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I asked this question in Math.S.E. but got no answer!

I'm going to introduce my middle school students to the distributive law in arithmetic, in a meaningful way so that they understand its importance and value.

I need some interesting examples and problems like the one below. Do you know any such example?

My example: In this figure, the solid arcs make a path. How long is it? Find the answer in less than one minute! No calculator allowed! $\pi$=3.14$.

enter image description here

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    $\begingroup$ Area of a split rectangle. $\endgroup$ – JP McCarthy Oct 31 '14 at 12:25
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People who are competent at mental arithmetic routinely use the distributive law when doing multiplication. If asked to evaluate $38\cdot14$, I would immediately break it up as $38\cdot10+38\cdot4=380+38\cdot2\cdot2$.

However, I think it's more important to focus on the meaning of the distributive law rather than on its applications. The meaning is of course that if I have $a$ of something in one hand and $b$ in the other, than I have $a+b$ in total, and this applies even if that "something" is $x$ of some object, thus we conclude that $ax+bx=(a+b)x$.

Maybe you're worried that once this is explained, students will feel it's so obvious that it couldn't possibly be of any use. Even in the mental arithmetic example I gave above, you don't have to be explicitly aware of the symbolic law $ax+bx=(a+b)x$ to apply that kind of reasoning, most people do it without thinking. You only really appreciate the value of the law when you do algebra.

I think most students should be able to solve equations of the form $ax=b$ even before taking algebra. Until you've taken algebra, though, an equation like:

$$7(5x+6)+2x=3x+18(5x+3)$$

seems utterly intractable. What's remarkable is that through simple application of distributivity, the above reduces to the form $ax=b$ and can then immediately be solved.

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Nothing calls for students to respect the law of distribution more than provide them with arithmetic problems that conduct their lives in school. You can simply build many of the issues that seem to make the distribution more feasible to perform routine procedure, such as:

6 × 15 = 6 × ( 10+5 ) = 60 + 30 = 90

also 24 × 98 = 24 × ( 100 - 2 ) = 2400- 48 =2352

or 13 × 103 = 13 × ( 100 + 3)= 1300 + 39 = 1339

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