The Sieve of Eratosthenes as an Aid to the Memorization of Prime Numbers

Just as memorizing sight words can help a kid read better and more confidently, there are tons of math facts that, if memorized, will make a kid's calculation work quicker and more confident. 

Our culture is well used to having kids memorize the multiplication tables through at least 10 (through 12 is better!), and certain formulas like the quadratic formula or the Pythagorean theorem, but it's so helpful to just know, when you're busy doing your algebra, say, if a number is a perfect square or a Pythagorean triple, etc. It builds confidence when a student is learning advanced math concepts, and it increases their speed and fluency, which they will VERY much appreciate whether they're working through a page-long proof or an SAT problem set!

When my kids were pretty little, we dedicated the first ten minutes of the first car trip of the day to memory work, and they memorized a lot of advanced concepts by rote then (most famously, the first 25 digits of Pi, a party trick that they still both often pull out over a decade later, lol!), but it's a better aid to learning and to memorization to have them, whenever possible, create for themselves the anchor chart that contains the information I want them to memorize.

So when I realized recently that my teenager has lost most of the prime numbers to 100, I pulled back out the same activity that she used to create her Prime number chart back when the kids memorized the primes to 100 the first time around back in 2016.

It's the Sieve of Eratosthenes!

Creating the Sieve of Eratosthenes is simple. All you need are a hundred chart and some colored pencils or crayons. This hundred chart has the numbers by rows, and this hundred chart has the numbers by columns. This hundred chart is blank, for some sneaky real-world handwriting practice writing the numbers to 100. 

To create the sieve, you simply start with the first Prime number, 2. Don't color it, but color all of its multiples. Bonus points if you unlock the pattern and color it that way! The next uncolored number is your next Prime number, 3, so leave it blank but color all its multiples. It makes a pretty pattern, too!

Carry on through 7, and by the time you've colored the last multiple of 7, you'll have colored every composite number through 100, and every uncolored number is a Prime. Your grid will look like this:

photo credit: Wikipedia

I think the patterns that it makes are beautiful and fascinating!

While you're working, it's best if you have a Ginger Gentleman supervise you:

While you're working, you also might notice that you have a sudden, inexplicable swarm of Asian lady beetles inside your home. Would the Ginger Gentleman like to meet one?

He very much would!

Please note: no Asian lady beetles were harmed in *that* particular encounter. When Matt got home and found the swarm and went for the vacuum cleaner, though, well...

The Sieve of Eratosthenes is a quick, enjoyable, non-rigorous enrichment activity for an older kid, best used for a review of Prime numbers or to construct a memory aid/anchor chart. However, you can actually also do this activity with quite young kids, since multiplication is the only skill required. It's fun and hands-on, the patterns are pleasing, and it gives kids a really interesting math concept to explore.

Here are some good books to use with younger kids in partnership with this activity:

To extend the fun, younger kids can play Prime Number Slapjack or color in a Prime path maze. If kids are a bit older and are ready to properly learn about Primes, composites, factor trees, and the factorization of Primes, this lesson and this lesson are excellent jumping-off points. 

We have a lot of wall space in our home, and my kids have always enjoyed making large-format posters, maps, and charts to put on our walls. A large-format hundred chart mounted on a wall lets kids have a different experience coloring it in mural-style, and would also allow room for kids to write each composite number's factors into those squares. Alternately, extend the hundred chart to 1,000 and keep sieving, although I wouldn't blame you for eventually pulling out the calculator!

Here are some books that older kids and adults would enjoy; completing a reading assignment (and perhaps even a response essay!) builds context and adds rigor to an otherwise simple activity, and is a good way to facilitate different ages/abilities working on the same project:

Here are some other math facts that a student could aid fluency by memorizing:

• fraction/decimal conversions
• PEMDAS
• Quadratic formula
• squares
• square roots (perfect square factors and simplified square roots to 100)
• Pi to several digits
• Pythagorean theorem
• Pythagorean triples
• triangle identities
• SOH CAH TOA

P.S. Want to follow along with my craft projects, books I'm reading, dog-walking mishaps, encounters with Chainsaw Helicopters, and other various adventures on the daily? Find me on my Craft Knife Facebook page!

How to Sew a Fibonacci Quilt

I originally published this tutorial on Crafting a Green World.

Quiltmaking is surprisingly mathematical. If you love to sew quilts, then whether you realize it or not, your geometry and trigonometry skills are probably on point!

Why not celebrate how mathematically beautiful a well-made quilt is by making a quilt out of one of the most beautiful mathematical sequences that we know so far.

Let's sew a Fibonacci quilt!

The Fibonacci sequence, named after the guy who first noticed it, is a series of numbers created by adding up the two previous numbers in the sequence. You're given 0 and 1, so add them together and the next number is also 1. 1 and 1 make 2, but then 2 and 1 make 3. 3 and 2 make 5, 5 and 3 make 8, and you can just keep going, ad infinitum.

To make the Fibonacci squares, use each of the Fibonacci numbers as the length of the sides of a square--leave out 0, because that doesn't make a square, of course. Piece them together in a spiral, much like a log cabin quilt block, and you'll have a Fibonacci rectangle that looks like this:

CC BY-SA 4.0

We're going to go up one more number in the sequence, all the way to 34, because that's the last number in the sequence that you can make from one continual piece of yardage. Here, then, will be the finished measurements of the quilt pieces that you'll need:Now, pretend that each of these squares is the finished measurement of a quilt block--wouldn't that make an absolutely beautiful quilt?

• 1"
• 1"
• 2"
• 3"
• 5"
• 8"
• 13"
• 21"
• 34"

I used a quarter-inch seam allowance on all of the pieces, so add a half-inch to each of these measurements when you cut your quilt pieces.

You will also need the following:

• one 34"x55" piece of backing fabric. I backed this quilt with nothing but another piece of quilting cotton, and I am in love with how light it is. Not every quilt has to be warm enough for winter--some quilts are destined for summertime naps on the couch!
• double-fold bias tape. You can make your own double-fold bias tape, but I buy mine from Laceking on Etsy.
• cutting and sewing supplies.

1. Pre-wash, measure, and cut fabric pieces. Don't forget to add 1/2" seam allowance to each measurement!

2. Piece the quilt. This Fibonacci quilt is easy to piece--just follow the above diagram, adding each piece in numerical order of its measurement. Be very strict about your 1/4" seams, and iron after every seam. I like to use a walking foot when I sew quilts, so if you're struggling to feed your fabric evenly, that might be worth checking out.

3. Put the front of the quilt with the back, wrong sides together. Pin it as much as you can stand to!

4. Sew double-fold bias tape all the way around the quilt. Miter the corners as you go to save time--I really like the first method shown in this video.

When you're finished, you'll have a lovely, light summer quilt that's both aesthetically and mathematically beautiful:

Interested in more cool math activities? Check out my list of even MORE fun Fibonacci sequence stuff!

Now get back to your sewing machine and get going!

How to Square Binomials and Trinomials using Area Models

Let's say that you have two lengths: a and b. You would like to know what area would be covered by a square whose sides are each of these lengths combined.

The equation for that is (a+b)^2.

But how do you actually multiply that?

The algebraic way is to use the FOIL method: First, Outside, Inside, Last. This gives you (a+b)(a+b)=a^2 + ab + ab + b^2, which you simplify to (a+b)(a+b) = a^2 + 2ab + b^2.

That's fine algebraically, and you should totally memorize it, but here's what you should VISUALIZE when you do this, because here's what makes sense:

Visualize sitting on the rug in your family room. It's a Friday afternoon, soooooooo close to the end of your school week, and you'd very much rather be done with school and go walk your dog or listen to your music, but your mother wants to do one final project together before she sets you free. She hands you and your sister the decanomial square manipulatives and asks you to set them up:

Stacking the area models is NOT part of setting them up, but is also nearly irresistable.

Fun fact: a decanomial square is the same concept as a binomial square or a trinomial square. Whereas a binomial square is (a+b)^2 and a trinomial square is (a+b+c)^2, a decanomial square is the representation of (a+b+c+d+e+f+g+h+i+j)^2. How would you like to factor THAT puppy without being able to visualize an area model to make sense of it?

Your mother asks you to choose two squares from the decanomial. You'll label one square as a^2, with sides measuring length a, and the other square as b^2, with sides measuring length b. The challenge is build an area model of a square whose sides are each length a+b; to complete this challenge you may use a^2, b^2, and two other rectangles of your choice.



Visually, it's not hard to find the rectangles that match to complete the square, but it might take a little longer to notice that these rectangles each have two sides that match the length of one of the squares.

The solution to the puzzle, then, is (a+b)^2 = a^2 + ab + ab + b^2. Since you have two rectangles labeled ab, you can simplify your equation to (a+b)^2 = a^2 + 2ab + b^2.

This is exactly what you get when you the FOIL method, but doesn't it make a little more sense to see it?

Unfortunately, your cruel mother now tells you that you have to add a THIRD square to your puzzle. She. Is. So. MEAN!

You add a third square and label it c^2, with sides measuring length c. The puzzle is now a trinomial square, (a+b+c)^2, and your challenge is, once again, to complete the square. Your mother does not tell you how many pieces you have to use to complete this trinomial square, so you make yourself a shortcut:

  Actually, this works algebraically!

The equation you've created is (a+b+c)^2 = a^2 + 2ab +b^2 + 2((a+b)c) + c^2. As long as you can complete the puzzle with area models that have sides that relate to lengths a, b, or c, you can translate the solution algebraically... but this isn't the simplest solution algebraically.

THIS is the puzzle solution that's also the simplest algebraically:

(a+b+c)^2 = a^2 + 2ab + b^2 + 2bc + c^2 + 2ac.

You can make infinitely bigger squares with an infinite number of terms. I mean, think of how many terms that decanomial square has, and yet it still follows the pattern you can see in the binomial and trinomial square.

And as soon as you memorize the binomial square and trinomial square equations that you created, you can go listen to music out on the back deck with your cat!

Girl Scout Cookie Booth Math: Equal It Out and Assign It Fairly

We've gone through several iterations of the cookie booth tracking form. This is the most recent!

Okay, you guys. This is why you studied algebra. You studied algebra when you were 13 because one day you'd be a Girl Scout troop leader, and your Girl Scouts would go mad for cookie sales, and you'd have to schedule nine kids into 40+ hours of booths every week, and then you'd have to figure out how to divide the booth sales fairly between everyone who worked, for all the various hours they worked, and you'd also have to make sure that your booth sales matched your booth inventory every time, and that your physical inventory matched what your database says you have.

And don't forget that you'd have to figure out how many of each type of cookie to even bring to every booth in the first place!

Functional literacy in math. That's what algebra gives you.

The most important thing is that whatever method you use, you MUST WRITE EVERYTHING DOWN. An accurate paper trail is key. Count your inventory. If it's off, note it. Count your money. If it's off, note it. Know exactly how many boxes of each variety of cookie you're taking to each booth. Know exactly how much starting cash you're taking. Know exactly what kids and what adults worked there, and exactly how long they worked. Know what you took in via credit cards. Know what you took in via cash.

The key to doing all that easily and efficiently is using a booth form. Some councils make their own, and you can find them if you Google, but if you know exactly what information you want on your form, and especially if your partner is a graphic designer (yay!), then you can make your own. 

At the top of my form I fill in the booth location, time, and date. My Girl Scout troop works so many booths that we might be back and forth between the same location twice in one day, or three times in a weekend.

Below that is space to write in all the kids and adults who worked, and the hours that they worked. We sometimes have booths that are six to eight hours long in a single spot, and have to be able to record kids coming and going all day for various hours, or even taking a break and coming back later. You also need to know all the adults who were present, if for no other reason than you know who to loop into the mass text if the numbers come out wrong.

Next come a series of boxes that are specific to what our troop needs. The first box is to record the amount of money in the Operation: Cookie Drop/Cookies for a Cause donation can. The second box holds tally marks that must be made every time someone runs a cookie donation on credit card, because our credit card processing company doesn't let you look at transaction details easily--if someone thinks that they might have forgotten a cookie donation credit card transaction, or if the final amount is off, I have to manually click through every single transaction to see its details, and it. Is. TEDIOUS.

The third box records the amount in our own troop donation can, although this is not incorporated into any other calculations. Early in the season, the kids were trying to collect enough donations to buy a box of Girl Scout cookies for every kid in our local Backpack Buddies program, and so we'd often use the box to record that, instead. If we do local donations again next year, I will likely include yet another box just for that.

The next couple of boxes are so that adults whose phones were used to run the credit card processing app can record their transaction totals. Adults come and go, too, so you need more than one box for this.

The final two boxes record the amount of starting cash and ending cash. Our cookies are all $5, so we start with $200.

2023 Update: Our cookies are now all $6 a box, so we go back and forth between starting with $200 or $300 while we try to decide which is better. Whatever number we use, we record it!

I devote a lot of space simply to the cookie inventory. Using a lot of space to keep the numbers separate helps avoid confusion, and the color coding and illustrations are visual reminders to help keep everything correctly sorted by type. The bigger spacing lets the kids, who often can't write tiny, also do inventory, and it leaves some room for calculations.

Notice that unlike a lot of booth tally sheets, mine doesn't include a place to record parents who take cookies from the booth supply to add to their kid's personal orders. I HIGHLY discourage doing this at cookie booths, because it's not the time and place and it's distracting, but in case of emergency I'll do it and just subtract what they take from the starting inventory of that booth. Another reason for all that space between the inventory number boxes!

Below the inventory section is where you walk through the two main booth calculations: the calculation of inventory, and the calculation of money. The first line is inventory, so you copy down what you sold of each type of cookie, then what you sold of Operation: Cookie Drop/Cookies for a Cause (include what you sold from both the donation can and by credit card). Add those together, multiply by the cost per box of cookies, and you'll see how much profit you earned.

The second calculation is just of the money. Copy down your ending cash, then your starting cash, and subtract them. Add to that your total credit card sales and the amount in the Operation: Cookie Drop/Cookies for a Cause can (do NOT include the amount on credit card here, because that's already included in your credit card sales). Does your final answer equal your final answer from your cookie inventory?

If yes, YAY!!!!!

If no...

1. Recount your cookies. And I mean REALLY recount them! Are you trying to count them in your car? Unload them all and spread out so you can see what you've actually got. Repack everything so that you have only one partially full case of each type and the rest are all full cases--multiple partial cases of the same type of cookie are inefficient to carry and a nightmare to count.
2. Recount your cash. Get someone else to count it this time, or ideally two someones.
3. Double-check your credit card transactions. Log out and then back into your processing app to make sure its transaction record is completely current. Make sure no other adults ran credit cards on their phone and just forgot to record it. If you use Digital Cookie, double-check that no adults accidentally ran cards through their kid's site instead of the troop's site. Look through each transaction and make sure you didn't charge someone twice. If a credit card transaction seems weirdly large, see if anyone remembers a customer buying a ton of cookies at once--if nobody remembers it, someone probably punched something in wrong during the transaction.
4. Recalculate all of your math. Do the inventory subtractions again, with a calculator if necessary. If you're counting by cases, again, use a calculator to make sure you've got the total number of boxes correct. 
5. If you can, compare your current total troop inventory to the total troop inventory that you should have. I can rarely do this, because I've got multiple booths going at one time, but if you've just done one booth, then a whole troop inventory might find that you mysteriously are missing a box or have an extra box that will match what you're over or under.

Here's what these booth forms look like in real life:

You can see we write all over them and alter them however we need to. Next year's updated form might include a Notes area, because there are often extra details we want to record...

...such as how on earth we can end up with $817, when all of our cookie boxes are FIVE DOLLARS EACH. I SUPER love it when everything equals out, and I have to admit that when we're over or under, even by a little bit, it drives me nuts.

ESPECIALLY when we're over by a number that doesn't even equal a box of cookies!

Thankfully, we mostly equal out:

But sometimes we don't, sigh:

But mostly we do:

So how do you assign all of these sold cookie boxes fairly to kids who worked varying hours at each booth?

You turn it into a rate-time-distance problem!

Distance = rate x time. Distance will represent how many cookie boxes a kid sold. Rate is how many boxes the booth sold per hour, and time is how many hours the kid worked.

To find the rate for a booth, add up the total number of hours every kid worked altogether. Let's say Kid A worked 2 hours, Kid B worked 3 hours, and Kid C worked 5 hours at a booth. Maybe they worked all together for part of the time, or maybe they all came in shifts--doesn't matter. Just add up the totals and you have 10.

Divide the total number of cookies sold at that booth by the total number of hours worked. Let's say that this booth sold 100 boxes of cookies. 100/10 = 10, so 10 boxes of cookies per hour worked is the rate for this booth.

Rate x time means that to find each kid's distance, or total boxes of cookies that kid, personally, gets credit for, all you have to do is multiply the number of hours they worked by the rate.

Kid A earned 2x10, or 20 boxes of cookies.
Kid B earned 3x10, or 30 boxes of cookies.
Kid C earned 5x10, or 50 boxes of cookies.

This also works with partial hours. If Kid A had worked 2.75 hours, then they'd have earned 2.75 x 10, or 27.5 boxes of cookies. You can't allocate partial boxes, so someone will have to gain a whole box or lose a whole box somewhere.

If you're mathy, like me, then you can eat up all your free time doing all this pleasant cookie booth math. But even if you're not mathy, you should be functionally literate in math, which means that you should be able to easily handle the calculations required. And, of course, it's very important to give ownership of this process over to the kids, whenever you're able. An elementary school student can do all of the inventory calculations with your oversight. A middle school student can run all of this math with your guidance. A high school student who's had enough Algebra 1 to know how to solve a rate-time-distance problem can do it all without you standing over their shoulder.

And what's more, they SHOULD be doing this. THIS is why we study math. We study it so that we have the tools to solve whatever problems come our way in life. We study it because it's important to be functionally literate in all areas. We study it because we want to be able to calculate, distribute, count, and audit.

We study it because one day we might find ourselves in the Girl Scouts, and Girl Scouts sell cookies!

P.S. Want to read more about Girl Scout cookie booth math and marketing? Here's my complete series (so far!):

• Girl Scout Cookie Booth Analytics, or, Math Helps Girl Scouts Sell More Cookies!
• My Very Own Girl Scout Cookie Booth Calculator: How Many Cookies to Order for Booth Sales
• Girl Scout Cookie Booth Math: Equal It Out and Assign It Fairly
• Girl Scout Cookie Marketing: Kid-Led and Kid-Made Signage and Campaigns

P.P.S. Want to know more about all the weird math I have my kids do, as well as our other wanderings and wonderings? Check out my Facebook page!

Homeschool Math: Resources for Art of Problem Solving's Introduction to Algebra: Chapter 1

The big kid's eighth-grade math spine is Art of Problem Solving's Introduction to Algebra. To her, it feels like a big step up from Math Mammoth 7, which she pretty much breezed through with little outside assistance, and she is working through the text quite slowly, much more slowly than she'd do in an organized class.

Our focus, though, is that she actually deeply understands the concepts as she goes, which is far more important than zipping through at a steady pace.

Part of that process is providing a lot more scaffolding to the concepts than AOPS provides. By that, I mean that if a kid doesn't understand a concept when it's explained one way, then I find a couple more ways to explain it. I find a visual way to explain it. I find a hands-on way to work through it. I find drill problems to practice and cement it. That gets harder as the math gets more advanced--not because there aren't visual or hands-on ways to explain any math concept, because there always are, but because visual and hands-on learning is often neglected for older kids.

Here, then, are my hard-won extra resources for Art of Problem Solving's Introduction to Algebra chapter 1, which covers the order of operations, distribution and factoring, an introduction to equations, exponents, fractional exponents, and radicals:

Order of Operations

• Order of Operations notebook pages. I use a LOT of material from Math Equals Love, and that's because it's excellent, relevant, and just the kind of visual, hands-on exploration that makes math make sense. Here, I used the practice problems with the kid so that she would have examples to put in her math notebook. We also discovered right away that her biggest issue with algebra is going to be writing out solutions step by freaking step. I still do not understand why she is balking at this, but she will actually erase a previous step and write the new one in its place rather than writing out the solutions. It's maddening. I've tried explaining to her every way I know how why writing out the solutions is important, but honestly, I think she's just decided to be stubborn about it. I now employ natural consequences in that if a problem is incorrect and she's written out the solution for it, I will mark exactly where she made her mistake and sometimes give her a hint about what to do next. If she's not written out the solution, I just mark it incorrect to try again, full stop.
• Order of Operations worksheets. Drill problems, if you need them!

Distribution and Factoring

• Distributive Property notebook pages. The section on distribution and factoring in this chapter is really just a review on combining like terms--they get into factoring for real in the next chapter. For this, then, I just gave the kid some more drill problems to practice combining like terms.
• Distributive Property worksheets. Drill problems, if you need them!
• Combining Like Terms worksheets. 

Introduction to Equations

• Evaluating Expressions question stack. This section mostly reviews the concept of an equation, so the kid didn't need any extra work here. I had this question stack activity in my back pocket just in case, though.
• Evaluating Expressions worksheets. Drill problems, if you need them!
• One-Step Equation worksheets.
• One-Step Equation word problem worksheets.

Exponents, Fractional Exponents, and Radicals

• The same Order of Operations notebooking pages, above, have a section on Negatives and Exponents that I think is necessary to review before beginning to learn exponent rules. I saved that section for the kid to do here.
• Exponent Properties. THIS handout was the game changer for the kid's understanding of exponent rules. Before we completed these handouts (she and I both worked them, then compared our answers and discussed), she did not understand exponent rules and could not remember them. After we completed these handouts and discussed them, she understood them, and they were easy for her to memorize. Here's a little of my work in progress with the handouts:

See how working out the solution means that the math rule makes sense? Math rules. Make. Sense. If you don't understand WHY a math rule works, then you better figure it out, because there's no point in memorizing it otherwise.

• Exponents Game. I don't usually make games and manipulatives anymore, as often they don't get enough use to justify the work and materials. I made this game, however, so that the kid would have some practice without having to write and write and write.
• Here the Exponent Rules are broken down more quickly, as a review. I'm holding onto this to present at another time, if the kid seems like she needs to explore the concept again.
• Exponent Rule Mistakes. We also didn't use these pages, but only because the kid was ready to move on. They're still a possibility if she needs to review the concepts again later.
• Radicals. The kid started off absolutely baffled by radicals, so we used every single one of the resources here, other than the ones on rationalizing the denominator. Explicitly working through these resources on factoring radicals, adding and subtracting radicals, and multiplying radicals is the only way that the kid was then able to understand the AOPS section on them. We also both completed the entire prime factorization chart in one evening, because she was enjoying it (!!!). Previously, both kids have also memorized all of the prime numbers under 100, and I will tell you that has made life incalculably easier for both of them.
• Two Methods of Prime Factorization. I taught the kid both of these methods, but we both tend to prefer the factor tree, I think because we have the primes under 100 memorized.
• Exponent worksheets. Drill problems, if you need them!
• Radical Expressions worksheets.

Enrichment

These resources don't always teach the specific Chapter 1 skills, but they're good for enrichment and to encourage a mathy mindset:

• Algebra the Beautiful. Read this for yourself so that you've got a ready store of fun facts about the different concepts. You can also use it as a jumping-off point to add some cross-curricular studies into your mathematical mix.
• Bill Nye's Solving for X. It's probably not going to teach a kid the concepts all by itself, but we LOVE Bill Nye!
• Cyberchase Algebra. This could well be too young for your kids, but mine LOVED Cyberchase when they were very little, so it's mostly a nostalgia watch with a teeny bit of sneaky algebra enrichment.
• Emmy Noether: The Most Important Mathematician You've Never Heard Of. Strong female role model!
• Math Without Numbers. This will be too advanced for most kids this age, but it's great for changing YOUR way of thinking about algebra, which will then help your kids.

P.S. Want to follow along with my craft projects, books I'm reading, road trips to weird old cemeteries, looming mid-life crisis, and other various adventures on the daily? Find me on my Craft Knife Facebook page!

Homeschool Math: How to Model the Pythagorian Theorem with the Decanomial Square

Neither kid has yet studied the Pythagorean theorem in math, but they DID both study Pythagoras a couple of weeks ago, and what better way to bring him to life than to model his most famous theorem?

And fortunately, if you focus on the Pythagorean triples, the Pythagorean theorem is also actually really easy to model, and quite accessible to even a younger learner. To two kids who've studied area, square numbers, and triangles, it's a snap!

First, of course, you need to build the decanomial square. Notice that the kid starts with the square, then adds the matching pieces in descending order:

When those are all placed, she finds the next largest square and carries on:

It's also good to model your work on gridded centimeter paper. This makes the translation between model and equation much clearer:

Find the square whose side matches side a. That square is a squared. The square whose side matches side b is side b squared. The hypotenuse is c. Once you've got the squares in place, you can start to do the calculation:

You can either work the calculation first, or find the square whose side matches the hypotenuse first. Either way, your work should match:

Although that's the only Pythagorean triple that's modeled in whole pieces in the decanomial square, you can use the decanomial square and/or Base 10 blocks and Cuisenaire rods to piece together the squares of the larger Pythagorean triples:

And to prove that a squared plus b squared really does equal c squared, break down the blocks that make up a squared and b squared--

--and put those pieces on top of c squared. You'll have to puzzle them together a bit, but in the end, they should fit perfectly!

There are lots of other fun ways to model the Pythagorean theorem, although since they also mostly rely on the 3, 4, 5 Pythagorean triple, it can get tedious if you do too many of them with the kids. It's more fun to make larger square models of the other triples to test, or to use the Pythagorean theorem in real-life situations.

Here are some other resources that we've enjoyed:

P.S. Want to follow along with my craft projects, books I'm reading, dog-walking mishaps, encounters with Chainsaw Helicopters, and other various adventures on the daily? Find me on my Craft Knife Facebook page!