1. Introduction to the Adjective/Noun Theme. © 2012 Math As A Second Language All Rights Reserved next #1 Taking the Fear out of Math

2. The basis of “Math as a Second Language” is that most students see numbers as quantities. If you ask students to tell you what the number 3 is, they might hold up 3 fingers. next © 2012 Math As A Second Language All Rights Reserved

3. In other words, we have seen 3 fingers, 3 apples, 3 tally marks, etc. but never “threeness” by itself. next © 2012 Math As A Second Language All Rights Reserved

4. next A quantity is a phrase consisting of an adjective and a noun. The adjective is a number, and the noun is the unit. Definition 3 fingers is a quantity in which the adjective is 3 and the noun (unit) is fingers. next © 2012 Math As A Second Language All Rights Reserved

5. next In a similar way, 3 inches is a quantity in which the adjective is 3, and the noun (unit) is inches. Definition As quantities, 3 fingers is not the same as 3 inches. However, as adjectives, the “3” in “3 fingers” means the same thing as the “3” in “3 inches”. Key Point © 2012 Math As A Second Language All Rights Reserved

6. next Definition Hence, at least in English grammar, it is rather vague for someone to say “This is a blue”. The above concept transcends mathematics. Although a blue pencil doesn’t look like a blue sweater, the adjective “blue” means the same thing in each case. © 2012 Math As A Second Language All Rights Reserved

7. next With the above concept in mind, our innovative approach to teaching basic mathematics, which we call “Mathematics as a Second Language”, is to introduce numbers in the same way that people from all walks of life use them; namely as adjectives that modify nouns. Our technique is to show that by using this concept, all of basic arithmetic can be done by just knowing the addition and multiplication tables from 0 through 9. © 2012 Math As A Second Language All Rights Reserved

8. next The greatest obstacle to this approach is the tendency for presenting numbers to students only in the form of adjectives. That is, we often will talk about 3 without reference to what noun 3 is modifying. 1 © 2012 Math As A Second Language All Rights Reserved note 1 In our opinion it is amazing how much clearer the various computations in both arithmetic and algebra become when students are allowed to visualize the adjectives as modifying nouns of their own choosing. Since the noun is usually omitted, we have to understand a few things about quantities. next

9. © 2012 Math As A Second Language All Rights Reserved When we write such apparently simple statements as 1 = 1, we are assuming that the 1 on one side of the equal sign is modifying the same noun as the 1 on the other side of the equal sign. First…

10. next © 2012 Math As A Second Language All Rights Reserved 1 inch ≠ 1 mile 2, next note 2 To negate a relationship, it is a common mathematical procedure to put a “slash mark” through the symbol that expresses the relationship. Thus, to negate a statement such as b = c, we would write b ≠ c, which we read as “b is not equal to c” or “b is unequal to c”. even though as an adjective the 1 that modifies “inch” means the same thing as the 1 that modifies “mile”. Secondly…

11. next © 2012 Math As A Second Language All Rights Reserved On the other hand, as adjectives 12 ≠ 1, but it is true that 12 inches = 1 foot. There are other interesting things that occur when we study the arithmetic of quantities that we will mention briefly here but explore in greater detail as the course unfolds. Thirdly…

12. next When we write that 3 + 2 = 5, we are assuming that 3, 2, and 5 are modifying the same noun. © 2012 Math As A Second Language All Rights Reserved note 3 Of course if the nouns are present, it is possible that 3 + 2 = 5 even if the nouns aren’t all the same. For example, 3 dimes + 2 nickels = 5 coins. However, if we are thinking in terms of the amount of money, 5 coins doesn’t mean the same things as 3 dimes and 2 nickels. On the other hand, if we are thinking in terms of the number of coins it does make sense to replace “dimes” and “nickels” by “coins” and write 3 coins + 2 coins = 5 coins. 3 dimes + 2 nickels = 40 cents, but as adjectives it is false that 3 + 2 = 40. 3 next

13. Why it is Important! For example, young students might be overwhelmed by an addition problem such as 3,000,000,000 + 2,000,000,000 because of the number of digits. The fact that 3 + 2 = 5 whenever 3, 2, and 5 modify the same noun is extremely important because it can be used to explain many things in a simple manner. © 2012 Math As A Second Language All Rights Reserved

14. next Based on how we add quantities, one does not have to know what a gloog is to know that… 3 gloogs + 2 gloogs = 5 gloogs. However, this problem is simply the place value version of 3 billion + 2 billion for which the answer is 5 billion because the 3, 2, and 5 are each modifying “billion”. © 2012 Math As A Second Language All Rights Reserved

15. next In demonstrating that… 3 dimes + 2 nickels = 40 cents, we changed dimes and nickels to a common denomination (cents). Something similar to this occurs in a beginning algebra course when students are asked to simplify 3x + 2x. We do not have to know what number x represents in order to know that 3 of them plus 2 more of them is 5 of them. © 2012 Math As A Second Language All Rights Reserved

16. next For example, to add 3/7 and 2/5, think of the problem as being written in the form 3 sevenths + 2 fifths. The same thing happens when we add fractions. © 2012 Math As A Second Language All Rights Reserved We cannot add the 3 and the 2 because they are modifying different units (sevenths and fifths). next

17. On a report card if you got 3 A’s and 2 B’s you do not say that you got 5 AB’s. You simply say that you got 3 A’s and 2 B’s. 4 © 2012 Math As A Second Language All Rights Reserved note 4 Schools have solved the problem of adding A’s and B’s by going to a 4.0 grade point scale. An A is worth 4 points and a B is worth 3 points. Without going into how the computation is formed, the student with 3 A’s and 2 B’s gets a GPA (grade point average) of 3.6. Report Card A Social Studies AScience AMathematics BLanguage BReading 4321 Grading Period

18. next 3 feet × 2 pounds = 6 foot pounds 5 The statement 3 × 2 = 6 is always true, but what the 6 modifies depends on what the 3 and the 2 are modifying. © 2012 Math As A Second Language All Rights Reserved 3 kilowatts × 2 hours = 6 kilowatt hours 3 hundred × 2 thousand = 6 hundred thousand note 5 When we multiply 2 quantities, we multiply the two adjectives (numbers) to obtain the adjective part of the product, and we multiply the 2 nouns (which we do my writing them side by side) to obtain the noun part of the product. next

19. Why it is Important! …students mechanically multiply the 3 by the 2 to obtain 6 and then annex the total number of 0’s to obtain 600,000. In doing multiplication problems of the form… 300 × 2,000 © 2012 Math As A Second Language All Rights Reserved However, as seen above, our adjective/noun theme gives us the answer in an easy to understand format.

20. next In terms of our adjective/noun theme, the reason is that the numerators are the adjectives and the denominators are the nouns. 6 In multiplying two fractions, we multiply the two numerators to obtain the numerator of the product, and we multiply the two denominators to obtain the denominator of the product. © 2012 Math As A Second Language All Rights Reserved note 6 The rule for multiplying two fractions might seem “self evident”. However, the “rule” doesn’t work when we add two fractions. Namely, we can only add the numerators (i.e., the adjectives) if they modify the same noun (i.e., denominator). next

21. However, using our above “rule”, when we multiply 3x by 2y, we multiply 3 by 2 to obtain 6 and we multiply x and y (which we may view as the nouns) by writing them side by side. In algebra, if we are given a problem such as 3x + 2y, students often want to add the 3 and 2, not recognizing that the 3 is modifying x and the 2 is modifying y. © 2012 Math As A Second Language All Rights Reserved next In other words… 3x + 2y ≠ 5xy, but 3x × 2y = 6xy.

22. It follows rather simply that… 3 tens + 2 tens = 5 tens. A rather nice way to have students see the difference between adding and multiplying is to have them compare how we add 3 tens and 2 tens with how we multiply 3 tens by 2 tens. © 2012 Math As A Second Language All Rights Reserved next However, 3 tens × 2 tens ≠ 6 tens. Rather, 3 tens × 2 tens = 6 “ten tens”.

23. And since ten tens is equal to a hundred we see that… 3 tens × 2 tens = 6 hundreds. 7 According to our rule, (multiply the adjectives and multiply the nouns)… 3 tens × 2 tens = 6 “ten tens”. © 2012 Math As A Second Language All Rights Reserved next note 7 Don’t confuse 3 tens × 2 tens with 3 × 2 tens. If we take 2 tens, 3 times (that is 3 × 2 tens) the answer is 6 tens. However, 3 tens × 2 tens = 30 × 20 = 600 = 6 hundred.