7



NUMERATION

1. Explain why 2546 and 2547 are not equal.

2. Let us return to the Alphabetia system for a moment. You have earned BCD Alphabetian dollars, and your friend has earned DDA dollars. You decide to invest into a new cattle business, that will cost ADAA dollars to start. Do you have enough money to start it? If not, how much more do you need? (You may use manipulatives if you wish).

3. A visitor from Elbonia requests to buy `15' donuts from Krispy Kreme. After much hassle, the clerk realizes that the Elbonian visitor really wants 12 donuts, and that Elbonia must use a place value system with a different base. What base was the Elbonian using? Defend your answer.

4. In this problem you will be working with a base 7 number system that uses letters instead of numbers. Add together the numbers CDE + BCF drawing place value blocks or a counting board to illustrate your answer. Convert your answer to a number in base 10.

5. Feel free to use blocks to solve part b. of this problem. You have to explain your solution. If you decide not to draw it, you have to explain how you performed the operations.

a. Imagine we had three fingers on each hand. Which base do you think we would be counting in? Write the first 15 numbers in the base from part a).

b. Still using the same basis, solve the following problem: You have 14 sheep, while your neighbor has 23 more. How many sheep do the two of you have together?

OPERATIONS

1. The clothes store at the mall has six employees (William, Brittany, Kyle, Michael, Sydney, and Alyssa). This week they worked 46, 29, 28, 36, 40, and 44 hours. The employees at are paid by the hour. Each employee is paid at a different hourly rate ($12, $14, $6, $17, $10, and $16). Figure out how many hours each employee worked this week. Also, determine each employee's hourly pay.

|1. |   |The employee that worked forty-six hours this week, worked fifty-three hours last week. |

| | |Last week, the employee earned $98 more than the amount the employee earned this week. |

|2. |   |Michael had the largest paycheck for the week. |

|3. |   |Brittany worked more than twenty-eight hours this week. |

|4. |   |This week, William worked the most number of hours. |

|5. |   |Brittany earns the most amount of money per hour. |

|6. |   |The employee that worked twenty-nine hours this week, worked thirty-five hours last week. |

| | |Last week, the employee earned $60 more than the amount the employee earned this week. |

|7. |   |Kyle earns more than $10 per hour. |

|8. |   |The employee that worked twenty-eight hours this week, worked thirty-four hours last week. |

| | |Last week, the employee earned $72 more than the amount the employee earned this week. |

|9. |   |Alyssa earned $216 this week. |

2. Your rich relative dies and leaves 1105 sheep behind. In her will she specifies that her sheep should be divided equally to her 4 nieces. How many will each niece get? Will there be any sheep left over? (Remember: you are NOT in base 10!)

Each of your 4 children loses 44 cows. How many cows did they lose all together? Use the area model to solve this problem.

3. Without using a formal algorithm (lining up) or using manipulatives (place value blocks, an abacus, or some other picture), solve the problem 42 – 26 in two different ways using invented strategies (either one you observed or made up yourself) Just show your strategies, do not write an explanation of how they work.

4. Consider the following strategy for solving 53 – 26 (this is a real method that was popular in the 15th century):

I can’t take 6 from 3. So instead, I subtract 6 from 10 and get 4. I then add this 4 on to the 3 to get 7 and I write this down. Then I add one to the 2, making it a 3. I subtract this 3 from the 5 to get 2 and write down 2 next to the 7. The answer is 27.

Demonstrate how/why this strategy really works by showing what the numbers really represent and what is really happening to them. (do not use the terms “borrow”, “carry”, “column”, “line up”, or “place” as these are artifacts of our formal method)

5. A student solved 62 – 39 as follows:

62 – 40 = 22

22 + 1 = 23

62 – 39 = 23

Justify each step in his calculation.

6. Michelle is given the problem 47+28 = ? to solve. Her

method is as follows: ``47 + 3 = 50; 50 + 25 = 75. The answer is

75.''

a) What type of strategy did Michelle use?

b) Solve this problem using a different addition strategy; state what type of strategy you are using.

7. A certain property of addition helps simplify this problem. State this property, and then use it to solve: 36 + 89 + 64

8. A student attempts to multiply 47 x 28 and does the following:

50 x 30 = 1500 and 3 x 2 = 6, so the answer is1500 – 6 = 1494.

Is this answer is correct? Explain. Draw a picture or use algebra to defend your answer.

9. When we divide 144 by 11 using long division, why do we say “11 doesn’t go into 1”, when the 1 stands for 100, and 11 does go into 100? What is going on here mathematically?

10. In the multiplication problem

34

x 15

170

34_

510

why is 34 shifted over by one place? Explain mathematically.

11. A sheet of poster paper measures 22 inches by 30 inches. A teacher wants to cut 30 smaller posters so that each of the students in her class of 30 will have one. If the smaller posters are to be 5 inches by 8 inches and she cuts the maximum number from each large sheet of poster paper, how many sheets of poster paper will she need?

12. Divide 4356 by 6 using the scaffolding algorithm. Explain WHY (not how) this algorithm works. Which model of division (repeated subtraction or partitioning) do you think the algorithm uses? Justify your answer.

13. Consider the following multiplication problem:

1

3 4

x 2 3___

1 0 2

+ 6 8__

7 8 2

Explain what the 1 is doing above the 3; why we multiply the 3's before adding the 1; why we don't first add 1 to 3 then multiply; why the answer is 782 and not 102 + 68 = 170.

FRACTIONS AND DECIMALS

1. If ********** is 1 2/3, show 1/2.

2. If ******* is 2/7, find 1/4.

3. 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.

4. 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.

5. Name a fraction close to 7/9. Name the fraction halfway between 7/9 and your fraction.

6. How would you explain to a friend that 3/5 + 1/4 cannot equal 4/9, using fraction sense and no calculations?

7. 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?

8. With whole numbers, multiplication makes things bigger. Multiplying 3 by 1/4, however, makes the answer smaller than 3. Explain why this happens.

9. In a small town, 1/4 of the women and 2/3 of the men are married. What fraction of the population is married?

10. 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).

11. 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)

12. If O O = 1 1/3, sketch 1.

13. **************** = 2 2/3, show 3/2.

14. Explain how you can order the following numbers from smallest to largest. Do not find common denominators: 121/98, 61/50, 122/99.

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. Use a discrete model of fractions to show that the following three fractions represent the same amount: 3/4, 6/8, (1 1/2) / 2

20. Let O O represent 4/5 of an unknown whole. Draw what 1 would look like. Explain.

21. Determine, without any computations, between which two natural numbers the sum 4 2/3 + 3 1/7 + 2 3/5 is located. Explain how you got the answer.

22. A student says:

a) To add 2 1/3 + 3 1/4, we add 2+3 and 1/3 + 1/4 separately and then add the two sums together to get the answer.

b) To multiply 2 1/3 x 3 1/4 we multiply 2 x 3 and 1/3 x 1/4 and add the products together to get the answer.

For both (a) and (b) answer:

Is the student's strategy correct? Use one of the models (area, length, discrete) for fractions to answer the question. If the strategy doesn't work, explain how you can correct it.

23. In a class 2/9 of the students are female and there are 15 more male students than females. How many students are there in the class?

24. Explain why shifting the decimal point in a number is equivalent to multiplying or dividing a number by a power of ten.

25. Explain how you would use manipulatives to

i) multiply 0.3 x 1.4

ii) add 0.33 + 0.456.

26. A recipe calls for 2 1/4 cups of flour and 1 2/3 cups of sugar. How much of each will you need if you i) triple the recipe? ii) halve the recipe? Do not use the standard algorithms for multiplication and division of fractions.

27. Let O O represent 4/5 of an unknown whole. Draw what 1 would look like.

28. 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.

29. Name a fraction between 2/3 and 3/4.

30. Which of the following fractions is farthest from 1: 2/3 3/4 4/5 5/6? Explain why.

31. Write each as a decimal:

i) fourteen and five hundredths ii) 92 hundredths iii)1 2/10 iv) 7 tenths v) forty-four hundredths vi) 7 tenths PLUS forty-four hundredths.

32. Use manipulatives to determine which decimal number is

larger

i) 52 hundredths or 731 thousandths

ii) 135.7 tenths or 1.74 tens

iii) 234 tenths or 2.34 tens.

33. How many tenths are in a hundredth? How many thousandths are in a hundredth? How many tenths are in 123.6 hundredths?

34. How would you explain why 0.2 and 0.20 are the same number?

RATIOS

1. An all-female college first began admitting men three years ago. Its goal for this, the third year, is to have a 2:1 ratio of females to males in the incoming freshman class. The admissions department has just reported a problem. When the college sent out letters of acceptance to this year’s freshman class, the ratio of females to males was 2:1. However, the ratio of females to males in the 1000 students who indicated they will be coming this fall is 7:3. If the college still wants to meet the 2:1 goal, how many more males will it need to accept to reduce the ratio of females to males to 2:1?

2. If the ratio of women to men at a particular college is 5:3, and there are 180 more women than men, how many students are enrolled at this college?

3. This morning, driving to work, you got stuck behind a truck that was driving 20 miles per hour. How late would this make you for work? (Note: not all the necessary information is given in the problem, and you have to make some additional assumptions.)

4. Ms. Anderson works on a straight 20% commission on her sales. Last week her sales were $3127. How much did she earn? Her boss offers her the option of changing her plan in the future to $100 a week plus 15% commission. Should she stay on straight commission or go for the new plan?

5. Last year, your employer had to cut your salary by 25% due to the bad situation in the company. This year, sales have picked up and your employer has raised your salary (from last year) by 25%. Is your salary now the same, lower, or higher than before your salary cut? Justify your answer. (You may want to look at a concrete example.)

6. When college bookstores purchase textbooks, they generally sell the books for 20% to 25% more than they paid for them. Let's say that you paid $65 for a book and the bookstore marked up the price by 25%. How much did the bookstore pay for the

textbook?

7. Let's say I am the president of a small company that has done very well in the last year. I have decided to share my good fortune with my hard-working and devoted employees. I announce that I will be giving a $1-an-hour raise to everyone. Two days later I become aware of some grumbling. A number of employees are complaining that this is not fair. I am stunned. I thought giving everyone the same raise was the epitome of fairness. Why are some employees grumbling? How would you explain this to me if you were one of my dissatisfied employees?

7. What is 10% less than 36?

8. Two dresses are for sale. The first was selling for $119 and is being marked 40%. The other was selling for $79.99 and is being discounted by 20%. Which dress costs less now?

9. If the ratio of nickels to dimes in my pocket is 3:2 and the ratio of dimes to quarters is 2:4 and the ratio of quarters to dollar bills is 2:3, how much money do I have in my pocket if there are 2 dimes?

10. In 2000, City A, with population 55,489 reported 214 violent crimes. In the same year, City B, with population 185,217 reported 639 violent crimes. Which city has the worse crime rate? Explain your answer in 1 sentence.

11. The length and width of a rectangle are each doubled. The area of the old rectangle is what percent of the area of the new rectangle?

NUMBER THEORY

1. Use divisibility rules to find a five digit number divisible by a) 6; b) 40.

2. What is the remainder when 1254632 is divided by 3? by 9? (Decide without actually dividing.)

3. Your friend tells you that if 2 divides a number and 10 divides a number, then 20 will divide that number. How do you respond?

4. 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?

5. If you have a pair of natural numbers a, b such that a < b, what is: the largest possible value for the GCF of a and b? the smallest possible value for the LCM of a and b?

6. The product of two numbers is 360 and the GCF is 6. Neither one of the numbers is 6. What are the numbers? Show all your work.

7. 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.

8. 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?

9. 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?

10. If a and b are different and relatively prime, what can you say about their LCM and GCF?

11. If a divides b, what can you say about their LCM and GCF?

12. If the sum of three numbers is even, is their product even, odd, or are both options possible?

13. How would you convince someone that the sum of any three consecutive numbers will be divisible by 3?

14. 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.

15. Explain why divisibility by 2 and 3 will guarantee divisibility by 6.

16. Write a divisibility rule for 15.

17. If a divides b and b divides c, will a divide c? How do you know?

18. If a divides b and a divides c, will a divide (b + c)? Explain your answer.

19. 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?

20. If you add together 25 consecutive numbers starting with an odd number, will the sum be even or odd?

21. 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?

22. 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?

23. The GCF of two numbers is 24 and the LCM is 480. What are the two numbers?

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