Tag Archives: math

Math With No Numbers

Can you do math without numbers? The answer is obviously yes.

Several years ago, I read about someone asking for more math problems without numbers and thought to myself, huh? What’s that even mean? What would it look like? From there, I discovered the vintage book Problems Without Figures by S.Y. Gillian.

Reading on, I discovered exactly what a math problem with no numbers looked like.

If you know the width of one stripe on a United States flag, how can you find the total width of the red stripes?

See what I mean? In order to answer the question, students need to know how many stripes are on the flag and how many are red. They need to understand that to find the total width, they’ll need to choose an operation. In this case, they need multiplication. And in order to answer the question, they’ll have to explain it, because the problem doesn’t tell you the width. The only way to answer is by explaining your process.

See how sneaky a numberless problem is? Sometimes numbers worm their way in there, and several of the problems in the original book did include a number or two. However, most of them were like the problem above. They made students really think about the process of solving the problem.

When students face a word problem, they often revert to pulling all the numbers out and “doing something” to them. They want to add, subtract, multiply, or divide them, sometimes without really considering which operation is the right one to perform or why. When you don’t have numbers, it sidesteps that problem. For students who freeze up when they see the numbers, this can be a really good way to get them to think about their process with math.

That’s been an increasing focus in the wake of Common Core to get kids to be able to show that they understand the math they do. This is a very old fashioned approach that does exactly that.

However, when I first read Problems Without Figures, I saw that Denise Gaskins, the author of the excellent Let’s Play Math, pointed out that it could really use a rewrite. Excited to give it a try before using it with my own kids, I did just that for the first few dozen problems and went on to use them off and on with my kids over the last few years.

Recently, I pulled out the book again and decided to give it a full facelift and publish it. Some of the problems just have updated language. However, for many others, updating didn’t seem to make a ton of sense. Take this gem:

I know the length of a field in rods and the width in feet, how can I find how many acres it contains?

Kids are barely familiar with acres today and rods are entirely bygone as a system of measurement. Some problems like this got rewritten. I added problems with meters, for example. However, some of the problems just needed a totally new take. I tried to add a lot more problems about figuring out how to navigate all the choices we have nowadays.

If you plan to leave approximately a 20% tip on your restaurant bill, what’s a quick way to calculate that amount?

Overall, this was a really fun project. I hope other people find it useful! You can find it on Amazon.

Different Paths, Same Endpoint

Way back in first grade, we started with MEP Math, which I adored, but which turned out to be all wrong for both my kids at that point. When both the kids were frustrated by MEP’s tricky problems, I pulled out Math Mammoth and tried that.

BalletBoy took to the Math Mammoth immediately. He liked that it was so straightforward. It made him practice a good bit, but he didn’t mind that, especially when I didn’t make him do all the problems. In fact, BalletBoy kept doing Math Mammoth all the way through fourth grade math. At that point, the Math Mammoth shine seemed to wear off. The order of topics got a little confusing for him. So we jumped ship. We tried a number of different things, including almost a full year’s worth of the Singapore program, Math in Focus, which was great, but also didn’t quite work for him.

In the end, we went back to MEP Math. BalletBoy finished out his elementary math doing MEP. This time, the tricky problems worked for him. Knowing that he was good at math that emphasized following strong examples and just getting in the practice, I got a vintage copy of Dolciani’s Pre-Algebra: An Accelerated Course to use with him. It was perfect and he did the whole book.

On the other hand, Mushroom found Math Mammoth just as stressful and confusing as he had found MEP Math. For several months, I didn’t make him do anything formal for math. He read living math books and played math games. At some point, I let him try Miquon Math and finally we had found something that clicked. Mushroom did so incredibly well with Miquon that I came to adore the program. Unlike BalletBoy’s Math Mammoth, this was a program that inspired me as a teacher. The number relationships and the huge flexibility of the Cuisenaire Rods as a learning tool was perfect.

When Mushroom ran out of Miquon books, he turned to Beast Academy, which unfortunately only had a few volumes out at the time. However, he did them all. He started talking about how much he loved math. While he never became a fast worker, he was sometimes an incredible problem solver with math. He could think creatively about it. I really credited that to Miquon. He thought about math in a much simpler, straightforward way than his twin.

When we ran out of Beast Academy books, he continued his eclectic math path. He did a lot of the Key to Math books, as well as some problem solving books, like the Ed Zaccaro books. I started him on Jousting Armadillos, which is a pre-algebra program. Unfortunately, the amount of writing focus in that program was all wrong for him. He finished it, but barely. We took a math break, then he started in on Jacobs’s Mathematics: A Human Endeavor, which started to re-invigorate his love of math, though he never quite regained it. Mushroom has a lot of anxiety about academics in general, even though he keeps making good progress.

Having liked Jacobs’s other books, I chose Jacobs for Mushroom’s algebra program. Since BalletBoy did so well with Dolciani’s pre-algebra, I assumed he would continue with Dolciani’s algebra program. However, partway through the year, BalletBoy hit a major snag with algebra and I hit a major snag in teaching it. Since I was loving Jacobs, we made the switch.

That means that, for the first time since the very beginning of first grade, my twins are heading into high school finishing the exact same math program, at more or less the same pace.

It’s fascinating to me how different their paths have been. BalletBoy continues to be a “get it done” math student for the most part. He sometimes gets very stuck in his thinking and I have to tell him to stop and try again the next day. He argues with me about math, only to realize he’s completely wrong when he tries to do the problem. Mostly he likes to do his work and he tends to score well, especially if the problem sets are repetitive. If he gets to one he doesn’t understand, he’s liable to skip it and happily go on to the next problem. Overall, he’s in very good shape for finishing algebra.

Mushroom meandered through so many different math concepts. He continues to be a slow worker. While he doesn’t like to admit it, he does better when he can get engrossed in a few very challenging problems instead of a lot of repetitive practice. He second guesses himself and refuses to move on until he understands, which can be good, but can also bring down his scores on tests.

Despite all these differences, Jacobs’s Elementary Algebra has been great for both of my students. It’s not a perfect program, but it has enough challenges and enough practice. It has engaging introductions and enough example problems. It’s really a thorough and great program. I’m also just thrilled to be back teaching the same math again!

I also think there’s something to be said here about letting kids take their own paths through math. It’s okay to take different ways through the material. In the end, you’re going to emerge in more or less the same place.

Numberless Problems for Today – Free Download

In my last post, I linked to a really cool vintage resource called Problems Without Figures. As I pointed out, I decided to update it so I could use it with my kids more easily. I didn’t update the whole work because it was very long. I did the first 150 problems, which is a little less than half, but I thought I’d share my update here with you. It’s free to download. For more about the types of problems, the reasons to use a book like this, and how to use the book, have a look at my previous post. Below, I’m going to talk a little more about the changes I made to the book and why. I can’t promise that every problem in here is perfect. I’m pretty sure I thought through everything, but I didn’t have a professional editor. Feel free to send me any corrections and do tweak any problems to make them make more sense for your kids if needed. Note that the nature of these problems is to include superfluous information, have answers that are impossible to get without more information, have multiple potential solutions or paths to a solution, and to have “trick” problems where the answer is much simpler than the problem implies. None of those things are necessarily errors. Again, see my previous post for more about these problems.

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I changed the problems in a couple of ways. A large number I left the same or simply updated a word or two to be less old fashioned (such as “shall”), including some of the old fashioned problems about farms and orchards and the like, simply because it’s nice to have different perspectives occasionally. I updated the language in a few problems. For example,  instead of buying marbles and penny candy, kids are buying Legos and other contemporary items. For others, I changed the situation or the measuring units for a problem to be more up to date but tried to keep the spirit of the question. However, for several problems, I couldn’t come up with a modern equivalent so I completely rewrote the problem to be more contemporary.

Doing this was an interesting process for me. It let me think about how math has changed over time. Much of the original book was out of date because of the situations it portrayed. It assumed the readers were familiar with agriculture and farm animals. However, much of what has changed is the math. There were several measurement units that are simply not used anymore, such as the rod as a measurement of length. Other measurement units are in use today and students may have a general sense of what a bushel or an acre is, but they are less a part of everyday life. In general, while measurement math is still a key component of everyday math and we have the complexity in the US of having to go between customary and metric measurements, this book drove home for me how it was an even bigger share of the everyday math people used a century ago. For one thing, we tend to deal with standardized sizes of things much more often. Doors, windows, pieces of paper, tablecloths, picture frames, etc. are all standardized. If we want to know how much land we own, we look it up instead of measuring. If we build something, we tend to follow preset directions instead of making them up ourselves with all the accompanying measurement challenges.

On the other hand, there are types of math that were never or almost never covered in the original book. For example, there were only a couple of problems involving averages. There were no problems involving estimation. There were no problems involving statistics or ratios. Very few problems involved fractions or percents, which surprised me. There were no problems involving permutations or combinations.

Life today is more complex with more choices. We tend to need to know more about combinations and permutations just to order off a menu or decide how to buy something or what to wear. We have to evaluate more complex data and statistics to understand a news article or a scientific claim on the internet. We’re used to bigger numbers. None of the numbers in the original book were very big. These days we’re used to hearing about numbers in the millions and billions. Our tax code and economy are much more complex, meaning that percentages come into play a lot more often. When the original book was written, people lived with a cash economy and prices were more straightforward. It’s different now.

When I tossed out those few problems from the original, I tried to replace them with ones that asked kids about making the sort of decisions that we often have to make using math these days. So there are problems about how to make basketball matchups, how to choose which toilet paper to buy, how to know if a statistic is reasonable, and other more modern conundrums than how many acres in your fields or how many fence posts you need.

Numberless Math

BalletBoy has always gravitated toward just getting his math done. Ideally, he likes to have a page of all the same sort of problem, let me remind him how to do them, and then just do them all. However, as he’s gotten older, this has meant a struggle for him to some extent. As the math gets more complex, with more to remember, not having a strong foundation in the whys of math has led to more and more difficulties for him. If you’ve been reading this blog for awhile, you may note that he’s jumped around programs in the last year as a result. I’ve been trying to honor the fact that he’s pretty good at getting the algorithms memorized while still helping him understand the whys. Finding the right approach hasn’t been easy. Most things have been too easy or too hard.

Recently, the author of the excellent Let’s Play Math blog pointed me to the vintage book Problems Without Figures, which you can find as a pdf here. I immediately fell in love a little bit. Many of them have numbers, but there’s usually missing information so they can’t be “solved.” Instead, the question is focused on the process. If given this and this, can you find that? How would you do it? What other information might you need? Many of the problems can’t be solved unless you know more information. Others are easy, but they require a lot of steps. A ton of them require that you move between different measuring units. Many of them are filled with superfluous information.

There are also several trick problems. My favorite, by far, is the one that asks how you can find how old a coin dated 56 B.C. is. Obviously, a coin couldn’t be dated that (think about it…). The author of the book suggests that these should be sprung on students. I’ve already given the kids a few of these but warned them to look out for them. They were delighted to discover them and felt very clever doing so. I think it’s really good for kids to realize that the answer isn’t always straightforward, that it might be easier or harder than they anticipate, or that they might need different information than what you’re given or expect to need.

In general, I like the focus of having kids doing math that’s not about getting “the answer.” It changes the focus of math and makes it feel more approachable for BalletBoy. It helps kids with their writing as well. Mushroom has been using the Arbor School algebra series, which requires a lot of writing. I wish we had been doing these for a little while in preparation because they really focus a student on writing out a clear, step by step set of instructions for solving a problem. However, the small nature of the problems makes it feel like a doable task. Mushroom has struggled with the writing in the Arbor School series because it asks that he summarize everything he learned and give his own examples. This is such a small chunk and so specific that it builds good logical writing and thinking skills. Basically, it’s a great thing for kids to do for math, writing, logic, and thinking skills. Explaining how to solve a problem is just good across the board.

You may notice that many of the problems in the book are outdated. In order to easily use it with my boys, I updated the language for us. My next post is about that process and I’ll link to the updated version if you’d like to use it.

Math Notebook

Since the boys were in kindergarten, we’ve done math on the white board or math on scratch paper or math with me scribing or math in workbooks or worktexts or with manipulatives. But when Mushroom reached pre-algebra this year I realized that what we had not done was math neatly laid out in a notebook. It was a mess.

However, I was patient. I gave Mushroom a special notebook for math to keep it separate for the first time from the rest of his written work and made him a special cover for it. Then I tried to instill in him to label the top of every page: the lesson number or “Scratch.” Then we got to simplifying expressions and I explained that you have to copy the expression at the start. He looked nigh on devastated. And the notebook was a mess.

But, hey, look at this! Just a month or so after starting to learn about how to keep his math notebook nice and neat, he did this:

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Math is so much about the process. However, there comes a point when the process is hurt by sloppiness. We try really hard to focus on what matters more than how it’s dressed in our schooling. So the quality of the writing matters more than the spelling, that you worked on art for an hour matters more than whether you ended up with a finished product, that you got the right answer matters more than if you forgot to write the units next to it. However, eventually, some of those things matter sometimes. I told Mushroom he had acquired a lifelong skill by being able to keep his math notebook neat and functional.

But I’m also glad I didn’t try to make him acquire this skill earlier. It was pretty painless at this point while it would have been difficult for him earlier. So I’m glad I waited for the right moment to worry a little more about how it looks.

Hands on Math: Manipulatives and More

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I’m a bit of a math manipulatives nut. The folks over at SecularHomeschool.com asked me to write a post for their new Soup to Nuts discussion group so my post is up today. Here’s the first little bit:

I remember the first time I encountered Cuisenaire rods in a graduate workshop. “Be sure you allow time for kids to play with them,” began the instructor, looking around at a room full of educators turning the tiny blocks into towers and patterns of stripes. As we knocked over towers and tried to pay attention to the instructions on how to use these colorful little things with students, we laughed. Even the adults were drawn to playing with their math.

I’ve since learned that there are a million ways to play with your math and hold it in your hands. It’s not a necessary step for absolutely every student, but for most, it makes math more fun, more tactile, and easier to understand. Math manipulatives can be a lifeline for some math strugglers, a shortcut to understanding for some thinkers, and a means to get to a deeper understanding for others. There are dozens of different products out there for both arithmetic and geometry and even an array of products for algebra. There are also ways to make math hands on by bringing it into the real world in other ways.

You can find the rest of my post as well as any discussion that arises from it here.

Probably Probability

We had never covered probability, so I felt like it was time to dive in with a little bit of a focus. Now that we’ve done a good bit of it, I feel like we would have been fun and totally possible to have done it a few years ago as well and then returned to it with a stronger focus before moving into pre-algebra. It’s one of those topics that’s not really covered in books for younger kids, yet kids are constantly encountering probability in their lives, in part because of games. I think it makes a huge amount of sense to cover it earlier.

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Throwing game sticks in an activity from the GEMS Guide.

Picture Books
There are a couple of good picture books for younger students about probability. The first is the MathStart book Probably Pistachio by Stuart J. Murphy. This is really not one of my favorites in the series, but it does introduce the concepts, especially about using probability to predict what comes next. It’s Probably Penny by Loreen Leedy is similar in concept, and readers who like Leedy’s classic picture book about measuring will recognize the same characters. However, my favorite was A Very Improbable Story by Edward Einhorn from Charlesbridge. This one is more clever and introduces a lot more basic vocabulary to start talking about probability, not only for what happens next but also for games.

Chapter Book
Of course there’s a Murderous Maths book for kids who are just a little older. Are You Feeling Lucky? is yet another excellent resource. We have been loving the Murderous Maths series. This one, like the one about shapes, asks readers to try a number of things out and I think it’s best when you do the activities with the book, such as flipping a coin or rolling a die. The book also covers some combinations math, which is nice for us as a follow up to some of the Beast Academy combinations math. If you haven’t yet discovered the Murderous Maths books, know that they often cover some surprisingly complex and difficult math, far beyond what kids would typically do before high school. However they cover it in such a friendly and humorous way that it feels approachable and enjoyable.

Activities
The GEMS guide In All Probability is an excellent one. It’s intended for students grades 3-5. I think the activities could be done more quickly for older kids who need an introduction to the subject as well and perhaps beefed up a little. There are five sets of activities that include flipping coins, making spinners, rolling dice, and making game sticks based on a Native American game. I really like the thought behind the GEMS Guides in general, however, as always, you have to adapt them to homeschool use since they’re really set up for a large class. In this book, some of the activities assume that you’ll gather lots of data from the games. Also, I wish the books were organized with the math more clear and more in depth. There is a teacher section in the back that explains the math behind the activities in more depth, but, for example, the number of chances in the Native American game sticks activity is tied to Pascal’s Triangle, yet it never mentions that. Still, I like the way the investigations are set up for real discovery math and Mushroom and BalletBoy both enjoyed building the spinners, making game sticks, and playing all the various games.

I wanted some pages to practice probability problems, so I had Mushroom do some of the pages from MEP Math that deal with probability. You can find them in the 5b book at the end of this section and the beginning of this one. An alternate source of probability problems and text could have been the NCERT 7th grade math book chapter on data, which covers a variety of concepts, including probability. You can find that here, if you’re interested.

Of course, since probability is in our lives so much, it’s good to look at other places it appears. We started with something greatly revered in our house: game shows. In case you didn’t know, the Husband won us the down payment on the rowhouse many years ago on a game show. Thus the game show’s exalted place in our hearts. It’s fun to look at the odds on nearly any game show. However, the classic game show problem is the Let’s Make a Deal problem, which has been written about a ton, most famously by Marilyn Vos Savant and Martin Gardner (if you’d like to be a math nerd and don’t know who Martin Gardner is, you need to remedy that, by the way). It’s great to actually watch the show and learn about the problem. I saw a great demonstration of it by Ed Zaccaro of the Challenge Math books at a conference once, using envelopes instead of doors. With three envelopes, it’s not clear which one to choose. However, with a hundred envelopes, it’s almost immediately clear. We tried that and talked a little about how probability is one of those things that can be tricky to think about sometimes. This was followed up on nicely by the Murderous Maths book when it talked about how pennies don’t have memories.

If you look up probability lessons on Pinterest, you’ll find lots of options. However, the classic probability lesson that I wanted to be sure to do was with M&M’s. There are many variations, but essentially you have students calculate the probability of drawing an M&M out of the bag. The more bags you calculate, the closer you can start to get to what is, presumably, the actual ratio in which M&M’s are actually produced. You’re finding the experimental probability, so this is a good activity to introduce this term. The theoretical probability can also be found just by looking up in what ratios the M&M/Mars company actually makes the M&M colors. It’s slightly different for each type of M&M, but you can find it easily online.

Finally, we simply tried to be more alert to probability in our lives, such as in weather forecasts, board games, video games, and random events. It’s nice when kids can see math in the real world, especially when it’s things that aren’t money. Overall, this was a good unit for us.