Homeschool Math Newsletter, Vol. 6
February 22, 2007

Winter greetings! It is Maria Miller here. Hope you are enduring the winter time - however dark, cold, or snowy it might be - soon the spring will come!

In this month's newsletter:

1. Did you know subtraction is addition, and division is multiplication?
2. Math Mammoth 7th grade (prealgebra) worksheets
3. Models for multiplying decimals
4. Tutoring
5. Tidbits

1. Subtraction is addition, and division is multiplication

In school math, we always study the FOUR basic operations.

But when you study abstract algebra, you get to concept of a group, or a ring, or a field - and all of those consist of a set of elements and one (group) or TWO (ring or field) operations.

Why are two operations enough in higher-level algebra? Where did they throw subtraction and division?

Addition and multiplication

Did you ever notice these similarities between addition and multiplication?

• Both addition and multiplication are commutative: a + b = b + a and ab = ba for all real numbers a, b. Division and subtraction are not.
• Both addition and multiplication are associative: (a + b) + c = a + (b + c). Division and subtraction are not.
• There exists an element so that adding it to a number doesn't change it (ADDITIVE IDENTITY). We call it zero: 0 + x and x + 0 both end up being just x.
• There also exists an element so that multiplying by it doesn't change the number (MULTIPLICATIVE IDENTITY). This is number one: 1x and x1 both are just x.

You might a lready know these well... but how about the INVERSE:

• For any real number x, there is a real number so that when you add the two, you will get the additive identity element or zero.(THE ADDITIVE INVERSE, or OPPOSITE)
• For any real number x (excepting zero), there is another real number so that when you multiply those, you will get the multiplicative identity element or one. (THE MULTIPLICATIVE INVERSE, or RECIPROCAL)

For example, for 8 we have −8: if you add them, you get zero. For −5.4 you have 5.4; if you add them, you get zero.

And with multiplication: For 8 we have 1/8. If you multiply them, you get 1. For −5.4 or −54/10, we have −10/54: if you multiply them, you get 1.

Continue reading how the four operations become two.

Math Mammoth 7th grade (prealgebra) worksheets ready

Math Mammoth pre-algebra sheets contain problems for beginning algebra topics such as expressions, linear equations, and slope - plus your typical 7th grade math topics such as integers, fractions, decimals, geometry, statistics, and probability.

You will find the problems are very varying as these worksheets have been created one by one (not script made).

Go download and enjoy the free sample sheets! - including a free downloadable Percent Fact Sheet.

3. Models for multiplying decimals

Someone asked,

how can you use models to multiply decimals?

Learning to multiply decimals, I feel, is built on students' previous understanding of multiplying whole numbers and fractions.

So models wouldn't necessarily be the focus, but instead relating decimals to fractions first, and learning from that.

Multiply a decimal by a whole number

Of course, when multiplying a decimal by a whole number, you could use the same models as for fractions: say you have a problem

2 × 0.34

You can use little hundredths cubes, or draw something that's divided to 100 parts.

BUT you can also just use fractions, and justify the calculation that way:

2 × 0.34 = 2 × 34/100 = 68/100 = 0.68.

OR you can explain it as repeated addition:

2 × 0.34 = 0.34 + 0.34 = 0.68.

I employ that idea in these lessons:

Multiply mentally decimals that have tenths and Multiply decimals that have hundredths

Multiply a decimal by a decimal

When students are learning to multiply a decimal by a decimal, they're on 5th or 6th grade perhaps. One of the most obvious ways to teach this is to use fractions:

Continue reading how to teach decimal multiplication

4. Tutoring

Sometimes personal one-on-one tutoring is needed or is the most convenient way to help your child in math.

In today's world, tutoring can be done over the internet so there is no need to leave to someplace, or to let a tutor enter your house.

The companies doing such are using interactive whiteboards to which the student and the tutor can both write, plus some sort of voice-over-internet mechanism, such as Skype.

This also cuts down the costs of the tutor, so tutoring can be offered at much lower prices than if the tutor travels to your house.

I have recently partnered with TutorNEXT company to promote their service on my site.

NORMALLY they offer personal tutoring for the price $199 per 8 hours. They also offer homework help, in which a tutor is available at certain times, or you can email the tutor your problem and get response in two hours.

But right now TUTORnext also has a SPECIAL offer to my visitors: you get 8 hours of personal one-on-one tutoring for only $99. Doing your math, that's $12.375 per hour!

So if this is of interest to you, click on this link ( http://www.tutornext.com/hsm123.php ) and sign up on that particular page.

This special offer is not visible if you enter the site via their home page, but only on that one special page.

5. Tidbits

• Recently, my daughter has greatly enjoyed magic square addition worksheets from Kidzone.Ws (scroll down to item #5 on the page) to practice adding single-digit numbers where the sum goes over 10, such as 7 + 7 and 9 + 5.

  I always tell her that when it's 9 and other number, then one (dot) of the other number "jumps" to go with the nine and makes ten. That way she has learned to add 9 + 6 or any other such sum.

  And if it's 6 + 7, I tell her to think of 6 + 6 which she knows by heart, and figure it out from there.

  I hope she gets to remembering these by heart eventually, but at this point I'm just glad she is able to figure them out by using these 'helping' ideas. And of course using those also teaches her important principles of mathematics.

• What's special about this number? - a list of numbers from 0 to 9999, and there's something special about almost all of them!

• You can now buy a hardcopy (already printed version) of any of the 16 "BLUE" series Math Mammoth books at Lulu.com - at the address www.lulu.com/mathmammoth/. The copies are black and white.

• An article about The Benefits of Memorizing Math Facts - "Quite simply, a lack of fluency in basic math fact recall significantly hinders a child's subsequent progress with problem-solving, algebra and higher-order math concepts."

• Ma and Pa Kettle math - a little fun video clip where Pa and Ma Kettle prove themselves good 'mathematicians'! He "proves" by long division that 25 ÷ 5 = 14, and she "proves" by multiplication algorithm that 5 × 14 = 25.

Till next time,
Maria Miller

How did you like this month's newsletter? Is there any topic you'd like to see in the next one? Contact me.