Math trick and its proof: square a number ending in 5

I will be hosting the blog carnival Math Teachers at Play next week. (You can send in submissions here.)

One submission I got about various multiplication tricks or shortcuts got me inspired to write a proof of the particular trick.

You could definitely use this in algebra class. First explain the shortcut or trick itself. Then ask students to prove it, or to explain WHY it works, using algebra.

You could also explain this to younger students as an additional "neat trick" and let them explore and play with it.

THE "TRICK"

If a number ends in 5, then its square can be calculated using this "trick" (I like to call it a shortcut because there's nothing magic about it):

Let's say we have 75 × 75 => Go 7 × 8 = 56. Then tag 25 (or 5 × 5) into that. You get 5625.

Let's say we have 35 × 35 => Go 3 × 4 = 12. Then tag 25 into that. You get 1225.

Let's say we have 115 × 115 => Go 11 × 12 = 132. Then tag 25 into that. You get 13,225.

Let's say we have 245 × 245 => Go 24 × 25 = 600. Then tag 25 into that. You get 60,025.

So, you simply take the digit or digits in front of the 5 and consider those as a number in itself. Multiply that by the next number. Then "tag" 25 to the answer you got in the previous step.

Photo by Lazy butterfly

PROOF

Any whole number that ends in five is of the form A + 5, and A is a multiple of 10. Since A is a multiple of 10, we can write A = 10b, where b is now some whole number. So, our number is of the form 10b + 5. Now, let's square it.

(10b + 5)(10b + 5) and we use the distributive property to multiply this out.

(10b + 5)(10b + 5) = 100b² + 50b + 50b + 25 = 100b² + 100b + 25

Now, notice those 100b's there. We can gather that as a common factor for the first two terms:

= 100 b (b + 1) + 25

This is now essentially in the form that the trick is using.

The trick says to take b, or the number formed by the digits in front of the 5. That corresponds exactly with our b! (For example, in number 645, b is 64. Our number 645 is 10b + 5, or 640 + 5.)

So we take b, multiply it by (b + 1) which is the next number, and also by 100, and lastly add 25.

Now, b × (b+1) is the part of the trick where you multiply the digits in front of the 5 by the next number. To "tag" 25 to those digits means you add 25 only after having multiplied the number by 100 so that it would end in "00". Once it ends in "00" you can add 25 (or any two-digit number) and it is the same as "tagging" 25 to the digits without the "00".

I hope this is clear enough.

By the way, I do not feel all students must learn this shortcut for finding the square of numbers ending in 5. It is a nice addition to one's mathematical knowledge, but not any necessity. However, it is useful as an algebra problem, and of course has been used as such over the course of centuries, I'm sure.

A math review or reference book for adults & students

I just stumbled upon the book Math on Call: A Mathematics Handbook:

Based on the reviews and from the "Look Inside" feature at Amazon it looks like a GREAT math reference book. I've been on the lookout for something like this, because I get emails from adults who want to review or refresh their basic math before going to college or some other course they're taking. So now I have something to refer them to.

The table of contents includes these chapters:

NUMERATION

• The Base Ten System
• Rational Numbers
• Irrational Numbers

NUMBER THEORY

• Factors and Multiples
• Powers and Roots

COMPUTATION

• Addition
• Subtraction
• Multiplication
• Division

ALGEBRA

• Algebraic Notation
• Properties
• Relations and Functions
• Equations
• Inequalities

GRAPHS AND STATISTICS

• Gathering Data
• Organizing and Summarizing Data with Statistics
• Displaying Data in Tables and Graphs
• Interpreting Data

GEOMETRY

• Elements of Geometric Figures
• Angles
• Plane Figures
• Solid Figures

RATIO, PROPORTION, AND PERCENT

• Ratio
• Proportion
• Percent

PROBABILITY AND ODDS

• Permutations and Combinations
• Probability
• Odds

ALMANAC

• Prefixes
• Suffixes
• Problem Solving
• Study Tips
• Using a Computer for Math
• Geometric Constructions
• Test-Taking Skills
• Handy Tables
• The Metric System
• The Customary System
• General Measurement
• Fraction/Decimal Equivalents
• Multiplication Table
• Square and Cube Roots
• Right-Triangle Ratios
• Prime Numbers Less Than 500
• Mathematical Symbols
• Number Patterns
• Number Systems

YELLOW PAGES

• Glossary of Mathematical Formulas
• Glossary of Mathematical Terms

INDEX

A total of 608 pages. I can hardly believe how good reviews it has!

I was unclear if Math on Call contained exercises or not, but I'm thinking not, since I didn't see anyone mentioning such in the reviews or the description.

Math Teachers at Play Carnival

There is lots of interesting stuff today at Math Teachers at Play Carnival #14 at Math Mama Writes. Quite a nice assortment - surely something for everyone!

I particularly recommend Three Applications of High, Abstract Math. We teachers can always use other ideas as to how mathematics is useful in real life.

Learn to recognize coins

Image by Stopnlook

In the past few days I decided it's time for my younger girl to start learning how to recognize coins and learn the cent-values of various coins. She's generally been doing kindergarten-1st grade addition and subtraction and I figured she's ready.

First, I got a pile of pennies, a pile of nickels, and a pile of dimes (no quarters at first). We played where I told her to make a certain amount, such as "Make 24 cents." She made it, I checked. Then she told me to make some amount. We just take turns.

It's like a game, and great fun for her! I figure it'll work the same with other kindergarten or 1st grade kids.

Since she did pretty good, I was able to introduce quarters the next day. I showed her that two of them makes 50, and we practiced making 62 or 58 or other such amounts that use 2 quarters.

Then I showed her a quarter and a nickel, and we figured out how much that was. That's always a difficult thing - to combine just one quarter with other coins. I soon asked her to make 30 cents, or 32 - and she was able to use the quarter and nickel combination to make 30 cents.

One important thing I've done is that I have deliberately NOT yet introduced the words "nickel", "dime", and "quarter". She's heard of the penny so much that she was fine with that. I want to scaffold the teaching so that we'll first learn the coin values, and later the customary names.

So, in essence, for now I've been calling them as "5-cent coin", "10-cent coin", and "25-cent coin". Has anyone else done that?

Review of Algebra Unplugged

Algebra Unplugged by Kenn Amdahl and Jim Loats is not a textbook, nor does it have any exercises. Instead, it is filled with verbose, often humorous explanations of algebra 1 concepts for those who would rather hear or read math explained in many words, instead of in a few symbols.

Algebra Unplugged also often explains the reasons behind some peculiar mathematical notations or terminology, and in general, tells the students WHY things are done the way they are done in your "Real Algebra Book".

Note: the cover image doesn't directly have to do with the contents of the book... the book is not about music, nor graffitis. I think it's just conveying the idea that just like musicians might "jam" freely after the practice, with this book you'll get to experience algebra "freely" after the class.

Excerpt:

The Associative Principle

Organizing your singers into sections won't affect how many are in your choir. Grouping them is creating associations. The associate law recognizes the benefits of confining your tenors to one easy-to-patrol area. Fifteen tenors in a corner is no different from fifteen scattered throughout room. It's just safer.

If all you're doing to a series of numbers is adding them, you can add them one at a time, or you can put them in groups, then add the totals of each group.

I certainly enjoyed reading through the book, and feel it can be very helpful for algebra students who feel "lost" and want to understand more thoroughly how to navigate in the maze of rules, symbols, terminology, and unknowns.

Here's another fun little excerpt to get you a "taste":

Why They Make You Factor

Factoring is, on the surface, a foolish waste of time. You will read many books and factor many polynomials before you have a clue why you are doing it. The next sentence in this book will save you two months of confusion.

We factor to take advantage of some neat properties of zero.

In addition and subtraction problems, zero is powerless. You can add or subtract zero to a number for the rest of your life and you won't change the number at all. But zero become an all-powerful super-hero once you move into the domain of multiplication.

When you multiply any number by zero, you get zero. A million times zero equals zero.

When you divide zero by any number, you get zero. Zero divided by a million equals zero.

But you can't multiply any other two numbers together and get zero. And you can't divide any other number by anything and get zero. If the answer to a multiplication problem is zero, one of the numbers you multiplied must also be zero. And, if the answer to a division problem is zero, your original numerator was zero.

That's why we factor.

By all means, click to Amazon and "click to look inside" the book to read some more and see if you like the style. I did - but I never know if others do, or if teenagers do.

The authors have definitely succeeded in keeping the conversation on a lighthearted level. They make light of mathematicians and mathematics and tenors (of course, some might not like that). The book often uses apples, bananas, rats, kangaroos, catbox, jabberwocky, etc. as variables, instead of always resorting to x, y, and z. The authors let Little Weenie Numbers show us the way when things get complicated (these are 1, 2, 3, 4, 5, 6 and other small, easy-to-handle numbers).

The only complaint I have is that the book could have given a bit more attention to graphing with more graphs and actual visual illustrations. The author Kenn Amdah confesses he is not a visual person, so that is why.

Check also for used copies. The book is listed at both Amazon and Barnes & Noble

The same authors have also written Calculus for Cats, a clear and entertaining introduction to the mysteries of calculus. It is written with similar goals in mind. It doesn't contain exercises. For some reason, the analogies to the world of cats didn't quite "jibe" with me as well as the writing in the algebra book. This doesn't mean that that book is not useful; it can be tremendously useful for people who feel intimidated or even dismayed by calculus. It just didn't feel quite as fun to read as the algebra book to me.

This book is also listed at Amazon and Barnes & Noble.

Multiplication Mountain songs CD - a review

It has been proven that music can help children memorize things. There is something about the music or the rhythm or both that changes or improves the way our brain works. I haven't studied this effect in-depth; I'm just familiar with the general idea.

So, using music to memorize math facts—or in this case, multiplication tables—is definitely a sound idea.



Multiplication Mountain is a CD by Hap Palmer, filled with songs for the multiplication tables. For each table, there is one song, but it is presented twice on the CD: once sung completely, and the second time without the answers sung. That way, the child can try to fill in the answers while listening to the song from the second track.

I found the songs to be pleasant and jolly, easy to listen to. The musical style varied somewhat so that one song was more jazzy, another was bluegrass, another a waltz, and so on. But they all were melodic, harmonic, and had a medium beat—easy to sing along. My favorite was the song for the sixes times table.

The songs are presented in a particular order: the easiest tables first. Learning all the tables is compared to climbing a mountain: you start at the bottom with the 2s, 5s, 10s, and 11s. You gradually climb up till you reach 7s as the top of the mountain — and then you reach for the sun (the 12s)!

You can listen to samples at www.myspace.com/happalmer (the songs for nines and twos) and at CdFreedom website.

Here's yet another sample of his music from YouTube. It's called Bean Bag Alphabet Rag. It's not from the Multiplication Mountain CD. I just put the link here in case you want to check out the musical and singing style of this artist.
Bean Bag Alphabet Rag by Hap Palmer .

Multiplication Mountain is not the only CD I have that has songs about multiplication tables. I have gotten one other CD in the past, but much disliked that one, because its musical style was rap with quite heavy beats - reminded me of heavy rock. Hap Palmer's music is quite the opposite of that, so I liked it much better (of course, all of us have VERY varying tastes of music).

My older child already knew the multiplication tables when I got the CD, so I could not test how much she would learn with it, and the younger one is not quite ready for that yet. But, in a nutshell, I feel the CD is well done and there is no reason why it wouldn't be effective in helping children to learn the tables.

I would definitely recommend it for anyone interested in trying musical approach to learning the tables and for elementary teachers & tutors in general. Also, if your child has ADD/ADHD or any learning disability, I would suggest that you try music-enhanced learning simply because of the special effect music has on our brain.

Here's a link to Multiplication Mountain CD at Amazon.com.

Disclaimer: I was given this CD for free to review it. I'm not receiving any monetary gain from the company, however.

Math Mammoth Grade 5 Complete Curriculum now available

163 pages
147 lesson pages


218 pages
179 lesson pages

5-A contents and samples
5-B contents and samples

Finally it is ready! I know some folks have been waiting for the 5-B part to get finished, and now it is here!

Math Mammoth Grade 5 complete curriculum consists of two student worktexts (A and B), a separate answer key for each, chapter tests and an end-of-year test, cumulative reviews, and an easy worksheet maker (Internet access required) to make extra practice worksheets when needed.

The two books (worktext part A and part B) for 5th grade deal with

• multi-digit multiplication and long division
• simple equations
• problem solving
• place value with large numbers and the judicious use of calculator
• all operations with decimals
• statistics and graphing
• all fraction operations
• geometry: classifying and drawing triangles & quadrilaterals; calculating the area of rectangles, triangles, parallelograms, and compound figures; surface area and volume of rectangular prisms
• introduction to integers
• introduction to percent

You can purchase the whole curriculum as a download, or as printed copies.

The downloadable curriculum is available at Kagi and at Currclick. The printed copies are made by Lulu (see links on this page, under the cover image).