Explain Equivalent Fractions

Doable deal on debt is right in front of us

The Hippocratic injunction should be something befuddled economists and warring politicians can agree on. With the nation struggling to recover from a devastating recession, unemployment stuck at crisis levels, financial markets spooked by the possibility of European defaults and consumers disinclined to consume, it makes no earthly sense to suck money out of the economy. As a practical matter, this means Republicans must swallow an increase in the debt ceiling and Democrats must accept painful spending curbs that kick in when the economy is off its sickbed. It means conservatives have to be patient in bringing expenditures down and progressives have to be patient in returning tax rates - even for the wealthy - to what many of us consider appropriate levels. Two years ago at a seminar, I heard a distinguished economic forecaster confidently explain how the recovery would proceed. While some usually reliable indicators were anomalous and contradictory, he said, the one thing he knew from the historical record was that sharp, deep recessions are followed by steep, roaring recoveries. By the second quarter of 2010, he said, growth would be as high as 4 percent and unemployment would be tumbling. Happy days would be here again.

Heavy Equipment Mechanic » U Teach Your Child Fractions 6: Addition

Mary mows her rear lawn in three hours. With a comparable mower, her younger brother can complete the job in four hours. After the lawn is mowed, Mary and her brother plan to see the fireworks display in the park, a ten minutes walk from their house. Mother has granted them permission to go to the park after the lawn is mowed. If Mary and her brother work together, will they be able to see the start of the fireworks display? The firework display starts two hours after the mowing begins.

This discussion centers on  the presentation and solution of this problem.

What is needed? Counting is one of the first operations a child learns. When she recites, for the first time, one, two, three, etc…, our faces light up with pride. Before long, she masters the counting procedure having learned to  point to an object and assign a number from the sequence, 1, 2, 3, … ; stopping after she assigns to the last object a number. For small counting chores, this method is very effective. However, when counting a large number of objects or counting objects from two or more large piles, this method is time-consuming and often ineffective.

We accelerate the counting process with addition. Addition allows us to count without physically pairing the numbers 1, 2, 3, … etc.  to each objects. We only need to know how many are in each pile in order to get the total.

How is it done?

Let’s take our lead from a child. A young boy describes to his friends the method he used to win his marbles. Boasting he says, ”I won 3 shooters and 12 marbles. Marbles are non-shooters. Shooters are special, so he counts them separately. What’s going on? The boy is counting the marbles that are alike in different piles. But sometimes we must group objects and count the contents of the group.

Now consider this example. The proprietor of a small pest control company owns 12 light trucks for his technicians and 4 cars for his salesmen.  Each truck has pest fighting equipment. When asked the size of his mobile inventory, he reclassifies the trucks and the cars in a common category with the answer, “I have 16 vehicles.”  The word, vehicle, strips the trucks and the cars of their designated function and places them in a broader category as things that are alike — namely motor vehicles. Once they are reclassified into a common category, he counted them. That is, they can be added.

These two examples highlight two principles in fraction arithmetic.

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