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This Lesson (Checking Equality) was created by by math_tutor2020(3822)
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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. ---------------------------------------------------------------- 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. ---------------------------------------------------------------- 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. ---------------------------------------------------------------- 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. ---------------------------------------------------------------- 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. ---------------------------------------------------------------- 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). ---------------------------------------------------------------- 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 https://www.algebra.com/algebra/homework/Exponents/Exponents.faq.question.1204752.html This lesson has been accessed 659 times. |
