Math Mammoth Early Geometry

Copyright 2008-2013 Taina Maria Miller. EDITION 3.0 All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, or by any information storage and retrieval system, without permission in writing from the author. Copying permission: Permission IS granted for the teacher to reproduce this material to be used for students, not for commercial resale, by virtue of the purchase of this book. In other words, the teacher MAY make copies of the pages to be used for students. Permission is given to make electronic copies of the material for back-up purposes only.

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Contents Introduction .......................................................................... 5 Geometry Games on the Internet .....................................

6

Basic Shapes ......................................................................

10

Playing with Shapes .........................................................

15

Drawing Basic Shapes .....................................................

16

Practicing Basic Shapes and Patterns ............................

19

Shapes Review....................................................................

22

Shapes ................................................................................

25

Right Angles ......................................................................

29

Surprises with Shapes ......................................................

31

Making Shapes .................................................................

33

Rectangles and Squares ...................................................

36

Some Special Quadrilaterals ...........................................

39

Geometric Patterns ..........................................................

42

Line Symmetry ..................................................................

45

Perimeter ...........................................................................

48

Problems with Perimeter ..................................................

51

Getting Started with Area ................................................

54

More About Area ..............................................................

56

Multiplying by Whole Tens ..............................................

60

Area Units and Problems ..................................................

64

Area and Perimeter Problems ........................................

68

More Area and Perimeter Problems ..............................

70

Three-Dimensional Shapes ..............................................

73

Solids 1 ................................................................................

75

Solids 2 ................................................................................

77

3

Review 1 .............................................................................

79

Review 2 .............................................................................

80

Geometry Review ..............................................................

82

Answers ...............................................................................

84

Printable Cutouts for Common Solids ............................

105

More from Math Mammoth ............................................

119

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Introduction Math Mammoth Early Geometry covers geometry topics for grades 1-3. These lessons are taken out from the Math Mammoth Complete Curriculum (Light Blue Series). The first lessons in this book have to do with shapes—that is where geometry starts. Children learn to recognize and draw basic shapes, and identify triangles, rectangles, squares, quadrilaterals, pentagons, hexagons, and cubes. They also put several shapes together to form new ones, or divide an existing shape into new ones. We also study some geometric patterns, right angles, have surprises with pentagons and hexagons, and make shapes in a tangram-like game. These topics are to provide some fun while also letting children explore geometry and helping them to memorize the terminology for basic shapes. The students also learn a little about symmetry—hopefully an easy and fun topic. In the latter part of the book, the emphasis is on two third grade concepts: area and perimeter. Students find the perimeters of polygons, including finding the perimeter when the side lengths are given, and finding an unknown side length when the perimeter is given. They learn about area, and how to measure it in either square inches, square feet, square centimeters, square meters, or just square units if no unit of length is specified. Students also relate area to the operations of multiplication and addition. They learn to find the area of a rectangle by multiplying the side lengths, and to find the area of rectilinear figures by dividing them into rectangles and adding the areas. We also study the distributive property “in disguise.” This means using an area model to represent a × (b + c) as being equal to a × b plus a × c. The expression a × (b + c) is the area of a rectangle with side lengths a and (b + c), which is equal to the areas of two rectangles, one with sides a and b, and the other with sides a and c. Multiplying by Whole Tens is a lesson about multiplication such as 3 × 40 or 90 × 7. It is put here so that students can use their multiplication skills to calculate areas of bigger rectangles. Then we solve many area and perimeter problems. That is necessary so that students learn to distinguish between these two concepts. They also get to see rectangles with the same perimeter and different areas or with the same area and different perimeters. Lastly we touch on solids, such as a cube, a rectangular prism, pyramids, a cone, and a cylinder, and study their faces, edges, and vertices. You can make paper models for them from the printouts provided after the answer key. Just print them out, cut out the shapes, fold the sides, and glue or tape the figures together. Alternatively you can buy them, usually made from plastic. Search on the internet for “geometric solids.” I wish you success with your math teaching! Maria Miller

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Geometry Games on the Internet I encourage you to use some of these free resources that can make geometry so much fun!

SHAPES Buzzing with Shapes Tic tac toe with shapes; drag the counter to the shape that has that amount of sides. http://www.harcourtschool.com/activity/buzz/buzz.html Shape Cutter Draw any shape (polygon), cut it, and manipulate the cut pieces. You can have the computer mix them up, and then try to recreate the original shape. http://illuminations.nctm.org/ActivityDetail.aspx?ID=72 Shifting Shapes Figure out what shape it is when viewed through a small opening! Click on the “eye” button to see it in its entirety. http://www.ictgames.com/YRshape.html Patch Tool An online activity where the student designs a pattern using geometric shapes. http://illuminations.nctm.org/ActivityDetail.aspx?ID=27 Polygon Matching Game http://www.mathplayground.com/matching_shapes.html Polygon Playground Drag various colorful polygons to the work area to make your own creations! http://www.mathcats.com/explore/polygons.html Interactive Quadrilaterals Drag the corners to play with squares, rectangles, rhombi, and more. http://www.mathsisfun.com/geometry/quadrilaterals-interactive.html Shapes Identification Quiz from ThatQuiz.org An online quiz in a multiple-choice format, asking to identify common two-dimensional shapes. You can modify the quiz parameters to your liking. www.thatquiz.org/tq-f/math/shapes/ Tangram puzzles for kids Use the seven pieces of the Tangram to form the given puzzle. Complete the puzzle by moving and rotating the seven shapes. http://www.abcya.com/tangrams.htm Logic Tangram game Note: this uses four pieces only. Use logic and spatial reasoning skills to assemble the four pieces into the given shape. http://www.mathplayground.com/tangrams.html

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Interactive Tangram Puzzle Place the tangram pieces so they form the given shape. http://nlvm.usu.edu/en/nav/frames_asid_112_g_2_t_1.html Tangram set Cut out your Tangram set by folding paper http://tangrams.ca/fold-set Online Kaleidoscope Create your own kaleidoscope creation with this interactive tool. http://www.zefrank.com/dtoy_vs_byokal/ SYMMETRY Symmetry Game Tell how many lines of symmetry a shape has. http://www.innovationslearning.co.uk/subjects/maths/activities/year3/symmetry/shape_game.asp Primary Resources: Mirror Images See images mirrored in a line. http://www.primaryresources.co.uk/online/symmetry.swf Primary Resources: Reflection Color the squares and reflect the given pattern in a line. http://www.primaryresources.co.uk/online/reflection.swf AREA AND PERIMETER Free Worksheets for Area and Perimeter Create worksheets for the area and the perimeter of rectangles/squares with images, word problems, or problems where the student writes an expression for the area using the distributive property. Options also include area and perimeter problems for irregular rectangular areas, and more. http://www.homeschoolmath.net/worksheets/area_perimeter_rectangles.php Everything you wanted to know about area and perimeter Short explanations of how to find the perimeter of simple shapes and the area of rectangles, followed by quizzes on three levels. In perimeter, level two, some side lengths are not given. In level three, you calculate the perimeter of compound shapes. In area of rectangles, level 1 has just rectangles, and levels 2 and 3 have compound shapes made of rectangles. www.bgfl.org/custom/resources_ftp/client_ftp/ks2/maths/perimeter_and_area/index.html Shape Explorer Find the perimeter and area of odd shapes on a rectangular grid. http://www.shodor.org/interactivate/activities/ShapeExplorer/ Math Playground: Measuring the Area and Perimeter of Rectangles Amy and her brother, Ben, explain how to find the area and perimeter of rectangles and show you how changing the perimeter of a rectangle affects its area. After the lesson, you will use an interactive ruler to measure the length and width of 10 rectangles, and to calculate the perimeter and area of each. http://www.mathplayground.com/area_perimeter.html

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Math Playground: Party Designer You need to design areas for the party, such as a crafts table, food table, seesaw, and so on, so that they have the given perimeters and areas. http://www.mathplayground.com/PartyDesigner/PartyDesigner.html BBC Bitesize - Perimeter A simple revision (review) “bite” for perimeter that includes short explanations and a few quiz questions. http://www.bbc.co.uk/schools/ks3bitesize/maths/measures/perimeter/revise1.shtml BBC Bitesize - Area Brief revision (review) “bites”, including a few interactive questions, about area: counting squares, area of rectangles, area of triangles, parallelograms, and of compound shapes. Includes an activity and a test. http://www.bbc.co.uk/schools/ks3bitesize/maths/measures/area/revise1.shtml Geometry Area/Perimeter Quiz from ThatQuiz.org An online quiz, asking either the area of perimeter of rectangles, triangles, and circles. You can modify the quiz parameters to your liking, for example to omit the circle, or instead of solving for area, you solve for an unknown side when the perimeter/area is given. http://www.thatquiz.org/tq-4/?-j201v-lc-m2kc0-na-p0 Perimeter Game from Cyram.org A simple online quiz for finding the perimeter of rectangles, triangles, or compound rectangles where not all side lengths are given. http://www.cyram.org/Projects/perimetergame/index.html FunBrain: Shape Surveyor Geometry Game A simple and easy game that practices finding either the perimeter or area of rectangles. http://www.funbrain.com/poly/index.html Area of Rectangle Drag the corners of the rectangle and see how the side lengths and areas change. http://illuminations.nctm.org/ActivityDetail.aspx?ID=46 XP Math: Find Perimeters of Parallelograms This online quiz shows you parallelograms and rectangles, and you need to calculate the perimeter, including typing in the right unit, and not using the altitude of the parallelogram. http://www.xpmath.com/forums/arcade.php?do=play&gameid=10 SOLIDS Identify solids Select the name and drop it on the correct solid. http://www.softschools.com/math/geometry/shapes/solids/games/ Geometric Solids Manipulate various geometric solids. Color the solid to investigate properties such as the number of faces, edges, and vertices. http://illuminations.nctm.org/ActivityDetail.aspx?ID=70

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2-D and 3-D Shapes Learn about different solids and see them rotate. http://www.bgfl.org/bgfl/custom/resources_ftp/client_ftp/ks2/maths/3d/index.htm Identify solids Click to identify the partially buried 3-dimensional shapes. http://www.primaryresources.co.uk/online/longshape3d.html Space Blocks Build with blocks to illustrate three-dimensional shapes. http://nlvm.usu.edu/en/nav/frames_asid_195_g_2_t_2.html

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Basic Shapes

These are circles. They don't have any corners. They are perfectly round!

These are triangles. They have THREE corners, and three sides.

These are rectangles. They have four “straight corners.” They look like books!

These are squares. Squares, too, have four corners, and each corner is “straight.” All sides of a square are the same length.

1. Color the circles yellow; the squares red; the triangles green and the rectangles blue. One shape will not be colored.

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So what are these shapes? They have four corners and four sides. But they don't have four straight corners, like squares and rectangles do. They are just four-sided shapes that are not squares nor rectangles. In mathematics we call them quadrilaterals. “Quadri” comes from quattuor, Latin for four. “Lateral” comes from lateralis, Latin for side.

2. Count how many corners each shape has.

a. _____

b. _____

c. _____

d. _____

e. _____

f. _____

g. _____

h. _____

i. _____

j. _____

3. a. In the shapes above, there is one rectangle. Mark it with R. b. Mark the other four-sided shapes with Q (for quadrilateral). c. Mark the one circle with C. d. Find another rounded shape that is not a circle.

4. a. Draw three dots anywhere in this space. Join them with lines. What shape do you get?

b. Draw again three dots anywhere in this

space, and join them with lines.

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5. Draw a line from dot to dot so that you divide the shape into two new shapes. Use a ruler. How many sides do the new shapes have? How many corners? a. The new shapes have _______ sides, and _______ corners. They are ________________________ b. The new shapes have _______ sides, and _______ corners. They are ________________________ c. The new shapes have _______ sides, and _______ corners. They are

quadrilaterals

d. The new shapes have _______ sides, and _______ corners. They are ________________________ e. The new shapes have _______ sides, and _______ corners. They are ________________________

Divide this shape, using one line, into a triangle and a pentagon (five-sided shape).

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13

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Playing with Shapes Cut out the shapes on page 13. Hint: if you have the download version of this curriculum, print the shapes page scaled in 140-150% and landscape, so that it prints on two sheets of paper. All the shapes will then be much bigger.

1. Make a big triangle with the four yellow triangles (marked with 1). 2. Take all six of the yellow triangles (marked with 1). Put them together to get a six-sided shape (a hexagon). 3. Use the two pink rectangles (marked with 2) and make a square. 4. Use one pink rectangle (#2) and two blue squares (#7) to make a square. 5. Can you make a bigger square than what you made in exercise 4, using any pieces you want? 6. Make a rectangle using two red triangles (#5). 7. Make a bigger rectangle using four red triangles (#5). 8. Put together two of the green triangles (#4) to get a four-sided shape. You can do this in two different ways! Each time you will get a parallelogram. 9. Put together the two slim rectangles (#3) to make a. a rectangle; b. an L-shape; c. an eight-sided shape. 10. Put together the two purple shapes (#6) to make a six-sided shape (a hexagon). You can do this in many different ways! 11. Put together the two purple shapes (#6) to make a four-sided shape (a quadrilateral). 12. A challenge: put together the two purple shapes (#6) to make a five-sided shape (a pentagon). 13. Make your own five-sided shapes (pentagons) using any shapes! Make many different ones. 14. Make your own six-sided shapes (hexagons) using any shapes! Make many different ones. 15. Make your own interesting shapes using any shapes. Have fun!

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Drawing Basic Shapes 1. Use a ruler to join the dots carefully with straight lines. What shape do you get?

a. triangle / square / rectangle /

b. triangle / square / rectangle /

other four-sided shape

other four-sided shape

c. triangle / square / rectangle /

d. triangle / square / rectangle /

other four-sided shape

other four-sided shape

f. triangle / square / rectangle /

e. triangle / square / rectangle /

other four-sided shape

other four-sided shape

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2. a. Draw four dots anywhere in this space. Join the dots with lines. Use a ruler!

b. This time try to draw four dots in this space so that you would get a rectangle.

What shape did you get? A square, a rectangle, or just a four-sided shape?

c. Draw a rectangle. This time, use a BOOK to draw straight corners.

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3. The shapes (a), (b), (c), and (d) below are four-sided shapes (quadrilaterals). In each shape, draw a line from one corner to the opposite corner. What kind of shapes do you get now? ______________________ Now draw another line from corner to corner in each shape, using the two other corners you didn't yet use. How many parts does each four-sided shape have now? _______ What kind of shapes are these parts? ______________________

a.

c. b.

d.

4. Choose a color for each shape, and color! Triangles are _________________.

Circles are _________________.

Squares are _________________.

Rectangles are _________________.

Other four-sided shapes are _________________.

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Practicing Basic Shapes and Patterns 1. In each figure, draw a straight line with a ruler from one black dot to the other black dot. Color the two new parts with different colors. Write inside each new shape a letter: S for square, T for triangle, R for rectangle, Q for other quadrilateral (four-sided shape).

2. Join each dot to a dot on the other side with straight lines (horizontal and vertical lines) so that you get a grid of squares. Use a ruler and draw neatly.

Then color the squares using to this pattern: (ye = yellow)

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3. Repeat the patterns to fill the grids.

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4. Here you can design your own patterns!

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Shapes Review 1. Draw three dots on the right. Connect the dots with straight lines. You have drawn a triangle (tri means three). It has _____ vertices (corners) and three sides. Draw two more triangles in the same space. They can overlap.

2. Draw FOUR dots on the right. Connect the dots with straight lines. You have drawn a quadrilateral (quadri means four; lateral has to do with sides).

It has ______ vertices (corners) and four sides. Draw two more quadrilaterals in the same space.

3. The figures on the right are a square and a rectangle. Can you tell which is which? Squares and rectangles are quadrilaterals because they have four sides. Draw at least one more square and one more rectangle into the picture, the best you can.

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4. Draw FIVE dots on the right. Connect the dots with straight lines. You have drawn a pentagon (penta means five). It has _____ vertices and _____ sides. Draw one more pentagon in the space.

5. Draw SIX dots on the right. Connect the dots with straight lines. You have drawn a hexagon (hex means six). It has _______ vertices and _____ sides. Draw yet one more hexagon in the space.

6. How is a circle different from all of the shapes above?

7. Continue the pattern, and color it with pretty colors!

8. Color all triangles yellow. Color all quadrilaterals green. Color all pentagons blue. Color all hexagons purple. Or choose your own colors for each kind of shape.

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9. Now, this is a challenge to check if you remember the words for different shapes. Don't look at the previous pages! You can use the “dot” method: first draw dots for the corners, then use a ruler to draw the lines connecting the dots. a. Draw here a big and a small four-sided shape. What are four-sided shapes called?

b. Draw here a skinny and a fat three-sided shape. What are three-sided shapes called?

c. Draw here a blue five-sided shape and a green six-sided shape. What are five and six-sided shapes called?

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Shapes

If a shape has three vertices (corners) and three sides, it is a triangle.

If a shape has FOUR vertices and four sides, it is a quadrilateral, or a four-sided shape. “Quadri” means four, and “lateral” refers to sides.

If a shape has FIVE vertices and five sides, it is a pentagon.

If a shape has SIX vertices and six sides, it is a hexagon.

Seven-sided figure = heptagon Eight-sided figure = octagon

Nine-sided figure = nonagon Ten-sided figure = decagon

1. Draw two pentagons here by drawing dots and connecting them with lines. Remember, your pentagons don't have to look “regular” or nice. You can draw them to look “funny,” too, as long as they have five sides and five vertices.

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2. What shape is formed if you place the bolded sides of the two figures together? You can trace the shapes and cut them out.

a. ____________________________________

b. ____________________________________

c. ____________________________________

d. ____________________________________

3. Draw a straight line or lines through the shape and divide it into other shapes!

a. a square and a rectangle

b. a triangle and a pentagon

c. three rectangles

d. two quadrilaterals

e. two parts that are

f. four triangles

that are not rectangles

g. four triangles

exactly the same shape

h. a triangle and a pentagon

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i. four quadrilaterals

4. Divide the pentagon and the hexagon into new shapes using one straight line. Notice: your line does NOT have to go from corner to corner. Write what new shapes you get. a.

b.

c.

d.

5. Continue the tilings so they fill the grids, and name what shape(s) are used in the tiling.

a. ________________________

b. ________________________

c. ________________________ and ________________________

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6. Design your own tilings here.

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Right Angles These look like corners, but in math we call them angles. Imagine sitting inside of each angle, and the walls going up around you in the shape of the “corner.” In which angle would you have lots of space to sit? In which angle would you have only a little space to sit? Find two “square corners.” In mathematics we call them right angles.

Sometimes we draw a round line (an arc) inside of the angle to mark it.

Right angles are marked this way.

Corners of books are examples of right angles.

1. Write how many angles each shape has. Write how many right angles each shape has.

a.

b.

c.

_____ angles

_____ angles

_____ angles

_____ right angles

_____ right angles

_____ right angles

d.

e.

f.

_____ angles

_____ angles

_____ angles

_____ right angles

_____ right angles

_____ right angles

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2. Draw the shapes below. First draw dots for the corners. Then connect those with lines.

a. a rectangle

b. a square

c. a triangle with

one right angle

d. a triangle with

e. a quadrilateral with

f. a pentagon with

no right angles

one right angle

one right angle

3. Continue this pretty pattern. Look carefully. Where in the pattern (not in the grid) can you find right angles?

4. Which of these shapes has to ALWAYS have a right angle? a) triangle

b) square

c) pentagon d) hexagon

e) rectangle

5. The shapes are divided into parts. Write how many right angles there are. a.

b.

_____ right angles in the big shape.

_____ right angles in the big shape.

_____ right angles in each part.

_____ right angles in each part. c.

d.

_____ right angles in the big shape.

_____ right angles in the big shape.

_____ right angles in each part.

_____ right angles in each part. 30

Surprises with Shapes 1. Connect the dots using a ruler. Be neat! What shape do you get? _______________________________

2. Draw a line from one corner to some other corner. This divides your shape into two new shapes. What shapes are they? _______________________________

3. Draw more lines from a corner to some corner so that the whole shape gets divided into triangles.

4. Connect the dots using a ruler. Be neat! What shape do you get? _______________________________

5. Draw a line from one corner to the opposite corner. Then repeat so that each corner gets connected to the opposite corner. You need to draw three lines to do that. 6. Decorate your shape now so that it becomes a SNOWFLAKE! ALL snowflakes have this basic shape.

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7. Connect the dots in the numbered order using straight lines. Be neat! What do you get? _______________________________

8. In the middle of that shape, another shape is formed. What is it? _______________________________

9. Also connect the dots in the order 1 - 4 - 2 - 5 - 3 - 1. What shape is formed now? _______________________________

10. Connect the dots 1-2-3 using a ruler. Then connect the dots a-b-c also. Be neat! What shape do you get? _______________________________

11. In the middle of that shape, another shape is formed. What is it? _______________________________

12. Also connect the dots in the order 1 - a - 2 - b - 3 - c - 1. What shape is formed now? _______________________________

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Making Shapes We can make new shapes from putting several shapes together. For example, these two triangles together form a square:

1. Cut out the shapes on the next page. What shapes can you use to make the given shapes? There may be several possible solutions. The figures below are smaller than the ones you will cut out.

a.

b.

c.

d.

e.

f.

2. Now, you do the same. Put together some shapes. Trace the outline of your combined shape on paper, and give that to your friend to solve. 3. The game you just played is very similar to the ancient Chinese puzzle called Tangram. Play an interactive tangram game online at http://nlvm.usu.edu/en/nav/frames_asid_112_g_2_t_1.html or http://www.abcya.com/tangrams.htm

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Rectangles and Squares 1. Continue these patterns that use rectangles and squares. a.

b.

c.

d.

Make your own patterns here!

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Josh counted how many little squares were inside this rectangle. He got 12 little squares.

2. Now you do the same. Count how many little squares are inside each rectangle.

a.

b.

______ little squares

c.

______ little squares

______ little squares

3. Draw rectangles so they have a certain number of little squares inside. Guess and check!

a.

b.

10 little squares

15 little squares

c.

d.

8 little squares Can you make two different ones?

12 little squares Can you make two different ones?

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4. Here is a pattern where several squares are inside each other. Continue the pattern. Use pretty colors.

5. Design your own pattern, where you start with a small rectangle in the middle, then draw bigger ones around it like in the pattern above.

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Some Special Quadrilaterals Some Special Quadrilaterals

Rectangles look like book covers. The corners are “straight.”

Squares are actually rectangles where each side is the same length.

1. Draw three different rectangles and three different squares.

2. Draw three quadrilaterals that are NOT squares nor rectangles.

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A rhombus is a quadrilateral where each of the four sides has the same length. A rhombus is also called a diamond-shape or a diamond in common language. The corners of a rhombus don't have to be “straight” like the corners of a square. But they can be. The plural of rhombus is rhombi.

3. You can make a rhombus by taking four popsicle sticks or pencils or other sticks of the same length. Arrange the four sticks into a diamond shape. Now, change it slightly to get another rhombus. Make a skinny one, a less skinny one, and so on. You can even make a square! You can also play with rhombi on this web page. Choose “rhombus.” Just drag the dots! http://www.mathsisfun.com/geometry/quadrilaterals-interactive.html 4. A square or a rhombus?

a. _______________

b. _______________

c. _______________

5. Is a square also a rhombus? Read the definitions again: Squares are rectangles (with straight corners) where each side is the same length. A rhombus is a quadrilateral where each of the four sides has the same length. So, is a square also a rhombus? Why or why not?

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d. _______________

6. Color all the rectangles green, squares blue, rhombi red, and other quadrilaterals yellow. Or, choose your own colors.

7. This is a tiling with rhombi. Continue it! Use pretty colors.

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Geometric Patterns 1. The design below is often seen in Greek vases. Continue it.

2. This is a pattern from an apron used by Kirdi people in Cameroon, Africa. Notice it uses PARALLELOGRAMS that are inside each other. Continue the coloring in the pattern.

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3. This is a geometric design found on a Greek vase. a. What two shapes are used in this design?

_______________________________ and _________________________________ b. Copy the design at least once in the empty shapes.

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4. Repeat the patterns to fill the grids.

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Line Symmetry These figures are symmetrical in relation to the dashed line. The line is called a symmetry line. What does that mean? Imagine that you folded the figure along the symmetry line. Then both sides would exactly meet. Or, if you placed a mirror along the symmetry line, you would see the other half of the figure reflected in the mirror. Many figures are not symmetrical at all. You cannot draw a symmetry line in them.

1. Is the line drawn a symmetry line for the figure? You can cut out the images and fold them along the dashed line to check.

a.

c.

b.

d.

e.

g.

h.

f.

i.

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Some shapes you can fold in two different ways so that the sides meet. The cross-shape on the right has two different symmetry lines.

2. Draw as many different symmetry lines as you can into these shapes.

b.

a.

c.

d.

e.

f.

3. Write the capital letters in which you can draw a symmetry line. Draw the symmetry lines in them.

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4. Draw a mirror image in the symmetry line to get a symmetrical figure.

a.

d. Continue the pattern. Then draw its mirror image.

b.

c.

e. Draw your own design and find its mirror image.

f. Draw your own design and find its mirror image.

5. Examining logos. Look for logos on food products, cars, stores, magazines, and so on. Find at least three logos that have symmetry. Sketch them below. Answer the questions a. and b. for each logo. a. Does the logo employ a square, a rectangle, a triangle, a circle, or some other basic geometric figure in some way? b. Does it have any symmetry?

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Perimeter Perimeter means the “walk-around measure,” or the distance you go if you walk all the way around the figure. The word comes from the Greek word perimetros. In it, peri means 'around' and metros means 'measure'. To find the perimeter of this rectangle, count the units as you go around the figure. You can think of running or hopping around the figure. The units are marked with little arrows in the picture. The top side is four units long. The right side is two units long. Make sure you understand that! So, what is the perimeter? _______ units Here it is trickier to count those little units. Be careful! How many units is the perimeter? _______ units

1. Find the perimeter of these figures. Your answer will be so many units. P means perimeter. a.

b.

c.

P = _______________

P = _______________

d.

e.

f.

P = _______________

P = _______________

P = _______________

P=

units

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2. Measure with a ruler to find the perimeter of these figures in centimeters. a.

b.

P = ____________ cm

P = ____________ cm

c.

3. Measure with a ruler to find the perimeter of these figures in inches. a.

b.

P = ____________ in.

P = ____________ in. You can trace the ruler below and tape it on an existing ruler or cardboard! Or cut it out after you have finished the neighboring page.

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To find the perimeter, simply add all the side lengths. How many units is the perimeter of the triangle on the right? It is 8 + 9 + 10 units, or _______ units. Often you need to figure out some side lengths that are not given. What side lengths are not given? The perimeter is _______ cm. Don't forget the unit of measurement in your answer. If the side lengths are in centimeters, the perimeter will be so-many centimeters. If the side lengths are “plain numbers” without any particular unit, then the perimeter is so-many units. 4. Find the perimeter. Notice: some side lengths are not given! Don't forget to use either “cm” or “in.” or “units” in your answer. a.

b.

c.

P = ________________

P = ________________

d.

e.

f.

P = ________________

P = ________________

P = ________________

6

P=

units

5. Find the perimeter.... a. ...of a square with 7-in. sides b. ...of a square with 13-cm sides

50

Problems with Perimeter The perimeter of a rectangle is 30 cm. Its one side is 9 cm. How long is the other side? We can write a “how many more” addition, or an addition with an unknown:

?

9 + ? + 9 + ? = 30 You could guess and check to solve it. But, there is also an easier way. Just think: the two sides, 9 and ? , form half of the perimeter. So, 9 + ? = 15.

9 cm

9 cm ?

Thinking either way, we can solve that ? = 6 cm. 1. Solve. Write an addition with an unknown for each problem. a. The perimeter of this rectangle is 20 cm. Its one

6 cm

side is 6 cm. How long is the other side? ?

_____________________________________ Solution: ? = ____________ b. The perimeter of this rectangle is 44 cm. Its one

?

side is 15 cm. How long is the other side? _____________________________________

15 cm

Solution: ? = ____________

c. The one side of this rectangle is 12 in. Its

?

perimeter is 82 in. How long is the other side? _____________________________________

12 in.

Solution: ? = ____________ d. The perimeter of this square is 12 in.

?

How long is its side? _____________________________________ Solution: ? = ____________

51

?

2. Solve. a. The perimeter of this square is 44 cm.

How long is the side of the square? ?

b. Find the perimeter of this square with

12-inch sides.

c. Find the perimeter of this L-shape. Notice that some

side lengths are not given.

3. The parking lot of a school is in the shape shown here. Each little square in the image has a side of 10 feet. What is the perimeter of the parking lot?

4. Kyle's house measures 25 feet wide and 35 feet long. What is its perimeter?

5. Mandy wants a rectangular garden with a perimeter of 18 meters. One side of the garden is 3 m. How long should the other side be?

52

6. Draw many different rectangles that all have a perimeter of 24 units. Then, write the side lengths of those rectangles in the table. Hint: The two sides of the rectangle form half of the perimeter, which is 12 units.

One side

Other side

Perimeter

3 units

9 units

24 units 24 units 24 units 24 units

Draw a shape here that is not a rectangle, and that has a perimeter of a. 8 units

b. 10 units

c. 14 units

53

Getting Started with Area How many little squares do you need to cover this rectangle? That is its area. Area has to do with covering, and it is measured in little squares, which we call square units. The area of this rectangle is ______ square units. 1. How many square units is the area of these figures?

a. The area is ______ square units.

b. The area is ______ square units.

c. The area is ______ square units.

d. The area is ______ square units.

You can use multiplication to find the area of a rectangle. Notice how there are rows and columns of squares! There are 3 rows, and 8 columns. We multiply 3 × 8 = 24. The area of this rectangle is 24 square units.

2. Write a multiplication to find the area. “A” means area.

a.

b.

c.

____ × ____ = _______

____ × ____ = _______

____ × ____ = _______

A = ______ square units.

A = ______ square units.

A = ______ square units.

54

3. Find the areas of these figures.

a. The area is ______ square units.

b. The area is ______ square units.

c. The area is ______ square units.

d. The area is ______ square units.

4. Find the areas.

a.

b.

The area is ______ square units.

The area is ______ square units.

5. Draw two rectangles or squares with an area of 16 square units.

6. Draw two rectangles with an area of 24 square units.

55

More about Area To find the area of this figure, we can divide the shape into two rectangles. We then use two multiplications, and add their results. 3 × 2 + 3 × 5 = 6 + 15 = 21 square units

Here, can you think how to use multiplication and subtraction to find the shaded area? Don't look at the answer (below) yet! Think first! It is 4 × 5 − 2 × 2 = 20 − 4 = 16 square units

1. Write two multiplications to find the total area.

a.

b.

___ × ___ + ___ × ___ = ________

______________________________

c.

d.

______________________________

______________________________

56

The total area of this rectangle is 3 × 8 = 24 square units. But notice: we can write the longer side of the rectangle as a sum (3 + 5). Then, its area would be written as 3 × (3 + 5). But if we think of it as two rectangles, we can write the area as 3 × 3 + 3 × 5. 3 So, thinking of it as a one rectangle or two rectangles, we get:

3 × (3 + 5) = area of the whole rectangle

3×3 + area of the first part

3+5

3×5 area of the second part

2. Write a number sentence for the total area, thinking of one rectangle or two. a.

___ × ( ___ + ___ ) = ___ × ___ + ___ × ___ area of the whole rectangle

area of the first part

area of the second part

b.

___ × ( ___ + ___ ) = ___ × ___ + ___ × ___ area of the whole rectangle

area of the first part

area of the second part

c.

___ × ( ___ + ___ ) = ___ × ___ + ___ × ___ area of the whole rectangle

area of the first part

area of the second part

d.

___ × ( ___ + ___ ) = ___ × ___ + ___ × ___ e.

___ × ( ___ + ___ ) = ___ × ___ + ___ × ___

57

3. Now it's your turn to draw the rectangle. Fill in. a.

3 × (2 + 4) = ___ × ___ + ___ × ___ area of the whole rectangle

area of the first part

area of the second part

b.

5 × (1 + 4) = ___ × ___ + ___ × ___ area of the whole rectangle

area of the first part

area of the second part

c.

4 × (3 + 1) = ___ × ___ + ___ × ___ area of the whole rectangle

area of the first part

area of the second part

d.

___ × ( ___ + ___ ) = area of the whole rectangle

3×2 + area of the first part

3×1 area of the second part

e.

___ × ( ___ + ___ ) = area of the whole rectangle

2×5 + area of the first part

2×2 area of the second part

58

4. Find the areas of the figures. a. Find the shaded area. Write a number sentence for the area. __________________________________________________ __________________________________________________

b. Find the shaded area.

Think what operations you can use this time. Write a number sentence for the area. ______________________________________________ ______________________________________________ c. Find the shaded area (not including

the school). Write a number sentence for the area. _____________________________________ _____________________________________ _____________________________________

The area of this shape is 32 squares. Your task is to write a number sentence for the area.

59

Multiplying by Whole Tens 1. Fill in the missing parts of the multiplication table of 10. Think of counting by tens!

9 × 10 = _________

14 × 10 = _________

19 × 10 = _________

10 × 10 = _________

15 × 10 = _________

20 × 10 = _________

11 × 10 = _________

16 × 10 = _________

21 × 10 = _________

12 × 10 = _________

17 × 10 = _________

22 × 10 = _________

13 × 10 = _________

18 × 10 = _________

23 × 10 = _________

There is a pattern: Every answer ends in _____. Also, there is something special about the number you multiply times 10, and the answer. Can you see that?

SHORTCUT To multiply any number by ten, write the number, and tag one zero on it. For example:

78 × 10 = 780 or 10 × 49 = 490

2. Multiply. a.

10 × 11 = ________

b.

56 × 10 = ________

10 × 99 = ________

c.

18 × 10 = ________

82 × 10 = ________ 10 × 0 = ________

Note: If the number you multiply by 10 ends in zero, you still need to tag one zero on the answer. For example: 30 × 10 = 300 3. Multiply. a.

10 × 5 = ________ 10 × 50 = ________

b.

10 × 90 = ________ 100 × 9 = ________

60

c.

17 × 10 = ________ 17 × 1 = ________

This rectangle illustrates the multiplication 7 × 20. It has 7 rows and 20 columns. We could COUNT the little squares to find its area. Or, we could solve 7 × 20 by adding 20 repeatedly. But here is yet a different way to think about it: Let's divide this big rectangle into TWO smaller rectangles that each are the size 7 × 10. Each of the two rectangles has an area of 7 × 10 = 70. So, in total their area is 70 + 70 = 140.

4. Solve. a. Solve 8 × 20 by dividing this rectangle into TWO equal parts.

Parts: ____ × ______ and ____ × ______. The total area is __________.

b. Solve 5 × 30 by dividing this rectangle into THREE equal parts.

Parts: ____ × _____ and ____ × _____ and ____ × _____. The total area is_______.

c. Solve 7 × 30 by dividing this rectangle into THREE equal parts.

Parts: ____ × _____ and ____ × _____ and ____ × _____. The total area is_______.

d. Solve 4 × 40 by dividing this rectangle into parts.

Parts: __________________________________________. The total area is_______.

61

We can solve multiplication problems, such as 5 × 60, by repeated addition.

5 × 60 = 60 + 60 + 60 + 60 + 60 (60 added five times)

5. Solve these multiplications by repeated addition. But also look for a pattern and a shortcut. Can you find it? a.

3 × 40 = ________

b.

2 × 80 = ________

c.

4 × 40 = ________

d.

5 × 30 = ________

e.

5 × 70 = ________

f.

3 × 80 = ________

Here's another idea for solving multiplication problems, such as 5 × 60. Notice: 60 is equal to 6 × 10, isn't it? So, to solve 5 × 60, we can multiply 5 × 6 × 10. And 5 × 6 × 10 is the same as 30 × 10. Then, 30 × 10 is just 30 with a zero tagged on the end of it... or 300. 6. Break each multiplication into another where you multiply three numbers, one of them being 10. Multiply and fill in. a.

7 × 90 = =

c.

e.

b.

7 × 9 × 10 63

4 × 80 = ____ × ____ × 10

× 10 = __________

= ______ × 10 = __________

6 × 40

d.

9 × 90

= ____ × ____ × 10

= ____ × ____ × 10

= ______ × 10 = __________

= ______ × 10 = __________

30 × 6

f.

80 × 3

= 10 × ____ × ____

= 10 × ____ × ____

= 10 × ______ = __________

= 10 × ______ = __________

62

Study the shortcut for multiplying by whole tens. Example 1. 6 × 20

Example 2. 90 × 7

Multiply 6 × 2 = 12. Tag a zero to 12, to get 120.

Multiply 9 × 7 = 63. Tag a zero to 63, to get 630.

7. Multiply using the shortcut. a.

7 × 70 = ________

b.

6 × 80 = ________

c.

40 × 7 = ________

d.

50 × 4 = ________

e.

70 × 3 = ________

f.

3 × 90 = ________

8. This rectangle is 7 units high and 80 units long. What is its area?

9. This rectangle is divided into 8 equal parts. What is the area of each small part?

10. Find the total area of this rectangle, and also the area of each little part.

11. Find the total area.

Figure out a way or two ways to solve 5 × 16 without counting all the squares.

63

Area Units and Problems Area is always measured in squares of some size. To find the area of a shape, we check how many squares are needed to cover the shape. Each side of this square measures 1 centimeter. It is a special square. It is called a square centimeter. We can use it to measure areas of other shapes. We need 6 square centimeters to cover this rectangle. So, its area is just that: 6 square centimeters. We abbreviate this as 6 cm2. The elevated 2 indicates the “squaring.” We can also use multiplication to find the area:

3 cm × 2 cm = 6 cm2 1. Write a multiplication for the area of each rectangle. Measure the sides of the rectangles in centimeters using a ruler. Don't forget the units (cm and cm2)!

b. a.

A = ____ cm × ____ cm = _____ cm2

A = ____ cm × ____ cm = _____ cm2

c. d.

A = _________________________

A = _________________________

64

Each side of this square measures 1 inch. It is also a special square. It is called one square inch, abbreviated as 1 sq. in. or 1 in2. We can use it to measure areas of other shapes.

2. Find the area of each rectangle. Measure in inches using a ruler. Don't forget the unit for the area.

a.

b.

A = ____ in. × ____ in. = ______ in2

c.

A = ____ in. × ____ in. = ______ in2

A = ______________________________________

65

The following pictures are not to scale. They show some other square units for area.

This is one square foot or 1 ft2.

This is one square meter, or 1 m2.

We need 8 square inches to cover this rectangle. So, its area is 8 square inches. We abbreviate this as 8 sq. in. or 8 in2.

If no particular unit of length is given for the sides of a rectangle, we just use the word “unit.”

Again, use multiplication to find the area:

The sides are 7 and 4 units, and the area is 28 square units.

4 inches × 2 inches = 8 square inches

3. Find the areas of the rectangles. Be very careful about the unit you need to use, whether square centimeters (cm2), square meters (m2), square inches (in2), or square feet (ft2).

a.

b.

A = ________________________

A = ________________________

c.

d.

A = ________________________

A = ________________________

66

4. Find the area of this children's playground.

5. Find the area of Margaret's garden.

6. Danny's room measures 4 m by 4 m. His brother Joe's room is 5 m by 3 m. Whose room is bigger in area? How much bigger?

7. A notebook measures 6 in. by 8 in. On its cover is a white rectangle. The white rectangle is 3 in. by 2 in. How many square inches is the white rectangle?

How many square inches is the shaded (pink) area?

67

Area and Perimeter Problems Sometimes it's easy to confuse perimeter and area. 



AREA has to do with covering the shape with squares. Your answer will be in square centimeters, square inches, square feet, square meters, or just square units.

Area: 4 cm × 8 cm = 32 cm2.

PERIMETER has to do with “going all the way around.” Your answer will be in some unit of length, such as centimeters, meters, inches, or feet.

Perimeter: 4 cm + 8 cm + 4 cm + 8 cm = 24 cm

1. Find the area and perimeter of the rectangles.

a.

b.

Perimeter = ______________________

Perimeter = ______________________

Area = ______________________

Area = ______________________

c. 4 in. wide, 2 in. tall

d. A square with 3 cm sides

Perimeter = ______________________

Perimeter = ______________________

Area = __________________________

Area = __________________________

2. Find the area and perimeter of this shape. Notice that one side length is not given. You need to figure that out. Area Perimeter

68

3. Find the area and perimeter of this shape. Notice that one side length is not given. You need to figure that out. Area Perimeter

4. This is a two-part lawn. a. Find the areas of the two parts.

_____________ and __________________ b. Find the total area.

c. Find the perimeter.

5. Find the total area of this rectangle, and also the area of each little part. Area of each part: Total area:

Can you draw these rectangles? Guess and check!

a. Draw a rectangle with an

b. Draw a rectangle with an

area of 39 squares, and a perimeter of 32 units.

area of 56 squares, and a perimeter of 36 units.

69

More Area and Perimeter Problems 1. a. Find the area for each part. _____________ and __________________ b. Find the total area.

c. Find the perimeter.

2. Make rectangles that have an area of 24 square units. Draw them in the grid. Write in the table their side lengths. One is already given.

first side second side Rectangle 1

2 units

12 units

area 24 square units

Rectangle 2

24 square units

Rectangle 3

24 square units

3. For each rectangle you made in #2, calculate its perimeter.

Rectangle 1

one side

second side

area

2 units

12 units

24 square units

Rectangle 2

24 square units

Rectangle 3

24 square units

70

perimeter units

4. Make rectangles that have a perimeter of 20 units. Hint: the two different side lengths add up to half of the perimeter.

Draw them in the grid. Write in the table their side lengths. One is already given.

first side second side perimeter Rectangle 1

2 units

8 units

20 units

Rectangle 2

20 units

Rectangle 3

20 units

5. For each rectangle you made in #4, calculate its area.

first side second side perimeter Rectangle 1

2 units

8 units

20 units

Rectangle 2

20 units

Rectangle 3

20 units

6. The image illustrates Jane's garden. a. Find the area of each part.

_____________ and __________________ b. Find the total area.

c. Find the perimeter.

71

area square units

7. Draw and fill in. a. Write a number sentence using the area of this

two-part rectangle.

___ × ( ___ + ___ ) = ___ × ___ + ___ × ___

b. Draw a two-part rectangle to illustrate this number

sentence.

4 × (3 + 5) = 4 × 3 + 4 × 5

c. Fill in the missing parts, and then draw a two-part

rectangle to illustrate this number sentence.

2 × (5 + 2) = ___ × ___ + ___ × ___

d. Fill in the missing parts, and then draw a two-part rectangle

to illustrate this number sentence.

___ × ( ___ + ___ ) = 3 × 2 + 3 × 1

a. Write a number sentence using the area of this two-part rectangle.

___ × ( ___ + ___ ) = ___ × ___ + ___ × ___ b. Sketch a rectangle to match

20 × (3 + 7) and find its area.

72

Three-Dimensional Shapes

This is a box. It is also called a “rectangular prism.”

A cube is a box, too, but all of its sides are equally long.

A cylinder has a circle on the bottom and at the top.

A ball or a sphere.

1. Are these things in the shape of a box or a cube? Underline the right choice.

b.

a.

box or cube

box or cube

d.

c.

box or cube

box or cube

h.

e.

box or cube

g.

f.

box or cube box or cube

box or cube

2. Find four things in your classroom or at home in the shape of a box. Put them in order from the smallest to the biggest. I found __________________________, _______________________________, _____________________________, and _______________________________. 3. Find two things in your classroom or at home in the shape of a cube, one smaller and one bigger. I found __________________________ and _______________________________. 73

4. Are these things in the shape of a cylinder or a ball? Underline the right choice.

c.

b.

a.

cylinder or ball

cylinder or ball

cylinder or ball

cylinder or ball

cylinder or ball

f.

e.

d.

g.

cylinder or ball

h.

cylinder or ball

5. Which shapes can roll on the floor? Underline. cylinder 6. Which shapes will slide, and not roll on the floor?

box

cylinder

cylinder or ball ball

box

cube

ball

cube

7. Find four things in your classroom or at home in the shape of a ball. Put them in order from the smallest to the biggest. I found __________________________, _______________________________, _____________________________, and _______________________________. 8. Find four things in your classroom or at home in the shape of a cylinder. Put them in order from the smallest to the biggest. I found __________________________, _______________________________, _____________________________, and _______________________________. 9. Name the basic shape.

a.

c. b.

74

d.

Solids 1

This is a box. It is also called a “rectangular prism.”

A cube is a box, too, but all of its sides are equal in length.

A pyramid has a pointed top. Its bottom shape can be any many-sided figure, such as a triangle, a rectangle, a square, or a pentagon.

A cylinder has a This is a circle on the bottom sphere, or just a ball. and at the top.

A cone has a pointed top, as well, but it has a rounded shape on the bottom.

1. Make a cube, a cylinder, a cone, and a pyramid using the cut-outs provided on the following pages. Your teacher will help you. 2. A face is any of the flat sides of a solid. a. Count how many faces a cube has.

_________ faces

What shapes are they? b. Count how many faces a box has.

_________ faces

What shapes are they? c. Count how many faces this pyramid has.

_________ faces

What shapes are they? d. Count how many faces a ball has.

_________ faces

How about the cylinder? It has three faces: the top and bottom circles are two faces, and the third face is “wrapped around” it. And the cone? It has two faces.

75

3. You might have seen safety cones on the street. They are used to mark off areas where people are not supposed to go. Can you think of other things in real life that are in the shape of a cone, or a part of them is a cone? _____________________________________________________ _____________________________________________________ (Hint: One thing that is cone-shaped tastes really yummy!) (Hint: Another thing you might see in birthday parties.)

4. Label the pictures with box, cube, cylinder, pyramid, or cone.

a. _________________________

b. _________________________

d. _________________________

e.

j. _________________________

_________________________

f.

_________________________

g. _________________________

c.

_________________________

h.

i.

_________________________

k.

_________________________

l.

_________________________

76

_________________________

Solids 2 You can make paper models of these solids with the help of the printable cutouts provided (see introduction).

Solids are shapes that don't just exist on paper: you can fill them with something, such as water or stones. We say they are three-dimensional shapes.

A rectangular prism. We also call it a box. Its faces are rectangles.

A square pyramid: its base (bottom) is a square.

A cube: all of its sides are of the same length.

A rectangular pyramid has a rectangle as its base.

A cylinder

A cone

A pyramid with a triangle at the bottom is called a tetrahedron.

Let's also study the parts of solids: faces, edges, and vertices.

A face is a flat side with area. The other face of the cone is “wrapped around” it.

An edge is a boundary “line” for the face. For the cone, the marked edge is its only one!

A vertex (pl. vertices) is the same as corner. The cone only has one!

77

1. Name the shapes, and find how many Edges, Vertices, and Faces they have. Shape

Name

E

V

F

Shape

a.

b.

c.

d.

e.

f.

Name

E

V

F

2. Which of these shapes belongs where in the table? (Sometimes you have two right choices.) Try to do this exercise without checking back!

Name of Shape

78

Faces

Edges

Vertices

6

12

8

6

12

8

5

8

5

5

8

5

4

6

4

2

1

1

3

2

0

Review 1 1. Divide the shapes using one straight line. Divide the shape A into a triangle and a five-sided shape. Divide the shape B into a square and a rectangle. Divide the shape C into a four-sided shape and a triangle.

3. Join these dots carefully with lines, from 1 to 2 to 3 to 4 to 1. Use a ruler.

2. Color the triangles orange, the rectangles red, the squares blue, and the little circles light blue.

What shape do you get? Divide your shape into two triangles.

4. How many corners are in this shape? (We call it a pentagon.) Measure its sides in centimeters.

79

Review 2 1. Connect the dots. Use a ruler! What shape do you get? ______________________________ 2. Choose one corner of your shape. Now draw a line (with a ruler) from that corner to some other corner so that you will divide the shape into a triangle and a pentagon.

3. Draw in the grid a square that has 4 little squares inside.

4. Draw in the grid a rectangle that has 18 little squares inside.

80

5. What is this shape called? ______________________________

How many faces does it have? _______

What shape are the faces? ______________________________

6. Sarah put together these two triangles. What new shape did she get?

→←

7. Label the pictures as box, cylinder, pyramid, or cone.

a.

_________________________

b.

c.

_________________________

81

_________________________

Geometry Review 1. a. Find the rhombi among these figures.

b. Find quadrilaterals that are not

rectangles nor rhombi.

2. Draw a quadrilateral that is not a rectangle.

3. Fill in. a. Write a multiplication for

b. Draw a rectangle that has the

the area of this figure.

area shown by the multiplication.

___ units × ___ units = ____ square units

4 × 5 = 20 square units

4. Find the perimeter and area of this rectangle. Use a centimeter ruler. Area: Perimeter:

82

5. Find the area and perimeter of these figures. a.

b.

Area:

Area:

Perimeter:

Perimeter:

6. Write a multiplication and addition for the areas of these figures. a.

b.

A = ________________________________

A = ________________________________

7. Multiply using the shortcut. a.

7 × 70 = ________

b.

6 × 80 = ________

8. Find the total area of this rectangle, and the area of each part. Area of each part: Total area: 9. Draw and fill in. a. Fill in the missing parts, and then draw a two-part

rectangle to illustrate this number sentence.

3 × (5 + 1) = ___ × ___ + ___ × ___ b. Fill in the missing parts, and then draw a two-part rectangle

to illustrate this number sentence.

___ × ( ___ + ___ ) = 4 × 2 + 4 × 3

83

c.

40 × 7 = ________

Math Mammoth Early Geometry Answer Key Basic Shapes, p. 10 1.

5. a. The new shapes have 4 sides, and 4 corners. They are squares . b. The new shapes have 3 sides, and 3 corners. They are triangles.

2. a. 3 f. 4

b. 5 g. 4

c. 5 h. 0

d. 0 i. 4

3. a.

R

b.

c.

C

d.

e. 6 j. 7

c. The new shapes have 4 sides, and 4 corners. They are quadrilaterals Q

d. The new shapes have 3 sides, and 3 corners. They are triangles.

It is an oval.

e. The new shapes have 3 sides, and 3 corners. They are triangles.

and

4. a. You get a triangle. b. You again get a triangle, unless you draw the three dots so that they are “perfectly aligned,” so that joining them you just get a line.

Puzzle corner:

84

Playing with Shapes, p. 15 1.

9. a.

b.

2. c. One possibility:

3.

or the two rectangles side-by side

10.

4.

or this combination in other positions

5. 11.

12. 6.

7.

8.

or

13. Answers vary. For example:

or

14. Answers vary. For example:

85

Drawing Basic Shapes, p. 16 4. Answers vary since the student can choose the colors. For example:

1. a. triangle

b. square

c. rectangle

d. other four-sided shape

e. square

f. other four-sided shape

Triangles are blue. Circles are yellow. Squares are purple. Rectangles are green. Other four-sided shapes are red.

1.

2. Answers will vary. 3. What kind of shapes do you get now? triangles How many parts does each four-sided shape have now? 4 What kind of shapes are these parts? triangles

a.

b.

c.

d.

86

Practicing Basic Shapes and Patterns, p. 19 1. The student's coloring will vary. As long as they are not 3. colored the same, it does not matter what color they are.

2.

4. Answers will vary.

87

Shapes Review, p. 22 1. Check the student's pictures. It has 3 vertices. 2. Check the student's pictures. It has 4 vertices. 3. Check the student's work. 4. It has 5 vertices and 5 sides. 5. It has 6 vertices and 6 sides. 6. A circle has no vertices or straight sides.

7. 8.

9. a. quadrilaterals b. triangles c. 5-sided is a pentagon and 6-sided is a hexagon

Shapes, p. 25 1. Answers vary. For example:

2. a. a pentagon

b. a quadrilateral (a kite)

c. a pentagon

d. a hexagon

3. Answers can vary. These are example answers.

a.

b.

c.

d.

or a vertical line in the middle.

e.

g.

h.

e.

f.

i.

4. Answers vary.

a pentagon and a quadrilateral a.

A triangle and a hexagon b.

a quadrilateral and a pentagon c.

two pentagons d.

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5. a. rectangle b. quadrilateral c. octagon and a square a. b.

c.

Right Angles, p. 29 1. a. 3 angles; 0 right angles d. 3 angles; 1 right angle

b. 4 angles; 4 right angles e. 4 angles; 2 right angles

c. 5 angles; 0 right angles f. 4 angles; 4 right angles

2. Pictures vary. Check the students' pictures. 3. There are right angles in the top figures.

4. b and e 5. a. Right angles in the big shape: 4. Right angles in each part: 1. b. Right angles in the big shape: 0. Right angles in each part 0. c. Right angles in the big shape: 1. Right angles in each part 0. d. Right angles in the big shape: 1. Right angles in each part 1.

Surprises with Shapes, p. 31 1. A pentagon:

3. Answers vary as it is possible to draw these lines in many different ways. One example:

2. A triangle and a quadrilateral. You can draw the one line from corner to corner in many different ways; here is one example: 4. A hexagon:

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5. & 6. Check students' work.

10. A six-pointed star:

7. A five-pointed star: 11. A hexagon. 12. A hexagon:

8. A pentagon. 9. A pentagon:

Making Shapes, p. 33

1. a.

d.

b.

c.

e.

f.

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Rectangles and Squares, p. 36 1.

2. a. 4 little squares

a.

b.

c.

d.

b. 20 little squares

c. 16 little squares

3. a.

b.

c.

d.

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4.

Some Special Quadrilaterals, p. 39 1. Answers vary. Here are some examples:

2.

4. a. a square

b. a rhombus

c. a square

d. a rhombus

5. Yes, a square is a rhombus, because all of its four sides have the same length. 6.

6. The rectangles are c, g, j, and l. The squares are h, k, and m. The rhombi are b, d, f, and l. Other quadrilaterals are a, e, and n.

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7.

Geometric Patterns, p. 42 1.

4.

2.

3. a. Circles and squares b.

Line Symmetry, p. 45 1. a. no b. yes

2. a.

c. yes d. no

e. no

b.

f. no

g. no

h. yes

i. yes

c.

d.

e.

f.

and many more... any diameter of a circle (a line through the center point) is its symmetry line. 93

3. You can draw a vertical symmetry line to the letters A, H, I, M, O, T, U, V, W, X, and Y. You can draw a horizontal symmetry line to the letters B, C, D, E, H, I, K, O, and X.

4. a.

b.

c.

d.

Perimeter, p. 48 1. a. 14 units d. 12 units 2. a. 16 cm 3. a. 6 in.

b. 12 units e. 18 units b. 16 cm

c. 12 units f. 24 units

c. 12 cm + 5 cm + 13 cm = 30 cm

4. a. 24 units b. 48 units c. 3 in. d. 42 cm e. 24 cm f. 11 in. 5. a. 28 in.

b. 10 in.

To find the perimeter, simply add all the side lengths. How many units is the perimeter of the triangle on the right? It is 8 + 9 + 10 units, or 27 units. Often you need to figure out some side lengths that are not given. What side lengths are not given? The perimeter is 24 cm.

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b. 52 cm

Problems with Perimeter, p. 51 1. a. 6 + ? + 6 + ? = 20 or 6 + ? = 10. The unknown ? = 4 cm b. 15 + ? + 15 + ? = 44 or 15 + ? = 22. The unknown ? = 7 cm c. 12 + ? + 12 + ? = 82 or 12 + ? = 41. The unknown ? = 29 in. d. ? + ? + ? + ? = 12 or 4 × ? = 12. The unknown ? = 3 in. 2. a. ? + ? + ? + ? = 44 or 4 × ? = 44. The unknown ? = 11 cm. b. The perimeter is 48 in. c. P = 12 cm + 4 cm + 8 cm + 6 cm + 4 cm + 10 cm = 44 cm 3. Just counting the units in the picture, the perimeter is 18 units. Since each unit is 10 feet, we get 18 × 10 feet = 180 feet. Or, you can count by tens as you count the units for the perimeter. 4. 120 feet 5. 6 m 6. Answers vary. In each rectangle, the two side lengths should add up to 12 units (half of the perimeter).

One side Other side Perimeter 3 units

9 units

24 units

1 unit

11 units

24 units

2 units

10 units

24 units

4 units

8 units

24 units

5 units

7 units

24 units

6 units

6 units

24 units

Puzzle corner: Answers vary, for example: 8 units:

10 units:

14 units:

95

Getting Started with Area, p. 54 1. a. 8 square units

b. 13 square units c. 8 square units d. 12 square units

2. a. 2 × 5 = 10 A = 10 square units.

b. 3 × 3 = 9 A = 9 square units.

3. a. 15 square units b. 12 square units

c. 6 × 3 = 18 A = 18 square units.

c. 10 square units

d. 17 square units

4. a. 32 square units b. 31 square units 5. The rectangles can be 1 × 16, 2 × 8, or 4 × 4.

6. The rectangles can be 1 × 24, 2 × 12, 3 × 8, or 4 × 6.

More About Area, p. 56 1. a. 3 × 3 + 3 × 5 = 24 c. 3 × 5 + 2 × 3 = 21

b. 2 × 5 + 3 × 3 = 19 d. 4 × 5 + 2 × 4 = 28

2. a. 4 × (2 + 5) = 4 × 2 + 4 × 5 b. 4 × (4 + 2) = 4 × 4 + 4 × 2 c. 5 × (3 + 4) = 5 × 3 + 5 × 4 d. 3 × (4 + 2) = 3 × 4 + 3 × 2 e. 2 × (3 + 3) = 2 × 3 + 2 × 3

96

3. a. 3 × (2 + 4)

=

area of the whole rectangle

3×2

+

area of the first part

3×4 area of the second part

b. 5 × (1 + 4)

=

area of the whole rectangle

5×1

+

area of the first part

5×4 area of the second part

c. 4 × (3 + 1)

=

area of the whole rectangle

4×3

+

area of the first part

4×1 area of the second part

d. 3 × (2 + 1)

=

area of the whole rectangle

3×2

+

area of the first part

3×1 area of the second part

e. 2 × (5 + 2) area of the whole rectangle

=

2×5 area of the first part

+

2×2 area of the second part

4. a. 3 × 3 + 3 × 6 + 3 × 4 = 39 square units b. 6 × 8 − 3 × 3 = 39 square units c. 7 × 4 + 5 × 3 + 7 × 4 = 71 square units or 13 × 7 − 5 × 4 = 71 square units Puzzle corner. 3 × 4 + 4 × 6 − 4 × 1 = 32 squares

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Multiplying by Whole Tens, p. 60 1.

6.

9 × 10 = 90 10 × 10 = 100 11 × 10 = 110 12 × 10 = 120 13 × 10 = 130

14 × 10 = 140 15 × 10 = 150 16 × 10 = 160 17 × 10 = 170 18 × 10 = 180

19 × 10 = 190 20 × 10 = 200 21 × 10 = 210 22 × 10 = 220 23 × 10 = 230

There is a pattern: Every answer ends in 0. Also, there is something special about the number you multiply times 10, and the answer. Can you see that? You simply add a zero on the end of the number. 2. a. 110, 560 b. 990, 180 c. 820, 0 3. a. 50, 500 b. 900, 900 c. 170, 17 4. a. Parts: 8 × 10 and 8 × 10. The total area is 160. b. Parts: 5 × 10 and 5 × 10 and 5 × 10. The total area is 150. c. Parts: 7 × 10 and 7 × 10 and 7 × 10. The total area is 210. d. Parts: 4 × 10 and 4 × 10 and 4 × 10 and 4 × 10. The total area is 160. 5. a. 3 × 40 = 40 + 40 + 40 = 120 b. 2 × 80 = 80 + 80 = 160 c. 4 × 40 = 40 + 40 + 40 + 40 = 160 d. 5 × 30 = 30 + 30 + 30 + 30 + 30 = 150 e. 5 × 70 = 70 + 70 + 70 + 70 + 70 = 350 f. 3 × 80 = 80 + 80 + 80 = 240 Multiply the numbers, then tack on the zero.

a.

7 × 90 = 7 × 9 × 10 = 6 3 × 10 = 630

b. 4 × 80 = 4 × 8 × 10 = 32 × 10 = 320

c. 6 × 40 = 6 × 4 × 10 = 24 × 10 = 240

d. 9 × 90 = 9 × 9 × 10 = 81 × 10 = 810

e.

f.

30 × 6 = 10 × 3 × 6 = 10 × 18 = 180

7. a. 490 b. 480 d. 200 e. 210

80 × 3 = 10 × 8 × 3 = 10 × 24 = 240

c. 280 f. 270

8. The area is 7 × 80 = 560 square units. 9. 7 × 10 = 70 square units 10. The total area: 8 × 30 = 240 square units. Area of each part: 8 × 10 = 80 square units. 11. The rectangle is divided into thirds. Each third has the area of 7 × 40 = 280 square units. The total area is then 280 + 280 + 280 = 840 square units. Puzzle corner. Answers may vary. You can add 16 repeatedly: 16 + 16 + 16 + 16 + 16 = 80 squares. Or, you could divide the rectangle into two parts, each having the area of 5 × 8 = 40. Then the total area is 80 squares.

Area Units and Problems, p. 64 1. a. A = 2 cm × 4 cm = 8 cm2

3. a. A = 4 m × 3 m = 12 m2

b. A = 6 cm × 3 cm = 18 cm2

b. A = 5 ft × 6 ft = 30 ft2

c. A = 8 cm × 2 cm = 16 cm2

c. A = 12 cm × 4 cm = 48 cm2

d. A = 4 cm × 3 cm = 12 cm2

d. A = 8 in. × 7 in. = 56 in2

2. a. A = 3 in. × 3 in. = 9 in2 b. A = 2 in. × 4 in. = 8 in2 c. A = 5 in. × 1 in. = 5 in2

4. A = 11 m × 4 m + 4 m × 4 m = 60 m2 5. A = 4 ft × 6 ft + 12 ft × 6 ft = 96 ft2 6. Danny's room is 16 m2. Joe's room is 15 m2. Danny's room is bigger by one square meter. 7. The white rectangle has the area of 3 in. × 2 in. = 6 in2. The pink area is 6 in. × 8 in. − 3 in. × 2 in. = 42 in2.

98

Area and Perimeter Problems, p. 68 1. a. perimeter 14 m; area 10 m2

4. a. 5 m × 4 m = 20 m2 and 10 m × 4 m = 40 m2. b. 60 m2. c. 38 m

b. perimeter 24 ft; area 36 ft2 c. perimeter 12 in.; area 8 in2

5. Area of each little part is 6 m × 10 m = 60 m2.

d. perimeter 12 cm; area 9 cm2

The total area is 6 m × 60 m = 360 m2.

2. a. You can divide the shape into four 4 m by 4 m squares, each having the area of 16 m2. The area is then

Puzzle corner. a. 13 × 3 rectangle.

16 m2 +16 m2 + 16 m2 + 16 m2 = 64 m2. The perimeter is 40 m. 3. For the area, divide the shape into two rectangles. That can be done in two ways. You could get 11 cm × 4 cm + 4 cm × 8 cm = 76 cm2. or 4 cm × 12 cm + 7 cm × 4 cm = 76 cm2. b. a 14 × 4 rectangle.

The perimeter is 4 cm + 8 cm + 7 cm + 4 cm + 11 cm + 12 cm = 46 cm.

More Area and Perimeter Problems, p. 70 1. a. 20 m × 9 m = 180 m2 and 30 m × 9 m = 270 m2 b. 450 m2 c. 118 m 2. first side second side

area

Rectangle 1

2 units

12 units

24 square units

Rectangle 2

3 units

8 units

24 square units

Rectangle 3

4 units

6 units

24 square units

Rectangle 4

1 unit

24 units

24 square units

99

3. one side second side

area

perimeter

Rectangle 1 2 units

12 units

24 square units

28 units

Rectangle 2 3 units

8 units

24 square units

22 units

Rectangle 3 4 units

6 units

24 square units

20 units

Rectangle 4

24 units

24 square units

50 units

1 unit

4. first side second side perimeter Rectangle 1

2 units

8 units

20 units

Rectangle 2

3 units

7 units

20 units

Rectangle 3

4 units

6 units

20 units

Rectangle 4

5 units

5 units

20 units

Rectangle 5

1 unit

9 units

20 units

5. first side second side perimeter

area

Rectangle 1

2 units

8 units

20 units

16 square units

Rectangle 2

3 units

7 units

20 units

21 square units

Rectangle 3

4 units

6 units

20 units

24 square units

Rectangle 4

5 units

5 units

20 units

25 square units

Rectangle 5

1 unit

9 units

20 units

9 square units

6. a. 30 m × 9 m = 270 m2 and 30 m × 6 m = 180 m2. b. 450 m2. c. 90 m

100

7. a. 3 × (5 + 2) = 3 × 5 + 3 × 2 b. 4 × (3 + 5) = 4 × 3 + 4 × 5

c. 2 × (5 + 2) = 2 × 5 + 2 × 2 d. 3 × (2 + 1) = 3 × 2 + 3 × 1

Puzzle corner. a. 9 × (20 + 30) = 9 × 20 + 9 × 30 b. 10 × 20 = 200 m2

Three-Dimensional Shapes p. 73 1. a. box b. cube c. box d. cube e. box

f. box

g. cube

h. box

2. Answers will vary. Please check the student's work. 3. Answers will vary. Please check the student's work. 4. a. ball b. cylinder c. ball d. cylinder

e. cylinder

f. cylinder

5. cylinder, ball 6. box, cube 7. Answers will vary. Please check the student's work. 8. Answers will vary. Please check the student's work. 9. a. ball

b. cylinder

c. box

d. cylinder

101

g. ball h. cylinder

Solids 1, p. 75 1. The teacher will assist the student in making the shapes from the cut-outs. 2. a. 6 faces. They are squares. b. 6 faces. They are rectangles. c. 5 faces. The bottom face is a rectangle. The other four are triangles. d. Just one face! 3. Examples: Ice cream cones, party hats, decorations on towers, a funnel. 4. a. cylinder b. cylinder c. pyramid d. cube e. cone f. box g. cone h. pyramid i. pyramid j. pyramid k. box l. cylinder

Solids 2, p. 77 Shape

Name

E

V

F

a.

cube

12

8

c.

cone

1

e.

pyramid

8

2.

Name of Figure

Shape

Name

E

V

F

6 b.

cylinder

2

0

3

1

2 d.

rectangular prism

12

8

6

5

5 f.

tetrahedron

6

4

4

Faces

Edges

Vertices

cube

6

12

8

rectangular prism

6

12

8

square pyramid

5

8

5

rectangular pyramid

5

8

5

tetrahedron

4

6

4

cone

2

1

1

cylinder

3

2

0

102

Review 1, p. 79

1. 3.

c: Answers can vary. For example:

It is a quadrilateral (or, to be more precise, a parallelogram).

4. 5 corners.

2.

Review 2, p. 80 1. A hexagon

2. Answers vary. For example:

3.

4.

5. A cube. It has 6 faces. The faces are in the shape of a square. 6. She got a quadrilateral (to be exact, a parallelogram). 7. a. box b. pyramid

c. cone

103

Geometry Review, p. 82 1. a. A, B, F, H, J

b. C, E, I, K, L

2. Answers vary. Check students' answers. 3.

a.

b.

7 units × 2 units = 14 square units

4 × 5 = 20 square units

4. a. Area 35 cm2

b. perimeter 24 cm

5. a. Area 12 square units; perimeter 14 units 6. a. A = 3 × 2 + 3 × 4 = 18 square units 7. a. 490

b. 480

b. Area 11 square units; perimeter 24 units b. A = 2 × 2 + 3 × 4 = 16 square units

c. 280

8. Area of each part: 9 × 10 = 90 square units. Total area 9 × 40 = 360 square units. 9. a. 3 × (5 + 1) = 3 × 5 + 3 × 1

b. 4 × (2 + 3) = 4 × 2 + 4 × 3

104

Rectangular Prism Cut-out (A Box)

105

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106

Cube Cut-out

107

[This page is intentionally left blank.]

108

Cylinder Cut-out It might be easier to use a toilet paper roll as a model for a cylinder than to cut and glue/tape this cut-out together. However, putting this together will help the student to understand that the “body” of the cylinder is in the shape of a rectangle.

109

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110

Rectangular Pyramid Cut-out

111

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112

Square Pyramid Cut-out

113

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114

Tetrahedron Cut-out

115

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116

Cone Cut-out

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