How to Teach Fractions

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In Year 2 of our school, I had what I call, "the fractions crisis". We were in RightStart, book D. Up to that point, we had breezed through the excellently written books*. One day we were going through lesson 83 when Hannah said, "Huh?" That threw me into a tailspin. Up to that point, we had breezed right through the phenomenal books. Her question had something to do with fractions (whether directly or indirectly). I started searching for answers for how to teach fractions. I searched books and the internet to no avail. I finally put up all the books in the armoir and closed it. Vain was the help of man. My crisis was not solved and I cried out to God. I decided that God had to be the God of the mathematics or else forget it all and send her to school. I had never said that before.

*[RightStart starts to worsen with Book E--too much information was thrown in one book--we did select portions of it and purchased their Geometry book as a reference, but began using our own homemade mathematics books to review and extend to higher levels of practical, useful mathematics. In Year 9 we used A Beka Algebra I video course. Hannah scored 100% on the final exam but I can say that without a basic background of understanding, there would have been big trouble--trouble as in memorizing formulas instead of understanding and enjoying mathematics. With A Beka, at two separate points we had to take some time to get understanding on new concepts (both related in some aspect to "systems of equations". Hannah went to two tutoring sessions the first time and the second time I consulted "Understanding Mathematics: From Counting to Calculus" by Keith Kressin--I shield from confusion by not showing her the book--I think about the issue and how to quickly and easily present the information correctly THE FIRST TIME and in a common sense way with plenty of white space on the page, no lined paper, just white printer paper when I am explaining. During the Algebra I course, I insisted on Hannah (1) remembering the way that we learned things (not changing to new methods) and (2) her being sure to show steps when calculating.)

I finally went to sleep and woke up the next morning with a thought, "I wonder if I can do something...?" I then sat down at the computer and began typing a teaching on fractions. Just as I was finishing and was going on to another part, Hannah walked in and said, "What is that?" I basically shrugged my shoulders and figured that I could test out what was written. In a matter of a few minutes the teaching was done - and permanent. I never had to teach her, or me, fractions after that. I can remember one memorable day afterwards when I asked Hannah how much milk was left in the refrigerator and she told me, "There is about five-eighths (5/8) of a gallon left." She was about five years old.

This fractions teaching was sent to me in a way that I could understand and that a five-year old could understand. The following section was typed from that same paper that I typed that fateful morning. There are only slight additions to help the reader to understand. Even if you have been frustrated in the past, I believe that you can understand this. Just be calm in your spirit and try to understand each step before proceeding to the next one (actually, you may want to read the whole teaching first and then go back and look at each step). What you read in the teaching below may not contain the current textbook definition for a fraction or "fractioning", but you can learn what a fraction is from this. Webster's 1828 define fractions thusly,

"FRACTION, n. The act of breaking or state of being broken..." (I am not sure why the act of breaking is listed as a noun).

When it says "fractioning" below, it means the act of breaking.

One thing that I have always done is use the proper mathematical terms for what we are doing, regardless of age. I was not afraid to use "numerator" or "denominator" and use it CONSISTENTLY (we did--and do--almost no memory work, we use and repeat basic things continually). My daughter started RightStart book A (kindergarten mathematics) after she could read at 3 years old. It was based on the abacus and understanding mathematics, not memorizing formulas. When we began writing our own mathematics books, I made sure that there was a constant review of "the four basic operations of mathematics--addition, subtraction, multiplication, and division". If I said, "What are the four basic operations of mathematics?" She would tell me, "Addition, subtraction, multiplication, and division." This was from very early on. We also reviewed fractions and other operations with simple worksheets--no reteaching, just simple exercises interspersed with whatever mathematical ground that is being covered (by God's grace we progress incrementally and consistently). Fractions not only include fractions like the ones on this page (with numerator and denominator), BUT ALSO decimals, and percents--ALL OF THESE ARE FRACTIONS/PIECES OF NUMBERS. At some point, I will look to add links for these additional teachings on decimals and percents at the bottom of this page (the additional teachings were greatly spread out to avoid confusion). But for now, the teaching that we used for learning what fractions are.


Fractioning

(This teaching is explained to the child, not given to him to read. For the first section below you can use a stick, chalkboard, etc. to explain)

Fractioning is breaking up or splitting up. To fraction something is to break it up. A fraction is a broken piece of a number. Fractions are broken numbers. Fractions tell you how many pieces a unit was broken into, and how many of those pieces you are working with.

Fractioning is the operation of breaking up an integer (a whole number) or integral (a whole thing like a stick). The unit is broken up into equal-sized pieces:   7, 2, 37, 15, 3, etc.

Each broken piece is a fraction (broken piece) of the whole thing. If a stick was broken into three pieces--

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And I took two pieces--

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Then I would have two out of the three pieces or 2/3 of the stick. One out of the three pieces (1/3) is left.

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If I decided I wanted the other piece, I would have three out of the three pieces (3/3) which would be the whole thing. I would have the whole stick--even though it was broken.

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Let us say that I wanted to break up each of the three pieces in two--

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Now, instead of having three thirds (3/3), I would have six sixths (6/6).




The Number Line
[This can be drawn on a chalkboard--this section
is only for if the child is already acquainted with the number line.]

We can show fractions on the number line. Look at this whole, unbroken unit--

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Now look below. How many pieces is this unit broken into? [Remember to count whole segments (the spaces between the lines), not the vertical lines. The answer is six (6)] That number is the denominator. Denominator means "he who names" or "he who gives a name". The denominator tells you how many total pieces there are.

What fraction are the three lines [teacher you can use an arrow instead] below pointing to? Three out of the six pieces or 3/6. The "3" is called the numerator. The numerator tells how many of the pieces that you have. There are six total pieces, you have three pieces.

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0                                                                               1



Fractioning up Strips of Paper

Let us fraction up strips of paper. [Make lengthwise strips out of clean white printer paper or any plain white paper with no lines on it. Take one, fold in half and tear. Give both pieces to the child. "You have 2 pieces of 2 or 2/2 [I think that I drew the 2/2 on the whiteboard next to us as we sat on the floor]." "What if I take one?" "How many do you have now?" "Right, you have 1 of 2 pieces or 1/2."]

[Take another strip. Fold it in half. Pointing to the two pieces, ask, "How many pieces are there?" (2). Fold it in half again. "How many pieces are there?" (4). "Let us tear these apart. If I give you all four, how many pieces do you have? 4 of 4 or 4/4. What if I take two? How many do you have then? 2 of 4 or 2/4.]

At some point I think we spent some time at the white board where I let her actually write the fraction that she had, e.g., if she had two out of the four pieces, she wrote 2/4 on the board, etc.

Keep going if the child wants to. This is the whole math class for the day. We did not have to do it again (my daughter is now ten years old. UPDATE: The years have gone on, she is no longer ten...).





Postscript

Concerning improper fractions, I thought that I would have to do a lesson with my daughter to show her how to do them. Before I could explain the lesson, she looked at my piece of paper that said 11/7 (eleven sevenths) and just said "That is one and four sevenths (1 4/7)."

I was going to add a part two to this page for improper fractions but I do not know if it is necessary. Improper fractions are improper because how can you have more pieces than you have? 7/7 means that there are seven pieces and I have seven of them. 11/7 means that something was broken up into 7 pieces and I have 11 of them. How can that be? It means that I have the seven and four more of the same size from another source, e.g., if two pies were cut into seven pieces each, I would have seven pieces from one pie and another four pieces from the second pie--I would have eleven sevenths.

I do not recommend that the reader look at the following update until he or she is very familiar with the above-information on "fractioning." Too much information at once can make a person forget what they knew.

Update: I saw the following in my logs: someone searched on, "how to teach a 8 year old what 2/3 of 12 is in fractions"

Our approach:

  1. Teach the "fractioning" teaching above.
  2. The next day review it (on board and/or with homemade worksheets, etc.. Maybe do this for a few days. Then approach this problem. Just

    A. put the problem on a chalkboard or blank 8 x 12 piece of paper

    2/3 of 12 = ______

    [Note: I use the horizontal dividing line, not the diagonal] and

    B. put a group of 12 objects nearby but somewhat out of sight. The student may be able to look at the problem and figure it out. If the child is not familiar with division or needs some help--

    Teacher: We need two-thirds (2/3; I would write it out). What is our denominator (he who names)?

    Student: Three.

    Teacher: Right. So we need three to break up/divide the 12 into three equal groups or piles. How many are going to be in each pile?

    Student: [student can sort them into three equal piles or divide (if they know how). Let him do it the way he needs to]. "There are four in each pile."

    Teacher: Right. We need 2 of the three piles. How many do we need?

    Student: "Eight."

    Teacher: Right. 2/3 of 12=____ Please write the answer.

    Student: [student draws the number 8]

    Teacher: Right, very good.

    I would do this several times and provide a sheet with additional practice problems, but not with pictures if the child already knows division. If the teacher decides to teach an algorithm (the short cut way), I might wait until the next day. I would give the student some problems to solve "the old way." If they do okay with that then I might say, "Good. Now I'm going to show you the short cut way." We often did this in those early days after my daughter understood what was happening with a new concept. She learned the short cut but understood what was actually going on with the procedure. It was emphasized that if she could not do a problem one way, she could figure it out another way.

    Before showing the short cut way, I would go back to that initial problem and let the student do it so the student would be oriented. I would then put it to the side. "Now I am going to show you the "short cut way." The following assumes that the teacher understands that (1) "of" is really x (times) (2) any whole number can be made into a fraction by putting it over the number 1.

    2 of 12 =      
    3       1

    2 x 12 =      
    3       1

    2 x 12 = 24
    3       1     3

    2 x 12 = 24 = 8
    3       1     3

    [Note the line between 24 and 3 is a dividing line. Fractions are divisions. I would just simply say that the line can mean division. 24 divided by 3 = 8. 2/3 of 12 = 8 which is what we learned when we first did the problem.]




Spread-out, simple work is important, no competing ideas. Dates of when I taught her various fractions concepts.













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