UNIT 3:
FRACTIONS
Understanding Fractions
Addition And Subtraction Of Fractions
Multiplication Of Fractions
Division Of Fractions
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UNIT 3: FRACTIONS
TABLE OF CONTENTS
Introduction
•! Aims and goals of Unit 3
•! Key things to look for throughout Unit 3
•! Structure of Unit 3 student work
•! Unit 3: Fractions alignment to national standards
o! National Council of Teachers of Mathematics (NCTM) standards
o! Common Core standards
Topics:
•! Understanding Fractions:
o! Introduction to understanding fractions (for parents/teachers)
o! What is a Fraction? discussion questions
o! What are Equivalent Fractions? discussion questions
o! Equivalent fractions story problems
o! Equivalent fractions direct operation problems
o! Equivalent fractions data collection and observations
o! Equivalent fractions teaching and hypothesis creation
o! Equivalent fractions hypothesis test and theorem
o! What is Simplifying Fractions? discussion questions
o! Simplifying fractions direct operation problems
o! Simplifying fractions teaching and hypothesis creation
o! Simplifying fractions hypothesis test and theorem
o! What is an Improper Fraction? discussion questions
o! What is a Mixed Number? discussion questions
o! Converting an improper fraction to a mixed number story problems
o! Converting an improper fraction to a mixed number direct operation
problems
o! Converting an improper fraction to a mixed number teaching and
hypothesis creation
o! Converting an improper fraction to a mixed number hypothesis test and
theorem
o! Converting a mixed number to an improper fraction story problems
o! Converting a mixed number to an improper fraction direct operation
problems
o! Converting a mixed number to an improper fraction data collection and
observations
o! Converting a mixed number to an improper fraction teaching and
hypothesis creation
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o! Converting a mixed number to an improper fraction hypothesis test and
theorem
o! End of topic tasks
o! Supplemental activities
•! Adding And Subtracting Fractions:
o! Introduction to adding and subtracting fractions (for parents/teachers)
o! Adding fractions story problems
o! Adding fractions direct operations problems
o! Adding fractions data collection and hypothesis creation
o! Adding fractions hypothesis test and theorem
o! Adding with mixed numbers story problems
o! Adding with mixed numbers direct operation problems
o! Adding with mixed numbers data collection and hypothesis creation
o! Adding with mixed numbers hypothesis test and theorem
o! Subtracting fractions story problems
o! Subtracting fractions direct operation problems
o! Subtracting fractions data collection and hypothesis creation
o! Subtracting fractions hypothesis test and theorem
o! Subtracting with mixed numbers story problems
o! Subtracting with mixed numbers direct operation problems
o! Subtracting with mixed numbers data collection and hypothesis creation
o! Subtracting with mixed numbers hypothesis test and theorem
o! End of topic tasks
o! Supplemental activities
•! Multiplying Fractions:
o! Introduction to multiplying fractions (for parents/teachers)
o! Multiplying a fraction and a whole number story problems
o! Multiplying a fraction and a whole number direct operation problems
o! Multiplying a fraction and a whole number data collection and
hypothesis creation
o! Multiplying a fraction and a whole number hypothesis test and theorem
o! Multiplying two fractions story problems
o! Multiplying two fractions direct operation problems
o! Multiplying two fractions data collection and hypothesis creation
o! Multiplying two fractions hypothesis test and theorem
o! Multiplying a fraction and a mixed number story problems
o! Multiplying a fraction and a mixed number direct operation problems
o! Multiplying a fraction and a mixed number data collection and
hypothesis creation
o! Multiplying a fraction and a mixed number hypothesis test and theorem
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Multiplying two mixed numbers
Multiplying two mixed numbers
Multiplying two mixed numbers
Multiplying two mixed numbers
End of topic tasks
Supplemental activities
story problems
direct operation problems
data collection and hypothesis creation
hypothesis test and theorem
•! Dividing Fractions:
o! Introduction to dividing fractions (for parents/teachers)
o! Dividing a fraction by a whole number story problems
o! Dividing a fraction by a whole number direct operation problems
o! Dividing a fraction by a whole number data collection and hypothesis
creation
o! Dividing a fraction by a whole number hypothesis test and theorem
o! Dividing a whole number by a fraction story problems
o! Dividing a whole number by a fraction direct operation problems
o! Dividing a whole number by a fraction data collection and hypothesis
creation
o! Dividing a whole number by a fraction hypothesis test and theorem
o! Dividing two fractions story problems
o! Dividing two fractions direct operation problems
o! Dividing two fractions data collection and hypothesis creation
o! Dividing two fractions hypothesis test and theorem
o! Dividing fractions and mixed numbers story problems
o! Dividing fractions and mixed numbers direct operation problems
o! Dividing fractions and mixed numbers data collection and hypothesis
creation
o! Dividing fraction and mixed numbers hypothesis test and theorem
o! End of topic tasks
o! Supplemental activities
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UNIT 3: FRACTIONS
Where does Unit 3: Fractions fit in?
The MATH INSPIRATIONS curriculum series is designed to aid student exploration,
discovery and mastery of mathematics with an emphasis on mathematical thinking
through the Discovery MethodTM. Unlike traditional curriculums which are divided by
grade level and not by topic, students in this series will explore and master specific
topics in their entirety before beginning another topic. The full K-8 curriculum is
divided into eight units of study including U1: Number Sense, U2: Basic Operations,
U3: Fractions, U4: Geometry, U5: Units of Measure, U6: Advanced Numbers, U7:
Advanced Operations and U8: Data. This book, Unit 3: Fractions, is designed to
follow a study of U2: Basic Operations and precedes a study of either U4: Geometry
or U5: Units of Measure.
Advanced Mathematics
algebra, advanced geometry, trigonometry, pre-calculus, calculus, statistics, financial math and beyond
U6: Adv. Numbers
integers, decimals,
ratios, proportions
U7: Adv. Operations
exponents, radicals,
order of operations
U4: Geometry
2-D & 3-D shapes,
transformations, area
U8: Data
collecting, representing
data, probability
U5: Units of Measure
units, lengths,
estimation, weights, time
U3: Fractions
fractions, equivalency,
fraction operations
U2: Basic Operations
addition, subtraction,
multiplication, division
U1: Number Sense
place value,
comparisons, patterns
Love of Learning
fun, discovery, play,
questioning everything
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What is the goal of Unit 3: Fractions?
This unit is designed to build upon the foundations of basic operations by exploring a
new number family, fractions. When students have a true mastery of fractions,
equivalency, and fraction operations, they will be fully prepared to dive deeper into
a study of other types of numbers and real-life applications such as geometry and
measurement. Unit 3 is also focused on growing student mastery of the Discovery
Method; including making observations, writing hypotheses and proving their own
theorems. All of which aid in the development of problem solving, pattern
recognition and analytical communication skills.
Key things to look for throughout Unit 3
•! Utilizing proper vocabulary
o! It is always important throughout all units to use proper vocabulary.
Refer to fraction parts as numerators and denominators as well as
identify different types of fractions by their names including improper
fractions and mixed numbers.
•! Finding new methods
o! A major focus of this unit must be on student’s discovery of new,
unique, personalized approaches and processes to each topic. Your
student will find amazing new ways to solve problems and prove their
ideas. Allow and encourage them to explore and find new methods and
then make them prove their methods’ effectiveness for all problems.
o! Remember also that it is ok to teach your student the way that
you learned how to solve each problem. However, you must wait
until the student has created and proved their own method first.
Then, use your way and any others you or the student is familiar with
to spur an even deeper discussion on the topic.
•! Detailed communication
o! Look for opportunities during discussions, hypothesis and theorem
writing and task problems for your student to communicate
mathematics in specific, detailed and structured ways. Ensure that
student hypotheses are organized, specific and detailed enough for any
situation. Also make an effort to talk “thinking” and logic throughout
the day, even outside of designated math time.
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General structure of the unit
•! Direct Operation vs. Story Problems
o! In the discovery piece of the Discovery MethodTM, students are required
to solve problems which they have never seen before. They will have
this opportunity in one of two different ways: direct operation problems
where they will be required to solve new math problems in their own
way using logic and problem solving skills and story problems where
they can use context with pictures and manipulatives to solve. We
encourage teachers to allow the student to choose which (or both)
approach they would prefer to use in this step.
•! Hypothesis Creation and Testing
o! Once students have solved the unknown problems in the discovery
phase, it is time for them to organize their work, look for patterns and
create a hypothesis. Students make a list of all the math sentences
they created in the direct operations and/or story problem pieces and
then look for the patterns among the data. These patterns should lead
to the creation of a detailed and clear procedure, or set of
instructions, called a hypothesis, which describes exactly how the
problem can be solved in the student’s own words and way of thinking.
After students create the hypothesis, they then test it by solving each
of the seven problems on the Hypothesis Test pages and prove each in
a second way (such as drawings or manipulatives) to check that their
hypothesis answer matches their second result. If they match for all
seven problems, the student’s hypothesis can be deemed a true
theorem. However, more likely they will need to add to or adjust their
hypothesis to accommodate all situations.
•! End Of Topic Tasks
o! The end of topic tasks serve as culmination activities which are
designed specifically to stretch student’s calculation and problem
solving skills. These tasks are a great way to assess what they have
learned and gives them a way to show off their skills. There are basic,
intermediate and advanced tasks for each topic. Teachers should
select tasks that are just beyond the ability of the student. There
may be overlapping ideas and concepts between tasks, however, each
task is designed to provide a unique perspective and application as well
as challenge students to grow as mathematical thinkers.
•! Supplemental Activities
o! There are several supplemental activities for each topic. Supplemental
activities are included in order provide greater enrichment to the
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learning process and make the student’s experience more interesting
and fun. They are separate from the Discovery MethodTM and end of
unit tasks, but are still designed to encourage the student to think and
perform at a high level, and also to show students different uses and
applications of mathematics. There is no specific order to these
activities other than they align with the topic. The activities can be
used at any time during their study of the topic.
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