Monday, 19 August 2013

Fraction Problems: Grades 3-5

Solve the problems in each set.  You may work alone, with a partner, or in small groups.


Problem Set 1: Different Representations of the Same Fraction
Think carefully about each situation and make a representation (e.g., picture, symbols) to represent the meaning of 3/4 conveyed in that situation.  (Lamon, 1999, pp. 30-31)

1.1. John told his mother that he would be home in 45 minutes.

1.2. Melissa had three large circular cookies, all the same size – one chocolate chip, one coconut, one molasses.  She cut each cookie into four equal parts and she ate one part of each cookie.

 1.3. Mr. Albert has 3 boys to 4 girls in his history class.

 1.4. Four little girls were arguing about how to share a package of cupcakes.  The problem was that cupcakes come three to a package.  Their kindergarten teacher took a knife and cut the entire package into four equal parts.
 1.5. Baluka Bubble Gum comes four pieces to a package.  Three children each chewed a piece from one package.

1.6. There were 12 men and 3/4 as many women at the meeting.

1.7. Mary asked Jack how much money he had.  Jack reached into his pocket and pulled out three quarters.

 1.8. Each fraction can be matched with a point on the number line.  3/4 must correspond to a point on the number line.

 1.9. Jaw buster candies come four to a package and Nathan has 3 packages, each of a different color.  He ate one from each package.

 1.10. Martin’s Men Store had a big sale – 75% off.

 1.11. Mary noticed that every time Jenny put 4 quarters into the exchange machine, three tokens came out.  When Mary had her turn, she put in twelve quarters.

 1.12. Tad has 12 blue socks and 4 black socks in his drawer.  He wondered what were his chances of reaching in and pulling out a sock to match the blue one he had on his left foot.

 Problem Set 2: Representing Different Fractions

2.1. Represent each of the following:

      a. I have 4 acres of land.  5/6 of my land is planted in corn.

      b. I have 4 cakes and 2/3 of them were eaten.

      c. I have 2 cupcakes, but Jack as 7/4 as many as I do.

2.2. The large rectangle represents one whole that has been divided into pieces.

   Identify what fraction each piece is in relation to the whole rectangle.  Be ready to explain how you know the fraction name for each piece.  (Inspired by the Balanced Assessment materials, 1999.)


      A ___        B ___         C ___         D ___        E ___         F ___        G ___          H ___

 2.3. What is the sum of your eight fractions?  What should the sum be?  Why?

2.4. Mom baked a rectangular birthday cake.  Abby took 1/6.  Ben took 1/5 of what was left.  Charlie cut 1/4 of what remained.  Julie ate 1/3 of the remaining cake.  Marvin and Sam split the rest.  Was this fair?  How does the shape of the cake influence your answer?


 2.5. If the number of cats is 7/8 the number of dogs in the local pound, are there more cats or dogs?  What is the unit for this problem?

 2.6. Ralph is out walking his dog.  He walks 2/3 of the way around this circular fountain.  Where does he stop?

Problem Set 3: Unitizing
Suppose these circles represent cookies.  (Lamon, 2002, p. 18)
O   O   O   O   O   O
O   O   O   O   O   O
O   O   O   O   O   O


3.1. Can you see ninths?  How many cookies will you eat if you eat 4/9 of the cookies?
O   O   O   O   O   O
O   O   O   O   O   O
O   O   O   O   O   O



3.2. Can you see twelfths?  How many cookies will you eat if you each 5/12 of the cookies?
O   O   O   O   O   O
O   O   O   O   O   O
O   O   O   O   O   O



3.3. Can you see sixths?  How many cookies will you eat if you eat 5/6 of the cookies?
O   O   O   O   O   O
O   O   O   O   O   O
O   O   O   O   O   O



3.3. Can you see thirty-sixths?  How many cookies will you eat if you eat 14/36 of the cookies?
O   O   O   O   O   O
O   O   O   O   O   O
O   O   O   O   O   O

 3.5. Can you see fourths?  How many cookies will you eat if you eat 3/4 of the cookies?
O   O   O   O   O   O
O   O   O   O   O   O
O   O   O   O   O   O




Problem Set 4: More Unitizing Problems

4.1. 16 eggs are how many dozens?  26 eggs are how many dozens?

 4.2. You bought 32 sodas for a class party.  How many 6-packs is that?  How many 12-packs?  How many 24-packs?

 4.3. You have 14 sticks of gum.  How many 6-packs is that?  How many 10-packs is that?  How many 18-packs is that?

4.4. There are 4 2/3 pies left in the pie case.  The manager decides to sell these with this plan:  Buy 1/3 of a pie and get 1/3 at no extra charge.  How many servings are there?

 4.5. There are 5 pies left in the pie case.  The manager decides to sell these with this plan:  Buy 1/3 of a pie and get 1/3 at no extra charge.  How many servings are there?

 4.6. Although “unitizing” is a word for adult (and not children), how might work with unitizing help children understand fractions?

 Problem Set 5: Keeping Track of the Unit

5.1. How do you know that 6/8 = 9/12?  Give as many justifications as you can.

5.2. Ten children went to a birthday party.  Six children sat at the blue table, and four children sat at the red table.  At each table, there were several cupcakes.  At each table, each child got the same amount of cake; that is they “fair shared.”  At which table did the children get more cake?  How much more?

      (a) blue table: 12 cupcakes
            red table: 12 cupcakes


      (b) blue table: 12 cupcakes
            red table: 8 cupcakes


      (c) blue table: 8 cupcakes
            red table: 6 cupcakes


      (d) blue table: 5 cupcakes
            red table: 3 cupcakes


      (e) blue table: 2 cupcakes
            red table: 1 cupcake


Problem Set 6: In Between

6.1. Find three fractions equally spaced between 3/5 and 4/5.  Justify your solutions.











6.2. We know that 3.5 is halfway between 3 and 4, but is 3.5/5 halfway between 3/5 and 4/5?  Explain.







6.3. Find three fractions equally spaced between 1/4 and 1/3.  Justify your solutions.











6.4. We know that 3.5 is halfway between 3 and 4, but is 1/3.5 halfway between 1/4 and 1/3?  Explain.



Problem Set 7: Variations on Fraction Tasks

7.1. What number, when added to 1/2, yields 5/4?  Write at least 5 different answers.








7.2. Write two fractions whose sum is 5/4.  Write at least 5 different answers.














7.3. Write two fractions, each with double-digit denominators, whose sum is 5/4.  Write at least 5 different answers.














7.4 Which of problems 7.1, 7.2, and 7.3 is the most “unusual”?  Why?



Problem Set 8: Modifying Fractions

8.1. What happens to a fraction if

      (a) the numerator doubles

      (b) the denominator doubles

      (c) both numerator and denominator double

      (d) both numerator and denominator are halved

      (e) numerator doubles, denominator is halved

      (f) numerator is halved, denominator doubles


8.2. What happens to a fraction if

      (a) the numerator increases

      (b) the denominator increases

      (c) both numerator and denominator increase

      (d) both numerator and denominator decrease

      (e) numerator increases, denominator decreases

      (f) numerator decreases, denominator increases


8.3. The letters a, b, c, and d each stand for a different number selected from {3, 4, 5, 6}.  Solve these problems and justify each answer.

      (a) Write the greatest sum: a/b + c/d



      (b) Write the least sum: a/b + c/d



      (c) Write the greatest difference: a/b - c/d



      (d) Write the least difference: a/b - c/d

Problem Set 6: In Between

6.1. Find three fractions equally spaced between 3/5 and 4/5.  Justify your solutions.











6.2. We know that 3.5 is halfway between 3 and 4, but is 3.5/5 halfway between 3/5 and 4/5?  Explain.







6.3. Find three fractions equally spaced between 1/4 and 1/3.  Justify your solutions.











6.4. We know that 3.5 is halfway between 3 and 4, but is 1/3.5 halfway between 1/4 and 1/3?  Explain.



Problem Set 7: Variations on Fraction Tasks

7.1. What number, when added to 1/2, yields 5/4?  Write at least 5 different answers.








7.2. Write two fractions whose sum is 5/4.  Write at least 5 different answers.














7.3. Write two fractions, each with double-digit denominators, whose sum is 5/4.  Write at least 5 different answers.














7.4 Which of problems 7.1, 7.2, and 7.3 is the most “unusual”?  Why?



Problem Set 8: Modifying Fractions

8.1. What happens to a fraction if

      (a) the numerator doubles

      (b) the denominator doubles

      (c) both numerator and denominator double

      (d) both numerator and denominator are halved

      (e) numerator doubles, denominator is halved

      (f) numerator is halved, denominator doubles


8.2. What happens to a fraction if

      (a) the numerator increases

      (b) the denominator increases

      (c) both numerator and denominator increase

      (d) both numerator and denominator decrease

      (e) numerator increases, denominator decreases

      (f) numerator decreases, denominator increases


8.3. The letters a, b, c, and d each stand for a different number selected from {3, 4, 5, 6}.  Solve these problems and justify each answer.

      (a) Write the greatest sum: a/b + c/d



      (b) Write the least sum: a/b + c/d



      (c) Write the greatest difference: a/b - c/d



      (d) Write the least difference: a/b - c/d

Problem Set 6: In Between

6.1. Find three fractions equally spaced between 3/5 and 4/5.  Justify your solutions.











6.2. We know that 3.5 is halfway between 3 and 4, but is 3.5/5 halfway between 3/5 and 4/5?  Explain.







6.3. Find three fractions equally spaced between 1/4 and 1/3.  Justify your solutions.











6.4. We know that 3.5 is halfway between 3 and 4, but is 1/3.5 halfway between 1/4 and 1/3?  Explain.



Problem Set 7: Variations on Fraction Tasks

7.1. What number, when added to 1/2, yields 5/4?  Write at least 5 different answers.








7.2. Write two fractions whose sum is 5/4.  Write at least 5 different answers.














7.3. Write two fractions, each with double-digit denominators, whose sum is 5/4.  Write at least 5 different answers.














7.4 Which of problems 7.1, 7.2, and 7.3 is the most “unusual”?  Why?



Problem Set 8: Modifying Fractions

8.1. What happens to a fraction if

      (a) the numerator doubles

      (b) the denominator doubles

      (c) both numerator and denominator double

      (d) both numerator and denominator are halved

      (e) numerator doubles, denominator is halved

      (f) numerator is halved, denominator doubles


8.2. What happens to a fraction if

      (a) the numerator increases

      (b) the denominator increases

      (c) both numerator and denominator increase

      (d) both numerator and denominator decrease

      (e) numerator increases, denominator decreases

      (f) numerator decreases, denominator increases


8.3. The letters a, b, c, and d each stand for a different number selected from {3, 4, 5, 6}.  Solve these problems and justify each answer.

      (a) Write the greatest sum: a/b + c/d



      (b) Write the least sum: a/b + c/d



      (c) Write the greatest difference: a/b - c/d



      (d) Write the least difference: a/b - c/d

Problem Set 6: In Between

6.1. Find three fractions equally spaced between 3/5 and 4/5.  Justify your solutions.











6.2. We know that 3.5 is halfway between 3 and 4, but is 3.5/5 halfway between 3/5 and 4/5?  Explain.







6.3. Find three fractions equally spaced between 1/4 and 1/3.  Justify your solutions.











6.4. We know that 3.5 is halfway between 3 and 4, but is 1/3.5 halfway between 1/4 and 1/3?  Explain.



Problem Set 7: Variations on Fraction Tasks

7.1. What number, when added to 1/2, yields 5/4?  Write at least 5 different answers.








7.2. Write two fractions whose sum is 5/4.  Write at least 5 different answers.














7.3. Write two fractions, each with double-digit denominators, whose sum is 5/4.  Write at least 5 different answers.














7.4 Which of problems 7.1, 7.2, and 7.3 is the most “unusual”?  Why?



Problem Set 8: Modifying Fractions

8.1. What happens to a fraction if

      (a) the numerator doubles

      (b) the denominator doubles

      (c) both numerator and denominator double

      (d) both numerator and denominator are halved

      (e) numerator doubles, denominator is halved

      (f) numerator is halved, denominator doubles


8.2. What happens to a fraction if

      (a) the numerator increases

      (b) the denominator increases

      (c) both numerator and denominator increase

      (d) both numerator and denominator decrease

      (e) numerator increases, denominator decreases

      (f) numerator decreases, denominator increases


8.3. The letters a, b, c, and d each stand for a different number selected from {3, 4, 5, 6}.  Solve these problems and justify each answer.

      (a) Write the greatest sum: a/b + c/d



      (b) Write the least sum: a/b + c/d



      (c) Write the greatest difference: a/b - c/d



      (d) Write the least difference: a/b - c/d

Problem Set 9: Reflection

9.1. Write a division story problem appropriately solved by division so that the quotient has a label different from the labels on the divisor and the dividend.  (What does “divisor” mean?  What does “dividend” mean?)

 9.2. Write a story problem appropriately solved by division that demonstrates that division does not always make smaller.

9.3. Is a fraction a number?  Explain.

9.4. Why are fractions called equivalent rather than equal?




















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