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Math Mammoth Early Geometry

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    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 inwriting from the author.

    Copying permission: Permission IS granted for the teacher to reproduce this material to be used forstudents, not for commercial resale, by virtue of the purchase of this book. In other words, the teacherMAY make copies of the pages to be used for students. Permission is given to make electronic copies ofthe 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 ...................................................................... 29Surprises 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

     

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    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 outfrom 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 torecognize 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 existingshape into new ones.

    We also study some geometric patterns, right angles, have surprises with pentagons and hexagons, andmake shapes in a tangram-like game. These topics are to provide some fun while also letting childrenexplore 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, andfinding 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 arectangle by multiplying the side lengths, and to find the area of rectilinear figures by dividing them intorectangles and adding the areas.

    We also study the distributive property “in disguise.” This means using an area model to representa × (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 theother 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 thatstudents 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 areasor 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, andstudy their faces, edges, and vertices. You can make paper models for them from the printouts providedafter 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 themup, and then try to recreate the original shape.http://illuminations.nctm.org/ActivityDetail.aspx?ID=72 

    Shifting ShapesFigure out what shape it is when viewed through a small opening! Click on the “eye” button to see it inits 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 QuadrilateralsDrag 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. Youcan 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 intothe 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 paperhttp://tangrams.ca/fold-set 

    Online KaleidoscopeCreate 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. Optionsalso 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 byquizzes on three levels. In perimeter, level two, some side lengths are not given. In level three, youcalculate the perimeter of compound shapes. In area of rectangles, level 1 has just rectangles, and levels2 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 howchanging the perimeter of a rectangle affects its area. After the lesson, you will use an interactive ruler tomeasure 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 theyhave 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, areaof 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 modifythe quiz parameters to your liking, for example to omit the circle, or instead of solving for area, yousolve 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 wherenot 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 solidsSelect 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 offaces, edges, and vertices.http://illuminations.nctm.org/ActivityDetail.aspx?ID=70

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    2-D and 3-D ShapesLearn about different solids and see them rotate. http://www.bgfl.org/bgfl/custom/resources_ftp/client_ftp/ks2/maths/3d/index.htm 

    Identify solidsClick 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

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

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

    These are triangles. They have THREEcorners, and three sides.

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

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

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    2. Count how many corners each shape has. 

    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. 

    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 mathematicswe call them quadrilaterals.“Quadri” comes from quattuor, Latin for four.“Lateral” comes from lateralis, Latin for side.

     a.  _____ 

     

    b.  _____ c.  _____ 

     

    d.  _____ e.  _____

    f.  _____ g.  _____ 

     

    h.  _____ i.  _____  j.  _____

    4. a. Draw three dots anywhere in thisspace. Join them with lines.

    What shape do you get?

    b. Draw again three dots anywhere in thisspace, 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 aruler. 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 atriangle and a pentagon (five-sided shape). 

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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 theshapes page scaled in 140-150% and landscape, so that it prints on two sheets of paper. All the shapeswill then be much bigger.

    1. Make a big triangle with the fouryellow 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 piecesyou 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 thisin 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 afive-sided shape (a pentagon).

    13. Make your own five-sided shapes (pentagons) using anyshapes! 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 /

    other four-sided shape

    b.  triangle / square / rectangle /

    other four-sided shape

    c.  triangle / square / rectangle /other four-sided shape

    d.  triangle / square / rectangle /other four-sided shape

    e.  triangle / square / rectangle /other four-sided shape

    f.  triangle / square / rectangle /other four-sided shape

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

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

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

    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. 

    4. Choose a color for each shape, and color! 

    Triangles are _________________. Circles are _________________. 

    Squares are _________________. Rectangles are _________________. 

    Other four-sided shapes are _________________. 

    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.

      b.

    c.  

    d.

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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 colorthe squares usingto 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 thesame 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 havefour sides.

    Draw at least one more square andone more rectangle into the picture,the best you can.

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

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

    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. Connectthe 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. 

    8. Color all triangles yellow.Color all quadrilaterals green.Color all pentagons blue.Color all hexagons purple. 

    Or choose your own colorsfor 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 andsix-sided shapes called? 

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    Shapes

    1. Draw two pentagons here by drawingdots and connecting them with lines.Remember, your pentagons don'thave to look “regular” or nice. Youcan draw them to look “funny,” too,as long as they have five sides andfive vertices. 

    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 verticesand five sides, it is a pentagon.

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

    Seven-sided figure = heptagonEight-sided figure = octagon

     Nine-sided figure = nonagonTen-sided figure = decagon

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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. 

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

    a. ____________________________________ b.  ____________________________________

    c. ____________________________________ d. ____________________________________

    a. a square and a rectangle b. a triangle and a pentagon c. three rectangles

    d. two quadrilateralsthat are not rectangles

    e. two parts that areexactly the same shape

    f. four triangles

    g. four triangles h. a triangle and a pentagon i. four quadrilaterals

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

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

    a. b.

    c. d.

    a.  ________________________ b.  ________________________

    c.  ________________________ and ________________________

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    6. Design your own tilings here. 

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    Right Angles

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

    These look like corners, but in math we call them angles.

    Imagine sitting inside of each angle,

    and the walls going up around youin the shape of the “corner.” 

    In which angle would you havelots of space to sit? 

    In which angle would you haveonly a little space to sit? 

    Find two “square corners.”In mathematics we call them right angles.

     

    Sometimes we draw around line (an arc) insideof the angle to mark it. 

    Right angles aremarked this way. 

    Corners of books areexamples of right angles. 

    a. b.  c.

     _____ angles

     _____ right angles 

     _____ angles

     _____ right angles 

     _____ angles 

     _____ right angles 

    d.  e.  f. 

     _____ angles

     _____ right angles 

     _____ angles

     _____ right angles 

     _____ angles

     _____ right angles 

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

    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. a rectangle  b. a square  c. a triangle withone right angle 

    d. a triangle with

    no right angles 

    e. a quadrilateral with

    one right angle 

    f. a pentagon with

    one right angle 

    3. Continue this pretty pattern. Look   carefully. Where in the pattern (not

    in the grid) can you find right angles?

    a. 

     _____ right anglesin the big shape. 

     _____ right anglesin each part. 

    b. 

     _____ right anglesin the big shape. 

     _____ right anglesin each part. 

    c. 

     _____ right anglesin the big shape. 

     _____ right anglesin each part. 

    d. 

     _____ right anglesin the big shape. 

     _____ right anglesin each part. 

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    Surprises with Shapes

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

     _______________________________

    2. Draw a line from one corner tosome other corner. This dividesyour shape into two new shapes.What shapes are they? 

     _______________________________

    3. Draw more lines from a cornerto some corner so that the wholeshape 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 theopposite corner. Then repeat so thateach corner gets connected to theopposite corner. You need to drawthree lines to do that. 

    6. Decorate your shape now so that

    it becomes a SNOWFLAKE!ALL snowflakes have this basicshape. 

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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 order1 - 4 - 2 - 5 - 3 - 1. What shape isformed 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 theorder 1 - a - 2 - b - 3 - c - 1.What shape is formed now? 

     _______________________________ 

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    Making Shapes

    1. Cut out the shapes on the next page. What shapes can you use to make the given 

    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 calledTangram. Play an interactive tangram game online athttp://nlvm.usu.edu/en/nav/frames_asid_112_g_2_t_1.html orhttp://www.abcya.com/tangrams.htm 

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

      shapes? There may be several possiblesolutions. The figures below are smallerthan the ones you will cut out. 

    a.  b.

     

    c.  d.

      e.  f.

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    Rectangles and Squares

    1. Continue these patterns that use rectangles and squares. 

    Make your own patterns here! 

    a.  b. 

    c.  d. 

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

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

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

    a.

     ______ little squares 

    b.

     ______ little squares 

    c.

     ______ little squares 

    a.

    10 little squares 

    b.

    15 little squares 

    c.

    8 little squaresCan you make two different ones? 

    d.

    12 little squaresCan 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, thendraw bigger ones around it like in the pattern above. 

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    Some Special Quadrilaterals

    1. Draw three different rectangles and three different squares. 

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

    Some Special Quadrilaterals 

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

    Squares are actually rectangleswhere each side is the same length.

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    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 itslightly to get another rhombus. Make a skinny one, a less skinnyone, 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? 

    5. Is a square also a rhombus? Read the definitions again: 

    So, is a square also a rhombus?

    Why or why not? 

    A rhombus is a quadrilateral where each of the foursides has the same length. A rhombus is alsocalled a diamond-shape or a diamond in commonlanguage.

    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.

    a. _______________ b. _______________ c. _______________ d. _______________

    Squares are rectangles (with straight corners)where each side is the same length.

    A rhombus is a quadrilateral where each of the foursides has the same length.

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    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

    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.

    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 placeda mirror along the symmetry line, you would see theother half of the figure reflected in the mirror.

     

    Many figures are not symmetrical at all.You cannot draw a symmetry line in them.

     

    a.b. c.

     

    d. 

    e.  f. 

    g.  h.  i.

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

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

    Some shapes you can fold in two different waysso that the sides meet. The cross-shapeon the right has two different symmetry lines.

     

    a.  b.

    c. d.

    e.

    f.

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

    5. Examining logos. Look for logos on food products, cars, stores, magazines, and so on. Find at leastthree 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 geometricfigure in some way?

    b. Does it have any symmetry?

    a. b. c.

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

     

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

     

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

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    Perimeter

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

    Perimeter means the “walk-around measure,” or the distance you go if you walk allthe way around the figure.

    The word comes from the Greek word perimetros. In it, peri means 'around' and metrosmeans 'measure'.

    To find the perimeter of this rectangle, count the units as you goaround the figure. You can think of running or hopping aroundthe figure.

    The units are marked with little arrows in the picture. The top side isfour 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

    a.

     

    P = u n i t s  

    b.

     

    P = _______________  

    c.

     

    P = _______________  

    d.  

    P = _______________  

    e.

    P = _______________  

    f.

    P = _______________  

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

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

    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. 

    a.

    P = ____________ cm 

    b.

    P = ____________ cm 

    c.

     

    a.

    P = ____________ in. 

    b.

    P = ____________ in. 

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    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. 

    5. Find the perimeter.... 

    a.  ...of a square with 7-in. sides

    b.  ...of a square with 13-cm sides

    To find the perimeter, simplyadd all the side lengths. 

    How many units is the perimeterof the triangle on the right?

    It is 8 + 9 + 10 units, or _______ units.

    Often you need to figure out someside 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.

    P = u n i t s

    a.

      6 

    P = ________________

    b.

    P = ________________

    c.

    P = ________________

    d.

     

    P = ________________

    e.

    P = ________________

    f.

     

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    Problems with Perimeter

    1. Solve. Write an addition with an unknown for each problem. 

    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, oran addition with an unknown:

    9 + ? + 9 + ? = 30

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

    Thinking either way, we can solve that ? = 6 cm.

    9 cm

    ?

    ?

    9 cm

    a. The perimeter of this rectangle is 20 cm. Its oneside is 6 cm. How long is the other side?

     _____________________________________

    Solution: ? = ____________

    ?

    6 cm

    b. The perimeter of this rectangle is 44 cm. Its one  side is 15 cm. How long is the other side?

     _____________________________________

    Solution: ? = ____________

    15 cm

    ?

    c. The one side of this rectangle is 12 in. Its  perimeter is 82 in. How long is the other side?

     _____________________________________

    Solution: ? = ____________

    12 in.

    ?

    d. The perimeter of this square is 12 in.How long is its side?

     _____________________________________

    Solution: ? = ____________

    ?

    ?

     

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    2. Solve. 

    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? 

    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 someside lengths are not given.

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    6. Draw many different rectangles that all have a perimeter of 24 units. Then, write theside 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

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    Getting Started with Area

    1. How many square units is the area of these figures? 

    2. Write a multiplication to find the area. “A” means 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 measuredin little squares, which we call square units. 

    The area of this rectangle is ______ square units. 

    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. 

    a. 

     ____ × ____ = _______  

    A = ______ square units. 

    b. 

     ____ × ____ = _______  

    A = ______ square units. 

    c. 

     ____ × ____ = _______  

    A = ______ square units. 

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    3. Find the areas of these figures.

    4. Find the areas. 

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

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

    a. The area is ______ square units. b. The area is ______ square units.

      c. The area is ______ square units. d. The area is ______ square units.

     a. 

    The area is ______ square units. 

    b. 

    The area is ______ square units. 

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    More about Area

    1. Write two multiplications to find the total area. 

    To find the area of this figure, we can divide the shape into tworectangles. 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 findthe shaded area? Don't look at the answer (below) yet! Think first!

    It is 4 × 5 −  2 × 2 = 20 − 4 = 16 square units 

    a. 

     ___  × ___ + ___ × ___ = ________

    b. 

     ______________________________

    c. 

     ______________________________

    d. 

     ______________________________

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

    The total area of this rectangle is 3 × 8 = 24 square units. But notice: we can write the longerside 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.

    So, thinking of it as a one rectangle or tworectangles, we get: 

    3 × (3 + 5) = 3 × 3 + 3 × 5

    area of thewhole rectangle

     area of thefirst part

     area of thesecond part

    3

    3 + 5 

    a.

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

    area of thewhole rectangle

     area of thefirst part

     area of thesecond part

    b.

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

    area of thewhole rectangle

     area of thefirst part

     area of thesecond part

    c.

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

    area of thewhole rectangle

     area of thefirst part

     area of thesecond part

    d.

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

    e.

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

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

    a.

    3 × (2 + 4) = ___ × ___ + ___ × ___ 

    area of thewhole rectangle

     area of thefirst part

     area of thesecond part

     

    b.

    5 × (1 + 4) = ___ × ___ + ___ × ___ 

    area of thewhole rectangle

     area of thefirst part

     area of thesecond part

     

    c.

    4 × (3 + 1) = ___ × ___ + ___ × ___ 

    area of thewhole rectangle

     area of thefirst part

     area of thesecond part

     

    d.

     ___ × ( ___ + ___ ) = 3 × 2 + 3 × 1

    area of thewhole rectangle

     area of thefirst part

     area of thesecond part

     

    e.

     ___ × ( ___ + ___ ) = 2 × 5 + 2 × 2

    area of thewhole rectangle

     area of thefirst part

     area of thesecond part

     

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    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 includingthe school). Write a number sentencefor the area. 

     _____________________________________

     _____________________________________

     _____________________________________

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

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    Multiplying by Whole Tens

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

    2. Multiply. 

    3. Multiply. 

    9 × 10 = _________

    10 × 10 = _________

    11 × 10 = _________

    12 × 10 = _________

    13 × 10 = _________

    14 × 10 = _________

    15 × 10 = _________

    16 × 10 = _________

    17 × 10 = _________

    18 × 10 = _________

    19 × 10 = _________

    20 × 10 = _________

    21 × 10 = _________

    22 × 10 = _________

    23 × 10 = _________

    There is a pattern: Every answer ends in _____. Also, there is something specialabout 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 

    a.  10 × 11 = ________

    56 × 10 = ________

    b. 10 × 99 = ________

    18 × 10 = ________

    c. 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 

    a. 10 × 5 = ________

    10 × 50 = ________

    b. 10 × 90 = ________

    100 × 9 = ________

    c. 17 × 10 = ________

    17 × 1 = ________

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    4. Solve. 

    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 TWOsmaller 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. 

    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_______.

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

    6. Break each multiplication into another where you multiply three numbers, one of them being 10. Multiply and fill in. 

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

    5 × 60 = 60 + 60 + 60 + 60 + 60 

    (60 added five times) 

    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.

    a. 7 × 90

    = 7 × 9 × 10

    = 6 3 × 10 = __________

    b. 4 × 80

    = ____ × ____ × 10

    = ______ × 10 = __________

    c. 6 × 40

    = ____ × ____ × 10

    = ______ × 10 = __________

    d. 9 × 90

    = ____ × ____ × 10

    = ______ × 10 = __________

    e. 30 × 6

    = 10 × ____ × ____

    = 10 × ______ = __________

    f. 80 × 3

    = 10 × ____ × ____

    = 10 × ______ = __________

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    7. Multiply using the shortcut. 

    8. This rectangle is 7 units high and80 units long. What is its area? 

    9. This rectangle is divided into8 equal parts. What is the areaof each small part?

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

    11. Find the total area. 

    Study the shortcut for multiplying by whole tens.

    Example 1.  6 × 20

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

    Example 2.  90 × 7

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

    a. 7 × 70 = ________ b. 6 × 80 = ________ c. 40 × 7 = ________

    d. 50 × 4 = ________ e. 70 × 3 = ________ f. 3 × 90 = ________

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

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    Area Units and Problems

    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)! 

    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 iscalled 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 

    a.

    A = ____ cm × ____ cm = _____ cm2 

    b.  

    A = ____ cm × ____ cm = _____

    cm2 

    c.

    A = _________________________  

    d.

    A = _________________________  

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

    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. 

    a.

    A = ____ in. × ____ in. = ______ in2 

    b.

    A = ____ in. × ____ in. = ______ in2

     

    c. A = ______________________________________  

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    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).

    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 thisrectangle. So, its area is 8 square inches. We

    abbreviate this as 8 sq. in. or 8 in2. 

    Again, use multiplication to find the area: 

    4 inches × 2 inches = 8 square inches 

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

    The sides are 7 and 4 units, and the area is28 square units. 

    a.

    A = ________________________

    b.

    A = ________________________

    c.

    A = ________________________

    d.

    A = ________________________

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    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 isa 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? 

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    Area and Perimeter Problems

    1. Find the area and perimeter of the rectangles.

    Sometimes it's easy to confuse perimeter and area.

    AREA has to do with covering the shape withsquares. Your answer will be in squarecentimeters, square inches, square feet, squaremeters, or just square units.

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

    Area: 4 cm × 8 cm = 32 cm2.

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

    a. 

    Perimeter = ______________________

    Area = ______________________

    b. 

    Perimeter = ______________________

    Area = ______________________

    c.  4 in. wide, 2 in. tall

    Perimeter = ______________________

    Area = __________________________

    d.  A square with 3 cm sides

    Perimeter = ______________________

    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  

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

    Area of each part: 

    Total area: 

    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.

    Can you draw these rectangles? Guess and check!

     

    a. Draw a rectangle with an  area of 39 squares, and

    a perimeter of 32 units.

    b. Draw a rectangle with an  area of 56 squares, and

    a perimeter of 36 units.

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    More Area and Perimeter Problems

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

    1. a. Find the area for each part.

     _____________ and __________________

    b. Find the total area.

    c. Find the perimeter.

    2. Make rectangles that have anarea of 24 square units.

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

    first side second side area

    Rectangle 1 2 units 12 units 24 square units

    Rectangle 2 24 square units

    Rectangle 3 24 square units

      one side second side area perimeter

    Rectangle 1 2 units 12 units 24 square units units

    Rectangle 2 24 square units

    Rectangle 3 24 square units

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

    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

      first side second side perimeter area

    Rectangle 1 2 units 8 units 20 units square 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.

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    7. Draw and fill in. 

    a. Write a number sentence using the area of thistwo-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.

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    Three-Dimensional Shapes

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

    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 _______________________________. 

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

     

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

     

    A cylinder has acircle on the bottomand at the top.

     

    A ball or a sphere.

    a.  

     box or   cube 

    b.

     box or   cube 

    c.  

     box or  cube 

    d.  

     box or   cube 

    e.  

     box or   cube  f.

     box or   cube 

    g.  

     box or   cube 

    h.  

     box or   cube 

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

    5. Which shapes can roll on the floor? Underline. cylinder box ball cube 

    6. Which shapes will slide, and not roll on the floor? cylinder box 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.  

    cylinder or   ball 

    b.

    cylinder or   ball 

    c.  

    cylinder or   ball 

    d.  

    cylinder or   ball 

    e.  

    cylinder or   ball 

    f.

    cylinder or  ball 

    g.  

    cylinder or  ball 

    h.  

    cylinder or  ball 

    a.

    b.

    c.d.

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    Solids 1

    1. Make a cube, a cylinder, a cone, and a pyramid using the cut-outs provided on thefollowing 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. 

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

     

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

     

    A cylinder has acircle on the bottomand at the top.

     

    This is asphere, or 

     just a ball.

     

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

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

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

    3. You might have seen safety cones on the street. They are used tomark off areas where people are not supposed to go. Can you thinkof other things in real life that are in the shape of a cone, or a partof them is a cone? 

     _____________________________________________________  

     _____________________________________________________  

    (Hint: One thing that is cone-shaped tastes really yummy!)

    (Hint: Another thing you might see in birthday parties.) 

    a.

     _________________________

    b.  

     _________________________ 

    c.  

     _________________________  

    d.  

     _________________________

    e.  

     _________________________ 

    f.  

     _________________________  

    g.  

     _________________________

    h.  

     _________________________ 

    i.  

     _________________________  

     j.  

     _________________________

    k.  

     _________________________ 

    l.  

     _________________________ 

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    Solids 2You 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 cube: all of its sidesare of the same length.

     

    A cylinder  A cone 

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

     

    A rectangular pyramidhas a rectangle as its base.

     

    A pyramid with a triangleat the bottom is calleda 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! 

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    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. 

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

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

    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. 

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    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 othercorner so that you will divide theshape into a triangle and a pentagon. 

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

     

    4. Draw in the grid a rectangle that

    has 18 little squares inside.

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    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.

     _________________________ 

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    Geometry Review

    1. a. Find the rhombi among these figures.

    b. Find quadrilaterals that are notrectangles nor rhombi. 

    2. Draw a quadrilateral that is not a rectangle. 

    3. Fill in. 

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

    Area: 

    Perimeter: 

    a. Write a multiplication forthe area of this figure. 

     ___ units × ___ units = ____ square units 

    b. Draw a rectangle that has thearea shown by the multiplication. 

    4 × 5 = 20 square units 

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    5. Find the area and perimeter of these figures. 

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

    7. Multiply using the shortcut. 

    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.

     

    Area:

    Perimeter: 

    b.

     

    Area:

    Perimeter: 

    A = ________________________________  

    a.

     

    A = ________________________________  

    b.

    a. 7 × 70 = ________ b. 6 × 80 = ________ c. 40 × 7 = ________

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

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    Math Mammoth Early GeometryAnswer Key

    Basic Shapes, p. 10 

    1.

    2. a. 3 b. 5 c. 5 d. 0 e. 6f. 4 g. 4 h. 0 i. 4 j. 7

    3. a. R b. and Q

    c. C d. It is an oval. 

    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.

    5.

    Puzzle corner:

    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.

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

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

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

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    Playing with Shapes, p. 15 

    1.

    2.

    3. or the two rectangles side-by side

    4. or this combination in other positions

    5.

    6.

    7. or

    8. or

    9. a. b.

    c. One possibility:

    10.

    11.

    12.

    13. Answers vary. For example:

    14. Answers vary. For example:

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    Drawing Basic Shapes, p. 16 

    2. Answers will vary.

    3. What kind of shapes do you get now?trianglesHow many parts does each four-sidedshape have now? 4

    What kind of shapes are these parts?triangles

    a. b.

    c. d.

    1.

    a. triangle 

     b. square

     

    c. rectangle  d. other four-sided shape 

    1. 

    e. square  f. other four-sided shape 

    4. Answers vary since the student can choose the colors.  For example:

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

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    Practicing Basic Shapes and Patterns, p. 19 

    1. The student's coloring will vary. As long as they are not  colored the same, it does not matter what color they are.

     

    2.

    3. 

    4. Answers will vary.

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    Shapes Review, p. 22 

    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.

    g. h. i.

    4. Answers vary.

    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. trianglesc. 5-sided is a pentagon and 6-sided is a hexagon

      e.

    or a vertical line in the middle. e. f.

    a.

     a pentagon anda quadrilateral

     b.

    A triangle and  a hexagon

    c.

     a quadrilateral and a pentagon

    d.

    two pentagons

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

    c.

    Right Angles, p. 29 

    1. a. 3 angles; 0 right angles b. 4 angles; 4 right angles c. 5 angles; 0 right anglesd. 3 angles; 1 right angle e. 4 angles; 2 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 e5. 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:

    2. A triangle and a quadrilateral. You can draw theone line from corner to corner in many differentways; here is one example:

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

    4. A hexagon:

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    Making Shapes, p. 33 

    1. a. b. c.

    d. e. f.

    5. & 6. Check students' work.

    7. A five-pointed star:

    8. A pentagon.

    9. A pentagon:

    10. A six-pointed star:

    11. A hexagon.

    12. A hexagon:

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    Rectangles and Squares, p. 36 

    1.

    2. a. 4 little squares b. 20 little squares c. 16 little squares

    3. a. b.

    c. d.

    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 

    Line Symmetry, p. 45 

    1. a. no b. yes c. yes d. no e. no f. no g. no h. yes i. yes

    2. a. b. c. d.

    e.

    f.

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

    1.

    2.

    3. a. Circles and squares 

     b.

    4. 

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    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 b. 12 units c. 12 unitsd. 12 units e. 18 units f. 24 units 

    2. a. 16 cm b. 16 cm c. 12 cm + 5 cm + 13 cm = 30 cm 

    3. a. 6 in. b. 10 in. 

    To find the perimeter, simplyadd all the side lengths. 

    How many units is the perimeterof the triangle on the right?

    It is 8 + 9 + 10 units, or 27 units.

    Often you need to figure out someside lengths that are not given.

    What side lengths are not given?

    The perimeter is 24 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. 52 cm

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    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).

    Puzzle corner: Answers vary, for example:

    8 units: 10 units: 14 units:

    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

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    Getting Started with Area, p. 54 

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

    2.

    3. a. 15 square units b. 12 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 b. 2 × 5 + 3 × 3 = 19c. 3 × 5 + 2 × 3 = 21 d. 4 × 5 + 2 × 4 = 28

    a. 2 × 5 = 10A = 10 square units.

     b. 3 × 3 = 9A = 9 square units.

    c. 6 × 3 = 18A = 18 square units.

    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

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    3.

    4. a. 3 × 3 + 3 × 6 + 3 × 4 = 39 square units b. 6 × 8 −  3 × 3 = 39 square unitsc. 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

    a. 

    3 × (2 + 4) = 3 × 2 + 3 × 4

    area of thewhole

    rectangle 

    area ofthe

    first part 

    area of thesecond part  

     b. 5 × (1 + 4) = 5 × 1 + 5 × 4

    area of thewhole

    rectangle 

    area ofthe

    first part 

    area of thesecond part

    c. 

    4 × (3 + 1) = 4 × 3 + 4 × 1

    area of thewhole

    rectangle 

    area ofthe

    first part 

    area of thesecond part  

    d. 

    3 × (2 + 1) = 3 × 2 + 3 × 1

    area of thewhole

    rectangle 

    area ofthe

    first part 

    area of thesecond part  

    e. 

    2 × (5 + 2) = 2 × 5 + 2 × 2

    area of thewhole

    rectangle 

    area ofthe

    first part 

    area of thesecond part

     

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    Multiplying by Whole Tens, p. 60 

    Area Units and Problems, p. 64 

    1. 

    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 = 160c. 4 × 40 = 40 + 40 + 40 + 40 = 160d. 5 × 30 = 30 + 30 + 30 + 30 + 30 = 150e. 5 × 70 = 70 + 70 + 70 + 70 + 70 = 350f. 3 × 80 = 80 + 80 + 80 = 240Multiply the numbers, then tack on the zero. 

    9 × 10 = 9010 × 10 = 10011 × 10 = 11012 × 10 = 120

    13 × 10 = 130

    14 × 10 = 14015 × 10 = 15016 × 10 = 16017 × 10 = 170

    18 × 10 = 180

    19 × 10 = 19020 × 10 = 20021 × 10 = 21022 × 10 = 220

    23 × 10 = 230There is a pattern: Every answer ends in 0. Also, thereis something special about the number you multiplytimes 10, and the answer. Can you see that?You simply add a zero on the end of the number.

    6.

    7. a. 490 b. 480 c. 280

    d. 200 e. 210 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 hasthe 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 therectangle into two parts, each having the area of 5 × 8 = 40.

    Then the total area is 80 squares. 

    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. 30 × 6= 10 × 3 × 6= 10 × 18 = 180

    f. 80 × 3= 10 × 8 × 3= 10 × 24 = 240

    1. a. A = 2 cm × 4 cm = 8 cm2 

     b. A = 6 cm × 3 cm = 18 cm2 

    c. A = 8 cm × 2 cm = 16 cm2 

    d. A = 4 cm × 3 cm = 12 cm2 

    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 

    3. a. A = 4 m × 3 m = 12 m2

      b. A = 5 ft × 6 ft = 30 ft2 

    c. A = 12 cm × 4 cm = 48 cm2 

    d. A = 8 in. × 7 in. = 56 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 m

    2.

    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. 

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    Area and Perimeter Problems, p. 68 

    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.

    1. a. perimeter 14 m; area 10 m2 

     b. perimeter 24 ft; area 36 ft2 

    c. perimeter 12 in.; area 8 in2 

    d. perimeter 12 cm; area 9 cm2 

    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

    16 m2  +16 m

    2 + 16 m

    2 + 16 m

    2  = 64 m

    2.

    The perimeter is 40 m. 

    3. For the area, divide the shape into two rectangles. Thatcan 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.

    The perimeter is4 cm + 8 cm + 7 cm + 4 cm + 11 cm + 12 cm = 46 cm. 

    4. a. 5 m × 4 m = 20 m2 and 10 m × 4 m = 40 m

    2.

     b. 60 m2.

    c. 38 m 

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

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

    Puzzle corner. a. 13 × 3 rectangle. 

     b. a 14 × 4 rectangle. 

    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

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    7.

     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 g. ball h. 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

    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

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    Review 1, p. 79 

    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

    1.

    c: Answers can vary. For example:  

    2.

    3.

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

    4. 5 corners.

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    Geometry Review, p. 82 

    1. a. A, B, F, H, J b. C, E, I, K, L

    2. Answers vary. Check students' answers.

    3.

    4. a. Area 35 cm2  b. perimeter 24 cm 

    5. a. Area 12 square units; perimeter 14 units b. Area 11 square units; perimeter 24 units

    6. a. A = 3 × 2 + 3 × 4 = 18 square units b. A = 2 × 2 + 3 × 4 = 16 square units

    7. a. 490 b. 480 c. 280

    8. Area of each part: 9 × 10 = 90 square units. Total area 9 × 40 = 360 square units.

    9.

    a.

    7 units × 2 units = 14 square units

     b.

    4 × 5 = 20 square units

    a.

    3 × (5 + 1) = 3 × 5 + 3 × 1

     b.

    4 × (2 + 3) = 4 × 2 + 4 × 3

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    Rectangular Prism Cut-out (A Box)

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    Cube Cut-out

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    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-outtogether. However, putting this together will help the student to understand that the “body” of thecylinder is in the shape of a rectangle.

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    Rectangular Pyramid Cut-out

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    Square Pyramid Cut-out

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    Tetrahedron Cut-out

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