I found this interesting online game:

# Math thinking

## Friday, January 11, 2008

## Wednesday, January 9, 2008

### Fibonacci

I think I'm enjoying learning about non-linear patterns a great deal because there are so many applications of this pattern in nature. Also, since I was in the sixth grade, I've always held an awe for anything relating to the Golden Section.

This last Christmas I actually knit a scarf for my brother that followed the sequence of stripping for the first several numbers in the sequence.

1,1,2,3,5,8,13,21....

When I was substitute teaching several years ago I had fifth grade class who was reviewing number patterns and sequencing... I put a number of different sequences up on the board. There was a student who was moderately autistic in the class and before I completed one of the sequences he blurted out "Fibonacci! Fibonacci!"

I spent some time fiddling and I built my own tiled visual model of the Fibonacci sequence. I think I spent too much time trying to make sure that the squares were positioned correctly. I truly feel that hands on manipulation of visuals is key to helping me grasp and understand concepts fully. I hate passive learning.

You can draw a spiral around the pattern of increases and it looks something like this.

A Fibonacci Spiral is similar to a "Golden Spiral" based on the Golden Ratio. However, if you try to apply the the Fibonacci design to a nautilus shell interior you see that the ratio of the growth increases do not fit.

This last Christmas I actually knit a scarf for my brother that followed the sequence of stripping for the first several numbers in the sequence.

1,1,2,3,5,8,13,21....

When I was substitute teaching several years ago I had fifth grade class who was reviewing number patterns and sequencing... I put a number of different sequences up on the board. There was a student who was moderately autistic in the class and before I completed one of the sequences he blurted out "Fibonacci! Fibonacci!"

I spent some time fiddling and I built my own tiled visual model of the Fibonacci sequence. I think I spent too much time trying to make sure that the squares were positioned correctly. I truly feel that hands on manipulation of visuals is key to helping me grasp and understand concepts fully. I hate passive learning.

You can draw a spiral around the pattern of increases and it looks something like this.

A Fibonacci Spiral is similar to a "Golden Spiral" based on the Golden Ratio. However, if you try to apply the the Fibonacci design to a nautilus shell interior you see that the ratio of the growth increases do not fit.

### Population statistics and making inferences

Just playing around with some statistics from the NEA ranking report on school aged population rankings.

It's just interesting to look at the stats.

If we know that:

Alaska (2002) has an estimated 30% of its population aged under 18

and 22.3 percent of the population is aged 5-17, then nearly eight percent of the population is still very young.

Mississippi had the highest population of 5-17 year olds at 35.9 percent. That's huge. That's over a third of the population. Is this normal? This isn't including the percent of the population under 5. Now talk about not having a large voting constituency compared to other states... though you might have a growing population of impressionable voters.

It's just interesting to look at the stats.

If we know that:

Alaska (2002) has an estimated 30% of its population aged under 18

and 22.3 percent of the population is aged 5-17, then nearly eight percent of the population is still very young.

Mississippi had the highest population of 5-17 year olds at 35.9 percent. That's huge. That's over a third of the population. Is this normal? This isn't including the percent of the population under 5. Now talk about not having a large voting constituency compared to other states... though you might have a growing population of impressionable voters.

## Tuesday, January 8, 2008

### Rational Numbers and Yarn

I have a few little problems I'd like to work out. Remember my muffler problem from a previous post? I probably should finish it before winter is over so I need to figure out if I have enough yarn, and what type of striping design or pattern I can create.

About the amounts noted above, I basically eye-balled these amounts, so there isn't 100% accuracy. That would entail unwinding all the yarn and then measuring it by the yard, which is of course, something I'm not willing to do.

I do know that according to the label there is 108 yards in every skein. So to get my approximate yardage for each skein I simply have to multiply the numbers and fractions above. I'll give a visual representation of solving for the moss green amount first. By the way I got lazy and wrote out the problem on paper and scanned it as an image rather than using PowerPoint or a graphics program.

Of course, I could have just divided 108 by 4. That would have been easiest, but for the sake of reinforcing knowledge I regained last week, I wanted to solve using fractions.

Now I want to make a muffler which is shorter than a scarf so I'm going to make it 5 feet in length. I want the wearer to be able to tie the scarf at least once. FYI - a normal length for a scarf is the height of the person who is going to wear it. From my estimation below it will take 40 inches of yarn to make a foot. Therefore, it will take 200 inches or 5.555 yards per row (actually 5.5555 with the decimal repeating).

I suddenly realized that .1111 with decimal repeating equals the fraction 1/9. (Mr. Grant's, my sixth grade teacher, efforts to drill the fraction decimal amounts into us didn't fail me). So this number is actually 5 5/9 yards or in fraction form 50/9.

Back to my calculations, as I noted in the graphic each row of striping in a Garter stitch pattern takes 4 rows of knitting. So I'm going to take my 5 5/9 or 50/9 yard number and multiply that by 4. I get 200/9 or 22 2/9 yards. So considering that I only have 27 yards of yarn I now know that I can only make 1 stripe of moss green on my muffler. Maybe it's enough for just the edging.

Gosh this is a lot more work that I thought, and of course, I'm an experienced enough knitter to be able to visually estimate what I can do, but now I can the problem thinking I employed here in the future to analyze more complicated problems when it comes to yarn estimation for patterns. I'm pretty stoked!

About the amounts noted above, I basically eye-balled these amounts, so there isn't 100% accuracy. That would entail unwinding all the yarn and then measuring it by the yard, which is of course, something I'm not willing to do.

I do know that according to the label there is 108 yards in every skein. So to get my approximate yardage for each skein I simply have to multiply the numbers and fractions above. I'll give a visual representation of solving for the moss green amount first. By the way I got lazy and wrote out the problem on paper and scanned it as an image rather than using PowerPoint or a graphics program.

Of course, I could have just divided 108 by 4. That would have been easiest, but for the sake of reinforcing knowledge I regained last week, I wanted to solve using fractions.

Now I want to make a muffler which is shorter than a scarf so I'm going to make it 5 feet in length. I want the wearer to be able to tie the scarf at least once. FYI - a normal length for a scarf is the height of the person who is going to wear it. From my estimation below it will take 40 inches of yarn to make a foot. Therefore, it will take 200 inches or 5.555 yards per row (actually 5.5555 with the decimal repeating).

I suddenly realized that .1111 with decimal repeating equals the fraction 1/9. (Mr. Grant's, my sixth grade teacher, efforts to drill the fraction decimal amounts into us didn't fail me). So this number is actually 5 5/9 yards or in fraction form 50/9.

Back to my calculations, as I noted in the graphic each row of striping in a Garter stitch pattern takes 4 rows of knitting. So I'm going to take my 5 5/9 or 50/9 yard number and multiply that by 4. I get 200/9 or 22 2/9 yards. So considering that I only have 27 yards of yarn I now know that I can only make 1 stripe of moss green on my muffler. Maybe it's enough for just the edging.

Gosh this is a lot more work that I thought, and of course, I'm an experienced enough knitter to be able to visually estimate what I can do, but now I can the problem thinking I employed here in the future to analyze more complicated problems when it comes to yarn estimation for patterns. I'm pretty stoked!

## Friday, January 4, 2008

### A number is a number is a number

It's been years since I had to think about classifying numbers, and to be honest, I really don't think that I understood what the teachers meant about rational numbers, real numbers... I knew that an integer was a number that could be expressed in either a positive or negative form, but that's about it. I don't think I was alone in my ignorance. If you can get maturing generation x'er like myself to understand these concepts after years of misconception; how come it's so difficult to teach this to middle schoolers and high school kids. Okay, there are numerous reasons, but still... if I can get this, most people can with the right help.

Here's an image I made to show the relationships of the following types of numbers (are they called types?):

It makes sense that kids, when then get into Middle School, they have difficulties understanding the other types of numbers outside of "whole numbers." They spent all of elementary school only focusing on how whole numbers function and are used. Honestly, I think it's because that's all that most elementary school teachers have been trained to teach (and feel comfortable with).

Here's an image I made to show the relationships of the following types of numbers (are they called types?):

- Real Numbers
- Algebraic Numbers
- Rational Numbers
- Integers
- Whole Numbers

Rational numbers are algebraic real numbers. Rational numbers include integers. Integers include both whole numbers and their negatives. All integers can be presented or expressed in their rational form. For example:

- 3 can equal 3/1 or 30/10 or 9/3
- 2.5 can equal 3/2,
- -5 can equal -5/1, etc

It makes sense that kids, when then get into Middle School, they have difficulties understanding the other types of numbers outside of "whole numbers." They spent all of elementary school only focusing on how whole numbers function and are used. Honestly, I think it's because that's all that most elementary school teachers have been trained to teach (and feel comfortable with).

## Thursday, January 3, 2008

### Curse of the bad math teacher

I had very difficult time understanding the concept of rational numbers. Well, I do get them... now, sort of.

Who didn't have a teacher at one point who totally blanked when it came to teaching math. I was learning about fractions in the late seventies early eighties and I was having an awful time with math in general. I had a teacher who told me... "That's okay, honey, you're a girl." She was insinuating that because of my gender I either wouldn't need math or was not predisposed at being good at math.

This really still burns me up even now.

Just thinking about what that teacher said makes me even more determined to fully understand what I missed out on or forgot. I could call her a number of things, but considering the old addage... if you don't have something nice to say... then don't say it at all, I won't. I remember though that this teacher was not particularly good at teaching fractions or how to add and subtract them. It took two years and an excellent teacher in the sixth grade to remedy a good part of the damage that was done to my math self-esteem in that grade.

The trick was really learning the power of decimals and understanding that a fraction could be expressed as a decimal by simply dividing the numerator or the top by the denominator or the bottom. This teacher actually made us memorize the fractions and their equivalent decimals as well as gave us various representations of fractions in circles, squares and rectangles and had us assign a decimal value to each of these pictures. I actually think that the visualization was extremely helpful to me.

However I don't think I fully understood all properties of rational numbers until later on...

To be continued.

Who didn't have a teacher at one point who totally blanked when it came to teaching math. I was learning about fractions in the late seventies early eighties and I was having an awful time with math in general. I had a teacher who told me... "That's okay, honey, you're a girl." She was insinuating that because of my gender I either wouldn't need math or was not predisposed at being good at math.

This really still burns me up even now.

Just thinking about what that teacher said makes me even more determined to fully understand what I missed out on or forgot. I could call her a number of things, but considering the old addage... if you don't have something nice to say... then don't say it at all, I won't. I remember though that this teacher was not particularly good at teaching fractions or how to add and subtract them. It took two years and an excellent teacher in the sixth grade to remedy a good part of the damage that was done to my math self-esteem in that grade.

The trick was really learning the power of decimals and understanding that a fraction could be expressed as a decimal by simply dividing the numerator or the top by the denominator or the bottom. This teacher actually made us memorize the fractions and their equivalent decimals as well as gave us various representations of fractions in circles, squares and rectangles and had us assign a decimal value to each of these pictures. I actually think that the visualization was extremely helpful to me.

However I don't think I fully understood all properties of rational numbers until later on...

To be continued.

## Wednesday, January 2, 2008

### Yarn Problem - Math Problem

I have several left over skeins of wool yarn from Christmas gift projects and I'd like to use them. I decided that I would make a muffler for my husband. I'm a visual learner and problem solver so I made a quick little illustration. By the way, it's really easy to make simple-simple graphics by simply using Powerpoint and the set of rough drawing tools. This is how I made the image below.

I knit fairly evenly and regularly, so I think my numbers of stitches per inch will work okay. So now my next steps are to calculate how much yarn I need and then make sure my stripe color design will work out. Note, I tested how much yarn it takes to make and 8 inch row. I was lazy and didn't feel like doing the entire foot. I could easily solve this and determine how much it takes to make a foot by dividing the amount it takes to knit the 8 inches (40 inches of yarn/2=20 inches) and multiplying it by 3. So it takes 60 inches to knit a one foot row. Accordingly:

Maybe I'll just make a 4 foot long scarf... after all it's just a muffler...I'll spend some time doing more calculations on the amount of yarn I'll need to make the muffler and then share them later.

I knit fairly evenly and regularly, so I think my numbers of stitches per inch will work okay. So now my next steps are to calculate how much yarn I need and then make sure my stripe color design will work out. Note, I tested how much yarn it takes to make and 8 inch row. I was lazy and didn't feel like doing the entire foot. I could easily solve this and determine how much it takes to make a foot by dividing the amount it takes to knit the 8 inches (40 inches of yarn/2=20 inches) and multiplying it by 3. So it takes 60 inches to knit a one foot row. Accordingly:

- It would take 180 inches (or 5 yards) to make make a one yard row
- It would take 360 inches (or 10 yards) to make a six foot row. Your average scarf is about 5-6 feet long.

Maybe I'll just make a 4 foot long scarf... after all it's just a muffler...I'll spend some time doing more calculations on the amount of yarn I'll need to make the muffler and then share them later.

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