Maria's Math News, January 2024

Hello again, and welcome to a new year!!!

Math Mammoth news
PEMDAS (grades 4-8)
A basic principle of math teaching: Know Your Tools
Doing math with AI
Just for fun!

1. Math Mammoth news

After a long wait, Grade 8-B, and the entire grade 8 is now available! 😃

Overview of topics

Math Mammoth Grade 8 covers various traditional pre-algebra and algebra 1 topics, plus some geometry and statistics. The main areas of study are:

exponent laws and scientific notation
geometry: geometric transformations, angle relationships, and volume
solving linear equations
introduction to functions
graphing linear equations and proportional relationships
irrational numbers, square and cube roots, and the Pythagorean Theorem
solving systems of linear equations (in two variables)
statistics: scatter plots and two-way tables

See contents and samples:
8-A samples 8-B samples

=> Read more and see purchasing info!

Printed copies are available from Lulu and with ISBN numbers, at Amazon and at other bookstores.

Homeschool Buyers Club has a sale for Math Mammoth bundles - for 40% off!

It runs from 1/10 through 1/18. Check the sale out here.

I will soon run my traditional January sale... starting in a week.

I got some feedback again, and a permission to share it, so here goes. 😊

Thank you, Maria, so much for your wonderfully comprehensive and engaging maths programme, Math Mammoth, which I have used to homeschool my children through their primary years. One of my children struggled with maths when attending school in the early days but is now excelling and doing maths two years ahead of her peers.

Thank you, also, for your recommendation for high school maths. We are launching into Foerster's Algebra next, and will additionally use David Chandler's Math Without Borders when we need more expertise assistance than I can provide.

My homeschool maths teaching has been such a success thanks to you!

Kind regards, Julia.

2. PEMDAS

I don't feel there's anything wrong with PEMDAS as such (to teach the order of operations), but this post points out a difficulty many students have: it's kind of hard for them to remember/understand the in "PEMDAS", multiplications and divisions are "on the same level" — multiplications are NOT to be done before all divisions, but from left to right.

For example, in 24 ÷ 4 × 2, we do 24 ÷ 4 first, not 4 × 2.

And it works similarly with addition and subtraction (the "AS" part of PEMDAS): In 24 − 4 + 5, we do the subtraction first.

This blogger suggests an alternative mnemonic - GEMA, which stands for:

Grouping symbols
Exponential operations (this includes roots)
Multiplicative operations
Additive operations

Interesting! Maybe I will even some day revise my materials to reflect it. (However, I do need to note... any kind of revision that affects the answers to questions is a process that I have to go about carefully these days, because Rainbow Resource Center is keeping some stock of MM books. I cannot just change some question, and make 100s of answer key books already printed to go obsolete overnight. This said, I am currently working on a revision for grade 3... not sure when it will be finalized... hoping for this spring but not sure.)

Maybe for PEMDAS, if we wrote it as PE(MD)(AS), it would help some? Or with colors:
PE[MD][AS]

M and D are the "multiplicative operations". Not really two separate things really because ANY DIVISION can be written as a multiplication.

An example. 24 ÷ 4 × 2 is the same as 24 × (1/4) × 2. Now it should be clear that we go from left to right, and 24 × (1/4) gets done first.

Similarly, ANY SUBTRACTION can be written as an addition (the addition of the opposite).

For example, 70 − 24 + 21 is the same as 70 + (−24) + 21. It's all additions now, so we go from left to right.

3. Principle 3: Know Your Tools

Continuing our series... this is another habit or basic principle of effective math teaching. 😃

A math teacher's tools are quite numerous nowadays.

First of all, of course, comes a black or white board, or paper — something to write on.

And the book/curriculum you're using.
Then the basic measuring and geometric tools, such as a ruler, a compass, a protractor.
Then we also have software, animations, videos, and activities.
There are supplemental workbooks, math "readers", additional worksheets, and such.
And don't mention all the manipulatives, such as an abacus, scales, algebra tiles, fraction tiles, and so on.
And then there are games, games, games!

The choices are so numerous it's daunting. What's a teacher to do?

Well, you just have to get started somewhere, probably with the basics, and then add to your "toolbox" little by little as you have opportunity.

There is no need to try to use tons of tools all at once. It's important to learn how to use any tool you might acquire. Quantity won't equal quality. Knowing a few "math tools" inside out is more beneficial than a mindless dashing to find the newest activity or manipulative to spice up your math lessons.

Basic tools

The board and/or paper to write on. Essential. Easy to use.
The book or curriculum. Choosing a math curriculum is often difficult for homeschoolers. But keep in mind that no matter what curriculum you're using, YOU as the teacher have the control. Don't be a slave to the curriculum. You can skip pages, rearrange the order in which to teach the material, supplement it, and so on.
Concerning manipulatives: I once saw a question asked by a homeschooling parent, on the lines, "What manipulatives must I use and when?" The person was under the impression that manipulatives are a "must".

Manipulatives are definitely emphasized in these days. They are often useful, but they're not the end goal of math education, and there is no need to "go hog wild" over them. The idea is to help students learn to do the math without them!

Some manipulatives that I have found very helpful in the early grades are:
- Playing cards — when my youngest was about 3, he spent hours just organizing the cards in number order and also by color. Naturally, once the child grows a bit more, you can use them for games — which there exist a lot of math games that can be played with regular playing cards (see Acing Math book for example).
- Something to illustrate place value, especially with tens and ones (for Kindergarten). I made my daughter ten-bags by putting marbles into little plastic bags.
- A 100-bead abacus (for K-2)
- Some kind of fraction manipulatives. You can even simply make circular fraction models out of cardboard.
Often, drawing pictures can take the place of manipulatives, especially from middle elementary and on up.
Measuring tools: rulers, scales, and measuring cups. These are essential in the early elementary grades in order to teach children about measuring units. They need that hands-on experience!
Geometric tools: a ruler, a protractor, and a compass. Again, I feel these are very necessary.

Acing Math by PEP nonprofit. Redistributed here under CC BY-NC 4.0.

Math games: I believe in math games — they are often very enjoyable for children and thus quite motivating. I especially recommend games for drilling basic facts. For example, here's a card game that's worth 1000 worksheets.

And here's an ebook for you with LOTS of math card games: Acing Math.

Math Mammoth Light Blue series and Blue series products include lists of online games and activities for each math topic. These games and activities are also found in the lesson plans, where they are handily matched for each lesson in the Math Mammoth curriculum.
Math software: I definitely recommend the usage of some math software in algebra and calculus. I won't delve into the possibilities here; suffice it to say, Desmos is a popular and free online graphing calculator.

4. Doing math with AI

I decided to let Bard check the answer to one of the geometry questions in grade MM 8-B.

I honestly expected it to get it right! It's not a difficult question at all. (But since then, I have heard that "everyone" knows that these bots are bad at math... so, maybe I'm just behind times. 😀)

But notice, it gives two different answers, and both are different from the one I got, which was 25.8 units.

Students, don't use these AI chatbots to do your math homework.

(click to enlarge)

5. Just for fun!

Thanks for reading! 🙂

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Till next time,
Maria Miller