dividing sausages

Fractions: 1/5 = 1/10 = 1/80 = 1?

[Feature photo is a screen shot from the video "the sausages sharing episode," see below.]

Fractions: 1/5 = 1/10 = 1/80 = 1?

How in the world can 1/5 be the same as 1/10? Or 1/80 be the same as one whole thing? Such nonsense!

No, not nonsense. This is real-world common sense from a couple of boys faced with a problem just inside the edge of their ability — a problem that stretches them, but that they successfully solve, with a bit of gentle help on vocabulary.

Here’s the problem:

  • How can you divide eight sausages evenly among five people?

Think for a moment about how you (or your child) might solve this puzzle, and then watch the video below.

What Do You Notice?

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playing cards

Fraction Game: My Closest Neighbor

[Feature photo above by Jim Larrison, and antique playing cards below by Marcee Duggar, via Flickr (CC BY 2.0).]

I missed out on the adventures at Twitter Math Camp, but I’m having a great time working through the blog posts about it. I prefer it this way — slow reading is more my speed. Chris at A Sea of Math posted a wonderful game based on one of the TMC workshops. Here is my variation.

Math concepts: comparing fractions, equivalent fractions, benchmark numbers, strategic thinking.

Players: two to four.

Equipment: two players need one deck of math cards, three or four players need a double deck.

How to Play

three-eighths-of-clubs

Deal five cards to each player. Set the remainder of the deck face down in the middle of the table as a draw pile.

You will play six rounds:

  • Closest to zero
  • Closest to 1/4
  • Closest to 1/3
  • Closest to 1/2
  • Closest to one
  • Closest to two

In each round, players choose two cards from their hand to make a fraction that is as close as possible (but not equal) to the target number. Draw two cards to replenish your hand.

The player whose fraction is closest to the target collects all the cards played in that round. If there is a tie for closest fraction, the winners split the cards as evenly as they can, leaving any remaining cards on the table as a bonus for the winner of the next round.

After the last round, whoever has collected the most cards wins the game.


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Quotable: Math Connections

ConnectedGearsJoBoaler

It turns out that the people who do well in math are those who make connections and see math as a connected subject. The people who don’t do well are people who see math as a lot of isolated methods.

— Jo Boaler
Math Connections

If you or your children struggle with math, Boaler’s non-profit YouCubed.org may help you recover your joy in learning.


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fraction pieces

Reblog: A Mathematical Trauma

Feature photo (above) by Jimmie via flickr.

My 8-year-old daughter’s first encounter with improper fractions was a bit more intense than she knew how to handle.

I hope you enjoy this “Throw-back Thursday” blast from the Let’s Play Math! blog archives:


Photo (right) by Old Shoe Woman via Flickr.

Nearing the end of Miquon Blue today, my youngest daughter encountered fractions greater than one. She collapsed on the floor of my bedroom in tears.

The worksheet started innocently enough:

\frac{1}{2} \times 8=\left[ \quad \right]

[Click here to go read the original post.]


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Princess in the Dungeon Game

Yet more fun from Rosie at Education Unboxed. I found these while looking for videos to use in my PUFM Subtraction post. Rosie says:

This is seriously embarrassing and I debated whether to put this video online or not because this is NOT my normal personality, but my girls made up this game and will play it for over an hour and ask for it repeatedly… so I figured someone out there might be able to use it with their kids, too.

If you know me, please don’t ever ask me to do this in public. I will refuse.

Princess in the Dungeon, Part 1 – Fractions with Cuisenaire Rods

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by Scott Robinson via flickr

How to Understand Fraction Division

photo by Scott Robinson via flickr

A comment on my post Fraction Division — A Poem deserves a longer answer than I was able to type in the comment reply box. Whitecorp wrote:

Incidentally, this reminds me of a scene from a Japanese anime, where a young girl gets her elder sister to explain why 1/2 divided by 1/4 equals 2. The elder girl replies without skipping a heartbeat: you simply invert the 1/4 to become 4/1 and hence 1/2 times 4 equals 2.

The young one isn’t convinced, and asks how on earth it is possible to divide something by a quarter — she reasons you can cut a pie into 4 pieces, but how do you cut a pie into one quarter pieces? The elder one was at a loss, and simply told her to “accept it” and move on.

How would you explain the above in a manner which makes sense?

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The Weird Number

A bit of number fun to start your week off right:

And speaking of those “scary” numbers out in the forest… Monday is Pi Day!
Have some irrational fun with these posts:

[Hat tip: Julie mentioned this video on the Living Math Forum.]


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Egyptian Math: Fractions

I have been enjoying James Tanton’s website. In this video, Tanton explains a foolproof method for creating Egyptian fractions:

See more posts on Egyptian math.


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Probability Issue: Hints and Answers

Alexandria JonesRemember the Math Adventurer’s Rule: Figure it out for yourself! Whenever I give a problem in an Alexandria Jones story, I will try to post the answer soon afterward. But don’t peek! If I tell you the answer, you miss out on the fun of solving the puzzle. So if you haven’t worked these problems yet, go back to the original posts. If you’re stuck, read the hints. Then go back and try again. Figure them out for yourself — and then check the answers just to prove that you got them right.

This post offers hints and answers to puzzles from these blog posts:

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Quotable: I Hate Fractions

“I hate fractions.”

“They probably hate you, too. The question is, which one of you will be master.”

— Jonathan, jd2718
“I hate fractions”


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dragon and moon

Hobbit Math: Elementary Problem Solving 5th Grade

[Photo by OliBac. Visit OliBac's photostream for more.]

The elementary grades 1-4 laid the foundations, the basics of arithmetic: addition, subtraction, multiplication, division, and fractions. In grade 5, students are expected to master most aspects of fraction math and begin working with the rest of the Math Monsters: decimals, ratios, and percents (all of which are specialized fractions).

Word problems grow ever more complex as well, and learning to explain (justify) multi-step solutions becomes a first step toward writing proofs.

This installment of my elementary problem solving series is based on the Singapore Primary Mathematics, Level 5A. For your reading pleasure, I have translated the problems into the world of J.R.R. Tolkien’s classic, The Hobbit.

[Note: No decimals or percents here. Those are in 5B, which will need an article of its own. But first I need to pick a book. I'm thinking maybe Naya Nuki...]

Printable Worksheet

In case you’d like to try your hand at the problems before reading my solutions, I’ve put together a printable worksheet:

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dwarfswolves

Narnia Math: Elementary Problem Solving 4th Grade

[Photo by armigeress.]

In 4th grade, math problems take a large step up on the difficulty scale. Students are more mature and can read and follow more complex stories. Multi-step word problems become the new norm, and proportional relationships (like “three times as many”) show up frequently. As the year progresses, fractions grow to be a dominant theme.

As a math teacher, one of my top goals is that my students learn to solve word problems. Arithmetic is (relatively) easy, but many children struggle in applying it to “real world” situations.

In previous posts, I introduced the problem-solving tools of word algebra and bar diagrams, either of which can help students organize the information in a word problem and translate it into a mathematical calculation. The earlier posts in this series are:

In this installment, I will continue to demonstrate the problem-solving tool of bar diagrams through a series of ten 4th grade problems based on the Singapore Primary Math series, level 4A. For your reading pleasure, I have translated the problems into the universe of a family-favorite story by C. S. Lewis, The Lion, the Witch and the Wardrobe.

Update

I’ve put the word problems from my elementary problem solving series into printable worksheets:

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Tangrams and Other Dissection Puzzles

tangram-cat[Photo by jimmiehomeschoolmom.]

One of the things I meant to do with my elementary math class (the one that got canceled due to low enrollment):

And then we would play around with Tangram puzzles, and perhaps make up a few of our own.

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Math History on the Internet

[Image from the MacTutor Archive.]

The story of mathematics is the story of interesting people. What a shame it is that our children see only the dry remains of these people’s passion. By learning math history, our students will see how men and women wrestled with concepts, made mistakes, argued with each other, and gradually developed the knowledge we today take for granted.

In a previous article, I recommended books that you may find at your local library or be able to order through inter-library loan. Now, let me introduce you to the wealth of math history resources on the Internet.

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Caution children at play

Math Games by Kids

Photo by Mike Licht, NotionsCapital.com.

The cold came back and knocked me flat, but there are compensations. The downtime gave me a chance to browse my overflowing bookmarks folder, and I found something to add to my resource page. Princess Kitten and I enjoyed exploring these games and quizzes from Ambleweb.

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Calculations

Subtracting Mixed Numbers: A Cry for Help

Photo by powerbooktrance.

Paraphrased from a homeschool math discussion forum:

Help me teach fractions! My son can do long subtraction problems that involve borrowing, and he can handle basic fraction math, but problems like 9  -  5 \frac{2}{5} give him a brain freeze. To me, this is an easy problem, but he can’t grasp the concept of borrowing from the whole number. It is even worse when the math book moves on to 10 \frac{1}{4}  -  2 \frac{3}{7} .

Several homeschooling parents replied to this question, offering advice about various fraction manipulatives that might be used to demonstrate the concept. I am not sure that manipulatives are needed or helpful in this case. The boy seems to have the basic concept of subtraction down, but he gets flustered and is unsure of what to do in the more complicated mixed-number problems.

The mother says, “To me, this is an easy problem” — and that itself is one source of trouble. Too often, we adults (homeschoolers and classroom teachers alike) don’t appreciate how very complicated an operation we are asking our students to perform. A mixed-number calculation like this is an intricate dance that can seem overwhelming to a beginner.

I will go through the calculation one bite at a time, so you can see just how much a student must remember. As you read through the steps, pay attention to your own emotional reaction. Are you starting to feel a bit of brain freeze, too?

Afterward, we’ll discuss how to make the problem simpler…

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Egyptian Fractions: The Answer Sheet

Alexandria JonesRemember the Math Adventurer’s Rule: Figure it out for yourself! Whenever I give a problem in an Alexandria Jones story, I will try to post the answer (relatively) soon afterward. But don’t peek! If I tell you the answer, you miss out on the fun of solving the puzzle. So if you haven’t worked these problems yet, go back to the original post. Figure them out for yourself — and then check the answers just to prove that you got them right.


The Secret of Egyptian Fractions

Alex made a poster of Egyptian-style fractions, from 1/2 to 9/10. Many of the fractions were easy. She knew that…

\frac{5}{10}  = \frac{4}{8}  = \frac{3}{6}  = \frac{2}{4}  = \frac{1}{2}

Therefore, as soon as she figured out one fraction, she had the answer to all of its equivalents.

She had the most trouble with the 7ths and 9ths. She tried converting these to other fractions that were easier to work with. For example, 28 has more factors than 7, making 28ths easier to break up into other fractions with one in the numerator.

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Egypt's Great Pyramid

The Secret of Egyptian Fractions

Photo from Library of Congress via pingnews.

Archaeology professor Dr. Fibonacci Jones came home from a long day of lecturing and office work. Stepping inside the front door, he held up a shiny silver disk.

“Ta-da!” he said.

Rhind papyrus

“All right!” said his daughter Alexandria. “The photos are here.”

They had to chase Alex’s brother Leon off the computer so they could view the images on the CD, but that wasn’t hard. He wanted to see the artifacts, too. Alex recognized several of the items they had dug up from the Egyptian scribe’s burial plot: the wooden palette, some clay pots, and of course the embalmed body.

Then came several close-up pictures of writing on papyrus.

Photo from MathsNet.net.

How to Write Egyptian Fractions

“I remember how to read the Egyptian numbers,” Alex said, “but what are these marks above them?”

Dr. Jones nodded. “I thought you would catch that. Those are fractions. The scribe places an open mouth, which is the hieroglyph ‘r’, over a number to make its reciprocal.”

“I know that word,” Leon said. “It means one over the number. Like, the reciprocal of 12 is 1/12, right?”

“That is right. 1/12 would be written as…”

The Rest of the Story

As I transcribed this article from my old math newsletter, I realized that it would require more graphics than I was willing to construct. LaTex does not handle Egyptian hieroglyphs — or at least, I don’t know how to make it do so. Instead, I decided to scan the newsletter pages and give them to you as a pdf file:

Right-click and choose “Save” to download:

The file includes a student worksheet for Egyptian fractions from 1/2 to 9/10.

Egyptian Fractions: The Answer Sheet

The answers are now posted.

To Be Continued…

Read all the posts from the January/February 1999 issue of my Mathematical Adventures of Alexandria Jones newsletter.


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Fraction Models, and a Card Game

Fraction cards

Models give us a way to form and manipulate a mental image of an abstract concept, such as a fraction. There are three basic ways we can imagine a fraction: as partially-filled area or volume, as linear measurement, or as some part of a given set. Teach all three to give your students a well-rounded understanding.

When teaching young students, we use physical models — actual food or cut-up pieces of construction paper. Older students and adults can firm up the foundation of their understanding by drawing many, many pictures. As we move into abstract, numbers-only work, these pictures remain in our minds, an always-ready tool to help us think our way through fraction problems.

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How to Read a Fraction

Fraction notation and operations may be the most abstract math monsters our students meet until they get to algebra. Before we can explain those frustrating fractions, we teachers need to go back to the basics for ourselves. First, let’s get rid of two common misconceptions:

  • A fraction is not two numbers.
    Every fraction is a single number. A fraction can be added to other numbers (or subtracted, multiplied, etc.), and it has to obey the Distributive Law and all the other standard rules for numbers. It takes two digits (plus a bar) to write a fraction, just as it takes two digits to write the number 18 — but, like 18, the fraction is a single number that names a certain amount of whatever we are counting or measuring.
  • A fraction is not something to do.
    A fraction is a number, not a recipe for action. The fraction 3/4 does not mean, “Cut your pizza into 4 pieces, and then keep 3 of them.” The fraction 3/4 simply names a certain amount of stuff, more than a half but not as much as a whole thing. When our students are learning fractions, we do cut up models to help them understand, but the fractions themselves are simply numbers.

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Solving Complex Story Problems II

[Oops! I found one more post from my old blog. It apparently slipped off the back of my metaphorical desk and has been sitting with the dust bunnies.]

Here is a math problem in honor of one of our family’s favorite movies

Han Solo was doing some needed maintenance on the Millennium Falcon. He spent 3/5 of his money upgrading the hyperspace motivator. He spent 3/4 of the remainder to install a new blaster cannon. If he spent 450 credits altogether, how much money did he have left?

[Modified from a word problem in Singapore Primary Math 5B. Stop and think about how you would solve it before reading further.]

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How Shall We Teach Fractions?

How did you fare on the Frustrating Fractions Quiz? With so many apparent inconsistencies, we can all see why children (and their teachers) get confused. And yet, fractions are vital to our children’s test scores — and scores are important to college admissions officers. What is a teacher to do? Must we tell our children, “Do it this way, and don’t ask questions”?

Parents and teachers are tempted to wonder if the struggle is worth it. After all, how often do you divide by a fraction in your adult life? If only we could skip the hard stuff…

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ordering-fractions-by-jimmiehomeschoolmom

Quiz: Those Frustrating Fractions

[Photo by jimmiehomeschoolmom.]

Fractions confuse almost everybody. In fact, fractions probably cause more math phobia among children (and their parents) than any other topic before algebra. Middle school textbooks devote a tremendous number of pages to teaching fractions, and still many students find fractions impossible to understand. Standardized tests are stacked with fraction questions.

Fractions are a filter, separating the math haves from the luckless have nots. One major source of difficulty with fractions is that the rules do not seem to make sense. Can you explain these to your children?

Start with an easy one…

Question #1

If you need a common denominator to add or subtract fractions…

  • Why don’t you need a common denominator when you multiply?

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reading-aloud-by-betsssssy

Reading to Learn Math

[Photo by Betsssssy.]

Do you ever take your kids’ math tests? It helps me remember what it is like to be a student. I push myself to work quickly, trying to finish in about 1/3 the allotted time, to mimic the pressure students feel. And whenever I do this, I find myself prone to the same stupid mistakes that students make.

Even teachers are human.

In this case, it was a multi-step word problem, a barrage of information to stumble through. In the middle of it all sat this statement:

…and there were 3/4 as many dragons as gryphons…

My eyes saw the words, but my mind heard it this way:

…and 3/4 of them were dragons…

What do you think — did I get the answer right? Of course not! Every little word in a math problem is important, and misreading even the smallest word can lead a student astray. My mental glitch encompassed several words, and my final tally of mythological creatures was correspondingly screwy.

But here is the more important question: Can you explain the difference between these two statements?

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7 Things to Do with a Hundred Chart


This post has been revised to incorporate all the suggestions in the comments below, plus many more activities. Please update your bookmarks:

Or continue reading the original article…


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How Old Are You, in Nanoseconds?

Conversion factors are special fractions that contain problem-solving information. Why are they called conversion factors? “Conversion” means change, and conversion factors help you change the numbers and units in your problem. “Factors” are things you multiply with. So to use a conversion factor, you will multiply it by something.

For instance, if I am driving an average of 60 mph on the highway, I can use that rate as a conversion factor. I may use the fraction \frac{60 \: miles}{1 \: hour} , or I may flip it over to make \frac{1 \: hour}{60 \: miles} . It all depends on what problem I want to solve.

After driving two hours, I have traveled:

\left(2 \: hours \right) \times \frac{60 \: miles}{1 \: hour} = 120 miles so far.

But if I am planning to go 240 more miles, and I need to know when I will arrive:

\left(240 \: miles \right) \times \frac{1 \: hour}{60 \: miles} = 4 hours to go.

Any rate can be used as a conversion factor. You can recognize them by their form: this per that. Miles per hour, dollars per gallon, cm per meter, and many, many more.

Of course, you will need to use the rate that is relevant to the problem you are trying to solve. If I were trying to figure out how far a tank of gas would take me, it wouldn’t be any help to know that an M1A1 Abrams tank gets 1/3 mile per gallon. I won’t be driving one of those.

Using Conversion Factors Is Like Multiplying by One

If I am driving 65 mph on the interstate highway, then driving for one hour is exactly the same as driving 65 miles, and:

\frac{65 \: miles}{1 \: hour} = the \: same \: thing \: divided \: by \: itself = 1

This may be easier to see if you think of kitchen measurements. Two cups of sour cream are exactly the same as one pint of sour cream, so:

\frac{2 \: cups}{1 \: pint} = \left(2 \: cups \right) \div \left(1 \:pint \right) = 1

If I want to find out how many cups are in 3 pints of sour cream, I can multiply by the conversion factor:

\left(3 \: pints \right) \times \frac{2 \: cups}{1 \: pint} = 6 \: cups

Multiplying by one does not change the original number. In the same way, multiplying by a conversion factor does not change the original amount of stuff. It only changes the units that you measure the stuff in. When I multiplied 3 pints times the conversion factor, I did not change how much sour cream I had, only the way I was measuring it.

Conversion Factors Can Always Be Flipped Over

If there are \frac{60 \: minutes}{1 \: hour} , then there must also be \frac{1 \: hour}{60 \: minutes} .

If I draw house plans at a scale of \frac{4 \: feet}{1 \: inch} , that is the same as saying \frac{1 \: inch}{4 \: feet} .

If there are \frac{2\: cups}{1 \: pint} , then there is \frac{1\: pint}{2 \: cups} = 0.5 \: \frac{pints}{cup} .

Or if an airplane is burning fuel at \frac{8\: gallons}{1 \: hour} , then the pilot has only 1/8 hour left to fly for every gallon left in his tank.

This is true for all conversion factors, and it is an important part of what makes them so useful in solving problems. You can choose whichever form of the conversion factor seems most helpful in the problem at hand.

How can you know which form will help you solve the problem? Look at the units you have, and think about the units you need to end up with. In the sour cream measurement above, I started with pints and I wanted to end up with cups. That meant I needed a conversion factor with cups on top (so I would end up with that unit) and pints on bottom (to cancel out).

You Can String Conversion Factors Together

String several conversion factors together to solve more complicated problems. Just as numbers cancel out when the same number is on the top and bottom of a fraction (2/2 = 2 ÷ 2 = 1), so do units cancel out if you have the same unit in the numerator and denominator. In the following example, quarts/quarts = 1.

How many cups of milk are there in a gallon jug?

\left(1\: gallon \right) \times \frac{4\: quarts}{1\: gallon} \times \frac{2\: pints}{1\: quart} \times \frac{2\: cups}{1\: pint} = 16\: cups

As you write out your string of factors, you will want to draw a line through each unit as it cancels out, and then whatever is left will be the units of your answer. Notice that only the units cancel — not the numbers. Even after I canceled out the quarts, the 4 was still part of my calculation.

Let’s Try One More

The true power of conversion factors is their ability to change one piece of information into something that at first glance seems to be unrelated to the number with which you started.

Suppose I drove for 45 minutes at 55 mph in a pickup truck that gets 18 miles to the gallon, and I wanted to know how much gas I used. To find out, I start with a plain number that I know (in this case, the 45 miles) and use conversion factors to cancel out units until I get the units I want for my answer (gallons of gas). How can I change minutes into gallons? I need a string of conversion factors:

\left(45\: min. \right) \times \frac{1\: hour}{60\: min.} \times \frac{55\: miles}{1\: hour} \times \frac{1\: gallon}{18\: miles} = 2.3\: gallons

How Old Are You, Anyway?

If you want to find your exact age in nanoseconds, you need to know the exact moment at which you were born. But for a rough estimate, just knowing your birthday will do. First, find out how many days you have lived:

Days\: I\:have\: lived = \left(my\: age \right) \times \frac{365\: days}{year}

+ \left(number\: of\: leap\: years \right) \times \frac{1\: extra\: day}{leap\: year}

+ \left(days\: since\: my\: last\: birthday,\: inclusive \right)

Once you know how many days you have lived, you can use conversion factors to find out how many nanoseconds that would be. You know how many hours are in a day, minutes in an hour, and seconds in a minute. And just in case you weren’t quite sure:

One\: nanosecond = \frac{1}{1,000,000,000} \: of\: a\: second

Have fun playing around with conversion factors. You will be surprised how many problems these mathematical wonders can solve.


[Note: This article is adapted from my out-of-print book, Master the Math Monsters.]


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fraction pieces

Improper Fractions: A Mathematical Trauma

Feature photo (above) by Jimmie via flickr. Photo (right) by Old Shoe Woman via Flickr.

Nearing the end of Miquon Blue today, my youngest daughter encountered fractions greater than one. She collapsed on the floor of my bedroom in tears.

The worksheet started innocently enough:

\frac{1}{2} \times 8=\left[ \quad \right]

Continue reading

Percents: Key Concepts and Connections

[Rescued from my old blog.]

Paraphrased from a homeschool math discussion forum:

“I am really struggling with percents right now, and feel I am in way over my head!”

Percents are one of the math monsters, the toughest topics of elementary and junior high school arithmetic. Here are a few tips to help you understand and teach percents.

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Fraction Division — A Poem

[Rescued from my old blog.]

Division of fractions is surely one of the most difficult topic in elementary arithmetic. Very few students (or teachers) actually understand how and why it works. Most of us get by with memorized rules, such as:

Ours is not to reason why;
just invert and multiply!

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