## Sunday, June 30, 2013

### Understanding Ratios Using Base Ten Blocks.

I had a student talk about cross multiplication like it was some kind of dark magic. Here is an explanation of WHY cross multiplication works.

You should be able to see it. The relationship is one to three. in the second set the relationship is still one to three...but it's two to six. You can play games where you get one and the student gets three. In that case you might have put a two down and the student might get out 3 twos.

But here we are looking at the relationship in a slightly different way.

You can see how it works: if I have the rectangle and one side or two sides of the rectangle...and this is what we are employing to problem solve. Concept three: we form rectangles to facilitate counting.

Now from there we can do problem solving in Geometry with triangles and other shapes..."this is to this as x is to this..."

For example 6 : 15 as 14 : x...of course you could cross multiply and divide but there is an easier way. If you can start using rectangles in you mind problems like this will be easy to see. And even if you can't, the concepts are still easier to understand. This also points out the need for mastering multiplication tables. People will say who need it now that we have calculators...well being able to punch a calculator will not help you recognize 6 and 15 are on the threes multiplication table. And people often lament that calculators don't help with fractions very much...

They do but only for computation not for understanding the math. Some kids can "see" right off the bat that on top we have 2 threes and 2 sevens and if we have 5 threes on the bottom we must have 5 sevens too. Others take more explaining. Adults always take more explaining.

Here I drew in the rectangles. Use you imagination to see the sides. The whole thing is six and one side is two the other side MUST be 3. The whole thing is fourteen and one side is 2 the other side HAS TO BE 7. If the "top" is 3 the bottom is 3, if the top is 7 the bottom has to be seven because the relationships are the same.

So if I know the sides are 7 and 5 the whole thing HAS TO BE 35. We love math. It isn't going to be different tomorrow. No exceptions. And you should be able to see the relationships are the same two to five...doesn't matter what it is.

2x/5x in this case we have 3 and 7...the logic also works no matter where we put the x.

Then we the student sees this, it makes perfect sense. It's not some kind of magic or a stupid rule to follow that just is.

Students need to see where the rules come from and that makes them easier to understand and then we can work on applications with triangles. We can also work on applications with triangles or rectangles and show how these rules work, but...

It's been my experience that having some experience here first makes a big difference there.

And here is some more covering this topic where we can do it the easy way or the hard way. And if we HAVE to do it the hard way at leaste we understand why it works. Taking the mystery out of the math is important.

This student has just about gotten the concepts mastered. He is a quiet boy so he doesn't say much but he did exclaim, "Ok I get it!" at one point and "that makes it easier to understand..." at another. You will find this response almost universal among students who just learned it the other way...using cross multiplication.

Here is a slightly different way of looking at it using different symbols:

In this method it becomes visually obvious that we are talking about multiples.

There is lots more we can do with ratios but that's enough for now...

### Simple Math In Chinese Using Manipulatives With DBoyz.

What's annoying to me is a lot of the kids their age can barelyadd numbers in English. I had a interesting conversation with a Chinese teacher who pointed out that yes there may be as many honor students in China as the USA has students but they also have as many failing students as we have students as well.

"Only the smart one comes to USA, no failing students get out..."

I had a similar conversation with a friend who is from Hong Kong. She did "okay" in math but a lot of her friends failed...and what did they fail? Algebra. Base ten blocks make math easy to see in any language. They are quite useful with the deaf and blind too.

Practice and patience is pretty much all it takes to learn another language...and like Chinese they aren't going to learn it all at once. It's the same with math...you didn't get bad at math in one day....you won't get good at math in one day. Chinese, English and Spanish with those three Languages you can communicate with most of the inhabitants on this planet. They got English by default, Chinese is proving to be a challenge but it's FUN and after Chinese Spanish will seem like a breeze. I was fortunate to go to a high school where I got Latin, 5 years of Spanish and a year of French JFSAG. But no Asian languages were offered...helping them learn Spanish will be MUCH easier than helping them learn Chinese. And Yes there are plans to get some of the Math Materials translated into Spanish. Right now the house of math is mono-lingual.

Anyway we are just having fun playing around with math in Chinese. They still have lots of work to do when it comes to writing symbols...lots of people have already noted that their stroke order is incorrect. Considering their age at that they only get an hour a week I'd say they are doing pretty well.

There is always room for impoverishment, and I have no doubt that they will get better as they progress. Right now they are finding fun that they can write a little bit in "code." I am told they need to master about 3,000 symbols...numbers are pretty straight forward in Chinese. Very simple rules. More uniform than English. Eleven is ten and one literally in Chinese...twelve is ten and two. Twenty is two tens...lol...in English we have a word for two tens: twenty.

The Mathematics is a Universal Language, learning some in Chinese is fun and will pay dividends in the future. I was going to attempt to do some simple math in Pinyin here but it's too much work...I think I will let the videos do the talking...and people who speak Chinese can cringe all they want.

Trying to write the inflections for even simple stuff like jia (add or plus) jian (subtract or minus) would be a little more effort than I want to put into a simple post like this. Eventually, I think I will discover free software that does some translating that I can just cut and paste...I'm sure it exists I just don't know where at the moment.

The Kit shown is called the numeral identification kit, or Number ID for short. It is great for REALLY young kids. like two and a half...if they are learning to talk they can learn the names of numbers just like they learn other words. It's just vocabulary. The how many part can be learned later and the fact that the six can be made up of a combination of other numbers can be learned later too.

Use the basic concepts and teaching techniques like the three period lesson and degree of difficulty to make the learning easier and fun...also don't forget to play, they learned a lot of Chinese playing Chutes and Ladders in Chinese and you can make up other games as well even if they aren't learning Chinese. Games and Play: that's

**HOW**they learn.

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## Saturday, June 29, 2013

### Factoring Forms Using Base Ten Blocks

Here is a quick video showing various ways of factoring a quadratic. Using the the blocks to show what we are talking about with each factored expression of

3x² + 17x + 24

Can be factored several ways. The most common way is a great example of economy of symbol.

Note how we are still all positive...keeping in mind degree of difficulty. Lets understand some basic concepts before we start adding negatives coefficients to the mix. Most text books just dive right in instead of taking it one step at a time.

The blocks take the fear out and make the symbols relate to something tangible in reality. They can actually see it.

Then when you show your students the other forms it's obvious. Then when it's time to start graphing these bad boys finding zeros and more points and the vertex and y-intercepts and such is easy because they already have this much under their belts instead of being inundated with it all at once.

The student in this video is shy and as soon as the camera goes on he chokes up...had the camera not been on he would have had an easier time answering my questions. Anyhow the action figures or lego men or whatever you have are handy to help get the point across.

Got the Star Wars effect going on here...a long time ago in a galaxy not far enough away they were still teaching math like idiots. Then suddenly math was cool...and the kids could see it and the symbols made sense because they could decode them and everybody was laid back and groovy.

For more get a password and head over to the house of math.

Learn to use your base ten blocks.

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3x² + 17x + 24

Can be factored several ways. The most common way is a great example of economy of symbol.

Note how we are still all positive...keeping in mind degree of difficulty. Lets understand some basic concepts before we start adding negatives coefficients to the mix. Most text books just dive right in instead of taking it one step at a time.

The blocks take the fear out and make the symbols relate to something tangible in reality. They can actually see it.

Then when you show your students the other forms it's obvious. Then when it's time to start graphing these bad boys finding zeros and more points and the vertex and y-intercepts and such is easy because they already have this much under their belts instead of being inundated with it all at once.

The student in this video is shy and as soon as the camera goes on he chokes up...had the camera not been on he would have had an easier time answering my questions. Anyhow the action figures or lego men or whatever you have are handy to help get the point across.

Got the Star Wars effect going on here...a long time ago in a galaxy not far enough away they were still teaching math like idiots. Then suddenly math was cool...and the kids could see it and the symbols made sense because they could decode them and everybody was laid back and groovy.

For more get a password and head over to the house of math.

Learn to use your base ten blocks.

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