Showing posts with label math. Show all posts
Showing posts with label math. Show all posts

Friday, March 12, 2021

I Made Rainbow Fibonacci Placemats, Because It's My Pandemic Hobby

 

I've been obsessed with rainbows (it's like alphabetizing for colors!) and math (so many patterns, so many puzzles, so many correct answers that it's possible to suss out!) forever, and teaching myself how to make my niece's Sierpinski triangle quilt also got me obsessed with how satisfying it is to sew absolutely perfect seams and angles and iron them just so and end up with a creation that's gloriously precise.

Add in a nice long podcast, preferably in the true crime genre, and nowhere else to be and nothing else to do but sew and listen, and I'm as happy as a clam can be while remaining socially isolated and terrified of the ongoing global pandemic...

And we actually DID need placemats! I Googled around for a while trying to find a pattern I liked, but I don't like anything floral or cutesy or too curvy, I'm not in the mood for something novelty or pop culture, and the only color schemes I really like are galaxy and rainbow, because I am ten years old. I was about to just put the whole project on the back burner when I suddenly thought, "Ooh, I could sew the Fibonacci sequence just like I did when I made my Fibonacci quilt, and stop on a number that's the perfect size for a placemat!"

"Let's see... how many different squares would that give me.... OMG SEVEN?!?!?!?"

And the rainbow Fibonacci placemat is born!

Don't those squares, sewn so precisely and ironed nice and sharp, soothe that tense pit in the bottom of your stomach?

And don't you find that concentrating on cutting and pinning the layers together and matching everything up just perfectly requires so much focus that there's no room for any actual original thoughts in your head?


So pretty! So orderly!


And isn't it nice to quilt such nice, straight lines while listening to the behind-the-scenes story of the Chippendale dancers?


And doesn't it make your kid's ravioli soup, ripe peach, and half a homemade pumpkin chocolate chip cookie look elegant instead of the result of you 1) cleaning out the freezer and 2) foisting off the last of the canned goods you panic-bought almost exactly one year ago today because it was practically the only thing still on the shelves at THE HARDWARE STORE (stay classy, Midwest!)?

Also, both that placemat and table are kinda dirty. I'd warn you not to look too closely, but I suspect you already knew I'm not landing on the cover of Good Housekeeping anytime soon...

I'm so happy with how these placemats turned out!




I wanted to make more, but we don't actually *need* anymore (especially not if I'm going to make myself a whole other set out of rainbow prints, ahem...), so I made a rainbow Fibonacci placemat listing in my Pumpkin+Bear shop, because making them for other people is just as fun!


Anyway, I suppose that mathematical rainbow-themed quilting is as good of a pandemic hobby as anything else!

Saturday, October 17, 2020

How to Remake a Puzzle

I originally published this tutorial on Crafting a Green World.

There are lots of cute ways to upcycle an old puzzle into something completely new, but what if what you'd really prefer is simply another puzzle?

I mean honestly, have you ever looked at the picture on the front of a puzzle and thought, "Huh. I could do better than that!"

Well, you absolutely can!

It's surprisingly easy and doesn't take a ton of artistic talent to remake a puzzle into one that's brand-new-to-you.  This is a great craft for kids or a great way to make an educational toy for a kid. Here, for instance, I've remade a very odd puzzle (I'll show you in a minute, but trust me--it's ODD!) into a puzzle of the digits of pi. What classroom wouldn't want to have that on the shelf in readiness for a rainy-day recess?

Here's how to remake a puzzle of your own!

Supplies Needed

An old puzzle to upcycle

I have a very firm Puzzle Policy, which consists of this: we buy puzzles only from thrift stores or yard sales, never spending more than a buck or two. We happily put together our puzzles, knowing that it's highly likely that there will be missing pieces. When there are, we don't care, because we still had the fun of putting together the puzzle and anyway, it only cost a couple of dollars; I'll recycle or upcycle that puzzle, then, with a clean conscience.

When the puzzle gods smile down upon us, however, it's a real treat, and then when I put the puzzle back in the box I'll securely tape it shut, tape on a note that reads "NO MISSING PIECES," and donate it to a thrift shop to move on to some other lucky soul. So even though my family loooooved putting together this very, very, VERY strange puzzle, it has two missing pieces, so recycled or upcycled it must be:

Acrylic paints and brushes

You'll be watering the paints down, so a little will go a long way.

Paint Pens

These can be pricey, but you don't have to buy a full set if you're planning on a limited color scheme. I got by using only black for this project.

Spray sealant (optional)

Aerosol sprays are the WORST, but if you feel like you have to seal your work--I would only if it's quite detailed and delicate--then you'd better use this instead of a brush-on sealant, which will go on too heavy and stick the puzzle pieces together.

Directions

1. Assemble the puzzle, turn it upside down, and make any necessary repairs. When my kids were younger and much more emotionally attached to their puzzles, I figured out how to make a pretty decent replacement for missing pieces--it's at least good enough to make the puzzle complete again, although of course you can tell the difference:

Here's my tutorial for how to repair a puzzle by recreating missing pieces. Give it a try!

2. Paint the puzzle with watered-down acrylic paint. Watercolors would work, but wouldn't be vibrant, and acrylic straight from the tube would be so thick that the puzzle pieces might stick together. Instead, use a paint palette to water down your acrylic paints, and use those to paint your puzzle.

Pro tip: to avoid the colors bleeding together, let an adjacent color dry before you paint right up next to it.

3. Add details with paint pens. Since the acrylics have been altered to behave more like watercolors, when you want to add detail to your painting, you'll want to use paint pens. These are spendy, but they're absolutely terrific for fine work, and you can add tons of detail that you'd otherwise need the tiniest paintbrush for.

After all the paint and glue has dried, the only thing left to do is work your brand-new puzzle!

As you can see, this puzzle is even more fun than working one whose only claim to fame is that you bought it from a store, and the more time that you put into it, the better it can look. I kept mine pretty plain, because my children are heathens and wouldn't appreciate it anymore if I'd spent fifteen more hours on it making it look cute.

You could add a ton more embellishments with the paint pens alone, and there's nothing to stop you from adding even more decorative elements, such as Swarovski crystals, glitter, or anything else that you fancy.

Thursday, September 3, 2020

Homeschool Math: Perfect Squares Hiding inside Area Models

Syd was simplifying radicals the other day, and not having a fun time of it. She was struggling to link the concept of factoring the radicand to simplifying, and I was trying, as usual, to think of hands-on manipulatives that might clarify the process. 

I did NOT find a way to model simplifying radicals using manipulatives, alas, but while I was playing around with the decanomial square I DID find a hands-on enrichment that kids who are first learning the concept of perfect squares might enjoy.

I like this little activity because it connects the mathematical definition of the perfect square with the Montessori-style sensorial skill of eyeballing it, or even measuring it by feel. Although you're technically not allowed to eyeball stuff as mathematical proof, pattern recognition via the senses is very important. That's how kids learn to read, for one thing, and it's how IQ tests are built, for another. 

Use this activity with a kid who's first learning, or reviewing, the concept of the perfect square. You can do it with paper area models that a kid can draw and color on, or you can do it, as I've done here, with the decanomial square model, which is extra fun because it has pieces you can manipulate. Kids could try to find the largest perfect square(s) that would fit inside the area model, or just find any perfect squares that would--whatever they find fun and you find helpful! Here are some models that show examples:

These first two are when I was still thinking I might figure out a way to model simplifying radicals. I LOVE combining manipulatives with a dry-erase board to help kids connect the model to the algorithm it represents.


For all these examples, I've pulled an area model from our decanomial square, and we're arranging the perfect squares on top of it, leaving, of course, a remainder since the area models aren't themselves perfect squares.









You can write algebraic equations with these, showing how to use the Order of Operations and/or solve for x. For example:

5 + 5 x 5 = 30

or

8^2 + 2^2 + y = 80

You just can't, you know, use them to model how to simplify radicals...

The search continues!

P.S. Here are the resources that I used to help both kids master radicals.

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Two Years Ago: We Went to Canada and Saw the World's Highest Tides in Fundy National Park
Three Years Ago: History of Fashion: How to Put on a Doric Chiton in Ancient Greece
Four Years Ago: Cruise to Alaska Day 7: Haines
Five Years Ago: Watercolor and Wood Burned Building Block Excess
Six Years Ago: Cardstock Covered Wagons on the Oregon Trail
Seven Years Ago: Demonstrating the Commutative Property of Addition using Cuisenaire Rods
Eight Years Ago: Cave Painting with Story of the World Ancient Times
Nine Years Ago: Giant Cardboard Constructions
Ten Years Ago: Have You Climbed a Tree Lately?
Eleven Years Ago: Where Small Things Will Soon be Sorted
Twelve Years Ago: Scooby Dooby Doo, Where are You?

Thursday, July 30, 2020

How to Homeschool Math: Our Curricula, Resources, and Activities for Middle School and High School (So Far!)

In a lot of ways, middle school and high school are GREAT years to homeschool math. Homeschool math curricula are written to the student in these grades, so most of your mentor time is simply keeping them on track and making sure they're completing their work correctly. 

This, though, seems to be when a lot of parents lose confidence. Yes, you can teach your kid how to multiply multi-digit numbers with your eyes closed (I hope! If not, just look it up--it's easy to re-learn). But if your kid needs help with quadratics, you might be at a loss, because you don't remember that nonsense AT ALL, and now you're back to thinking that you're terrible at math, you don't like it anyway, and you probably shouldn't be teaching your kid, anyway.

I'm not going to tell you whether or not you should be teaching your kid, but I AM going to tell you that middle school math and high school math are NOT beyond you. The only reason you think they are is that your kid's independence has meant that you've stopped following along with them in their math curriculum, and of course you're going to be lost when they ask you to help with some random skill that you haven't reviewed the background for. 

So keep following along in your kid's math, even if they can follow it themselves these days. And even when they don't need your help, keep doing enrichment with them. All those fun games and activities and craft projects and puzzles that inspired them to love math in the elementary years are still fun and inspirational, and being older and knowing more skills just means that they can be exposed to even cooler games and activities and craft projects and puzzles! Sometimes math is going to be a real slog for them, because some of those skills ARE hard and DO take a lot of grit to master, and while that grit is also a very important lesson, kids also need to be reminded that math is fun and exciting. 

That's what YOU'RE for!

The kids' curriculum for middle school math was the same as for elementary school mathMath Mammoth's Light Blue Series. After a kid finishes level 7, she's ready for Algebra!

For part of middle school, we also used this text as a combined math/science/history study:



This is Will's curriculum for high school math:

Here are some other textbook, reading, and viewing resources that we used:

Depending on your kid's interest and attention span, middle and high school open you up to the whole range of books and documentaries geared to a general audience. You are going to be shocked at how many really fun non-fiction books on mathematical topics there are! There are histories, biographies of mathematicians, cool things that people have done using math--seriously, do a catalog search at your library! It's going to be awesome! There are also tons of documentary series and high-quality feature films--think about the first time you watched something like Primer or The Astronaut, and how inspired you were. Now imagine that you're a teenager with a billion opportunities in front of you. It's really exciting!

I don't have a list of must-have manipulatives and enrichment resources for these grades, because there aren't really any "must-haves." That doesn't mean that middle and high school kids DON'T need manipulatives, however--just the opposite! Middle and high school kids have the sophistication to use anything in the real world that interests them, or any high-quality tools and supplies that might help them engage in their passions or inspire them to try something new. Here are some of the super random things that my kids have enjoyed and that have given them a deeper understanding of mathematical concepts:



Here are some of the other exciting stuff that we've done in middle and high school math. Just like with elementary math, some of these projects were inspired by what the kids were currently studying or their interests at the moment, and some were cool projects that I presented to them or that we did as a family. I like both ways--if the kids are studying geometry, for instance, that's the perfect time to introduce some really wild and fun geometry activity, but it's also the perfect time to introduce some really wild and fun very much NOT geometry activity, too. Get those brains ranging deep and wide!


  • Archimedes' Method of Exhaustion to Calculate PiWe're kind of collectively obsessed with pi in our homeschool. I'm also a huge history nerd, so I was all about showing the kids how Archimedes approximated pi by calculating the perimeters of of inscribed and circumscribed regular polygons on a circle... and then I had the kids try it!
    • Level this down to a challenging elementary activity by simply showing a kid how to draw circumscribed and inscribed regular polygons on a circle.
    • This activity also works as-is for high school.
  • Map ColoringUsing the greedy algorithm makes map coloring surprisingly systematical (and fun!), but it's still a tricky little challenge.
    • Mathematical map coloring can be used as-is as a challenging elementary activity.
    • Mathematical map coloring can be incorporated into geography and history studies at the high school level. There are always maps to color!

  • Model Algebraic Equations with a Decanomial SquareThis is a terrific introductory exploration of how to write and solve algebraic equations. 
    • Level this down to a challenging elementary activity by using it as an extension after a kid learns how to calculate area.
    • Level this up to high school by asking the kid to simplify the equations they create.



  • DIY Binomial Square/Trinomial Square ManipulativeThis manipulative is related to the binomial cube/trinomial cube, below. Your kid might encounter binomial squares in their math curriculum, but even if they don't, I am a BIG fan of introducing it here. It's great practice in modeling formulas that might seem abstract (and therefore scary), and it's fun!
    • Level the binomial square/trinomial square manipulative down to a challenging elementary activity by playing with it after the decanomial square.

  • Binomial Square from a Decanomial SquareI love to present the same concept several different ways and at lots of different times. This activity uses the decanomial square to build a binomial square, and then explore creating an equation that represents that configuration. I like it even better than the manipulative, above, because there are so many more choices a kid can make.

  • Area Models to Square Binomials and Trinomials. Before (or while) you teach the FOIL method to square a binomial, let kids actually see what they're doing! THIS is how FOIL makes sense!
    • Level the area models to square binomials and trinomials down to a challenging elementary activity by exploring it without connecting it to the FOIL method.

  • Fibonacci SequenceMiddle school is a great time to explore the Fibonacci Sequence for fun! Kids can create their own Fibonacci Sequence models, and then play with pattern creating using them. 
    • Level Fibonacci Sequence exploration down to elementary by providing the models as manipulatives, and letting kids concentrate on the pattern creating.
    • Level Fibonacci Sequence exploration up to high school by including more reading/viewing resources that discuss extensions and applications.

  • Fraction Division with Cuisenaire RodsIf all you know about dividing fractions is to invert the divisor and then multiply, you don't *really* understand dividing fractions. Model it with Cuisenaire rods, and you can see what you're actually doing!
    • Level this to high school by using it as a review when a kid can't immediately recall the correct algorithm.
  • Fraction Multiplication Model Sun CatchersHere's a really fun project that's as much art as it is math. As long as you choose your sample problems correctly, you'll make a model that will both demonstrate exactly what you're doing when you multiply fractions, AND you'll have a beautiful sun catcher!
    • Level this down to a challenging elementary activity by pre-printing the model blanks and encouraging more open exploration of the ways that one can combine two colors rather than specifically making multiplication models.
  • Geometry Nets with Building ToysConvert a kid's building toys into polyhedra models, either challenging kids to come up with specific polyhedra or encouraging open exploration to see what polyhedra are possible. Here are some building toy sets that work well with this activity:
 


  • Level this activity down to elementary by introducing the concept of geometry nets during free play, and encouraging kids to create the nets for some of the Platonic solids.

  • Graph Candy. Here's a delicious way to review fractions, decimals, percents, and ratios, as well as how to make graphs and pie charts. 

  • Graph the digits of piDon't believe that pi goes on forever? You will after you've graphed it as far as you can stand! This also makes a beautiful art project, even a permanent installation. It can be a fun activity to do over several days, eventually taking over an entire wall or spreading out across the whole house.
    • Level graphing the digits of pi down to elementary by introducing it after learning the concept of graphing. Use Cuisenaire rods as the graph pieces!

  • Mark Circles in the Snow. A fresh snowfall is the perfect time to take math outdoors! Here, the kids practice their geometry knowledge by marking a giant circle in the snow. Grab a meter stick or tape measure to take some measurements, and head inside, or snuggle up to a bonfire, to calculate the radius, diameter, and circumference.


  • Zometool Crystallizations. The sky's the limit when you figure out how to tile a basic shape in both two and three dimensions. This is a surprisingly tricky logic puzzle, and it's terrific for building patterning intuition.
    • You can use this activity as-is in both elementary and high school.
  • TangramsI first introduced tangrams to the kids when they were very small and we were studying China, but tangrams are such a sophisticated manipulative that you can easily challenge yourself with them no matter how old you are.
    • Definitely use tangrams as-in in high school!
  • DIY Binomial Cube/Trinomial Cube Manipulative: This is a visual, tangible model of the binomial cube and trinomial cube. Kids can help build it, or simply manipulate it after it's built. You can create pattern cards or work through it verbally. You can use it as a bridge to teach kids the formula or let them figure out a workable formula through their own experimentation.
    • Level the binomial cube/trinomial cube manipulative down to a challenging middle school activity by calling it a "puzzle," the object of which is to create a perfect cube.
    • Level the binomial cube/trinomial cube manipulative down to an elementary activity by offering it as free play.
I don't blog about every single thing that I do with the kids, so here are some of the Pinboards where I collect even more middle and high school math resources. Some I've done, some I haven't, but I think they all look pretty cool!
I often tell people that I LOVED homeschooling the early years--I mean, who wouldn't? The picture books! The mud puddles! The field trips to the fire station! But the thing about me that you've probably sussed out is that I need a LOT of mental stimulation--like... a LOT--and homeschooling those little kids didn't always provide it to me. There was a whole summer when I think I fell asleep every single afternoon while reading Syd whatever Rainbow Fairy book she'd presented to me that day. Anyway, homeschooling big kids isn't like that. They can do puzzles and work projects and try activities that genuinely grown-up humans think are cool, too, and don't have all the answers for. That Zometool crystallization activity, for instance? I STILL think about the crystal that I created, and I still kinda think that I accidentally found the cure for cancer or unlocked another dimension with it...

...until, you know, I took it apart so that we could do something else cool with the pieces.

P.S. Want to see what else we do that's cool? I share resources, works in progress, and more updates on our adventures daily over at my Craft Knife Facebook page.