Math 123
Math 123
Test 3 review sheet
Also use this sheet to review for the final
Number theory. Know prime factorization. Be able to answer questions about the number of factors a certain number has, and to conjecture which numbers have exactly 2, 3, 4, 5 factors etc. Be able to check if a number is prime. Know how to find GCF and LCM, and how to use them in contextual problems. Make sure you go over the packet you turned in, especially the questions you had trouble with, for example, how you know that a perfect square has an odd number of factors; how tiling a rectangle is related to the number of factors; or that two numbers always have 1 as a common factor. You do not have to know the pool problem from the packet. Also know the six problems from the first number theory worksheet.
Fractions. Understand the meaning of fractions; the part-whole, division, ratio interpretations of fractions; be able to represent fractions using area, number line, and set models; understand how the size of one whole affects the meaning of a fraction. Know how to represent fractions using manipulatives, like in the activities we did in class. Know how to compare fractions. Understand what equivalent fractions are and how and why we can reduce fractions. Be able to add simple fractions using manipulatives, and to multiply fractions using the area model. Understand mixed numbers and improper fractions. Be able to clearly explain the algorithms for operations on fractions, including “invert and multiply.” Know why “1/3 of 1/2” is written as 1/3 x 1/2. Be aware of common mistakes that children make when working with fractions (examples from the reading).
To study for the test, go over all assignments from class. Some additional problems to consider:
Number theory
1. What is the largest square that can be used to fill a 6 by 10 rectangle? What is the largest square that can be used to fill a 12 by 18 rectangle? What is the largest square that can be used to fill an x by y rectangle?
2. What can you say about two numbers if their sum is odd and product is even? Justify your answer.
3. How are the GCF and LCM related to each other?
4. A child has a large supply of dominoes, each of which measures 32 millimeters by 56 millimeters. She wants to lay them out to form a solid square, and she wants them all to be laid out horizontally. What will be the dimensions of the smallest square that she can form, and how many dominoes will it require?
5. (challenging) The LCM of two numbers is 280 and the GCF is 5. What are the two numbers? If the LCM of two numbers is 90 and the GCF of the same two numbers is 5, what are the two numbers? The GCF of two numbers is 24 and the LCM is 480. What are the two numbers? Are these the only two that will work? If so, why? If not, why not?
6. Find a number that has a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6.
7. If 2 divides a number and 3 divides a number, then 6 divides that number, but if 2 divides a number and 4 divides a number, 8 does not necessarily divide the number. What makes the difference in the two examples?
8. Find a number, or show that it doesn't exist, between 100 and 200 that has:
i) exactly one factor ii) exactly two factors iii) exactly three factors.
What do all the numbers that have exactly three factors have in
common?
9. If a divides b, what can you say about their LCM and GCF?
10. A four-digit number is divisible by 6, and the sum of the digits is 9. What is the number? Are there others? Verify your answer.
11. If you are checking for factors to determine whether a number is prime or composite, what is the largest factor you need to check and why?
12. If you add together 25 consecutive numbers starting with an odd number, will the sum be even or odd?
13. Explain how we can find the LCM of two numbers. How can you sure that the number you find is the least common multiple? How would you find the LCM of three or more numbers? Explain on examples: first consider 42 and 56, then 12, 42 and 56, and
finally 14, 42 and 56. What do you notice about the three examples? Can you find a shortcut in the last one?
14. Kathy and her friend Marilyn like to walk around the college track for exercise. Since they walk at different rates, they start off together but do not stay together during the walk. Kathy takes 6 minutes to complete one lap, and Marilyn takes 8 minutes.
If they walk for about an hour and a quarter, how many times will they be at the starting place at the same time?
Fractions
1. The students in the Douglas School have left for their annual school trip. One-third of the students went to Washington D.C. One-third of them went to Annapolis. One-fourth of them went to Williamsburg, and the remaining 100 students went to Monticello. How many students are in the class? How many went to each place?
2. One jar is half full of vinegar and another jar twice its size is one quarter full of vinegar. Both jars are filled with water and the contents are mixed in a third container. What part of the mixture is vinegar?
3. A man spent 1/3 of his money and then lost 2/3 of the remainder. He was left with $12.00. How much did he start with?
4. A recipe for chocolate chip cookies follows:
1 ¼ cps flour
½ cp sugar
½ t salt
½ C butter
6 oz chocolate chips
1t vanilla extract
1 egg
½ t baking powder.
This recipe makes 4 dozen cookies. How much of each ingredient would you need if you wanted to make 10 dozen cookies?
5. Determine which is bigger without converting to decimals, finding the LCM, or drawing a diagram:
a. 9/11, 13/15
b. 7/12, 13/28.
6. Without finding common denominators, using a picture, converting to decimals, or using multiplication or division rules, tell how you can order the following fractions from smallest to largest: 23/18, 6/7, 17/8, 18/29.
7. If * * * * * * * * * * is 1 2/3, show 1/2.
8. If * * * * * * * is 2/7, find 1/4.
9. Use a diagram to illustrate that 3/5 is equivalent to 6/10. Explain, more generally, what we mean by simplifying (reducing) fractions, and why this can be done.
10. How would you explain to a friend that 3/5 + 1/4 cannot equal 4/9, using fraction sense and no calculations?
11. A student suggests that to multiply 2 1/3 x 3 1/4 you can multiply 2 x 3 and 1/2 x 1/4 and then add the results. Do you agree with the student? Can you justify your answer?
12. With whole numbers, multiplication makes things bigger. Multiplying 3 by 1/4, however, makes the answer smaller than 3. Explain why this happens.
13. Consider the following problems:
(i) 3/4 + 2/5
(ii) 3/4 – 2/5
(iii) 3/4 x 2/5
(iv) 3/4 / 2/5
Write a word problem corresponding to each.
Solve each problem using one of the models for fractions (you can also use and draw manipulatives).
14. How would you explain to someone that 3 2/3 is equivalent to 11/3? (saying "3*3+2=11" will not count as an answer)
15. Without actually doing any computation (use your fraction sense), determine whether a) 5 5/8 + 4 3/42 is greater than 10 or less than 10
b) 7/8 + 3/4 + 1/16 is greater than 2 or less than 2.
c) 8 1/2 - 2 2/3 is between 5 and 5 1/2 or between 5 1/2 and 6.
16. An analysis of first-year students at a college revealed that 1/4 of first-year women were from homes where both parents were professionals. Of these, 3/5 were interested in the same professions as one or both of their parents. If this latter group is made up of 18 students, how many first-year women were there?
17. If you need 1 3/4 yards of fabric to make a skirt, how many skirts can you make with 10 yards of material? Solve the problem both algebraically and with a diagram. Explain how you deal with the remainder.
18. For each of the following, justify your reasoning.
a) Name three fractions between 1/8 and 1/9.
b) Is 10/13 closer to 1/2 or to 1?
c) Name a fraction closer to 1/2 than to 5/12.
19. Let O O represent 4/5 of an unknown whole. Draw what 1 would look like. Explain.
20. Let
X X X X
X X X X
X X X X
X X X X
X X X X
X X X X
represent 6/5 of an unknown whole. Draw 1/4 of the whole. Explain.
21. Name a fraction between 2/3 and 3/4.
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