What's in the name "Math Mammoth"?

Well, folks, you might be in for a surprise, but when I was thinking what to name my math program.... You know, first of all, English isn't my native language. Secondly, I'm not super creative when it comes to naming math programs.

I chose "mammoth" because it sort of rhymes with "math". I thought people would be able to REMEMBER it easily! You know, let's say a person stumbles on my website, and days later they try to remember what was it called? Maybe, just maybe, the woolly animal would have made a connection, even if an amusing one, in their mind.

So there you have it. There are no hidden implications. It's not ancient math, nor "humongous" in any sense. The math in "Math Mammoth" is pretty normal and logical.

All this was spurred by a really enjoyable and fun review of my books by Mary Grace at Books and Bairns. She really has a knack for writing!

The New Year 2009

I hope you all have had some happy family time (or otherwise) during these Christmas days! Now, I'm already going to turn your thoughts towards the changing of calendar year.

We're about to change from 2008 to 2009. If you'd like to have some mathematical fun with a new year's theme, check out MathNotation's Get Ready for Happy 41*7^2.

Basically, what you do for this "game" is try to find something special about the number 2009. Like the title of Dave's post tells us, 2009 factored is 41 × 7². So one thing you can do is ask the kids to factorize 2009.

Then, it's just up to you - or the students - to find anything interesting or special about the number 2009. Maybe they can explore the remainders when 2009 is divided by various numbers and find something that sounds "special". Maybe they can explore what kinds of sums they can make with it (it's 1004 + 1005 for example). Or, how about this sum: 2009 = 777 + 29 + 92 + 209 + 902. Or, you could simply ask students to write 2009 as a sum of four (or five or seven) whole numbers that are as close together as possible.

Read also Dave's post.

Yet another way to play a game with 2009 is explained at Math Forum's 2009 Year Game. here you will use the digits from 2009 to form all counting numbers from 1 to 100.

But, most importantly I want to wish you a prosperous & good year 2009! Mathematically and otherwise.

Solving direct and inverse variations in chart form

Dave Marain recently featured my blog on his, and now it just so happens I get to promote his, because I really liked his post about learning direct and inverse variations.

He uses a beagle problem with interesting numbers:

"Three beagles can dig 4 holes in five days. How many days will it take 6 beagles to dig 8 holes?"

The solution is actually quite easy — just think how there are exactly twice as many beagles and also twice as many holes. Dave shows a chart that can help youngsters grasp the solution.

I want to go one step further with this. Let's try a little more awkward numbers:

"Three beagles can dig 7 holes in eight days. How many days will it take 5 beagles to dig 9 holes?"

I want to show you how a "chart" approach will still work. This situation describes joint variation, because there is both inverse and direct variation involved: the number of days it takes to dig the holes varies inversely with the number of beagles (the more beagles, the less days it takes), and varies directly with the number of holes (the more holes, the more days it takes).

In each step on our chart, we change ONE variable (either the number of beagles, the number of holes, or the number of days), keep ONE variable unchanged, and figure out how the third variable changes. You need to carefully think if that third variable is multiplied or divided — if it is in direct or inverse variation.

For example: if the number of beagles is halved, and there are the same amount of holes, how will the number of days change?
Or: if the number of holes is quadrupled, and there are the same amount of beagles, how will the number of days change?

Let's start with the situation given in the problem.

Beagles       Holes     Days
----------------------------------
3             7         8

We will want to find out how many days it takes ONE beagle to dig ONE whole, and then use that as a "stepping stone" to find how many days it takes 5 beagles to dig 9 holes .

So, we first change the chart so we have only ONE beagle, digging the same amount of holes. This, of course, TRIPLES the amount of days.

Beagles       Holes     Days
----------------------------------
3             7         8
1             7         24

Next, we find out how long it takes this one beagle to dig just ONE hole. Now, the amount of days is divided by 7.

Beagles       Holes     Days
----------------------------------
3             7         8
1             7         24
1             1        24/7 = 3 3/7

Then, to the question. How many days will it take 5 beagles to dig 9 holes? First, let's increase the amount of beagles to 5, digging the one hole. That will slash the amount of days by 5:

Beagles       Holes     Days
----------------------------------
3             7         8
1             7         24
1             1        24/7
5             1       (24/7)/5 = 24/35

Lastly, we increase the number of holes from 1 to 9, so that the amount of days will also increase 9-fold:

Beagles       Holes     Days
----------------------------------
3             7         8
1             7         24
1             1        24/7
5             1        (24/7)/5 = 24/35
5             9       9 * 24/35 ≈ 6.17

You could also go through this in some other order. But the beauty of this chart approach is that it will work for any numbers. So, kids who have hard time with joint variation formulas might be able to use such chart approach successfully for all these types of work problems.

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Joint Variation: The formula

For completeness sake, I'll also solve this problem of mine using the formula. Let d be the number of days, h be the number of holes, and b be the number of beagles. And, k is the constant of variation. We know that d varies directly with the number of holes (the more holes, the more days it takes). Also, d varies inversely with the number of beagles (the more beagles, the less days). So:

d = k * (h/b)

To solve for k, we plug in the values from the original situation (Three beagles can dig 7 holes in eight days).

8 = k * (7/3) from which k = 24/7.

Then we have our formula ready to use:

d = (24/7) * (h/b)

The question was: how many days will it take 5 beagles to dig 9 holes? So b is 5 and h is 9:

d = (24/7) * (9/5) = 216/35 ≈ 6.17 days.

Use that brain

You know the old saying, "Use it or lose it." It definitely applies to our brain, as well, and not just our muscles (it's been proven). Using the brain can even help prevent dementia in seniors!

So, if you feel that your brain isn't getting the exercise it needs amidst all the routine housework, kids schoolwork, holiday preparations, and just life, bookmark the following link list.

Then, every day go to one of the sites on the list, and do one puzzle, riddle, or whatever it might be. There are plenty of sites on the list to keep you busy for a while. It says "50 games" but some of the links go to sites full of games:

50 Fun, Free Web Games to Make Your Brain Smarter, Faster, Sharper

For example, promise yourself to have some brain gym time just before or after you finish reading your favorite blogs or news sites or check email in the morning. It should work!

TIMSS 2007 results are in

TIMSS - or Trends in International Mathematics and Science Study - is an international mathematics and science assessment that is conducted every four years. The results for 2007 are in now.

Happily, the United States has improved! The average math scores for both fourth-graders and eighth-graders have risen since 1995, the first year the test was administered. Most of those gains took place among the lowest-performing students. That could be a result, analysts say, of the increased focus on bringing up America's struggling students without as much attention to those at the top. (See US students improve in math.)

I was trying to find a chart that would list the average student achievement by countries, but I couldn't find it directly online; only in the downloadable documents. So I took screenshots from the full report to post the charts here, for your convenience. Note: Not all countries of the world participate in the TIMSS.

4th Grade Student Achievement:


8th Grade Student Achievement:


Quoting from the press release:

CHESTNUT HILL, MA (12-9-08) – Students from Asian countries were top performers in math and science at both the fourth and eighth grade levels, according to the most recent reports of the Trends in International Mathematics and Science Study (TIMSS), released today by the study's directors Michael O. Martin and Ina V.S. Mullis of Boston College.

In mathematics, at the fourth grade level, Hong Kong SAR and Singapore were the top performing countries, followed by Chinese Taipei and Japan. Kazakhstan, the Russian Federation, England, Latvia, and the Netherlands also performed very well. In mathematics achievement at the eighth grade, Chinese Taipei, Korea, and Singapore were followed by Hong Kong SAR and Japan. There was a substantial gap in average mathematics achievement between the five Asian countries and the next group of four similarly performing countries, including Hungary, England, the Russian Federation, and the United States.

You can download and read the executive summary here. It explains some more details about the study, including various factors associated with higher achievement in mathematics.

Download the full report from TIMSS website here.

Origami

Origami is the ancient art of paper folding.

Yesterday I was asked by my daughter to make a paper boat (after reading a Curious George book where he made them). Well, that was easy for me because I learned that when I was a child.

Then she asked me to make a paper bird that she saw in the Adventures of Penrose book. I've never even attempted to make one before. The book had "instructions" but those images just looked quite cryptic. So I turned to the Internet. I found several sites that had picture instructions, but they all made me just puzzled; there would always be one step that I couldn't see how they did it.

Then, I found this site: www.origami.org.uk. They had ANIMATED instructions on how to fold a flapping bird! Finally I was able to make one!



Just thought I'd pass this one, in case some of you are interested in the paper folding stuff. It can be pretty fun for the kids.

Word problem

A word problem:

Luisa has four times as much money as Mary. If Luisa has $240 more than Mary, how much money do they have in all?

Mary has less. Let Mary's money be represented by ONE BLOCK. Then, Luisa's money would be FOUR BLOCKS.



Now, the problem says the DIFFERENCE in their monies is $240. The difference is also three blocks.

So, three blocks is $240.
Then one block is $80.

The question was, how much money do they have in all? They have 5 x $80 = $400 together.

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The same can be solved with algebra. Instead of blocks, we use x. Mary has x, Luisa has 4x. The difference 4x − x is $240. As an equation:

4x − x = $240
3x = $240
x = $80
Then, together they have 4x + x = 5x, which is 5 x $80 = $400.

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You can use the exact same kind of reasoning and block model to solve any similar word problem where one thing is so many times as another, and the actual difference is also given. Try this on your own:

A daddy elephant weighs 7,000 pounds more than his child. Also, he weighs three times as much as his child. How much does the small elephant weigh?

Thanksgiving contest winners - 2

It's time to publish the winners for category 1 of Math Mammoth Thanksgiving contest. I got a lot of very nice testimonials, and in fact I'm still going through them.

Some of the stories of how people have used Math Mammoth were actually quite touching. In one case, it seems Math Mammoth saved the entire homeschool (see the first winning entry below)! In others, my books have made a big difference.

So without further ta-doo let's announce some winners!

1st Prize: $50 credit to get Math Mammoth books

There are two winners.

1. Coleen Posey

Excerpt from her entry:

"I do not exaggerate when I tell you that we were on the verge of giving up on homeschooling and putting our kiddo in school because math was ruining our entire day.

That was not what I wanted to do, so back into "research mode" I went. Enter Math Mammoth. I don't even remember how I stumbled onto the website, but I soon realized it looked like exactly what we needed. What attracted me the most was the clear, concise, to the point explinations and pictures that seemed to explain the comcepts in a way that I (and all of those other programs) just couldn't seem to communicate. I was also thrilled by the no frills layout. There were no cardstock puppets, colored bears, or snap cubes to drag out every day (and fish out of the baby's mouth after math class). There was just a simple, clear explanation, followed by exercises that directly reinforced that explanation.

Now for the great part. I started my second grader with the first grade book, because even though she was up to grade level in things like time, money, and geometry, I knew she needed to pretty much start over in the areas of computation and place value. The first day I gave her the book, she literally did 20 pages or more. She felt confident with the stuff in the beginning, as much of that was review. But she said several times, "I love this math book!" I have NEVER heard that in this house, no matter how simple a math page might have been for her."

Read her entry in its entirety.

2. Tracey R.

Excerpt from her entry:

"...but after several lessons hit a point where he wasn't understanding something again, and then we'd be back to the tears/avoidance. ...

But earlier this fall I was getting very worried. He was now 12 and the age to be in sixth grade, and he was still struggling with long division and advanced multiplication. On top of that, he'd never memorized his times tables because he could add large numbers quickly in his head. He wants to be a chemical engineer, and that's just not very likely for someone who struggles with elementary math!
...
Someone else mentioned Math Mammoth, which was a program I'd never heard of. By the end of that day I'd downloaded the Sample Pack and was trying to figure out where to place him! And within a week I'd bought the complete Blue pack.

...for about two weeks, I had my son go through the multiplication and division books. We started out with both of us looking at the computer. I printed out some of the instructional pages with memorization tips for him to keep in a notebook. Then, we'd scroll through the lessons, around 30 pages a day, only stopping on things that were things he'd struggled with. I'd print out those pages and he'd read them and then try some problems. If he got them right, we'd move on.

He also went to one of the websites mentioned on Maria's blog in order to help him memorize the times tables. He still really didn't see the need to memorize them at first. But when I pointed out to him advanced multiplication was a whole lot easier, with fewer steps, if you didn't have to add up every single individual product in the bigger problem, he realized he needed to memorize them and buckled down! That was also happening during those two weeks.

He's about through all the Math Mammoth books, and surprisingly, Singapore 6B as well. I don't think he's quite ready for Singapore's New Elementary Math yet, so he'll be doing Life of Fred books, supplemented by Math Mammoth worksheets through algebra.

So, thanks to Math Mammoth, he's gone from struggling with 3rd grade math, to doing above-level math, in just about a month. I'd always wondered how some homeschooling moms I know who'd taken their kids out of government schools when they fell behind caught them up, and now I know one good way to recommend."

Read her entry in its entirety.

2nd Prize: $30 credit to get Math Mammoth books

Tricia Huebschman

"... My son found your worksheets very easy to follow with directions that made sense to him. This is a huge advantage in homeschooling as I have 3 kids to teach, all at different levels. We still worked together at times, but he found that he could work independently as well. Explanations are clear and concise. The problems kept him engaged, but did not get too repetitive. I appreciated that some of the sections were difficult as it gave me a chance to work through multi-step problems together. From a teaching standpoint, I like that I can choose a subject to focus upon."

Read her entry in its entirety.

Jeannie Gambill

Excerpt:

"...I have resisted the temptation, and her begging, to omit some of the practice problems, and I'm beginning to see progress in her ability to recall facts. For example, she's always insisted on using the multiplication chart when working problems, so when we began the times tables chapter and she was asked to fill in boxes on the chart, she found it familiar & fun. Each day, she was asked to fill in more & more. Each day she grew crankier about doing it. But we persisted throughout the chapter. My fingers were crossed, but I wasn't sure if this would help either. Before giving her the chapter test on times tables, I let her take a few days to work on another workbook. She was asked to complete multiplication facts (in algebraic form, yeah!) and then plot a graph using the numbers. I was so shocked and overjoyed that she did it quickly and WITHOUT the multiplication chart! All those facts just came from her little brain! Thank you, thank you! I tried three different curricula last year. This is the first one that has made a difference. We are loyal fans!"

Lauri Bolland

Excerpt:

"... I ordered the Blue Addition/Subtraction 2-A, downloaded, and printed that very day.

Well, it turned out to be just what Gracie needed! You teach math topics efficiently and interestingly, with no extraneous fussiness. After doing a few weeks of your math (slowly - 2 pages per day) Gracie asked to know a bit more about the writer. We went online so she could read your "about" page, and was so pleased to put a name and face to your math books. You see, she discovered that you like patterns, and believes you hide patterns all through your math worksheets. In fact, she now notices lots of math patterns quickly, and makes extra effort with her math worksheets to make sure Mrs. Miller doesn't "trick" her. Her ability to see patterns makes learning math facts so much easier. It never occurred to me (or my Engineer husband) to explain math the way you do. You seem to be speaking her "math language" and - finally - math makes sense to her."

Read her entry in its entirety.

Maria Wright

Excerpt:

"...Last year, I bought Add & Subtract 2A&B (Blue series) because my 1st grader was having trouble memorizing math facts with Horizons. I was so pleased with the format and ease of use that this year I am using Grade 2 of the Light Blue series as my math curriculum, and it is working very well for us. We had tried other curriculums that relied heavily on memorization and drill (Saxon and Horizons). We also tried Miquon and Singapore Math, which did not have enough drill and were difficult to follow and teach. Math Mammoth seems to have just the right combination between theory and practice."

Read her entry in its entirety (It's the entry by maria05.)

Michele Call

Excerpt:

"...My fifth grader has always done well with math, but we hit a wall with Singapore. My younger ones were academically immature for their grade, and were understanding nothing. I had long since looked to the math mammoth website when I needed worksheets. I found that more and more often I needed to print out the mammoth worksheets to explain their math, in place of the day's assignment in Singapore. So, because the price was so reasonable, and since I liked the worksheets (particularly for place value-I personally learned so much on the website about place value, which had always seemed intuitive for me, but was not at all intuitive for my second grade son) I chose to buy the set.

I put my daughter in the gold series, fifth grade. I started my other two, if I remember right, in the first addition and subtraction book. The second grader may have been in the second one if I remember right. Anyway, it worked much better than Singapore. I really liked the fifth grade curriculum, although I would have very much preferred to have the full fifth grade curriculum, rather than the gold book. The books for the younger ones worked well also. There was a ton of practice, but it wasn't just boring, rote drill. "

Read her entry in its entirety

Dian Dewi

Excerpt:

"...Bottom line is, this is a very good program. However, be prepared to adjust its scope and sequence according to the student's needs. It is a mastery program, so it sometimes includes more advanced concept in the same chapter as introductory lesson. Do not feel compelled to finish off a chapter in one go. Children need time to solidify a concept. When a child is stuck, move on to other topics while reviewing the previous topics lightly to prevent children from forgetting what have been taught before.

I've mentioned the weakness of a mastery program above. However, a mastery program like Math Mammoth has its own strength. Apart from being logical and conceptual, its scope and sequence is easy to tweak. In fact, this is one factor which makes me use math mammoth. Not all math programs are easy to tweak. I found it hard to tweak a scripted program like Right Start. When I tried tweaking it, I had to go through the whole sets, made note and then taught based on the methods it employs. I guess a spiral program would be more difficult to tweak as well.

I personally do not like programs which make me slave to it. I like to be able to tweak it with ease to match my student's needs.And Math Mammoth fits the bill in that respect. "

Read her entry in its entirety (entry by mom2alma)

Safiyah

Excerpt:

"...The textbooks are pretty much self teaching, with little or no preparation required for each lesson. Since the subjects are taught topically, they can be easily supplemented with other curriculum, though I don't feel the need to supplement as there are plenty of practice questions, drills, word problems, and links to online practice games to reinforce topics learnt. There is enough "meat" in this program to challenge the gifted, and enough practice to encourage the struggler. Each concept is first explained in its longest form, so the children really get a grasp of how it works. The problem is simplified in steps until towards the end of the chapter, the concept is taught in the traditional method. This seemed to us at first to be a long way around, and somewhat confusing, but once we reached the final outcome, it made sense to teach it that way, and the concept is well understood. This is why I really like the program."

Read her entry in its entirety

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There were also other winners with smaller winning amounts:

Amy Burt
Cynthia C, Alaska
Shellie
Rebecca Dickenson
Anita Dillahunty
Lori Thogersen
Raina McGrath
Jenni Thames
Denise

Winners have received an email from me.

Thanks again, for all participants!