Shelving the Blog

You may have noticed that our blog hasn't had much new material recently. This is mainly because we've been focusing more on social media websites and our newsletter. For this reason, we are currently shelving this blog with the option to reactivate it in the future. In the meantime, we suggest that you follow us on Google+ or sign up for our newsletter.


Decimal Addition: A Short Introduction

Decimal Addition

The first time your students are faced with decimal numbers might be a bit shocking to them, but actually there’s not much difference between operations with decimals and operations with whole numbers. Generally speaking, these operations need to be done more carefully –if they are not, the students’ results might differ greatly from the correct answer. You can find exercises and drills to help your students practice the content of this lesson at the Decimals page of our website

In this article, we want to deal with the most basic operation of all, that is, decimal addition. First of all, let’s see a very simple sum: 
Simple Decimal Sum: 1.2 + 2.1

As you can see, decimal addition is performed exactly as integer addition, except for the placement of the decimal comma in the result. The comma’s position is probably the most important thing to consider when dealing with decimal addition. Each addend’s comma must be aligned with the rest (see image):

Correct Alignment for Decimal Addition

Notice that the addends’ digits are also aligned around the decimal comma. This remains true even when the number before the comma is a zero:
Decimal Addition: 0.22 + 25.1

A “trick” that could help to align the digits, is to fill the empty spaces with zeros, until both numbers are the same length before and after the decimal:

A visual help for decimal addition.

In this case, we must remember that except for the first zero before the decimal, none of these zeroes is normally written, and serve only as a “crutch” until the student is able to work without them.
Until now we have only seen sums with no regrouping. The principle is the same behind integer addition with regrouping, while also considering the placement of the decimal. So, 6.6 + 5.5 becomes:

Decimal Addition with Regrouping

If both addends are between zero and one, we operate in the same way:
Decimal Addition with Regrouping

It’s the same even when we add a number greater than one, with a number between zero and one:
Decimal Addition with Regrouping

These are the basics of decimal addition. Obviously, the student must have a good understanding of integer addition beforehand, since decimal addition is an extension of it. Once the student notices the relationship between them, and learns the details of decimal addition, s/he should have no problems, as they apply to any number, regardless of how many digits it has:
Complex Decimal Addition

We hope you find this lesson useful. You can find more articles, drills and free math goodies at our site, Math-Drills.com.


Is Facebook Limiting What You See?

Help us reach our goal of increasing the number of people who see our posts from 15% to 50% or more! Get notifications today!

Is Facebook Limiting What You See?

The short answer to the question posed in the title of this article is YES. You may or may not realize that when you "Like" a page on Facebook, this doesn't necessarily mean that you are going to get any posts from that page. For example, when Math-Drills.com posts something on its Facebook page, usually only 10 to 20 percent of people who "Like" our page get to see our post. This is mainly because an algorithm decides how many posts it will show you from all of the pages that you "Like" and the friends that you have. The more friends and liked pages you have, the less likely you will see anything from one particular page or person unless you tell Facebook what you want. Here is an example of the number of views our post on October 6 got compared to the number of people who like the page:

Comparison of Total Likes and Views on October 6 Post

If you feel that you would like to see all of the posts from a page (like the Math-Drills.com Facebook page) or a friend, or conversely, if you would like to see less, the power is already in your hands. There are two very simple things you can do to make your Facebook feed the way you like it.

Get Notifications

This is simply the easiest way to find out when there are new page posts or friend posts on Facebook without filling up your feed. Simply hover your mouse over the "Liked" button on any page and click on "Get Notifications." If you want to stop getting notifications, of course, you would click so it is unchecked. If you want to see all updates from Math-Drills.com, we highly recommend that you turn on "Get Notifications" from the Math-Drills.com Facebook page.

How to turn on notifications from Math-Drills.com.

Show in News Feed

You can also control what posts get shown and how often they get shown in your feed. The "Show in News Feed" option is just below the "Get Notifications" option. If you choose to have it checked, then click on "Settings" and you will see the options as shown in the picture below. Choose between "All Updates," "Most Updates" or "Only Important" depending on what you want to see in your feed.

How to show Math-Drills.com in your news feed.

Hopefully, you found this short article helpful and that you will choose to get notifications from us or show all of our posts in your news feed. We don't post a lot actually, at most once or twice a day, so you can be sure your notification list won't be full of us! We generally post new math worksheet announcements, information about our website such as design changes and usage tips, and a few math ideas that we find interesting.


Base Four Blocks or Thinking Outside the Cube

Base Four Blocks or Thinking Outside the Cube

On a recent visit to a local thrift store, I discovered a cardboard box labelled, "Multi-Base Arithmetic Blocks." This in itself was interesting because I figured they were some sort of base ten blocks set until I noticed the additional text, "Additional Base 4." Because it was in a display case, I asked the clerk if I could look at the box. To my delight, I found a wooden set of blocks very similar to base ten blocks, but using a base of four instead.

The box containing the multi-base arithmetic base 4 blocks.

A pile of base 4 arithmetic blocks.

Even the uninitiated know of other base number systems such as binary (base 2), hexadecimal (base 16) and the Babylonian numbering system (the latter being what we use for seconds and minutes with a base of 60). Many people, however, have difficulty conceptualising numbers in other base systems, and this is probably due, in part, to the fact that we use the same numerals for base number systems under 10 (the decimal system). In the binary system, for example, 0's and 1's are used. In the base 4, or quaternary system, the numbers: 0, 1, 2, and 3 are employed.

After the initial excitement and purchasing the block set for a low price of $5.99, I wanted to find out more about these blocks and their history and find out what other additional bases were available. The information proved to be difficult to find which was surprising with the amount of information available on the Internet. Apparently, the manufacturer, Tiger Toys Ltd of Petersfield in England, is no longer in business. One of the only references to Tiger Toys was a short article about a 2011 reunion of Tiger Toys employees who had worked for the company in the 1960's. Searches on EBay resulted in no results for anything similar to these arithmetic blocks. From other sources, it seems that these blocks were available in different bases up to ten, and that different base number systems were routinely taught in schools in the past. With no additional information, it was time to focus on the more interesting aspect of the base four blocks, actually using them.

If you are familiar with expanded numbers in the base ten system, you probably recognise that the number 4567 can be represented as (4 × 103) + (5 × 102) + (6 × 101) + (7 × 100). Notice that the powers of ten format clearly shows us the base number used, in this case 10. To represent numbers in the base 4 system, the powers of 4 are used. The number, 12234 (the subscript 4 indicates the base 4 numeral system) for example is represented as (1 × 43) + (2 × 42) + (2 × 41) + (3 × 40).

Converting between different base numeral systems takes a little effort, but it is fairly straight-forward. In the case of converting from a base four system to a decimal system, one could just multiply out the number in expanded form. In our previous example, (1 × 43) + (2 × 42) + (2 × 41) + (3 × 40), multiplying out would result in (1 × 64) + (2 × 16) + (2 × 4) + (3 x 1) = 64 + 32 + 8 + 3 = 107. The reverse of this is converting a decimal number to a base four number. In the case of 107, keep dividing the number by 4 until you end up with a quotient less than 4. 107 ÷ 4 = 26 R 3; 26 ÷ 4 = 6 R 2; 6 ÷ 4 = 1 R 2. Starting from the last quotient then using the remainder values, you get 1, 2, 2, 3 or 1223 which was our original number in the base four system.

If that confused you, then the base four blocks are for you! Let's model the number 33334 using the base four blocks (this is the largest number you can model with these blocks without getting "creative").

The base 4 number, 3333, represented with base 4 blocks.

Students should be able to easily see the divisions in the blocks and find out that the small cubes are worth 1, the rods worth 4, the flats worth 16 and with a little help maybe, the large cubes are worth 64. If you have a set of base ten blocks, students can exchange their blocks for the base ten blocks. For example, trading 1 large block, 2 flats, and 1 rod (64 + 32 + 4 = 100) for a 100 flat gets rid of many of their blocks to start with. Grouping together the two remaining large blocks (128) and two unit cubes (2) enables the student to exchange for a 100 flat and three 10 rods. The final blocks add up to 25 (1 flat, 2 rods, 1 unit) and can be exchanged for two 10 rods and 5 units from the base 10 set. All told, students will end up with 100 + 100 + 30 + 20 + 5 = 255.

Working within the base four system with the blocks is just as easy as working with base 10 blocks. The only different rule is that piles of blocks must be exchanged for larger blocks in groups of 4 rather than groups of 10.

So, what is the point of learning another base number system. This could easily result in some confused students if they don't already have strong skills in place value and the decimal numbering system, but for those students who have it mastered and need some more challenge, teaching other base number systems and getting them to work with them can have certain advantages. The biggest advantage, in my estimation, is in the computer programming field. An understanding of hexadecimal, binary, octal (base 8), Base64 and other numbering systems give students a huge advantage. Even though base 4 numbering systems aren't really in practical use, extending skills from one numbering system to another is always a valuable brain-building activity.

Feel free to comment on this post, especially if you have worked with other multi-base arithmetic blocks or know where to purchase multi-base arithmetic blocks in different base numbers.


Math Worksheets on Any Device and Any Size

Math Worksheets on Any Device and Any Size

Recently at Math-Drills.com, we updated the design of the website to be responsive to different screen and browser window sizes. What this means is that you should be able to access our free math worksheets from any device or screen size, including iPhones, iPads, Android based phones and tablets and any other device with an internet browser. We asked our Facebook users to test out our design on their devices and here are some of the comments we got:

"I have a Pantech mobile phone and I rarely go to outside website links as they are impossible to navigate with my silly little phone. I was curious and pleasantly surprised, the format of your site fits in my little 3x3 screen and I could easily navigate through your site! I even discovered useful geometry worksheets for my HS junior!"

"I opened it on my iPad. It looks just like it does on my pc. I even found another worksheet that I want to use soon, expanded decimal numbers in Decimals and Percents. I even pinned it from my iPad! I don't think I can print from my iPad though, but I can save documents to my dropbox to open at school."

"I am using a motorola xoom running android jellybean. I love the layout and the ease of navigation of the mobile site from my tablet. I have downloaded and printed from my tablet different worksheets so far. I never had problems with the desktop site at all though. If you are using android you should not have an issue seeing or printing the worksheets. I have even downloaded them and viewed them using Adobe and the Aldiko ereader app."

Desktop Computers

Although many of the changes we made were intended to make Math-Drills.com more accessible on mobile devices such as tablets and phones, there are also benefits for those using the website on desktop computers.

The first benefit is there is no longer a need for horizontal scrolling if you have a small browser window. Maybe you have a small monitor, or you have a big monitor and like to have several windows open and visible at the same time; either way, you will see the contents of the website without having to scroll sideways no matter how wide you choose to make your window.

The second important benefit is for those who might find the text a little small to read. Just hold down the <Ctrl> key (on Windows computers) and spin the wheel mouse. On other operating systems, zoom in using the preferred method for that operating system. The contents of the window should fit nicely, but you will be rewarded with larger text. You can even override the website font using your browser settings if you require a special font.

Screen cap of the Patterning Math Worksheet using OpenDyslexic font and
a narrow screen size. Font settings overridden in Firefox browser.

If you change the size of your browser window while the website is open, it will change with you. If you happen to make it really narrow (i.e. under 700 px), the navigation menu on the right will move down to the bottom of the page.

Mobile Phones and Tablets

Screen capture of Patterning Math
Worksheets page on Android based
phone. Right menu is at bottom of
page now.
Most of our mobile users arrive at Math-Drills.com using an Apple iOS or Android based system. In the past, it wasn't so easy to navigate around the website, but that has changed. When you load the website on any device with a screen width greater than 700 pixels, you will see the normal desktop website. Under 700 px, you will see a different layout while still retaining all of the content. The major difference is that the worksheet navigation links will move to the bottom of the page rather than remaining on the right side of the page.

Scroll down to see navigation menu.
On mobile devices with a screen width under 700 px, you will most likely not see the pdf file on the worksheet page; instead you will see a thumbnail image. Android and iOS both make it difficult to print a worksheet from this page, so we've hidden it on small screens. For Android users, you will most likely be prompted to download the pdf worksheet immediately. For iOS users, you will have to tap on the worksheet thumbnail image to open the pdf. Either way, once the pdf file is downloaded and opened, you can print it or zoom in for some mental math. For frequent Android users, you can set up your mobile browser to automatically download the math worksheet without a prompt.

Printing from a mobile device depends a lot on what printers you have available. For iOS users, you should be able to print to AirPrint printers or use an app to connect your phone to your other printers. For Android users, you may be able to use Google Cloud Print or an app specific to your printer.

Worksheet page as it is seen on small screens. Tap on the
image to open the pdf worksheet or swipe from the top to see
the downloaded worksheet on Android devices.

A Few Tips

Some mobile browsers (e.g. Dolphin on Android) include options to "auto-fit pages" or something similar. This will probably cause some issues if you have it enabled.

Make sure you have a pdf reader app installed on your device. Apple iOS includes a pdf reader built into Safari, so navigating Math-Drills.com with Safari is the best way to go. On Android devices, we suggest installing the Adobe Reader app for best results.

If you end up with a "watermark" on your printouts when using Google Cloud Print, you may have to go into your printer settings and delete all of the possible watermarks, so none will show up when you print.

Crowd Testing

Now that the website design is responsive, it doesn't mean it is perfect. Please try browsing the website from your various mobile devices and let us know what you think. Are you able to see everything clearly? Can you find the menus? Is there anything that would make things easier? Please let us know by replying/responding to this blog post or by commenting on our Facebook or Google+ pages.