SOLUTION: I'm trying to help my niece. How can I show 1/3+1/4 as a diagram? Thanks!

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Question 1208933: I'm trying to help my niece. How can I show 1/3+1/4 as a diagram? Thanks!
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52787)   (Show Source):
You can put this solution on YOUR website!
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        Use the watch dial.

1/3 of an hour is 20 minutes. 1/4 of an hour is 15 minutes. The sum 1/3 + 1/4 is 20 minutes + 15 minutes = 35 minutes, or 7/12 of an hour. In fractions, it corresponds to fraction addition + = work with the common denominator = = , giving you the same value as the watch dial gives visually.

        Then you can continue your explanations this way.

Draw first vertical rectangle with the horizontal base of 1 cm and vertical height of 4 cm. Draw second vertical rectangle with the horizontal base of 1 cm and vertical height of 3 cm. Draw third vertical rectangle with the horizontal base of 1 cm and vertical height of 12 cm. First rectangle represents 1/3 of the third rectangle. Second rectangle represents 1/4 of the third rectangle. +------------------------------------------------------------+ | So, first rectangle and second rectangle represent | | fractions 1/3 and 1/4, which you want to add. | +------------------------------------------------------------+ Divide/separate each of the three rectangle by horizontal segments 1 cm of height each. These small rectangles all have the same measure: this common measure is 1/12 of the measure of the great rectangle. Then you see that first rectangle is 4 times the common measure, while the second rectangle is 3 times the common measure. The sum of the first and the second rectangles is 4+3 = 7 times the common measure. This corresponds to fraction addition + = work with the common denominator = = , giving the same value as you got using the diagram.

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Solved, answered and explained.

These two explanations are just enough.

It is the way on how my grandmothers and grandfathers were taught more than 100 years ago.

Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!

Tutor ikleyn's approach of using a watch dial (i.e. analog clock face) is clever. And perhaps the best way to solve this type of problem. This assumes that schools are still teaching students how to read analog clocks in an era of increasing reliance on digital clocks instead.

One thing to point out about her solution is that 35/60 = (7*5)/(12*5) = 7/12.
Just in case if the person asking the question is wondering why 35 minutes is 7/12 of an hour.

Or you can look at it like this:
1/3 of an hour = (1/3)*60 = 20 minutes which represents the "4" on the clock
1/4 of an hour = (1/4)*60 = 15 minutes which represents the "3" on the clock
Adding the clock face values gives 4+3 = 7
There are 12 values on the clock face, so it yields 7/12.

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If your niece's teacher requires the use of round pizzas or pies for the fraction diagram, then it could look like this:

The portion in blue represents 1/3 aka 4/12.
To go from 1/3 to 4/12, multiply top and bottom by 4.
Notice each third is split into 4 pieces to get 3*4 = 12 pieces total.
What's so special about 12? It's the LCM of 3 and 4.

The yellow portion is 1/4 = 3/12.
Multiply top and bottom by 3.
Each original quarter piece is split 3 ways.

Once everything is in a common denominator (12), we can add the fractions.
Add the numerators only. Leave the denominator as 12.

1/3 + 1/4 = 4/12 + 3/12 = (4+3)/12 = 7/12

In other words,
4 blue + 3 yellow = 7 shaded pieces out of 12 total, that leads to 7/12

You can think of it like 4 pieces + 3 pieces = 7 pieces, where you'd replace each copy of "pieces" with "/12" or "twelfths".
I apologize if the diagram seems a bit cluttered.
There might be a more efficient way of displaying this.

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Another approach is to use a fraction tape diagram.

Compared to the previous method, I prefer this method better.
This is because it's easier (for me at least) to draw rectangles than to divide up a circle equally.
Here is what the diagram could look like.

The 1st row represents 1/3.
The 2nd row has us rewrite 1/3 as 4/12. Notice the vertical alignment matches perfectly to help visualize 1/3 = 4/12. Put another way: The blue 1/3 in the top row is split into 4 equal smaller pieces, and the same is done to the other unshaded pieces in the top row.
The 3rd row represents the fraction 1/4.
The 4th row has us rewriting 1/4 as 3/12.

The last row of the diagram has us adding the blue smaller pieces and yellow smaller pieces (blues from row 2, yellows from row 4).
This addition is valid because we have a common denominator (12). The color coding is the same as the previous diagram.