Tag Archives: math

Long Division

Mushroom actually made this long division problem for himself because he was enjoying doing these so much.
Mushroom actually made this long division problem for himself because he was enjoying doing these so much.

I wrote about math lately at the Rowhouse, but since long division is the heart of fourth grade math woes, I wanted to post what we’ve done because I’m feeling in a good place about it.

I felt a real frustration that there didn’t seem to be any books specifically about long division for kids.  I looked and looked, but everything I found was little more than practice, not a full book about division.  That really surprised me since I know this is a topic that a lot of kids need a second stab at.  We did like reading the section on division in the Murderous Maths book Awesome Arithmetricks, but I wished there was a book of conceptual problems that would build up understanding of division of larger numbers.  Or even a single living math story book that would present some of the concepts.

Up until recently, the only division the boys had done were division drills, division with zeros, and partial sums.  In the partial sums method, you break the dividend into two or more parts that are easy to divide.  For example, in 287 ÷ 7, you would make it (280 ÷ 7) + (7 ÷ 7).  This is a nice lead in to traditional long division and in some ways is harder since there’s not a step by step algorithm.  This way is introduced in the final books of Miquon really clearly.  I felt like the kids were reasonably proficient with the drills and the partial sums, but that left the dreaded columns of long division.

To introduce the long division algorithm, we turned to the our old friends the Cuisenaire rods.  While we won’t get rid of them any time too soon, I realized this is one of the last major topics we can cover with them.  I feel a bit teary just thinking about it actually.  We have loved these rods so much and used them for so many things.  It’s not that they have been out on the table every day for the last five years, but they have played a role in every major math topic we’ve covered since kindergarten.

Over at Education Unboxed, one of the best free resources for elementary math there is, there are several great videos about doing long division with Cuisenaire rods.  I had to borrow a few extra tens from a friend to help make as many exchanges as we needed in order to illustrate these ideas.  Basically, you think of multiplication as the area of a rectangle, so the divisor and the quotient are the two measurements of the side of the rectangle.

After using this method for a few days, we have gotten into a pretty good place with long division.  The kids feel like they can tackle any problem without the rods.  When they get tripped up, such as Mushroom did with a problem that didn’t have any tens in the quotient, we can go back to the rods and model the problem and clearly see the mistake.

Here’s hoping that this good streak with math holds.

Math Lately

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Mushroom plays a symmetry game in Beast Academy.

We have bounced around with math so much more than I anticipated this year.  I thought I’d post about all the various things we’ve ended up using and where we landed.  Math continues to be one of my favorite subjects to teach.  I was never a “math person” as a kid.  It was always boring because it was too easy in elementary school and then boring because it was too hard later on.  But since then I’ve learned to appreciate math, especially when I unexpectedly ended up teaching it when I was still a school teacher.  I really want my kids to be challenged by it and to know it’s more about solving puzzles and asking questions than about doing sums, though you can’t solve the problems and answer the questions without first learning how to do all those sums and procedures.

BalletBoy’s Math

Math Mammoth Light Blue Series Grade 4-A Worktext (Revised) | Main photo (Cover)The boys have done separate maths since first grade, when BalletBoy happily plugged on and Mushroom became completely disenchanted with the whole idea of math.  For BalletBoy, Math Mammoth has been his spine for what seems like eons and he began the year by wrapping up the last bits of 4a.  Math Mammoth is not exactly exciting math.  The pages can be crowded and the pace can go slow, but for a child who just wanted to plug away at math and get it done, it was a perfect program.  However at the end of 4a, when I went to print packets from 4b, he had a sudden desire to do something completely different.  Meanwhile, the Singapore Challenging Word Problems, which had been his supplement for almost two years, also became…  not too hard exactly, but too confusing.  I often found the wording on the problems confounding such that I didn’t even know what they wanted.  What had always been slightly annoying became too much to interpret when coupled with harder math.

Key to Fractions complete set workbooks only | Main photo (Cover)So it was time to change things up.  We tried to switch back to MEP, which we have used for brief periods since first grade, but it wasn’t right for him either.  I knew the Key to Math books would be a breeze for him, so I had him do the first and second books of several series, including Fractions, Measurement, and Decimals.  If you don’t know this program, it has short, gentle workbooks about specific topics.  The look of the texts is nicely pared down and I like how they have an incremental approach that isn’t as overwhelming as Math Mammoth’s.  However, the biggest topic in fourth grade math, long division, was still missing from his education.  We had to find something else.Math in Focus Grade 4 Student Book A | Main photo (Cover)

After a lot of consideration, we landed on Math in Focus.  He had already covered most of the topics in the second half of their fourth grade syllabus, so he’s working on completing most of 4a right now, including .  He says he likes it so much that he’d like to do the fifth grade program next.  In case you’re not familiar with it, Math in Focus is “the other” Singapore math.  When I looked at it ages ago, I liked it a lot less than Singapore Primary Math, but I think I didn’t give it a fair enough shake.  It has the look of a more American style program, but the math is much more similar to Primary Math.  I really appreciate, coming from using Math Mammoth, how pared down the amount of problems is to the ones that really matter.  We also found Fan Math’s Process Skills in Problem Solving, which we like so much better than the Singapore Challenging Word Problems.  The problems are much more clear and there is much better modeling of solutions and information on how the solutions are arrived at in the back.

Mushroom’s Math

Beast Academy 4A Math Guide | Main photo (Cover)Mushroom is my real math lover.  At the start of the year, I had him working on Beast Academy and the Key to Math books on Fractions and Decimals.  He’s now in Beast Academy 4a and I’ve been dragging things out in the hope that 4b will be out (it was supposed be by late February or early March, but I’m still waiting!!!).  I cannot sing Beast Academy’s praises enough.  The program didn’t work for BalletBoy when he tried it when it was newer, but it has been a boon for Mushroom.  The story in the comic book style text is always funny and very well done.  We especially like the recurring elements such as the little beasts rivalry with the bots, the way Grogg finds bizarre solutions to problems, and the multiple personalities (and alliteration) of Professor Grok.  The tricky, thought-provoking problems in the text are also great for encouraging kids to really delve in and think.  They’ve been good for teaching Mushroom patience with his math.

Spectrum Math Gr. 4 | Main photo (Cover)After he finished some of the Key to books, I felt that with Beast’s slow release pace, he needed to do some really basic fourth grade math review, so I bought him a Spectrum math practice book.  It’s not a real curriculum, but he just needed to practice traditional algorithms.  It’s been a mixed bag.  I think it’s good for him to do this and it has been mostly very easy, but there have been a few things, including the long division algorithm, that he really needed to learn and other things, such as stacked multi-digit multiplication, unit conversion, and names of shapes (memorization of lists is just not his strong suit so “hexagon” is like new information every single time), that he really needed to review.  Some days it’s good for when he needs to escape the frustration of an especially tricky Beast problem, but other days, he has been known to scream, “This is not math!  This is just adding numbers and stuff!”  Well, at least I know he gets that math is more than this.  If I could go back, I might buy the fifth grade book instead of the fourth, since it presumably would have a slightly trickier range of numbers for him to practice.

Extra Stuff

Awesome ArithmetricksAs is the case every year, we use a lot of extras for math.  This year, probably the two most used extras have been the Murderous Maths books and Hands on Equations.  Murderous Maths is a great resource that presents math that’s both easy and difficult (or, as the books would probably say…  diabolical!) with a sense of humor.  The explanations are often really clear and clever and they touch on ways to see numbers and math that most elementary math texts don’t bother with.

Hands-On Equations Learning System | Main photo (Cover)We did the first level of Hands on Equations last year, but I put it away for awhile and pulled it out again to finish the second and third levels.  It’s not a difficult program by any means, but I didn’t want to run through it all in a couple of weeks, so we’ve been doing a lesson once a week or less to draw it out.  The system they present is really very ingenious and some of the tricks they employ have really grown on me over time.  At first I felt like it might be too simplistic, but I now see how they are slowly introducing the basics of algebra one skill at a time.  There’s not much too the program other than a laminated scale and some dice and game pieces, but I think it’s probably worth the cost of the homeschool kit to see how they’ve laid out these lessons.  Between Hands on Equations and Dragonbox, I feel like the kids are going to go into algebra in a couple of years with a really firm grasp of basic concepts to give them a head start.

Three Math Programs at Once

This is Mushroom using Cuisenaire rods and Hands-on-Equations pieces to work variables puzzles in Beast Academy.  He was supremely happy as he did it.  It made the puzzles easier, but once he had done them this way, doing them without the manipulatives was a breeze.

variables

If anyone out there is interested in the Hands on Equations program, I highly recommend it.  Both the boys enjoyed it and, as you can see, it occasionally comes out as a visual reminder of the principles they learned.  As well, I highly recommend sticking Cuisenaire rods and base ten blocks in there to use, which allows for bigger numbers and more complexity as well as a more clear visual representation of the numbers.

Math Forks in the Road

Mushroom is rapidly nearing the end of Miquon.  I predict he’ll be finished with Purple within the month and that’s if we draw it out.  BalletBoy has recently finished Math Mammoth 3.  In the meantime, we’ve been trying out Beast Academy.  BalletBoy likes the graphic novel textbook, but the workbook isn’t right for him.  Everything in it is either too easy or too hard.  I haven’t felt like he’s gotten a lot out of it, so I don’t think he’ll be continuing other than to read the textbook for fun.  If you’re not familiar with Beast Academy, Tinderbox has an excellent review of it here.  Essentially, through a graphic novel about monsters, it introduces math in a very conceptual way and the more difficult practice problems often practically invite frustration.  They want you to try and fail and try again and have an epiphany.  Shockingly, it has been working for Mushroom, which is re-emphasizing the realization I’ve been having lately that he is actually pretty good at conceptual math thinking, even if his calculation skills lag behind.

The other day, we were covering the triangle inequality in Beast Academy and Mushroom wasn’t getting it, so I pulled out our constant friends the Cuisenaire rods.  See how the triangle on the left works because the sum of the two shorter sides are longer than the long side?  But the triangle on the right can never work.

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For a split second, when I didn’t see this activity on Education Unboxed to link it, I thought I had made up a new use for the rods, but nah, I found it somewhere else.  Sometimes I think the rods are pure magic.  They really can be used to teach nearly any math.

What’s next for math at the Rowhouse?  I don’t totally know.  Mushroom will continue Beast Academy, but he needs the ability to switch away when he gets frustrated.  The spiral, jumpy, non-threatening nature of Miquon worked so well for him that we have to find a way to recreate some part of it.  We have several books like this one which should help us use the rods, but we need something else.  I know that Singapore, Math Mammoth and MEP aren’t right for him and it seems silly to begin Right Start only to have it run out on us soon thereafter.  What we do is still a bit up in the air.  For BalletBoy, after a lot of discussion between him and myself, we’ve decided he needs more practice with third grade concepts so he’s going to do MEP 3b, which will be a lot of review and a few new things, alongside the Singapore Challenging Word Problems 3, which we’ve done a little in, but not much.

I feel very unsure about math right now and am worried we’re playing hopscotch with programs a little too much.  I’m trying to be mindful of the need to stick with a sequence to help us keep gaps at bay.  On the other hand, I feel like when we do stick too closely with a single program, both boys have trouble honing their math thinking, not to mention that they get bored and frustrated.  It’s definitely a time of some self-doubt here.

Weird Math

We’ve been doing “fun” math word problems here lately.  I think nearly everyone in homeschooling already knows about Ed Zaccaro’s great series of books, including Primary Challenge Math, but in case you don’t, his books are wonderful.  I have been looking forward to using Primary Challenge Math for awhile.  It explains a type of problem, then gives three different levels of problems about it.  It’s hard to say exactly what makes it so appealing.  The math isn’t so different from what’s in many quality word problem sets.  However, the way Zaccaro lays it out is fun and engaging and not overwhelming at all.  The boys have enjoyed this.

Second, we have done a few problems from the bizarre book The Book of Perfectly Perilous Math.  The problems in this book all involve monsters and impending doom.  If you can solve the problem, you’ll survive and if not, then you’re done for.  Each problem also includes a math lab to explore, which is fun.  We got this at the start of the year and have only done a few, but it’s been a neat thing to have on hand.  Some of the problems, such as about probability and statistics, will be better suited to fourth grade anyway.

Finally, I have finally taken out an old favorite of mine, Louis Sachar’s Sideways Arithmetic from Wayside School.  I adore this book and used it for fun when I was teaching sixth grade math.  I haven’t seen people mention using it as a resource, which makes me sad.  More people should know about it!  The opening chapters give math problems where each letter stands for a digit 0-9 and you have to figure out what each is by looking at the whole problem.  For example, four + eight = twelve and elf + elf = fool.  It’s good mathy fun.  The rest of the book gives amusing logic problems.  It is already a big hit here.

Polyhedra in the Meantime

Here’s another post that languished without me getting around to posting it for a couple of weeks.  But hey, now it gets to have a pretty picture of some geometry adorning our Christmas tree, so it’s more festive now anyway.

BalletBoy has hit a bit of a math wall.  He’s been slowly trudging his way through Math Mammoth Blue.  He made it through nearly all of the parts of the various special topic books, like Clocks and Measurement, and through both the Place Value 3 and Multiplication 1 books, which are the heart of the third grade sequence, along with Division.

Then we hit a snag.  He knows his facts, but memorization has never been his strong suit.  He can run through the whole multiplication table with the flash cards in about six minutes.  But it’s not good enough to get him through division, which is so slow it’s at frustration level for him.  He’s only made it through half of the book and the second half will be even slower going.  Thus the the need for some breaks.

Cover of: Sir Cumference and the Isle of Immeter by Cindy NeuschwanderCover of: Primary Grade Challenge Math by Edward Zaccaro

In the meantime, while we run drills and practiced things, we’ve had more of a chance to look at some of our other resources.  For one thing, we revisited geometry and make a little booklet of the polygons with cute smiley faces.  Then we did a shape scavenger hunt and made a little picture of them all outlined.  The tiles in the Metro are all hexagons, you know.  We also found areas and perimeters and played around with graph paper.  Some of the popular Sir Cumference books have been read recently here for this, though we’ve also been reading The Number Devil aloud as well, so math read alouds have abounded.

Also for geometry, we made the polyhedra models we’d skipped in Math Mammoth earlier.  I went to Worksheet Works to print out a bunch.  The kids made cubes, tetrahedrons and octahedons.  I made a truncated icosahedron, otherwise known as a buckyball.  When we colored them, they came out so pretty that the Husband said they should be Christmas ornaments, so we made some more for the tree.  That’s an icosahedron decorating the tree there.

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We played around with measuring a little more and have especially enjoyed doing a few of the measuring dot to dot puzzles found at Math Pickle.

We’ve also finally started doing Ed Zaccaro’s Primary Challenge Math.  I saw Ed Zaccaro speak at a conference last year and just loved all the things he had to say about math and problem solving.  I had been concerned that some of the stuff in Primary Challenge Math might be too challenging, but it has turned out to be the opposite.  It’s perfect fun math for my kids.  As we’ve gone back to the Singapore Challenging Word Problems a little more, I’ve found that one of Zaccaro’s strategies, to plug in easier numbers for more challenging ones, has worked really well for BalletBoy.

Math App Explosion

We have an absurd number of math apps on our iPad and I thought I’d give a rundown of what we like and find useful.  If you’re like me, you’re always adding more apps, especially when they’re free or only a dollar.  A few of these are more, but they’re all ones we found to be worth the price.

Rocket MathRocket Math
This is the math app we’ve had the longest and we love the designer, who also made Stack the States.  The format of making your own rocket and testing it is appealing.  The math is mixed.  I wish it was easier to customize the math to focus on different skills and skip others, but overall, it’s a fun app worth buying.

Sushi Monster
I like that this app asks you to start with the answer and find the numbers that make it instead of vice versa.  There is an addition and a multiplication option.  It’s a simple game, but the graphics are offbeat and cute.  Plus, you can’t beat the free price.

Math Motion Wings
This game asks kids to visualize multiplication and numbers as you fly a bird around.  It’s not about knowing the exact answer quickly, it’s about comparing values and knowing which is greater.  There’s some fun bits where you get to customize your own bird nest as you go along.

Marble Math
The graphics on this game are just okay, but it’s a clever little concept.  The marble in the maze needs to hit certain numbers before the hole opens up and lets you win.  Almost everything about the game, from the color of your marble to the whether you tilt the screen or use your finger to the math practiced is customizable in this app, which makes it good for more specific practice.

Space Math
If you remember Math Blasters, then this is pretty much the same concept, shooting things and all.  Simple but fun as a way to practice basic math facts.  There’s a lite version and a full version.  We have so many others that we haven’t sprung for the full version.

Numbers League
This game plays off of comics and superheros and has one of the best looks to any of the apps listed.  It’s also a more involved game that feels more like a game in places than a math drill app.  However, math is still at the heart and there’s a lot of good basic math fact practice built into this app.

Dragon Box
This app is set up like the popular puzzle apps with different levels that build on each other.  The goal is to learn, through little cards and pictures, how to isolate unknowns in algebra equations.  This game was incredibly fun for the kids and has been good for referring back to.  I wish it had more levels and more puzzles, especially for the somewhat high price, but it’s still a really innovative app – one of the most innovative that I’ve seen.

24
This is an extremely simple app, intended for the iPhone, that lets you simply play a round of 24, where you use the four basic operations to make the number 24 from the four numbers the card gives you.  There are absolutely no settings.  You draw and play eight cards and the game times you.  I wish there were some settings (it would be nice to be able to choose how many dots you want for your challenge, for example), but at 99 cents, it’s worth buying.

Math Evolve
This is one of the newest buys at our house.  It’s basically another Math Blasters style app where a character swims through the right answer, but it has more levels as you “evolve” your character.  The kids like it and it practices basic facts.

Pick a Path
This was a quick free app where you pick a path with different numbers and operations to try and reach a target number.  The levels are limited and they get too difficult for Mushroom and BalletBoy too quickly (multiplication and division by fractions and decimals are introduced just a few levels in).  I wish there were more boards though because it’s a clever little puzzle.

Coop Fractions
I discovered this little app by accident.  The chickens lay eggs that you have to catch in a moving nest that you place on a number line.  It’s a great game for teaching fractions values and has a bunch of options of how to use it.  I do find the look on the chickens’ faces a bit disconcerting as they lay their eggs, but as it’s a free one, I’m not going to complain.

Math Board
Despite the price tag, this is the best app I’ve seen for just straight math drills on the iPad.  I’ve mentioned it before, but I really like how it lets you customize exactly what you want to drill.  There’s a “chalkboard” space to work out longer problems and an explanations function for more difficult ones.

Miquon and Cuisenaire Love

After using a hodgepodge of half a dozen things for math this year, Mushroom has settled into a regimen of mostly Miquon with the other resources, including Singapore Challenging Word Problems, Math Mammoth, MEP, and Right Start games, taking a clear supporting role.  After dallying for so long to find his stride and taking some diversions for other topics, he’s just now starting Miquon Green, which is essentially the second half the second grade curriculum.  But no worries.  I’d rather go slow and feel like he gets it than rush ahead.

I feel like I’ve really hit the Miquon stride with him.  It’s so very different from the mastery approach I take with BalletBoy’s Math Mammoth.  Instead of covering a topic from head to toe, not releasing until a child can recite it in their sleep, Miquon gives you little pieces of the puzzle in digestible chunks.  A child doesn’t have to master multiplication by fractions.  Instead, Miquon allows them to become confident with the easiest problems first.  I can really see the ways in which this ought to help Mushroom down the line see his way to solving math problems with mental math and common sense instead of by pulling out paper and pencil to do a complicated algorithm (or, heavens forbid, a calculator).

I’ve also found a good balance of really using the Annotations to build supporting activities for the lab sheets using the Cuisenaire rods and a white board.

Speaking of those terrific C-rods, a poster on the Well-Trained Mind Forum has created an invaluable set of user friendly videos about how to use the rods to teach, well, nearly everything.  Whether you’re a Miquon user or just Cuisenaire curious, please go check it out here at Education Unboxed on Vimeo.  It’s an amazing free resource, one of those things that makes me feel like the homeschool community is so generous and cool.  Here, this one on square numbers sums up what’s so nifty about using C-rods and Miquon with kids to teach concepts used reserved for older kids:

I Love Math Books

I heard about the I Love Math series through the Living Math website awhile back, but I didn’t immediately bite and buy them.  Despite their praise-filled reviews, I couldn’t quite get a handle on what these looked like and what they might contain.  Because they were sold as Time Life book sets about twenty years ago, I suspect you’re unlikely to find them in many libraries (they weren’t in ours).  However, I finally bought a couple of them – Look Both Ways and Do Octopi Eat Pizza Pie? and was glad I did.

Each book has a lot of different things all around a central theme.  The theme in Look Both Ways is cities.  There are mathy poems and stories that take place in cities, math puzzles and problems, and math games.  Different parts of the book are done by different illustrators and writers in somewhat different styles.  One page is photos of buildings encouraging readers to find different shapes and look at symmetry.  Another page is cartoon animals in a story about directions through the city.  The table of contents as well as small notes at the bottom of some of the pages explain what math is being covered by various activities, such as odd and even numbers, addition and subtraction, geometrical shapes, money and so forth.  In some ways, we’re already past these, which is too bad.  I think they’re mostly K-3rd grade math and some of the things in them will be too simple for Mushroom and BalletBoy.  However, they’re appealing to kids and have a sit down and browse feel.  They introduce some solid concepts so I may get more, especially for Mushroom.  You know, as if we didn’t have enough living math books already!

Mushroom’s Math

I posted before about how Mushroom has split off for math.  We’re doing a whole spiraly, roundabouty, wibbly wobbly timey wimey curriculum.  And while I usually adore things that are wibbly wobbly timey wimey, I would much prefer to do something more straight up Asian mastery style.  But, hey, you take the kid you’ve got, right?

We’re doing a hodgepodge of things.  He keeps a math journal, in which I put quick drills, a few “challenge” problems, some money math, some catalog math, and a whole bunch of other things.  He’s continuing through Miquon and is almost done with all of the Red book and moving into the proper second grade Blue book.  He’s also playing games a lot more and doing well at them.  We have the Right Start Games, and while I knew the Right Start Curriculum wasn’t for me as a teacher, we’re enjoying the games a lot.

Below you can see a selection of the sort of math journal pages we’ve done so far, in case you’re curious.

 

But here is the lovely thing.  He seems to be less afraid to do math and put out an answer.  I have discovered he actually likes having a math “drill.”  He understands that it’s practice of ideas he already learned and seems to get that if he makes a mistake, then it’s just a mistake, not a total failure of concepts.  What an amazing thing!

And he occasionally makes some great leaps.  We read The Greatest Guessing Game, which is a Young Math Book (I know I’ve said it before, but I’ll say it again: I adore the Young Math Books!) about division and Mushroom began dividing things left and right.

He has been doing more real life math more comfortably.  The other day, after doing arrays in Miquon, he arrayed the Christmas cookies.  He added up the money he had spent for Christmas so far as well.

But here is the best bit.  I apparently checked out a book quite a long time ago from the library and then lost it (it seemed to have fallen into a funny corner of the car during transport).  It was an early 80’s title with slightly psychedelic monster illustrations called Crazy Creature Number Puzzles.  It’s possible for me to not known when it came from because our public library does not assess any fines on juvenile materials.  I know, you’re jealous, right?  Anyway, I discovered it from who knows when and instead of returning it (I’ve probably accidentally renewed it too, anyway), I made Mushroom do some of the problems in it.  They’re easy enough for K-2nd grade math, though they’re tricky enough that it might take you a minute or two to figure them out (well, not most of them, but a few of them took me a minute or two).  But he did them!  And then he said to me when I assigned some more, “I really like those Crazy Creature Number Puzzles.”  Yes, that is a direct quote that I have not made up.  Really, this was his biggest hurdle, being able to be patient enough to sit and think through a problem and try different solutions until he got the right answer.

So I’m feeling pleased with the less curriculum approach right now.

PS – Sorry if the image sizes are a little wonky.  There’s some sort of image sizing bug on my blog!