Math: assorted recent topics

As discussed in a previous post, the Singapore Math books serve as a backbone of our math curriculum. It is largely an individually-paced program (though small cohorts often form and work together), allowing students to progress at a rate that makes sense for each topic. We have Math class four times a week, and we generally devote one or two sessions to Singapore work, which allows for questions, small group break-out sessions, and targeted group lessons. 

With the rest of our class time, we do a variety of activities that benefit the entire group, regardless of where they happen to be in their particular math books. In this post, I wanted to share a few of the things we've done in recent weeks apart from Singapore work. 

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We regularly work on problem solving as a class. First we'll go through a problem together, and then we'll break off to solve similar problems in small groups or individually. The problems themselves are sometimes slightly ridiculous ones pulled from various places (you may recall this blog post from October), including some of my own design. For example: 

For this one, kids worked in groups of two to three, picked a spot and attempted to calculate the volume of the space. After some measurements and calculations, the kids had an approximate volume of the space. They then turned to the internet to try and find inexpensive sources of plastic balls. It turns out there are a lot of options: 

You can get anything on the internet. 

One enterprising young student even found an online ball pit calculator, complete with tips on how to construct your pit. 

Taking measurements to convert the middle school airlock into a ball pit (which is almost certainly a fire code violation).

Now I just need to organize a fundraiser to raise capital for the venture... 

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Sometimes, someone will ask a question that lends itself to an impromptu mini-lesson. 

For example, someone had a question about how to break down a problem from their Singapore book. As a class, we spent a period really breaking down a word problem. 

First, I wrote it up on the board as written in the book.

Then, we talked through it. Determining what the question is asking can sometimes be deceptively complicated, so parsing out what is being asked is crucial. Also, there are some vocabulary words that are used infrequently in everyday life, but are important to basic math. Here's what the board looked like after we finished: 

It looks particularly complicated because we explored a few of different ways to approach the problem. 

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We often devote a day to playing math-related games. Sometimes they're classic games like Uno, Blokus, or Othello, and other times, they're Summers-Knoll classics, such as Factor Tag. 

A new favorite that we've played is "The Card-Forehead Game." We obviously need to work on a flashier title, but the game itself is really fun. 

You play in groups of three. Two players face each other, then each take a playing card and hold it up to their own forehead (so they can't see it, but their opponent can). The third person then generates a simple math problem on the spot based on the cards that each player holds up. 

For example, if Nick holds up a 10, and Ella holds up a 7, the third person could say, "Nick minus Ella equals 3." 

Both players have to quickly think about what their card must be if that is true. The third player also gets practice by generating the problem in the first place, and making sure they know the correct answer. 

Here's a quick video of the game in action:  


It's a fun, easy way to get practice in the four operations, firming up math facts along the way. 

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This post has grown beyond my expectations, so I'll cap this entry here. Look for a follow-up installment in upcoming weeks!

Math: Problem Solvers!

Can you see the patterns emerging?

As mentioned a couple of weeks ago, Singapore Math is the backbone of our Math program. However, we also do whole-group inquiry and instruction in our Math group. One frequent group activity is using story problems to develop and practice new problem-solving strategies. 

One such strategy: Making an organized list.

Here's a recent problem we tackled together, starting out as a group, and finishing individually:

This type of problem gets kids thinking about a couple of different things. First of all, it quickly drives home the importance of devising some sort of methodical system for keeping track of your work. You can attempt to do this sort of problem in your head, but you will quickly find that you are either repeatedly coming up with duplicate combinations (actually, they're permutations, but we'll get to that in another lesson), or that you're missing some entirely. 

Instead, it becomes necessary to document your work, and to attempt to do so in an organized manner. In this case, we started by making a chart listing the fish costs, then methodically working from largest numbers to smallest: 

Using the Glass Fish as our basis, we can see a pattern emerging with the remaining numbers in each permutation. Can you see it? (Within each set of numbers for the Glass Fish, the Gouramis go down by one. Once you've populated your chart to a certain point, you can begin to notice patterns, which can inform the choices that you try. 

We did this problem largely together as a whole group. Kids have since had opportunities to try similar problems on their own. Here's another that suggests that you should employ the same strategy ("make an organized list"). 

Kids almost always buy in when there are goofy drawings involved. 

Here, we see the Weebles, the Wobbles, and the Widgets, all working together to rescue poor Wally Widget. (An aside: Before passing this out, I stressed that I want them to become confident problem-solvers in real world math situations. Then I gave them this problem, which is patently absurd. They laughed.)

And here they all are working together. 

Here's how we all got started, 

...and here are some final results: 

The more we've practiced, the more consistently kids have been able to spot patterns early in a problem, then use them to solve them. 

Math: A Singapore Math primer

Math groups at Summers-Knoll are taught four times a week at 8:55 AM. Almost the entire school (the 7/8s have Math later in the day) has Math at the same time so that kids at different points in math can join math groups that are at an appropriate level. These levels are determined by where each individual student is in their Singapore Math books. 

As I'm sure many of you know, Singapore Math is the backbone of our Math program. You can read a good primer about the program here. It is an article well worth your time, as there are aspects of the program that are unfamiliar to many parents and families. 

For example, the Singapore Bar Model! I've had a number of parents come up to me over the last few years saying, "I tried to help my kid do this problem last night, but I realized that I'd have to teach them algebra to do it." These parents have benefited from learning about the Singapore Bar Model, a useful way of thinking about and taking apart problems. 

Check out this slideshow to see how the Bar Model is taught and used throughout the program. Read through the whole thing. It seems simple at first (because it is!), but it can quickly become confusing if you skip ahead before understanding the earlier steps. 

This is true of the Singapore system in general: As a general rule, racing through the books is ill-advised. Singapore Math strives to teach students why things work in math, rather than just teaching an algorithm to use. It cultivates a deeper sense of understanding, which makes learning higher math concepts more natural later in their education. 

One way that Singapore does this is adding an additional pictorial step, such as the bar models. 

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Here are a few other things to keep in mind when working with your child in Singapore Math: 

The 4B Textbook and Workbook. They differ in important ways!

There are two books at each level: A Textbook and a Workbook. They differ in an important way: 

- The Textbook introduces topics, teaches strategies to approach problems, and exercises to practice. 

- The Workbook only provides additional practice problems. There is no instruction in the workbook. 

This means that, on balance, the Textbook is far more important than the Workbook! 

I drew a helpful cartoon to reinforce this in class the other day: 

Here's how the Textbook and the Workbook interact. A child starts out working in the textbook, which introduces a new concept. Here's an example from book 4B, introducing the concept of symmetry:

Most students will work through these two pages on their own (though I will also do direct instruction, depending on student needs and the complexity of the concept). Then, they'll get to a little arrow at the bottom of the page. Here's a close-up: 

This means that the student can turn to Exercise 42 in the Workbook to find more problems of this type for additional practice. 

Here's Exercise 42 in the Workbook: 

Your child's math assignment is very individualized to the work I've seen them do in the classroom. For some topics, I might assign the Textbook lesson, as well as the corresponding Workbook exercises. However, in other cases, I may opt to have a student skip the Workbook exercise if they've clearly demonstrated repeated mastery of the concept in class. (This becomes a useful contract with students: "I won't make you do busywork just because the book suggests it. But that means that when I do think you need to do the extra practice, you'll know that I really feel you need it.") 

If your child is in my math group, you should start hearing about weekly math assignments, usually given in the form of a sticky note placed in their books. 

Of course, we do more than just Textbook and Workbook in our math groups, which we will discuss in a future post! 

Math: Fermi Problems

Enrico Fermi, the noted physicist and architect of the atomic age, was also renowned for his uncanny ability to formulate fairly accurate estimates to seemingly impossible questions, often in his head. These exercises are now known as "Fermi Questions," or "Fermi Problems," and they're used in a variety of contexts, from classrooms to job interviews. In the classroom, they build problem-solving and estimation skills. With Fermi Questions, the methodology of the solution is far more important than the actual answer. In many cases, the exact answer is unknowable, but it is possible to calculate a reasonable estimation. The goal of a Fermi Question is to estimate an answer that's within one order of magnitude of the actual answer. 

Today, my math group worked on a classic Fermi Question: How many piano tuners are there in New York City? The only piece of information that the students were given is that there are about 10,000,000 people living in New York City. (The actual population is closer to 8,000,000, but Fermi Questions involve a lot of rounding.) 

Students worked in small groups, generating a lot of discussion and debate. Of course, the first instinct of many students was to ask for a computer to search for more information. I encouraged them to use their brains and each other instead.

Here are some helpful guidelines for approaching Fermi questions:

Focus on smaller problems
Estimate when necessary
Remember to round
Make realistic assumptions and state them
Include units

As a challenge, consider this problem yourself! How would you approach it? 

After much discussion, the questions began to flow, each requiring a reasonable estimate: 

"If there are 10,000,000 people, how many of them have pianos?" 

"How often is a piano tuned? Some may be tuned monthly, others never..."

"How many pianos can a person tune in a day? Does the tuner take a lunch break?"

"How many days a year does a piano tuner work?"

Discussion ideas for home: 

Work through this problem together! Ask your student how they approached the problem. What smaller problems and calculations were required? 

Stumped? Click here for one possible solution, complete with the thought process. One of our groups today came to roughly the same conclusion!



Here's a huge list of Fermi Questions (along with a bit of educational rationale at the end). Try a few out!