook 7.1 G-75

7 Investigating Area of Parallelograms INVESTIGATION Two rectangles with the same area and perimeter are congruent. Are all parallelograms with the same area and perimeter congruent? A. Find all possible combinations of base and height for a parallelogram with area 24 cm 2. The base and height are whole numbers of centimetres. Start with base 1 cm and search systematically. Base (cm) Height (cm) B. On 1 cm grid paper, draw all rectangles that have area 24 cm 2 and sides that are whole number of centimetres. C. Find the perimeters of all rectangles in part B. Length (cm) Width (cm) Perimeter (cm) D. Are all rectangles also parallelograms? Explain. E. Copy these parallelograms onto 1- cm grid paper. Draw the next two parallelograms in this pattern. B C B C B C A D A D A D Figure 1 Figure 2 Figure 3 F. Measure the sides to the nearest millimetre and find the perimeter (P) and the area (A) of the parallelograms in the pattern. Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 P = P = P = P = P = A = A = A = A = A = Do these parallelograms have the same area? Do these parallelograms have the same perimeter? G. Look at your answers in parts C and F. Can you find two parallelograms that have the same perimeter and area but are not congruent? Which ones? Use part E to confirm your answer. G-76 Blackline Master Measurement Teacher s Guide for Workbook 7.1

8 Triangles Right triangles Acute triangles Obtuse triangles Blackline Master Measurement Teacher s Guide for Workbook 7.1 G-77

9 Subtracting Area Dianne wants to find the area of the shaded part. She notices that the large rectangle can be divided into three parts, two of which can be joined to form a small rectangle: 3 cm 3 cm 1 cm 2 cm = 2 cm + The area of the large rectangle is 4 cm 2 cm = 8 cm 2. The area of the small rectangle is 3 cm 2 cm = 6 cm 2. So the area of the shaded part is 8 cm 2 6 cm 2 = 2 cm Complete the chart. Show all the steps in your answers. Parallelogram Area of large rectangle Area of small rectangle By subtraction Area of parallelogram By formula of area = 27 cm = 7 cm 2 = (9 7) 3 cm 2 = 2 3 cm = 6 cm 2 base = 2 cm height = 3 cm area = 2 cm 3 cm = 6 cm Find the area: G-78 Blackline Master Measurement Teacher s Guide for Workbook 7.1

10 Area of Parallelogram and Triangle Parallelogram A parallelogram is a quadrilateral with two pairs of parallel sides. Any pair of parallel sides can be chosen to be the bases. The distance between these two parallel sides is the height. bases height The height is measured along a line perpendicular to the bases. This line can be drawn anywhere. In these pictures, the thick black line is one of the bases and the dashed line is the height. Area of a parallelogram = base height Triangle A triangle is a polygon with three sides. Any side of a triangle can be the base. Draw a perpendicular from the vertex opposite the base to the base. The distance from the vertex to the base along that perpendicular is the height. Sometimes the height is outside the triangle. In these pictures, the thick black line is the base and the dashed line is the height. height base Any triangle is half of a parallelogram with the same base and height. Area of a triangle = base height 2 Blackline Master Measurement Teacher s Guide for Workbook 7.1 G-79

11 Area of Trapezoid A trapezoid is a quadrilateral with exactly one pair of parallel sides. trapezoids not trapezoids The parallel sides are called bases. The distance between the bases is called the height. The height is measured along a line perpendicular to the bases. base 1 height base 2 Sometimes the height is outside the trapezoid. In these pictures, the thick black lines are the bases and the dashed line is the height. You can make a parallelogram from two copies of the same trapezoid. The base of the parallelogram is the sum of the bases of the trapezoid, and the height of the parallelogram is the height of the trapezoid. Area of trapezoid = height (base 1 + base 2) 2 G-80 Blackline Master Measurement Teacher s Guide for Workbook 7.1

12 Triangles to Circles 1. How many triangles are there for every 3 circles? Guess, check and revise to find the ratio : a) b) triangles to circles = : 3 triangles to circles = : 3 c) d) triangles to circles = : 3 triangles to circles = : 3 e) triangles to circles = : 3 2. In which parts above are there more triangles than circles? How can you tell that from the ratio? There are more triangles than circles in parts,, and because 3. The ratio of circles to triangles is 94 : 93. Are there more circles or triangles? How do you know? Blackline Master Measurement Teacher s Guide for Workbook 7.1 G-81

13 Circles G-82 Blackline Master Measurement Teacher s Guide for Workbook 7.1

14 Facts About Circles A circle with centre P: all the points on the circle are the same distance from P. A radius: any line segment joining a point on the circle to its centre. The plural of radius is radii. The radius (r): the distance between any point on a circle to P; the length of a radius. centre radii A diameter: a line segment between two points on a circle passing through the centre of the circle. The diameter (d): the distance between any two opposite points on a circle, measured through the centre of the circle; the length of a diameter. A diameter is twice as long as a radius, so d = 2r diameter An angle that has its vertex at the centre of a circle is called a central angle. The sum of the central angles is 360. Example: a + b + c = 360 c a b The distance around a circle is called the circumference (C). The ratio C : d is the same for all circles. When this ratio is converted to a unit rate and we look at it as a number, we see that it is special it has an infinite number of digits after its decimal point and there is no pattern in these digits, so you cannot predict which digit is next. A Greek letter π (pronounced pie ) is used to identify this number. To the nearest hundredth, π = To 5 decimal places, π = Another good approximation is To three decimal places, the fraction 7 7 = 3.143, and it is often used to approximate π. C : d = π : 1, so C = π d = πd = 2πr. C The area of a circle of radius r is π r 2 = πr 2. Here is an acronym that helps to remember the value of π rounded to 6 decimal places. The number of letters in each word in order gives the digit. How I wish I could calculate pie! Blackline Master Measurement Teacher s Guide for Workbook 7.1 G-83