Grade 6 Practice Problems: Numbers
Examples from Standards Revision and GLEs
6N-44) Answer: 31 rungs
A firefighter is standing on the middle rung of a ladder. She moved up 9 rungs. The smoke got worse, so she moved back down 13 rungs. When the smoke cleared, she went up 19 rungs to the top. How many rungs does the Ladder have? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-43) Answer: This problem has four answers: 12 boxes of three and 0 boxes of four; 8 boxes of three and 3 boxes of four; 4 boxes of three and 6 boxes of four; and 0 boxes of three and 9 boxes of four
A gardener has 36 poinsettia plants to ship to local florists. He ships them in boxes of 4 plants, or boxes of 3 plants. How might he ship all 36 plants if all boxes shipped must be full? Find as many solutions as you can. Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-42) Answer: There are only three possible ways to score 96. Michelle had 20 correct, 2 incorrect, and 18 omitted; or 22 correct, 7 incorrect, and 11 omitted; or 24 correct, 12 incorrect, and 4 omitted.
A multiple-choice test contains 40 items. A correct response earns 5 points, but 2 points are deducted for every incorrect response. Nothing is given nor deducted if the question is unanswered. Michelle scored 96. How might she have gotten this score? Find as many solutions as you can. Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-41) Answer: 37 miles
Along a straight path from Raleighville to Urbantown the Jones family passes through Sierra and Timeless. The distance from Raleighville to Urbantown is 85 miles. From Raleighville to Timeless is 20 miles, and from Raleighville to Sierra is 48 miles. How far is it from Sierra to Urbantown? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-40) Answer: $20 profit
An antique dealer bought an old clock for $30 and sold it for $40. He later bought it back for $50, and then resold it for $60. Did he make money or lose money after completing these transactions? How much? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-39) Answer: The last 6 people in line cannot take the ride before it closes.
At the amusement park, there is a ride through the Haunted House. Each boat holds 8 people, and there are 3 boats. The ride takes 10 minutes. There are still 150 people in the line, and the ride closes in exactly one hour. Will all the people be able to ride through the Haunted House? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-38) Answer: David 7 coins; Mike 1 coin
David and Mike went fishing. David brought 5 hoagie sandwiches and Mike brought 3 hoagie sandwiches. As they were getting ready to eat, their friend, Jerry, joined them. However, Jerry did not bring any hoagies with him. After lunch, during which the three men shared the 8 hoagies equally, Jerry put down 8 coins of equal value to pay for his share of the lunch. How should David and Mike share the 8 coins? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-37) Answer: 3 (Cars #24, #48, #72)
Dominic's Automobile Agency has 80 new cars lined up in their lot. Every third car has a vinyl roof. Every fourth car has a stick shift, and every eighth car has a security system. Audrey wants to buy a car with all three of these options. How many cars at Dominic's will meet her needs? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-36) Answer: 106
Georgette is writing the page numbers on a handwritten edition of a book that contains 250 pages. How many times will she write the numeral 2? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-35) Answer: 220
In a bowling tournament, Elton scored 188, 212, 214, 196, and 200 in the first five games. In order to quality for the semi-final round, he must average at least 205 for the six games. What is the least he can bowl in his final game to qualify? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-34) Answer: 1 or 3
In a game at the County Fair, each color marker has a different point value: black = 2, green = 3, and red =5. Howard has a score of 55 points. He has 12 green markers. How many red markers does Howard have? Is there more than one solution? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-33) Answer: 4 cars
In our school's parking lot one morning, Janice noticed that all but three of the cars parked were made by General Motors, all but three of the cars were made by Chrysler, all but three of them were made by Ford, and all but three of them were made by Toyota. What is the minimum number of cars parked in the school parking lot? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-32) Answer: Yes - saved $72.60
Julie's birthday is May 1. She wants to buy a new camera that costs $70.00 as a present to herself. She decides to save 1 cent on January 1st, 2 cents on January 2nd, 3 cents on January 3rd, 4 cents on January 4th, and so on. Will she have enough to buy the camera on her birthday? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-31) Answer: 11 cassettes
Keith wants to join a cassette club. The Liberty Cassette Club has an offer of a $20 membership initiation fee and only $6.20 per cassette. The Patriot's Cassette Club has no membership initiation fee, but charges $8.10 per cassette. How many cassettes must Keith buy to make joining the Liberty Cassette Club the better deal? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-30) Answer: There are 12 different answers to this problem. Day 1,2,3: 1,2,12; 1,3,11; 1,4,10; 1,5,9; 1,6,8; 2,3,10; 2,4,9; 2,5,8; 2,6,7; 3,4,8; 3,5,7; 4,5,6
Last week, Laura and her dad went fishing for three days. They caught a total of 15 fish. They caught a different number of fish each day. How many fish did they catch each day? Find as many possible solutions as you can. Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-29) Answer: 800,249
I have six digits. My hundred-thousands digit is 1 less than my ones digit, 8 more than my thousands digit, twice my tens digit, and 4 times my hundreds digit. What number am I? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-28) Answer: Either $6-$6-$6 or $5-$6-$7
Matt is selling baseball cards at a local flea market. Rookie cards sell for $5, Champion cards sell for $6, Old-Timer cards sell for $7, and Hall-of-Famer cards sell for $9. Renee bought 3 cards and spent $18. What cards did she buy? Find as many possible solutions as you can. Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-27) Answer: 14 plants
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.
6N-26) Answer: 7; prime-numbered dates greater than 5 (i.e. 7, 11, 13, …)
Mrs. Taylor has a new way of giving homework problems. If the date is divisible by 2, she gives 2 problems. If the date is divisible by 3, she gives 3 problems. If the date is divisible by 5, she gives 5 problems. Otherwise, she doesn't give any problems on that day. On the 6th of the month, she gave 5 problems, since 6 is divisible by both 2 and 3. How many problems did she assign on the 10th? On which dates would there be no problems assigned? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-25) Answer: $2
On Monday, Mary, Joyce, Gwen, and Robin bought a gift for Phyllis that cost $40. They shared the cost of the gift equally. On Tuesday, Anita asked to join group. They agreed to share the cost of the gift five ways, instead of four. How much must Anita give each of the other four girls? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-24) Answer: $25
On the quiz show "WHO WANTS TO BE A BILLIONAIRE?" 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.
6N-23) Answer: 40 outfielder cards
Renee collects baseball cards. She has 240 cards in her collection. For every outfielder card, she has 5 cards of players from other positions. How many cards with outfielders does she have? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-22) Answer: there are 5 possible answers: 31 and 32; 30 and 33; 29 and 34; 28 and 35; 27 and 36.
The sum of two whole numbers is 63. The difference between the numbers is less than 10. Find as many solutions as you can. Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-21) Answer: 26 checks
The local bank charges a monthly service charge of $3.50 plus 8 cents for each check. Last October, Michelle's total charges were $5.58. How many checks did she write? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-20) Answer: 9 days overdue
The local library fine schedule for overdue books is as follows: 25 cents per day for each of the first three days; 10 cents per day thereafter Sandra paid a fine of $1.35. How many days overdue was her book? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-19) Answer: number 15
The players on the Hawks hockey team wear the numbers from 1 through 18 on the backs of their jerseys. On opening night, as the players were being introduced, Maura noticed that the players were standing in nine pairs. She also noticed that the sum of the numbers on the jerseys of each pair was a perfect square (an integer times itself). The goalie wore number 1. With what numbered player was he paired? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-18) Answer: 9 packages of plates and 7 packages of cups
The Stewarts are buying cups and plates for their huge annual family picnic. Cups come in packages of 54, whereas plates come in packages of 42. How many packages of each must the Stewarts buy to have the same number of cups and plates? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-17) Answer: there were 1200 students in all; 400 went to Washington, D.C.; 400 went to Annapolis; 300 went to Williamsburg; 100 went to Monticello.
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? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-16) Answer: 41 students
There are 23 students in the school orchestra. There are 25 students in the school band. Seven of these students are in both. How many students are there in both band and orchestra? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-15) Answer: wagon #1 will have 8+7, wagon #2 will have 4+5+6, wagon #3 will have 1+2+3+9
There are 9 bags of aluminum cans to be taken to the recycling center. The bags contain 1, 2, 3…8, 9 pounds of cans, respectively. They are going to place the bags into three wagons so that each wagon will carry the same weight. How should they do it? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-14) Answer: $50.26
Mrs. Kim needs to feed her baby 4 ounces of baby formula every three hours. A 16-ounce can of formula is $3.59. How much will it cost Mrs. Kim to feed her baby for a week? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-13) Answer: Friday
Today is really your busiest day! You have to meet your buddy Mark for lunch at the neighborhood deli, get to the Art Museum, and go to the dentist. In addition, you promised to visit your sick aunt. The deli is closed on Monday, and the art museum is open only on Mondays, Wednesdays, and Fridays. Your dentist has office hours on Thursday, Friday, and Saturday. Your sick aunt 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.
6N-12) Answer: 132
When a man went to visit a friend, he realized he had forgotten the apartment number. He does remember the apartment is on the first floor where the apartments are numbered 101 through 150. He also knows the apartment number is an even number and a multiple of 3 and 11. What is the correct apartment number? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-11) Begin with the number 1,234. Rearrange two adjacent numbers at a time. (“Adjacent” means “next to one another”.) What is the fewest number of steps it will take before you can write the number 4,321? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-10) How many rectangles are in a 2x3 grid? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-9) Janice has a 5-card hand containing only 7s, 8s, and 9s. The sum of the numbers in her hand is 41. How many 7s, 8s, and 9s does Janice have? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-8) Luis and some friends are sitting around a table. The art teacher hands Luis a box with 7 crayons in it. He takes 1 crayon and passes the box to the next person, Maria. She takes 1 crayon and passes the box to the next person. The box is passed around until all 7 crayons have been given out. Luis took the first crayon and also the last one. How many people are sitting around the table? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-7) Penny sells shoes at the local store and recently set up a display to help sell shoes. One third of the shoes in the display are high-tops. One third of the shoes are boots, and one sixth of the shoes are flats. Eight of the shoes are sandals. How many shoes are in the display? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-6) Place 20 pennies on the table in a row. Now replace every fourth coin with a nickel. Now replace every third coin with a dime. Now replace every sixth coin with a quarter. What is the value of the 20 coins now on the table? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-5) Sally was given 5 cents for each test she passed. But she had to return 10 cents for each test she failed. At the end of 3 months she passed four times as many tests as she failed and she had earned 20 cents. How many tests did she pass? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-4) Some children were seated around a table having milk and cookies. There were 18 cookies in all. Each child received the same number of cookies. How many children were sitting around the table? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-3) Ted has violet, yellow, pink, and red flowers at home. The ratio of violet flowers to yellow is 3 to 5. The ratio of yellow to pink is 2 to 4. The ratio of violet flowers to red is 2 to 5. There are 6 violet flowers. How many flowers of each color is Ted growing. Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-2) There are 32 people at a party. Someone suggests each person shake hands with everyone else once as they leave. How many handshakes are there? Explain in detail how you found your answer using words, numbers, and/or pictures.
6N-1) They are putting a fence around Mr. Rogers's farm. Each fence post has 1/4 of its length buried in the ground. Six feet of the post is above the ground. How long is each fence post? Explain in detail how you found your answer using words, numbers, and/or pictures.
Expectations & Examples of Numbers from the 2008 math Standards Revision 9draft) - Grade 6
Compare and order non-negative rational numbers in both fraction and decimal forms using number lines, ordered lists, or the symbols <, >, or =.
Identify and represent ratios as comparisons of part-to-part and part-to-whole relationships, and solve problems involving ratios.
Identify and represent rates and solve problems involving rates.
Represent percents in various forms using numbers, pictures, models, or circle graphs and solve problems involving percent.
Translate among the fraction, decimal, and percent forms of a number.
Solve contextual problems using ratios, rates, or percents and verify the reasonableness of the solution.
Identify the ratio of the circumference to the diameter of a circle as the constant p , and recognize 22/7 and 3.14 as common approximations of p . Write the prime factorization of any whole number using exponents when appropriate. |
Expectations and Examples of Operations from the 2008 Math Standards Revision (draft) – Grade 6
Represent multiplication and division of positive fractions and decimals using numbers, pictures, or models, and translate among representations.
Use estimates of fractions and decimals to approximate products and quotients and to determine the reasonableness of solutions. Example: Therefore, since ½ x 1 is ½, the product of Multiply and divide non-negative fractions using the standard algorithms. Multiply and divide non-negative decimals using the standard algorithms. Describe the effect of multiplying or dividing a number by 0, 1, a number between 0 and 1, and a number greater than 1. Solve contextual problems involving fraction and decimal computations and justify the solution.
Apply the commutative, associative, and distributive properties, and use the order of operations to evaluate mathematical expressions. Examples:
Describe and use strategies for mental computation involving addition, subtraction, multiplication, and division of non-negative decimals and fractions. |
is a little less than ½, and 7/6 is a little more than 1;
should be about ½.
Examples of Number Sense from the 2006 GLEs – Grade 6
Explain or illustrate integer values using words, pictures, models, and symbols. Explain the meaning of integers and gives examples. Locate the additive inverse of a given integer on a number line. Order different representations of fractions, decimals, and/or percents. Show and determine equivalence between non negative integers, fractions, decimals, and percents using words, pictures, models, and symbols. Order integers, fractions, decimals, and/or percents and explain why one number is greater than, less than, or equal to another. Explain when a fraction, decimal, or percent of one whole is not the same as the same fraction, decimal, or percent of a different whole. Illustrate and explain the commutative, associative, and identity properties of addition and multiplication and the zero property of multiplication on non negative decimals and fractions. Use addition and multiplication properties to assist in computations. Determine whether a computation is reasonable based on application of the commutative, associative, and identity properties of addition and/or multiplication. Write or show and explain ratios in part/part and part/whole relationships using words, objects, pictures, models, and/or symbols. Represent equivalent ratios using objects, pictures, or symbols. Represent equivalent percentages using objects, pictures, and symbols. Express or represent percent as a ratio based on 100 equal size parts of a set. Explain ratio and percents and give examples of each. Create a ratio equivalent to a given ratio to determine an unknown value for a dimension or a number of events or objects. Explain or show the meaning of multiplying and dividing non negative fractions and decimals using words, pictures, or models. Explain the effect of multiplying a whole number by a decimal number. Explain why multiplication of fractions involves multiplying denominators. Demonstrate how multiplication and division with decimals affects place value. Explain remainders of a division problem in a given situation. Translate a picture or illustration into an equivalent symbolic representation of multiplication and division of non negative fractions and decimals. Select and/or use an appropriate operation to show understanding of addition, subtraction, multiplication, or division of non negative rational numbers. Find the sums or differences of non negative fractions or decimals. Find sums or differences of decimals or fractions in real world situations. Use the least common multiple and the greatest common factor of whole numbers to simplify or compute with fractions. Calculate sums of two numbers with decimals to the thousandths or three numbers with decimals to hundredths. Calculate difference between numbers with decimals to thousandths. Complete multiple step computations requiring addition and/or subtraction. Select and use appropriate strategies and tools from among mental computation, estimation, calculators, manipulatives, and paper and pencil to compute in a given situation. Explain why a selected strategy or tool is more efficient or more appropriate than another strategy or tool for a situation. Describe strategies for mentally adding and/or subtracting non negative decimals and fractions. Explain whether estimation or exact calculation is appropriate in situations involving addition and subtraction of non negative decimals and fractions. Use a variety of estimation strategies prior to computation to predict an answer. Use estimation to verify the reasonableness of calculated results. Compute to check the reasonableness of estimated answers for a given situation. Explain an appropriate adjustment when an estimate and a computation do not agree. Explain or describe a strategy for estimation involving addition and subtraction of non negative decimals and fractions. |