Homeschooling Corner: Math ideas and manipulatives for younger grades

Archimedes

When you love a subject and your kids love it, too, it’s easy to teach. When you’re really not sure how to approach the subject or your kids hate it, it gets a lot trickier. (See: spelling.)

So I thought I’d make a list of some of our favorite math related materials–but please remember, all you really need for teaching math is a paper and pencil. (Or less–Archimedes did math with a stick and some sand!)

Countables

Little ones who are just learning to count and add benefit from having something concrete they can hold, touch, and move around when thinking about concepts like “two more” or “two less.”

You can count almost anything–pebbles, shells, acorns, pennies, Monopoly money, fingers–but having a box of dedicated, fun, colorful countables on hand is useful. My favorites:

Abacus. The abacus has the lovely advantage that all of its counters are on rods and so don’t get scattered around the room, stepped on and lost. I made my own abacus (inspired by commenter Dave‘s abacus) out of a shoe box, plastic beads, pipecleaners, and tape. You can count, add, subtract, multiply, divide, etc., on an abacus, but for your purposes you’ll just need to learn addition and subtraction.

Different abaci have different numbers and arrangements of beads. If your kids are still learning to count/mastering addition and subtraction up to ten (standard kindergarten goals,) I’d use an abacus with 9 beads per string. (Just like writing numbers, after you get to nine on the “ones” string, you raise up one bead on the “10” string.)

We adults tend to take place value for granted (“it’s obvious that we use the decimal system!”) but switching from column to column can be confusing for young kids. There’s no intuitive reason why 11 doesn’t = 2. The abacus helps increase awareness of place value (typically taught in first grade) because you simply run out of beads after 9 and have to switch to the next row.

Once kids have the basic idea, you can switch to a more advanced abacus like the Soroban. The top bead on the Soroban is worth 5, so students count 1-2-3-4, then click the 5 bead and clear the unit beads, then add unit beads to the five to count 6-7-8-9, then click one bead in the tens column and clear all of the beads in the unit and five column. My apologies if it sounds complicated; it really isn’t, it’s just a little tricky to put into words.

You can get abacus workbooks; I have not used any so I cannot review them but they look fun. Rather, I just use the abacus as a complement to the other math problems we are already doing. (I have read Mr. Green’s How to Use a Chinese Abacus, which was the only book my library had on the subject. It is a very good introduction aimed at adults.)

Counting Penguins

There is nothing magical about penguins; I just happen to like them. The set has 100 penguins in ten sets (distinguished by color) plus ten “ice bars” that hold ten penguins each. (Besides addition and subtraction,) I find these useful for introducing and visualizing multiplication , eg, 3 rows of 5 penguins = 3×5.

Counting Cubes

For bigger numbers, we have a bag of 1,000 interlocking cubes. Kids will want to just plain build with them, like Legos, which is fine–a fun treat after hard work. You can easily use these to represent 1s, 10s, and 100s (it takes a while to assemble a full 1,000 cube,) and to represent operations like 3x3x3, helping bridge both place value and multiplication. Legos work for this, too, though you’ll probably want to sort out ones that are all the same size and shape.

 

Pattern Blocks

(I think I’ve been incorrectly calling these tangrams, though the principles are similar.)

These pattern blocks are a family heirloom, sent to me by my grandmother upon the birth of my first child. I played with them when I was a child; my siblings played with them; now my children play with them. Someday I will pass them on to my grandchildren… but you can also get them on Amazon. (We use these with a book of pattern block activities that hails from the 80s; I am sure there are many good books of a similar nature published within the past couple of decades.

Apparently there are workbooks with pattern block activities aimed all the way up to 8th grade, but I have not read them and cannot comment on them.

Cuisenaire rods

We didn’t use cuisenaire rods when I was young, but I think I would have liked them. Similar to the tangrams pattern blocks, there are lots of interesting workbooks, games, and other activities you can do with these.

Building toys

Open-ended building toys (Legos, Tinker Toys, blocks, magnetic tiles) come in almost endless forms and can be used to build all sorts of geometric shapes.

Fractions

Fraction blocks and fraction circles are both handy.

Games

Almost any kids’ board game can be transformed into a math game by adding cards with math problems to be solved before completing a turn or using math dice. Your local games shop can help you find dice with numbers higher than six, or you can just tape paper onto an existing cube to make a custom die of your liking (like an + and – die). There are also tons of fun logic games; I pull these out whenever kids start getting restless.

Books

There are so many great math books, from Sir Cumference to Penrose, that I can’t hope to list them all. I encourage you to check out your library’s selection. Here are a few of my favorites:

The Adventures of Penrose the Mathematical Cat (plus sequels) makes a very pleasant enrichment portion of our daily maths. Each day we read one of Penrose’s stories (on subjects like Fibonacci numbers, primes, operations, etc) and do a short, related math activity.

Penrose is probably most appropriate for kids in mid to late elementary, not little ones just learning to count and add. (Note: the first story in the book was about binary, which flew over my kids’ heads.) Sir Cumference is more appropriate for younger learners.

Mathematicians are People, Too: biographies of great mathematicians. I’m not keen on the title, but my kids liked the chapter on Archimedes.

Balance Benders These workbooks come in different levels, from beginner to expert. Each puzzle presents students with a drawing of a balance with shapes on either side, and asks them to figure out, from a choice of answers, which statements about the shapes are true, eg “One circle equals two squares” after viewing a balance with two circles and four squares. (We also do logic puzzles and picture sudoku.)

 

Textbooks

I am not recommending any textbooks because I don’t have any idea which is the best. We don’t use a pre-packaged curriculum, because they tend to be expensive–instead I’ve just picked up a whole bunch of different math texts at the second hand shop and been gifted some lovely hand-me-downs from relatives. At this point I might have too many math books… I use 3 or 4 interchangably, depending on exactly which concepts we’re covering and whether I think the kids need more practice or not. I recently lucked into a volume of the “What your X Grader Needs to Know” series, and it gives a very nice overview of grade-level math expectations (among other things.)

Incidentally, the local public school math expectations appear to be:

Kindergarten: Reliably add and subtract the numbers 0-10; add small numbers to numbers between 10 and 20; be able to write all of the numbers from 0-20; count to 100.

1st grade: Place value; add and subtract one and two digit numbers with no regrouping.

2nd grade: Add and subtract multiple two an three-digit numbers.

I think they only explain regrouping in third grade.

In my experience, kids can do a lot more than that. These aren’t the standards I use in my classroom. But if you’re struggling to get your kindergartener to concentrate on their math worksheets, just remember: professional teachers don’t actually expect all that much at these ages. (And my kids don’t like doing a bunch of worksheet problems, either.)

Don’t sweat it. Do a few problems every day, if you can. Try teaching the same material from different angles, if necessary. Don’t be afraid to pull out pencil and paper and just make up a few problems and work through them together. Make patterns. Play games. Relax and have fun, because math at these ages really is beautiful.

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Homeschooling Corner: Introducing Mr. Poop & Custom Dice

I happened to have a poop-shaped pinata sitting around (Why? Look, sometimes these things just happen) of the pull-the-flap-on-the-bottom variety rather than the smash-it-with-a-bat kind, so I decided to add a little fun to our day by filling Mr. Poop with school-related ideas written on strips of paper. Give Mr. Poop a shake and a scrap of paper flutters out–today’s idea was to design your own game, which the kids are working on now.

I’ve decided to incorporate the Cub Scout handbooks–which have lots of useful information about subjects like first aid, water safety, civics, history, etc.–into our rotation. (The Cub Scouts have a different handbook for 1st, 2nd, 3rd, and 4th graders.) Today we learned about knots–mostly square knots–complemented with The Camper’s Knot Tying Game. Knots are practical for anyone, but also good practice for kids with fine motor difficulties.

Over in Professor Astro Cat, we’re collecting space dust, keeping a moon journal (the eclipse was well-timed for this) and made impact craters in the sandbox. The book recommends spreading out newspaper indoors and using flour or cocoa powder, but sand, outside, is much easier to clean up. (Walmart sells beautiful colored sand for like $4 a bag. I sprinkled some green on top of the regular brown sandbox sand to simulate Earth’s surface.)

Custom Dice

There are lots of interesting dice–math dice, fraction dice, letter dice, place value dice, etc. Customized dice are easy to make: just take a cube (you probably have a building block or letter cube or some Legos lying around,) cover it with paper, and write whatever you want on the faces. (Note it is probably best to write on the paper before applying tape, as many pens won’t write properly on tape.) I have a custom die with +,-, <, and division signs on it that I use along with custom “numbers larger than six” dice for math games. (“Looks like you rolled 5,000,000,000 divided by 7,000!”) (For smaller kids, you may want to stick to + and -.)

I’m still trying to work out good ways to teach history. I’ve got some rudimentary ideas, but I’ll save them for later.

Everything I’ve Read about Food, Summed up in One Graph:

A few years ago I went through a nutrition kick and read about a dozen books about food. Today I came across a graph that perfectly represents what I learned:

Basically, everything will kill you.

There are three major schools of thought on what’s wrong with modern diets: 1. fats, 2. carbs (sugars,) or 3. proteins.

Unfortunately, all food is composed of fats+carbs+proteins.

Ultimately, the best advice I came across was just to stop stressing out. We don’t really know the best foods to eat, and a lot of official health advice that people have tried to follow actually turned out to be quite bad, but we have a decent intuition that you shouldn’t eat cupcakes for lunch.

Dieting doesn’t really do much for the vast majority of people, but it’s a huge industry that sucks up a ton of time and money. How much you weigh has a lot more to do with factors outside of your control, like genetics or whether there’s a famine going on in your area right now.

You’re probably not going to do yourself any favors stressing out about food or eating a bunch of things you don’t like.

Remember the 20/80 rule: 80% of the effect comes from 20% of the effort, and vice versa. Eating reasonable quantities of good food and avoiding junk will do far more good than substituting chicken breast for chicken thighs in everything you cook.

There is definitely an ethnic component to diet–eg, people whose ancestors historically ate grain are better adapted to it than people who didn’t. So if you’re eating a whole bunch of stuff your ancestors didn’t and you don’t feel so good, that may be the problem.

Personally, I am wary of refined sugars in my foods, but I am very sensitive to sugars. (I don’t even drink juice.) But this may just be me. Pay attention to your body and how you feel after eating different kinds of food, and eat what makes you feel good.

Grace Under Fire or Fire with Fire?

Let’s suppose you’re going about your business, trying to do something nice for a friend/loved one/relative who needed help, when suddenly they get mad at you.

You’re blameless, of course.

You try to defend yourself, but the other person grows increasingly hostile, accusatory, and paranoid, so you attempt to deescalate by leaving.

They call you to “work things out,” but your attempts to explain your side don’t work and they get mad and start insulting you, ranting about other relatives, and dredging up old grudges and grievances going back a decade or two.

At this point, do you respond by calling them a childish jerk who throws a temper tantrum when they don’t get their way, or do you attempt to take the high road, responding as well as you can to the substance of their complaint?

Note that this is someone whom you care about and will be seeing again, so just telling them to “fuck off and die” isn’t an option.

If you turn on the insults, there’s the possibility that they will just say, “See, I knew you were the kind of person who says hurtful things!” and your relationship will be further damaged.┬áBut if you take the high road, there’s the chance that they will think their behavior was justified, or not realize just how entirely out of line you think they are.

Now, we can all come up with high-falutin’ philosophy–and philosophy tends to come up with, “Always take the high road.”

But does that actually work?