Lesson Checking Equality

Source code of 'Checking Equality'

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A classmate claims that 4/7 and 7/12 are equal.
How can we prove or disprove this claim?

I'll go over 4 methods.

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Method 1

Rewrite each fraction so that they have the LCD 7*12 = 84.
4/7 = (4/7)*(12/12) = 48/84
7/12 = (7/12)*(7/7) = 49/84

In short
4/7  = 48/84
7/12 = 49/84

Imagine somehow you were able to cut a pizza into 84 slices.
48 slices represent 48/84 = 4/7 of the pizza.
49 slices represent 49/84 = 7/12 of the pizza.

The two fractions are NOT equal since we have a different number of slices.
Therefore, the classmate is incorrect with his/her claim.

Another way to represent fractions is using a tape diagram which is perhaps a better method depending on the student's preference.
Or you can split up a rectangle into a grid of smaller rectangles.

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Method 2

The cross multiplication rule has us go from A/B = C/D to A*D = B*C

Use that rule to have these steps
4/7 = 7/12
4*12 = 7*7
48 = 49
The two sides are close, but they do not match perfectly.

The last equation being false leads to 4/7 = 7/12 being false.

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Method 3

Use long division or a calculator to find these decimal values
4/7 = 0.57142857142858
7/12 = 0.58333333333333
Each value is approximate.

Let's say we rounded each to 3 decimal places
4/7 = 0.571
7/12 = 0.583
It becomes immediately clear that while the two fractions are close, they are not the exact same.

Therefore, 4/7 = 7/12 is false.

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Method 4

If A = B then A-B = 0.
This rule is very useful to check equality of two values. 
The two values A and B do not have to be fractions.

Grab a calculator to type in 4/7 - 7/12
The approximate result would be -0.01190476190477 depending how your calculator rounds the display.

Since 4/7 - 7/12 = 0 is false, it leads to 4/7 = 7/12 being false.

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Let's elaborate further on method 4.

This simple idea of A = B leading to A-B = 0 is very powerful to check equality of other math objects (not just fractions).

For example, let's say your teacher wrote on the board that 2*sqrt(2)+3*sqrt(2) = 6*sqrt(2)
where "sqrt" means "square root".
At first glance the equation may look correct. 
But how can we check this claim?

By using A-B = 0 of course.
A = 2*sqrt(2)+3*sqrt(2)
B = 6*sqrt(2)
Type that into the calculator to find those items subtract to -1.4142135623731 approximately.
The difference is nowhere close to 0.
We cannot chalk this up to rounding error. 
The two values A and B are not the same.

Instead it should be 2*sqrt(2)+3*sqrt(2) = 5*sqrt(2)
Subtract left and right hand sides to find the difference is 0 or very very close to it (due to slight rounding error).

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Extending the idea further to check functions.

We've established we can check equality of numbers. 
Now we can check whether two functions are the same or not.

2+3 = 5 leads to 2+3-5 = 0
2x+3x = 5x leads to 2x+3x-5x = 0

So the general template would be
f(x) = g(x) leads to f(x)-g(x) = 0

With this section, a graphing calculator is needed.
The idea is to plot the curve f(x)-g(x) and see if it produces a flat horizontal line over the x axis. Use a different color for the curve compared to the axis if possible.

Or you can produce a table of values. A function curve is simply a visual representation of the collection of (x,y) points.
If for every x input the y input is 0 (or very close to it), then f(x)-g(x) = 0 is true and it leads back to f(x) = g(x) being true for all x in the domain.

An example of this in action is shown in method 4 of this problem
<a href="https://www.algebra.com/algebra/homework/Exponents/Exponents.faq.question.1204752.html">https://www.algebra.com/algebra/homework/Exponents/Exponents.faq.question.1204752.html</a>