Category: Teaching Geometry

Finding Area – Classroom Exercises

Area is All Around Us!  Simple Geometry Classroom Activities


One challenge that students often face is realizing that math DOES actually relate to the real world and that they will actually use the information they are learning at some point in their life. Area happens to be one of those topics that students struggle to understand the reasoning behind.  Before introducing the topic of area, you may want to ask your students some engaging questions such as: “If you wanted to put new tile down on the floor, how would you know how much to buy?” or “If you wanted to put wallpaper on the walls of the classroom, how much wallpaper would you buy?”. These questions will be sure to get them thinking about how to calculate these answers.

Primary Geometry All Around

Here’s a fun way to integrate math, technology and language in one project for your youngest learners. Mathematically, students learn about solid figures and their properties. Technologically, they use digital cameras and work with word processing programs to insert pictures, word process and format documents. Students practice writing original thoughts and grammatically correct sentences as they describe the objects they have discovered.

Start the project by having students take digital pictures of solid figures in their everyday lives. They can either do this at home or (if they do not have a digital camera) at school. The following chart lists the common solids primary students learn about and some everyday items children would be familiar with. Hopefully, your students will find numerous examples of each.




Rectangular Prisms


-ice cream cone

-construction cone

-cone used for sporting events

-party hat



-scoop of ice cream


-play block

-sugar cube


-tissue box

-cereal box

-stick of butter

-pack of gum

-juice box

-soup can


-dowel rod

-stove pipe

-rolling pin


Once students have taken their pictures, visit the computer lab so students can create their “Book of Geometric Solids.” Students can organize their books in a variety of ways. However, each picture should have one to three sentences to serve as a caption. Captions must identify the type of solid the object is and must tell something about that object. For example, the student inserts a picture of a soccer ball. He then writes, “A soccer ball is an example of a sphere. I play soccer every Saturday morning. It is my favorite sport.”

This activity not only allows students to discover math in their everyday lives, but it also helps develop multiple academic skills.

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Math Manipulatives in Middle School

Most teachers are very comfortable using manipulatives with their youngest students. However, fewer and fewer manipulatives are utilized as students enter middle school, and rote memorization of rules becomes the focus. Yet, Middle School Learners benefit from the use of manipulatives just as much as younger students do. Students find that decimals, fractions and integers are significantly different from the whole numbers they have worked with up to this point. Manipulatives help students explore these new concepts, communicate their thoughts, share examples and truly understand mathematics. Below are three manipulatives every middle school math teacher should be utilizing to help her/his students reach their full potential.  (Pic: Cusinaire Rods in a Staircase Arrangement;  In Wikipedia.  Retrieved March 26, 2013 from http://

Base-Ten Blocks – Yes, everyone uses base-ten blocks (units [1], longs [10] and flats [100]) with primary students to help them learn place value, counting and operations.

However, base-ten blocks are the perfect tool to teach decimal places as well. Simply reversing the value of each piece (flats [1]; rods [ ]; units [ ]) allows students to explore smaller numbers in a hands-on fashion.

Fraction Towers– Fraction towers are possibly the best manipulative for middle school students. Fractions are often the most challenged of all middle school math concepts, usually because students do not understand fractions. They are taught to memorize rules, which make no sense to them. Towers allow students to easily compare fractions and complete various operations. Working with fractions in a concrete, visual and hands-on way makes fractions less intimidating for students.

Color Chips – Color chips are most often used for statistics and data analysis topics. They are wonderful tools for those topics. However, another great way to use them is with integers. The red side is negative, while the yellow side is positive. You can even take a permanent marker and draw + and – signs on the chips. Students can model adding, subtracting, multiplying and dividing using the chips.

You don’t always need to have the physical materials. Visit the National Library of Virtual Manipulatives at: to see online versions of these and many other math manipulatives.

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Geometry Math Stars for Christmas!

Learn and Practice Geometry by Making Your own Family Christmas Tree Star Ornaments!  Great for decorating your Christmas Tree or for making a Gift to give to a friend or family member.

Materials Needed:

Colored Markers or Crayons

Glue or Scotch Tape


Colored Yarn or String


Colorful Recycled Paper (tissue boxes, flyers, old wrapping, etc.)

Directions for Christmas Star Cutout:

1.Cut out the outline.

2.Cut along all heavy lines.

3.Score plain lines on the front.

4.Score dotted lines on the back.

5.Fold triangles upword along plain lines.

6.Fold triangles downword along dotted lines.

7.Glue or tape tabs to form small tetrahedrons.

8.Continue until you have your Geometry Christmas Star Tetrahedron.

Everyone has their own Star!  Everyone has their own Inner Light!  With Favorite colors, draw your stars, or print this page and cut the pattern of the star out.

Inside each shape on the side of the star, write your name and birthday.  Or – write the names of each member of your family (If you Wish, your family / family Tree Star can be made of relatives (close or extended), friends and/or adopted family – as long as you write each name on each.  You could make many stars – a pretty star for each member of your Christmas Tree, or write everyone’s name on the same Star.

Color and Decorate each one with colored markers, crayons, sparkles, gluing pieces of recycled Christmas paper from last year.  You can also write happy words all over your star like Love, Divine Wisdom, Infinity, Pure Spirit, Fun and Harmony!

Remember to Decorate your Stars!

Glue Yarn / Strong at the top  into a 2 inch loop and tie a knot at the end

Hang your Christmas Geometry Math Stars and Decorate your Tree at Home or in your Classroom!

Have a Merry Christmas and Happy Holiday!

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Reference for Star Tetrahedron Geometry Template: “The Ancient Secret of the Flower of Life,” Vol. 2, by Drunvalo Melchizedek

Drawing with Pythagoras

“Pythagorean Theorem?!!”…I know what you are thinking…..(You can say to your class)…..”OMG  Teacher…..What could we possibly ever use this for, in the future of our lives?“……Well, in this article, we talk about real life examples of using the Pythagorean Theorem.  First let’s learn a basic calculation, corresponding to the diagram here to the <<left, followed by a bit of Pythagorean history.  The basis of the theorem is that the Area of the 2 Squares of the 2 Lines that form the Right Angle of a Right Triangle are the equivalent ( = ) Area of the Square that opposes them in the Triangle. If you look at the diagram here to the << left, the Areas of “a” added to “b” = the Area of “c” (the Square of the opposite line (or “leg”). That opposite line (or  “c” “leg”) is called the hypotenuse. Also, if one of the lines “a” of the right triangle is 4 inches and the other line “b”  is 6 inches, we can calculate how long the hypotenuse is, or the “third leg”. Letting a = 4, b=6, and c= the length of the hypotenuse.  (4)^2 + (6)^2 = c^2. Accordingly, 4 x 4 =16, and 6 x 6 = 36. Thus, 16 + 36 = 52. The square root of 52 is approximately 7.21 , hence the length of the hypotenuse or “third leg“ of the right triangle, is 7.21 inches.

The Pythagorean Theorem is named after the Greek mathematician Pythagoras. Many believe the first discovery and proving of this ancient math theorem came before Pythagoras, but since no tangible account of this has yet been documented, it is named as such. If that was true, however, we wonder what another name of the theorem would have been, and from what country and nation? (a fun question to ponder).  The Pythagorean Theorem can be used with any shape and for any formula that squares a number. And, in fact, the area of any shape can be computed from any line segment squared.  (reference also for Diagram Above).   Apparently, even the teenage Brainiac Lisa Simpson from The Simpsons television series knows all about Pythagoras.  On this site, there are some cool diagrams showing the differences of Lines (Segments / Legs), Radius, and Area:  And, to assist in figuring out these math scenarios, this site offers tabs for entering in numbers to calculate the square root of the numbers in question:

Finally, here are a few Real Life Examples:

Meet Me at The Corner:
Let’s say Stephanie and Maria are meeting at the Hiking Trail Entrance on the corner of Saanich Rd. and Cedar Rd. One phones the other on her mobile phone and asks, “How long will it be before you arrive at the entrance?”… “Well, let’s estimate by first finding out how far away we are from each other.” In present time, Stephanie is on Saanich Rd. to and is 10 miles away. Meanwhile, Maria is on Cedar Rd. and is 4 miles away.  How far away from one another are Stephanie and Maria?  The distance between them = a^2 + b^2 = c^2 or, respectively: 10^2 (10 squared) + 4^2 (4 squared); or respectively, 100 +16 = 116 miles. The square root of 116 is 10.77. Thus, Stephanie and Maria are 10.77, almost 11 miles away from one another. Hence, they figure they will be another hour on their bicycles to meet one another to go hiking.

Firefighters Needing to Know Height of a Building:
3 Firefighters receive a call to help Ann rescue her cat Tia from the Oak Tree outside her window. The tree is about 3 stories tall, and the Tia, after chasing a squirrel, is stuck on a branch at about the height of 2 stories of her house. The height to the branch may be 20 feet, and the firefighters have to put the ladder about 10 feet away from the Oak Tree in order to go around Ann‘s shed. How long of a ladder do the firefighters need in order to rescue Ann‘s cat? a^2 + b^2 = c^2 or, respectively: (20)^2 + (10)^2 = 2^2, the length of ladder required. 400 + 100 = 500. The square root of 500 is approximately 22.37. The firefighters extend their expandable ladder to be approximately 23 feet, whereas they need at least 22.37 feet to safely reach the Oak branch. Ann’s cat Tia is rescued and All are Happy!

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Teaching Math: A Breeze when Incorporating Favorite Interests

Teaching math in the Elementary classroom can be a challenge, as all elementary math teachers know, but when favorite activities of students are incorporated into the curriculum, it can be a breeze. When students have a required learning skill to attain, and they incorporate their favorite interests, then there is inspiration – and the student becomes “self-motivated”.    This is a great First Item to address with the rounds of new students returning for the fall season.  Find out Learners’ Favorites, and keep the list in a special file.

Then, as the year progresses, if a student is having a challenge learning a particular new math lesson, Teachers can refer to the student’s personal file of “Favorites Activities List”.  At this time, then, introducing the association and how that interest relates to the new lesson.

Example No. 1:  Young Matthew enjoys playing or watching the game of baseball. That is included in their list of favorite activities.  November rolls around, and the lesson of drawing shapes in geometry arises, but Matthew is not grasping the concepts.  If looking down onto a Baseball Diamond from an aerial perspective, the shape of Square is easily seen in the formation of the 4 bases on the ball field.  As well, the shape of the bases individually, is a square.

Then show in sequence what happens when the player runs to first base, second, third and fourth, demonstrating the making of a Straight Line 4 times, and in consecutive order. Within each corner, while the player stands on the base, the player looks down at both straight lines that connect, and the player can then see a perfect Right Angle of 90 degrees.  Drawing a line across from the base to the left to the base t the right demonstrates a perfect Right Triangle.

Suddenly a light is switched on in the child’s brain, and Matthew is on the way to understanding the concept of geometry.  Not only do they understand it on paper in 2-D form, but now in 3-D form, in the context of a baseball game, in a real life scenario.

Example No. 2:
  Ashley likes archery.  Archery is included in her favorite activities list.  While imaging and practicing her archery skills, she sees concentric circles – one inside the other.  When a line is drawn from her bow to the target, she demonstrates a perfect straight line.  Hence, she has a different yet equally effective association of a favorite interest to relate to the concepts in geometry –  as Mathew’s love of baseball.  Imagine now that Susie, not only is attaining the required skills the in geometry lesson, but is also having fun while doing it, and has developed self-motivation and interest in learning math.

In these examples, both sides of the Brain are exercised, (Left Right Brain Learning and Thinking) new neural connectors and dendrites grow, and you have encouraged the growth of a healthy developing young brain.

Start the school year off right, and find out what your students’ favorite activities are. Keep the lists on file, and refer to them from time to time during the school year.  You may be surprised at the effectiveness of this subtle teaching tool.

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