For Students

For Parents and Teachers

 

Home > Course Descriptions and Sample Final Exams > 22M:012 > Final Exam

22M:012 Theory of Arithmetic-Final Exam

 

1. Suppose a shipment of milk to a local grocery consists of individual gallons of either skim milk or whole milk. Suppose the shipment contained 40% more skim milk than whole milk. If the total shipment consisted of 48 gallons altogether, how many gallons of each type were contained in the shipment?

 

2. Give a description, based on any of the manipulative or geometric models (colored disks, number line arrows, mail time, etc.) we discussed, of the how you might explain the fact that

- 6   -  - 2  = - 4

to children who are first learning about the integers and the arithmetic operations.

If you use the number line model be sure to indicate whether you are using the missing addend model or an alternative model (e.g. the take-away model).

Any diagrams you use must be accompanied by a correct written identification of what the model is showing (e.g. the number line model of the missing addend approach, the set model with red/black chips, etc.).

3. Give an induction proof for the following sum of the first n odd natural numbers:

 

1 + 3 + 5 + 7 + …+ ( 2 n - 1 ) = n 2  

where n is any natural number 1, 2, 3, 4,  …

4. How can you use the number line model of whole numbers and whole number arithmetic to illustrate the idea of the greatest common divisor of two natural numbers a and b, GCD(a, b)? For example, we have GCD(108, 144) = 36. What might be a number line interpretation of this fact?

Your explanation should be simple enough that you might use it in an elementary school class studying math and should apply to GCD(a,b) no matter what natural numbers a and b are. You are welcomed to draw on real-life examples as long as they accurately support the mathematical ideas used in defining the GCD.
 

 

                                                                       

5. Write two different explanations of how one can describe the quotient 45 / 9. In the first use the measurement, also known as the repeated subtraction model, in the second the partition, also known as the sharing model. You do not need to include real life applications illustrating these models (unless you feel including those helps to illustrate the mathematics). Your answers should only be around two to four sentences each (or less!).

 

i.   Measurement or repeated subtraction model:

 

 

 

 

 

 

ii.   Partition or sharing model:

 

 

6. This problem invovles the numbers 35 and 100 (among others). The numbers 35 and 100 were essentially chosen at random, there is no special significance to those choices other than we chose to use them.

 

Consider the part of the number line consisting of all the numbers starting at 0 and ending at 35.

|___|___|___|_______________________________|  

0    1      2     3…                                                          35

(image that there are marks “|” for each of the whole numbers from 0 to 35).

If you divide this part of the number line into one hundred equal-sized pieces, you would expect that each piece has length 35/100.

We use the number line defintion of fractions in this class, and that definition gives the number 35/100 as a dot on the line between 0 and 1 defined in a precise way.

Using the number definition of 35/100, explain why or not the following statement is correct or is not correct.

"If you divide the above part of the number line (consisting of all the numbers starting at 0 and ending at 35)  into one hundred equal-sized pieces, then each piece has length 35/100."

 

 

 

7.

i.  Give an example of a fraction which does not have a terminating decimal expression, and give its nonterminating decimal expression. Explain briefly, using ideas or results we have discussed in class or that appear in the course text, why it is true that it has no terminating decimal expression.

 

 

 

 

 

ii.  Give an example of a fraction which has a terminating decimal expression, and give its terminating decimal expression. Then express this same fraction as a nonterminating repeating decimal expression (with non zero repeating part).

 

 

 

 

iii. Describe the difference or differences between a rational and an irrational number. What are the key features which would allow you to determine if a given number was rational or irrational? To earn credit, your answer must be more informative than something like, "An irrational number is a number which is not a rational number".

8. Give an example of a real-world application problem, suitable for discussion with elementary school students, which uses the idea of proportion or proportional quantities in an essential way for its solution. And solve the resulting problem. Your answer must not be anything of the form, "One person asks another person to solve the proportion
2/4 = 6 / n for n, and the other person figures it out." Your problem must instead involve a realistic application which might occur outside of the classroom setting, but still be understandable to an elementary school student.

9. Give an example of a real-valued function f [x] of a real variable x for which the graph is a line in the x-y plane which contains the points (1 , 4) and (-3 , 8).

10. 
i. Show how to solve the gardener problem below using a problem solving strategy such as one of the Polya strategies mentioned in the text. If you use guess and check your answer must include at least three different guesses and checks, and that strategy should show how your guesses led to additional guesses.

Your solutions should state clearly which problem-solving strategy is being used and it must be clear that the solution actually used that strategy.

Put your part i. SOLUTION in the space below (state solution strategy being used)

 

 

 

 

ii. Show how to solve the sequence of letters problem below using a problem solving strategy such as one of the Polya strategies which is DIFFERENT from the one you used above. If you use guess and check (which you should not do if part i. was solved using guess and check) your answer must include at least three different guesses and checks, and that strategy should show how your guesses led to additional guesses.

Your solutions should state clearly which problem solving strategy is being used and it must be clear that the solution actually used that strategy.

Put your part ii. SOLUTION in the space below (state solution strategy being used)

 

 

 

 

Copyright © 2006 The University of Iowa All Rights reserved