## Preface

This page contains some quizzes that I wrote and used in my Pre-algebra classes (in 1996). They were intended to examine the mastery, or lack thereof, of “problem solving” concepts of my students, as recently studied in class. Each problem is based on work discussed a few days earlier. So as presented, the wording of some will seem a little cryptic or confusing at first. This was intentional, done to see if the students were reflecting back on facts and structures covered in the homework. Background explanation will be provided to you in such cases. The inspiration for each of the items came from different sources: the textbook, the problem solving section of the NCTM journal *Mathematics Teaching in the Middle School*, and other places which I’ve forgotten. A second feature that will soon become apparent as you scroll down this page is that the quizzes come in pairs, each parallel to the other. This is because I had many sections of students and I needed to have different versions (to avoid you-know-what).

**Quiz: Problem Solving (A)**

1. The evil warlord, Grossout, got even meaner and nastier after you helped the princess solve the problem last time. So he raised his demand. Now she must tell him the next four squares (after your last answer) whose digit sum is not a perfect square. Help save the princess again. [See below for original.]

2. Kendra’s little cousin also collects shells, but her collection is smaller than Kendra’s. Counting the shells by 4’s or 5’s leaves three shells left. But counting by three’s leaves 2 left over as before. How many shells are in this collection? [See below for original.]

3. Find at least three solutions for the alphabetic when the sum is now 10,431 . [See below for original.]

4. Use the pattern ABBCCCDDDD… . (a) What letter is in the 151st position? (b) What position does it occupy within its own letter group?

5. On his birthday Jaime received $5 from his grandfather. “Gee whiz!” he said. “Now I have three times as much as I would have had as if I had spent $15 for that CD.” How much did Jaime have before receiving the money? [See below for original.]

tt(4/27/96)

**Quiz: Problem Solving (A’)**

1. The evil warlord, Grossout, got even meaner and nastier after you helped the princess solve the problem last time. So he raised his demand. Now she must tell him the first perfect square larger than 999 whose digit sum is *not* a perfect square. Help save the princess again. [See below for original.]

2. Kendra’s little cousin also collects shells, but her collection is smaller than Kendra’s. Counting the shells by 3’s or 4’s leaves only one shell left. But counting by five’s leaves 2 left over as before. How many shells are in this collection? [See below for original.]

3. Find at least three solutions for the alphametic when the sum is now 10,422 . [See below for original.]

4. Use the pattern ABBCCCDDDD… . (a) What letter is in the 121st position? (b) What position does it occupy within its own letter group?

5. On his birthday Jaime received $6 from his grandfather. “Gee whiz!” he said. “Now I have three times as much as I would have had as if I had spent $14 for that CD.” How much did Jaime have before receiving the money? [See below for original.]

tt(4/27/96)

**Original problems for A/A’**

** 1. The evil warlord, Grossout, locked Princess Stunning in the dungeon. Before he lets her out, she must find the first perfect square number
larger than 100 where the sum of the digits is not a perfect square. Help the princess by finding the number.
**

[Menu of Problems. MTMS, March/April 1996, p. 720]

2. Ken has between 3 and 100 shells in his collection. If he counts his shells 3 at a time, he has 2 left over. If he counts them 4 at a time, he has 2 left over. If Ken counts them 5 at a time, he still has 2 left over. How many shells does Ken have in his shell collection?

2. Ken has between 3 and 100 shells in his collection. If he counts his shells 3 at a time, he has 2 left over. If he counts them 4 at a time, he has 2 left over. If Ken counts them 5 at a time, he still has 2 left over. How many shells does Ken have in his shell collection?

[Menu of Problems. MTMS, March/April 1996, p. 720]

3. The alphametic referred to came from the UCSMP textbook Transition Mathematics, p. 306. It was “U + CAN + DO + THIS = 10440″, presented in vertical format. As given in the text, there are quite a number of solutions. Then by altering the sum to other numbers one can produce additional equivalent exercises.

3. The alphametic referred to came from the UCSMP textbook Transition Mathematics, p. 306. It was “U + CAN + DO + THIS = 10440″, presented in vertical format. As given in the text, there are quite a number of solutions. Then by altering the sum to other numbers one can produce additional equivalent exercises.

** **

**5. Karen found $2 on the sidewalk. “Wow!” said Karen. “Now I have five times as much as I would have had if I had lost $2!” How much money did Karen have before she found the $2?
**

[Menu of Problems. MTMS, March/April 1996, p. 729]

**Quiz: Problem Solving (B)**

1. At the 7th grade’s Candy Sale at break, 4 Hershey’s kisses and 7 Rice Krispy Treats cost 75 ctvs., and 7 Hershey’s kisses and 4 Rice Krispy Treats cost 90 ctvs. How much change would I get from a ¢5 bill if I bought just 30 kisses?

2. The Mensa toy store is having a special sale of used toys. Everybody is confused about the prices. (Remember the clothing store case.) At this sale, a truck costs $24, a cart costs $19, and a ball costs $19. How much would a bucket (like a child uses at the beach) cost under the Mensa system?

3. Susan gave Karen an amount of money equal to one-half of what Karen had in her purse. Then Karen gave Susan an amount equal to twice of what Susan had left. If now Susan has $18 and Karen has $12, what did each girl have before this sequence of strange transactions took place? [HINT: all numbers are whole numbers.] Also, who made a profit in this deal, and how much?

tt(5/1/96)

**Quiz: Problem Solving (B’)**

1. At the 7th grade’s Candy Sale at break, 5 Hershey’s kisses and 6 Rice Krispy Treats cost 85 ctvs., and 6 Hershey’s kisses and 5 Rice Krispy Treats cost 80 ctvs. How much change would I get from a ¢5 bill if I bought just 20 kisses?

2. The Mensa toy store is having a special sale of used toys. Everybody is confused about the prices. (Remember the clothing store case.) At this sale, a truck costs $21, a cart costs $17, and a ball costs $17. How much would a shovel (like a child uses at the beach) cost under the Mensa system?

3. Susan gave Karen an amount of money equal to twice of what Karen had in her purse. Then Karen gave Susan an amount equal to one-half of what Susan had left. If now Susan has $15 and Karen has $7, what did each girl have before this sequence of strange transactions took place? [HINT: all numbers are whole numbers.] Also, who made a profit in this deal, and how much?

tt(5/1/96)

**Original problems for B/B’**

** 1. If 6 pieces of licorice and 5 red-hots cost 49¢, and 6 red-hots and 5 pieces of licorice cost 50¢, how much change will you get back from a $1.00 bill if you buy 20 red-hots?
**

[The Mensa Puzzle Calendar. January 6, 1996] Special note: in the quiz problems I am assuming Salvadorean currency. So “ctvo. = centavos” and “¢ = colon”.

2. Our local clothing store is going out of business. And no wonder. A coat used to cost $30.00; a dress, $45.00; and socks, $45.00. How much would a blouse cost under this system?

2. Our local clothing store is going out of business. And no wonder. A coat used to cost $30.00; a dress, $45.00; and socks, $45.00. How much would a blouse cost under this system?

[The Mensa Puzzle Calendar. February 29, 1996] Special hint: the solution of this problem lies in the LETTERS of the words

*coat*,

*dress*, etc.; specifically the vowel-consonant concept. Is that enough help?

3. Steve gave Keith an amount of money equal to the amount in Keith’s pocket. Keith then gave Steve an amount equal to what Steve had left. Keith then had $14 and Steve had $8. How much money did Keith originally have in his pocket?

3. Steve gave Keith an amount of money equal to the amount in Keith’s pocket. Keith then gave Steve an amount equal to what Steve had left. Keith then had $14 and Steve had $8. How much money did Keith originally have in his pocket?

[Menu of Problems. MTMS, March/April 1996, p. 729]

**Quiz: Problem Solving (C)**

1. Donald Duck drove his car at a uniform speed from Duckton to Drake City. At 2 p.m. he was one-eighth the way to his destination, and at 5 p.m. he was three-fourths of the way. What fractional part of his trip had he traveled by 4 p.m.? And what times did he start and finish this trip? [Draw a sketch of the “highway”.]

2. The number 350 is the product of several pairs of numbers that could represent the ages of persons in a particular family. Using only the reasonable pairs, state the kinds of relationships that could exist (i.e. from babies to grandparents and inbetween). [Make a factor T-chart.]

3. A child glues together 105 cubes with 1-cm edges to form a solid,rectangular brick. If the perimeter of the base is 20 cm, what is the height of the brick? [Draw a sketch of the this object.]

tt(5/12/96)

**Quiz: Problem Solving (C’)**

1. Mickey Mouse drove his car at a uniform speed from Tune Town to Ytictar. At 2 p.m. he was one-fourth the way to his destination, and at 5 p.m. he was seven-eighths of the way. What fractional part of his trip had he traveled by 3 p.m.? And what times did he start and finish this trip? [Draw a sketch of the “highway”.]

2. The number 330 is the product of several pairs of numbers that could represent the ages of persons in a particular family. Using only the reasonable pairs, state the kinds of relationships that could exist (i.e. from babies to grandparents and inbetween). [Make a factor T-chart.]

3. A child glues together 66 cubes with 1-cm edges to form a solid, rectangular brick. If the perimeter of the base is 28 cm, what is the height of the brick? [Draw a sketch of the this object.]

tt(5/12/96)

### Original problems for C/C’

** 1. Jack climbed up the beanstalk at a uniform rate. At 2 p.m. he was one-sixth the way up and at 4 p.m. he was three-fourths the way up. What fractional part of the entire beanstalk had he climbed by 3 p.m.?
**

[Menu of Problems. MTMS, March/April 1996, p. 730]

2. Two adults have ages in years whose product is 770. What is the age of the younger adult? Does your answer depend on how

2. Two adults have ages in years whose product is 770. What is the age of the younger adult? Does your answer depend on how

*adult*is defined?[Menu of Problems. MTMS, March/April 1996, p. 730]

3. A child glues together forty-two cubes with 1-cm edges to form a solid, rectangular-faced brick. If the perimeter of the base is 18 cm, what is the height of the brick, in centimeters?

3. A child glues together forty-two cubes with 1-cm edges to form a solid, rectangular-faced brick. If the perimeter of the base is 18 cm, what is the height of the brick, in centimeters?

[Menu of Problems. MTMS, March/April 1996, p. 730]

**Quiz: Problem Solving (D)**

1. Porky Pig really loves eating popcorn, but lately he’s been getting a bit “pudgy” (fat). So his doctor put him on a peculiar diet. Porky could eat one piece of popcorn on the 1st day of any given month, two pieces on the 2nd day, three on the 3rd day, and so on during the month. The next month the whole process would begin anew. How many pieces of popcorn will Porky pop into his “pucker” (mouth) during the first six months of this year?

2. Bugs Bunny has three types of coins in his bunny-bag: nickels, dimes, and pennies. Thirteen of them are nickels, 3/7 of them are pennies, and 20% of them are dimes. How much money does Bugs have in his bunny-bag?

3. A rock-n-roll singer was so bad that one-fifth of the crowd left after the first song. After the 2nd song, one-half of the remaining crowd also departed. When the 3rd song was ended, one-third of those people got up and “hit the road” too. If there were now only 160 die-hard music lovers remaining, how many people paid good money for “nuthin’ much at all”? [Work hint: there were < 1000.]

tt(5/16/96)

**Quiz: Problem Solving (D’)**

1. Petunia Pig (Porky’s sweetheart) really loves eating popcorn, but lately she’s been getting a bit “pudgy” (fat). So her doctor put her on a peculiar diet. Petunia could eat one piece of popcorn on the 1st day of any given month, two pieces on the 2nd day, three on the 3rd day, and so on during the month. The next month the whole process would begin anew. How many pieces of popcorn will Petunia pop into her “pucker” (mouth) during the six months from last November to theend of this past April?

2. Yosemite Sam has three types of coins in his saddlebags: nickels, dimes, and pennies. Eleven of them are pennies, 40% of them are dimes, and 2/7 of them are nickels. How much “loot” (money) is Sam “a-totin’ around in them there saddlebags”?

3. A rock-n-roll singer was so bad that one-fourth of the crowd left after the first song. After the 2nd song, one-third of the remaining crowd also departed. When the 3rd song was ended, one-half of those people got up and “hit the road” too. If there were now only 160 die-hard music lovers remaining, how many people paid good money for “nuthin’ much at all”? [Work hint: there were < 1000.]

tt(5/16/96)

**Original problems for D/D’**

1. As well as I can recall now, the popcorn problems were not based on any particular practice exercise, but rather the general concept of adding expressions like “1 + 2 + 3+ … + n”, using the famous triangular number formula

### T_{n} = ½n(n + 1)

Click here for more information about triangular numbers.

2 & 3. At the moment I can’t find among my files the problems that were the motivators for these two items. But as they are solvable as they stand, I will defer my search for the original sources to another day.

**Quiz: Problem Solving (E)**

1. Kermit the Frog is competing in the famous jumping contest of Calavarass County. He took 5 leaps, each one 1½ times as long as the preceding one. What was length of his first leap if his total distance came to 422 cm? [Work hint: all jumps are whole numbers.]

2. Fred Flintstone currently has $256 in his bank account and being the jolly fellow he is, spends $6 from it each week at his Moose Lodge meeting. On the other hand, his wife, Wilma, has $136 in her kitchen’s cookie jar, and is saving $4 every week to buy herself a nice tiger skin dress later on. After how many weeks will they have the same amount of money? What will that amount of money be? [Make a week-by- week chart to see their respective weekly progress.]

3. [Three Coins Revisited] I have three kinds of coins in my pocket: quarters, dimes, and nickels. a) 20% of them are type “x”; b) 3/7 of them are type “y”; and c) thirteen of them are of type “z”. If the total value of my money is $8.40 , how many coins of each kind do I have?

tt(5/25/96)

**Quiz: Problem Solving (E’)**

1. Jiminy Cricket is competing in an insect jumping contest. He took 5 leaps, each one 1½ times as long as the preceding one. What was length of his first leap if his total distance came to 211 cm?

[Work hint: all jumps are whole numbers.]

2. Barney Rubble currently has $248 in his bank account and being the jolly fellow he is, spends $5 from it each week at his Moose Lodge meeting. On the other hand, his wife, Betty, has $128 in her kitchen’s cookie jar, and is saving $5 every week to buy herself a nice tiger skin dress later on. After how many weeks will they have the same amount of money? What will that amount of money be? [Make a week-by-week chart to see their respective weekly progress.]

3. [Three Coins Revisited] I have three kinds of coins in my pocket: quarters, dimes, and nickels. a) 40% of them are type “x”; b) 2/7 of them are type “y”; and c) eleven of them are of type “z”. If the total value of my money is $9.30 , how many coins of each kind do I have?

tt(5/25/96)

**Original problems for E/E’**

1 & 3. The same is true for this pair of problems. So again I will just let sleeping dogs lie and not get too disturbed about my references until a more serendipitous opportunity arises. Check back at a later time if you’re curious. Sorry.

2. This one was based on a problem in an old MATHCOUNTS handbook. I even used the concept with some classes of 4th grade students prior to

1996, employing the names of John and Mary. Go to Up-down Match Game to see how this was done.

Good way of explaining, and nice paragraph to obtain facts about my presentation subject,

which i am going to present in academy.