Grade 4 math arkansas 04 + answers

PART II Released Mathematics Items CALCULATOR NOT PERMITTED — ITEMS 1–8 AR4M02N107NXXX-026A AR4M01P102Y012-012B Which number will make the equation true? Mr Preddy put the chart below on the board He asked his students, “If the pattern continues, which pair of numbers would come next?” What answer should the students have given? (10 + n) – (2 × 3) = * A B C 10 D 16 20 A 55, 10 * B 60, 12 C 75, 15 D 90, 18 PART II Released Mathematics Items AR4M01D202Y110-110C The school librarian made a graph showing the number of books checked out each month by students in the fourth grade In April, there were 45 books checked out How many books should be displayed in the graph? A 05 B 06 * C 09 D 45 AR4M02M303NXXX-160A AR4M02N203NXXX-067A Last week, Mark’s pet bunny weighed 12 ounces This week, it had gained more ounces How much does it weigh now? * A pound B ounces C 16 pounds D 48 ounces Kisha visited the Aerospace Education Center in Little Rock with her family She bought a toy airplane for $4.88 in the gift shop She gave the salesperson $5.00 How much change should Kisha receive? 21 * A $0.12 B $0.22 C $1.12 D $1.88 PART II Released Mathematics Items Use the table below to answer question Use the figure below to answer question Hector’s Table Approx Miles From Little Rock State Park Pinnacle Mountain Woolly Hollow Village Creek Crater of Diamonds Bull Shoals 10 40 90 100 120 AR4M02G103Y088-088A Which figure is the flip image of Figure A above? AR4M02D102Y053-053B * Hector made the table above for his family to compare the approximate distance from their home in Little Rock to several state parks How many more miles is it to Village Creek State Park than to Woolly Hollow State Park? A 40 * B 50 C 80 D 90 Use the calendar below to answer question Sun MAY Wed Thu Mon S Tue 10 16 21 15 S 22 28 29 14 Fri Sat 11 12 13 17 18 19 20 23 24 25 26 27 30 31 AR4M02P102Y086-086B Jane has swim practice on the first and third Monday of each month What will be the date of her first swim practice in June? A 1st * B 5th C 15th D 29th 22 PART II Released Mathematics Items CALCULATOR PERMITTED — ITEMS 9–40 AR4M01D301NXXX-089C Cliff brought cupcakes to share with his classmates He brought 12 chocolate and 13 vanilla What are Jane’s chances of getting a chocolate cupcake? A 01 out of 25 B 01 out of 12 * C 12 out of 25 D 12 out of 13 Use the figure below to answer question 12 AR4M02G106Y167-167C 12 Annie made a pattern block design using hexagon and trapezoids How many equilateral triangles will cover this design? (You may use your pattern blocks to help you find the answer.) AR4M02P101NXXX-085D 10 Mrs Andrew’s class collected canned food for a local food drive They wanted to sort the cans before they donated them Which of the following is not a way they could group the cans? A size B weight C type of food * D A B * C 12 D 14 Use the information below to answer question 13 total number of cans AR4M02M301Y009-009C SCHEDULE FOR FEEDING SNAKES AR4M02N104NXXX-105C 10:00 A.M 12:00 P.M 2:00 P.M 4:00 P.M 11 Marty is about to begin reading the 250-page book she just checked out from the library She plans to read 25 pages per night At this rate, how many nights will it take her to finish her book? A B * C 10 D 20 13 Sandra’s fourth-grade class went to the Little Rock Zoo on a field trip Her group arrived at the snake exhibit at 10:25 A.M How long will the group have to wait to see the next snake feeding? A hours B 35 minutes * C D 23 hour, 35 minutes hours, 35 minutes PART II Released Mathematics Items Use the figure below to answer question 14 Use the chart below to answer question 15 AR4M02G107Y169-169C Pattern Chart A B 12 12 24 24 48 ? ? AR4M02P102Y055-055D 15 Mrs Gordon put the chart above on the board What pair of numbers would come next in the pattern? 14 The figure above is made with cubes which are cm3 What is its volume? A cm3 B 12 cm3 * C 18 cm3 D 21 cm3 24 A 30, 60 B 30, 96 C 48, 60 * D 48, 96 PART II Released Mathematics Items Use the figure below to answer question 16 AR4M02M302Y165-165A 16 How long is the rod above, to the nearest half inch? * A in B in C 11 cm D 11 cm AR4M01N102NXXX-042B 18 A runner can burn 1,280 calories in 60 minutes Which of the following number sentences would you use to find out how many calories can be burned in 30 minutes of running? Use the diagram below to answer question 17 A * B AR4M01G201Y005-005C 17 The basketball court above is 94 feet long and 50 feet wide What is the perimeter of this basketball court? A 0,144 ft B 0,188 sq ft * C D 1,280 × 30 = 1,280 ÷ = C 21,380 − 30 = D 1,280 × = AR4M02D201NXXX-051B 19 Melissa is first in line to go on the pony ride at the state fair There are brown ponies, tan ponies, and black ponies What is the probability that she will get to ride on a black pony? 0,288 ft 4,700 sq ft 25 A * B C D PART II Released Mathematics Items AR4M02M301NXXX-080B AR4M02N202NXXX-016B 20 Kathy is wearing a heavy coat, scarf, mittens, and a hat Which temperature is it most likely outside? 23 What number comes next in the pattern? 1, 2, 4, 7, 11, A 35°C A 15 * B 35°F * B 16 C 68°C C 18 D 68°F D 22 AR4M01M103NXXX-040A AR4M02G104Y040-040D 21 One morning it took Carlos 167 seconds to brush his teeth How many minutes and seconds did it take him to brush his teeth? 24 Which of the following has the most lines of symmetry? * A minutes and 47 seconds B minutes and 67 seconds C minutes and 17 seconds D 16 minutes and seconds AR4M0P102NXXX-122D 22 Ashley is visiting her grandparents for the weekend Her mom packed pair of black pants, pair of tan shorts, and shirts: white, blue, green, and yellow How many different outfits can she make from the clothes her mom packed? A B C * D 26 A B C *D PART II Released Mathematics Items AR4M02M103NXXX-157A 27 Jim’s fish tank holds 40 quarts of water How many gallons of water does that equal? Use the chart below to answer question 25 The Population of Pulaski County, Arkansas 1960–2000 B 20 Population C 40 1960 242,980 D 60 1970 287,189 1980 340,613 1990 349,660 2000 361,474 AR4M02G106NXXX-168C 28 Mr Taylor told his math students to guess what pattern block he was holding He gave the following clues: • • • 25 According to the table above, which of the following is a true statement about the population of Pulaski County, Arkansas? * B 10 Year AR4M02D102Y061-061B A * A What is it? It has remained the same It has increased every ten years C It has increased every five years D It has decreased every ten years It has sides All of its sides are equal It has no right angles A triangle B hexagon * C rhombus D trapezoid AR4M01N101NXXX-077C 26 Mrs Kramer wrote the following number on the board: 9,463,187 AR4M02P105NXXX-149B 29 Ms Smith wants to cover the floor of her sun porch with tiles Which of the following is the correct formula for finding the area of the porch? What is the value of the in this number? A 00,006 B 06,000 * C 60,000 D 63,000 27 A l+w=A * B l×w=A C 2l + 2w = A D l+l+w+w=A PART II Released Mathematics Items AR4M02G204Y045-045A 32 I am Sam the Robot My head is a hexagon My legs are right triangles My body is a trapezoid Which robot am I? Use the table below to answer question 30 Lengths of Snakes at City Zoo Black Rat Snake 72" King Snake 48" Worm Snake 11" Scarlet Snake 15" Brown Snake 13" Ring-neck Snake 15" * AR4M02D302Y050-050B 30 The table above shows the lengths of the snakes at City Zoo What is the mode of these lengths? A 6" * B 15" C 29" D 57" AR4M02P105NXXX-148D 31 What is the rule for the pattern below? 14 17 34 37 A Add 2, then add in a repeating pattern B Double each number C Double the first number, then double the next number, then subtract * D Double the first number, then add three to the next number 28 PART II Released Mathematics Items Use the table below to answer question 33 Use the chart below to answer question 35 AR4M02D102Y100-100B Super Cinema Theaters Adult admission Children Large popcorn Small popcorn Large soda Small soda AR4M02N203Y042-042B 33 Alicia took a survey of the fourth-graders in her class to find out which of the four snacks above they liked best Which of the following statements is true about the data Alicia collected? A 35 Mr Dunbar took his five-year-old daughter to the movies After paying for the admission for each of them, he bought a large popcorn and small sodas How much did he spend? No fourth-graders liked yogurt * B More fourth-graders like cheese and crackers than chips C Fourth-graders like chips better than yogurt D More fourth-graders prefer yogurt to fruit $6.25 $3.75 $4.00 $3.00 $1.89 $0.99 A $14.98 * B $15.98 C $16.88 D $17.98 AR4M02G103NXXX-041A 36 Pinnacle Mountain has a height of 1,011 feet What would be its height in yards? AR4M02M101NXXX-093A 34 Sandra brought a bottle of soda to share at her class party What is most likely the size of the bottle? * A liters B 20 liters C milliliters D 20 milliliters * A 337 yards B 975 yards C 1,014 yards D 3,033 yards AR4M02P104NXXX-142D 37 What number does n represent in order for the statement below to be true? 6×3=2×n 29 A B C * D PART II Released Mathematics Items AR4M02M301NXXX-095B AR4M02N103NXXX-033C 38 Pat’s coach told the team to arrive at the ballpark 45 minutes before their baseball game The game begins at 6:00 P.M What time should Pat arrive at the ballpark? 39 Alex went to a movie His ticket cost $5.75, and he bought a popcorn and soda for $3.50 His change was $10.75 What amount of money did he take to the movie? A 4:45 P.M A $ 9.25 * B 5:15 P.M B $10.00 C 5:30 P.M * C $20.00 D 6:45 P.M D $21.25 Use the chart below to answer question 40 AR4M02D202Y108-108A 40 Molly made the pictograph above to show how many books were read by all of the fourth-grade students at Burton Elementary Students in Ms Jarrell’s class read a total of 250 books How many books should Molly display in the row representing Ms Jarrell’s class? * A 10 B 25 C 250 D 275 30 PART II Released Mathematics Items MATHEMATICS OPEN-RESPONSE ITEM A AR4M01D301NXXX-162R A Each of the four counselors at Camp Lake View must choose one camper from his/her group to go waterskiing Each counselor will put the names of all his/her campers in a bag One name will be drawn from the bag to select the “winner.” Each group consists of the following: ● ● ● ● Lonnie’s group: 10 boys and girls Maria’s group: boys and 10 girls James’s group: boys and girls Tonya’s group: boys and 12 girls Which of the four groups offers the best chance that a boy will win? Explain your answer using words and/or numbers Which of the four groups offers the best chance that a girl will win? Explain your answer using words and/or numbers BE SURE TO LABEL YOUR RESPONSES AND RUBRIC FOR MATHEMATICS OPEN-RESPONSE ITEM A SCORE B DESCRIPTION The student earns points The response contains no incorrect work The student earns – ½ points The student earns – ½ points The student earns ½ – ½ points or shows some minimal understanding No understanding is shown Blank – No response A score of “B” will be reported as “NA” (No Attempt – Zero Score) 31 PART II Released Mathematics Items Solution and Scoring Part 1: (2 points possible) point for correct answer of Lonnie’s group, AND point for correct and complete work or explanation ● Give credit for the following explanations or equivalent: o “Lonnie’s has more boys than any other group and all groups have the same number of kids (or 18 kids),” or o “They are the only group with more boys than girls,” or o “They have the greatest number of boys and the fewest number of girls of any group,” or o “10/18 are boys,” or “There are 10 chances out of 18.” OR ½ point for correct, but incomplete work or explanation ● Example: “Lonnie’s group has the most boys (or more boys) than any group Note: Do not give credit for the following incomplete explanation ● “Lonnie’s group has more boys than girls.” (Doesn’t say “only group.”) Part 2: (2 points possible) point for correct answer of Tonya’s group, AND point for correct and complete work or explanation ● Give credit for the following explanations or equivalent: o “Tonya’s has more girls than any other group and all groups have the same number of kids (or 18 kids),” or o “They have the greatest number of girls and the fewest number of boys of any group,” or o “12/18 are girls,” or “There would be 12 chances out of 18.” OR ½ point for correct, but incomplete work or explanation ● Example: “Tonya’s group has the most girls (or more girls) than any group.” Note: Do not give credit for the following incomplete explanation ● “Tonya’s group has more girls than boys.” (So does Maria’s group.) 32 PART II Released Mathematics Items MATHEMATICS OPEN-RESPONSE ITEM B AR4M02G201Y132-132R B A group of students is helping to paint the basketball court at their neighborhood recreation center They need to find the area of the court in order to buy enough paint They measured the court and found the length to be 80 feet and the width to be 50 feet What is the area of the court? Explain your answer using words, numbers, and/or pictures They need to tape the outer edges before painting They need to find the perimeter so they will know how much tape they need How much tape will they need? Explain your answer using words and/or numbers BE SURE TO LABEL YOUR RESPONSES AND 33 PART II Released Mathematics Items RUBRIC FOR MATHEMATICS OPEN-RESPONSE ITEM B SCORE B DESCRIPTION The student earns points – Answers must have correct labels The student earns points The student earns points The student earns point or shows some minimal understanding Example: A and/or P are reversed with work shown or explained No understanding is shown Blank – No response A score of “B” will be reported as “NA” (No Attempt – Zero Score) Solution and Scoring Part 1: (2 points possible) point for correct answer of 4,000 (with or without square feet), AND point for correct and complete procedure for finding area shown and/or explained ● Give credit for the following explanations or equivalent: o Area is found by multiplying the length by the width The length is 80 feet and the width is 50 feet Therefore, the area is 4,000 (square feet), or o A = 80 x 50 = 4,000 (square feet), or o Correct procedure (l x w = A) to find area is used, but answer is incorrect due to a calculation or copy error Note: Do not give credit for copying the formula l x w = A only Part 2: (2 points possible) point for correct answer of 260 (with or without feet), AND point for correct and complete procedure for finding perimeter shown and/or explained ● Give credit for the following explanations or equivalent: o Perimeter is found by multiplying the length and the width times two, then adding these two numbers Two times 80 is 160 Two times 50 is 100; therefore 160 added to 100 is 260 feet, or o P = 2l + 2w = 2(80) + 2(50) = 160 + 100 = 260 (feet), or o Diagram correctly labeled on all four sides (l = 80, w = 50), or o Correct procedure (l + l + w + w = P) to find perimeter is used, but answer is incorrect due to a calculation or copy error Note: Do not give credit for copying the formula l + l + w + w = P or 2l + 2w = P only 34 PART II Released Mathematics Items MATHEMATICS OPEN-RESPONSE ITEM C AR4M02M102NXXX-161R C Barb and William are each growing sweet potato vines Barb’s plant measures one-and-a-half feet long William’s plant measures 24 inches long Whose plant is longer? Explain your answer using words and/or numbers How much must William’s plant grow before it is one yard long? Explain your answer using words and/or numbers BE SURE TO LABEL YOUR RESPONSES AND RUBRIC FOR MATHEMATICS OPEN-RESPONSE ITEM C SCORE B DESCRIPTION The student earns points – Answers must have correct labels The student earns points The student earns points The student earns point or shows some minimal understanding No understanding is shown Blank – No response A score of “B” will be reported as “NA” (No Attempt – Zero Score) Solution and Scoring Part 1: (2 points possible) point for correct answer of William’s plant is longer ● Note: Do not give credit if there is clear evidence of incorrect procedure AND point for correct conversion of Barb’s ½ foot plant into 18 inches or William’s 24 inch plant into feet shown and/or explained ● Give credit for any of the following explanations or equivalent: o 12 + = 18 and 24 >18, or o 24/12 = and >1 ½, or o “Twenty-four inches equals two feet, so William’s plant is longer,” or o “His plant is longer since a foot and a half would only be eighteen inches.” ● Do not give credit for the following incomplete explanations: o “His plant is inches longer,” or o “His plant is longer since 24 inches is longer than ½ feet.” 35 PART II Part 2: Released Mathematics Items (2 points possible) point for correct answer of “12 inches” or “1 foot.” Label must be correct AND point for correct procedure shown and/or explained ● Give credit for any of the following explanations or equivalent: o There are 36 inches in a yard William’s plant is 24 inches long The difference is 12 inches, or o yard = feet and he has feet Therefore, it needs to grow foot, or o yard = 36 inches 36 - 24 = 12 inches OR ● Answer is incorrect due to a calculation or copy error, but correct procedure is used o Example: 36 - 24 = 11 inches ♦ Note: Credit will be received for the procedure, but not for the answer 36 PART II Released Mathematics Items MATHEMATICS OPEN-RESPONSE ITEM D AR4M02P102Y124-124R D Dwight and his brother, Jim, decided what they would like to eat for breakfast every day for a month They each began filling in the calendar above with their favorite breakfast foods If Dwight’s pattern on the calendar above continues, how many times during the month will he have eggs? Explain your answer using words, numbers, and/or pictures If both Dwight’s and Jim’s patterns continue, how many days during the month will both boys have the same thing for breakfast? Explain your answer using words, numbers, and/or pictures BE SURE TO LABEL YOUR RESPONSES AND 37 PART II Released Mathematics Items RUBRIC FOR MATHEMATICS OPEN-RESPONSE ITEM D SCORE B DESCRIPTION The student earns points The student earns points The student earns points The student earns point or shows some minimal understanding No understanding is shown Blank – No response A score of “B” will be reported as “NA” (No Attempt – Zero Score) Solution and Scoring Part 1: (2 points possible) point for correct answer of Dwight eats eggs 10 times, AND point for correct and complete procedure shown and/or explained ● Give credit for the following explanations or equivalent (Answer might not be “10.”): o “He has eggs on the 2nd ,5th ,8th, 11th, 14th, 17th, 20th, 23rd, 26th, and 29th;” or o “He has eggs on the 2nd and every third day after that;” or o 31/3 = 10 r1; or o Correct number of times eggs are eaten each week (1 + + + + 2); or o Calendar is correctly filled out indicating days when Dwight eats eggs (Dates not have to be circled.) Note: Do not give credit for incorrect or incomplete procedures: ● Student continues the pattern past May 31st (Incorrect procedure This usually follows an answer of “He has eggs 11 times.”) ● “The pattern is CEW,” or “I did the pattern.” (Incomplete procedure.) 38 PART II Part 2: Released Mathematics Items (2 points possible) point for correct answer of they will eat the same things 11 times, AND point for correct and complete procedure shown and/or explained ● Give credit for the following explanations or equivalent (Answer might not be “11.”): o “They will both have – ♦ Cereal on the 1st, 7th, 13th, 19th, 25th, and 31st; and ♦ Waffles on the 6th, 12th, 18th, 24th, and 30th,” or o “They will eat the same things on the following dates: 1, 6, 7, 12, 13, 18, 19, 24, 25, 30, and 31,” or o “They will both have cereal on the first day of the month and every sixth day after that They will both have waffles every sixth day,” or o Correct number of times they have the same breakfast each week (1 + + + + 2), or o Calendar is correctly filled out indicating days when they eat the same thing (Dates not have to be circled.) 39 PART II Released Mathematics Items MATHEMATICS OPEN-RESPONSE ITEM E AR4M02N107Y130-130R E Jake used the 100’s chart above to solve a mystery number puzzle with the following clues: ● It is a two-digit number greater than 23, but less than 50 ● It is an even number ● Two, four, and twelve are some of its factors ● The sum of its digits is a two-digit number What is the mystery number? Use words, pictures, and/or numbers to explain your answer RUBRIC FOR MATHEMATICS OPEN-RESPONSE ITEM E SCORE DESCRIPTION Correct answer of 48 Work and/or explanation is correct and contains: • Elimination Clue by Clue Method: Numbers, charts, lists, calculations and/or words clearly show or explain the procedure of elimination Clues and may be combined At a minimum, the numbers remaining after performing Clue (24, 36 and 48) or Clue if reversed (28, 38, 46 and 48) are listed OR • “48 is the number that fits the Clues” Method: Four clues are addressed or repeated Work is shown for Clue AND Clue Elimination may be partial or missing o Example: “It’s 48 because it’s between 24 and 49, it’s even, 2, and 12 are factors because 48/2 = 24 and 48/12 = 4, and the sum of its digits is a 2-digit number since + = 12 40 PART II Released Mathematics Items Correct answer of 48 Work and/or explanation is incomplete, but is beyond repeating clues • Elimination Method is shown or explained (for at least Clues and 2), but elimination using other clues is missing or incomplete in some respect Three to four clues are addressed or repeated o Example: Verbal description of elimination clue by clue, but the numbers remaining after performing Clue (or Clue if reversed) are not listed o Example: 23-50 are listed, with all numbers crossed out except 48 without differentiation between clues or explanation o Example: Chart lists numbers between 23 and 50 and eliminates odd numbers Clue and/or Clue are addressed or repeated OR • “48 is the number that fits the Clues.” Method is incomplete, but some work is shown for either Clue OR Clue Three to four clues are addressed or repeated Elimination of other numbers is partial or missing: o Example: “It’s 48 because it’s between 24 and 49, it’s even, 48/2 = 24 and 48/12 = 4.” • Answer of 24, 36 and 48 Response explains process of elimination by going through Clues 1, and 3, with 24, 36 and 48 remaining (Does not proceed far enough – Clue is not addressed.) OR • Answer of 28, 38, 46 and 48 Response explains process of elimination by going through Clues 1, and 4, with 28, 38, 46 and 48 remaining (Does not proceed far enough – Clue is not addressed.) OR • Correct answer of 48 Three to four clues are addressed or repeated with no supporting work for Clue and no supporting work for Clue o Example: “It’s 48 because it’s between 24 and 49, it’s even, 2, 4, and 12 are factors and the sum of its digits is a 2-digit number.” OR • Correct answer of 48 One to two clues are addressed or repeated, with or without work or elimination shown and/or explained o Example: “It’s 48 because it’s between 24 and 49 and it’s even.” • Answer of 48 No clues are addressed, no work is shown or explained, or work contains vague, but not incorrect, procedure OR • Answer is an even number between 23 and 50 (≠ 48) At least one clue is addressed o Example: “It’s 44 because it’s greater than 23 and less than 50.” o Example: “It’s 24 because 2, 4, and 12 are its factors.” No understanding shown o Example: 48 with obvious use of incorrect procedure o Example: An even number (≠ 48) between 23-50 with no clues addressed: “It’s 44.” o Example: Any odd number B Blank – No response A score of “B” will be reported as “NA” (No Attempt – Zero Score) 41 PART II Released Mathematics Items Solution and Scoring Answer: The mystery number is 48 Explanations may be: ● Numerical – o Clue 1: All numbers from 24 to 49 (or 23 to 50) are listed o Clue 2: All odd numbers are eliminated: 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48 remain o Clue 3: All numbers that not have two, four and twelve as factors are eliminated: 24, 36 and 48 remain o Clue 4: All numbers whose digit sum is not a two-digit number are eliminated: only 48 remains Note: Some students perform Clue before Clue (after Clues and 2) ♦ Example: 28, 38, 46 and 48 remain since the digit sum is a 2-digit number Then, the only number remaining with factors of 2, and 12 is 48 OR ● A chart – The numbers are listed and eliminated clue by clue The clues are differentiated in some way OR ● Verbal – The response contains a correct and complete verbal explanation of the procedure using elimination clue by clue, with at least 24, 36, and 48 mentioned (or 28, 38, 46, and 48 if Clue is performed before Clue 3) o Example: “First, I omitted all of the numbers that were not greater than 23 or less than 50 Then I omitted all of the odd numbers, since it said the number was even Then I crossed out all of the numbers that 2, 4, and 12 did not go into, leaving 24, 36 and 48 Then I saw which of these numbers had a 2-digit sum and it was 48.” 42 PART II Released Mathematics Items Mathematics Reference Sheet Grade Use the information below, as needed, to answer questions on the Mathematics test foot = 12 inches cup = ounces (oz) kilogram = 1000 grams yard = feet pint = cups liter = 1000 milliliters quart = pints gallon = quarts pound (lb) = 16 ounces (oz) 43 ... number? A 00,006 B 06,000 * C 60,000 D 63,000 27 A l+w=A * B l×w=A C 2l + 2w = A D l+l+w+w=A PART II Released Mathematics Items AR4M02G204Y 045 - 045 A 32 I am Sam the Robot My head is a hexagon My... What number comes next in the pattern? 1, 2, 4, 7, 11, A 35°C A 15 * B 35°F * B 16 C 68°C C 18 D 68°F D 22 AR4M01M103NXXX- 040 A AR4M02G104Y 040 - 040 D 21 One morning it took Carlos 167 seconds... “It’s 48 because it’s between 24 and 49 , it’s even, 2, and 12 are factors because 48 /2 = 24 and 48 /12 = 4, and the sum of its digits is a 2-digit number since + = 12 40 PART II Released Mathematics