Teacher Instructions: Fraction Book Cover

Grade Level: 3 - 5

Task: Fraction Book Cover

Standard: Number Sense and Operations

Design a cover for a book about fractions using fraction pieces. Make the cover
attractive. Then find the total value of the pieces you used to create your design.
Represent your solution numerically in the simplest terms possible, as well as visually
by showing us what your cover would look like .

Context – From the Task Author: This task was assigned to students after they had
considerable practice using fraction pieces, adding fractions , and changing improper
fractions to mixed numbers. Students were encouraged to check their work using a
fraction calculator. Students could also be encouraged to change their fractions to
decimals and to check their solutions this way as well.

What the task accomplishes…
This is an assessment of the concepts taught in addition of fractions, equivalent
fractions, improper fractions, and mixed numbers.
This activity allows for student exploration using various fractional pieces. It provides
a different way of adding fractions, and allows the student to develop a method that
works best for him/her.
It also helps the student see relationships of equivalent fractions and to make
connections.

What students will do…
The student will create and solve his/her own problem.
The student will demonstrate his/her proficiency in adding fractions, finding
equivalent fractions, writing fractions correctly, using and writing improper fractions
and mixed numbers, converting improper fractions to mixed numbers, and explaining
his/her work using the language of mathematics. Use of fraction calculators affirms
the students' work.

Time Required: This task takes approximately one hour.

Interdisciplinary Links: Students could explore how different book covers are made.
They could also create book covers for other subjects. Students could create geography
book covers, and figure out the relative sizes of states or countries, or create nutrition
pyramid book covers and figure out the fractional part of each layer of the food pyramid.

Teaching Tips…
• In lessons prior to this assessment, the students can make various pictures with
teacher direction. For example, make a mushroom that equals 5/8, a sailboat that
equals 5/6, a cat that equals 1 1/4, etc.
• Students will want to show what they can make on their own.
• It is helpful for each student to have their own set of fraction pieces with which to
work.
• A great book to help students understand the idea of using fraction pieces for
making pictures is Ed Emberly's Picture Pie 1 & 2. This book does not use all of the
fraction pieces in a set, but it helps those students who need a point of reference to
get started.

Suggested Materials:
Sets of fraction pieces (halves, thirds, fourths, sixths, eighths, etc.)
Drawing paper
Colored pencils, crayons, etc.
Fraction calculators, such as TI Explorer or the Casio Fraction Mate

Possible Solution…
The solutions are endless since students can create any figure they wish.
Students can demonstrate different solutions to determine the value of their creation.

Benchmark Descriptors:
The benchmark descriptors and rubric are designed to help the teacher analyze
student thinking and understanding at each of the four performance levels.
The descriptors are generalizations of what student work could look like.
It is not possible to anticipate every answer a student can give, so in scoring student
work the teacher must use these generalizations to come to their own conclusions
as to where a student is performing on the assessment.
It is recommended that teachers create their own task specific rubric by listing the
specific math skills that would make up each section of the four performance levels.

Novice
Although the student has created a fractional picture, s/he cannot add or trade the
fraction pieces correctly, and cannot simplify and/or change an improper fraction to a
mixed number.

Apprentice
The student has created a fraction picture and has added the pieces correctly, but
cannot simplify and/or change an improper fraction to a mixed number.
The picture may also have been created to conveniently equal a whole number.

Practitioner
The student has created a fraction picture and added correctly. However, the
student has not been able to explain the process s/he went through to do this.
There may also be a minor mathematical error.

Expert
The student has exhibited mastery of all the mathematical processes required and
shows an understanding of the concepts involved in his/her explanation.

APS Mathematical Standards…
The math standards stated for this task are aligned to the APS Draft Standards 2000.

Strand – Number Sense and Operations:
Students will demonstrate number sense through experiences with meaningful
mathematical problems that focus on number meaning, number relationships, place
value concepts, relative effects of operations, and multiple representations to
communicate sound mathematical thinking.

Benchmark (K – 5): The student will understand place value of whole numbers,
compose and decompose whole numbers, understand the operations and their effects
on numbers and solve problems with fluency and a variety of methods.

Performance Standards:
Third Grade:
• Reads, writes, and uses conventional fraction words and notation and links them
to their pictorial representations.
• Explains that when all fractional parts are included, such as four-fourths, the result
is equal to the whole and to one.
• Adds and subtracts unit fractions that have the same denominator, using
manipulatives.
• Explains common equivalents, especially relationships among wholes, halves,
fourths, and eighths, as well as wholes, thirds and sixths.
• Uses fractions to solve everyday problem situations.

Fourth Grade:
• Explains that equal fractions of a whole have the same area but are not necessarily
congruent.
• Explains how the "size of the whole" affects the size of the fraction (e.g. Is ½ of $1
better than ¼ of $100?).
• Explains the meaning of the numerator and denominator in fractional notation.
• Identifies and constructs models to represent equivalent fractions, mixed numbers,
and improper fractions.
• Adds and subtracts fractions with common and uncommon denominators using a
variety of strategies (e.g. manipulatives, numbers, and pictures).
• Compares proper, improper, and mixed fractions to fractions and whole numbers
using a variety of strategies (e.g., manipulatives, numbers, pictures, and a number
line).
• Uses fractions to solve everyday problem situations.

Fifth Grade:
• Uses addition and subtraction of mixed numbers with common denominators in
problem solving situations.
• Uses fractions and decimals to help solve everyday problem.
• Explains that the size of a fraction is based on the relationship between the
numerator and the denominator and is dependent on the size of the whole.
• Estimates and solves problems involving addition and subtraction of fractions, and
justifies the reasonableness of the solution.

Strand - Global Mathematical Processes:
Students will understand and use mathematical process.

Benchmark (K - 12): The student will use problem solving, reasoning and proof,
communication, connections, and representation as appropriate in all mathematical
experiences.

Performance Standards:
Grades Kindergarten through twelve:
• Develop resourcefulness and perseverance in problem solving in mathematics and
other disciplines.
• Recognize when to use previously learned strategies to solve new problems.
• Develop and use strategies for solving given problems.
• Monitor and reflect on the process of mathematical problem solving.
• Make and investigate mathematical conjectures and use them successfully in
developing and evaluating mathematical arguments and proofs.
• Use the concept of counterexample to test the legitimacy of an argument.
• Develop a logical sequence of arguments leading to a valid conclusion or solution to
a problem (statement/reasons, proof, informal proof, and algebraic steps ).
• Work in teams to share ideas, to develop and coordinate group approaches to
problems, and to share from each other in communicating findings.
• Relate applications to mathematical language in various modalities.
• Communicate mathematical thinking coherently and clearly to others.
• Analyze and evaluate mathematical thinking and strategies of others.
• Identify and connect functions with real-world applications.
• Identify how seemingly different mathematical situations may be essentially the
same (e.g. the intersection of two lines is the same as the solution to a system of
linear equations ).
• Investigate and explain the mathematics required for various careers.
• Recognize and apply mathematics in contexts outside the mathematics course.
• Develop a repertoire of mathematical representation that can be used purposefully,
and appropriately interchangeably (e.g. pictures, written symbols , oral language,
real -world situations, and manipulative models).
• Select, apply, and translate among mathematical representations to solve
problems.
• Use representations to model and interpret physical, social, and mathematical
phenomena.