Category arithmetic

Feb 13 2018

What Should Students Learn about Fractions?

ESSENTIAL FRACTION SKILLS

Not everyone loves fractions, but fraction skills are important for a number of reasons.

  • It helps to be fluent with fraction skills when you take algebra, trigonometry, and calculus courses. When you solve for an unknown in algebra, the answer doesn’t always turn out to be a whole number. One reason that cross multiplication is difficult for many students is that the problems inherently involve fractions. In calculus, simple polynomial anti-derivatives naturally involve fractions (like the integral of x squared, which is one-third of x cubed plus a constant).
  • We also use fractions in science courses like chemistry and physics. So if a student is struggling with fractions in math courses, these struggles also impact science courses. On the other hand, if a student takes the time to master fractions in math, this becomes an asset when fractions appear in other courses. Decimals and percentages (which are basically just other forms of fractions where the denominator is a power of ten) are very common forms of fractions in science and engineering.
  • Fractions are actually pretty common. Almost every day, we see something happen a few times, and start to wonder how often that happens. Maybe your cell phone is doing something funny, or maybe you notice a quirk in somebody’s personality. If you give attention to this, you’ll realize that it’s common to wonder, “How often does that happen?” Many people just guess at it, but their guesses aren’t always realistic (especially when something obviously isn’t extremely common, but they say something like 90% of the time). If you have a small sample, like it happened 3 out of the last 8 times, you can use the fraction (in this case, 3/8) as a projection. You might convert 3/8 to a percentage and see that it equates to 37.5%, for example.
  • If you have a ruler marked in inches, you will see fractions in many common measurements. The fraction will be something like 5/12, 7/8, 3/4, or 1/2. If you have a good feel for fractions, it helps to interpret these numbers. For example, if you measure the diameters of two different balls, and one ball is 3/8″ while the other is 1/3″, can you tell which is bigger just from the numbers?

With the importance of fractions in mind, following is my list of essential fraction skills, based on my experience helping students learn how to apply their math skills in science classes and laboratories.

  • Visual association. Students should be able to draw pie slices to represent fractions, or should be able to write down a fraction to represent a pie slice.
  • Terminology. You can’t discuss fractions with anybody or understand a lecture about fractions if you don’t understand what the different words and phrases mean, like numerator, denominator, reciprocal, common denominator, reduced fraction, greatest common factor, decimal, percentage, proper fraction, improper fraction, mixed number, ratio, and proportion.
  • Reducing fractions. Students should be fluent in reducing fractions down to their simplest form. For example, 8/12 reduces to 2/3 if you divide the numerator (8) and denominator (12) both by 4.
  • Common denominators. Given two different fractions, like 4/7 and 3/5, students should be able to find a common denominator.
  • Mixed numbers. Students should be able to convert between improper fractions and mixed numbers.
  • Addition and subtraction. Students should be able to find a common denominator in order to add or subtract fractions.
  • Multiplication. Students should be able to multiply fractions. (It’s easier than addition or subtraction.)
  • Reciprocals. Students should know how to find the reciprocal of any fraction or whole number.
  • Division. Students should know that dividing two fractions is equivalent to multiplying by the reciprocal of the second fraction.
  • Decimals. Students should be able to convert fractions into decimals or decimals into fractions.
  • Percentages. Students should be able to convert decimals into percentages or percentages into decimals.
  • Repeating decimals. Students should be familiar with repeating decimals and how they relate to fractions.
  • Word problems. You know you understand the concepts well and can apply them when you can solve a variety of word problems that involve fractions.

CHRIS MCMULLEN, PH.D.

Copyright © 2018 Chris McMullen, author of the Improve Your Math Fluency series of math workbooks

Newest releases (in math):

  • Fractions Essentials Workbook with Answers
  • 300+ Mathematical Pattern Puzzles

Improve Your Math Fluency. Build fluency in:

  • arithmetic
  • long division
  • fractions
  • algebra
  • trigonometry
  • graphing
  • physics
Dec 7 2016

How to Use Algebra in Arithmetic

algebra-arithmetic

USING ALGEBRA IN ARITHMETIC

It’s interesting that algebra can be useful in arithmetic.

Since arithmetic is a prerequsite to learning algebra, most algebra courses don’t focus on how to apply algebra to do arithmetic.

However, you can apply your algebra skills to arithmetic problems.

Following is an example.

Suppose that you would like to multiply 58 times 58, but don’t have a calculator (or cell phone) handy.

One way to do this, which is often taught in elementary school, is basically to multiply 8 times 58, then multiply 50 times 58, and add these together.

Algebraically, what your elementary school teacher taught you is that 58 × 58 = (50 + 8) × 58 = 50 × 58 + 8 × 58.

You may not have realized it at the time, but your teacher was applying the distributive property of mathematics.

When you solve the problem this way, first you multiply 8 × 58 to get 464. Then you multiply 50 × 58 to get 2900. Finally, you add these together to get 3364.

Now let’s see how algebra can supply an alternative solution.

I’d prefer to work with nice round numbers. Hey, 60 is a round number close to 58. If I could work with a round number, like 60, and a small number, like 2, that would make the arithmetic much simpler to do without a calculator.

So let’s write 58 as 60 minus 2.

Then the problem is 58 × 58 = (60 – 2) × (60 – 2).

Use the f.o.i.l. method from algebra. Recall that the f.o.i.l. method stands for “first,” “outside,” “inside,” “last.”

One example of the f.o.i.l. method from algebra is (x + y)(x + y) = x² + xy + xy +y² = x² + 2xy +y².

Let’s apply this to the arithmetic problem. We can think of (60 – 2) as (x + y), where x = 60 and y = –2.

58 × 58 = (60 – 2) × (60 – 2) = 60 × 60 + 2 (60)(–2) + (–2) × (–2).

Look what we did. We wrote the original problem, 58 × 58, in terms of three simple multiplication problems: 60 × 60, 2 × 60 × (–2), and (–2) × (–2).

58 × 58 = 3600 – 240 + 4 = 3364.

Note that (–2) × (–2) = +4 because the two minus signs cancel out.

CHRIS MCMULLEN, PH.D.

Copyright © 2015 Chris McMullen, author of the Improve Your Math Fluency series of math workbooks

Newest release:

  • 300+ Mathematical Pattern Puzzles

Improve Your Math Fluency. Build fluency in:

  • arithmetic
  • long division
  • fractions
  • algebra
  • trigonometry
  • graphing
Jul 16 2015

Multiplication Facts 0-10 Memory Tips

Multiplication Fives

MULTIPLICATION FACTS 0-12

When students learn multiplication, it starts out easy.

Too easy, as 0 times anything equals 0 (e.g. 0 x 8 = 0, while anything times 1 equals itself (e.g. 4 x 1 = 4).

But it doesn’t stay easy. Most students struggle to remember multiplication facts where one of the numbers is 6, 7, 8, or 9.

However, there are some tips to aid in memorization.

For one, mirror images don’t matter. The word for this is commutative.

This means 2 x 8 = 8 x 2, for example. Both equal 16.

So if you know 6 x 8 = 48, you also know that 8 x 6 = 48. You don’t need to memorize both.

One of the tricky multiplication facts is 7 x 8. But it’s easy if you remember the trick.

Remember 5678. These are 5 thru 8 in order. 56 = 7 x 8. Piece of cake, huh?

The 9’s are easy. The answer is one decade less than multiplying by 10, then make the two digits add up to 9.

For example, 9 x 7 is in the 60’s (because 10 x 7 = 70, one decade less is 60). It’s 63 since 6 + 3 = 9.

Another example is 5 x 9. It’s in the 40’s (one decade less than 5 x 10 = 50). It’s 45 since 4 + 5 = 9.

To do the trick, after you figure out the decade (by multiplying by 10 and then subtracting 10), subtract the tens digit from 9 to get the units digit.

For example, consider 9 x 6. Multiply 10 x 6 to get 60, and subtract 10 to make 50 (one decade less). Now subtract 5 (the tens digit of 50) from 9 to get 4. Therefore, 9 x 6 = 54.

Once you get the hang of it, this makes remembering the 9’s easy. Try out all the 9’s to get some practice.

If you’re good at doubling numbers quickly, try writing 6 as 2 x 3.

Then 6 x 7 = 2 x 3 x 7. If you know 3 x 7 = 21, double 21 to get 42.

Similarly, for 6 x 4 = 2 x 3 x 4, start with 3 x 4 = 12 and double 12 to make 24.

You can use the doubling trick for the 8’s, too. Just double the number 3 times.

For example, consider 5 x 8. Double 5 three times: 10, 20, 40. So 5 x 8 = 40.

Try 8 x 6. Double 6 three times: 12, 24, 48. Therefore, 8 x 6 = 48.

It works with the 4’s, also. Just double twice.

With 4 x 9, double 9 twice: 18, 36. So 4 x 9 = 36.

That leaves 7 x 7 = 49. You should know 7 x 9 from the 9’s trick.

You can make 7 x 6 and 7 x 8 from the doubling tricks. (The latter you can also know from the 5678 trick.)

5 and under are easier. So to complete the 7’s, you really just need to memorize 7 x 7 = 49.

This covers the 6 thru 9’s, which tend to be the trickier multiplication facts.

The 10’s and 11’s are easy. For the 10’s, just add a zero, as in 8 x 10 = 80 or 6 x 10 = 60.

For the 11’s, with 1 thru 9 just double the digits, like 3 x 11 =33 or 11 x 8 = 88. Get 11 x 10 = 110 from the 10’s trick (add a zero). Then you just need to memorize that 11 x 11 = 121 to complete the 11’s.

You can get the 12’s by doubling the 6’s. For example, knowing that 6 x 5 = 30, you can find that 12 x 5 = 60 by doubling 6 x 5.

Do you know any other tips for remembering multiplication facts 0-12? If so, please share them in the comments.

CHRIS MCMULLEN, PH.D.

Copyright © 2015 Chris McMullen, author of the Improve Your Math Fluency series of math workbooks

  • arithmetic facts
  • multi-digit arithmetic
  • long division with remainders
  • fractions, decimals, and percents
  • algebra and trigonometry
Jan 10 2015

Practice Arithmetic with Geometric Dice

Dice

MATH DICE

My daughter recently received a huge pack of cool geometric dice in several different colors:

  • An icosahedron (20-sided polyhedron) with the numbers 1 thru 20.
  • A dodecahedron (12-sided polyhedron) with the numbers 1 thru 12.
  • A decahedron (10-sided polyhedron) with the numbers 0 thru 9, and another with the numbers 0 thru 90.
  • An octahedron (8-sided polyhedron) with the numbers 1 thru 8.
  • A cube (6-sided polyhedron) with the numbers 1 thru 6.

These dice turned out to be really handy for learning addition and multiplication facts.

ARITHMETIC DICE GAMES

You can easily practice addition facts and multiplication facts with these dice.

Here are some examples:

  • ADDITION/MULTIPLICATION. Roll two decahedra, marked 0 thru 9. Add or multiply the two numbers to practice addition or multiplication facts 0 thru 9.
  • SMALLER NUMBERS. Roll two cubes, marked 1 thru 6. Add or multiply the two numbers to practice addition or multiplication facts 1 thru 6. The cubes let students focus on the smaller numbers first, before working with 7, 8, and 9. (If you want more basic practice, find tetrahedra—4-sided polyhedra—marked 1 thru 4.)
  • FOCUSED FACTS. Roll one decahedron, marked 0 thru 9. For example, suppose you want to practice your multiplication table of 4’s. Simply multiply the die by 4. This lets you concentrate on a single number’s addition or multiplication facts at a time.
  • 11 THRU 20. Advance to dodecahedra or icosahedra to practice the facts 1 thru 12 or 1 thru 20.
  • SUBTRACTION. Practice subtraction facts using an icosahedron and a decahedron. Be careful to subtract the smaller number from the larger number; sometimes, the number on the 10-sided die will be larger. (Advanced students who are learning about negative numbers can use these to sometimes subtract the larger number from the smaller number.)
  • TENS. Multiply powers of 10 using one decahedron with 0 thru 90 and another with 0 thru 9. Or roll one decahedron and multiply that by 10 for more basic tens practice.
  • POWERS. Roll a tetrahedron and a decahedron together to learn about powers. Let the tetrahedron serve as the exponent.
  • FRACTIONS. Roll four decahedra to learn about fractions. These will give you the numerators and denominators of two fractions. Then you can add them, multiply them, divide them, compare them (figure out which is bigger), or subtract them (but first find out which is larger).
  • DICE WAR. If you have several dice to divide equally, you can play dice war with a friend. Each player rolls two dice. Either add or multiply the numbers (choose one before the game begins). The higher sum or product collects both dice.

HANDS-ON GEOMETRY

Another cool thing about using a variety of geometric dice to play math games is that kids get to hold various geometric solids in their hands, see how they look, get a feel for them, and after much use remember how many sides each shape has.

Better than just being told or shown what a dodecahedron is… hold one in your hands, roll it, play with it for months. Then you’ll ‘know’ that solid when you hear its name. (It helps when someone learns and uses the correct names while using the dice.)

Many of these dice packages are sold with role-playing games in mind, but there is no reason that you can’t use them for math practice instead.

TERMINOLOGY

  • Polyhedron: a three-dimensional solid.
  • Polygon: a two-dimensional object, not a solid; it’s flat.
  • Polyhedra is plural, polyhedron is singular.
  • Dice is plural, die is singular.

CHRIS MCMULLEN, PH.D.

Copyright © 2015 Chris McMullen, author of the Improve Your Math Fluency series of math workbooks

Nov 11 2014

Singing Arithmetic

Singing

SINGING

Sound can be a useful learning tool.

I saw this firsthand a few weeks ago when I met a couple and their daughter at a local restaurant.

Their daughters were learning arithmetic, history, language, science, and other facts by singing.

They were using a special curriculum that included songs for many basic things that students learn in various subjects.

One of the girls sang a few different songs and it was quite impressive how much she had learned from singing.

My daughter enjoys singing, too. Recently, I heard her singing some of her multiplication facts.

You can simply sing facts in order, like the table of fours: 4 times 1 equals 4, 4 times 2 equals 8, 4 times 3 equals 12, etc. Or you can add an occasional phrase here and there, especially if it rhymes.

It’s good for patterns, too, like 5, 10, 15, 20, 25, etc.

Most children learn the alphabet song. There are many other songs available to help with learning.

I recall listening to the Schoolhouse Rock songs when I was a kid.

Most people learn better one way than another. So using a variety of teaching strategies helps each student learn through his or her strength.

But I think it’s also important not to let every student rely only on his or her strengths. It’s important to develop the other learning styles, too.

We become better students not just by focusing on improving our strengths, but also by improving our weaknesses.

Singing arithmetic isn’t for everybody, though. I’m sure nobody wants to hear me sing my multiplication facts—or anything else, for that matter. 🙂

Copyright © 2014 Chris McMullen, author of the Improve Your Math Fluency Series

Aug 20 2014

Creative Arithmetic

7 times 7

CREATIVITY IN MATH

It’s wonderful to see young kids as they first learn new math concepts.

Very often, they see things in a different way than what we’re accustomed to, and different from how math is often taught.

I believe that such creativity should be encouraged. Isn’t math more exciting when you think of your own way to do something and it works out, compared to being told what to do and how to do it?

Sometimes, though, one method does have an advantage over another method. In that case, it may help to challenge the student with a situation where his or her method doesn’t work out. When the child sees this firsthand, he or she will be more likely to embrace a different method.

Other times, one method is much more efficient than another. Again, it helps when the child can see a specific example where his or her method is inefficient, and to understand why it matters.

Where possible, encouraging a little creativity may be a good thing, even in mathematics. That creativity can help to engage the student in math. Looking way ahead, if the child ever gets far in math, creative problem-solving can become a valuable skill. So it would be a shame not to foster some creativity along the way.

MULTIPLICATION EXAMPLE

I’ve seen many young kids come up with neat math ideas. My daughter (six) recently thought of a neat way of looking at multiplication. I will use this as an example.

To start out, she was presented only with the definition of multiplication—rather than starting out with a table of multiplication facts. I wanted her to first understand what multiplication means on her own terms and then proceed from there.

She was given a few examples of what multiplication means. For example, 3 times 2 means to add three two’s together (or to add two three’s together). With a few examples like this, she quickly understood what a multiplication problem meant.

The next step was to give her multiplication problems and ask her to figure them out. So, for example, she figured out that 3 times 2 was six by adding three two’s together (and then she could see that adding two three’s together produced the same result).

Now here is where things became really interesting.

My daughter was asked, “What is 6 times 3?”

Surprisingly, after a while, she answered, “9 plus 9 is 18.”

I was obviously curious about this, but decided to be content that she answered the problem correctly (even though her solution seemed questionable) and see how this would play out before investigating this.

A couple of days later, she was asked another question. “What is 5 times 4?”

This time, she said, “10 plus 10 is 20.”

Evidently, it wasn’t a fluke. There must be some method to this madness. So now I asked her a few questions about this to figure out what she was doing.

It turns out that she was visualizing pyramids in her head. She was solving the multiplication problems by grouping numbers together in pyramids.

Below is a picture showing how she worked out 6 times 3.

6 times 3

She started with the bottom row: She knew that 6 times 3 meant to add six three’s together, so she started with six three’s. Then she grouped the three’s into two sets of nine’s, and added the nine’s to make 18.

The following figure shows how she figured out 5 times 4.

5 times 4

Again, she began with four five’s, regrouped them into two pairs to create two ten’s, and added the ten’s to make 20.

Eventually, she will learn and practice her multiplication facts to become fluent in multiplication (that’s the goal, anyway). She will also learn that her solution isn’t efficient if, for example, she wants to multiply 9 times 8—or, worse, 35 times 24. But for now, she has made multiplication her own, and her method works fine for simple multiplication facts.

My daughter doesn’t realize it, of course, but she has the basis here for factoring. For example, she’s writing 6 x 3 as 9 x 2, which both boil down to 3 x 3 x 2.

Copyright © 2014 Chris McMullen, author of the Improve Your Math Fluency Series (which, by the way, is focused on practice and drills—which build fluency—and not on the creative learning aspect that I described in this article)