MATH MEASUREMENT AND GEOMETRY

Transcription

1 Students: 1. Students choose appropriate units of measure and use ratios to convert within and between measurement systems to solve problems. 1. Compare weights, capacities, geometric measures, time, and temperature within and between measurement systems (e.g., miles per hour and feet per second, cubic inches to cubic centimeters). Weights How many ounces are in 105 pounds? A new born seal pup weighs about 45 pounds at birth. In the first four days of life, the pup gains about 56 ounces a day. How many pounds will the average pup gain in one day? Mein filled a one-cup measuring cup with sand. She said it weighed 8 ounces. Is she right? Capacity The owner s manual of Car A states that it has a capacity of 13.2 gallons of gasoline. Car B has a capacity of 13 gallons 1 quart. Which car has the greater gasoline capacity? The gasoline tank of a minivan holds 18 gallons. How many quarts is this? 23

2 Geometric measure Convert 80 miles/hr =? feet/sec. ( FW) Convert. 20 oz/min =? qts/day ( FW) cube side = 5cm, what is the volume in cm 3? Change 1 in 3 to cm 3. Convert 16 feet into inches and centimeters. Time A light-year is the distance that light travels in one year. Light travels about 186,000 miles per second. Find the length of light-year in miles. Temperature Change 72 F to C. On a January day in Buffalo, New York, the temperature dropped to -5 F. Find this temperature in degrees Celsius. 24

3 Know the following approximations. ( FW) a) 1 meter 1 yard (baseball bat) b) 1 cm 1/2 inch (width of a fingernail) c) 1 km.6 miles d) 1 kg 2.2lbs (a textbook) e) 1 liter 1 quart f) 1 gram (1 paper clip g) 1mm (thickness of a dime) 2. Read and construct scale drawings and models. Read scale drawings The scale of a map is 1 inch:152 miles. The distance between two cities on the map is 2 1/4 inches. What is the actual distance between the cities? Explain what it means when a scale drawing shows a scale of 1 = 10 Construct scale drawings Chu has a bedroom 8 feet by 12 feet. Make a scale drawing using 1/4 inch = 1 foot. 25

4 A car model is being built with a scale of 2 in.:5 ft. The actual length of the car is 12 ft. What is the length of the model? If you double the area of a square with side equal to 3 feet, what would be the new area? *3. Use measures expressed as rates (e.g., speed, density) and measures expressed as products (e.g., persondays) to solve problems, checking units of the solutions; and use dimensional analysis to check the reasonableness of the answer. A highway patrol officer is monitoring traffic with a radar gun. The speed limit is 45 mph. The first exit is 3.6 miles up the road. His radar picks up a speeding car averaging 68 mph. Traffic prevents the officer beginning his pursuit. He begins 30 seconds after the speeding car has passed. How fast does he need to go to catch the speeding car before it goes off the highway? 102cm/sec would be how many mph? Jose receives a paycheck for 12 days of work in the amount of $ How much did he earn per day? 26

5 Students: 1. Routinely use formulas for finding the perimeter and areas of basic twodimensional figures and for the surface area and volume of basic threedimensional figures. 2. Students compute the perimeter, area and volume of common geometric objects and use these to find measures of less common objects; they know how perimeter, area, and volume are affected under changes of scale. Two-dimensional figures Find the area of the following 2-D figures: Triangle Trapezoid Parallelogram Triangle: base 9cm, height 2.6cm Trapezoid: base.3km and.5km, height.25km Parallelogram: base.7cm side.4 cm and Height.35 cm Rectangle Find the perimeter of a rectangle whose length is twice its width. The width is 8 cm. Square Find the area of a square with a perimeter of 28 feet. Circle A diameter of a circular field is 600 feet. One fourth of the field is to be planted in corn. How many square yards will be planted with corn? 27

6 Use formulas for finding the surface area and volume of basic threedimensional figures Prisms What is the volume of a cube with sides of length 8m? What is the surface area of a box with length 10:, width 8 and height 4.5? Cylinders A drum is closed on top and bottom. The diameter of the drum is 20 cm. The height is 27 cm. Find the approximate surface area of the drum. The diameter of a can of paint is 8 inches and the height is 10 inches. Find the volume 2. Estimate and compute the area of more complex or irregular two-and threedimensional figures by breaking them up into more basic geometric objects. Find the area of a regular hexagon 5 meters on each side. 28

7 Find the area. 14 cm A= 5 A = Compute the length of the perimeter, the surface area of the faces, and the volume of a 3-D object built from rectangular solids. Students understand that when the length of all dimensions are multiplied by a scale factor, the surface area is multiplied by the square of the scale factor and the volume is multiplied by the cube of the scale factor. 4. Relate the changes in measurement under change of scale to the units used (e.g., square inches, cubic feet) and to conversions between units (1 square foot = 12 square inches, 1 cubic inch = 2.54 cubic centimeters). These 3-D objects are made of cubes with sides = 2 cm. What is their surface area and volume? If you double the sides of the cubes, what will happen to surface area and volume? One square foot = square inches. 1 cubic inch = cubic centimeters 29

8 Students: 3. Students know the Pythagorean Theorem and deepen their understanding of plane and solid geometric shapes by constructing figures that meet given conditions and by identifying attributes of figures. 1. Identify and construct basic elements of geometric figures (e.g., altitudes, midpoints, diagonals, angle bisectors and perpendicular bisectors; and central angles, radii, diameters and chords of circles) using compass and straight-edge. Altitudes Construct the altitude of an equilateral triangle with sides equal to 10 cm. Midpoints Find the midpoint of line segment AB. Diagonals A B Angle bisectors Bisect an angle of 45. Perpendicular bisectors Construct a line perpendicular to a given line through a point on the line. Construct a line perpendicular to a given line through a point not on the line. Parts of a circle Use a compass to draw a circle with center N. Draw, name, and label the radius, diameter, central angles and a chord. 30

9 2. Understand and use coordinate graphs to plot simple figures, determine lengths and areas related to them, and determine their image under translations and reflection. *3. Know and understand the Pythagorean Theorem and use it to find the length of the missing side of a right triangle and lengths of other line segments, and, in some situations, empirically verify the Pythagorean Theorem by direct measurement. Graph points (2,3), (2,1), (5,1), and (5,3). What is the area of this quadrilateral? Slide the quadrilateral to the right 3 spaces and up 2 spaces. List the new coordinates. Reflect this figure over the y axis. What are the coordinates of the reflection image? Find length of the diagonal of a 9 by 12 rectangle. Find c. Find a. 6 c a Determine whether the triangle with sides of 9mm., 12mm., and 16mm. is a right triangle. What is the side length of an isosceles right triangle with hypotenuse 72? ( FW) 31

10 Verify the Pythagorean Theorem by direct measurement. *4. Demonstrate an understanding of when two geometrical figures are congruent and what congruence means about the relationships between the sides and angles of the two figures. Using a 3,4,5 right triangle, verify the Pythagorean Theorem. If two triangles are congruent, how many pairs of corresponding parts do they have? ABCD EFGH What else do you know? D C H G A B E F 5. Construct two-dimensional patterns for three-dimensional models, such as cylinders, prisms and cones. Draw nets for cylinders, cones and prisms. Draw 4 different nets that fold into cubes. Draw the net of a rectangular prism that has a height of twice its width, and its length is three times its height. Given a polyhedron, identify the vertices, edges, and faces. 32

11 *6. Identify elements of threedimensional geometric objects (e.g., diagonals of rectangular solids) and how two or more objects are related in space (e.g., sky lines, the possible ways three planes could intersect). Name all the lines that are skew to DE. Parallel to DE. G F C B E A D What is the difference between a point, line, and a plane? 33

12 34