MSIS GEOMETRY AND MEASUREMENT. Day 3

Transcription

1 MSIS GEOMETRY AND MEASUREMENT 1 Day 3

2 AGENDA DAY 3 9:00 Geometry and Measurement G.6 10:15 Break 10:30 Geometry and Measurement G.7 12:00 Lunch 1:00 Geometry and Measurement G.8 2:45 Wrap Up

3 AARON SAYS MORE INFORMATION IS NEEDED TO FIND THE VOLUME OF THE PRISMS. IS AERON MISTAKEN? CAN YOU CALCULATE THE VOLUME OF THE PRISMS?

4 THE TANK, SHAPED LIKE A RECTANGULAR PRISM, IS FILLED TO THE TOP WITH WATER. Will the beaker hold all the water in the box? If yes, how much more will the beaker hold? If not, how much more will the cube hold than the beaker? Explain how you know.

5 PROBLEM A rectangular tank with a base area of 24 cm2 is filled with water and oil to a depth of 9 cm. The oil and water separate into two layers when the oil rises to the top. If the thickness of the oil layer is 4 cm, what is the volume of the water?

6

7 Read Grade 6 Introduction to CC Math Standards to look for the Geometry

8 GRADE 6 Four Critical Areas Connect ratio and rate to whole number multiplication and division Division of fractions and extending to rational number system including negatives Writing, interpreting, and using expressions and equations Develop understanding of statistical thinking

9 SOLVE REAL-WORLD AND MATHEMATICAL PROBLEMS INVOLVING AREA, SURFACE AREA, AND VOLUME. 6.G.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; Apply these techniques in the context of solving real-world and mathematical problems.

10 DISCOVERING THE FORMULA FOR FINDING THE AREA OF RIGHT TRIANGLES. 1. Using a ruler, draw a diagonal (from one corner to the opposite corner) on shapes A, B, and C. 2. Along the top edge of shape D, mark a point that is not a vertex. Using a ruler, draw a line from each bottom corner to the point you marked. (Three triangles should be formed.) 3. Cut out the rectangles. Then, divide A, B, and C into two parts by cutting along the diagonal, and divide D into three parts by cutting along the lines you drew. What shapes did you get? 4. How do the areas of the triangles compare to the area of the original shape?

11 Calculate the area of each triangle using two different methods. Figures are not drawn to scale.

12 PROBLEMS USE GRID PAPER TO a. Find the area of a triangle with a base length of three units and a height of four units. b. Find the area of the trapezoid shown below using the formulas for rectangles and triangles.

13 ANOTHER EXAMPLE PROBLEMS A rectangle measures 3 inches by 4 inches. If the lengths of each side double, what is the effect on the area? The area of the rectangular school garden is 24 square units. The length of the garden is 8 units. What is the length of the fence needed to enclose the entire garden?

14 PROBLEM The 6th grade class at Hernandez School is building a giant wooden H for their school. The H will be 10 feet tall and 10 feet wide and the thickness of the block letter will be 2.5 feet. The truck that will be used to bring the wood from the lumberyard to the school can only hold a piece of wood that is 60 inches by 60 inches. What pieces of wood (how many pieces and what dimensions) are needed to complete the project? How large will the H be if measured in square feet?

15 SOLVE REAL-WORLD AND MATHEMATICAL PROBLEMS INVOLVING AREA, SURFACE AREA, AND VOLUME 6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = lwh and V = bh to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems

16 VISUALIZING AND MANIPULATING TO UNDERSTAND VOLUME Students need multiple opportunities to measure volume by filling rectangular prisms with blocks and looking at the relationship between the total volume and the area of the base. Through these experiences, students derive the volume formula (volume equals the area of the base times the height). Students can explore the connection between filling a box with unit cubes and the volume formula using interactive applets such as the Cubes Tool on NCTM s Illuminations.

17 VISUALIZING AND MANIPULATING TO UNDERSTAND VOLUME In addition to filling boxes, students can draw diagrams to represent fractional side lengths, connecting with multiplication of fractions. This process is similar to composing and decomposing two-dimensional shapes.

18 Which prism will hold more 1 in. x 1 in. x 1 in. cubes? How many more cubes will the prism hold?

19 A ¼ in. cube was used to pack the prism. How many ¼ in. cubes will it take to fill the prism? What is the volume of the prism? How is the number of cubes related to the volume?

20 Practical Application Cube-shaped boxes will be loaded into the cargo hold of a truck. The cargo hold of the truck is in the shape of a rectangular prism. The edges of each box measure 2.50 feet and the dimensions of the cargo hold are 7.50 feet by feet by 7.50 feet, as shown below.

21 KELLY WANTS TO WRAP 20 GOLF BALLS, EACH IN A CUBE- SHAPED BOX, TOGETHER IN ONE LARGER BOX. WHICH ARRANGEMENT WILL USE THE LEAST WRAPPING PAPER? Build a box with 20 cubes Sketch each box, label dimensions, find area of each face and the total surface area Display all boxes on chart paper Label which arrangement has the largest surface area and which has the smallest.

22 WAIT A MINUTE How can the boxes have the same volume of 20 cubes and have different surface areas? Discuss with your table group how students in 6 th grade may respond to the above question.

23 PROBLEM SKETCH AND SOLVE a. Determine the volume of a rectangular prism length and width are in a ratio of 3:1. b. The width and height are in a ratio of 2: 3. The length of the rectangular prism is 5 ft.

24 SOLVE REAL-WORLD AND MATHEMATICAL PROBLEMS INVOLVING AREA, SURFACE AREA, AND VOLUME 6.G.3 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

25 PLOTTING FIGURES ON THE COORDINATE PLANE Determine the area of both polygons on the coordinate plane, and explain why you chose the methods you used. Then write an expression that could be used to determine the area of the figure.

26 1. Plot and connect the points A (3, 2), B (3, 7), and C (8, 2). Name the shape and determine the area of the polygon. 2. Plot and connect the points E(-8, 8), F (-2, 5), and G (-7, 2). Then give the best name for the polygon and determine the area.

27 PROBLEM On a map, the library is located at (-2, 2), the city hall building is located at (0, 2), and the high school is located at (0,0). Represent the locations as points on a coordinate grid with a unit of 1 mile. What is the distance from the library to the city hall building? The distance from the city hall building to the high school? How do you know? What shape is formed by connecting the three locations? The city council is planning to place a city park in this area. How large is the area of the planned park?

28 SOLVE REAL-WORLD AND MATHEMATICAL PROBLEMS INVOLVING AREA, SURFACE AREA, AND VOLUME 6.G.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems

29 EXAMPLES Describe the shapes of the faces needed to construct a rectangular pyramid. Cut out the shapes and create a model. Did your faces work? Why or why not? Create the net for a given Prism or pyramid and then use the net to calculate the surface area.

30 CLASSIFY EACH NET AS REPRESENTING A RECTANGULAR PRISM, A TRIANGULAR PRISM, OR A PYRAMID. RESET BUTTON.

31

32

33 WHAT THREE-DIMENSIONAL FIGURE WILL THE NET CREATE? What is the volume of the figure? What is the surface area of the figure?

34 A box needs to be painted. How many square inches will need to be painted to cover every surface? Draw and label a net for the following figure. Then use the net to determine the surface area of the figure.

35 List the measurements of three different rectangular prisms that each have a surface area of 72 square units?

36 Sketch a net of this pizza box. It has a square top that measures 16 inches on a side, and the height is 2 inches. Treat the box as a prism, without counting the interior flaps that a pizza box usually has.

37 What is least amount of surface area possible on a rectangular prism with a volume of 64 cubic inches?

38 6.EE.9 6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.

39 Each week Quentin earns $30. If he saves this money, create a graph that shows the total amount of money Quentin has saved from week 1 through week 8. Write an equation that represents the relationship between the number of weeks that Quentin has saved his money, ww, and the total amount of money in dollars that he has saved, ss. Then, name the independent and dependent variables. Write a sentence that shows this relationship.

40

41 USING PRECISE MATHEMATICAL LANGUAGE Remember that according to van Heile, language is the basis for understanding and communicating about geometry. Before we can find out what a student knows we must establish a common language and vocabulary.

42 Read Grade 7 Introduction to CC Math Standards to look for the Geometry

43 GRADE 7 Four Critical Areas Develop understanding of and applying proportional relationships Develop understanding of operations with rational numbers and working with expressions and linear equations Draw inferences based on population samples SOLVE PROBLEMS INVOLVING SCALE DRAWINGS INFORMAL GEOMETRIC CONSTRUCTIONS AREA, SURFACE AREA, VOLUME

44 DRAW, CONSTRUCT, AND DESCRIBE GEOMETRICAL FIGURES AND DESCRIBE THE RELATIONSHIPS BETWEEN THEM. 7.G.1 Solve problems involving scale drawings of geometric figures, such as computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

45 Students determine the dimensions of figures when given a scale and identify the impact of a scale on actual length (one-dimension) and area (two- dimensions). Students identify the scale factor given two figures. Using a given scale drawing, students reproduce the drawing at a different scale. Students understand that the lengths will change by a factor equal to the product of the magnitude of the two size transformations. Scale drawings of geometric figures connect understandings of proportionality to geometry and lead to future work in similarity and congruence. As an introduction to scale drawings in geometry, students should be given the opportunity to explore scale factor as the number of time you multiple the measure of one object to obtain the measure of a similar object. It is important that students first experience this concept concretely progressing to abstract contextual situations. Provide opportunities for students to use scale drawings of geometric figures with a given scale that requires them to draw and label the dimensions of the new shape. Initially, measurements should be in whole numbers, progressing to measurements expressed with rational numbers. This will challenge students to apply their understanding of fractions and decimals. Students should move on to drawing scaled figures on grid paper with proper figure labels, scale, and dimensions. Provide word problems that require finding missing side lengths, perimeters or areas. For example, if a 4 by 4.5 cm rectangle is enlarged by a scale of 3, what will be the new perimeter? What is the new area? Or, if the scale is 6, what will the new side length look like? Or, suppose the area of one triangle is 16 sq units and the scale factor between this triangle and a new triangle is 2.5. What is the area of the new triangle? Reading scales on maps and determining the actual distance (length) is an appropriate contextual situation.

46 Julie showed you the scale drawing of her room. If each 2 cm on the scale drawing equals 5 ft, what are the actual dimensions of Julie's room? Reproduce the drawing at 3 times its current size.

47 DRAW, CONSTRUCT, AND DESCRIBE GEOMETRICAL FIGURES AND DESCRIBE THE RELATIONSHIPS BETWEEN THEM. 7.G.2 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle

48 Draw an isosceles triangle with only one 80 angle. Is this the only possibility or can you draw another triangle that will also meet these conditions? Can you draw a triangle with sides that are 13 cm, 5 cm, and 6 cm? Draw a quadrilateral with one set of parallel sides and no right angles.

49 DRAW, CONSTRUCT, AND DESCRIBE GEOMETRICAL FIGURES AND DESCRIBE THE RELATIONSHIPS BETWEEN THEM 7.G.3 Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

50 EXAMPLE Using a clay model of a rectangular prism, describe the shapes that are created when planar cuts are made diagonally, perpendicularly, and parallel to the base.

51 John and Joyce are sharing a piece of cake with the dimensions shown in the diagram. John is about to cut the cake at the mark indicated by the dotted lines. Joyce says this cut will make one of the pieces three times as big as the other. Is she right? Justify your response.

52 Three vertical slices perpendicular to the base of the right rectangular pyramid are to be made at the marked locations: (1) through AB, (2) through CD,, and (3) through vertex E. Rectangular pyramid Based on the relative locations of the slices on the pyramid, make a reasonable sketch of each slice. Include the appropriate notation to indicate measures of equal length.

53 SOLUTION

54 SOLVE REAL-LIFE AND MATHEMATICAL PROBLEMS INVOLVING ANGLE MEASURE, AREA, SURFACE AREA, AND VOLUME. 7.G.4 Know the formulas for the area and circumference of a circle and solve problems; give an informal derivation of the relationship between the circumference and area of a circle. Note: Know the formula does not mean memorization of the formula. To know means to have an understanding of why the formula works and how the formula relates to the measure (area and circumference) and the figure. This understanding should be for all students.

55 FINDING RELATIONSHIPS: DIAMETER, RADIUS, CIRCUMFERENCE AND AREA OF A CIRCLE This is the students initial work with circles. Knowing that a circle is created by connecting all the points equidistant from a point (center) is essential to understanding the relationships between radius, diameter, circumference, pi, and area. Students can observe this by folding a paper plate several times, finding the center at the intersection, then measuring the lengths between the center and several points on the circle, the radius. Measuring the folds through the center, or diameters, leads to the realization that a diameter is two times a radius.

56 FINDING RELATIONSHIPS: DIAMETER, RADIUS, CIRCUMFERENCE AND AREA OF A CIRCLE Given multiple-size circles drawn on grid paper, students should explore the relationship between the radius and the length measure of the circle (circumference) finding an approximation of pi and ultimately deriving a formula for circumference. String or yarn laid over the circle and compared to a ruler is an adequate estimate of the circumference.

57 FINDING RELATIONSHIPS: DIAMETER, RADIUS, CIRCUMFERENCE AND AREA OF A CIRCLE Students measure the circumference and diameter of several circular objects in the room (clock, trashcan, doorknob, wheel, etc.). Students again lay string or yarn over the circumference of circles drawn on a grid to find the relationship between the diameter and the area of a circle by using grid paper to estimate the area

58 FINDING RELATIONSHIPS: DIAMETER, RADIUS, CIRCUMFERENCE AND AREA OF A CIRCLE Students understand the relationship between radius and diameter. Students also understand the ratio of circumference to diameter can be expressed as pi. Building on these understandings, students generate the formulas for circumference and area.

59 FINDING RELATIONSHIPS: DIAMETER, RADIUS, CIRCUMFERENCE AND AREA OF A CIRCLE Students organize their information and discover the relationship between circumference and diameter by noticing the pattern in the ratio of the measures. Students write an expression that could be used to find the circumference of a circle with any diameter and check their expression on other circles

60 THESE WERE EXAMPLES OF INDUCTIVE REASONING Students measure the circumference and diameter of several circular objects in the room (clock, trashcan, doorknob, wheel, etc.). Students organize their information and discover the relationship between circumference and diameter by noticing the pattern in the ratio of the measures. Students write an expression that could be used to find the circumference of a circle with any diameter and check their expression on other circles

61 The illustration shows the relationship between the circumference and area. If a circle is cut into wedges and laid out as shown, a parallelogram results. Half of an end wedge can be moved to the other end a rectangle results. The height of the rectangle is the same as the radius of the circle. The base length is the circumference. The area of the rectangle (and therefore the circle) is found by the following calculations

62 FINDING RELATIONSHIPS: DIAMETER, RADIUS, CIRCUMFERENCE AND AREA OF A CIRCLE This is the students initial work with circles. Knowing that a circle is created by connecting all the points equidistant from a point (center) is essential to understanding the relationships between radius, diameter, circumference, pi, and area. Students can observe this by folding a paper plate several times, finding the center at the intersection, then measuring the lengths between the center and several points on the circle, the radius. Measuring the folds through the center, or diameters, leads to the realization that a diameter is two times a radius.

63 FINDING RELATIONSHIPS: DIAMETER, RADIUS, CIRCUMFERENCE AND AREA OF A CIRCLE Given multiple-size circles drawn on grid paper, students should explore the relationship between the radius and the length measure of the circle (circumference) finding an approximation of pi and ultimately deriving a formula for circumference. String or yarn laid over the circle and compared to a ruler is an adequate estimate of the circumference.

64 PROBLEM The seventh grade class is building a mini golf game for the school carnival. The end of the putting green will be a circle. If the circle is 10 feet in diameter, how many square feet of grass carpet will they need to buy to cover the circle? How might you communicate this information to the salesperson to make sure you receive a piece of carpet that is the correct size?

65 If possible, find the radius of a circle where the area of the circle and the circumference of the circle are equal. Is there more than one possible answer?

66 The side of square DEFG,, EF=2 cm, is also the radius of circle,c-f.. What is the area of the entire shaded region? Provide all evidence of your calculations.

67 SOLVE REAL-LIFE AND MATHEMATICAL PROBLEMS INVOLVING ANGLE MEASURE, AREA, SURFACE AREA, AND VOLUME 7.G.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

68 IN THE TRIANGLE, WHAT IS THE DEGREE MEASURE OF ANGLE ABC?

69 USE THE DIGITS 1 THROUGH 10 (WITHOUT REPEATING ANY NUMBER) TO COMPLETE THE SCENARIOS BELOW:

70 WHAT IS THE GREATEST VOLUME YOU CAN MAKE WITH A RECTANGULAR PRISM THAT HAS A SURFACE AREA OF 20 SQUARE UNITS?

71 SOLVE REAL-LIFE AND MATHEMATICAL PROBLEMS INVOLVING ANGLE MEASURE, AREA, SURFACE AREA, AND VOLUME 7.G.6 Solve real-world and mathematical problems involving area, volume and surface area of twoand three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

72 The diagram below represents a solid of uniform cross-section. All the lines of the figure meet at right angles. The dimensions are marked in the drawing in terms of x Write simple formulas in terms of x for each of the following: (a) the volume of the solid; (b) the surface area you would have to cover in order to paint this solid (c) the length of decorative cord you would need if you wanted to outline all the edges of this solid.

73 THE FOLLOWING PRISM IS MADE UP OF 27 IDENTICAL CUBES. WHAT IS THE GREATEST POSSIBLE SURFACE AREA THE PRISM CAN HAVE AFTER REMOVING CUBES FROM THE OUTSIDE?

74 GRADE 8 Three Critical Areas Formulating and reasoning about expressions and equations, solving linear equations, solving systems of linear equations Concept of function and using functions to describe quantitative relationships ANALYZING 2 AND 3-D SPACE AND FIGURES DISTANCE ANGLE SIMILARITY CONGRUENCE PYTHAGOREAN THEOREM

75 UNDERSTAND CONGRUENCE AND SIMILARITY USING PHYSICAL MODELS, TRANSPARENCIES, OR GEOMETRY SOFTWARE. 8.G.1 Verify experimentally the properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to line segments of the same length. b. Angles are taken to angles of the same measure. c. Parallel lines are taken to parallel lines.

76 UNDERSTAND CONGRUENCE AND SIMILARITY USING PHYSICAL MODELS, TRANSPARENCIES, OR GEOMETRY SOFTWARE 8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them

77 IS FIGURE A CONGRUENT TO FIGURE A? EXPLAIN HOW YOU KNOW.

78 DESCRIBE THE SEQUENCE OF TRANSFORMATIONS THAT RESULTS IN THE TRANSFORMATION OF FIGURE A TO FIGURE A.

79 UNDERSTAND CONGRUENCE AND SIMILARITY USING PHYSICAL MODELS, TRANSPARENCIES, OR GEOMETRY SOFTWARE 8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

80 TRANSLATION

81 REFLECTION:

82 ROTATION:

83 UNDERSTAND CONGRUENCE AND SIMILARITY USING PHYSICAL MODELS, TRANSPARENCIES, OR GEOMETRY SOFTWARE 8.G.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

84 IS FIGURE A SIMILAR TO FIGURE A? EXPLAIN HOW YOU KNOW

85 DESCRIBE THE SEQUENCE OF TRANSFORMATIONS THAT RESULTS IN THE TRANSFORMATION OF FIGURE A TO FIGURE A.

86 UNDERSTAND CONGRUENCE AND SIMILARITY USING PHYSICAL MODELS, TRANSPARENCIES, OR GEOMETRY SOFTWARE 8.G.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so

87 UNDERSTAND AND APPLY THE PYTHAGOREAN THEOREM. 8.G.5 Explain a proof of the Pythagorean Theorem and its converse.

88 ATTEMPT 1

89 ATTEMPT 2

90

91 PYTHAGOREAN THEOREM 8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in realworld and mathematical problems in two and three dimensions.

92 Jane is hoping to buy a large new television for her den, but she is not sure what size screen will be suitable for her wall. The is because television screens are measured by their diagonal line. The following 42-inch screen measures 32 inches along the base. What is the height of the screen? Show how you know. What is the area of the screen in square inches? Jane would like to have a screen 40-inches wide and 32-inches high. What screen size, in inches, will she need to buy? Show your work

93 What is the ratio of the areas of the two squares? Show your work. If a second circle is inscribed inside the smaller square, what is the ratio of the areas of the two circles? Explain your reasoning.

94 PYTHAGOREAN THEOREM 8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system

95 FIND THE DISTANCE BETWEEN THE TWO POINTS ON THE COORDINATE PLANE.

96 FIND THE LENGTH OF SIDE X IN THE DIAGRAM BELOW

97 SOLVE REAL-WORLD AND MATHEMATICAL PROBLEMS INVOLVING VOLUME OF CYLINDERS, CONES, AND SPHERES. 8.G.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems