Shapes are everywhere! Look around your classroom. How many shapes can you see on your desk? How many can you see at the smartboard or chalkboard? Are there shapes on the walls and ceiling? What are some of the shapes you see?
You might find it easy to spot circles, squares, rectangles, and triangles. And you’re sure to find parallel lines, perpendicular lines, and angles. Are there any other shapes in your classroom that you don’t know the names for yet?
INVESTIGATE!
Are there more polygons or polyhedrons in your classroom?
polygon: a shape with three or more straight sides and angles.
three-dimensional (3-D): something that appears solid and can be measured in three directions.
polyhedron: a 3-D shape.
line segment: two connected points that create a line.
POLYGONS PRETTY MUCH EVERYWHERE
Your classroom is a good place to find shapes and patterns. It’s also a good place to learn about polygons and three-dimensional (3-D) shapes called polyhedrons!
Any shape made up of at least three line segments, or line pieces, is called a polygon. The word poly means “many” in the ancient Greek language. When a shape is made of many lines, it is called a polygon!
There are a couple of rules for polygons. First, they have to have at least three lines. Second, all the lines need to connect. Third, none of the lines can be curved. Think about the triangles we discussed in the last chapter. Are they polygons? Do they follow the three rules? How about squares and rectangles?
Type of Polygon |
Number of Sides |
Triangle |
3 |
Quadrilateral |
4 |
Pentagon |
5 |
Hexagon |
6 |
Heptagon |
7 |
Octagon |
8 |
Nonagon |
9 |
Decagon |
10 |
complex: having many parts. The opposite of simple.
rhombus: a shape with four equal sides that has opposite equal acute and obtuse angles and opposite equal parallel sides.
trapezoid: a quadrilateral with a pair of parallel sides.
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DID YOU KNOW? Is a star a polygon? Yes! It is called a complex polygon. It is also called a pentagram. |
Check out the chart to see the many kinds of polygons. You might have heard of a few of them already!
Look at the chart as a shape. How many sides does it have? What kind of polygon is it? A rectangle is a quadrilateral because it has four sides. Try to find other examples of quadrilaterals in your classroom.
Squares and rectangles are both examples of quadrilaterals. Another example of a quadrilateral is a shape called a rhombus. A rhombus is in the shape of a kite. It also looks similar to a diamond shape. A trapezoid is another kind of quadrilateral.
FIND THE WORD CLUES
A tricycle has three wheels. A triangle has three sides. A triceratops has three horns. What do you think the word “tri” means? Three! You can find clues about what a word means from the parts of the word. Quad means “four” and pent means “five.” What are some other clues you can find in the words for different polygons?
parallelogram: a four-sided shape with opposite parallel sides.
vertices: the corners where the lines of a polygon meet.
two-dimensional (2-D): something that appears flat and can only be measured in two directions.
width: the measure of something from one end to the other, or how wide something is.
depth: how deep something is, the measurement that gives a shape 3-D qualities.
height: the measure of how tall an object is.
Squares, rectangles, and rhombuses are all examples of quadrilaterals and parallelograms. They are quadrilaterals because they are shapes with four sides. They are parallelograms because their opposite sides are parallel.
All of these shapes have four sides. The line segments connect at the vertices, or corners, to make the shape. The four-sided shapes are in many objects you see every day.
POLYGONS IN 3-D!
We know paper as flat rectangles or squares. A piece of paper looks like a polygon! You can cut it into all sorts of shapes and sizes. But paper is flat. It is two-dimensional, or 2-D. In fact, all 2-D objects are flat. They are good to draw and cut, though! Circles, triangles, and pentagons are all 2-D. These are shapes that are made with lines and have length and width. But they do not have depth or height—they’re flat.
origami: the ancient art of paper-folding.
The ancient Japanese art of folding paper into shapes is called origami. Origami is the process of folding paper to make a 3-D shape out of a 2-D shape. Origami uses just one square of paper for each finished shape. That one sheet of paper can be folded into thousands of different objects. Many geometric shapes emerge when the paper is folded over and over again. A blue piece of paper can turn into a beautiful crane or a sturdy boat. Shapes that are 3-D are called polyhedrons.
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DID YOU KNOW? A regular piece of paper cannot be folded more than seven times! The thickness of each fold requires lots of energy for the next fold. When it is folded seven times, it becomes very hard and crumbles! |
If a shape has depth, it is 3-D. You can draw 3-D shapes if you add extra lines to make the shapes look like something you can hold in your hand. The extra lines make the shapes have depth.
Shapes that are 3-D almost look like they could pop off the paper! These are also good shapes for making models of out of playdough, cut paper, or cardboard.
cube: a 3-D shape with six faces, each with four sides, like a square.
face: the flat side of a 3-D shape.
edge: the line where two faces come together.
tetrahedron: a 3-D shape with four faces, each with three sides, like a triangle.
A polyhedron that might be familiar to you is found in freezers—ice cubes! A cube has six faces. Each face has four sides, like a square. A cube has 12 edges along those sides, where the faces meet. It also has eight vertices, or corners.
A polyhedron that has fewer faces, sides, and vertices than a cube is the tetrahedron. Each tetrahedron has four faces shaped like triangles, with three sides each. This shape also has four vertices and six edges. You can see a tetrahedron on the next page.
Your desk is a large, flat rectangle. But look at the side of the tabletop. It isn’t actually flat like a piece of paper, is it? The tabletop has depth. While the top of the table’s flat surface may be 2-D, the tabletop itself is 3-D.
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DID YOU KNOW? A tetrahedron is a 3-D shape made of four 2-D triangles. |
GREAT PYRAMIDS!
Pyramids have a square base and four sides that look like triangles. Ancient Egyptians are well known for building pyramids. Pyramids were also built in Peru and other areas of the world. In Egypt, the pyramids were used as tombs! The leader, known as a pharaoh, was buried in the pyramid when he died. More than 130 pyramids have been found in Egypt. A boy king was discovered in one of them. King Tutankhamun was a pharaoh when he was only nine years old!
prism: a solid shape whose two end faces are similar, equal, and parallel and whose sides are parallelograms.
If the whole tabletop is 3-D, it isn’t really a rectangle. Yes, the surface is a rectangle. But the whole top is a rectangular prism. This is the name for a 3-D rectangle. The “prism” part adds depth to the object.
Type of Polyhedron |
Number of Vertices |
Number of Edges |
Number of Faces |
|
Tetrahedron |
4 |
6 |
4 |
|
Cube |
8 |
12 |
6 |
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Octahedron |
6 |
12 |
8 |
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Dodecahedron |
20 |
30 |
12 |
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grid: a network of evenly spaced horizontal and perpendicular lines.
dilation: the motion of something getting larger or smaller.
rotate: to turn like a wheel around a fixed point.
What is a 3-D triangle called? That is a triangular prism! Think of a slice of pizza. The thick crust and layers of sauce and toppings give the triangle depth, turning it into a 3-D shape. That’s a 3-D snack!
TRANSFORMERS!
No matter what shape an object is, it can always transform. A transformation is the way a shape can move along a grid or graph. These movements include translation, which is when a shape slides over to the left or right. There is also reflection, which is when a shape has a mirror image. Dilation is when shapes get larger or smaller. Have you noticed that your pupil, the black center of your eye, gets larger when the lights are dim? That’s dilation!
Lastly, shapes can rotate. Rotation is when shapes move around in a circle. These turns make it move around, like a carousel or ferris wheel.
Look around your classroom. From paper to desks to chairs to smartboards and blackboards, geometry can be found in your classroom! What other shapes and patterns can you find around your desk?
CONSIDER AND DISCUSS
It’s time to consider and discuss: Are there more polygons or polyhedrons in your classroom?
POLYGON PILE-UP
SUPPLIES
* paper
* pencil
* objects around the classroom
* ruler or stencil
* scissor
* glue or tape
Polygons are found in the classroom, but we can cut polygons out of paper, too. Polygons might seem like basic circles, rectangles, and squares, but the more sides you add, the more complex the polygons become! What if you join two or more polygons together to make a new shape or a design? Take a look below!
1Take a piece of paper and trace a shape. Look at the polygon chart at the beginning of Chapter 2. Pick a polygon! Or two! Or five!
2You can also use an object in your classroom to trace shapes. For example, use a rectangular eraser to trace a small rectangle. What can you use to make a circle? A triangle? Use scissors to cut the traced or drawn shapes.
3Take all of the polygons and make a design. Or a robot! Many things are made up of many different shapes. What can you create from your polygons?
4Tape or glue your shapes together. Fold your design. Is there symmetry? If not, you can add more shapes to make the design symmetrical.
TRY THIS! You can create everyday items with the polygons you cut. But can you create an invention? Design something with your shapes that has never been made before. Give your invention a name and write about what it can do. Present it to the class!
SUPPLIES
* feet, legs, hands and arms
* partner (or 2 or 3!)
You don’t have to find dice to see what a cube looks like. You don’t have to hunt for a tetrahedron either. You can make these 3-D shapes with a few friends, using arms and legs. And suddenly you unite and become shapeshifters!
1First, make a two-person cube. With a partner, sit on the floor. Face one another. Place your legs out in front, straight on the floor. Now, spread your legs like a “V.”
2Connect your feet with your partner’s feet. You made a square! That is just one face of the cube though.
3Next, open your arms. They are parallel to your legs. Touch your fingertips to your partner’s fingertips. A two-person cube!
4Ask two more friends to join in. One sits where you and your partners’ feet meet on one side, and the other sits where you and your partner’s feet meet on the other side. They can extend their legs in a “V” shape as well. Their arms will open too. Their fingertips meet yours and then your partner’s fingertips. A four-person cube!
5With two people, you can make a tetrahedron. Each person raises their right arm. Lean in and touch fingertips. It almost looks like a tent.
6Stretch your left arm into the armpit of the person next you. Don’t tickle them! You’ve made a tetrahedron with four triangular faces.
TRY THIS! If you have four people or more, there are many shapes that can be made. One person can act as the photographer and take a picture of the shapes your group is making. Another person can be the director who guides you to make an interesting shape or pattern. You can plan to make a letter. You may even want to spell a word! Think of the letter “M.” Consider how many legs it has: four. Two people can lay down, making parallel lines for the left and right sides of the “M.” Then two more people can form a “V” in the center of the two parallel lines to complete the “M.” What words can you spell?
SUPPLIES
* sheets of paper
Shapes make up everyday objects. We can cut shapes from paper, but also make new shapes through folding. Let’s focus on rectangles. With some flips and folds, we will have something to place on our heads!
2Fold the top down to the bottom edge. Fold the top corners down to the line that runs down the center.
4Turn the paper over. Do step 3 again. Open the hat to shape it. You are done!
TRY THIS! A hat takes only a few steps to complete. How about a frog?
2Fold both top corners to the opposite edges of the paper. See where the diagonal creases meet in the middle? Fold the paper backward and then open it.
4Fold the top triangles up to the tippy top point. Fold the sides to the line in the center.
6Turn over the frog! Draw on some eyes. To make it jump, press on the lower back.
SUPPLIES
* paper
* cube template
* markers and crayons
* scissors
* glue or tape
Cubes aren’t just for plopping into a cup of juice or water to make it cold! Cubes are also fun to draw on. How? Find out below.
1Go to this website to print the cube template and cut it out of the paper. http://babbledabbledo.com/wp-content/uploads/2015/02/Doodle-Cube-Template-.pdf
KEYWORD PROMPTS
doodle cube template 
2Decorate the cube with shapes and lines. Try to do continuous designs. Those are ones you can see on more than one face of the cube.
3Cut out the cube. Fold it along the lines shown in the template. Tape or glue the sides together along the edges.
4Make several cubes and create a portrait of someone you know. Your final artwork might not look exactly like the person, but does it represent certain parts of their personality?
TRY THIS! Team up with classmates who have cubes. Build an object or a tower with many cubes. What colors and designs do you all have in common? What is different? Get ideas from seeing others’ cubes. You might add to your cube, or design it further. You can use other arts and crafts, such as glitter glue, stickers, and sequins.
Cubism is a type of art that was popular more than 100 years ago. Artists sketched and painted many shapes and cubes in their pictures. This type of art was abstract and modern. It was new and many people didn’t like it! Pablo Picasso and Paul Cézanne were two artists who introduced shapes and cubes into their art. From then on, the art world changed shape!
You can see paintings in the cubism style at this website.
KEYWORD PROMPTS
Tate cubism
SUPPLIES
* square- or rectangular-shaped paper
* scissors
* pencils, markers, or crayons
Sentences, words, and letters all follow patterns. This is especially true for rhymes and syllables. Poems have patterns based on their words and the way they are written. What if you made a poem about a shape, and then the paper was in the shape of the shape that you wrote about? That almost sounds like a tongue-twister!
1Decide on a polygon shape you want your poem to be in. Cut your paper into that shape.
2On a piece of scratch paper, work on a poem about the shape that you chose. You can try to make your poem rhyme or simply write a free verse, which is a poem that doesn’t rhyme.
Diamond Pointy, four-sided Sparkling, digging, wishing Ring, star, window, clown eyes Winking, spinning, waiting Transparent, sharp Diamond
3When you are happy with your poem, write your final draft on your shape. Your poem will reflect the shape of the paper it’s written on!
TRY THIS! Try the poetry writing on a polygon, but then fold your polygon-shaped poem into an origami shape! Use the hat or frog origami instructions on The Stop, Drop and Fold project pages, or research a new animal or object to fold.
SUPPLIES
* objects in the classroom, such as a ruler, piece of paper, book, lunchbox, crayon box, and more
* pencil
* math journal
Get ready to explore! You do not need a magnifying glass or even a telescope. You will be exploring your classroom. There is plenty to discover!
1Gather objects in the classroom or from your desk or backpack. Line up your objects on your desk.
2In your math journal, make a chart like the one below.
Draw the object |
Rotate one quarter |
Rotate one half |
Rotate three quarters |
Rotate all the way (full circle) |
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3Draw an object in the left box. Then, rotate it one-quarter of the way around. That is like when the clock has the second hand on the 15 (one quarter past the hour). Draw what the object looks like.
4Continue with the rotations and draw what each object looks like. What happens with each shape when it is turned one quarter? One half? Three quarters?
TRY THIS! Draw each object and then draw its reflection. Draw an object that has symmetry. For example, draw a butterfly. Now fold the paper in half. Is one half the reflection of the other half? Improve the drawing so it is an exact reflection.
