Rates © Math As A Second Language All Rights Reserved next #6 Taking the Fear out of Math miles per hour

next © Math As A Second Language All Rights Reserved Introduction next As an introduction to this section, consider the following exchange of words between a state trooper and a motorist. State Trooper: Sir, do you realize that you were driving at a speed of 75 miles per hour? Motorist: That’s impossible! I’ve only been driving for 20 minutes.

© Math As A Second Language All Rights Reserved The point of the previous story is that speed is neither a distance nor a time. It is a rate. next For example, knowing that a driver’s speed was 75 miles per hour does not tell you how far he drove or how long he was driving. It just tells you that if he maintained that constant speed, he would travel 75 miles each hour. 1 note 1 This is true with respect to any rate. For example, if apples are being sold for 80 cents per pound, it just means that every time you buy another pound it costs you another 80 cents. It does not tell you how many pounds you bought nor does it tell you how much you spent. next

© Math As A Second Language All Rights Reserved An easy way to recognize a rate is that it is usually expressed as a phrase that consists of two nouns, separated by the word “per”. next Thus, for example, suppose you are buying pens that are being sold at a rate of 2 for 5 dollars (that is, 2 pens per 5 dollars). This tells us that every time you spend another 5 dollars you get 2 more pens.

© Math As A Second Language All Rights Reserved Suppose now that two types of pens are on sale. One type is being sold at a rate of 2 pens for $5 and the other type is being sold at a rate of 3 pens for $8. next

© Math As A Second Language All Rights Reserved next We want to decide whether 2 pens for $5 is a better bargain than 3 pens for $8. Some students may decide that 3 pens for $8 is the better bargain because 3 pens is more pens than 2 pens. Should that happen, it should be pointed out that the $8 you must spend for the 3 pens is more than the $5 you spend for the 2 pens. Hopefully, this will help the students understand the concept of unit pricing.

© Math As A Second Language All Rights Reserved next One way to explain unit pricing is to point out to the students that it is only fair to compare the number of pens one can buy for a certain amount of money if the two amounts of money are the same. To do this, they should recall that 40 is a common multiple of 5 and 8, 5, 10, 15, 20, 25, 30, 35, 40, 45… 8, 16, 32, 40, 48, 56, … and therefore, the comparison should be based on spending $40. next

© Math As A Second Language All Rights Reserved next An easy way to help students visualize the situation is that since 5 × 8 = 8 × 5 = 40, if 8 people each buy 2 pens for $5, collectively they will have purchased 16 pens for $40. That is, each of the 8 people pays $5 and receives 2 pens in exchange. In other words, a rate of 2 pens for $5 is the same rate as 16 pens for $40 And if 5 people each buy 3 pens for $8, each of the five will receive 3 pens. That is, a rate of 3 pens for $8, is the same rate as 15 pens for $40.

© Math As A Second Language All Rights Reserved next The above discussion can easily be translated into a direct connection with equivalent fractions. Suppose that we had not yet developed the language of fractions. We could have decided to represent “pens per dollar” by writing pens / dollar. In this way, we would write 2 pens for $5 as 2 pens / 5 dollars.

© Math As A Second Language All Rights Reserved next To indicate that if we spend $5 eight times, we would get 2 pens eight times; we could write… If we now omit the nouns in the above string of equations, we obtain our previous result… 5 dollars 2 pens 5 dollars × 8 2 pens × 8 == 40 dollars 16 pens × 8 2 × 8 == next

© Math As A Second Language All Rights Reserved next Although the number line is a better model than tally marks for visualizing fractions, the fact is that students may feel more comfortable using tally marks. Therefore, students can visualize equivalent common fractions in terms of tally marks. Visualizing Fractions Using Tally Marks

next © Math As A Second Language All Rights Reserved next In this interpretation we view 2 / 5 as meaning that we are taking 2 tally marks from each group of 5. We may picture this as… Thus, every time we have 5 more tally marks we take 2 of them.

© Math As A Second Language All Rights Reserved next Thus, for example, we see that… 2 out of 5 Do you see the pattern? 4 out of 10 6 out of 15 8 out of 20

© Math As A Second Language All Rights Reserved next Every time we add another row of 5 tally marks, we cross out 2 of them. Thus, if we write 12 rows, we will have 12 rows with 5 tally marks each and we will have crossed out 2 tally marks from each of the 12 rows. In the language of common fractions… × 12 2 × 12 == 60 24

next © Math As A Second Language All Rights Reserved next Very often students internalize the above steps by rote memory. Notice however, how much more real this becomes when we visualize it as at a rate of 2 pens for $5 one can buy 24 pens for $ × 122 × 12 == 6024 More generally, introducing the concept of rates “personifies” the concept of what a fraction really means.

next © Math As A Second Language All Rights Reserved Using Equivalent Fractions to Compare Rates next Suppose, for example, that we wanted to compare the size of 2 / 5 and 3 / 8. Our last example, gives us a clue as to how we can compare two rates in the same way that we compared the size of two fractions. 2 note 2 The word “size" is a bit vague. A more precise way is to recognize that a common fraction names a rational number. Thus, when we say that one fraction is greater than another we mean that it represents a greater rational number.

next © Math As A Second Language All Rights Reserved next To compare the size of the two adjectives, we have to assume that they modify the same noun. For example, we know that 3 dimes is more money than 4 nickels, but 3 is not greater than 4. Let’s see how this works in terms of fractional parts of a circle.

© Math As A Second Language All Rights Reserved next 360 has many divisors, which is why the ancient Greeks decided that it would be nice to divide a circle into 360 equal parts, each of which was called 1 degree (1 o ). 5 and 8 are among these divisors.

© Math As A Second Language All Rights Reserved next Since 1 / 5 of a circle is 72 o (that is, 360 o ÷5), we see that… 2 / 5 of a circle is equal to 144°. 72 o 0o0o 144 o

next © Math As A Second Language All Rights Reserved next Since 1 / 8 of a circle is 45 o (that is, 360 o ÷8) we see that… 2 / 8 of a circle is equal to 90 o. 45 o 3 / 8 of a circle is equal to 135 o. next 90 o 135 o 0o0o

next © Math As A Second Language All Rights Reserved next In other words, 2 / 5 and 3 / 8 are fractional parts of a circle, but they are a whole number of degrees. And while it may not be obvious that 2 / 5 of a circle is greater than 3 / 8 of a circle, it is obvious that 144 o exceeds 135 o by 9 o. 45 o 90 o 135 o 0o0o 72 o 0o0o 144 o

next © Math As A Second Language All Rights Reserved next While it is convenient to think in terms of 360, it isn’t necessary. Notes For example, a smaller common multiple of 5 and 8 is 40. That is, we may think of a corn bread as being sliced into 40 equally sized pieces. corn bread

next © Math As A Second Language All Rights Reserved next 2 / 5 of the corn bread = 2 / 5 of 40 pieces = [(40 ÷ 5) × 2] pieces = 16 pieces

© Math As A Second Language All Rights Reserved next And 3 / 8 of the corn bread = 3 / 8 of 40 pieces = [(40 ÷ 8) × 3] pieces = 15 pieces

© Math As A Second Language All Rights Reserved next Comparing the results, we see that if the corn bread is sliced into 40 equally sized pieces, 2 / 5 of the corn bread exceeds 3 / 8 of the corn bread by 1 piece. 3/83/8 2/52/5 15 / / 40

next © Math As A Second Language All Rights Reserved next We could have obtained the same result without having to refer to a corn bread. In terms of equivalent fractions, since 5 × 8 = 40 we can multiply numerator and denominator of 2 / 5 by 8 and the numerator and denominator of 3 / 8 by 5 to obtain… × 8 2 × 8 == × 5 3 × 5 == next 16 fortieths exceeds 15 fortieths by 1 fortieth.

© Math As A Second Language All Rights Reserved next Not only is 2 / 5 greater than 3 / 8, but it exceeds it by 1 part per In terms of rates, for example, if you buy pens at the rate of 2 pens for $5, then for every $40 you spend, you will get one more pen than you would have gotten if you had bought the pens at a rate of 3 pens for $8. note 3 Stated in a mathematical form, we have shown that 2 / 5 – 3 / 8 = 1 / 40. But notice how much more intuitive it would be for beginning students to visualize this in the form of 16 pieces exceed 15 pieces by 1 piece. next

© Math As A Second Language All Rights Reserved next 40 is the least common multiple of 5 and 8. However, any multiple of 40 will also be a common multiple of 5 and 8. In the particular case of a circle we used 360 as the common multiple and in the language of equivalent fractions what we showed was… × 72 2 × 72 == × 45 3 × 45 == next

© Math As A Second Language All Rights Reserved next Since 144 (360ths) exceeds 135 (360ths) by 9 (360ths), we see that 2 / 5 exceeds 3 / 8 by 9 / 360. While 1 / 40 and 9 / 360 may not look alike, they are equivalent because… × 9 1 × 9 == 360 9

next © Math As A Second Language All Rights Reserved next To Compare Two or More Common Fractions If the common fractions have the same denominator, the one with the greater numerator names the greater rational number. For example, since 3 is greater than 2, 3 / 7 is greater than 2 / 7. next

© Math As A Second Language All Rights Reserved If they have different denominators, replace them by equivalent common fractions that have the same denominators. Then, the one with the greater numerator names the greater rational number. For example, to compare 7 / 9 with 8 / 11, observe that 99 is a common multiple of 9 and 11. Hence, we may rewrite… next × 11 7 × 11 == × 9 8 × 9 == next

© Math As A Second Language All Rights Reserved To see what we just did is equivalent to how we compare rates, suppose Store A is selling pens at a rate of 7 for $9, and Store B is selling the same pen at a rate of 8 for $11. What we have shown is that for every $99 you spend, you get 5 more pens by buying them at Store A than if you had bought them at Store B. Since 77 is 5 more than 72, 7 / 9 is greater than 8 / 11 by 5 / 99. next

© Math As A Second Language All Rights Reserved Suppose now that Store A is selling pens at a rate of $7 for 9 pens instead of 7 pens for $9, and Store B is selling pens at a rate of $8 for 11 pens instead of 8 pens for $11. Using the same arithmetic, we once again see that 7 / 9 is greater than 8 / 11 by 5 / 99. This time, however, 77 / 99 means $77 for 99 pens, and 72 / 99 means $72 for 99 pens. This means that for every 99 pens you buy, you would spend $5 less at Store B. next Caution

next © Math As A Second Language All Rights Reserved In more detail… next Thus, we see the importance of reading comprehension in distinguishing dollars / pen from pens / dollar. In other words, the more dollars per pen, the less pens per dollar. 9 pens 7 dollars 9 pens × 11 7 dollars × 11 == 99 pens 77 dollars 11pens × 9 8 dollars × 9 = = 99 pens 72 dollars 11 pens 8 dollars next

We will now look at addition and subtraction of fraction in the next presentation. © Math As A Second Language All Rights Reserved Rates 50 mph