Grade 6 Practice Problems: Strategies
Expectations from Standards Revision and Strategy Description
6S-8f) Strategy: Guess, check and revise Answer: X=9; Y=2; Z=1
One day Ben' teacher put a very unusual math problem on the board. It looked like this: XX + YY = ZYZ (XX refers to a two digit number with the ten's digit equal to the unit's digit). His teacher asked the class to find the values for X, Y, and Z in the problem. She told the class that if a letter was used more than once. It had the same value each time it was used. What are the values for X, Y, and Z? Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-8e) Strategy: Guess, check and revise Answer: 26
I'm thinking of a number. Twice the number minus 18 equals 34. What is the number? Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-8d) Strategy: Guess, check and revise Answer: 10
I'm thinking of a number. Three times the number plus 23 equals 53. What is the number? Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-8c) Strategy: Guess, check and revise Answer: 6
I'm thinking of a number. Multiply the number by itself and then by itself again and you will get 216. What is the number? Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-8b) Strategy: Guess, check and revise Answer: Jane Jones, Emily Robinson, Sherrie Smith, & Kathy Brown
Eight children divided 30 books. Jane took 1 book, Emily 2, Sherrie 2, and Kathy 4. Bob Smith took as many as his sister. Ken Brown took twice as many as his sister, Jim Jones took 3 times as many as his sister, and Tom Robinson 4 times as many as his sister. What is the last name of each of the 4 girls? Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-8a) Strategy: Guess, check and revise Answer: 16 problems were correct
At a math contest 20 problems were given. Each correct answer earned 5 points, and 2 points were deducted for each incorrect answer. Lenore answered all the problems, receiving a score of 72. How many correct answers did she have? Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-7d) Strategy: Draw a picture Answer: Farmer Joe would at 20.48 a dozen compared to Farmer John at $1.20
There are 2 chicken farmers who are constantly trying to outsell each other with their eggs. Farmer John planned to sell the first egg for 10 cents, the second for 20 cents, the third for 30 cents, and so on. Farmer Joe decided to sell his first egg for 1 cent, the second for 2 cents, the third for 4 cents, the fourth for 8 cents and so on, doubling the cost of each additional egg. Which farmer will earn more money selling one dozen eggs? Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-7c) Strategy: Draw a picture Answer: Ten is the greatest number of tents that could have been set up.
A group of 30 Scouts set up their tents for the campout. In each tent there were exactly the same number of scouts. Later another group of 20 scouts arrived. They did not set up any tents, but instead divided themselves into equal size groups so that they could share the tents already set up. If there were an equal number of Scouts in each tent, what was the greatest number of tents that could have been set up. Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-7b) Strategy: Draw a picture Answer: Jennifer took 15 seconds between each pair of flags. Six groups of 15 equals 90 seconds.
At the State Cross-Country Meet there is a circular track with 6 flags spaced around at an equal distance apart. At the race yesterday, the 3 fastest runners lined up at the finish line facing the first flag, and started to race all the way around the track Jennifer took 30 seconds to get to flag number 2. How long did it take her to get all the way around the track if she kept up the same pace? Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-7a) Strategy: Draw a picture Answer: First: Lightning; Second: Speedy; Sid Third: Secretariat; Fourth: Citation; Fifth: Seattle Slew
Five famous horses raced yesterday at the track. Citation finished 1 length ahead of Seattle Slew. Speedy Sid finished ahead of Citation but behind Lightning. Secretariat finished 4 lengths ahead of Seattle Slew and 1 length behind Speedy Sid. What was the finish place of each horse? Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-6e) Strategy: Act out the problem or use objects Answer: 7
Mr. Murray challenged his class with this puzzle. He placed 3 pieces of paper on the table. On the sheet on the far left he stacked up 3 circular disks, each one smaller than the one below it. His challenge was to move the 3 disks, one at a time, to the paper on the far right. Oh yes! He had one rule: a larger disk could never be stacked on top of a smaller disk. What is the least number of moves needed to accomplish this task? Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-6d) Strategy: Act out the problem or use objects Answer: 4 boys and 3 girls
There is a family in which each boy has as many sisters as brothers. But each of the girls has twice as many brothers as sisters. How many girls and boys are in the family? Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-6c) Strategy: Act out the problem or use objects Answer: They could get 20 prizes
There are 4 boys and 5 girls standing outside of the new music shop. The sign in the window offers a prize to every couple (1 boy and 1 girl) that enters the store. How many prizes can the 9 people get? Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-6b) Strategy: Act out the problem or use objects Answer: The number cards would read: 2,3,1,2,1,3
There are two 1's, two 2's, and two 3's. There is one digit between the two l's, there are two digits between the two 2's, and there are three digits between the two 3's. Holding number cards, can you duplicate this sequence of numbers? Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-6a) Strategy: Act out the problem or use objects Answer: 28 handshakes
All the members of an exclusive country club are very polite. Whenever they meet each other, they always shake hands, address each other by their full names, and inquire about the health of the club member, his wife, and children. At a recent meeting, 8 members of the club were present. If each member shook hands with every other member once, and only once, how many handshakes would have taken place? Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-5e) Strategy: Make a table, chart, or organized list Answer: In this case, the best buy is to purchase three 5-pound boxes and one 3-pound box, a total of 18 pounds at $24.24, to get the 17 pounds he needs.
Gardener Lee wants to put fresh grass seed down on his front lawn. Grass seed is available in three-pound boxes and in five-pound boxes. A three-pound box costs $4.50, and a five-pound box costs $5.58. Gardener Lee needs 17 pounds of the grass seed. How many of each size box should he purchase to get the best buy? Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-5d) Strategy: Make a table, chart, or organized list Answer: Mary - Chris: 1 - 48; 2 - 24; 3 - 16; 4 - 12; 6 - 8; 8 - 6; 12 - 4; 16 - 3; 24 - 2; 48 - 1
Mary and Chris were playing a factors game. Mary would name a factor of 48 and Chris would give the other factor. Make a table to show the different ways Mary and Chris could play the 48 game. Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-5c) Strategy: Make a table, chart, or organized list Answer: make a grid with 1-6 along the top row and 1-6 along the left side filling in all products row by row: 1,2,3,4,5,6; 2,4,6,8,10,12; 3,6,9,12,15,18; 4,8,12,16,20,24; 5,10,15,20,25,30; 6,12,18,24,30,36
Tweedle Dee and Tweedle Dum were playing a dice game. They needed to know all the possible products they could get by rolling two dice. What products are possible? Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-5b) Strategy: Make a table, chart, or organized list Answer: Tables-Stools-Legs: 8-0-32; 7-1-31; 6-2-30; 5-3-29; 4-4-28; 3-5-27; 2-6-26; 1-7-25; 0-8-24
Mr. Banks works in a shop that produces 4-legged desks and 3-legged stools. Nine customers ordered 8 items each. Each order was different. How many legs are needed for each of the customers? Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-5a) Strategy: Make a table, chart, or organized list Answer:BGY; BYG; GBY; GYB; YBG; YGB
Lavar has three markers- one blue, one green, and one yellow. If he arranges them in a row, show the different arrangements he could make. Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-4c) Strategy: Look for a pattern Answer: a) wagon - things that have wheels; b) shoes - words that end in s or things you wear; c) rooster - words that end in r; d) 1700 - add 500; e) 26 - five times the term number plus one
For each of the following, tell in your own words what the pattern rule is. Then write the next term in the sequence. a) Bicycle, car, airplane, roller-skates, __ Pattern rule: b) shirts, ties, socks, jackets, __ Pattern rule: c) river, paper, mother, strainer, __ Pattern rule: d) 200, 700, 1200, __ Pattern rule: e)6,11,16,21, __ Pattern rule: Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-4b) Strategy: Look for a pattern
Answer
Number of Points |
1 |
2 |
3 |
4 |
5 |
6 |
... |
N |
Number of Segments |
2 |
3 |
4 |
5 |
6 |
7 |
... |
N+1 |
In how many segments do "n" points divide a line segment? Explain in detail how you found your answer using words, numbers, and/or pictures.
Number of Points |
1 |
2 |
3 |
4 |
- |
- |
... |
N |
Number of Segments |
2 |
3 |
4 |
- |
- |
- |
... |
N+1 |
6S-4a) Strategy: Look for a pattern Answer: 10 line segments; 15 line segments; 5 line segments
Two points can be connected with 1 line segment. Three points (put them around a circle) can be connected with 3 line segments. Four points can be connected with 6 line segments. How many segments connect five points. Predict, then check, the number of line segments that will connect six points. Predict, then check. the number of line segments that will connect ten points. Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-3e) Strategy: Use logical reasoning Answer: From first to last: #4, #5, #7, #1, #2 and #3, and #6
The main event at the auto races had seven entries. In what order did the cars finish? a) The driver of car #1 was the only one wearing green. b) Car #6 blew a tire and finished last. c) Car #2 and car #3 crossed the finish line together. d) Car #4 beat car #7 by two lengths. e) Only one car finished ahead of car #5. f) The winning car had an even number. g) The driver of car #2 saw green on the driver of the car ahead of him. h) Car #7 finished two lengths ahead of car #1. Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-3d) Strategy: Use logical reasoning Answer: Your busiest day is Friday
Today is going to be your busiest day! You have to meet your friend Sue for lunch at the neighborhood deli, get to the Art Museum, and go to the doctor. In addition, you promised to visit your sick uncle. The deli is closed on Monday, and the art museum is open only on Mondays, Wednesdays, and Fridays. Your doctor has office hours on Thursday, Friday, and Saturday. Your sick uncle can have visitors only on Friday and Saturday. What day is your busiest day? Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-3c) Strategy: Use logical reasoning Answer: Jeannie is the Chemist, Wendy is the Announcer, and Sara is the Doctor
Wendy, Jeannie, and Sara live next to each other. They work as a chemist, a radio announcer, and a doctor. Find each woman's occupation from these clues. a) Jeannie lives in the middle apartment. b) When Sara goes away, her cat is fed by the radio announcer. d) The chemist taps on Wendy's wall when her stereo is too loud. Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-3b) Strategy: Use logical reasoning Answer: Keith is the dancer, William the singer, Bruce the writer, and Dave the artist
Keith, William, Bruce, and Dave all practice fine arts. One of the men is an artist, one is a singer, one is a writer, and the other a dancer. Find the interest of each man using the clues.
a) Keith and Bruce listened while the singer made his debut.
b) Both William and the writer have had their portraits done by the artist.
c) The writer, whose biography of Dave is a best-seller, is planning a biography of Keith.
d) Keith and Bruce do not know each other. Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-3a) Strategy: Use logical reasoning Answer: Debbie own the parakeet or the goldfish(b); Debbie own the parakeet (c); Marty does not own the cat (b); Marty owns the dog (b) and Debbie own the parakeet (c); Debbie owns the parakeet, Marty own the dog, Leo own the goldfish, and Linda owns the cat
Debbie, Marty, Leo, and Linda own four pets: a dog, a cat, a parakeet, and a goldfish. 1) Debbie's pet does not have four legs.
2) Leo owns the goldfish.
3) Marty went to the pet show with the cat's owner. Select the best answer: What conclusion(s) can you draw from statement 1?
a) Debbie owns the parakeet or the dog.
b) Debbie owns the parakeet or the goldfish.
c) Debbie owns the goldfish.
What conclusion(s) can you draw from statements 1 and 2?
a) Debbie owns the goldfish.
b) Debbie and Marty came to the pet show in the same car.
c) Debbie owns the parakeet
What conclusion(s) can you draw from statements l and 3?
a) Leo and Linda are brother and sister.
b) Marty does not own the cat
c) Debbie owns the parakeet
What conclusions call you draw from statements 1, 2, and 3?
a) Linda owns the goldfish.
b) Marty owns the dog.
c) Debbie owns the parakeet
d) Debbie owns the cat Tell who owns each pet.
Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-2e) Strategy: Work backward Answer: He saw 39 sea animals altogether
Claude kept track of what he saw at Point Defiance Zoo. He saw twice as many sharks as dolphins. The number of whales was 6 less than the number of sharks. There were 5 times as many sea lions as there were whales. He saw 20 sea lions. How many did he see altogether? Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-2d) Strategy: Work backward Answer: There are 24 pupils in the class
Half the students in Mrs. Feryn's class are boys. Half the boys have blue eyes. Half the blue-eyed boys have blond hair. There are 3 boys with blue eyes and blond hair in Mrs. Feryn's class. How many students are in that class? Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-2c) Strategy: Work backward Answer: $25
On the quiz show "What Do You Know?" each question is worth four times as much as the previous question. The fourth question is worth $1,600. How much was the first question worth? Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-2b) Strategy: Work backward Answer: 36
Mr. Carroll sells orchid plants. One day he found that 8 of his plants had died. So he bought the same number of plants as were still living. He then divided all the plants into equal packs and sold them to 7 customers. Each customer bought 4 orchid plants. How many had he started with? Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-2a) Strategy: Work backward Answer: 16
Jesse was out playing golf last Thursday. On the first six holes, he lost 4 golf balls. On the next six holes he lost one half of the number of golf balls he had then. On the last six holes, he only lost 2 more golf balls. He finished with 4 golf balls. How many did he start with? Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-1b) Strategy: Simplify Answer: 36
The nine directors of the Whacky Widget Corporation always open their annual board meetings with a special ceremony in which each director shakes hands with each of the other directors. How many handshakes take place? Explain in detail how you found your answer using words, numbers, and/or pictures.
6S-1a) Strategy: Simplify Answer: 108
How many palindromes are there between 0 and 1000? Explain in detail how you found your answer using words, numbers, and/or pictures.
Expectations of Core Processes from the 2008 Math Standards Revision (draft) - Grade 6
Extract and organize mathematical information for a given purpose, such as making conjectures or drawing conclusions. Identify relevant mathematical information in a problem situation and select and use the strategy to solve a problem. Justify the solution process or verify the reasonableness of a solution. Use appropriate representations, symbols, and informal and formal mathematical language to communicate mathematical thinking coherently and clearly. Analyze and compare different mathematical strategies that could be used to solve a problem. |
GUESS, CHECK, AND REVISE (6S-8 …)
Guessing and checking is helpful when a problem presents large numbers or many pieces of data, or when the problem asks students to find one solution but not all possible solutions to a problem. When students use this strategy, they guess the answer, test to see if it is correct and if it is incorrect they make another guess using what they learned from the first guess. In this way, they gradually come closer and closer to a solution by making increasingly more reasonable guesses. Students can also use this strategy to get started, and may then find another strategy which can be used.
DRAW A PICTURE (6S-7 …)
For some students, it may be helpful to use an available picture or make a picture or diagram when trying to solve a problem. The representation need not be well drawn. It is most important that they help students understand and manipulate the data in the problem.
ACT IT OUT OR USE OBJECTS (6S-6 …)
Some students may find it helpful to act out a problem or to move objects around while they are trying to solve a problem. This allows them to develop visual images of both the data in the problem and the solution process. By taking an active role in finding the solution, students are more likely to remember the process they used and be able to use it again for solving similar problems.
MAKE AND USE AN ORGANIZED LIST, TABLE, CHART OR GRAPH (6S-5 …)
Making an organized list, table, chart or graph helps students organize their thinking about a problem. Recording work in an organized manner makes it easy to review what has been done. Students keep track of data, spot missing data, and identify important steps that must yet be completed. It provides a systematic way of recording computations. Patterns often become obvious when data is organized. This strategy is often used in conjunction with other strategies.
LOOK FOR A PATTERN (6S-4 …)
A pattern is a regular, systematic repetition. A pattern may be numerical, visual, or behavioral. By identifying the pattern, students can predict what will "come next" and what will happen again and again in the same way. Sometimes students can solve a problem by recognizing a pattern, but often they will have to extend a pattern to find a solution. Making a number table often reveals patterns, and for this reason is frequently used in conjunction with looking for patterns.
USE LOGICAL REASONING (6S-3 …)
Logical reasoning is really used for all problem solving. However, there are types of problems that include or imply various conditional statements such as, "if.. then," or "if.. then.. else," or "if something is not true, then...” The data given in the problems can often be displayed in a chart or matrix. This kind of problem requires formal logical reasoning as a student works his or her way through the statements given in the problem.
WORK BACKWARD (6S-2 …)
To solve certain problems, students must make a series of computations, starting with data presented at the end of the problem and ending with data presented at the beginning of the problem.
SOLVE A SIMPLER OR A SIMILAR PROBLEM (6S-1 …)
Making a problem simpler may mean reducing large numbers to small numbers, or reducing the number of items given in a problem. The simpler representation of the problem may suggest what operation or process can be used to solve the more complex problem.