Did you know you have been doing math since you were born? Geometry can be found everywhere in our world and we begin noticing it even as babies! Geometry was there in the bars of your crib, the round bowl of mushed peas you threw on the kitchen floor, and even within the space taken up by the liquid in your cup.
WORDS TO KNOW
geometry: the math related to shapes, surfaces, points, lines, and solids.
point: a spot in space or on a line.
lines: many points in a row that make one length.
length: the measure of something from one end to the other, or how long something is.
mathematics: the study of ideas related to numbers. Mathematicians study mathematics.
Geometry is a type of mathematics that we find all around us. In fact, geometry shows us the size of the earth and the size or amount of everything that’s on the earth. Wherever you are right now as you read this, you are surrounded by geometry!
The word “geometry” has four syllables. Try saying it: Ge-om-et-ry. Let’s take a look at all the parts of this word. Ge is an ancient Greek word meaning “Earth.” The syllables “-metry” are an ending, or suffix. It shows that the word “geometry” has to do with a system. It comes from the ancient Greek word metria. That word is similar to the word “measure,” which is one of the definitions of metria. When you measure something, you are figuring out how big something is, how much space it takes up, or how much it weighs.
Look across the room at an object. What do you see? How far away do you think it could be? Think about that object’s shape. What is the size? These questions and thoughts are about geometry. You see—you are a mathematician, a person who studies math! Let’s learn how geometry was part of our past.
GEOMETRY AND HISTORY
People started using geometry when our human ancestors went to work. They did not get in their cars and drive to the office or go to the store to take care of customers as your parents might do today.
Ancient humans had very different lives. They made tools for hunting. They made tools for cooking meals. Ancient humans built boats and homes.
tally: to count the number of something.
scale: a tool used to weigh objects.
angle: the space between two lines that start from the same point, measured in degrees.
pattern: a series of repetitive connections and designs.
wavelength: the distance between two waves.
BCE: put after a date, BCE stands for Before Common Era and counts years down to zero. CE stands for Common Era and counts years up from zero. This book was published in 2017 CE.
solar calendar: a yearly calendar based on how long it takes the earth to move around the sun.
pyramid: a shape with a square base and triangles for sides that meet at a point.
square: a shape with four equal sides and four right angles.
triangle: a shape with three sides.
Measuring and building were happening thousands and thousands of years ago. Our ancient relatives counted and tallied. They made scales and used weights. All this work was related to geometry!
This was true about their art, too. Art is full of curves and lines. Ancient people carved shapes, curves, and angles in caves, in rock, and on trees. And what about music? There are patterns in sounds as well, and in the wavelengths that each sound makes invisibly in the air. All this geometry was happening around the world, in the Americas, Australia, Asia, the Middle East, and Africa.
In 4200 BCE, ancient Egyptians created a solar calendar. To do this, they looked at the stars and made measurements and found patterns.
The ancient Egyptians also did some large construction projects in the years after this calendar was made. This is when they built the ancient pyramids! That is some BIG geometry.
Mesopotamia: an area of ancient civilization between the Tigris and Euphrates Rivers in what is now called Iraq, Kuwait, and Syria.
chariot: a small cart with two wheels and a platform, pulled by horses.
right triangle: a triangle with a right angle.
rectangle: a shape with four sides and four right angles. The sides across from each other are equal.
right angle: an angle that measures exactly 90 degrees, as at the corner of a square or rectangle.
degree: a unit of measurement of angles.
Pyramids are enormous in size, but the invention of the wheel was just as important! In Mesopotamia, the wheel came about in 3500 BCE, but not for transportation. It was used as a potter’s wheel to make round bowls and other containers. Eventually, wheels were stuck on a chariot to be ridden into battle!
The ancient Babylonians in Mesopotamia also worked with triangles called right triangles. They used rectangles and squares in their construction projects. These shapes all have right angles. Patterns, angles, and shapes were in their plans for cities.
In 2500 BCE, we can find geometry in India being used to plan big cities with streets and buildings. The Harrapan people, who lived in the Indus River Valley, made pottery with shapes and lines carved into the clay.
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DID YOU KNOW? The Babylonians were able to track Jupiter across the sky and make detailed records of the planets. |
culture: a group of people and their beliefs and way of life.
circa (c.): around that year.
Way across the Atlantic Ocean, in the region that is now Mexico and Central America, more people were using geometry. In about 1400 BCE, the ancient Olmec people made well-planned maps for cities and gardens. They also created a calendar.
In China in 550 BCE, scholars wrote a math book called the Chou Pei Suan Ching. This book discussed many geometry concepts, including right angles.
By 300 BCE, a mathematician named Euclid (c. 300 BCE) of Greece had finished a geometry textbook that we still consider useful today. Euclid came from Alexandria in Egypt. He studied math for many years and wrote his book, The Elements.
Geometry ideas came about in many cultures. They also spread from culture to culture as people moved around. Many cultures used geometry even before they officially called it geometry. They used shapes and angles to make plans for their cities, to build their homes, to measure time and the movements of planets, and to create art.
HYPATIA OF ALEXANDRIA
Hypatia of Alexandria (c. 350–415) was the first woman to teach geometry. She learned from her father, Theon of Alexandria, when they wrote about a new version of Euclid’s book. Hypatia became head of the Platonist school at Alexandria, where she taught mathematics and science. She is the first woman in history recognized as a professional mathematician.
numerical value: a term to show numbers and their worth.
rhythm: a pattern of beats.
harmony: a pleasing blend of sounds.
PATTERNS IN OUR WORLD
We study geometry today for many of the same reasons the ancient civilizations did. In nature, there are numerical values all around us. Think about a garden of flowers. Each flower has a certain number of petals. Seeds in sunflowers follow a pattern. At the zoo, you might notice patterns in a tiger’s stripes as well as on a turtle’s hard shell.
Music has patterns, too. Think of your favorite beat and its sound, rhythm, and harmony. Each note creates a sound wave that follows a pattern.
In our neighborhoods, we use directions and move along with the help of maps. Builders construct homes and other buildings with different shapes. Angles and measurements are present in the objects we see, touch, and make.
There are shapes everywhere we look! There are also shapes that we cannot see. For example, the wireless networks we use for cell phones and the Internet have shapes and patterns. If you could see the wavelengths coming from a cell phone tower to one of these devices, they’d be circular!
Many patterns can be created with your fingers and hands. Did you know you can make shapes in order to communicate with others? This is called sign language. Try signing the letters for the word “love.” The letter “L” is made by pointing your pointer finger straight up and your thumb straight out. The other three fingers are down. Now try the letter “O” by connecting your fingertips with the tip of your thumb to make an “O” shape. The next letter is “V,” which looks just like a peace sign, or bunny ears! Then there is “E.” Curl all of your fingers together inward and connect them with your thumb across the tips. L-O-V-E!
You can learn how to sign the alphabet with this video!
KEYWORD PROMPTS
ASL ABC Smart Hands 
If geometry describes everything around us, does it describe us as well? Yes! You can find geometry not just with your body, but also on (and in!) your body.
In Explore Shapes and Angles!, we will take a field trip through our own bodies, our rooms, and outside to find the circles, squares, rectangles, triangles, and all different types of shapes, angles, and patterns. We’ll discover what makes each shape and angle special and why some are stronger than others. We’ll do lots of fun activities to find these shapes and use them in our own constructions. First, let’s find geometry on our own bodies. Get ready to bend, twist, and move!
Every mathematician keeps a notebook with all of their calculations and observations. Choose a notebook to keep as your math journal. As you read through this book and do the activities, use a chart like the one below to help you keep track of your observations, data, and designs.
What I did: |
Made different shapes by holding hands with people in different ways. |
What I observed: |
Our hands together made many different shapes. |
What I learned: |
Shapes and patterns are easy to find on our own bodies. |
Questions I have: |
Can I make more shapes with more people? |
When doing an activity, remember that there is no right answer or right way to approach a project. Be creative and have fun!
Each chapter of this book begins with an essential question to help guide your exploration of shapes and angles. Keep the question in your mind as you read the chapter. At the end of each chapter, use your math journal to record your thoughts and answers.
INVESTIGATE!
How many different shapes can you make with your hands and fingers?
SHAKE HANDS ON IT!
SUPPLIES
* hands
* partners
You can make patterns by drawing and making objects. You can also make patterns with your own body! Find a partner and see what shapes you can make with your hands.
1Shake hands with one person, two people, three people, then four people.
2What do your palms and thumbs do when you shake hands with another person? Can you spot any shapes when your hands are together?
3Stand face to face with a partner. Shake your right hand with their right hand. Hold on while you shake your left hand with their left hand. What letter did you make with your arms?
4Try making handshake patterns with more than one partner. Everyone extend their right hands forward as if to give a regular handshake, but instead, have all the fingertips meet in the middle.
5Then, everyone curl fingers at the same time, keeping the fingertips together. What do you notice?
TRY THIS! Make a group of five people. Everyone grab each other’s wrists to create a shape in the center. What other shapes can you make? How many sides do your shapes have?
