Middle school students should be developing a sound grasp of rational number concepts. They often need support when their whole number intuition and experience (adding and multiplying make numbers get larger, subtraction and division smaller) is replaced by sometimes unexpected outcomes. A wealth of practice with many different contexts will help middle school students develop understanding of the result of operations on rational numbers, and extend their understanding of inverse operations. Use the problems below to encourage discourse and promote understanding.

 1)      Which is closer to 1: 6/5 or 5/6? Explain your answer in words and/or pictures.

2)      True or false?   106.32  > 106.319 Explain your answer in words and/or pictures.

3)      True or false? 6/7 = .67  Explain your answer.

4)      True or false:  2 ½ x 3 ¼ = 6 and 1/8  Explain your answer.

5)      Which expression represents the largest value?  

               A) 29 + ½          B) 29 – ½         C) 29 x ½        D) 29 ÷ ½

6)      Given two points on a number line, a and b, where is a + b? a – b? a x b? a/b?

7)      If x = – ¼ , which is the least? Show your work.    

             A) 1 – x           B) x – 1            C) 2x             D) 1/x

8)      Which expression has the least value? Explain. 

A) 4.6507 – 5.196      B | -2 – 1 |       C) (-3) cubed          D) (-5/6) ÷ 1/36

9)      Evaluate  ½ (7) + ¼ (7 ½ ). Express your answer as a fraction or mixed number in simplest form. Is your answer reasonable? Why or why not?

10)   Evaluate 3 ¾ ÷ 7 ¼ . Express your answer as a fraction or mixed number in simplest form. Then state why your answer is reasonable.

11)   If 50% of a number is 18, what is ½ % of the number?

12)   The length of Kelsey’s garden is 4 and 5/8 feet. Find the perimeter of Kelsey’s garden if the width is 2 and 7/8 feet shorter than the length.

13)   A new section of freeway will be 6 and 3/5 miles long.  So far, 1/6 of the new section has been completed. How many more miles must be built to complete the project?

14)   Zarina likes to walk around Pelican Park. The park is in the shape of a square, 7/10 miles on each side. One morning she walked around the park 3 ½ times before stopping to rest. If her goal was to walk 10 miles that day, did she fall short of or exceed her goal? By how much?

15)   Water flows out of a shower head at a rate of 1 and 2/3 gallons per minute. How much water will be used for a 7 ½ minute shower?

16)   A sheet of paper is 8 ½ inches wide and 11 inches long. The sheet is printed with a margin 1 and 2/3 inches wide on each side and on the top and bottom.  Find the perimeter of the printed part of the page.

17)   Jill is working on plans for a 12-acre housing development. A park will cover 2 and 7/9 acres and paved areas will take 1 ¾ acres. How many acres are left for home sites?

18)   A railroad construction crew can lay 1 ¾  miles of track in one day. At that rate, how many days will it take to lay 35 miles of track?

19)   Mike ran in a 3 1/6 (three and one sixth) – mile relay race. He ran 2/3 of the race. How far did Mike run? Express your answer as a fraction or mixed number in simplest form.

20)   A circle and a rectangle have the same area. The circle has a diameter of 14 cm and the rectangle has a width of 11 cm.  Find the perimeter of the rectangle. (Use pi = 22/7).

21)   A recipe calls for 2 ¼ cups of flour and 1 ¾ cups of sugar. How many cups of sugar are needed if you only have 1 ¾ cups of flour? Express your answer as a fraction or mixed number in simplest form.

22)   Two thirds of the students in the pre-algebra class did the bonus problem on the test. Of those, 3/8 did the problem correctly. If there are 24 students in the class, how many did the problem correctly?    

23)   One turn of a screw tightens it 3/16 inch. How many turns will tighten it 1 ¾ inches?

24)   Dillon has a collection of 72 books. Of the books, 2/9 are math books, 1/6 are science books, and the rest are romance novels. How many more romance novels does he have than science books?

25)   Julio bought 45 books. He gave 2/5 of the books to the library and then gave 2/3 of the remainder to his friends. How many books did he have left?

26)   If a television uses 3/20 kilowatt hours of electricity per hour, how long will it take to use 14 kilowatt hours of electricity?

27)   Jenny refinished a wooden table. She used 1/3 can of varnish for a first coat, ¼ can for a second coat, and 1/6 can for a third coat. If the can held 48 ounces, how many ounces were left?

28)   One sixth of the students at a local college are seniors. The number of freshmen is 2 ½ times that amount. What fraction of the students are freshmen? If the college has 3,000 students, how many freshmen are there?

29)   A brochure is 3/16 of an inch thick. How many brochures can be stacked to fit in a box 12 inches high?

30)   Ty had $20. He spent 3/8 of it on a book, 2/5 of it on a present, and 1/3 of the remainder on a pen. How much did the pen cost?

31)   A box of laundry detergent contains 35 cups. If you use 1 ¼ cups per load, how many loads of laundry can you wash with one box?

32)   Roger has a 28 pound block of ice for his snow cone stand. If each snow cone requires 2/3 pound of ice, how many snow cones can Roger make?

33)   A cabinet has shelves that are 12 ½ inches apart. On one shelf, Brad stacked a CD player that is 6 ¼ inches high on top of an amplifier that is 3 1/8 inch high. How much space is left above the CD player?

34)   An aquarium holds 6 ¼ gallons of water. The water level has dropped to 4/5 of this amount. How much water should be added to fill the aquarium?

35)   A turtle walked ½ mile at the rate of 1/5 mile per hour. How long did it take?

36)   A high speed printer prints a page in 1/6 second. How long will it take Esther to print her 30 page History report on this printer?

37)   Jean owns a ¼ acre orchard. Two fifths of the orchard is planted in grapefruit trees. What fraction of an acre is planted in grapefruit trees?

38)   The distance a bicycle travels with each turn of its wheels is about 3 and 1/7 times the tire diameter. The tires on Mike’s bicycle have a diameter of 24 ½ inches. How far does it travel with each turn of the wheels?

39)   A recipe for 2 dozen cookies calls for 1 and 1/3 cups of flour. How much flour would be needed to make 5 dozen cookies?

40)   A bike helmet is on sale for 2/3 of the regular price. If the sale price is $44, what is the regular price?

41)   Jill made a sauce in cooking class She used ½ cup of milk, 2/3 cup of cream, and ¼ cup of water. How much less water was used than milk and cream combined?

42)   Our math class is having a party. One student said he could bring a pitcher of punch that holds 18 ½ cups. If the cups we have can hold 5/6 of a cup of punch, how many servings of punch will we have?

43)   Karen promised Sara that she would give half of her remaining money to Sara after giving 7/8 of it to her favorite charity. If Karen gave $672 to her favorite charity, how much money will Sara receive?

44)   Pete the Painter can paint 5/6 of a room in 90 minutes. How many minutes will it take Pete to paint the entire room?

45)   During a walk on the beach, Sam recorded that 4/5 of the people were 18 years of age or older. Half of the remaining people were under 12 years old. If there were 20 children under 12 years old, how many people did Sam count at the beach?

46)   After Jason spent 7/10 of his money and Marcus spent 5/6 of his money, they each had $15 left. How much money did they spend altogether?

47)   Suzanne and Jane made some cookies. They gave 1/3 to their friends, ate ¼ of the remaining cookies, and froze the rest to eat later. If they froze 27 cookies, how many cookies did they make altogether?

48)   The school secretary bought a large batch of pencils. She gave 9 to the principal and then gave 20% of what was left to the first grade teachers. She kept ½ of what was left and divided the other ½ among four grade level teams, each of whom got 21 pencils. How many pencils were in the original batch?

49)   Gary works at Ghosts and Goblins, a Halloween store. He puts orange, black, and white Halloween candies in boxes. After he fills a box, 1/4 is orange, 1/6 is black, and 21 are white. How many candies does Gary put in each box?

50)   Benjamin found a bag of money. He decided to give 1/2 of the money to his best friend, 1/4 of the money to his sister, 1/8 of the money to his mother, and 1/16 of the money to his dad. The amount of money that was left was $1.25. How much money was in the bag when Benjamin found it?