Math Mammoth Geometry 2 giveaway

Recently I offered one of my customers a free book for finding an error in the answer key... she declined and wished someone else could have it who is in need.

So I'll give one electronic copy of Math Mammoth Geometry 2 book (PDF file) to one randomly chosen lucky winner.

To participate, you need to do two things:
1) Leave a comment in this blogpost, explaining how and where you'd use this book, AND
2) Email me so I have your email address.

If you have no use for this geometry material, please don't participate, and let someone win who can use it.

If I choose a winner who didn't email me, then I cannot contact them. In that case I will have to choose another winner.

I'll first choose a few best entries based on how/where they'd use the book, and then choose randomly among those.

This 'contest' closes and winner will be chosen whenever I notice (in checking my mail) that there are at least 10 participants. So hurry up!

Free math tutoring

David Freeling from TutorTalk.org wants to let us know that as a part of a promotion to launch his site, he's offering FREE tutoring on every Friday.

Here is some information from David:

Free tutoring will apply to specific classes, labeled as Free. Students should visit www.TutorTalk.org in advance and register, because space is limited. Free lessons cover fundamental math, reading and writing skills and are suitable for a wide range of ages, but especially middle and high school. Curriculum focuses on problem-solving, and kids are encouraged to play along from home and be prepared with a calculator, pen and paper. Students will also be able to bring in their own homework problems for small group tutoring, in classes of up to 5 students.

The site also says, "TutorTalk Classes are dynamic, interactive small-group classrooms, led by a qualified teacher, and hosting up to five students. Students work together and chat to solve onscreen problems from the comfort of their own homes, with the help and hints of the instructor."

Homeschooling by the Numbers

My dh found an interesting "infographic" about homeschooling statistics. Click the image to enlarge.

Integer games online

I have updated the list of integer games online at my site. Now the games are organized into sections of:

• ordering integers, 
• addition & subtraction of integers, 
• multiplication & division of integers, and
• all operations with integers. 

Happy playtime!

How to use an abacus with Math Mammoth

Recently I've received several questions about abacus usage within Math Mammoth curriculum. Here is what I wrote and added to my FAQ on the site.

The only way the abacus is used in my books is where each bead counts as one. Nothing fancy. It is NOT used like Chinese, Russian, or any of the other abaci where one bead might count as 5, 10, or 100.

A 100-bead abacus or school abacus simply contains 10 beads on 10 rods, a total of 100. In the school abacus, each bead simply represents one. The 100-bead abacus lets children both "see" the numbers and use their touch while making them.

First and foremost, the abacus is used in the place value section in 1st grade where children learn about tens and ones (numbers up to 100). We use it to show clearly how 45 is made up of 4 tens and 5 ones, for example.

Secondly, you can use the abacus with addition and subtraction problems in 1st and 2nd grades. For example:

• Show the child additions and subtractions with whole tens. For example, to solve 50 + 20, first make 50 on the abacus. Then add 20 more.
• Add a two-digit number and a single-digit number. For example, to solve 23 + 5, first make 23 on the abacus. Then add five beads.
• Show some "shortcuts" in addition or subtraction. For example, to solve 34 + 20, first make 34 on the abacus. To add 20, add two whole rows of beads. Then the student checks how many whole tens and how many individual beads is the total.
  
  Or, to solve 85 − 20, first make 85. Then pull back two whole rows of beads.
  
  Or, to add 23 + 44. First make 23. Then make 44 on using the five lowest rows of the abacus. Have the child now count the whole tens (6), and the individual beads from the two rows (3 + 4). This shows adding the tens separately, and adding the ones separately. From this you can graduate to making first 23, then adding 4 full rows of beads for 40, and then adding 4 individual beads from the same row as the 3 beads.

The purpose is mainly to help children to visualize two-digit numbers, and to add and subtract two-digit numbers.

The goal in my books is to drop the abacus by 3rd grade. Even before that, students use visual models, and from those go on to the abstract. The quicker the child can use visual models, and then do the math problems without any models, the better.

At Amazon you can find Melissa & Doug Classic Wooden Abacus for around $12. An abacus where the beads alternate colors by fives is even more useful (but may be out of stock).

Browse Amazon's abacus selection here. Other stores carry abaci as well.

You can also use this virtual abacus. Or, make your own abacus. Just don't make it exactly like they show on that web page but instead use 10 bamboo skewer with 10 beads in each so you get a 10 x 10 abacus.

Order of operations / PEMDAS

PEMDAS does not cover matrices...
Photo courtesy of Stuartpilbrow

Someone asked me recently...

Could you please share with me your opinion of the "Please Excuse My Dear Aunt Sally" simplifying expressions. Any feedback could you give will be appreciated. Thank you.

This "PEMDAS" rule is a mnemonic for order of operations:

Please = Parenthesis
Excuse = Exponents
My = Multiplication
Dear = Division
Aunt = Addition
Sally = Subtraction

There's nothing wrong with using a mnemonic to remember the order of operations. However, one has to bear in mind that

1. This rule is not all-inclusive. It omits for example square roots. But the rule is good for all elementary grades. (Square roots would be on the same level or rank with exponents, by the way.)
    
2. The rule doesn't spell out the fact that in reality multiplication and division are "on the same level" or rank. This means that if you have several multiplications and divisions, you do them from left to right, and not "multiplication first, then division".
   
   For example: 60 ÷ 5 × 4. You go from left to right, and first do 60 ÷ 5 = 12. Then you multiply 12 × 4 = 48.
   
   If you want 5 × 4 to be done first, it needs to be in parenthesis: 60 ÷ (5 × 4). Here, first do 5 × 4 = 20, and then 60 ÷ 20 = 3.
   
   Similarly, addition and subtraction are on the same level: if both exist in an expression, they are to be done from left to right.
   
   An example: Simplify the expression 2 × 5 − 6 + 8.
   
   1: Multiply 2 × 5 = 10.
       The expression is now 10 − 6 + 8.
   
   2. Subtract. 10 − 6 = 4.
       The expression is now 4 + 8.
   
   3. Add. 4 + 8 = 12.
   
   So, perhaps it's more illustrative to lay out the PEMDAS rule like this:
   
   Please = Parenthesis
   Excuse = Exponents
   My Dear = Multiplication & Division
   Aunt Sally = Addition & Subtraction
   
   ...and say it with little pauses at the commas: Please, Excuse, My Dear, Aunt Sally.
   
   
   
3. Fraction line as a division symbol generates implicit groupings or parenthesis in the fact that anything written on top or bottom of the fraction line is to be done before the division. It's as if the whole numerator and denominator were inside parenthesis.
   
   So...
   
   
   
   

   4 × 32
   --------
   7 − 22
   

   
   means the same as (4 × 3²) ÷ (7 − 2²). Simplifying this is left as an exercise for the reader. The answer is 12.
   

See also a detailed lesson on order of operations at Purplemath and Wikipedia's note on mnemonics.

Estimation methods

A typical estimation problem:
guess how many paper clips are in a jar?
Photo by Dean Terry

Here recently I was asked a question about estimation methods:

My daughter's homework was to estimate the number of 953 divided by 18, using front-end estimation, using rounding to estimate, and using compatible numbers to estimate. What are the differences between these three methods? Do we get the same result? Thanks!

You definitely don't get the same results as these three methods are quite different!

1. Front-end estimation means you keep the "front" or first digit of each number, and make the other digits to be zeros.
   
   So, 953 ÷ 18 is estimated to 900 ÷ 10 = 90.
   
   Another example: 56 × 295 would be estimated as 50 × 200 = 10,000.
   
   
2. Rounding means you round the numbers, usually to their biggest place values, but sometimes you can round "creatively". In any case, the numbers you round to should be easy to work with mentally.
   
   So, in 953 divided by 18 we round 953 to nearest hundred, and 18 to nearest ten. 953 ÷ 18 becomes 1,000 ÷ 20 = 50.
   
   Another example: 56 × 295 would become 60 × 300 = 18,000.
   
   An example of rounding "creatively": with 24 × 32 you can round 24 to 25 (to the "middle five"), and 32 to 30. The estimated result is 25 × 30 = 750. The exact result in this case would be 768 so the estimation was fairly close.
   
   Another principle to keep in mind with when using rounding to estimate is that if you have an addition or multiplication problem, it's best to round one number down, the other up, in order to minimize the rounding error. If you have a division or subtraction, it's best to round both numbers "the same direction", either up or down.
   
   
3. Compatible numbers means finding numbers that are close to the numbers in the problems but such as are easily to work with mentally.
   
   So, 953 ÷ 18 could be estimated as either 960 ÷ 20 or 1000 ÷ 20, depending on your mental division skills.
   
   960 ÷ 20 = 96 ÷ 2 = 48.
   
   Another example: estimating 56 × 295 depends, again, on your mental multiplication skills. You could try to leave 56 as it is, and make 295 to be 300, to get 56 × 300 = 16,800. Or, you might make the numbers to be 60 and 300 (the same as in the rounding method) and get 18,000.

To compare all three methods, we check the exact result of our problem, which is 953 ÷ 18 = 52.944444444... From this we can see that the rounding method was most accurate in this case (it gave us 50), and front-end estimation did really bad (it gave us 90).

In the other example, the exact result is 56 × 295 = 16,520. Again, rounding (18,000) or compatible numbers (16,800 or 18,000) method did best, and front-end estimation the worst (10,000).