Hands-On Geometry is a book we used in our school once a week for the last seven weeks as part of our out-of-the-textbook learning time.  It was awesome for the 5th grader, 7th grader and ME!

Using only a compass and a straight edge, the seven chapters of this book take your children through geometric 2-dimensional constructions.  They learn how to correctly create perpendiculars, parallels, kites, triangles, quadrilaterals and more. And all I had to do was copy the worksheets!

Each chapter consists of four to seven worksheets that include step-by-step directions for completing one of the 2-D constructions.  The directions were so clear that it didn’t matter that I never remembered doing this even in high school geometry.  Man, if only geometry HAD been this “hands-on” in high school, I might have enjoyed it more!  But this isn’t even intended for high schoolers!  It’s written for upper elementary and middle school children.

I’ve been asked many times where I find living/fun/logical/hands-on math to do on “the other two days” of the week when we step away from our Saxon textbooks.  Here’s a perfect example!

If you’re interested in learning more about our living math days, you might read this post.  Also, under the “Categories” section in the right sidebar, click on “Living Math” to find several related posts.

Most of you know we only do Saxon math three days a week.  The other two days of math time are spent doing living math, logic, critical thinking activities and/or math games.  Over nine weeks of last semester, one day a week was spent with Mahayla (7th grade) doing the lessons from a Prufrock Press book called Moving Through Dimensions.  It a very hands-on, out of the text book approach to teaching 3-dimensional geometry concepts.  Take a look at some of the fun we had together.

For a homeschooling family, it’s kind of expensive at nearly $40.00.  (I got it free for review.)  But Mahayla (and I) learned a lot – and both of us loved the hands-on, critical thinking approach.  One of the neatest things we learned was how to transfer 3-dimensional objects into 2-dimensional drawings and vice-versa.  I was also amazed at how simple very in-depth concepts became after the explanations and activities.  Hey, even I understand polycubes and Sierpinski Triangles now!  Even though the learning was from a text of sorts, I still consider the lessons to be living math.  We were learning skills and doing projects that will easily translate into real world tasks.

Taking several sized lids, I asked the kiddos to trace five different lids on a piece of plain legal-sized paper.  Using a centimeter tape measure, I asked them to measure the circumference and diameter of each lid and jot that down on their papers beside the appropriate drawing.

Using their measurements, I held my fingers on the tape measure to show the circumference and diameter of each lid.  I asked them to think about what they noticed that was similar about each of the measurements.   (The diameter is always about 1/3 of the circumference.  You can show this by folding the tape measure in thirds each time.)

Once they saw this “almost 1/3″ measurement concretely, I told them there was a way to figure out the circumference accurately every single time.  All they need to know is the diameter of the circle.  I secretly held a calculator and asked them to give me the diameter of several of their lids.  I would multiply the diameter by Pi (3.1416) and get their circumference measurement every time.

This, piqued their interest of course, and they couldn’t wait to find out the magic trick.  So, I showed them the formula C = (Pi)d (I don’t know how to make the Pi sign on the computer!) and allowed them to use the calculator to find several circumferences around the house.

They were so excited to tell their Dad that they will be able to help him decide what size tire he needs to buy the next time a giant tractor tire needs to be bought. 

I hope you aren’t getting bored with all the fun math we’ve been doing from Hardhatting in a Geo-World.  I just can’t help sharing what has been a GREAT math & science unit!  If you are getting bored, you’ll be glad to know this is the last post!    The other posts can be found here and here.

We did one last lesson on bridges.  This demonstrated the importance of strong anchors for a suspension bridge to be strong.  (Yes, that’s our art table.  We needed to place thumbtacks somewhere and the art table did the job.  Yes, those are baby feet standing on the art table.  He wants to be part of every activity we do!)

We also did a couple of thought-provoking activities concerning geometric shapes.  Do you know if you’re a square, tall rectangle or short rectangle?  We do!  Just measure your height and arm span to find out.  Equal measurements mean you are a square.  Longer height than arm span makes you a tall rectangle, while shorter height than arm span makes you a short rectangle.

And this activity involved predicting shapes when circles were attached and cut in various ways.

As of January 5th, we’re back on our regular math schedule – three days per week of textbook, two days of living math/problem solving/logic.  I’ll continue to share living math ideas, but they won’t come quite as frequently.  Until then, happy math lessons!

After completing the activity and filling in the chart provided, I had the kiddos show their findings with homemade graphs.

The book talked a little about why some bridges were stronger than others, but we wanted to know more.  I found this very cool site linked from THINK’s blog.

We also decided to complete her Bridge Challenge.

Caleb’s first attempt (not so strong.)

Caleb’s second attempt (much stronger.)

Mahayla’s bridge.

Tomorrow – we’re off to do some Christmas problem solving!  You can find some links here.  Happy Christmas learning!

We are having SO much fun with this book!  A little break from the regular math schedule has turned out to include such exciting math and science that my kids are actually asking to do more. 

Here are some photos from one of this week’s investigations….

Cylinder structures – Using various sized cardboard tubes and homemade paper cylinders, the kids had to estimate and find out how many books each cylinder would hold.  Their estimations and actual results were kept on a data sheet provided in the book.  We then went on to talk about why cylinders are so strong, why certain heights might be better than others and where in creation and the man-made world we find the use of cylinders.

I’ve also recently discovered a wonderful blog called Think! that offers weekly math/science/art related challenges for children.  To go along with the math investigations, I gave the kids this challenge.  Using only 12 straws, 12″ of tape and a pair of scissors, they had to create a structure that would hold an egg at least 1″ off the ground.  No tape was allowed to adhere the structure to the surface or the egg to the structure.  Fun!

Caleb’s structure.

Mahayla’s structure.

Today, math was in the form of a cookie baking day in preparation for our annual cookie baking and exchange party at my mom’s house tomorrow.  I’ll probably post pictures and few recipes soon.  Have a wonderful weekend!

Whether you have extra M&M’s, Skittles, Smarties, Runts, or just a big bunch of assorted candy bars, take a few minutes to turn those sweets into more than cavities.  Even little bitty ones can sort candy, place the pieces on a blank grid, then color a bar graph.  As the kiddos get bigger, all the more fun they can have graphing their candy.  Here’s a post from last year about how we used M&M’s to do all sorts of graphs.  This year, we did some candy graphing, but not quite as in depth as last year.

And don’t forget all those autumn leaves!  Make a nature walk even more productive by collecting a sack of leaves to sort and graph.

Literal or concrete graphs are not only fun, but help younger ones understand better what a graph on paper represents.

After making the concrete graphs, my children jotted the information and used centimeter grid paper to design their own graphs.

Happy graphing!

We started with a review discussion about the purpose of leaves and why they change colors.  After a bit of struggle getting everyone to pay attention during the discussion, I announced there would be a test later in the day.  (Yep, that meant I had to come up with some sort of test on leaves!  In all that spare time I have, you know.  Lucky for me, I was able to find this quiz online.)  I don’t do tests very often, but it was just the thing needed today to turn their attention back toward our discussion.

Math was next on the list.  Using a few of the leaves each, the kids had to find the perimeter of the leaves in inches and centimeters.  Then had to find the area in square inches, square centimeters and an unknown square unit.  Yarn and a measuring tape made finding the perimeter easy.

The area in square inches was found using one-inch tiles.  A one-inch grid could be used as well.

We used cemtimeter grid paper for finding the area in square centimeters.  (Simply trace the leaf and count the full squares inside the leaf.  We estimated the half and quarter filled squares to add more area.  For instance, if there were two halves, we would count them as one.  If there was a square about 3/4 full and another 1/4 full, we combined them as one.)

Graph paper was used to find an unknown square unit.  We talked about smaller units making for more accurate measurments.

Finally, we used Melissa’s idea of creating leaf bursts for an art activity to add to the nature notebooks.

I’m trying to keep up with my promise to show you what some of our unconventional math lessons look like around here.  Today I want to share how I plan a math lesson around a piece of literature.  It’s really very easy!

Like most everything I do, I grab a pencil and a pad of paper before I do anything else because I’m always writing notes.  If I don’t write something down as it pops in my head, it’s usually gone – never to return again!

So with pen and paper in hand, I grab a math book and quickly read through it.  I jot down math concepts and activity ideas that jump out at me.  The more you do it, the easier ideas start jumping out at you, I promise.  Sometimes, I’ll only write a couple of ideas – which means one day’s worth of lessons is all we’ll do before that book goes back on the shelf (or back to the library.)  If a lot of ideas are written, I might plan a few days worth of lessons around the book.

If NOTHING pops out at me, but I really like the book, I’ll do one of two things:

1.  We’ll just read the book together for fun – no math strings attached.

2.  I’ll do a quick google search to see if any good lessons are floating around in cyberspace just waiting for me to grab.

This week we read Sir Cumference and the Isle of Immeter.  I found a worksheet in my geometry folder that sent Mahayla on a hunt through the house measuring the perimeter, area and volume of various objects.

Caleb and I pulled out the one-inch tiles and measured the perimeter and area of things around the school room.  After a while, he tried his hand at measuring the length and width of the objects with the tiles, then multiplying the numbers to find the area.  Finally, he had to use a ruler to find the perimeter and area of many things.

Not only did this simple little lesson create excitement, but it allowed the kids to practice measurement, perimeter, area, volume, mental addition & multiplication and more complex math formulas such as 2l + 2w=p, l x w=a and l x w x h=v.  I could’ve done the same sort of teaching from the textbook, but I’m rather sure this lesson will stick a little better!

A couple weeks ago, I forced gave the moms at co-op the opportunity to try their hands at this process and they were VERY GOOD at it!!  Here is a great list of math ideas they came up with on many various math titles.  The activities are in no particular order – some will be appropriate for little ones, while others will be appropriate for jr. high.  I just asked the moms to pick a book and try to think of ideas that would be good for their own children.

The cute math clip art came from Philip Martin.

1.) Living math really is thoughtful, meaningful math.

2.) Living math isn’t hard to implement at all.

Here are a few of our January/February activities with links…

This Grab the Candy Game allows the kids to practice coordinate graphing and record keeping.  Valentine Battleship is another version of a coordinate graphing game.

How Many Different Ways Can You Make the Word Snow? is a quick activity to encourage problem solving and critical thinking.  This Heart Paths sheet is similar, but requires an extra “rule” to make the kiddos think a bit harder.

These Candy Heart Patterns are a little harder than they first seem.  I gave these to my 5th grader as “fun filler” and she realized half-way through that she hadn’t thought the pattern through completely!

Glyphs are pictures children make based on a legend or key of codes.  Not only does it reinforce problem solving, critical thinking and organization skills, but mapping as well.  Here’s a link to the Snowman Glyph activity we used.

These Symmetrical Heart Masks reinforced the concept of symmetry, as well as creativity.

This President’s Day activity is going to require charting and creating a graph.

Even this worksheet of Valentine Word Problems is a fun, seasonal way to change the pace from a textbook.

There are a few more activities we’ve done or have planned to do, but I’ll let you find them for yourself at MathWire – an awesome site for living math ideas!

When the activities are quick, I’ll do several in one day.

As for living literature, we’ve checked out this series of books from the library to go along with our Medieval Unit.  Some of them are better than the others, but they’ll go along well with our unit.

Sir Cumference and the First Roundtable: A Math Adventure

Sir Cumference and the Sword in the Cone: A Math Adventure

Sir Cumference and the Great Knight of Angleland

Sir Cumference and the Dragon of Pi

Sir Cumference and the Isle of Immeter

I’d love to hear about your living math learning!  If you share on your blog, please let me know so I can come visit!  Happy math times!