1. Multiplying Mixed Numbers © Math As A Second Language All Rights Reserved next #7 Taking the Fear out of Math 1313 3 1212 2 ×

2. Let’s again begin by using a “real world” example… © Math As A Second Language All Rights Reserved next How much will you have to pay for candy in order to buy 2 1 / 2 pounds, if the candy costs $4.50 per pound?

3. next © Math As A Second Language All Rights Reserved next Stated in its present form, the problem is a “simple” arithmetic problem. Namely, at $4.50 per pound, 2 pounds would cost $9 and a half pound would cost half of $4.50 or $2.25. Therefore, the total cost is $11.25. Notice that in the language of mixed numbers, we have found the answer to… 2 1 / 2 pounds × 4 1 / 2 dollars per pound.

4. © Math As A Second Language All Rights Reserved next Notice that because the plus sign is “missing”, it is easy to overlook the fact that when we multiply 2 1 / 2 by 4 1 / 2, we must use the distributive property. For example, there is a tendency by students to multiply the two whole numbers to get 8 and the two fractions to get 1 / 4. 1 However, since 2 1 / 2 is greater than 2, we know that 2 1 / 2 × 4 1 / 2 is greater than 2 × 4 1 / 2, and since 2 × 4 1 / 2 = 9, we know that 2 1 / 2 × 4 1 / 2 is greater than 9. note 1 Learning by rote presents a tendency to confuse how we multiply two mixed numbers with how we add two mixed numbers. next

5. ► This validates our observation that the product of 2 1 / 2 and 4 1 / 2 is greater than 9. © Math As A Second Language All Rights Reserved Notes next We have multiplied 2 1 / 2 pounds by 4 1 / 2 dollars per pound and obtained 11 1 / 4 dollars as the product. next ► In addition to the fact that our result validates that the product is greater than 9, we have learned to use the distributive property to get the exact answer (4 1 / 2 × 2 1 / 2 = 11 1 / 4 ).

6. ► Notice that if you were the store owner and believed that 2 1 / 2 × 4 1 / 2 = $8 1 / 4, you would have “short changed” yourself by $3 on this transaction. © Math As A Second Language All Rights Reserved Notes next ► So even if it does seem “natural” or “logical” to multiply the whole numbers and multiply the fractions; it just doesn’t work in the real world!

7. © Math As A Second Language All Rights Reserved next Whether it is more difficult to multiply the “correct” way is not the issue. The issue is that if we want to multiply mixed numbers, we have to pay attention to the distributive property. (2 1 / 2 × 4 1 / 2 ) next = (2 × 4) + (2 × 1 / 2 )+ ( 1 / 2 × 4) + ( 1 / 2 × 1 / 2 ) = 8 + 1 + 2 + 1 / 4 = 11 1 / 4

8. next © Math As A Second Language All Rights Reserved next In schematic form, we may represent the distributive property as follows… 2 1/21/2 1/41/4 + × 4 8 × 1/21/2 + × 1 2

9. © Math As A Second Language All Rights Reserved next Notice that the area model may be used to help visualize the distributive property (just as we did in our discussion of whole number multiplication). 2 1 / 2 4 1 / 2 2 × 4 = 82 × 1 / 2 = 1 4 × 1 / 2 = 2 1 / 2 × 1 / 2 = 1 / 4 next 9 + 2 1 / 4 10+11/411/4 11 1 / 4 = next

10. © Math As A Second Language All Rights Reserved next The “big” rectangle has dimensions 4 1 / 2 by 2 1 / 2. 2 1/21/2 4 1/21/2 2 × 4 = 82 × 1 / 2 = 1 4 × 1 / 2 = 2 1 / 2 × 1 / 2 = 1 / 4 next Its area is the sum of the areas of the four smaller rectangles inside. That is, the total area is 8 + 1 + 2 + 1 / 4 = 11 1 / 4.

11. © Math As A Second Language All Rights Reserved The shaded rectangles show the region that’s represented by (4 × 2) + ( 1 / 2 × 1 / 2 ); which is the region that is represented by when we say “multiply the whole numbers and multiply the two fractions” (the regions in white represent the error in computing (4 × 1 / 2 ) + (2× 1 / 2 ) in this way). 2 1/21/2 4 1/21/2 2 × 4 = 82 × 1 / 2 = 1 4 × 1 / 2 = 2 1 / 2 × 1 / 2 = 1 / 4

12. next © Math As A Second Language All Rights Reserved next If we prefer not to use the distributive property, we may convert both mixed numbers to improper fractions and solve the problem that way. In other words… 2 1 / 2 × 4 1 / 2 = 5 / 2 × 9 / 2 = 11 1 / 4 = (5×9) / (2×2) = 45 / 4

13. To generalize the above result, the recipe for computing the product of two mixed numbers by using improper fractions is… © Math As A Second Language All Rights Reserved next Convert the answer from an improper fraction into a mixed number. Solve the resulting improper fraction problem. Rewrite the mixed number problem as an equivalent improper fraction problem. next

14. In this case… © Math As A Second Language All Rights Reserved Notes next It is a good idea to have students estimate the answer even before they do the actual computation. 4 < 4 1 / 2 < 5 next 2 < 2 1 / 2 < 3 × 8 < ? < 15 Thus, any answer that is 8 or less or is 15 or greater must be incorrect. next

15. Remember that the actual arithmetic involves only the adjectives. The noun that the answer modifies has to be determined by the actual problem. © Math As A Second Language All Rights Reserved next That is, there are many “real world” problems that can be solved by knowing that 4 1 / 2 × 2 1 / 2 = 11 1 / 4, Adjective/Noun

16. © Math As A Second Language All Rights Reserved next For example… If the problem was to find the area of a rectangle whose length is 4 1 / 2 inches and whose width is 2 1 / 2 inches, the answer would be 11 1 / 4 square inches. Adjective/Noun 4 1 / 2 inches 11 1 / 4 square inches 2 1 / 2 inches

17. next © Math As A Second Language All Rights Reserved next Our example then would be… 4 1 / 2 inches × 2 1 / 2 inches = (4 1 / 2 × 2 1 / 2 ) inch-inches = 11 1 / 4 inches 2 = 11 1 / 4 square inches

18. Applying the same adjective/noun theme to another example… © Math As A Second Language All Rights Reserved next If an object moved at a constant speed of 4 1 / 2 miles per hour for 2 1 / 2 hours, it would travel 11 1 / 4 miles during this time. 4 1 / 2 miles/hour 11 1 / 4 miles 2 1 / 2 hours next

19. In our next section we will discuss the process of dividing one mixed number by another mixed number. © Math As A Second Language All Rights Reserved 6 2 / 3 ÷ 3 3 / 4 = ?