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I'm a first year student at a university college, and I'm currently teaching 7th grade for a few weeks as a part of my education. I'm struggling a bit with finding models and strategies to develop an understanding of fractions within the class, specifically when multiplying a fraction with a fraction, and dividing fractions by a fraction.

Up until now I've been using a lot of squares to try to visualize the the parts, dividing them horizontally and vertically, to show how we get to the denominator and which numerator we end up getting.

So my question is this, do you guys know and strategies or visual models that are good at developing a profound understanding of fractions that I can use in class? Either through use of the blackboard, or assignments and activities? Have you been taught fractions through a method that you in retrospect found to have been particularly effective?

Edit: Thank you all so much for your feedback. I've been able to use a lot of your insight in practice, and although I have a long way to go yet as a teacher, this has aided immensely in my development.

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  • $\begingroup$ I suspect the time spent on squares and visual aids and the like is actually confusing the kids more than just computational practice. $\endgroup$ – guest Oct 19 '18 at 1:05
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This depends on what meaning you give to fractions. One approach is to give fractions the meaning of shares or portions of some unit e.g. a rectangle.

Multiplication is mostly equal to the word of. Examples:

$\frac{5}{8}=\frac{5}{8}\cdot 1$ are five eights of the unit.

$\frac{2}{3}\cdot\frac{3}{4}$ means two thirds of $\frac{3}{4}$. Show a rectangle, divide it into 4 equal parts and mark three of them red. Now, take two thirds of the red area.

  • Variant 1: Divide the red area into three equal parts. Hey, it's already divided into three equal parts. Mark two of them blue. The blue area is $\frac{2}{3}\cdot\frac{3}{4}$ of the rectangle. If you ignore the red part, you can see, that it's also $\frac{2}{4}=\frac{1}{2}$ of the rectangle.
  • Variant 2: Divide each of the quarters into three equal parts. How many small parts is the rectangle composed of? It's $3\cdot 4=12$, so we're talking twelths here. In each of the red quarters, mark two of the three parts blue. How many are those? It's $3\cdot 2=6$. So, what's the blue share of the full rectangle? It's $\frac{6}{12}=\frac{1}{2}$. That's the same as variant 1, so our thinking was right.

If you write the numbers in different colors, you can show the students by this generalizable example, that $\frac{a}{b}\cdot\frac{c}{d}=\frac{a\cdot c}{b\cdot d}$ and that they can cancel out the 3s as they are not needed in the end of each variant.

All this should be constructed as much as possible.

Division

In contrast to multiplication, we work formally here, revisiting the technique of inverse operations.

Example: $\frac{5}{7}:\frac{2}{3}=\frac{a}{b}$ and we want to know, what $a$ and $b$ are. The inverse operation to division is multiplication. $\frac{a}{b}$ is the fraction, for which $\frac{2}{3}\cdot\frac{a}{b}=\frac{5}{7}$. We know now, that then $2\cdot a$ must be $5$ and $3\cdot b$ must be 7, but that's not possible for integers $a$ and $b$. But expanding the fraction is our friend here. We just expand $\frac{5}{7}$ by $2$ and by $3$, then it's no problem anymore to find integers $a$ and $b$:

$$\frac{2}{3}\cdot\frac{a}{b}=\frac{5}{7}=\frac{5\cdot 2}{7\cdot 2}=\frac{5\cdot 2\cdot 3}{7\cdot 2\cdot 3}$$

So now we have $2\cdot a =5\cdot 2\cdot 3$, which means $a=5\cdot 3$, and $3\cdot b=7\cdot 2\cdot 3$, which means $b=7\cdot 2$. So:

$$\frac{5}{7}:\frac{2}{3}=\frac{5\cdot 3}{7\cdot 2}=\frac{5}{7}\cdot\frac{3}{2}$$

Compare the leftmost and the rightmost side and you can see:

One divides by a fraction by multiplying with the reciproke of this fraction.

Let them do another example this way on their own. Let them countercheck their results by multiplication.

Important notice Use lots of colors. Every number or associated variable gets the same color.

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  • $\begingroup$ Reciproke = reciprocal? $\endgroup$ – Daniel R. Collins Sep 21 '15 at 5:51
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"Fraction Bars" may be helpful to you. It's a virtual space for exploring bar-shaped fraction models.

Fraction Bars Page

On that page you'll find a link to launch the software as well as a software guide which includes some appropriate activities.

A version of this software has been used in classes to prepare 7th grade teachers in Georgia to (among other things) teach fraction and ratio concepts using a heavily representation and inquiry-based approach. You can find some of the recommended investigations here on the InterMath website. Many of the activities are standalone. You may find the software helpful, though, for some of the investigations.

The software is free to use, with no strings and also no user support. I don't use the software in my own research (I'm writing my dissertation on a different topic), but I would still be interested to hear whether you found it of use (I am one of the software developers).

Cheers and wellwishes!

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Not sure if you're still teaching, but the really right answer to this question in terms of fraction content is to read the work of Hung-Hsi Wu.

Here's a brief summary of some practical implications.

  • Fractions are not food, nor are they shapes that you color or shade. Fractions are numbers.
  • To teach the fact that fractions are numbers, use a number line from 0 to 3 with the whole numbers marked and perhaps thirds marked as well. Students can then practice counting by whole numbers (0,1,2,3), thirds (1 third, 2 thirds, 3 thirds, 4 thirds, 5 thirds), and mixed numbers (2 thirds, 1, 1 and 1 third, 1 and 2 thirds, etc.). [This may seem trivial, but it is an epiphany for many kids.] Have them notice that $\frac{5}{3}$ and $1\frac{2}{3}$ share a point on the number line, which is why they are equal.
  • If you restrict yourself to number lines from 0 to 1 or shapes where the implied whole numbers range from 0 to 1, then students can get 100% correct answer while completely ignoring what 0 and 1 are in that context, implying that fractions are ~meaningless in their mind. That's why the number line must go from 0 to 3.
  • Explain that mathematicians need fractions to name quantities between consecutive whole numbers.
  • Continue to use the number line to show equivalent fractions, fraction and whole number comparisons, addition, subtraction, multiplication, division, etc.
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You might want to check some of virtual manipulatives sites:

National Library of Virtual Manipulatives

Virtual Manipulatives - Glencoe

Math Playground

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Arguably, fractions provide a good first-pass at thinking abstractly, without having to develop complete intuition by manipulatives. (Priming students for the algebra that comes next.) As a community college lecturer teaching this in our remedial classes, I give a single visual example of each operation, and then we need to focus on practicing the mechanics of those operations.

But one thing I do hammer on throughout the semester is estimations for double-check and self-correction purposes (which most of my students find overwhelming, baffling, and frequently never heard of before). So a lecture later there's a beautiful opportunity when multiplying/dividing mixed numbers to practice more estimation double-checks (round and operate on whole numbers), which hopefully serves to bolster confidence that the process is giving reasonable results.

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