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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10 thoughts on “Homeschooling Corner: Math ideas and manipulatives for younger grades

  1. I’ve tried two well reviewed math curricula for my eldest. He absolutely hated Math Mammoth, which seemed to over-explain simple concepts (and may be more appropriate for less mathematically inclined youngsters), but is doing wonderfully with Saxon Math. I have been quite impressed with the pacing of the Saxon 5/4 book, which starts with addition, builds purposefully through subtraction and multiplication, and beyond. At some point, I’d like to introduce a traditional abacus, since I’ve read many great things about them, but they seem complicated, and I have no experience with them myself.

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    • You’d do fine with an abacus. Just start with one with 9 beads per string. The strings work like columns in regular math, so let’s say you have four strings with 2-4-0-7 beads pushed up per string. That represents the number 2,407. To add or subtract, you just slide the beads up or down. So to add 2, you’d just slide up two more beads in the units column; to add 20, you’d slide two beads in the tens column. There’s no need to introduce regrouping right away.

      To add 3 to 7, you slide up two beads in the units column, (abacus now reads “9”) realize you’re out of beads, and move over to the tens column and slide up one bead. (abacus now reads “19.”) Since you actually only want to add 1, not ten, you then slide down the 9 beads in the units column (because 10-9=1) and the abacus now reads “10”.

      To add 4 to 7, again slide up 2 beads in the units, (abacus reads “9”) realize you’re out of beads, and slide up one bead in the tens column. (“19”) This time you actually want to add 2, not 10, so since (10-8=2) slide down 8 beads in the units column. You get 11.

      Adding bigger numbers works the same way, just with more columns like in regular math; subtraction is just reversing the process.

      Multiplication gets tricky because you have to remember which column you’re using and there’s a bit of holding numbers in your head; it’s fun but more efficient to use paper.

      Once a kid has mastered the 9 bead abacus, then you can think about the fancier kinds, but I wouldn’t worry about those until later.

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  2. Patter blocks are amazing. I can directly tie my interest in math and problem solving to those little guys. There was an exhibit in the Smithsonian that my family attended when I was a kid that had a table of them. You were supposed to fill up various shapes using only multiples of 8. I was so intently focused on it that the old lady in charge let my family stay in the exhibit for longer than we had tickets for so I could work through the various infill shapes. I was able to complete the task with these elongated diamonds, upon which she told me that most adults who came through couldn’t even do it. I’ve since realized she was probably exagerating quite a bit, but none the less, I was hooked on becoming an engineer there after. (I became an architect instead)

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  3. I ran across a file I saved on home schooling I bet you’ll like it. Al Finn is gone but it’s backed up op the internet archive so I copied the url for that. I quit reading Al Finn because he’s Jewish and I got tired of his all hate Putin, all the time screeds but he had some great stuff.

    When I read about homeschooling I thought back to this article by Al Finn. I hope you ,will read this it’s an amazing educational system for raising children linked from Al Finn. I wish I had been educated this way. I’ll post a link to the idea about the method.

    http://www.see.org/garcia/e-ct-6.htm
    http://www.see.org/garcia/e06.htm
    http://www.waldorfanswers.org/Waldorf.htm

    You could do a lot of this without buying their materials.

    https://web.archive.org/web/20150425155334/http://alfin2100.blogspot.com/2012/01/how-old-should-children-be-before-you.html

    Al Finn’s old site had a lot of info on child education and links to free resources.

    https://web.archive.org/web/20101018005535/http://alfin2100.blogspot.com/search/label/homeschool

    Here’s his newer stuff.

    https://web.archive.org/web/20160213130032/http://alfin2100.blogspot.com/search/label/competence

    Might help you with your kid. I particularly liked the ideas at

    http://www.see.org/garcia/e06.htm

    where you teach your kids how to make baskets, stone tools, how to use levers, … If you taught a kid all this they would be so competent compared to all the others they wouldn’t need “esteem training” they would KNOW they were far above the rest.

    One of the best things about this kind of education is kids can’t really do anything these days. If a kid knew a few knots, how to make baskets or cook a little something. just little stuff I bet it would make life feel so much more in control. Today children are taught very little practical.

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