2012-07-30

Multi-Color Fraction Circles

There are some new and revised fraction circles at Math-Drills.com.

Fraction circles have been used for ages to assist students in learning fraction concepts. This is mainly because a circle can be divided into any number of equal size pie slices. Try slicing a rectangle into seven equal sized sections and you'll see why circles are so convenient.

Not only did we improve on the old fraction circles, we added a new choice: the multi-color fraction circles.

 First page of the multi-color fraction circles. These fraction circles continue up to twelfths.

In these fraction circles, we've kept the pie slices with the same numerators the same color. For example, every fraction slice with a 4 as a numerator (4/4, 4/5, 4/6, 4/7, etc.) is colored pink. This might be useful for comparing fractions with the same numerators in a question such as, "Which is greater: 7/8 or 7/11?" Of course, there are certainly other imaginative uses for these fraction circles.

Ideas for Using Fraction Circles

Here are ten teaching ideas you can employ with fraction circles.

1. If you use fraction circles as manipulatives, try to get them printed on transparencies/overhead slides. Not only will they be more durable, they will be translucent to allow overlapped items to be partially visible.
2. Use fraction circles for comparing and ordering fractions. If necessary, cut the fraction circles up into separate circles or even into separate fractions. With separated pieces, it should be easy enough to see which fraction of two given fractions is greater in size. Giving students this visual memory will encourage them to remember the relative sizes of various fractions. They will also start to recognize equivalent fractions.
3. Using the black and white fraction circles, you can easily compare fractions if you make a paper and a transparent version. Color the paper copy with pencil and the transparent version with a non-permanent marker, compare, rinse/erase, and repeat.
4. Operations with fractions usually require the the fraction circles be separated. To add two fractions together, continue the circle started by the first fraction with the second fraction. If the sum is less than a full circle, then find a section that is the same size using the remaining fraction circles. For example, adding 1/3 and 1/2 together makes a partial circle and should compare quite nicely with 5/6. To subtract using paper black and white versions, overlap the two numbers to be subtracted with the subtrahend on top and with one edge (radius) lined up. Draw a line at the end of the subtrahend's other edge and find another fraction circle section that is the same size as the remainder of the section not overlapped. This assumes that the difference is positive. For negative differences, you might have to flip the circles over.
5. Use the multi-colored fraction circles in simple probability experiments. A paper clip bent out and a pencil or compass point turns the fraction circle into a simple spinner. Just hold the pencil or compass point on the center of the circle, hook the paperclip over the point and flick the other end of the paper clip to spin it.
6. If you print multiple copies of the fraction circles, multiplying fractions with whole numbers can be accomplished through repeated addition and consolidating the fraction circle pieces into wholes (if you have a multiplication question that gets that high, e.g. 1/7 x 5 won't but 3/4 x 9 will).
7. Thinking about simpler skills, modeling fractions is easily accomplished with the black and white versions. Start with the segmented ones that are labeled, then the unlabeled ones, then try to see if the student can model fractions on the 1/1 whole circle (i.e. without guidelines to help). They can check how close they were by holding a segmented circle and their answer up to a light, or by using a transparent answer sheet.
8. Use the unlabeled versions for recognition. Have students identify fractions either still intact as full circles or sections of those circles cut out. The more visual information they have, the better they will understand what fractions are later on.
9. For students who breeze through everything else, give them some challenges like, how many different ways can you add three fractions together to get one whole. Have them show their work and/or make a poster showing the different ways they made one whole.
10. Use the fraction circles in games. This is limited by your imagination, but here is one idea. Give each team a full set of fraction circles cut into segments and with magnets on the back (to attach to a magnetic chalk or white board). Give students a question, such as, "This fraction is greater than 5/6 but less than 7/8." Either give each team a turn or make it a race. Keep score, and of course, try to make it a tie so everyone feels good at the end :-)

We thought of ten ideas to use our fraction circles, do you have any other ideas? Comment on this post to share them if you do.

2012-07-28

Patterning with Pascal's Triangle

Blaise Pascal (1623-1662) is considered to be one of the great mathematicians. Although he didn't discover "Pascal's Triangle"; it was known to Islamic, Indian and Chinese mathematicians many years before; he investigated its properties to a greater extent which is why it is currently named after him.

The number in each cell of Pascal's Triangle is found by adding the two numbers above it, except for the top cell which is assigned a value of 1. For the cells on the edge, only one cell is visible above each one; the other can be assumed to be zero.

Math-Drills.com now includes a printable of Pascal's Triangle both in a filled out form (first 12 rows) and in a blank form. Below is a thumbnail image of the filled out version. You can access it directly by clicking on the link in the caption.

 Pascal's Triangle Printable found at Math-Drills.com.

Below are a few ideas for elementary and middle school students.

1. Have students fill out the blank version to practice addition and to notice patterns that develop.
2. Show students the filled out version and have them search for patterns. Some of these patterns are quite easy to find, for example, the second diagonals contain a counting sequence. Other patterns are not so easy to see, for example, the sum of the cells in each row is a power of 2. The fifth row sums as follows: 1 + 4 + 6 + 4 + 1 = 16 which is 2 to the fourth power.
3. Have students color various multiples. For example, coloring the multiples of 2 or even numbers should result in an interesting pattern, especially if the triangle has been extended beyond 12 rows. With an infinite number of rows, you actually end up with a fractal named Sierpinski's triangle.
4. For even more ideas, please see All You Ever Wanted to Know About Pascal's Triangle.

We hope you enjoy our version of the Pascal's Triangle and use it in your home school or classroom.

2012-07-18

Opening Multiple Worksheets Quickly *TIP

Have you ever felt the need to open a number of math worksheets all at once from Math-Drills.com?

When you click on a worksheet link, the worksheet opens in a new tab in your browser. To open a subsequent worksheet, you need to click on the tab that contains the worksheet links then click on the next worksheet that you want. These extra clicks are quite unnecessary as there is an easier way to open all of the worksheet that you want at once.

This takes a little bit of coordination, but after you get the hang of it, you will be opening math worksheets like crazy! With your free hand (i.e. the one that isn't controlling the mouse) hold the "CTRL" key on your keyboard while you click. If you aren't clicking, you might want to let the key go as it may cause other odd things to happen (e.g. holding the CTRL key and spinning the wheel on your mouse will zoom in and out).

If you've gone and tried this already, you will note that the worksheets still open in a new tab, but they open behind the active window. You won't lose your place and you can continue clicking until you've opened all the worksheets that you want.

One word of warning ... if you have a "slow" computer opening too many tabs may cause some performance issues. In this case, try not to open more than 5 to 10 tabs at a time.

 A screen capture of the Safari browser showing multiple worksheets opened in a matter of seconds. The addition page remains on top while all the worksheets open in behind.

Once you have all of those tabs open, you can visit each one and print, save, or display the worksheet, or you can close the tab if you changed your mind.

An extension of this would be to open multiple index pages. For example, say you wanted to print addition, subtraction, multiplication and division worksheets. When you visit Math-Drills.com, hold down the CTRL key while you click on each of those pages from the navigation menu at the top of the page. Each page will load in its own tab for quick use.

This is not limited to Math-Drills.com, of course, this trick works anywhere you want to open multiple pages quickly. For example, you can open all of the news stories you want to read quickly from Google News using this method.

Happy Clicking!

2012-07-14

Summer Olympics Math Opportunities

The London 2012 Summer Olympic Games starts July 27, 2012. With this great event, comes great opportunities to integrate mathematics into the lazy hazy days of summer and have great fun while doing it.

Numbers are a large part of the Olympic games and that gives parents, teachers, and other educators opportunities to use those numbers to improve the math skills of their children and students. Below are some ideas on how you can start your own Olympics math program at home this summer.

Chart the Medal Standings

The medal standings is what most people pay attention to and the source of national pride among much of the population. Charting the medal standings can be accomplished in a number of ways from paper to objects to computers. Be creative to ensure that there is a good balance between work and play.

Let's start with paper. Pictographs are easy enough and work quite well to compare medal counts. You could create a pictograph to show the medal count for your own country by using bronze, silver and gold as the categories. How about creating a pictograph that compares the total medal count of the top ten countries? You could also make a more complicated pictograph using three different colors to show bronze, silver, and gold medals for the top ten countries. All of this could also be accomplished quite nicely on bar graphs (or triple bar graphs to show all three types of medals).

Using objects can make this activity a little more impromptu and possibly integrate some motivators. Maybe you could slice up some carrots (gold), radishes (silver), and parsnips (bronze) and model the medals with them. You could also make a chart with movable ribbons, so the medal count could be increased each day. If you have a bit of a sweet tooth, there's nothing like little round candies to show medal counts.

For the more advanced, computers can make medal standing stand out nicely. Here's one I made using OpenOffice to show the medal standings in the 2008 Beijing Summer Olympics.

The advantage of using a computer is that the numbers can easily be updated every day and a new chart printed.

There is no lack of decimals in the Olympic Games. Timing is done with computer precision and a single hundredth of a second can make the difference between gold and silver medals. Most people have access to a digital watch that will time to the nearest hundredth of a second (although the signal from the eyes to the brain to the finger to stop the timer often causes problems). It should be easy enough to time your child or student to run 100 meters, record this number and compare it to other numbers such as their favorite athlete's time. What is the difference between Usain Bolt's time of 9.69 s and your child's time?

Decimals can be modeled, written in words, read out loud, expanded, compared, ordered (although the broadcasters usually do that for you), added, subtracted, multiplied (how long would it take to run 500 m if the average speed was 10.52 s per 100 m?), divided, etc. Ask questions about decimals and if you don't get the right answer, see if you can teach your child a thing or two.

How many things can you do in a hundredth of a second, a tenth of a second, one second, ten seconds, etc.? These types of questions will help your child conceptualize the idea of time and make decimals make more sense.

Making Numbers Make Sense

Assuming numbers make sense sometimes leads us to forget that children need experiences to understand how the world works and by extension how math works. Perhaps you're watching a cycling event at the Olympics. How fast are those cyclists actually going? Jump in the car and see! How much weight are those weightlifters lifting? Don't lift it all at once, but see if you can come up with an approximation.

There are a number of interesting articles on Olympic numbers that give a different spin on things like medal counts. Media articles can be a great source of information to include in mathematical conversations, especially if they are written intelligently. Here is an example:

Be Creative!

The key to including a little math into summer via the Olympic Games is creativity. I've only touched on a few ideas, and there are probably thousands more. Look for opportunities as they present themselves. Search for information on the Internet. For example, you could look up the medal counts for your country for the last ten years and use that information to estimate what the final medal count will be this year. Make a game of it by having everyone in the family or classroom make their own estimates and see who was closest in the end. Maybe the winner will get their own gold medal.

Now it's your turn. If you have an idea how to integrate mathematics and the Olympics, please comment below. If you're looking for something a little more formal, we stumbled across this nice collection of Olympic Related Math Activities.

2012-07-01

\$50 in Math Stuff Contest Winners!

Thanks to everyone who submitted their entry for our contest to win \$50 in math stuff by including Math-Drills.com on a Pinterest board. In no particular order, here are the winners:

Math Worksheets and Activities
Mrs. Phelps Math-Drills Pages