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!
Selasa, 30 Desember 2008
Jumat, 26 Desember 2008
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.
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.
Minggu, 14 Desember 2008
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:
I want to go one step further with this. Let's try a little more awkward numbers:
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.
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.
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:
Lastly, we increase the number of holes from 1 to 9, so that the amount of days will also increase 9-fold:
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.
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.
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.
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
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.
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.
Jumat, 12 Desember 2008
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!
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!
Rabu, 10 Desember 2008
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:
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.
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.
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.
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.
Sabtu, 06 Desember 2008
Word problem
A word problem:
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.
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.
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?
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.
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:
3x = $240
x = $80
Then, together they have 4x + x = 5x, which is 5 x $80 = $400.
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?
Senin, 01 Desember 2008
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!
1. Coleen Posey
Excerpt from her entry:
2. Tracey R.
Excerpt from her entry:
Jeannie Gambill
Excerpt:
Lauri Bolland
Excerpt:
Maria Wright
Excerpt:
Michele Call
Excerpt:
Dian Dewi
Excerpt:
Safiyah
Excerpt:
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!
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
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!
Rabu, 26 November 2008
Thanksgiving contest winners 1
I have had a fantastic turnout to my Thanksgiving contest - nearly 110 entries! Thanks to all that participated!
I have spent a lot of time reading the responses, and there are some among them that are just jewels - people's testimonials of how Math Mammoth has helped the children or even saved the entire homeschool.
In fact, while reading through, I decided to cast aside my earlier rules about the number of winners, and so there will be MORE winners than what I announced originally.
This post is about the results for the category 2: Give your excuses for not using Math Mammoth yet -- because these were simply easier to judge. Category 1 winners will be published soon after this.
I hope you will enjoy reading through the entries.
TOOT TO DOO DOO!
1. Robin G.
Excerpt from her entry:
2. Shawn Orton
1. Anna Odle was full of questions. Here's a sample of them:
2. Kalyani made a chart of opposing and promoting thoughts that I thought was very original.
Read her entry in its entirety
Besides these, I decided to grant a bunch of $10 prizes (entries not published here).
Winners will receive an email about how to claim their prize.
I have spent a lot of time reading the responses, and there are some among them that are just jewels - people's testimonials of how Math Mammoth has helped the children or even saved the entire homeschool.
In fact, while reading through, I decided to cast aside my earlier rules about the number of winners, and so there will be MORE winners than what I announced originally.
This post is about the results for the category 2: Give your excuses for not using Math Mammoth yet -- because these were simply easier to judge. Category 1 winners will be published soon after this.
I hope you will enjoy reading through the entries.
TOOT TO DOO DOO!
1st Prize: $50 credit to get Math Mammoth books
There are two winners.1. Robin G.
Excerpt from her entry:
... Finances have been very tight for us -- at times nonexistent! My husband suddenly became ill at the beginning of this year and has been unable to work since then. Thankfully, he had short and long-term disability insurance through his now former employer. We do have an income each month, but it is significantly less than when he worked (and even that was much lower than most people consider it possible to live on!). We sold our house in September and have been trying to build another one debt free with the help of friends and family. It is a very small one room cabin style house with no electricity, simply because we cannot afford it. Now we do not have a house payment (and are currently living with my parents until our cabin is ready), but my husband's health insurance and medical expenses total more than our house payment was. I do not want to seem to be complaining, because we know we are very blessed. Right now, we simply cannot afford to buy homeschool curriculum. ...
Read this entry in its entirety
2. Shawn Orton
This year my class is struggling kids,
Whose math has never been the best
Their current book is full of words,
That put comprehension to the test.
One glance at the book, they're overwhelmed!
They need to learn in smaller bites,
But to create the worksheets they would need
Would take many of my days and nights.
And there's the bane of teachers all,
The school does not provide the means
To purchase supplies we feel would help,
And are wallets already are sadly lean.
To my rescue, will you come?
With green or gold books for middle school,
I'll cross my fingers and hope for the best,
To win would definitely be really cool.
2nd Prize: $30 credit to get Math Mammoth books
Again, two winners.1. Anna Odle was full of questions. Here's a sample of them:
1. I am happy with my 1st graders curriculum and his ability to do it. Is your curriculum a supplement? Or is it complete on it own, or perhaps both are true? I did not see if there was a teacher's book to go with a students work. If I download the book do I get the whole thing, or am I just downloading sections of the program?
2. My 3rd grader does not claim to enjoy math and struggles with the amount of work he is expected to do each day. He does not like to copy problems, which he does in his current book. I can see that your downloads would eliminate the need for him to copy his own work, he would just be filling in worksheets. But do I need a different book from the Blue Series for him or does it cover all 3rd grade material? It seemed like a couple of books overlapped the 3rd grade level. Where do I start with him?
3. ...
Read her entry in its entirety
2. Kalyani made a chart of opposing and promoting thoughts that I thought was very original.
| The devil... | The angel... | The result? Me thinks... |
| Indian math is supposed to be good, right? What makes me think Math Mammoth will add more value? | I've grown up studying Indian math. It is good but I had a tough time coping. I don't want my daughter to go through any of the math-phobia I had (and still have). I want material that looks fun, that patiently builds concepts, and that is smart enough for her. | Let me think about it a little more and decide... |
| There are tons of free worksheets on the internet why am I thinking of spending money (paying in dollars!!) to get some? | Thats what I've been doing so far search and search download whatever I can manage to get and serve it up to my daughter. Is this random selection doing her any good? One day she has a problem set too tough for her level the next day it is a breeze not at all challenging. | May be I should spend just ten minutes more and search a little longer... maybe there is a good worksheet site I'm missing... ..maybe... |
| I've never bought anything from overseas on the Internet. Can I do it? Will it work? Will I mess it up horribly? | That's a real concern. But I have to do it some day! | Maybe when I feel a little more brave and adventurous. Let me first check with my Bank if my credit card will work for this. |
Read her entry in its entirety
3rd Prize: $20 credit to get Math Mammoth books
Here we have several winners. Their entries are posted below.I am NOT a formal math teacher; rather, I am a regional instructional technology facilitator so do not directly teach children. I do, however, often teach technology integration to math teachers who seem to me to be in a rut, constantly trying to perk their students who are not so interested.
I usually provide activities, links, and hot lists for these teachers via the eight school districts technology directors, my home district math coordinator, and my web site. I do have a budget, but it is generally taken by salaries for my assistant and myself, travel to the large ends of my coverage area, and supplies and logistical purchases. I never have the luxury of buying specific things for my region.
I think math is important enough to be a major concern to us all. Your products might be a tool I can use to reach teachers to acquire for their classrooms and schools if they are of that much interest to them. Since I do know good instruction and hope I am able to recognize good or great materials, I would like to receive copies of the title series for pre-algebra and first algebra so I can present them to math teachers in my region for their evaluation.
Sincerely,
David Cox
Ok,
I suppose I should enter the contest.
I contemplated it.
But, I decided I'd enter it as much as I promise myself to research which package I'm going to buy.
I am economical minded. I want the best and most for my money. Therefore, I want a package deal. I go to the page for package deals. I decide I want the one that comes with free bonus Soft-Pak.
Then, I wonder which package Light Blue? or the blue, golden, or green and then I see the the Blue series, the Golden series, and the Green series books for 100!
However, I don't see a side by side comparison chart. Just a list. I'm confused. I wish there was a chart to indicate what books are in each package for comparison. I also need to figure out what 5-A from the light blue series is.
That's it. My time is up and the kids need instruction. The math book like I used in school is working for them. I return to using it. I will come back later and chart what workbooks are in what package and what workbooks cover the same topics. I will take time to read more of the emails. Perhaps I should buy a workbook to see the quality of the book too. 100 bucks is a lot to spend. I need time to research it.
However, I never make it back to make a chart that compares the series.
I end up still using the same old school text books.
That is why I haven't bought. I'm still trying to comparison shop and understand each program. I think a online chart would help. Which series goes over addition, fraction, money, etc...
Is there any comparison between the series?
However, it is time that keeps me from buying or writing a real winning entry for Thanksgiving Contest. It might be a better program for my kids.
Fawn
Dear Maria,
I have never used a Math Mammoth book. I have used some of the free samples online, though! And, each time I go through the free samples, I vow to order the whole download of all the books! I love the visuals the worksheets include. I love the suggestions for teaching concepts. When I plan a lesson and use these worksheets, it is always successful. I really love everything about Math Mammoth yet; I still have not purchased the download.
I teach special education and my students are in 6th and 7th grade. Their math skills vary from adding single digits to beginning algebra. Math Mammoth could help me met each of their needs individually. Math Mammoth could make me a more successful teacher. Math mammoth could make math make more sense to my students. Yet, each time I decide to order Math Mammoth, my personal budget will not allow it. And, our school budget is frozen due to the state of California’s budget crisis.
Until the day I can actually order Math Mammoth, I am thankful that some free worksheets are available to me. Thank you Maria Miller, and one day, I will use all the Math Mammoth books.
Sincerely,
Janice Giglio, Special Day Class teacher
Dear Maria
I am very impressed by your work and inspire to do such a program in mathematics at my own end.
But unfortunately my family and professional commitments bar me to do so and I regret for doing nothing inpite of knowing the dire need in my country.
Maria, in Pakistan we have a very low literacy rate and public school conditions are far from pathetic, what about the students and their learning.
We are developing items for conceptual understanding of concepts in English, Mathematics and Science. Your website has been a great source in helping developing questions for facilitating difficult concepts and terminologies.
Maria I would like to extend my profound gratitude in creating such a website where free material is available that helps individuals to benefit in the most ethical and moral manner.
We have developed a booklet for prospect teachers (who are unaware of such items) so that they gear themselves and also acquaint to the creative ways of questions and answers.
Thank you a bundle and look forward to your comments/suggestions . This will be very beneficial in development of more material (support and self-explanatory).
Regards
Bina Nadeem
Dear Maria,
CONTEST or NO CONTEST I am delighted to share my views on your Math Mammoth workbooks & worksheets.
Let me be frank with you that so far I couldn't buy your books despite all my willingness. Reason being simple, I am in Yemen and I don't have a CREDIT CARD to buy your books on line.
As far as series are concerned, I would love to go for all the series: Blue, LB, Golden & Green. This I am saying based on the content displayed on your website and some free downloads that you offered in the past. They are really amazing.
Back home in India, I have recommended this site to many of our friends & teachers and they might have used your resources for teaching and learning maths for their kids.
In my case I happen to download your free workbooks for our kids and they were so excited to do the worksheets. My children are in class -1 & Class-3 and I really miss the complete sets for them.
I myself have been an outstanding student throughout my school & college years and math used to be my choicest subject.
Your course materials & the collections of sums are really interesting & exciting for the kids specially during their formative years. The prices of the books are also quite reasonable.
You're doing an outstanding work in this area of spreading learning through mathmammoth. I really appreciate and would recommend all your books & WSs for every parent & the children.
I am trying to get an INTERNATIONAL CREDIT card in Yemen through which I will be able to buy your books on-line and once I get it, I will buy all your sets on the very first opportunity for my kids.
They love your WSs.
Thanks & best regards,
Udal Ram
Republic of Yemen.
Dear Maria,
I would really like to enter the contest for one (or more) of your Mammoth Books. I like all of them, but especially the light blue books. I have been especially impressed by the variety of concepts and activities that I have seen in the samples, and your explanations of these concepts are excellent.
Why can't I buy your Mammoth Books? I am a Religious Sister. I belong to the Franciscan Sisters of Mary Immaculate. A Religious Sister takes vows of poverty, chastity and obedience. This means that we have no money at our disposal, nor do we use credit cards, which generally excludes me from ordering ANYTHING from the Internet. I would receive permission to purchase materials if I could use them in my assigned work and had gift certificates or vouchers. If I were to win this contest, I feel that this would greatly help my ability to teach various levels of mathematics, and have a source of supplemental materials to support the curriculum in place in the school I work in.
Thank you. I really appreciate this opportunity, even if I shouldn't win. I think it is very generous to give those of us who for some reason are unable to purchase your materials a possibility to acquire them.
May God reward you,
Sister Valentine Curry
Besides these, I decided to grant a bunch of $10 prizes (entries not published here).
Winners will receive an email about how to claim their prize.
Senin, 24 November 2008
Dividing decimals
I feel students need to get grounded conceptually in this topic. So many times, all they learn about decimal division are the rules of how to go about decimal division when using long division, and it becomes an "empty" skill - a skill that lacks the conceptual foundation.
So for starters, we can do two different kinds of mental math division problems.
This decimal division lesson taken from my Decimals 2 book illustrates these two kinds of mental division problems.
Please also see the lesson on dividing decimals by decimals, from my Math Mammoth Decimals 2 book.
So for starters, we can do two different kinds of mental math division problems.
- Division by a whole number - using mental math
Here it is easy to think, "So much is divided between so many persons".
0.9 ÷ 3 is like "You have nine tenths and you divide it between three people. How much does each one get?" The answer is quite easy; each person "gets" 0.3 or three tenths.
And... remember ALWAYS that you can check division problems by multiplication. Since 3 × 0.3 = 0.9, we know the answer was right.
0.4 ÷ 100 turns out to be an easy problem if you write 0.4 as 0.400:
0.400 ÷ 100 is like "You have 400 thousandths and you divide it between 100 people; how much does each one get?" The answer is of course 4 thousandths, or 0.004. Check: 100 × 0.004 which is 100 × 4/1000 = 400/1000 or 0.400 = 0.4.
Here are some more similar ones:
0.27 ÷ 9
0.505 ÷ 5
0.99 ÷ 11
...and you can make more, just think of the multiplication tables. - Division where the quotient (answer) is a whole number
This time it helps to think, "How many times does the divisor go into the dividend?" In these types of mental math problems, the answer ends up being a whole number. (Of course the teacher has to plan these problems just right.)
For example, 0.4 ÷ 0.2. Ask, "How many times does 0.2 fit into 0.4?" The answer is, 2 times. So 0.4 ÷ 0.2 = 2. Again, we can check it by multiplying: 2 × 0.2 = 0.4.
Other similar division problems to solve mentally:
1 ÷ 0.5
3 ÷ 0.5
0.09 ÷ 0.03
0.9 ÷ 0.1
2 ÷ 0.4
1 ÷ 0.01
...and so on.
This decimal division lesson taken from my Decimals 2 book illustrates these two kinds of mental division problems.
Towards the general case
After the student is familiar with the two special cases above, we can go forward and study decimal division problems in general. Even here, we will divide the problems into two classes, depending on whether the divisor is a whole number or not.- The divisor is a whole number.
For example, 3.589 ÷ 4 or 0.1938 ÷ 83. These can simply be solved by long division as they are. Just put the decimal point in the same place in the quotient as where it is in the dividend.
The "stumbling block" may come when the division is not even (this also leads into the study of repeating decimals). Generally, you can continue the division indefinitely by tagging zeros to the dividend, such as making 3.589 to be 3.589000. Then when you've continued the division as long as you wish (or as long as the book tells you to do it), cut the decimal off at a desired accuracy and round it.
Typical problem in a textbook would say, "Do 2.494 ÷ 3 and give your answer with 3 decimal digits." For this, you need to do the long division until the fourth decimal digit - so as to be able to round to 3 decimal digits. Since 2.494 does not have four decimal digits, you tag a zero to it to make it have so (2.4940).
Fortunately, this process is not generally difficult. It's the second case that's more of a problem. - The divisor is not a whole number.
Here, we do something quite special before dividing, and turn the problem into one where the divisor is a whole number. Then, the actual division is done like explained above.
I say this is special, because this special thing that we do is based on a very important general principle of arithmetic:
If you multiply both the dividend and the divisor by some same number, the quotient won't change.
Let's see it in action with some easy numbers:
1000 ÷ 200 = 5
100 ÷ 20 = 5
10 ÷ 2 = 5
Each time both the dividend and the divisor change by a factor of ten, but the quotient does not change.
We can also try it using a factor of 3 (or any other number):
8 ÷ 2 = 4
24 ÷ 6 = 4
72 ÷ 18 = 4
Let's try one more time, with a factor of 2:
30 ÷ 6 = 5
15 ÷ 3 = 5
7.5 ÷ 1.5 = 5
3.75 ÷ 0.75 = 5
H hopefully by now you have convinced the student(s) of this principle. Now we can apply it to those pesky decimal division problems.
This image shows how the decimal division problem 0.644 ÷ 0.023 can be changed into another problem, with a whole number divisor, and with the same answer.
In each step, we multiply both the dividend and the divisor by 10. This, of course, is the same process as moving the decimal point.
Many textbooks only show the student the "trick" of moving the decimal point... but don't show him what that idea is based on.
An example
To solve 13.29 ÷ 5.19, we need to first change the problem so that the divisor 5.19 is a whole number. We multiply both the dividend and the divisor by 10 as many times as needful to accomplish that:
13.29 ÷ 5.19
= 132.9 ÷ 51.9
= 1329 ÷ 519, and now off you go to do long division... I'm not saying it's the easiest long division problem in the world, since the divisor is 519. Let's try an easier one.
2,916 ÷ 0.02
= 29,160 ÷ 0.2
= 291,600 ÷ 2 and now you can do the long division.
Of course, in reality you can also multiply by 100 instead of taking two steps of multiplying by 10. But students can start out by multiplying by 10 as many times as needed.
Please also see the lesson on dividing decimals by decimals, from my Math Mammoth Decimals 2 book.
Jumat, 21 November 2008
Off-Road Algebra
Here's an algebra resource that should interest (at least some) boys: Off-Road Algebra is a unit-study that revolves around the world of off-road motorcycle racing. The lessons fit pre-algebra and algebra, or approximately 9th grade.
For each of the 30 lessons, you view a video, then solve a problem. You'll also get printable explanations and solutions to all problems.
Glancing over the problems, they seem to cover a wide array of topics, such as miles per gallon, velocity, slope of ramps, GPS coordinates, decibels, acceleration, turn angles, lap times, and so on.
Off-Road Algebra
And it's all free (sponsored by Learning.com and Aha!Math) - so thanks, HotChalk and Learning.com!
Rabu, 19 November 2008
Decimal multiplication
This is a tough topic... in a sense. It is not difficult at all, if you just follow the rule given in your math textbook, because the rule is pretty straightforward:
Understanding the rule for decimal multiplication is actually fairly simple, because it comes from fraction multiplication. But, I will propose here a little different way of explaining all this.
First, look over this decimal multiplication lesson that is taken from Math Mammoth Decimals 2 book.
It talks about how 0.4 × 45 is like taking 4/10 part of 45. The same applies if you have 0.4 × 0.9 - you can think of it as taking 4/10 part of 0.9.
Can you see now why the answer to 0.4 × 0.9 has to be smaller than 0.9?
Or, turn it around: 0.9 × 0.4 is taking 9/10 of 0.4, and so the answer has to be smaller than 0.4 (slightly smaller).
Thinking this way, it shouldn't be a big surprise that 0.9 × 0.4 equals 0.36. (The student needs to have a solid grasp of decimal place value prior to this so he can immediately see that 0.36 is smaller than 0.4.)
Now, once your student is comfortable with this idea (as explained in the lesson), then you can proceed on with the explanation based on fraction multiplication. See, we're taking it one step at a time!
Comparing fraction multiplication and decimal multiplication
(I have not yet written a lesson about this for my books, but will do so for the Light Blue 5-B.)
Remember, decimals are fractions.
Let's take an easy example first.
Another example:
One more time:
So... when you write decimals as fractions, the denominators are powers of ten that have as many zeros as there are decimal digits in the decimal number. When you multiply, those denominators get multiplied, and you get another power of ten that has as many zeros as there were in the factors. That, in turn, translates being a decimal number with as many decimal digits as there were decimal digits in the factors.
(In case you don't know: powers of ten are the numbers 101, 102, 103, 104, 105, and so on. Written without the exponential notation these are 10; 100; 1000; 10,000; 100,000; and so on.)
- To multiply decimal numbers, multiply them as if there were no decimal points, and then put as many decimal digits in the answer as there are total in the factors.
Understanding the rule for decimal multiplication is actually fairly simple, because it comes from fraction multiplication. But, I will propose here a little different way of explaining all this.
First, look over this decimal multiplication lesson that is taken from Math Mammoth Decimals 2 book.
It talks about how 0.4 × 45 is like taking 4/10 part of 45. The same applies if you have 0.4 × 0.9 - you can think of it as taking 4/10 part of 0.9.
Can you see now why the answer to 0.4 × 0.9 has to be smaller than 0.9?
Or, turn it around: 0.9 × 0.4 is taking 9/10 of 0.4, and so the answer has to be smaller than 0.4 (slightly smaller).
Thinking this way, it shouldn't be a big surprise that 0.9 × 0.4 equals 0.36. (The student needs to have a solid grasp of decimal place value prior to this so he can immediately see that 0.36 is smaller than 0.4.)
Now, once your student is comfortable with this idea (as explained in the lesson), then you can proceed on with the explanation based on fraction multiplication. See, we're taking it one step at a time!
Comparing fraction multiplication and decimal multiplication
(I have not yet written a lesson about this for my books, but will do so for the Light Blue 5-B.)
Remember, decimals are fractions.
Let's take an easy example first.
0.5 × 0.7 is solved with fractions like this:Notice the denominators 10 and 10 got multiplied to produce the denominator 100 for the answer, and so the answer written as a decimal has two decimal digits.
(5/10) × (7/10) = 35/100 = 0.35
Another example:
0.384 × 2.91
= (384/1000) × (291/100)
= (384 × 291) / (1000 × 100)
= 111744 / 100000
= 1.11744
The denominators 1000 and 100 have as many zeros as as you have decimal digits in the number. The denominator of the answer is 100,000 — with 5 zeros — so the answer as a decimal has five decimal digits.
One more time:
0.45 × 1.3
= (45/100) × (13/10)
= (45 × 13) / (100 × 10)
= 585 / 1000
= 0.585
So... when you write decimals as fractions, the denominators are powers of ten that have as many zeros as there are decimal digits in the decimal number. When you multiply, those denominators get multiplied, and you get another power of ten that has as many zeros as there were in the factors. That, in turn, translates being a decimal number with as many decimal digits as there were decimal digits in the factors.
(In case you don't know: powers of ten are the numbers 101, 102, 103, 104, 105, and so on. Written without the exponential notation these are 10; 100; 1000; 10,000; 100,000; and so on.)
Senin, 17 November 2008
Thanksgiving contest
I feel very thankful for my book sales, for every one of them. So thank you, all my customers!
As a token of appreciation, I thought I'd host a little contest along these lines. I'll call it Math Mammoth Thanksgiving contest. Participate, and you may win some of my books!
Now, I will make you work a little for it, because this is a writing contest. You enter the contest by answering the question below.
There are two "categories": one for those folks who have already used Math Mammoth books, and another for those who haven't. Either way you can participate.
The writing prompts are as follows.
I will choose the winners based on the BEST RESPONSES on the day after Thanksgiving, and then email the winners. Some of the winning entries will also be posted online on this blog.
Since these are downloadable products, I won't limit the winners to one. Instead, there will be TEN winners in each category. That'll give you a good chance to win, I hope!
Each winner will get to choose Math Mammoth books for free, worth the specified amount below.
As a token of appreciation, I thought I'd host a little contest along these lines. I'll call it Math Mammoth Thanksgiving contest. Participate, and you may win some of my books!
Now, I will make you work a little for it, because this is a writing contest. You enter the contest by answering the question below.
There are two "categories": one for those folks who have already used Math Mammoth books, and another for those who haven't. Either way you can participate.
The writing prompts are as follows.
- You have bought and used Math Mammoth books before:
Tell me your experience with the books, good or bad. Mention which book or books this is about, explain how you used them, the kids/students age or grade, their past "math experience", and how did the teaching and learning go. - You have never used a Math Mammoth book:
Explain WHICH book or product interests you most (see the website). Then explain in several sentences the reasons and factors that keep you from buying it at this time. In other words, give your "excuses" for not buying. You can explain how much you perhaps love your current math curriculum, cite economics, and so on. Just let your fingers fly on the keyboard!
I will choose the winners based on the BEST RESPONSES on the day after Thanksgiving, and then email the winners. Some of the winning entries will also be posted online on this blog.
Since these are downloadable products, I won't limit the winners to one. Instead, there will be TEN winners in each category. That'll give you a good chance to win, I hope!
Each winner will get to choose Math Mammoth books for free, worth the specified amount below.
- Two winners - choose books worth $50.
- Three winners - choose books worth $20.
- Five winners - choose books worth $10.
Selasa, 11 November 2008
Voting begins for Homeschool Blog Awards
The polls are up and ready for the 2008 Homeschool Blog Awards. There are 24 categories. My blog has been nominated for the "Best Curriculum or Business Blog" category.
You can go vote following this link.
Honestly, for me the biggest value in this is not actually the idea of competing for the awards, but the fact that I can donate prizes. (That has marketing value... more people find out about my books.)
But anyhow, it's a fun contest and gives us all lots of good blogs to check out!
You can go vote following this link.
Honestly, for me the biggest value in this is not actually the idea of competing for the awards, but the fact that I can donate prizes. (That has marketing value... more people find out about my books.)
But anyhow, it's a fun contest and gives us all lots of good blogs to check out!
Senin, 10 November 2008
A story of a teacher and a boy
You may have read this story before, but I got it today in an email from a friend. It's a good one to pass along! (And sorry, it's not math related... just a real good story. It's fiction, but could be true - and is well worth reading.)
The 5th Grade Teacher
As she stood in front of her 5th grade class on the very first day of school, she Told the children an untruth. Like most teachers, she looked at her students and Said that she loved them all the same. However, that was impossible, because there in the front row, slumped in his seat, was a little boy named Teddy Stoddard.
Mrs. Thompson had watched Teddy the year before and noticed that he did not play well with the other children, that his clothes were messy and that he constantly needed a bath. In addition, Teddy could be unpleasant. It got to the point where Mrs. Thompson would actually take delight in marking his papers with a broad red pen, making bold X's and then putting a big 'F' at the top of his papers.
At the school where Mrs. Thompson taught, she was required to review each child's past records and she put Teddy's off until last. However, when she reviewed his file, she was in for a surprise.
Teddy's first grade teacher wrote, 'Teddy is a bright child with a ready laugh. He does his work neatly and has good manners... He is a joy to be around...'
His second grade teacher wrote, 'Teddy is an excellent student, well liked by his classmates, but he is troubled because his mother has a terminal illness and life at home must be a struggle.'
His third grade teacher wrote, 'His mother's death has been hard on him. He tries to do his best, but his father doesn't show much interest, and his home life will soon affect him if some steps aren't taken.'
Teddy's fourth grade teacher wrote, 'Teddy is withdrawn and doesn't show much interest in school. He doesn't have many friends and he sometimes sleeps in class...'
By now, Mrs. Thompson realized the problem and she was ashamed of herself. She felt even worse when her students brought her Christmas presents, wrapped in beautiful ribbons and bright paper, except for Teddy's. His present was clumsily wrapped in the heavy, brown paper that he got from a grocery bag. Mrs. Thompson took pains to open it in the middle of the other presents. Some of the children started to laugh when she found a rhinestone bracelet with some of the stones missing, and a bottle that was one-quarter full of perfume. But she stifled the children's' laughter when she exclaimed how pretty the bracelet was, putting it on, and dabbing some of the perfume on her wrist. Teddy Stoddard stayed after school that day just long enough to say, 'Mrs. Thompson, today you smelled just like my Mom used to.'
After the children left, she cried for at least an hour. On that very day, she quit teaching reading, writing and arithmetic. Instead, she began to teach children. Mrs. Thompson paid particular attention to Teddy. As she worked with him, his mind seemed to come alive. The more she encouraged him, the faster he responded.
By the end of the year, Teddy had become one of the smartest children in the class and, despite her lie that she would love all the children the same, Teddy became one of her 'teacher's pets.'
A year later, she found a note under her door, from Teddy, telling her that she was the best teacher he ever had in his whole life.
Six years went by before she got another note from Teddy. He then wrote that he had finished high school, third in his class, and she was still the best teacher he ever had in life.
Four years after that, she got another letter, saying that while things had been tough at times, he'd stayed in school, had stuck with it, and would soon graduate from college with the highest of honors. He assured Mrs. Thompson that she was still the best and favorite teacher he had ever had in his whole life.
Then four more years passed and yet another letter came. This time he explained that after he got his bachelor's degree, he decided to go a little further. The letter explained that she was still the best and favorite teacher he ever had. But now his name was a little longer, the letter was signed, Theodore F. Stoddard, MD.
The story does not end there. You see, there was yet another letter that spring. Teddy said he had met this girl and was going to be married. He explained that his father had died a couple of years ago and he was wondering if Mrs. Thompson might agree to sit at the wedding in the place that was usually reserved for the mother of the groom. Of course, Mrs. Thompson did. And guess what? She wore that bracelet, the one with several rhinestones missing. Moreover, she made sure she was wearing the perfume that Teddy remembered his mother wearing on their last Christmas together.
They hugged each other, and Dr. Stoddard whispered in Mrs. Thompson's ear, 'Thank you
Mrs. Thompson for believing in me. Thank you so much for making me feel important and showing me that I could make a difference.'
Mrs. Thompson, with tears in her eyes, whispered back. She said, 'Teddy, you have it all wrong. You were the one who taught me that I could make a difference. I didn't know how to teach until I met you.'
(For those of you who don't know, Teddy Stoddard is the Dr. at Iowa Methodist in Des Moines that has the Stoddard Cancer Wing.)
******
Warm someone's heart today. Pass this along. I love this story so very much, I cry every time I read it.
Just try to make a difference in someone's life today? Tomorrow? Just 'do it'.
Random acts of kindness, I think they call it!
Rabu, 05 November 2008
Math Mammoth Grade 5-A Complete Worktext
Finally! Grade 5-A is available for the LightBlue Series (the complete curriculum series). The part A of 5th grade focuses on
- multi-digit multiplication and long division
- simple equations
- problem solving
- place value with large numbers and the judicious use of a calculator
- all operations with decimals
- statistics and graphing
Please see the table of contents and samples for a complete lesson list, and read more info here.
Minggu, 02 November 2008
Another problem solving book
I know there already exist books that teach problem solving and contain lots of word problems. Well, there is a new one on the block, now: Solving Math Problems by John R. Dixon. It contains math problems with extremely detailed solutions for middle school, high school, and (some) for college level. He's made available a superb collection of samples from his book. Each sample has three completely solved problems. There's a sample file for pattern recognition problems, counting problems, word problems, optimization problems, and fun or "recreation" problems.
Even if you're not planning to buy any math books, I encourage you to check his samples and read them through, just for your own learning (and of your students). It isn't often that one finds such detailed expositions on how to solve word problems. I'm a firm believer in the "apprentice" principle, when it comes to learning problem solving. In other words, one of the best ways to learn to solve problems is to observe an "expert" doing so. (This is why I've often solved word problems here on my blog.) These example solutions can be very enlightening to parents and teachers who might not had much opportunity to witness the thinking that goes into solving even simple problems. For those who can solve word problems easily, such thinking seems to come easily and becomes automated. But for others, it doesn't, and so reading up well-written detailed solutions can help you.
In fact, it would be even more enlightening to witness and expert problem solver solve a problem from scratch and follow his thinking through the incorrect solution paths, as well. You know, all of us, when solving a problem, might (or will) start working on the problem and go down a path that will eventually be a dead end. Good problem solvers monitor their progress on a "meta" level, and "turn back" from those dead ends when they notice them to be so - and try to take a fresh approach.
All too often, youngsters simply give up when such happens! Wrong solution paths are valuable as well. They're just about inevitable when you tackle a non-routine problem!
The book is offered as a softcover for $19.95 and as a download (ebook) fo $9.95. The author claims it's written for middle and high school math teachers, but I feel he shouldn't market it that way — the book has a lot of value for us parent-teachers as well!
Sabtu, 01 November 2008
Multiplication family group
The information below is from another Maria, namely MariaD from NaturalMath.com. I'm posting it here with her permission, as it might interest some of my readers.
Hello!
My name is MariaD, and I love multiplication. Natural Math is starting a research and development family group about this topic. You are cordially invited! Please forward this invitation to other families who may want to join.
There are three main benefits. You receive individual family math coaching. You access a community of other parents sharing questions and ideas. And you contribute to a beautiful and much needed web resource for the future. There are two main responsibilities. At least weekly, you will run custom family math activities you select. As needed, you will talk with me or other group members about your activities. We can talk by email, chat, voice, or face-to-face in Cary, North Carolina, USA. At this early stage, we need active group members. If you plan to be a quiet fly on the wall, please wait until the next round of development. Time estimate is that the group will provide your family at least an hour a week of math and community activities.
Multiplicative reasoning is the capstone of arithmetic: it ties all the parts together. It is the cornerstone of algebra and the basis of pattern thinking. It is also one of the most badly taught areas of math. People spend a lot of effort and many years on times tables, division, fractions, and proportions. Still, many struggle with these multiplication topics for the rest of their lives. I am a strong believer in multiplication. A kid who "gets" multiplicative reasoning will probably be just fine with algebra and math in general. Based on this faith, I've spent more than twelve years collecting, researching and creating multiplication-related lore.
My collection includes psychology of multiplication. It explains why 7*8 and 6*7 are hard to memorize without gimmicks, or how doubles relate to our innate sense of health, beauty and order. There are tidbits about multiplication from histories of many cultures: Ancient Greek music of the spheres, and medieval Chinese secret finger codes for trades. The collection has a lot of modern children folklore. It includes rhymes, finger tricks for times nine and all times tables beyond five, silly pictures and jokes. There are all kinds of contraptions: abacuses, mirror books, bead strings, and Napier bones. There is software: powerful Excel, or small applets for a kaleidoscope, a snowflake creator, or a base two calculator. There is cutting-edge as well as classic research: hundreds of articles, conference presentations and books. Some of these are actually useful, but most are ever read by just a handful of academia people. Speaking of which, there are also people in my collection! Among our contemporaries, there are parents, researchers, designers, and writers who love multiplication, too. This collection of multiplication stuff, and people, can help us start.
I envision a "multiplication planet" map, connected by a web of many paths. Each family can start at a different entry point, depending on their goal. If you want to memorize times tables in three hours, your will probably trek through algebraic shortcuts, memory tools and work with patterns. If you want to have rich, deep experiences connecting many human endeavors, you will also visit algebraic shortcuts. But then you will travel to geometric explorations, history-centered projects, or psychological experiments. If you want arts and crafts, you'll head for drawing, cutting, or computer animation activities. This first stage of research has five main goals for the map.Please contact me if you are interested.
- Develop and find major multiplication activities to put on the map. As all Natural Math activities, they will be centered on creating something.
- Develop paths between activities, following each family's travels.
- Find out what kinds of families use each path, and for what. Use this knowledge to start a guide for new families joining us.
- Find out what support people need in their journeys.
- As we do all of the above, plan web tools that can help us do it better.
--
Cheers,
Maria Droujkova, PhD
Kamis, 30 Oktober 2008
A contest
Choosy Homeschooler is holding a little contest where you can win $50 worth of my Math Mammoth books.
Click here to enter.
This, in fact, has given me an idea to hold a contest of my own. I think I'll do it as a Thanksgiving contest, sometime in November.
Click here to enter.
This, in fact, has given me an idea to hold a contest of my own. I think I'll do it as a Thanksgiving contest, sometime in November.
Senin, 27 Oktober 2008
Roman numerals and other number systems
I recently created a worksheet generator for Roman numerals. Feel free to use it.
Roman numerals is not any major topic in the math curriculum. They are still used in clocks, to number chapters of a book, write year numbers, and such, so students need to study them, even if just briefly. Fortunately it isn't that difficult a topic.
Many youngsters might, in fact, be interested in learning about different number systems that have been used in various civilizations over the centuries.
I just posted about writing in math class, and this topic would make for an excellent writing project that connects math, writing, and history. Then you wouldn't be doing it just for the sake of learning the math, or the historical facts, but also to practice writing a report or an essay.
It's probably the easiest to work into the curriculum if you're homeschooling, because classroom teachers may have to just kind of scurry by the Roman numerals on into the next topic. But even if you're a teacher, consider printing out a few interesting articles about the various number systems, such as the Mayan, the Egyptian, or the Babylonian, and then giving these printed articles as extra reading to kids who might be interested in such.
Ideas for a writing project on number systems
Some resources
Use these web sites to get you started.
Roman Matching Game
Drag the Roman numerals to the corresponding Arabic numerals. If you win the next game will be faster. See if you can beat the clock!
Roman Numerals - Wikipedia
An article explaining the usage, origin, and a chart of Roman numerals.
Numbers
A comprehensive website about various number systems, such as the Egyptian, Babylonian, Chinese, Greek, Roman, Mayan, and Arabic.
Numeral Systems - Wikipedia
Wikipedia article on numeral systems, which contains links to Hindu-Arabic systems, Asian numerals, Alphabetic numerals such as Greek or Hebrew, and other systems including Mayan, Roman, and Babylonian.
Mayan Mathematics
Roman numerals is not any major topic in the math curriculum. They are still used in clocks, to number chapters of a book, write year numbers, and such, so students need to study them, even if just briefly. Fortunately it isn't that difficult a topic.
Many youngsters might, in fact, be interested in learning about different number systems that have been used in various civilizations over the centuries.
I just posted about writing in math class, and this topic would make for an excellent writing project that connects math, writing, and history. Then you wouldn't be doing it just for the sake of learning the math, or the historical facts, but also to practice writing a report or an essay.
It's probably the easiest to work into the curriculum if you're homeschooling, because classroom teachers may have to just kind of scurry by the Roman numerals on into the next topic. But even if you're a teacher, consider printing out a few interesting articles about the various number systems, such as the Mayan, the Egyptian, or the Babylonian, and then giving these printed articles as extra reading to kids who might be interested in such.
Ideas for a writing project on number systems
- A report on Mayan, Babylonian, Egyptian, or Chinese numbers. Even younger students could probably write a few sentences and give a few examples of their numbers. You can easily find articles to print about them on the Internet. Wikipedia is a good starting point.
- Another writing idea is to study several number systems at once, and write a "comparison report" where you compare these other systems to our current number system, which is called the Indian-Arabic number system.
Some main points for such comparison are:
* How many symbols are used?
* Are they used additively? Or is it a positional system?
Or a mixture of both?
* What are the bases used? (could be 10, 20, 60)
* How easy is it to perform the basic four operations? - Yet a third way is to write a report about base 2 numeral system and systems with other bases. This would work best with middle and high schoolers, and should appeal to any computer science minded folks, BTW, because computers use base 2 in their "internal workings".
Some resources
Use these web sites to get you started.
Roman Matching Game
Drag the Roman numerals to the corresponding Arabic numerals. If you win the next game will be faster. See if you can beat the clock!
Roman Numerals - Wikipedia
An article explaining the usage, origin, and a chart of Roman numerals.
Numbers
A comprehensive website about various number systems, such as the Egyptian, Babylonian, Chinese, Greek, Roman, Mayan, and Arabic.
Numeral Systems - Wikipedia
Wikipedia article on numeral systems, which contains links to Hindu-Arabic systems, Asian numerals, Alphabetic numerals such as Greek or Hebrew, and other systems including Mayan, Roman, and Babylonian.
Mayan Mathematics
Sabtu, 18 Oktober 2008
Writing and math
This post is inspired by Denise's recent coverage of the topic.
Combining writing and math, or using writing in math class is an interesting topic (to me anyway). Most kids probably feel that they are far from each other. Now, if all you do is plow through calculation problems, then of course you're not writing, but at some point students need to be able to write complete solutions to math problems. They need to be able to communicate their thoughts clearly.
Here are some ideas for you to get started with writing in math class. Please use your judgment in using these; for some kids it might be counterproductive - for example, if the child hates writing so much that math class is his/her only "refuge" from it.
If you are interested in trying this out, there are many more resources on writing in math here, compiled by Denise.
And feel free to share your experiences with math and writing, if you have any.
Combining writing and math, or using writing in math class is an interesting topic (to me anyway). Most kids probably feel that they are far from each other. Now, if all you do is plow through calculation problems, then of course you're not writing, but at some point students need to be able to write complete solutions to math problems. They need to be able to communicate their thoughts clearly.
Here are some ideas for you to get started with writing in math class. Please use your judgment in using these; for some kids it might be counterproductive - for example, if the child hates writing so much that math class is his/her only "refuge" from it.
- Younger students might enjoy making their own "mini math book" with some calculation problems and a few word problems. (My daughter constantly makes math problems for her stuffed animals : ) )
- Math journaling. This means that the student writes a short entry into a separate journal book at the end of some lessons, or every few days. These entries could explain a new concept just learned (with a picture), give a problem and its solution, or perhaps talk about a mistake and how it was corrected, etc.
The entries could also be more personal and deal with feelings or wishes towards math. You can find more ideas and math journal prompts here. - Making your own math dictionary. This could be especially helpful with geometry. I've written about this before; check it out.
- Writing word problems and solutions to them. A student could first simply rewrite a word problem from the math book by changing the numbers and/or the wording in it. Later you could ask him to make problems of his own, for example a word problem that employs addition and multiplication, or a word problem about shopping, going to the hairdresser, or some other real-world situation.
Students need to learn to write readable and understandable solutions to math problems, anyway. This becomes more and more important as they advance. Here's a guide of how not to write math solutions. It's written for students who take part in math competitions, but you can see the basic ideas clearly from the examples they give, even if you don't understand all the math. - A complete essay on a math topic. Try a topic that is not usually covered in school math, such as Fibonacci numbers or fractals.
If you are interested in trying this out, there are many more resources on writing in math here, compiled by Denise.
And feel free to share your experiences with math and writing, if you have any.
Rabu, 15 Oktober 2008
Nominations begin for Homeschool Blog Awards
There are 24 categories. A blog has to get at least three nominations before it will enter the voting stage. Nominations are accepted till the 24th of October.
So if you like my blog... feel free to nominate it.
Yours truly is also offering prizes for some of the winners. You can see the full prize list here.
Senin, 13 Oktober 2008
Coffee Shop
I couldn't resist playing this fun game and you and your kids might not resist it either.
Basically you first buy some inventory to make coffee, adjust your coffee recipe (more or less milk, more or less sugar, etc.), define a price, and go selling. These funny animated folks pass by your coffee stand and voice their opinions, whether it's needing more milk or is too pricey or good quality for the price, etc.
At the end of each day you'll see a graph of your earnings and of your reputation. Then you go shopping for more inventory and adjusting your recipe and price. The play continues for 14 days. And the weather changes, too.
Hoodamath.com/games/coffeeshop.php
My daughter wanted to keep the coffee price way too low, like $1.65 or $2.00 and in no time was almost running out of money... : ) My hubby played it for hours.
The game practices decision making and analyzing several variables - critical thinking in other words.
Basically you first buy some inventory to make coffee, adjust your coffee recipe (more or less milk, more or less sugar, etc.), define a price, and go selling. These funny animated folks pass by your coffee stand and voice their opinions, whether it's needing more milk or is too pricey or good quality for the price, etc.
At the end of each day you'll see a graph of your earnings and of your reputation. Then you go shopping for more inventory and adjusting your recipe and price. The play continues for 14 days. And the weather changes, too.
Hoodamath.com/games/coffeeshop.php
My daughter wanted to keep the coffee price way too low, like $1.65 or $2.00 and in no time was almost running out of money... : ) My hubby played it for hours.
The game practices decision making and analyzing several variables - critical thinking in other words.
Minggu, 12 Oktober 2008
Math Mammoth Blue Series for grades 1-3
For Grades 1-3
16 Blue Series books
Price: $34 (download)
All in all these books contain about
760 lesson & practice pages.
Buy at Kagi
Addition 1
Subtraction 1
Add & Subtract 2-A
Add & Subtract 2-B
Add & Subtract 3
Place Value 1
Place Value 2
Place Value 3
Multiplication 1
Division 1
Clock
Measuring
Early Geometry
Money
Canadian Money
European Money
Introduction to Fractions
Learn more!
Sabtu, 11 Oktober 2008
News article: Math skills suffer in the U.S.
I found this an interesting read; the study it talks about has found how undervalued math and math skills are in the States. As a result, very few kids want to pursue it.
Math Skills Suffer in U.S., Study Finds
QUOTE:
"The United States is failing to develop the math skills of both girls and boys, especially among those who could excel at the highest levels, a new study asserts, and girls who do succeed in the field are almost all immigrants or the daughters of immigrants from countries where mathematics is more highly valued."
Math Skills Suffer in U.S., Study Finds
QUOTE:
"The United States is failing to develop the math skills of both girls and boys, especially among those who could excel at the highest levels, a new study asserts, and girls who do succeed in the field are almost all immigrants or the daughters of immigrants from countries where mathematics is more highly valued."
Jumat, 10 Oktober 2008
Worksheet news
Some worksheet-related news from HomeschoolMath.net site:
- Grade 5 worksheets - ready-made worksheets, yet different (randomly generated) each time.
- Decimal worksheets generator just got better. Now you can let the number of decimals vary randomly in the problems. Also includes a bunch of ready-made worksheets that you generate just by clicking on links.
- Addition worksheet generator got better also. Now you can set the range individually for addends 3-6, and randomly switch all the addends (previously only addends 1 and 2). This page also has some links to click on to make worksheets readily, without actually bothering with the generator itself.
Minggu, 05 Oktober 2008
Decimals worksheet generator
UPDATED! My Decimal worksheets generator just got better. Now you can vary the number of decimals randomly in the problems.
Also, the page now includes a bunch of ready-made worksheets that you generate just by clicking on links - what could be easier than that? Of course you can still use the generator to tailor-make worksheets to your exact needs.
This generator makes worksheets for decimal addition, subtraction, multiplication, and division.
Also, the page now includes a bunch of ready-made worksheets that you generate just by clicking on links - what could be easier than that? Of course you can still use the generator to tailor-make worksheets to your exact needs.
This generator makes worksheets for decimal addition, subtraction, multiplication, and division.
Selasa, 30 September 2008
Another algebra problem - or is algebra needed?
Updated!
There are some marbles in Box A and Box B. If 50 marbles from Box A and 25 from Box B are removed each time, there will be 600 marbles left in Box A when all marbles are removed from Box B. If 25 marbles from Box A and 50 marbles from Box B are removed each time, there will be 1800 marbles left in Box A when all marbles are removed from Box B. How many marbles are there in each box?
Again, this is from Singapore and teachers have told students not to use algebra to solve this question. However, any form of heuristic tools are allowed to facilitate the students in solving the questions.
I'd like to point out that I feel it's a good problem, but students might benefit from some "preparation". You could set up a preparation problem like this:
Jar A has 100 marbles and jar B has 40 marbles. You will start removing marbles one by one from jar A, but by 2's from jar B. How many marbles are left in jar A when jar B is empty?
What if you remove 2 marbles at a time from jar A and one marble at a time from B, then how many marbles are left in A when B is empty?
Then we can vary the numbers, for example let jar A have 250 and B have 90. Or, let A have 250 and B have 400 and see what happens! Lastly, change the number of marbles removed to 25 and 50 as in the real problem.
Now, the information given is "reversed" in the original problem because there we know how many marbles will be left and don't know how many marbles were there in the beginning. This does make the problem more difficult, obviously.
Solution:
When I saw this problem, my head automatically "saw" another usable unknown as "how many times do we scoop out marbles from each box?" You see, in scenario 1 we scoop out 50 marbles at a time from A and 25 from B, but the scooping is done the same number of times. So I called that n.
That automatically headed me down the "algebra" route... I wanted to write stuff using n:
There are 50n + 600 marbles in A, and 25n in B.
Then in situation 2, we don't do the same amount of scoopings... so the n is not the same. This time, we take 50 marbles at a time from B until it's empty, so B got emptied in double time as compared to situation 1. So... we have n/2 or half as many scoopings taking place.
Thus there are (n/2)*25 + 1800 in A, and (n/2)*50 in B.
Now you can get an equation that solves it by setting the number of marbles in A equal to number of marbles in A from the two situations:
50n + 600 = (n/2)*25 + 1800
Some solving... n = 32.
Therefore there are 50*32 + 600 = 2200 marbles in A, and 25 * 32 = 800 marbles in B.
Now, as far as solving it heuristically without using algebra... it was really hard for me at this point, since my head only wanted to consider the problem this one way. I started visualizing in my head two jars and two hands picking the marbles out... First one hand picks 50 each time out of the jar with more marbles and 25 out of the jar with less marbles until jar B runs out. Then the other way around: picking 25 out of A while taking 50 out of B, until B runs out.
Then I "saw" that the amount of marbles actually taken out from A in situation 1 was FOUR times the amount of marbles taken out from A in situation 2. (Just comparing how many marbles got taken out from A.)
This is because in 1, we took two times as many marbles out each time (50 vs 25), and also because in 1 it takes double that long (double the amount of scoopings) than in situation 2 (because we're timing all this by how quickly B runs out, and B runs out in half a time in situation 2).
We also know that the first time we took out 1200 more from jar A than in situation 2. So, that 1200 is 3/4 of the marbles taken out in sit 1. From which it's easy to get that 1200/3 * 4 = 1600 is the number of marbles taken out from A, in situation 1.
And that solves it then because now we know that there were 1600 + 600 = 2200 in A. And, since 1600 marbles were taken out from A by 50s, it means it was done 32 times. So B had 32 * 25 = 800 marbles.
I definitely think algebra is the easier way to go ... less brain strain for sure!
You can find yet other solution methods in the comments.
There are some marbles in Box A and Box B. If 50 marbles from Box A and 25 from Box B are removed each time, there will be 600 marbles left in Box A when all marbles are removed from Box B. If 25 marbles from Box A and 50 marbles from Box B are removed each time, there will be 1800 marbles left in Box A when all marbles are removed from Box B. How many marbles are there in each box?
Again, this is from Singapore and teachers have told students not to use algebra to solve this question. However, any form of heuristic tools are allowed to facilitate the students in solving the questions.
I'd like to point out that I feel it's a good problem, but students might benefit from some "preparation". You could set up a preparation problem like this:
Jar A has 100 marbles and jar B has 40 marbles. You will start removing marbles one by one from jar A, but by 2's from jar B. How many marbles are left in jar A when jar B is empty?
What if you remove 2 marbles at a time from jar A and one marble at a time from B, then how many marbles are left in A when B is empty?
Then we can vary the numbers, for example let jar A have 250 and B have 90. Or, let A have 250 and B have 400 and see what happens! Lastly, change the number of marbles removed to 25 and 50 as in the real problem.
Now, the information given is "reversed" in the original problem because there we know how many marbles will be left and don't know how many marbles were there in the beginning. This does make the problem more difficult, obviously.
Solution:
When I saw this problem, my head automatically "saw" another usable unknown as "how many times do we scoop out marbles from each box?" You see, in scenario 1 we scoop out 50 marbles at a time from A and 25 from B, but the scooping is done the same number of times. So I called that n.
That automatically headed me down the "algebra" route... I wanted to write stuff using n:
There are 50n + 600 marbles in A, and 25n in B.
Then in situation 2, we don't do the same amount of scoopings... so the n is not the same. This time, we take 50 marbles at a time from B until it's empty, so B got emptied in double time as compared to situation 1. So... we have n/2 or half as many scoopings taking place.
Thus there are (n/2)*25 + 1800 in A, and (n/2)*50 in B.
Now you can get an equation that solves it by setting the number of marbles in A equal to number of marbles in A from the two situations:
50n + 600 = (n/2)*25 + 1800
Some solving... n = 32.
Therefore there are 50*32 + 600 = 2200 marbles in A, and 25 * 32 = 800 marbles in B.
Now, as far as solving it heuristically without using algebra... it was really hard for me at this point, since my head only wanted to consider the problem this one way. I started visualizing in my head two jars and two hands picking the marbles out... First one hand picks 50 each time out of the jar with more marbles and 25 out of the jar with less marbles until jar B runs out. Then the other way around: picking 25 out of A while taking 50 out of B, until B runs out.
Then I "saw" that the amount of marbles actually taken out from A in situation 1 was FOUR times the amount of marbles taken out from A in situation 2. (Just comparing how many marbles got taken out from A.)
This is because in 1, we took two times as many marbles out each time (50 vs 25), and also because in 1 it takes double that long (double the amount of scoopings) than in situation 2 (because we're timing all this by how quickly B runs out, and B runs out in half a time in situation 2).
We also know that the first time we took out 1200 more from jar A than in situation 2. So, that 1200 is 3/4 of the marbles taken out in sit 1. From which it's easy to get that 1200/3 * 4 = 1600 is the number of marbles taken out from A, in situation 1.
And that solves it then because now we know that there were 1600 + 600 = 2200 in A. And, since 1600 marbles were taken out from A by 50s, it means it was done 32 times. So B had 32 * 25 = 800 marbles.
I definitely think algebra is the easier way to go ... less brain strain for sure!
You can find yet other solution methods in the comments.
Minggu, 28 September 2008
An algebra problem
This question was set in one of the renowned primary school from Singapore. Given to me by "anonymous" to solve.
Solution:
Let A be the initial amount Andy has, and P the initial amount Peter has.
Then we know that A = P + 200. We're going to use that later, but for now I'm going to write it all in terms of A and P.
Andy gives 60% of his money or 0.6A to Peter.
Peter has now P + 0.6A.
Andy has now 0.4A.
Peter then gives 25% of his money to Andy. But this isn't 0.25P because Peter doesn't have P dollars anymore because Andy already gave him some. It's 0.25 (P + 0.6A) that he gives back to Andy.
Peter has now 0.75(P + 0.6A)
Andy has 0.4A + 0.25 (P + 0.6A)
In the end, Peter has $200 more than Andy. That gives us a way to write an equation:
0.75(P + 0.6A) = 200 + 0.4A + 0.25 (P + 0.6A)
It's now simple to solve this equation of two variables by first substituting A = P + 200. The rest of it is just basic manipulations.
0.75(P + 0.6(P + 200)) = 200 + 0.4(P + 200) + 0.25(P + 0.6(P + 200))
0.75(P + 0.6P + 120) = 200 + 0.4P + 80 + 0.25(P + 0.6P + 120)
0.75(1.6P + 120) = 280 + 0.4P + 0.25(1.6P + 120)
0.75(1.6P + 120) = 280 + 0.4P + 0.25(1.6P + 120)
1.2P + 90 = 280 + 0.4P + 0.4P + 30
0.4P = 220
P = 550
and so A = 750
Check:
First Peter had $550, Andy $750.
Then Andy gives $450 to Peter.
Peter has now $1000, Andy $300.
Then Peter gives $250 to Andy.
Peter has now $750, Andy $550.
Andy has $200 more than Peter. Andy gives 60% of his money to Peter. Peter then gives 25% of his money to Andy. In the end, Peter has $200 more than Andy. How much did Andy have at first?This is a great problem to solve with algebra. Why don't you try it first, before reading further? It sounds kind of interesting... first one guy has $200 more than the other, and in the end it's reversed.
Solution:
Let A be the initial amount Andy has, and P the initial amount Peter has.
Then we know that A = P + 200. We're going to use that later, but for now I'm going to write it all in terms of A and P.
Andy gives 60% of his money or 0.6A to Peter.
Peter has now P + 0.6A.
Andy has now 0.4A.
Peter then gives 25% of his money to Andy. But this isn't 0.25P because Peter doesn't have P dollars anymore because Andy already gave him some. It's 0.25 (P + 0.6A) that he gives back to Andy.
Peter has now 0.75(P + 0.6A)
Andy has 0.4A + 0.25 (P + 0.6A)
In the end, Peter has $200 more than Andy. That gives us a way to write an equation:
0.75(P + 0.6A) = 200 + 0.4A + 0.25 (P + 0.6A)
It's now simple to solve this equation of two variables by first substituting A = P + 200. The rest of it is just basic manipulations.
0.75(P + 0.6(P + 200)) = 200 + 0.4(P + 200) + 0.25(P + 0.6(P + 200))
0.75(P + 0.6P + 120) = 200 + 0.4P + 80 + 0.25(P + 0.6P + 120)
0.75(1.6P + 120) = 280 + 0.4P + 0.25(1.6P + 120)
0.75(1.6P + 120) = 280 + 0.4P + 0.25(1.6P + 120)
1.2P + 90 = 280 + 0.4P + 0.4P + 30
0.4P = 220
P = 550
and so A = 750
Check:
First Peter had $550, Andy $750.
Then Andy gives $450 to Peter.
Peter has now $1000, Andy $300.
Then Peter gives $250 to Andy.
Peter has now $750, Andy $550.
Mean & mode freebie download
This free lesson about mean and mode will get you a foretaste for my upcoming 5-A Complete Curriculum from the LightBlue Series.
Download it here:
Mean, Mode, and Bar Graphs - lesson for 5th grade.
In the lesson I highlight the idea of mode versus mean (average) and when you can calculate the mean. Students also graph the data in bar graphs.
I didn't include the median because elementary lessons on mean, median, and mode tend to concentrate on the calculation aspect only, and I didn't want that. In this lesson they at least get to graph the data and think if mean (average) is "calculable". So I decided to postpone the median till 6th grade...
But here are some other lessons on these topics. Even with these you can see how much the actual calculations dominate the lessons.
Using and Handling Data
Simple explanations for finding mean, median, or mode.
www.mathsisfun.com/probability
Mode of a Set of Data
A very simple and clear lesson with examples and interactive quiz questions.
www.mathgoodies.com/lessons/vol8/mode.html
Finding the Mean, Median, and Mode
A great lesson with interactive quiz questions in the end. It also explains briefly the different uses for mean, median, and mode. After all, why do we have three different numbers for the central tendency of the data set?
www.algebralab.org/lessons/lesson.aspx?file=Algebra_StatMeanMedianMode.xml
Mean, Median, and Mode
Lesson on how to calculate mean, median, and mode for set of data given in different ways. Also has interactive exercises.
www.cimt.plymouth.ac.uk/projects/mepres/book8/bk8i5/bk8_5i2.htm
GCSE Bitesize Mean, mode and median lessons
Explanations with simple examples.
www.bbc.co.uk/schools/gcsebitesize/maths/data/measuresofaveragerev1.shtml
Measures Activity
Enter you own data and the program will calculate mean, median, mode, range and some other statistical measures.
www.shodor.org/interactivate/activities/Measures/
Landmark Shark Game
You're dealt five number cards, and using that as your data set you need to choose which of the range, median, or mode is the largest number.
media.emgames.com/emgames/demosite/playdemo.html?activity=M5A006&activitytype=dcr&level=3
Train Race Game
Calculate the median and range of travel times for four different trains, then choose a good train to take based on your results.
www.bbc.co.uk/education/mathsfile/shockwave/games/train.html
Download it here:
Mean, Mode, and Bar Graphs - lesson for 5th grade.
In the lesson I highlight the idea of mode versus mean (average) and when you can calculate the mean. Students also graph the data in bar graphs.
I didn't include the median because elementary lessons on mean, median, and mode tend to concentrate on the calculation aspect only, and I didn't want that. In this lesson they at least get to graph the data and think if mean (average) is "calculable". So I decided to postpone the median till 6th grade...
But here are some other lessons on these topics. Even with these you can see how much the actual calculations dominate the lessons.
Using and Handling Data
Simple explanations for finding mean, median, or mode.
www.mathsisfun.com/probability
Mode of a Set of Data
A very simple and clear lesson with examples and interactive quiz questions.
www.mathgoodies.com/lessons/vol8/mode.html
Finding the Mean, Median, and Mode
A great lesson with interactive quiz questions in the end. It also explains briefly the different uses for mean, median, and mode. After all, why do we have three different numbers for the central tendency of the data set?
www.algebralab.org/lessons/lesson.aspx?file=Algebra_StatMeanMedianMode.xml
Mean, Median, and Mode
Lesson on how to calculate mean, median, and mode for set of data given in different ways. Also has interactive exercises.
www.cimt.plymouth.ac.uk/projects/mepres/book8/bk8i5/bk8_5i2.htm
GCSE Bitesize Mean, mode and median lessons
Explanations with simple examples.
www.bbc.co.uk/schools/gcsebitesize/maths/data/measuresofaveragerev1.shtml
Measures Activity
Enter you own data and the program will calculate mean, median, mode, range and some other statistical measures.
www.shodor.org/interactivate/activities/Measures/
Landmark Shark Game
You're dealt five number cards, and using that as your data set you need to choose which of the range, median, or mode is the largest number.
media.emgames.com/emgames/demosite/playdemo.html?activity=M5A006&activitytype=dcr&level=3
Train Race Game
Calculate the median and range of travel times for four different trains, then choose a good train to take based on your results.
www.bbc.co.uk/education/mathsfile/shockwave/games/train.html