Math 121 - Modesto Junior College



Math 105 Name _____________________

Final Exam December 15, 2004

There are 25 problems on this exam. Obviously, you can’t spend the same amount of time on each problem as on the midterm exams. The instructions have been modified to reflect this, so please follow them carefully. Your strategy is to do as many of the problems as you can, starting with the easiest ones and skipping the harder, longer ones until the end. After all, in a given 15 minute period you can do either one hard problem and earn five points or five easier ones and earn twenty-five points. You should first build up a solid foundation score, then finish it off by doing the harder ones.

For #1-11, draw an appropriate diagram (or two or three if necessary to show a progression of ideas on your overhead projector). Then give a brief explanation of what’s going on.

Example: Show 6 + 3 using the set model of addition.

Sample Answer:

Start with 6 marbles. Add in 3 more marbles. Now you’ve got nine

marbles. So 6 + 3 = 9

1. Show 437 + 685 using place-value cards. (Be sure to explain exchanging)

2. Show –23+7 using the number line/kinesthetic model of addition.

3. Show [pic] using the Cartesian Product model of multiplication.

4. Show [pic] using the repeated subtraction model of division.

5. Show [pic] using the partition model of division.

6. Subtract 11 from –5 using +’s and –’s.

7. Show [pic] on a number line.

8. Show [pic] on a number line.

9. Show why –13 is less than –8 even though 8 is less than 13.

10. Subtract [pic] from [pic] using fraction strips. (Be sure to explain what the LCD represents and why you need it.)

11. Multiply 3 by 4.56 using place value cards. (Be sure to explain exchanging.)

For #12-14, demonstrate your ability to use the instructional algorithms for arithmetic. Note, as in the example, what each line represents.

Example: Add 23, 37, and 52 using the instructional algorithm for addition.

Sample Answer: [pic]

For #12-14, see instructions on previous page.

12. Add 5876 + 7218 + 493 using the instructional algorithm for addition.

13. Multiply 5.2 by 6.74 using the instructional algorithm for multiplication, appropriately modified for decimals. Then explain in one sentence why it is “legal” to “scoot” the decimal in three places in the answer, as you’ve done.

14. Divide 12731 by 43 using the Scaffold Algorithm. Express your answer according to the notation of The Division Algorithm ([pic]).

15. (a) Complete this definition: Two sets are equivalent if ____________________________.

(b) Use your definition to prove that the sets F and R below are equivalent

[pic]

(c) Use your definition to prove that the two line segments to the right

are equivalent.

16. Use the method of prime factorizations to find GCD(1980, 1050) and LCM(1980, 1050).

17. Use the Euclidean algorithm to find GCD(1980, 1050) and use that answer to find

LCM(1980, 1050).

18. (a) The irrational numbers are dense in the real numbers. Explain what this means.

(b) The principal square roots, [pic], are not dense in the set of real numbers. Explain what this means.

19. Find three fractions between the two below in three different ways then explain why there are infinitely many fractions between these two.

[pic] and [pic].

20. Prove that [pic] is irrational.

21. (a) State the definition of an infinite set. (b) Then give an example of an infinite set and verify that it really is an infinite set by showing how it satisfies the definition. (c) Also, give an example of a finite set and show why it is not infinite based on this definition.

22. At a meeting of registered nurses, 86% were women. There were 35 male nurses in attendance. How many female nurses were at the meeting?

For Problems 23-25, It’s time to make another trip to ToonTown. Recall that since cartoon characters only have eight fingers, everything in ToonTown is done in base eight. That includes decimals and fractions!

23. (a) Which is bigger, [pic] in Modesto or [pic] in ToonTown? Explain.

(b) Which is bigger, [pic] in Modesto or [pic] in ToonTown? Explain.

24. How does a teacher in ToonTown explain to a student how to combine the following fractions? (Be sure to discuss the LCD. Do all work in base eight.)

[pic]

25. Suppose you go into a store in ToonTown with a 25% off coupon. Now in ToonTown, 25% means, similar to us, 25 per 100, but recall that the 25 and 100 are in base eight. Sales tax is charged in ToonTown at a rate of 7%. Also recall that their money isn’t dollars, it’s Disneys. If you buy a new frying pan to hit Popeye on the head with, and if it’s regularly priced at 25 Disneys, how much to you pay total? Round your answer to two octal places. (octal – like “decimal” but “deci-“ means ten!)

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12 + 100 = 112

3 + 7 + 2 = 12

20 + 30 + 50 = 100

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