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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