*Please Note: This post contains affiliate links which support the work of this site. Read our full disclosure here.*

This set of problems can be used in a lot of different ways and will give your kids a context for using and applying the formula.

If Pythagorean theorem is still new for your kids, you may want to start with these introductory lessons.

These real life examples provide further context and give space for kids to draw pictures and show their thinking.

Once they feel more confident, this set of Pythagorean theorem word problems task cards can help reinforce their understanding. Plus, it provides an opportunity to practice critical thinking and problem solving!

–>Pssst! Do your students need help breaking down and solving word problems? You might like this set of editable word problem templates! Use with any grade and any type of word problem:

Pythagorean Theorem Word Problems Set:

This download includes a set of 6 task cards. Each card includes a word problem that requires kids to use the Pythagorean theorem to solve.

Some of the problems ask for the length of one of the sides of the given right triangle, while a couple of the problems are actually two-step problems. This means using the Pythagorean theorem is a necessary step, but doesn’t provide the final answer.

Also included is a recording page. On this page, kids have space to draw a picture (ideal for these types of problems) and show their thinking. Then there is space for them to write the final answer.

Finally, there is a teaching tips page with examples and ideas for how to use the task cards, plus an answer key, making this super easy on you, the teacher!

To Prepare the Task Cards:

First, print the page of task cards. I suggest printing on card stock and laminating them for durability.

Cut out the cards, then print a recording page for each student.

Now you’re ready to decide how to use them!

How to Use the Pythagorean Theorem Task Cards:

Task cards are a fun way to practice because they can be used in whole class or small group games, or as individual review.

To play as a whole class or small group, use them in a game of scoot. This means each card is at a different “station” around the room.

Kids then (probably in small groups of 2-3) make their way around the room, solving each problem. You set a timer for each station and then call out, “Scoot!” to let kids know it’s time to move to the next problem.

You could also include the entire set as a single math station. Then when kids get to the station, they work through as many task card problems as they can, recording their answers.

This could also be used with small groups who need more review or intervention with you, or as individual practice at home as homework.

However you work the problems into your math routine, I hope it helps your kids see the benefits of math in real life!

Want to grab this free set of Pythagorean theorem word problems? Just click the link below to grab them in my shop!

{Click HERE to go to my shop and grab this set of Pythagorean Theorem Word Problem Task Cards!}

More Pythagorean Theorem Resources:

• Proving the Pythagorean Theorem with LEGO Bricks
• Real Life Pythagorean Theorem Practice

The post {FREE} Pythagorean Theorem Word Problems Task Cards appeared first on Math Geek Mama.

*Please Note: This post contains affiliate links which help support the work of this site. Read our full disclosure here.*

Learning the Pythagorean Theorem: 

Kids are usually exposed to the Pythagorean Theorem in middle school or Pre-Algebra. Although reciting the theorem is not difficult:

a² + b² = c², where a and b are the sides of a right triangle and c is the hypoteneuse

actually understanding what that means and applying it to solve problems is quite a different matter.

To solve problems using Pythagorean Theorem, kids will need to understand squaring numbers and finding square roots.

They also need to understand what a right triangle is and how to evaluate and solve equations.

But even if you give kids all the necessary numbers and show them step by step how to solve an equation using Pythagorean Theorem, it will just be a bunch of meaningless numbers to them.

It will seem like a pointless exercise, where once again math is completely irrelevant to their everyday life.

We don’t want kids to feel that way, though! Especially not about a fundamentally important and useful theorem such as this.

Pythagorean Theorem Practice in the Real World: 

To fight against this notion, I’ve created a simple set of problems that give kids practice with the Pythagorean Theorem, but more importantly show them why this theorem is beautiful and necessary.

We can use the Pythagorean Theorem to find lengths and distances of all sorts of questions in our everyday life.

And when we show kids that it is relevant, then practicing the Pythagorean Theorem will not feel pointless and boring!

This set of Pythagorean Theorem practice pages includes 3 problems for kids to visualize, think about and solve using the theorem.

The first discusses the layout of a baseball diamond, and challenges kids to find the distance from second base to home. Kids who love baseball will enjoy this challenge, and it will hopefully spark other math questions for them as they consider a game of baseball!

The next problem is about television sets. This is important for anyone wanting to make the right decision about which t.v. to buy, because you need to know if it will fit in the space that you have.

In this problem, kids learn that t.v.’s are measured by their diagonal, and have to find the length of a given television set.

Finally, there is a slightly more challenging problem, which will really require kids to think about the whole situation.

This problem involves a firetruck with a ladder of only 100 feet long. Because the truck is parked 25 feet away from the building, they have to determine how high the ladder can actually reach.

This problem is based on several articles about the safety of high rise apartments and condos, when people realize that a fire truck ladder can’t actually reach the top stories. [Read more here]

Although this is often shocking and concerning news to people, there are other fire safety precautions in place, making those very high condos perfectly safe in the event of a fire.

Using the Pythagorean Theorem Practice Pages:

Each problem in this download includes a one page snapshot for kids to learn and solve the problem. It starts with relevant and interesting facts. They may or may not need these as they solve the problem.

In the middle section, kids draw a sketch of the situation and label all the measurements. It’s important for kids to see a picture of what they’re thinking about so they know where the right angle is, and therefore what side is the hypotenuse.

Finally, in the bottom space kids write out an equation and solve it to answer the given question.

Depending on your kids and your situation, you could put kids into 3 small groups and let them work together to solve one of these problems.

Or you could assign everyone the same problem and let them work on it individually (in class or at home).

But I would certainly pick at least one of the problems to discuss as a class. Take time to draw a sketch of the situation. Then have kids label everything they know. Finally, ask kids how the Pythagorean Theorem helped them to solve the problem.

Proving the Pythagorean Theorem:

Another fun way to explore this theorem is to challenge kids to prove it. This might seem daunting to you, or above the ability of your students, but it’s really quite doable when you use geometry.

Further explaining and visualizing the theorem will help kids not only remember it, but deepen their understanding of it.

Find a quick and easy way to prove Pythagorean Theorem in this post.

And of course, enjoy exploring math in the real world with this problem set!

When your kids are finished and confident, challenge them to create their own example problem using the Pythagorean Theorem!

{Click HERE to go to my shop and grab the FREE Pythagorean Theorem Practice Pages!}

Fun Books to Read Along with Pythagorean Theorem: 

• What’s Your Angle, Pythagoras? by Julie Ellis
• Pythagoras and the Ratios: A Math Adventure by Julie Ellis

The post Real World Pythagorean Theorem Practice {FREE} appeared first on Math Geek Mama.

*Please Note: This post contains affiliate links which help support the work of this site. Read our full disclosure here.*

Over the years, I’ve come up with lots of fun ways to celebrate this day with my kids. We’ve sung silly Pi Day songs, and created sweet friendship cards to spread the Pi Day joy.

This year is no exception!

We’re a little early with this project, because the weather was beautiful, warm and not raining for the first time in what feels like weeks. So I jumped at the chance to get outside and create one of a kind math art as we begin to think about Pi Day.

Get Outside: Sidewalk Chalk Pi Day Art Project

The goal of this project was simple: briefly introduce my kids to circles and circle measurements, as well as using a compass, and then let them create unique chalk art.

So before letting them loose, we sat together in the driveway and I explained what a compass is, and how to use it. We talked about what makes a circle a circle, and measuring the diameter and circumference of a circle.

Then I explained that if you take the circumference of a circle and divide it by the diameter, it will equal the number pi. No matter how big or how small your circle.

Yes, I explained all of this to my 7, 8 and 9 year old.

Even though these concepts might normally be taught in middle school or high school Geometry, if kids understand how to measure and divide, they can understand how to find the number pi.

My kids were amazed that this would hold true for any and all circles! I love seeing their amazement and wonder!

Related: Learn what makes a triangle a triangle.

Then I explained their project for the day. The only requirements are that they use the compass to create a variety of overlapping circles. Any size, any quantity.

Then they can color all the different parts in any way they liked.

They loved this idea, and thought immediately of a color by number type of picture.

So yes, the end result will look like a color by number, but without the numbers!

Ready to get started with your own Pi Day art project? Here’s what you need.

Materials Needed for this Pi Day Art Project:

• Chalk compass*
• Colored chalk*
• Black construction paper (if you’d rather do this inside with chalk and paper)
• Printable lesson (optional, depending on the age of your kids)

*You could also do the same thing with pencil and paper, and color the circles in with colored pencils.

As I said, the weather was beautiful and I was looking for a chance to get outside. So we did this on the driveway with sidewalk chalk.

Alternatively, this would also make a lovely art project using chalk on black construction paper. This is probably what we’ll do on the actual Pi Day, now that my kids are familiar with the compass and the idea of the project.

Once you have all the supplies, kids simply draw various circles. It is a little tricky to use a compass on the driveway and have perfect circles come out. But just encourage kids to do their best. It doesn’t have to be perfect!

After kids are satisfied with their circle design, they can color it in however they’d like.

The result is a beautiful, unique design!

What I loved about this project is that all of my kids were able to participate and learn something, at their own level.

Even my 5 year old was able to learn how to use and practice making circles with the compass!

Now if you want to kick the learning up a notch, you can download the free printable math lesson to go along with this art project.

There are 2 pages included. The first is for younger kids and introduces the vocabulary terms and challenges them to measure the diameter of each of their circles.

The second is a more in depth investigation, in which older kids can discover the number pi on their own.

It includes a table to record the circumference measurements, diameter measurements and the ratio for each of their circles. Hopefully they will notice then that each time the number is close to a pi approximation: 3.14.

Then they take the average of all of their ratios, which will hopefully be an even closer approximation of the number pi.

You can then discuss their findings together as a group.

I hope you enjoy this fun, hands on Pi Day Art Project! If your kids have a fun, unique piece of art to share, tag me on Instagram so I can see it!

Happy Pi Day friends!

{Click HERE to go to my shop and grab the FREE Pi Day Art Project Recording Pages!}

Even MORE Pi Day Ideas:

• Ultimate List of Pi Day Resources
• Activities for “Sir Cumference and the Dragon of Pi” by Cindy Neuschwander
• Race to Pi: Simple Pi Day Card Game
• Hands On Pi Day Investigation for Middle School and Beyond

This post is part of the 28 Days of STEAM series at Left Brain Craft Brain! Check out the full collection of integrated learning ideas here.

The post Simple Pi Day Art Project {Free Printable!} appeared first on Math Geek Mama.

I even showed it to her and still she didn’t know. So, now I’m trying to explain how cool they are and how useful they are for learning. I was very articulate:

You know, you use these rubber bands and can stretch them across the pegs to make shapes and stuff. Then you can ask, ‘what shape is this?’ or say, ‘Can you make a triangle?’ And you can play games and … explore.

Surprisingly she took some home. Probably thinking about what a weird sister she has.

Now I’m going to do what I wasn’t able to do for her. I’ve created a list of what a teacher can do with a geoboard. Stay around to the end to get the free geoboard activity cards!

And just in case you thought geoboards were just for elementary students, I’ll be sure to highlight how you can use geoboards with your middle school students too.

*Please Note: This post contains affiliate links which help support the work of this site. Read our full disclosure policy here.*

This is a guest post from Danielle of Blessedly Busy.

In this article, I will highlight 3 different ways kids can learn math with geoboards. Get a quick overview in this video, and then read the complete article below:

How to Explore Using a Geoboard

Exploring can be as simple as handing them the materials and saying, “What shapes can you make?”

But I like using a more spontaneous approach. Before I went to bed, I put the geoboards in the middle of the table with cups of rubber bands. The kids didn’t even notice until breakfast was almost over. Hunger probably has a way of making you unaware of your surroundings.

Then one kid picks one up. “What do you do with this?” I show them how the rubber bands fit on the pegs and suddenly all of the kids want one and are busy making shapes and designs.

A morning of exploration commenced.

For the most part I let things unfold how they would. But when I was sensing a lull in their exploration I might ask a question like:

• Can anyone make a circle?
• What’s the biggest shape you can make? The smallest?
• What shape can you make that has the least number of sides? The most?
• Can you make a picture out of many shapes?

This is an activity the whole family, no matter their age, can participate in. Older students will bring a lot more knowledge to the table, but there will still be something to discover and fun to be had.

For exploration: give students time to explore how they will. Ask open ended questions that have multiple answers.

How to Analyze Using a Geoboard

Ask students to make a picture or design. It needs to be a little complicated, with rubber bands crossing each other. If they bring back something too simplistic, tell them they have to use at least 10 rubber bands.

Then, they analyze it. Here is an example:

Ask students what shapes they can find. Older students should be using higher vocabulary. If they say they found 5 triangles, ask what kind of triangles? My design has a lot of isosceles right triangles.

If you are working with more than one student, you can switch and see if you get the same answers.

Once you start analyzing designs, you’ll start to see the parts that make up a shape. I noticed that in each of my squares there are two triangles.

At first it looks like my design doesn’t have any rectangles (that are not squares, that is) but then I start seeing the two squares touching and suddenly I see several rectangles (at least 12).

For Analyzing: create a design and see how many parts and shapes you see.

How to Challenge Using a Geoboard

Since writing about STEM for Homeschoolers I’ve been thinking a lot about how engineers test limits. I generally don’t think like an engineer, but to challenge students I’m willing to give it a try.

So, let’s push the limits of what our students know.

Examples of challenge questions:

• Can you make a:
  • triangle
  • right triangle
  • isosceles right triangle
  • a shape with no sides
• How many triangles fit in a square?
• How many shapes can you make with only a square and a triangle?

To challenge students: push them a little past what they know.

How to Practice Using a Geoboard

The best way to practice…anything? Play! Here are three very simple games that only require a geoboard, rubber bands, and a list of shapes.

(You can use the geoboard activity cards in the download below)

Game 1:

Give a description of a shape and the student makes that shape. Examples: a triangle, a 3 sided shape, an isosceles right triangle.

Game 2:

Give students a list of shapes. They pick any 5 they want and put them on their geoboard. (They can use a shape more than once if they like). When you call out a shape, if they have it, they can take it off their board.

They win when their board is empty. (I numbered my list and used the random number generator function on my ti-84 calculator so I didn’t have to cut up a list. Yes, I am lazy).

Game 3 (for 2 players):

One player creates a shape on their geoboard (without showing their partner) and the other asks yes or no questions until they can guess the shape.

Where to Get a Geoboard

Of course you can get a geoboard from Amazon or any school supply store. They are relatively inexpensive and I have found even the cheap plastic ones to hold up well to students play. But there are two more options.

If you are the DIY type, check out this homemade Geoboard.

If the thought of rubber bands going flying across your dining room (or classroom) is just too much for you, there are also several apps or online geoboards you can use.

Geoboard Activity Cards to Get You Started!

If you love the idea of exploring shapes and more with Geoboards but are struggling to come up with engaging challenges, grab this free set of geoboard activity cards!

I suggest printing them on colored card stock and laminating for durability, then you can use them as a starting point again and again.

You could even use them with partners or groups. Have each student draw a card and complete the challenge. They then switch geoboards with a partner and the partner has to guess what their challenge was.

This download includes 30 activities, some appropriate for young kids, and some much more challenging.

For more hands on math with geoboards, be sure to check out my complete resource book for grades 5-8!

And be sure to grab this free set of Geoboard challenges to get started!

{Click HERE to go to my shop and grab the free geoboard activity cards!}

More Math Ideas with Geoboards:

• Plotting Points on a Geoboard
• Finding the Midpoint Using Geoboards
• Exploring Triangles with The Greedy Triangle

The post Geoboard Activity Cards {FREE Geometry Challenge} appeared first on Math Geek Mama.

*Please Note: This post contains affiliate links which help support the work of this site. Read our full disclosure policy here.*

Exploring Geometry with Technology

Oftentimes in math class, students are simply given a formula and told to just use it. Ample time and practice is given to make sure students know how to “plug in the numbers” and solve it correctly.

But is this really teaching students to understand the formula or the math they’re doing? Is it going to ignite a love and excitement about the math they’re learning, or cause them to see how this formula can be used to solve problems?

Hopefully you can agree with me that, no, it will not.

In my experience, this often teaches kids to memorize a formula, with no real understanding of what all the letters and numbers mean.

This leads to incorrect solutions, because they plug in a number wrong, or use the wrong values. Then they have no idea that their solution makes no sense in the context of the problem, because they never really understood it.

(I would also like to add that, for most formulas, it’s not even necessary to have them memorized, because kids receive a formula sheet on standardized tests.)

So today, I want to share just one example of a way that you can make a boring, abstract formula (area of a circle) more meaningful to students, and more engaging!

Simple Area of a Circle Activity

This area of circles investigation shows students exactly what it is they’re trying to figure out (the area inside a circle) and will cause them to ask important questions.

For example, “Why is the area of a circle so much harder to find than a rectangle?”

It is also a great introduction for Calculus students who are being introduced to the area under a curve.

Depending on the age and abilities of your students, you may even want to begin by drawing various rectangles on graph paper and finding the area, just to review area, how to find it, and how it compares to the area formula.

Materials Needed for this Area of a Circle Lesson:

This will take just a little bit of prep, but not much.

To get started, students will need some graph paper, a pencil and a calculator.

You may also want them to have some circular objects so that they can draw more exact circles. This is the part that may require some prep–or you could simply have each student bring 1-2 circular objects from home that day!

Or, you could give each student a compass, and they can draw each circle after marking the radius by hand.

Then, students can work through the investigation on their own.

How to Complete the Circles Lesson:

To begin, they will draw different sized circles on their graph paper.

Then then count whole squares (not cut off by the boundary of the circle) to find an “inner” area (a slightly under estimation) and “outer” area of the circle (a slightly over estimation).

They then use their calculator to find the average of the two areas.

Finally, they use the formula for the area of a circle to calculate a precise area and compare their results.

Once complete, you can then discuss their findings, and talk about the area formula and how they think it was derived.

You can also ask if anyone has a suggestion for an estimation that may be closer to the exact area, then test it out.

You may also want to discuss the number pi and what that means so they can further understand the area formula.

And hopefully this gives you some good ideas for digging into all those other math formulas a little deeper!

{Click HERE to go to my shop and grab the Area of a Circle Activity!}

**Psst! Did you enjoy this lesson? It’s a sample of my ebook, Graphing Calculator Activities: Geometry. See all the book details here!**

You may also like this hands-on Geometry lesson, exploring the angles in triangles.

Plus, don’t Miss the rest of the Math+Technology Series:

• Exploring Integer Multiplication with a Calculator
• Understanding Systems of Equations with a Graphing Calculator
• Engaging Data Analysis Lesson
• Fun Probability Experiment with Dice and a Calculator

The post Hands-On Area of a Circle Activity {FREE} appeared first on Math Geek Mama.

*Please Note: This post contains affiliate links which help support the work of this site. Read our disclosure policy here.*

Materials Needed for this Measuring Angles Activity:

• Card stock (for durability)
• Scissors
• Protractors for students
• And I suggest you laminate them to be used again and again, but this is optional

Preparing the Angle Measure Puzzles:

To get started, print out the set of angle puzzles (click the link at the end to get them from my shop). I suggest printing on card stock and then laminating them so they will last a long time.

I printed on colored card stock, just for fun, but white works perfectly well.

Once they’re ready, cut each puzzle into two pieces.

Then they’re ready to use!

To complete, students simply measure angles with a protractor, then match the angle to the correct degree measure.

Ways to Use this Measuring Angles Activity in the Classroom:

First, these would make a great math center. Have the puzzle pieces prepped and in a ziplock bag with some protractors, and it’s an easy way to get some additional practice.

You could also use them as a fun, quiet activity for early finishers.

Another idea is to use these as a way to practice estimating angle measures using benchmarks. Students would first match the puzzle pieces based on how it relates to 90 degrees or 180 degrees. Then, they would go back to measure and check their answers.

Finally, I think it would be fun to put students in small groups and let them race each other to measure and match all the angles correctly. In this case, you will need a complete set of puzzles for each group. Then give each student a protractor and see which group can correctly match all 15 first.

And whether you use these in groups or with individuals, I would also have students classify the angles after measuring into acute, obtuse or right angles.

And if you’re looking for more ideas to learn about angles, angle measures, and classifying angles, check out this free activity pack to go along with the book, Sir Cumference and the Great Knight of Angleland!

This is such a fun book, and the lesson pack includes a fun measuring angles project that your students might enjoy.

And if you’re learning about the angle measures of a triangle, check out this fun, hands-on lesson!

{Click HERE to grab this FREE Measuring Angles Activity from my shop!}

The post Measuring Angles Activity: Simple and Fun Puzzle Set {FREE} appeared first on Math Geek Mama.

*Please Note: This post contains affiliate links which help support the work of this site. Read our full disclosure policy here.*

To assemble this set of foldable 3D shapes: 

I highly recommend printing the nets on stock paper rather than regular paper. I have tried it both ways, and regular paper is just too flimsy.

After printing, simply cut out the 3d shape nets on the solid lines.

If you’ve printed on white card stock as I have, take some time to get creative and color or decorate the shapes before assembling them! This could make a great math art project!

If your students are older, have them label the different parts of the shape (face, edge, base) before assembling to use as a reference throughout their study of shapes!

This will help them see and learn math vocabulary.

Then, carefully fold each tab so that it can be used to glue the shape together, and fold each side of the shape.

Finally, glue each side together. I suggest adding a few dabs of glue to each tab and then hold it in place for a few seconds before gluing the next tab. This will help make sure it stays together.

You can also use tape if that’s easier for you (or you don’t have glue).

Once all the shapes are assembled, you can use them however you like, depending on the age of your kids!

Learning ideas for foldable 3D shapes:

• Discuss math vocabulary such as polyhedron, face, edge, prism, etc.
• Compare the shapes by counting number of faces and edges or other characteristics
• Use them to go on a 3D shape hunt: find the shapes in real life
• Compare the different pyramids and then compare them to the great pyramids of Egypt (combining math and history!)
• Use the nets to learn about surface area and volume
• Use them along with the book, Sir Cumference and the Sword in the Cone to learn about Euler’s formula (see more free resources to use with the book here!)
• Or simply as a fun math art project!

This set of geometric nets is FREE to download and contains the following three dimensional shapes:

• Cube
• Rectangular Prism
• Triangular Prism
• Cylinder
• Triangle Based Pyramid
• Square Based Pyramid
• Pentagon Based Pyramid
• Cone

In addition, this set includes a foldable copy (with tabs) to allow students as young as preschool build their own models, as well as geometric nets without tabs to be used for any other purpose in the geometry classroom!

For instance, let students measure the nets to explore surface area with these FREE surface area lessons (surface area of prisms and cylinders and surface area of pyramids and cones)!

{Click HERE to go to my shop to get the foldable 3D shapes Printable Pack!}

Hope you find these helpful and FUN as you explore three dimensional shapes!

Looking for more 3D shapes fun? Try one of these resources:

• 3D shapes games printable pack
• Exploring 3D shapes with a 3D drawing pen
• 3D shapes and their nets worksheets

The post Foldable 3D Shapes {FREE Printable Pack!} appeared first on Math Geek Mama.

*Please Note: This post contains affiliate links which help support the work of this site. Read our full disclosure here.*

This lesson includes two parts, showing students properties of angles that they will need to know in Geometry. These are important, but if you simply tell students they’re true, they will likely forget, and then have no way to figure it out again.

If you let them learn firsthand, however, they can always recreate the situation to figure it out again in the future if they forget.

So I hope you will find this investigation fun and useful!

Materials Needed to Explore Interior and Exterior Angles in Triangles:

• Construction paper
• Scissors
• Student handout (free download, click the link below)

The first lesson is intended to show students that the sum of the interior angles in a triangle (any triangle) is equal to 180 degrees.

That sounds simple enough, and again, you could just tell students this. But I think letting them see it and prove it themselves is more fun (and hopefully more meaningful to them!).

To use the lesson, students start by cutting out multiple triangles from their construction paper.

Encourage them to cut different sizes and types of triangles (acute, obtuse, large, small) so they have a variety.

Then have a discussion about the angle measurements. Do they think the sum of the angles will always be the same? Or will it be greater for larger triangles?

Students will also need to know that a straight line is equal to 180 degrees. If they don’t know this, have them look at their protractor to see a straight line measurement.

Then students can begin the investigation. The purpose is for them to “cut off” the angles in each triangle and line them up on a straight line to find their sum.

In each case, no matter the triangle, the three angles will fit on the line to equal 180 degrees.

*It is important that students label or mark their interior angles before cutting them off, however, because cutting will form new triangles. If they haven’t marked the original, they will  not know which angles they were supposed to be measuring.*

After completing and discussing this  first lesson, you can move on to exterior angles.

Understanding Exterior Angles in Triangles:

This lesson will probably require a bit of explanation before you let students begin.

An exterior angle is formed when one of the sides is extended. This creates two angles that are equal to 180 degrees, the adjacent interior and new exterior angle.

The goal of this second lesson is to show students that the sum of the two remaining interior angles (opposite the exterior angle) are equal to the exterior angle.

It is fairly easy to see why this is true:

The sum of the 3 interior angles is equal to 180 degrees, and the sum of the exterior angle and adjacent interior angle is equal to 180 degrees.

Therefore, the difference between 180 and the adjacent interior angle is equal to the two opposite interior angles. (180 – adjacent interior angle = sum of the two interior angles)

Likewise, the difference between 180 and the adjacent interior angle is equal to the exterior angle (because they form a straight line). (180 – adjacent interior angle = exterior angle)

Therefore, the two opposite interior angles and the exterior angle must be equal.

So to help show this to your students, they will cut off the interior angles and compare them to whatever exterior angle they form with their triangle.

I suggest they trace the side of the triangle that forms an exterior angle to help them see which angles they’re comparing.

Again, they should mark the interior angles, then cut them off to line up with the exterior angle.

They should see, no matter what exterior angle they form, or what their triangle looks like, that the two opposite interior angles are equal to the exterior angle.

I hope I’ve explained this in a way that’s clear. It is really easier to see that it’s true than to explain it in words (I think). Which, obviously, is why I think it should be shown to students as well, especially younger students who are being exposed to angles, angle measures and properties for the first time.

I’ve included some teaching explanations in the download as well, and an answer key for the discussion questions.

Most of all, I hope this guided lesson is a fun and helpful introduction to the interior and exterior angles in triangles, that helps your students understand and make sense of geometry!

{Click HERE to get a copy of the “Interior and Exterior Angles in Triangles” lessons!}

Want more fun, hands-on Geometry ideas? Try one of these FREE lessons!

• Exploring Triangles with The Greedy Triangle
• Understanding Volume of 3D shapes
• Understanding the Surface Area of Prisms and Cylinders
• Exploring 3D shapes with a 3D pen

The post Understanding the Angles in Triangles appeared first on Math Geek Mama.

*Please Note: I received this pen and was compensated for my time in writing this post. But all ideas and opinions are my own. Read our full disclosure here.*

Before receiving and trying out this pen, I was completely ignorant to the fact that such a thing even exists. So to have one and play with it and learn together with my kids has been so much fun!

Learning About 3D Shapes with Atmosflare 3D Pen:

To get started, I made some 3D shapes with the pen. The kids were so excited to watch as I formed the shapes, constantly asking questions and wondering what I was going to do next.

After completing each shape, I would ask questions about it such as, “What would you call this shape?” “What shapes do you see on it?” and “How many sides does it have?”

Once they were dry and firm, I would let the kids hold and examine them, and make observations of their own.

I then created other shapes so that we could compare them.

Two main shapes that I wanted them to compare were a triangular prism and a square based pyramid, as these are often confused.

This allowed for good discussions about the shapes of the faces because the triangular prism has two bases that are triangles, and faces that are rectangular, and it does not come to a point.

A pyramid, on the other hand, as one square base and all the faces are triangles, and comes to a point.

After drawing and playing with our 3 dimensional shapes, I let my kids try out the pen to see what they could create!

Tips for Using the Atmosflare 3D Pen:

*Please note: This product says it is intended for ages 14+. I allowed my 6 and 7 year old to use it with close supervision, but you should use it at your own risk.*

Using the pen takes a bit of practice. It’s not that the pen is difficult, but obviously, I’m not used to drawing in mid air, so I had to try different things and techniques to achieve what I wanted. (And as you can see from my pictures, the shapes are not perfect, but it worked! )

When first starting, you need the ink to adhere to something, so we started on a poster board and drew up from there. Then you can start at different parts of your artwork and build up and out!

Also note that you have to keep the pen moving slow and steady as you draw. If you stop, the ink gets stuck in the tip, but if you go too fast, your creation will not hold together.

And lastly, the ink is a little sticky, so I suggest drawing a little, then letting it set and then continuing on.

If you’ve drawn something that is held together well (i.e. the ink is not too thin), it should be fairly long lasting! We still have our 3D shapes to look at, play with and discuss!

Other Ideas for Learning with a 3D Drawing Pen:

• Learn about architecture and recreate famous buildings
• Learn about proportion and scale modeling
• Practice measurement by measuring the height and width of your art work

Have you ever used a 3D drawing pen? What other ideas do you have for learning with a 3D pen? Amazingly, you can get this pen for only $29.99 on their website or at Toys R Us!

You can also follow Atmosflare on social media: Facebook, Pinterest, Twitter and Instagram!

 Looking for more resources to explore 3D Shapes? Try one of these:

• 3D Shapes Game Bundle
• Printable Nets for 3D Shapes
• Exploring Euler’s Formula with “Sir Cumference”
• Exploring the Volume of 3D Shapes
• Form 3D Shapes out of play-doh

The post Learning About 3D Shapes with Atmosflare 3D Pen appeared first on Math Geek Mama.

*Please Note: This post contains affiliate links which help support the work of this site. Read our full disclosure here.*

Materials Needed for this Hands On Pi Day Investigation:

• Circular objects
• Measuring tape or ruler
• String (if you’re using a ruler to measure)

To Complete the Investigation:

The student handout included guides students through each step.

First they will need to measure the circumference and diameter of various circular objects (I like to use fun food such as pie or cookies, but anything circular will work ).

If students are not familiar with these measurements, take time to go over them before you start.

They then record the answers in their table and calculate the ratio.

Hopefully if students measure accurately, their ratio should be approximately the number pi, and their average should be an even closer approximation.

After completing the table and discussion questions, discuss the results together as a group!

Also included in this download are teaching tips.

And if you like, this lesson would go along perfectly with the book, Sir Cumference and the Dragon of Pi. (You can also get more lessons to use with the book when you subscribe to my free email newsletter!)

I hope you have a fantastic Pi Day celebration, and learn all about circles, measurement, pi and irrational numbers!

{Click HERE to go to my shop to get your copy of the hands on Pi Day investigation!}

Looking for more Pi Day ideas? Check out this ever growing list of videos, lessons and games for Pi Day fun!

The post Pass the Pi: A Hands On Pi Day Investigation appeared first on Math Geek Mama.