Manipulatives That Stand the Test of Time

I've had the idea for this post in my head for quite a while, as my kids get older (upper elementary and middle school--yikes!) and I see that although we've used some manipulatives just a time or two, or a month or two, or even a year or two, other manipulatives have been part of our regular rotation since the children were in preschool.

There are even a couple of manipulatives that I never purchased or spent the time making, thinking they'd be used only sporadically, and have simply wished that I owned several times a year since the kids were young. Silly me!

Cuisenaire Rods

You probably know by now that these and Base Ten blocks are my FAVORITE math manipulatives. We have used them endlessly. The kids have less patience for them now that they know that whenever I whip them out, there's generally a much quicker algorithm waiting for them as soon as I'm done, but nevertheless, they're vastly useful when a kid is having trouble understanding a concept. Two days ago I used them to help Syd understand decimal multiplication--let the ten bar equal one, and you've got several decimal options available to model. And seven years ago, I used them to help Syd understand addition within 5!

Here are some other ways that we've used them over the years:

• commutative property of addition
• introducing multiplication
• long division
• modeling multiplication and division as arrays
• repeated addition (and beginning multiplication)

As with most of the other manipulatives that I'll show you, I've occasionally expanded our collection over the years, as we've needed them for more and more advanced work. In a perfect world, I'd have enough to build an entire decanomial square from them, but we're at the point now where we have enough to do most of what we want to do.

Base Ten Blocks

This is the other math manipulative that I do not know what I would do without. I consider them indispensable, and again, I have expanded our collection several times, especially with more thousand cubes. These blocks, which consist of a 1cm cube, a 10 cm bar, a 100 cm flat, and a 1,000 cm cube, are brilliant for leading a kid into an intrinsic, whole-body understanding of the Base Ten system. You have to have them to understand what you're truly doing when you add or subtract multi-digit numbers. You have to have them simply to model numbers. We use Base Ten blocks and Cuisenaire rods as a single tool, with the Base Ten blocks representing all the powers of ten and the Cuisenaire rods often (although not always--for borrowing and carrying, you should only use Base Ten blocks) to represent all the numbers in between.

Here's some of what we do with them:

• building big numbers
• building decimal numbers and again with the younger kid!
• hands-on rounding
• modeling long division (also with Cuisenaire rods)
• multi-digit multiplication using area models
• subtraction with borrowing across zeroes

Decanomial Square

We came to the game a little late on this one, as it took me literally years to decide that I wasn't going to find exactly the decanomial square that I wanted. Instead, I had Matt make one for me. 

Anyway, you can start using this one as soon as your kid starts learning the multiplication table (or before, of course, as a puzzle or to model repeat addition as area), because it works quite well as a visual multiplication table of area models. However, it's also great for all kinds of area and perimeter studies, as well as how to form equations, and is an easy demonstration of the Pythagorean theorem.

Here's just a little of what we've done with it so far.

You can use the decanomial square with Base Ten blocks and Cuisenaire rods, and I am constantly in search of a cheap Montessori-style pink tower so that I can also use it to demonstrate cubes.

Zometools

These are the priciest of our manipulatives, a cost that I absorbed by making this the kids' big combined Christmas present one year when I'd had a good winter in my pumpkin+bear etsy shop. Although we started using these just for free play to build for fun--which we still do!--these building materials should see the kids through high school, and they're used in universities, and even grad programs, to model some amazingly sophisticated concepts.

Here's some of what we've done with them:

• bubbles and three-dimensional figures (there's a whole mathematical thing revolving around the figures that bubbles form inside prisms)
• crystallizations (ie. three-dimensional pattern building)
• modeling molecular structure (you need the molecule kit for this, just so that you have the cards)
• stellations (or two-dimensional pattern building)

Hundred Grid and Number Tiles

This, on the other hand, is easily handmade (here's how to make the grid, and here's how to make the tiles), although we do have a set of number tiles made for an overhead projector that we use on our light table. You can use the hundred grid and number tiles from the time a kid is wee, simply for naming numbers, ordering them, and counting on. You can use them for skip counting. You can use them for multiplying. If you make a second set of number tiles to match a multiplication table, you can use them for memorizing the table and for playing various multiplication games. 

Here's some of what we've done with them:

• counting with pennies
• hands-on rounding
• Multiplication Touch game
• number matching
• Roll to a Hundred game (number recognition, sequencing, counting, and addition concepts)
• skip counting with coins (money math)

We haven't used the hundred grid lately, although I won't rule out finding another use for it in a later unit. Even if we don't, though, using it from the age of three through the fifth grade is a good amount of use!

Pattern Blocks

We used pattern blocks a ton in preschool and lower elementary, but it's actually only recently that it's occurred to me how useful they can be in upper elementary and the middle grades. We recently used them in Will's lesson on congruent and similar figures, and considering the fact that they've been using these same simple shapes since they were wee, that's a pretty darn good run!

Here's some of what we've done with them:

• building on templates (following directions, shape matching, logical thinking)
• symmetry and similar figures
• free play and creative play (sometimes on the light table!)

Here are some outside links to activities that we've done but that I haven't blogged about, myself:

• mandalas (or snowflakes, if it's winter)
• fractions
• holiday-themed templates for little ones
• measurement
• guided art game (similar to these DIY art dice that we also enjoy)

If you, too, are really into hands-on work for older kids, I'd love to know what manipulatives you use and how you use them.

And if you see a Montessori-style pink tower going for cheap, let me know immediately!

Homeschool Math: Graphing Candy

There's a lot to cover with fractions, decimals, and percents, but fortunately, since they're all ways of representing the parts of a whole, your kiddos can review all of them together.

For our hands-on math enrichment lesson one day, my goal was to show the kids concrete representations of fractions, decimals, and percents, to have them practice the conversions, and to let them explore how they can model these relationships.

If that sounds dry (and it does), then let me explain to you that it was actually super fun, on account of...

CANDY!!!

We went to the grocery store and picked out several small bags of assorted candies, the kind that naturally come in various colors or flavors. I was surprised that it was impossible to find the small bags of M&Ms that I remember from the checkout aisle as a kid, but we ended up happy enough with Mike & Ikes, Reese's Pieces, Sour Patch Kids, and Sweet Tarts.

I bought a package of each for each kid, because I also thought that it would be interesting to compare each kid's results.

After I had made each kid promise not to eat any candy until they'd finished counting, I gave the kids their various little boxes and bags of candy and lots of little dishes. For each candy, each kid had to do the following:

1. Count the total number of pieces of candy in the package.
2. Count the total number of each kind of candy.
3. Calculate, for each kind, the fraction, decimal, and percentage of that kind to the total number of pieces.
4. Create a pie chart to represent the relationships between the candies.

Here is the page of Will's calculations. She doesn't have the neatest handwriting, but at least, after years of nagging, it is at last legible... to her:

And here are their pie charts!

I was interested to see that the ratios between each type of candy didn't remain consistent between the two packages that the kids studied, in any candy that they studied.

If we did this again, I'd make a chart ahead of time for the kids to record their results before graphing, and make it clear that I expected organized results that everyone, not just the kid recording the results, could read.

Extension activities that I'd also consider the next time:

• Use the leftover candy for candy science experiments. There are a ton online, invented in large part by parents desperate to use up Halloween candy, but they're all pretty cool.
• Or use the leftover candy for baking or decorating. Most of the candy that we used would taste good in cookies or brownies.
• Buy several packages of one candy, graph them all, then use the results to calculate an average of each type of candy in the mix. This would make a good STEM Fair project.
• Have the kids write up their results in an essay.

The kids LOVED this activity, and were super invested in the process. Sure, it made for a sugar-filled day and evening, but it was a small price to pay for such intense math practice, conducted fuss-free!

Here are some other fun ways to play with fractions, decimals, and percents:

• Make Rice Krispy Treat fractions. 
• Play edible chess. 
• Make fraction flags. 
• Play Spiraling Decimals. 
• Make pizza box fractions. We got a couple of small pizza boxes from our favorite pizza place to do this project. 
• Play Roll a Whole.
• Make fraction art on a hundred grid. 
• Write your name on a hundred grid. 
• Do color-by-fraction pictures.
• Play First to 50.
• Use dominoes as fraction manipulatives.
• Make play dough fractions.

And here are just a few resources that we've also enjoyed:

Work Plans for the Week of November 2, 2015: All the Arts!

Last week was a great school week, with a combination of wild-and-woolly hyper-elaborate projects, field trips, and just enough seat work to calm things down every now and then. Frankly, this week looks like much of the same!

Will is gearing up for another spelling bee season, so for memory work this week, Wordly Wise is on hiatus in favor of the Scripps 2015-2016 spelling list. Other memory work includes Mandarin vocabulary for Will, ballet practice for Syd, and cursive. Books of the Day include more books on World War 2, including a couple of living books, a couple of Magic Tree House fact finders, and one more book on volcanoes. I'll be refocusing on having Syd read aloud to me each day from her book, as last week's work on her History Fair project has made it clear to me that she needs my one-on-one attention to encourage correct pronunciation and complete comprehension.

And here's the rest of our week!

MONDAY: Math Mammoth is still fractions for Will and multi-digit multiplication for Syd--restful chapters, since they both feel pretty confident about the material. Our only other assignment for this day is having the kids rehearse their History Fair presentations several times, as today has both our regular volunteer gig and our homeschool History Fair! I'm really looking forward not only to my own children's presentations, but also to seeing the presentations of all the other children in our group. The kids always enjoy this, too, and it's amazing how much history content is covered, and how much children can learn when they're learning from other children.

TUESDAY: One of my favorite things about Girl Scouts is the ownership that the children have of their own fundraising money. Back in the summer, we held a Girl Scout meeting to let the children make proposals about ways to spend their Girl Scout cookie earnings; the children prepared presentations on their proposals, the other children asked thoughtful, supportive questions about each one, and then they all voted on what we should do. One of the winners was this day's field trip to a nearby children's museum, complete with the rock wall experience, and lunch spent at an indoor playground across the street from the museum. It'll be a fabulous time for all of the children, as well as sneaky STEM, creative, and gross motor enrichment. It's great to be a Girl Scout!

WEDNESDAY: The kids have a bunch of extracurriculars on this day--Magic Tree House Club, ballet class, AND LEGO Club!--so although we only have room in our school day for one additional assignment, I've made that assignment more elaborate than usual. Inspired by this hundred grid fraction art lesson, I designed my own for the children. It will incorporate symmetry and equivalent fractions, and, for Will, decimals, percentages, and simplifying fractions. It will also require close reading of the instructions, which is another thing that I want Syd to work on, and creative expression, which Will always needs more outlets for.

THURSDAY: We are in the end-game of our World War 2 unit! By this day, we'll have completed my main goals for the unit--interviewing a World War 2 veteran, visiting Pearl Harbor, and completing History Fair projects--so on this day, we'll review the entire war using Story of the World, and then all that really remains is the atom bomb, the aftermath, and a comprehensive review of the geography of the war.

There's no real academic reason for having the kids do more how-to-draw books on this day, other, of course, than what they clearly provide in terms of fine-motor skills and creative expression. The truth, though, is simply that although the children adore working from these books, they also never seem to choose any from our library shelves to do on their own. Assigning the books, then, is a sure-fire path to a happy morning.

FRIDAY: An organized rock collection is something that neither child has ever come to on her own, but it's something that I think they would both enjoy. Organizing and categorizing the rocks and coral that they collected on our Hawaii vacation, then, will be both useful and a good way for me to gauge their enthusiasm about perhaps continuing that work with the rest of their giant box of rocks. It could be the basis of an entire geology unit!

Although we've already had one Girl Scout activity this week, it also happens that both children have recently finished the badges they've been working on--Detective for Syd, and Comic Artist for Will. This means that it's time to get to work on a new badge! I do have a few more activities that I want the kids to do relating to each badge, but those are beyond the scope of the badges, themselves.

Syd is living and breathing the Nutcracker these days, of course, and has yet another practice on this day, but a few weeks ago I signed Will up for a series of girls-only podcasting workshops, and I'm thrilled at how she's taken to them. From what I can tell, the children involved in the workshop really have ownership of their work, and Will has displayed excellent teamwork, has composed original work to record for the podcast, has read the work of others ("I had to record one poem FOUR times," she told me, "because I had to work on my breathing and enunciation"), and has been learning how to use all the other audio equipment involved in this production.

SATURDAY/SUNDAY: Saturday has shaped up to be a day of ridiculousness this autumn. Don't tell the children, but it's 100% another school day! Syd has both her regular ballet class AND a Nutcracker rehearsal, and Will has classes in ice skating and Mandarin. Sunday, then, is our only day fully at home this week, and hopefully we'll get to spend it fully AT home. If nothing else, I need to work on some autumn landscaping, and Matt and the kids need to winterize the chicken coop!

As for me, if I want to buy the children Christmas gifts this year (and I do!), I really, really, REALLY need to spend this week adding seasonal items to my etsy shop, futzing with the SEO tags on a few of my listings, and otherwise getting my shop in shape for holiday shopping. I also, of course, have writing assignments to complete, some mending to do, and there is an absolutely massive bunch of overripe bananas on the kitchen counter that must become banana bread today, if not sooner.

It's going to be a wonderful week!

Hands-on Rounding, or, I cannot Make This Concept Any Clearer

Rounding is just non-intuitive.

Not only do you have to wrap your head around the fact that you're going to pretend that a number is a different number, but you have to remember that even though it's called rounding DOWN, you don't round down by subtracting 1 the same way that you might think that you round up by adding 1 (that's not *really* how you round up, but if you're nine, that might be how you've explained it to yourself).

Oh, and rounding up if the digit is 5? That doesn't actually make sense; it's just a convention to memorize.

I wanted Syd to understand the concept of rounding, not just memorize a rule that doesn't make sense to her, so using this post on hands-on rounding as a spine, we went over it again from the beginning.

The post doesn't mention using a hundred board, but we've relied on our hundred board a lot throughout Syd's entire math education, so once again we began a lesson by having Syd build it:

I've got three sets of these 1-100 tiles, two DIY sets and one for the light table/overhead projector, so I could leave the 100 grid intact and give Syd a second set of tens to use in making a number line between two consecutive tens:

This part was all review, as she had no trouble with any of it.

Syd built a number line between two consecutive tens a few times, each time with me handing her the numbers in between at random. We then reviewed rounding to the nearest ten by noting on each number line which consecutive ten a number fell between, and which consecutive ten it was closer to.

Numbers ending in 5, of course, round up merely through convention, as we also discussed.

To end the activity on this day, Syd and I played a game with two 10-sided dice. One of us was "Rounded Up" and the other was "Rounded Down;" we each took turns rolling the two 10-sided dice, making a number from it, then deciding if that number rounded up or down and giving a point to the correct category.

Although the points go to the correct category regardless of whose turn it is, when it's your turn to roll, you can do your best to build a number that gives you the point. It's tricky, though, because you can't always build a number that goes your way!

I let this lesson settle for a few days while Syd worked on different concepts in her Math Mammoth, then we had another lesson on rounding to the nearest hundred and thousand.

Syd didn't seem confident in building a number line between two consecutive thousands, so I had her do so using the hundred flats from our Base Ten blocks as physical markers, with each hundred written on a torn piece of paper--use what you've got! I then used a domino to mark a place on the line--9,600, say--and gave Syd another domino to use as a game piece to hop to each thousand:

The thousand with the fewest number of hops to get to it, obviously, is the thousand that the number is closer to.

After this, we turned to our light table, and I asked Syd to build another number line between two consecutive thousands:

I used the dry-erase marker to modify the 0-100 tiles to make them into numbers in the thousands (30, for instance, became 5,300), then asked Syd to place them on the number line. She found this a LOT harder than working with a number line between two consecutive tens. Her number line got so wonky, in fact, that I abandoned the tiles work, gave her a different number line--9,000 to 10,000--and asked her to simply draw in the hundreds in between.

As she did so, I noticed that she did not have 9,500 in the middle--actually, it was very close to 9,000---so I asked her to check her spacing. She was VERY not happy with this, because she knew then that she'd made an error, and she HATES making errors.

Aside: That's an attitude that we work on a lot, by the way. I see it as a very destructive type of thinking that will keep her from establishing a strong and persistent work ethic, so I speak constantly of the fact that knowledge and skill do not enter one's brain by magic, but instead require a lot of training to achieve. When one has the correct answers immediately, one is not learning, but reviewing. Learning is the space in which we try to spend much of our school days, I tell her, and we know that we are in that space if we are struggling and making mistakes.

Hopefully one day that will sink in!

Because Syd was tipped off that she'd made an error, all of her will became focused not on finding and correcting her error, but instead on defending it to the death. She insisted her spacing was perfect. I noted that there was only a finger's width of space between each integer up to 9,500, but that the others on the way to 10,000 were really far apart. She decided to deliberately misunderstand me, and began screaming that I wanted her to write everything with one finger in between, and that was impossible. I excused myself to my room, and told her that she should choose another activity, and we'd finish the lesson later. I then read quietly for an hour while Syd screamed the house down in the next room.

Of course, eventually one must finish one's schoolwork if one wants an hour of screentime, and one DOES want this hour, so much, much, much later, Syd went back to work with a good attitude renewed:

I was still pretty over it, so Matt finished the review with her, and then they played the same dice game, this time with four 10-sided dice, with the goal, again, to decide if the number rounded up or down to the nearest thousand:

Afterwards, I quizzed Syd by throwing out more numbers, and she was able to round them all correctly! She was also SUPER happy to do so, because she knew that she was getting them all correct.

Have I mentioned that this child exhausts me? Oh, my word, she exhausts me!

But, of course, so does the other one...

Work Plans for the Week of August 10, 2015: Math, Maps, and Many Animals

We had a great week last week! Mind you, a day in our house is never what you'd call tantrum-free, but the majority of each day contained content children who worked hard on their responsibilities, played together, played outside, and didn't give me too much of a hard time.

This chalkboard helps, I think:

I've been compiling it on Sundays, as I'm finishing up the next week's lesson plans, and then during the week, I really only have to update the daily chore list and erase the special events as they pass and the memory work as it's successfully memorized.

For memory work, daily practice on Mango Languages is a constant, as daily guitar practice for Syd will be when I can sort out her lessons. Mango Languages can be a little frustrating, as it's designed for adults and therefore moves quickly, but I continue to explain to the children that I do not expect perfection, or speedy progress, just the consistency that will allow their brains to master the material. And they do seem to have settled into their daily practice:


I'm studying Hawaiian, too, as I've never learned an Austronesian language before. Will does some Hawaiian, but also likes to explore the other languages, but after trying Mandarin Chinese, Syd seems happy to stay with Hawaiian for now. Here's she is comparing her pronunciation to the native Hawaiian speaker's:

Written work also went well last week, and has tended to go much better since I made the commitment to myself that when the children were working at our school table, I would be working there, as well, generally on my computer, which allows me to answer the random ready-reference questions that always come my way, but also with a white board and dry-erase marker at hand, tools that are essential for easily demonstrating math calculations. By these ages the children ought to be able to do their written work independently, but if working in companionable silence with me helps them focus, then I'm not going to protest--especially as I always have plenty of my own work to do!

Having a chick at the table to assist is also essential, apparently:

Anyway, this week's memory work includes Mango Languages, Will's World War 1 poem (Syd has mastered hers, but Will's is more challenging), the countries involved in World War 2, spelling words, and Hawaii state symbols. Spelling words and the poem can be knocked off the list by proving to me that they've been mastered, and we'll move on from World War 2 countries to World War 2 leaders next week, whether or not the kids have the countries down--this won't be the last time that we study World War 2, so I really just want the children to be familiar with these facts, not necessarily have them memorized cold.

The children also have an assigned book to read each day, and this week's books include picture books of butterfly and frog metamorphosis, living books on Hawaii and World War 2, a biography of Marie Curie, and some titles for STEM enrichment or that are related to the kids' current Girl Scout badges in progress.

Here's our week!


MONDAY: We're starting the week with some hands-on math enrichment. Syd loves fractions, so on this day she'll be exploring a set of fraction blocks and activity cards that I checked out for her from our local university's education library. For Will, I put together this model for multiplying and dividing by powers of ten, and printed out a multiplication by powers of ten worksheet for her to complete using the model. I laminated the strips that you pull through the model, so with a dry-erase marker, she can write each problem down on the strip, physically manipulate it to show the process, then erase it and begin again with the new problem. I think this will make the process very clear very quickly, as Will is excellent with patterns.

We don't follow a formal spelling curriculum; I prefer to keep an ear out for any words that a child needs help spelling, then create a list that contains that word and words that are spelled following similar rules or that have similar sounds. This week, for instance, the spelling list contains "oo, ue, ueue, ew" words. Next week's list, I already know, will contain "wh, w" words. Some weeks I use the spelling word list as cursive copywork, but this week I've created a word search; when finished, the children will keep the page in their binders as a study sheet.

I wanted the children to be able to recognize a few specialized terms in our brief mosaics unit, mostly to help them remember that there is a science even to artworks. I won't have them memorize these terms, but I do want them to practice their infographic-making skills to create a reference sheet that we can display.

We have our weekly volunteer gig at our local food pantry today; while I stock the pantry and assist customers, the kids can be found weeding and harvesting in the garden, repackaging bulk food, rinsing storage bins, snacking, helping themselves to seed packages to plant in our home gardens, or reading or playing quietly. They keep themselves entertained!

TUESDAY: Math is back to Math Mammoth today; Syd has a lesson on estimation, and then some review, and Will has word problems. She will be SO excited to move away from decimals next week!

Time spent outside on lovely summer days has allowed the children to become interested, again, in insects, so I have a few activities that we can build on if they seem to enjoy them. One is this BumbleBee Citizen Scientist Program, which will involve the children hunting down a bumblebee to photograph, and will give them the opportunity to research an identification for it, and to have that identification confirmed by an expert in the field. We'll see how it goes.

For our World War 2 study this week, the children will be labeling a world map with the countries that participated in World War 2, using these World War 2 notebooking pages. This lesson won't have a lecture component, but will simply be a useful reference for the rest of the unit.

Syd has a playdate planned for this day, and you might remember that I count even playdates as part of our scheduled day, since playdates aren't exactly the same thing as free time, and preserving the children's free time is very important to me.

WEDNESDAY: Syd has Math Mammoth again on this day, but for Will I created worksheets to teach her how to use this model to multiply decimals. Her worksheets simply consist of decimal multiplication problems, each followed by enough clip art hundred flats (you can also use a blank hundred grid) to model the problem, and space below to write the answer.

Syd is currently working on her Girl Scout Detective badge, so she and I will be making invisible ink using lemon juice, and playing with invisible ink pens that require a black light to reveal their writing--lots of good science there! Will is working on the Comic Artist badge, so she'll be studying comic strips and creating her own, for lots of good visual arts and creative writing practice. See why I love the Girl Scouts so much?

I've been sitting on this book that I need to review, and since Mango Languages has recently gotten the children interested in the variety of world languages that exist, I thought we'd give reading a chapter out loud a go. We'll read it together, looking up the pronunciation of the French words that we encounter using an online dictionary. We'll continue that next week if the kids seem to enjoy it, and if they don't seem to, I'll just pass it to them to finish reading on their own. Regardless, it's a nice little intro to the French language.

Will has often expressed interest in creating a butterfly collection, and I finally feel confident that I've done enough research to choose the most humane way to go about it. I've found some Youtube videos that provide a good overview of the process of collecting, euthanizing, preparing, and displaying insects, and I've purchased all the relevant supplies, so I'll have the kids watch these videos on this day, and if they're still enthusiastic about it, we can begin!

THURSDAY: Syd has her Math Mammoth, but Will has a review of multiplying and dividing by powers of ten and multiplying by decimals. If she doesn't have the process down cold by this day, then we're just going to move on anyway and revisit it in several months.

Our geography assignment for this week is simply a coloring page of Hawaii state symbols and facts, which the children can color while listening to traditional Hawaiian music streaming on Spotify. These facts will then be added to their memory work to review until they've mastered them.

Our homeschool group's playgroup is on this day, and that tends to take up most of the afternoon. Will has her horseback riding lesson following it (we've switched stables recently, and I think it was a good decision), and while she's in her lesson, Syd and I do a project together. Last week, we started a blog for her, and this week, I think we're going to work on some decorations for her bedroom.

FRIDAY: I purposefully stock Friday with work that the children can do independently; throughout the week I encourage them to work ahead on Friday's work so that they can have that day off, and even if they don't do it, assigning work that doesn't require my input still gives me a bit of a break on this day. Their grammar comes from library copies of Exploring Grammar (Syd) and Mastering Grammar (Will), with laminating pages over the workbook pages so that the children can complete them using dry-erase markers.

Froguts is a frog dissection ipad app that's realistic and humane--yay!

We now have a decent approximation of many of the basic fossil cleaning tools used in a paleontology lab--x-acto knife, dental pick, test tube brush, paintbrush--although I still need to buy some super glue, and as I recently also bought several specimen boxes, we've been able to get some good work done on cleaning and preparing our fossil finds from our 2014 dinosaur dig. Fossil prep is a LOT of work, however, and I anticipate that it'll be a long time before our specimen boxes are complete. In addition to this work, Will is studying a college-level intro to paleontology textbook, and then discussing each chapter with me.

We've got some other fun plans this week, of course--the Perseid meteor shower, an afternoon at the splash park, perhaps weekend swimming at the YMCA, and definitely the drive-in movie--but that encompasses our work week!

Tutorial: Roll-up Felt Hundred Mat

We have a laminated cardstock hundred grid that works with our number tiles, and I've recently realized that I can make some sturdy yet disposable hundred grids pretty easily from cardboard record album covers, but I was also wanting a hundred grid that would look nicer longer, and be easier to toss in my backpack for trips to the library or park to do schoolwork.

So I made one!

You make this hundred mat around the number tiles that you already own, so first you've got to make them. We have a set of clear one-inch number tiles that were great for matching to a printed hundred grid when the kids were littler, but now that they're older (AND have lost a couple of those tiles, sigh), I've made them a couple of sets of upcycled cardboard number tiles that are the same size, so they can be interchangeable. I've long wanted to make a set of number tiles out of Scrabble tiles, but I think their inevitable loss might break my heart, so...

Measure out the grid AROUND the tiles, so that when your kiddo puts a number tile on the grid, they'll still be able to see the border around the tile. A little extra space is always good for those fumbly kid fingers, anyway. For me, this meant measuring at one inch + 1 centimeter, and I still did a lot of double-checking:

Notice that I measured and drew this grid in chalk. I didn't end up making a mistake in my measuring, but if I did--chalk equals second chances!

When you've got the grid drawn out, get out a tiny paintbrush and some fabric paint, and carefully paint over your grid:

And that's it! You can roll up your mat, you can wash it, and you can play with it on the carpet:

I remember when Syd was first learning how to find numbers on the hundred grid, and now we're using it for multiplication and division. Ah, time...

Mental Math with Singapore 2B

I may have Willow start doing all her math in her head.

For a few weeks now, Willow and I have been working on adding one- and two-digit numbers using carrying. We visualized the process by setting up the equations with Base 10 blocks--

--then trading up and physically carrying the ten bars over to the next place before adding everything together, and finally moving into the conventional method of carrying while working arithmetic problems on paper.

 There was a lot of grumping, a lot of Base 10 block pattern building--

--and a lot more grumping, and the concept just didn't catch on. I'm pretty sure that the concept didn't catch on not because the kid didn't understand the concept, but rather because she was too busy grumping and pattern building to put the concept into her brain, but nevertheless, that's a lot of grumping and a lot of repetition just to learn (or NOT learn) carrying.

So we're just not going to learn that--not exactly, at least--right now.

In related news, last week my local indie teaching supplies store, which is normally too expensive for me to shop in, had their mid-summer sale, bringing their prices down to slightly more typical levels, and so I came home with the following:

• one postage stamp collector's album for Willow
• one book of world map post-its
• one book of hundred grid post-its
• one book of multiplication table post-its
• a set of coin rubber stamps
• the practice books for Singapore 1B and Singapore 2b

I avoid math curriculums because I have no problem teaching elementary math, but I do like the idea of having a logical order to work with, and I do like some of the mathematical concepts that Singapore math, in particular, teaches, so the reasoning behind purchasing just the practice books is that I can pretty easily figure out the concept that's being practiced, teach it on my own, and reinforce it with the books.

Singapore 2B has a lot of review in it for Willow, but I'm not opposed to drill and repetition, and there are a few key concepts in it that she hasn't explored yet, so that's why we settled there for her. Singapore 1B looks like a good fit for Sydney overall, so yay there for her.

My point with this digression is that, right there in the Singapore 2B practice book, front and center just before a money review and some new exploration of fractions, is a series of mental math practice problems that involve a different method of solving one- and two-digit addition problems, mentally, WITHOUT carrying. Here's what it looks like:

First, she's rounding the one-digit number to the nearest ten, and finding the difference, which is an easy subtraction problem. She's going to hold that number in her head. Second, she's adding the ten to the two-digit number, using skip-counting. Third, she's subtracting the difference from that new number, and it's another easy subtraction problem to the answer!

Now, do not even get me started on whether or not this method takes more steps than simply carrying the ten (it does), because for the little miss, that's not the point. The point for the little miss is, apparently, the fact that the steps are broken down into simple problems that she can do in her head. While the connection to the concept of what you're physically doing with the numbers isn't as clear, to me, as it is with carrying on paper, especially as visualized using Base 10 blocks, it IS still connected to the concept, just in a different way that seems to appeal more to the miss.

So this method, combined with simple "counting on" when the units to be added will add up to less than ten, gives Willow all the tools that she needs to mentally add one- and two-digit numbers to each other, abilities that we've been practicing with a homemade deck of laminated number cards:


And thus we can put aside that dreadful carrying altogether, to be brought back up at some point in the near future, preferably after the little miss has forgotten her grumps about it!

Skip Counting with Coins on the Hundred Grid

Our family is going to be participating in Lemonade Day in our town this Spring, and to that end I've reintroduced the money math unit that we first began some months ago when Willow got interested in coin collecting for a while.

My goal is (taking extra care with pricing, of course) for Willow to be able to do all the math and make all her own change at the girls' lemonade stand--whether or not this goal is realistic, I have no idea!

First step: skip counting by each of the coin denominations, followed immediately by keying the counting to the denomination. In other words, I want both girls to be able to easily skip count by fives, tens, and 25s at least up to 100, to count by hundreds up to at least 1,000, and to recognize that skip counting by fives, say, is the same as counting nickels.

The girls have been creating their own skip counting reference sheets using, of COURSE, our ubiquitous hundred grid. For nickels, for example, one day's schoolwork was simply to count off the fives and color in each five in their hundred grid. The next day's schoolwork (and the next!) was to memorize the fives, until they could recite it easily.

When a girl had her fives down cold, I gave her a new hundred grid, asked her to put a nickel down on each five, and then use the chart as a reference to solve a page of math problems:

The problems are all just iterations of how much a certain number of nickels equals. To solve the problem, the kid can either skip count over that many nickels, or just count over that many nickels, and then move aside that nickel. The number underneath is the correct answer!

Once the kiddos have all the skip counting and coin denominations memorized, I'm going to send them through the math drills in our Kumon money math workbook as well as some fun projects from my Money Math pinboard. And then when Lemonade Day comes around, providing I can convince the children not to price their lemonade at 63 cents or $1.07 or something else that will require a child to do twenty minutes of abacus work for every transaction, I think we may just have it made!

P.S. It's just occurred to me that I should also teach them to count by tens when beginning at 5. AND I should be mixing more subtraction drills into the prep work, especially two-digit subtraction.

Or I could just encourage the girls to price everything they make at one dollar?

Homeschool Math: Our DIY Roll to a Hundred Game

I have a strict Montessori-style policy that I do not discourage my children from doing academic work that is "too easy" for them. If Willow wants to play the Jump Start Kindergarten CD-Rom that I checked out from the public library for Sydney, then that's okay. If Sydney wants to spend the afternoon working color-by-number pages, then that is totally fine by me. If Willow suddenly develops a passion for those old First Chapter Books that she first read two years ago, and reads them all again, then good for her!

It's an important part of my homeschool philosophy that repetition reinforces skills, internalizes concepts, and builds the feelings of mastery that reward children for learning, and the confidence to take on more learning challenges.

Therefore, although our DIY Roll to a Hundred Game highlights skills that both my girls have already learned, we LOVE this game! It's excellent reinforcement for number recognition, sequencing, counting, and addition concepts. The unpredictable nature of the roll of the die prepares the girls for future lessons on statistics, graphing, and averages. The coloring requires fine motor skills, and is also graphing, and pattern-building.

Oh, and the game is based on a die, so the little one can win as often as the big one does, hallelujah.

To play Roll to a Hundred, you will need:

• a copy of a Hundred Grid for each person. You can use either a labeled hundred grid, or a blank hundred grid that the child labels for herself--this turns the potentially tedious activity of labeling a hundred grid into a useful activity that a child might choose to do for herself, by the way!
• one die
• crayons

1. Decide who goes first. The first player has the advantage, so it's important to remember to take turns.

2. During her turn, a player rolls the die--

--and then colors the same number of squares as pips on the die:

3. Change crayons each time so that you can see each individual roll on your hundred grid, and the first person to reach one hundred--

Wins!!!

We play such that you have to roll to reach 100 exactly--waiting for that perfect roll gives everyone time to catch up and makes the game a little more exciting.

Ways to modify the game:

• Use two dice, or a 20-sided die, etc.
• Play on a 200 number grid. 
• Play on a number line.
• Play Roll to Zero, where the game is subtraction!
• Multiply each roll by two.
• Assign a different mathematical operation to each number: One must be subtracted, Two gets doubled, Three gets added to the previous roll, Four gets divided by two, etc.
• Have everyone graph their rolls to see how many times each person rolled each number.

It's certainly not a bad way to spend a rainy afternoon!

Sydney Masters the Hundred Grid

We're big fans of the hundred grid around here--it's such a versatile math tool! You can use it to help you add and subtract, with skip counting, with coin calculations. Because Sydney likes puzzles, she also occasionally enjoys using our big laminated hundred grid with some overhead transparency number tiles that, even though I bought them at different times and places, happen to be exactly the same size--score!

I especially like the transparency of the overhead tiles, because if a child is still working on number recognition (and, as I learned while assisting Sydney through the most tedious three games of BINGO ever played at their 4-H club holiday party this week, we ARE still working on number recognition!), then that transparency allows instant self-correction.

During our most recent play with this board, I witnessed Sydney unlock one of the patterns implicit in the number grid. No more random seek-and-find for her--watch this girl go!

I was pretty thrilled that I was there to see it happen.

Syd has a lot of focus, and although she wearied a bit of the task near the end, she kept working, because she wanted to see it through:

Success!

Pretty proud kid, right?

We have an old garage sale BINGO set of our own, and I think that we'll be playing a lot of fun at-home BINGO games this week, because not only is Syd clearly ready for number recognition up to 100, but I'm not taking her near another BINGO party game until she has it down--geez Louise, what a nightmare!

How Many Pennies in One Dollar?

I re-branded the hundred grid as a dollar grid, so let's see...

This many!

It looks so cool that I considered having Sydney glue the pennies down to the grid so that we could keep it...

And then I thought, "Heck, no! I'm not wasting a whole dollar!"

The girls have recently wearied of all this talk of coins and commerce, so our math is moving on to geometry. Geomags, constructive triangles, spirograph, geoboard, patterns blocks and mirrors--that ought to hold us for a while!