CARDSTOCK MODELING Math Manipulative Kit. Student Activity Book

Transcription

1 CARDSTOCK MODELING Math Manipulative Kit Student Activity Book

2 TABLE OF CONTENTS Activity Sheet for L.E. #1 - Getting Started Activity Sheet for L.E. #2 - Squares and Cubes (Hexahedrons) Activity Sheet for L.E. #3 - Triangles and Tetrahedrons Activity Sheet for L.E. #4 - Volume and Surface Area Activity Sheet for L.E. #5 - Platonic Forms Activity Sheet for L.E. #6 - Comparing Interior and Exterior Angles Activity Sheet for L.E. #7 - Proving Formulas Activity Sheet for L.E. #8 - Modeling the Planets Glossary

3 Activity Sheet for Learning Experience #1 Name GETTING STARTED In the spaces provided below, write your word, two true statements about your word, and then one fib that is believable enough to fool your classmates. 1. word: fib: 2. word: fib: 3. word: fib: 4. word: fib: 5. word: fib: 6. word: fib: 7. word: fib: 8. word: fib: 3

4 Activity Sheet for Learning Experience #1 page 2 9. word: fib: 10. word: fib: 11. word: fib: 4

5 Activity Sheet for Learning Experience #2 Name SQUARES AND CUBES Answer the questions below based on the square template provided in the teacher s manual, not the image below. Assumption: Distance should be measured from intersection to intersection or from line to line. Exterior edge Interior edge Distance between interior and exterior edge. Interior angle Exterior angle Questions 1. Length of exterior edge. 2. Length of interior edge. 3. Distance between exterior and interior edge. 4. Perimeter of the interior shape. 5. Interior angles. 6. Exterior angles. 7. Sum of interior angles. 8. Sum of exterior angles. Metric Measurement Customary 9. Are the interior squares and the exterior squares similar? Explain. 5

6 Activity Sheet for Learning Experience #2 page Is the shape of the template a regular polygon? Explain how you arrived at your answer. 6

7 Activity Sheet for Learning Experience #2 page 3 Follow the directions listed below to create the panels for a hexahedron (cube). 1. Place the template on the surface of a lightweight card. 2. Trace the outside edge of the template and place a mark through the holes with a pencil. 3. Remove the template and cut along the outside edge that was traced. 4. Place a straight edge or ruler through the center of the interior points. Connect the points with lines that are parallel to the outside edges. The resulting lines should be 8 mm from the outside edges. Press hard with a pencil when lines are drawn. You will be folding these lines later. 5. Punch holes in the panels with hand paper punch. The hole should be about ¼ round. Each hole should be centered over each interior corner. 6. An opening must now be cut to the punched holes. The cuts will form a notch for the rubberband connection. Make two cuts toward the center of each hole. Each cut should be perpendicular to its nearest exterior edge. 7. Fold along each line connecting the holes. Fold the impression made by pencil. 8. Repeat steps 1-7 until six (6) panels are made. 7

8 Activity Sheet for Learning Experience #2 page 4 Create a model of a hexahedron using the six (6) square panels and rubberbands. Cube (Hexahedron) Shape Polygon(s) used for faces Number of faces Number of corners Number of edges Number of faces at each vertex Cube (Hexahedron) 8

9 Activity Sheet Learning Experience #3 Name TRIANGLES AND TETRAHEDRONS Answer the questions below based on the triangle template provided in the teacher s manual, not the diagram below. Assumption: Distance should be measured from intersection to intersection or from line to line. Exterior edge Distance between interior and exterior edge. Interior edge Interior angle Exterior angle Questions 1. Length of exterior edge. 2. Length of interior edge. 3. Distance between exterior and interior edge. 4. Perimeter of the interior shape. 5. Interior angles. 6. Exterior angles. 7. Sum of interior angles. 8. Sum of exterior angles. Metric Measurement Customary 9. Name the triangle shown in the template? 10. Explain how you arrived at your answer. 9

10 Activity Sheet for Learning Experience #3 page 2 Follow the directions listed below to create the panels for a tetrahedron. 1. Place the template on the surface of a lightweight card. 2. Trace the outside edge of the template and place a mark through the holes with a pencil. 3. Remove the template and cut along the outside edge that was traced. 4. Place a straight edge or ruler through the center of the interior points. Connect the points with lines that are parallel to the outside edges. The resulting lines should be 8 mm from the outside edges. Press hard with a pencil when lines are drawn. You will be folding these lines later. 5. Punch holes in the panels with hand paper punch. The hole should be about ¼ round. Each hole should be centered over each interior corner. 6. An opening must now be cut to the punched holes. The cuts will form a notch for the rubberband connection. Make two cuts toward the center of each hole. Each cut should be perpendicular to its nearest exterior edge. 7. Fold along each line connecting the holes. Fold the impression made by pencil. 8. Repeat steps 1-7 until four (4) panels are made. 10

11 Activity Sheet for Learning Experience #3 page 3 Create a model of a tetrahedron using the four (4) triangle panels and rubberbands. Tetrahedron Shape Polygon(s) used for faces Number of faces Number of corners Number of edges Number of faces at each vertex Tetrahedron 11

12 Activity Sheet for Learning Experience #4 Name VOLUME AND SURFACE AREA 1. Find the volume of the cube created with the panels. a a a Cube = a 3 Show your work. Volume = 2. Find the volume of the pyramid created with the panels. Pyramid = 1/3 Bh or Bh 3 NOTE: The capital B in this formula is indicating the area of the base is multiplied by the height and divided by 3. Show your work. Volume = 12

13 Activity Sheet for Learning Experience #4 page 2 3. Use the chart below to find the surface area of the cube, tetrahedron, and pyramid. Platonic form Cube Shape of base Area of base Other Surfaces Area of Other Surfaces Surface Area Tetrahedron Pyramid 4. Find the difference between the surface area of the tetrahedron and the pyramid. Difference = Show why the difference is present. 13

14 Activity Sheet for Learning Experience #5 Name PLATONIC FORMS Create the panels for one or more of the forms shown below. Assemble the panels with your group to form one of the platonic forms below. Octahedron Dodecahedron Icosahedron Complete the chart below on each form that you or your classmates have created. Shape Octahedron Polygon(s) used for faces Number of faces Number of corners Number of edges Number of faces at each vertex Dodecahedron Icosahedron 14

15 Activity Sheet for Learning Experience #5 page 2 Follow the shortcuts below to find the number of edges and vertices of the dodecahedron, icosahedron, and octahedron. To find the number of edges: Form # of faces # of edges on each face # of faces that share edge Dodecahedron ( x ) / = # of faces # of edges on each face # of faces share edge # of edges Form # of faces # of edges on each face # of faces that share edge Icosahedron ( x ) / = # of faces # of edges on each face # of faces share edge # of edges Form # of faces # of edges on each face # of faces that share edge Octahedron ( x ) / = # of faces # of edges on each face # of faces share edge # of edges 15

16 Activity Sheet for Learning Experience #5 page 3 To find the number of vertices: Form # of faces # of vertices on each face # of faces that meet at each vertex Dodecahedron ( x ) / = # of faces # of vertices on each face # of faces that meet at each vertex # of vertices Form # of faces # of vertices on each face # of faces that meet at each vertex Icosahedron ( x ) / = # of faces # of vertices on each face # of faces that meet at each vertex # of vertices Form # of faces # of vertices on each face # of faces that meet at each vertex Octahedron ( x ) / = # of faces # of vertices on each face # of faces that meet at each vertex # of vertices 16

17 Activity Sheet for Learning Experience #6 Name COMPARING INTERIOR AND EXTERIOR ANGLES Refer back to the activity sheet for Learning Experience #2 and #3. Record the measurement of the interior and exterior angles of the square and triangle. Then Use the panels of the pentagon, hexagon, and octagon and find the interior and exterior angles. Find the sum of the exterior angles. Polygon Triangle Number of sides Interior angle Sum of interior angles Exterior angle Sum of exterior angles Square Pentagon Hexagon Octagon Compare the sums of the exterior angles of each polygon. What do you conclude from your data? What other patterns do you see in the chart? 17

18 Activity Sheet for Learning Experience #6 page 2 How would you find the measure of the exterior angles of a seven-sided polygon? What would be the measure of the interior angles of this seven-sided polygon? 18

19 Activity Sheet for Learning Experience #7 Name PROVING FORMULAS The area of a square = bh. Cut out 9 square panels. Form a larger square in a 2 x 2 pattern using 4 panels. If we count each square as 1, then each side of the larger square equals 2. Area = bh = 2 x 2 = 4 You are able to see the 4 squares and why the area of the larger square is Add the remaining squares to your larger square. Draw this square and explain how you find the area. 19

20 Activity Sheet for Learning Experience #7 page 2 2. Using your square panels, find the actual area of the larger square (in centimeters) with 2 squares that create each side. Area = Round to the nearest whole number = 3. Using your square panels, find the actual area of the larger square with 3 squares that create each side. Area = Round to the nearest whole number = 4. How does the number sequence 1, 4, 9, 16, 25, 36 relate to the larger squares you have been creating with the smaller square panels? 5. What would be the next four (4) numbers in the sequence? 20

21 Activity Sheet for Learning Experience #7 page 3 The formula for the area of a rectangle = bh. The formula for the area of a triangle = ½ bh. Let s find out why. On a piece of graph paper, measure a rectangle that is 22 cm x 19 cm. Use 4 triangle panels to create a larger equilateral triangle. Trace this equilateral triangle on the graph paper. Cut out the triangle that was traced and place it on the table. You are left with 2 right triangles. Place the right triangles on top of the equilateral triangle that was cut out. 6. Explain the results. 7. What do these results prove about the formula for the area of a triangle? 21

22 Activity Sheet for Learning Experience #7 page 4 The formula for the area of a square = bh The formula for the area of a parallelogram = bh How can that be? The two shapes are different. Let s find out why. Using 2 triangles panels, create a parallelogram. Trace the parallelogram on a piece of graph paper and cut it out. 8. What does parallel mean? 9. What is a parallelogram? After cutting out the parallelogram, draw a straight line up from the bottom right hand corner to the upper exterior edge. You will see a right triangle is formed. Cut along that line. Right triangle Place the right triangle piece on the opposite side of the parallelogram to form a square. 10. What do these results prove about the formula for the area of a parallelogram? 22

23 Activity Sheet for Learning Experience #7 page 5 The area of a trapezoid = h/2 (b 1 + b 2 ). This formula looks a little more complicated, but we can rewrite it so it is a little easier to figure out: (b 1 + b 2 ) h 2 To take apart this formula so we can understand it, form a trapezoid with 3 triangle panels. Trace trapezoid on graph paper. b Measure the lengths of bases of the trapezoid. + = b 1 b Divide the sum of the bases by 2 to get an average length. b 1 / 2 = b 1 + b 2 (average length) 13. Multiply the average length by the height of the trapezoid. x = (average length) (height) (area of trapezoid) 14. Check your work: Find the area of one of the triangles and multiply by 3. Show your work. 23

24 Activity Sheet for Learning Experience #7 page 6 Another way to look at the area formula of a trapezoid is to, again, form a trapezoid using 3 triangles. Trace the trapezoid on graph paper. Draw a line segment from the bottom left corner to the top right corner. Two triangles are formed. b1 h I II h b2 The area of a trapezoid = the sum of the areas of the two (2) triangles. ½ bh + ½ bh 15. Area of triangle I = 16. Area of triangle II = 17. Area of trapezoid = Using the distributive property, the one half and the height (h) can be factored out from each term to form the formula for area of trapezoid to = ½ h (b 1 + b 2 ) 24

25 Activity Sheet for Learning Experience #7 page 7 Create a parallelogram with two (2) trapezoids made with triangle panels. 18. Find the area of the parallelogram using the formula bh. Show your work. 19. Find another way to find the area of the parallelogram. Show your work. 20. Find another way to find the area of the parallelogram. Show your work. 25

26 Activity Sheet for Learning Experience #7 page Using the formula for area of a rectangle, complete the chart below. Length Width Area x x+3 y y Use the algebraic expressions to complete the area of a triangle. X + 2 X + 3 Area = 23. Use the algebraic expressions to complete the area of a trapezoid. X + 1 X X + 3 Area = 26

27 Activity Sheet for Learning Experience #8 Name MODELING THE PLANETS 1. The diameter of the planetary bodies is shown in the chart below. Rewrite each of the diameters in scientific notation. Diameter of Planetary Planetary Bodies Bodies (km) Sun 1,391,000 Mercury 4878 Venus 12,104 Earth 12,756 Mars 6,787 Jupiter 142,984 Saturn 120,660 Uranus 51,118 Neptune 49,528 Pluto 2,274 Diameter rounded to nearest hundred. Diameters written in scientific notation 27

28 Activity Sheet for Learning Experience #8 page 2 2. To create a model of the planets using a cube, change the diameter of the planetary bodies into 1/1000 of the actual size (megameters). Diameter of Planetary Planetary Bodies Bodies (km) Sun 1,391,000 Mercury 4878 Venus 12,104 Earth 12,756 Mars 6,787 Jupiter 142,984 Saturn 120,660 Uranus 51,118 Neptune 49,528 Pluto 2,274 1/1000 size (megameters) Round reduced size to the nearest tenth. Using the information in the last column above, round to the nearest tenth as the measurement for the size of the panels to create cubic model of each planetary body (models will not be spherical). Be sure to have 8 mm edges for the attachment of rubberbands. First, create a template of your panels so each panel of your cube will be measured accurately. i.e. Mercury = mm + 8 mm = 16 mm or 1.6 cm 4.9 cm = 6.5 (total length of side) 8 mm 8 mm Using cardstock, measure the size for six (6) panels to create a cube model of the planetary body assigned to your group. 6.5 cm 28

29 Activity Sheet for Learning Experience #8 page 3 3. In the chart below, the distance between the planets is shown. Move the decimal to the place indicated with scientific notation. Planetary Bodies Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto Distance from Sun (km) Using the number that is 1/100,000,000 the actual distance in kilometers, find the location on a map from where your school is (your school will represent the Sun) to where the next planet (Mercury) would be located. Continue for the rest of the planets. Sun Your school Sun Your school Sun Your school Sun Your school Sun Your school Sun Your school Sun Your school Where would Mercury be located? Where would Venus be located? Where would Earth be located? Where would Mars be located? Where would Jupiter be located? Where would Saturn be located? Where would Uranus be located? 29

30 Activity Sheet for Learning Experience #8 page 4 Sun Your school Sun Your school Where would Neptune be located? Where would Pluto be located? 5. Again using the number that is 1/100,000,000 the actual distance from the Sun, find out where the planets would be located in relation to each other. If Mercury were where your school is If Venus were where your school is Where would Venus be located? Where would Earth be located? If Earth were where your school is If Mars were where your school is If Jupiter were where your school is If Saturn were where your school is If Uranus were where your school is If Nepture were where your school is Where would Mars be located? Where would Jupiter be located? Where would Saturn be located? Where would Uranus be located? Where would Neptune be located? Where would Pluto be located? 30

31 Activity Sheet for Learning Experience #8 page 5 From creating this model of the solar system, list the various things you learned about the size of the solar system (size of planets and distance between them). 31

32 GLOSSARY Area Base Centimeter Concentric Congruent Corner Cube Customary Diameter Difference Distance Dodecahedron Edge Equilangular Equilateral Equilateral triangle Exterior angle the measure of how much surface is covered by a figure. The number of square units a figure contains. This is often found by multiplying length and width. the lower side or face of geometric figure. a metric unit of length; 100 centimeters equals one meter. having a common center, such as two circles, one inside the other. used to describe identical geometric shapes. They are the same in size and shape. Their sides are equal in length and their angles are equal in measure. point where line segments meet. a prism that has six square faces. commonly practiced, used or observed. a straight line running from one side of rounded geometric figure through the center to the other side. the amount by which one quantity is greater or smaller than another. the length of space between two objects. a three-dimensional geometric figure with 12 equal pentagonal faces meeting in threes at 20 vertices. the segment where two faces of a solid figure meet. where all the angles in a shape have the same number of degrees of measure. where all the sides of a shape are the same length. a triangle with three congruent sides, and three equal angles. an angle on the outside of a polygon formed between a side and an extension of an adjacent side. 32

33 Face Form Hexagon Hexahedron Icosahedron Interior angle Interior line segment Intersection Lateral Line segment Model Octahedron Panel Parallel Parallelogram Pentagon Perimeter Perpendicular a flat surface of a three-dimensional figure. three-dimensional model of a shape. a polygon with six sides. a three-dimensional geometric figure that has six plane faces, for example, a cube. a three-dimensional geometric figure having 20 sides or faces. the angle formed between two adjacent sides of a polygon and lying in its interior. The sum of the interior angles of any polygon is equal to the number of its sides minus two and multiplied by 180º. the inside line segment in a diagram. where two lines meet or cross. pertaining to the side or sides. part of a line consisting of two endpoints and all the points between them. a representation of an object made to a larger or smaller scale than the original. a 3-dimensional geometric figure that has eight faces. flat part/side of three-dimensional forms. lines in the same plane that do not intersect. a four-sided plane figure in which both pairs of opposite sides are parallel and of equal length and opposite angles are equal. a polygon that has five sides. the linear distance around an object or figure. Also the boundary of a closed plane figure. two lines intersecting to form right angles. 33

34 Platonic solids Point Polygon Polyhedron Prism Proportional Pyramid Quadrilateral Radius Rectangle Regular polygon Right angle Scientific Notation Square Surface area Template Tetrahedron consist of the five regular polyhedrons: cube, tetrahedron, octahedron, icosahedron, and dodecahedron. an exact place or position in space represented by a dot. a closed figure formed from line segments that meet only at their endpoints. Polygons form the faces of a polyhedron. a three-dimensional geometric figure with many flat surfaces, formed by joining edges of polygons to enclose a region of space. a three-dimensional figure that has two congruent and parallel faces that are polygons. The remaining faces are parallelograms. having equivalent ratios. a polyhedron whose base is a polygon and whose other faces are triangles that share a common vertex. two-dimensional geometric figure with four sides. the segment or the length of the segment from the center of a circle to a point on the circle. a polygon having four polygon sides and four right angles. a polygon in which all sides have the same length and all angles have the same number of degrees of measure. an angle with a measure of 90 degrees. short way of expressing large numbers. A way of expressing a given number as a number between 1 and 10 multiplied by 10 to the appropriate power. a polygon with four right angles and four equal sides. the total area of the faces (including bases) of a solid figure. a pattern or mold with one or more shapes used to guide the manufacture or drawing of objects with a similar shape. a three-dimensional geometric figure that has four faces. 34

35 Trapezoid Triangle Vertex Vertices Volume a quadrilateral that has two parallel sides. a polygon with three sides. the corner point of an angle, polygon, or solid, where several edges meet. plural of vertex. the amount of space occupied by a three-dimensional object as measured in cubic units. This is often found by multiplying length, width and height. 35