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# #1 t quantities vs. numbers adjective noun

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Stressing the importance of using the concept of quantities in arithmetic. We call the number part of a quantity the adjective and the unit is our noun. For addition, using real-world examples, it is necessary for two quantities to have the same noun if you want to add them together. We illustrate this concept with, for examples, different denominations.
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### #1 t quantities vs. numbers adjective noun

1. 1. What’s My Noun? Adjective/Noun next Taking the Fear out of Math © Math As A Second Language All Rights Reserved Quantities vs. Numbers
2. 2. In our previous lesson, we emphasized the importance of this fact… nextnext Even though addition tables tell us that 3 + 2 = 5, the fact is that 3 dimes plus 2 nickels is neither 5 dimes nor 5 nickels. next © Math As A Second Language All Rights Reserved When we add two quantities the correct result is the sum of the adjectives (numbers) only if the nouns (the units) are the same.
3. 3. nextnext To find the amount of money, we converted both quantities to a common unit and then added. 3 dimes + 2 nickels = 30 cents + 40 cents nextnext 10 cents = next © Math As A Second Language All Rights Reserved
4. 4. nextnext For example, the value of 2 nickels is 1 dime. Hence, we may restate the problem in the form… 3 dimes + 2 nickels = 3 dimes + 1 dime = 4 dimes1 Note… There may be more than one common unit. note 1 In our adjective/noun theme we do not distinguish between singular and plural. The fact is that while “dime” and “dimes” are different nouns they represent the same unit. next © Math As A Second Language All Rights Reserved
5. 5. nextnext And even though 40, 4, and 8 are different adjectives (numbers), 40 cents, 4 dimes and 8 nickels all describe the same quantity. 3 dimes + 2 nickels = 6 nickels + 2 nickels = 8 nickels We could also have used nickels as the common unit, in which case we could have replaced 3 dimes by 6 nickels to obtain… © Math As A Second Language All Rights Reserved
6. 6. A Preview of Coming Attractions At first glance, our emphasis on the adjective/noun theme might seem like little more than just a novelty, but as we will see throughout the study of arithmetic, this theme can greatly improve students’ ability to internalize all of arithmetic. next © Math As A Second Language All Rights Reserved
7. 7. We will discuss this in greater detail as the course progresses, but for now let’s focus on just one aspect of how the adjective/noun theme simplifies arithmetic algorithms that often befuddle students. next © Math As A Second Language All Rights Reserved
8. 8. nextnext Too often the “numerator” is introduced as a synonym for “top” and “denominator” as a synonym for “bottom”. Note… On the Terms Numerator and Denominator © Math As A Second Language All Rights Reserved This obscures the fact that the numerator is the adjective and the denominator is the noun and leaves many students confused when they are asked to add fractions. top bottom numerator denominator adjective noun = =
9. 9. nextnext When adding two fractions, students feel it is more natural to add the two numerators to obtain the numerator of the sum and to add the two denominators to obtain the denominator of the sum. For example, they would prefer that adding 1 /2 + 1 /2 would mean to do the following… © Math As A Second Language All Rights Reserved 2 4 1 2 1 2 + = 1 + 1 2 + 2 = …which is a result that they will most likely recognize as being incorrect”.
10. 10. nextnext However, once the proper definitions are given for numerator and denominator, these students will quickly realize that this is not the correct way to add fractions. Namely, when they are called upon to compute a sum such as 6 nickels + 2 nickels, they would add the two adjectives (6 + 2) but then keep the common denomination (nickels). Even though it is true that a nickel and a nickel is a dime, in no way would they have felt that the answer was 8 dimes. © Math As A Second Language All Rights Reserved
11. 11. Guess My Noun next Guess My Noun is a fun way to reinforce the notion of the adjective/noun theme and how 3 + 2 = 40 can be a true statement. next For example, if there are certain facts you want the students to know (such as the fact that 7 days = 1 week) you might ask them to supply the nouns for… 7 ______ = 1 ______. © Math As A Second Language All Rights Reserved
12. 12. next If we want to emphasize that there are 12 months in a year, fill in the blanks for the missing nouns in… 12 ________ = 1 ______. Notice that there can be more than one correct answer. next 12 inches = 1 foot nextnextnext months year 12 eggs = 1 dozen eggs © Math As A Second Language All Rights Reserved
13. 13. next You might wonder why 12 was chosen rather than 10 for the number of inches in a foot. Such a question can lead to the “discovery” of whole number fractional parts. next For example, since “teen” means plus ten, one might naturally assume that the first teen should come after ten. That is, the number we call eleven should have been called “oneteen”. next © Math As A Second Language All Rights Reserved So why does the first teen come after twelve not ten?
14. 14. next As surprising as it might seem, the concept of ten was not considered to be important until the advent of place value. next Until that time, people preferred to avoid the need for using fractions whenever possible. Therefore, since 12 had more proper divisors than 10, it meant that by having a foot consist of 12 inches, more fractional parts of a foot would be a whole number than if there had been 10 inches in a foot. © Math As A Second Language All Rights Reserved
15. 15. next Aside from the “folk lore” values of these examples, it might be reassuring to students for them to know that hundreds of years ago people were already learning how to invent nouns that would minimize the need for using fractions. © Math As A Second Language All Rights Reserved
16. 16. next You may prefer additional examples, and you should feel free to create problems of your own choosing. Students also may want to create their own problems to challenge their classmates. next © Math As A Second Language All Rights Reserved Depending on the grade level you can ask more difficult questions by having the students add different quantities such as… 2 _____ + 12 ______ = 1 ______feet inches yard next
17. 17. next Examples such as 3 + 2 = 40 are too sophisticated for children in grades K – 2. Instead, colored rods of different lengths may help to reinforce the adjective/noun theme at the lower grade levels. nextnext © Math As A Second Language All Rights Reserved
18. 18. next From a different perspective, you could see that… next 1 blue rod = 12 red rods 1 blue rod = 6 green rods next © Math As A Second Language All Rights Reserved
19. 19. nextnext 1 blue rod = 4 yellow rods 1 blue rod = 3 white rods next 1 blue rod = 2 purple rods © Math As A Second Language All Rights Reserved
20. 20. next Therefore… next 2=1 3=1 4=1 6= 1 nextnext © Math As A Second Language All Rights Reserved
21. 21. nextnext 12 1 2 3 4 6 6 4 4 3 3 3 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 In summary… nextnextnextnextnextnext © Math As A Second Language All Rights Reserved
22. 22. next You could give your students examples in adding quantities such as… next 1 white rod 1 purple rod+ 1 green rod = nextnext In later grades, the above equality could become a visual model for showing that 1 /3 + 1 /6 = 1 /2 (that is, 1 third + 1 sixth = 1 half). nextnext © Math As A Second Language All Rights Reserved
23. 23. next Another example might be… next 5 red rods 3 yellow rods+ 1 white rod = nextnext (that is, 5 twelfths + 1 third = 3 fourths) nextnext © Math As A Second Language All Rights Reserved
24. 24. next If time is limited, assign them as homework under the heading of such phrases as “Fun With Math” and make sure that students know that it is just for fun and that they will not be graded. Rather they should be encouraged to work on the problems and share their results with the class (as time permits). Guess My Noun © Math As A Second Language All Rights Reserved
25. 25. Guess My Noun next We know that you are under pressure to cover a certain amount of prescribed content, and that as a result you may feel that there is no time for such “games” in your class. However, our approach to helping students to better internalize mathematics hinges on their thorough grasp of adding quantities using the adjective/noun theme, and the previous examples are fun ways in which to learn. © Math As A Second Language All Rights Reserved