Unit 1: Teaching Number and
Ope ration Sense Rational Numbers |
Connected Math: Bits and Pieces II
Using Rational Numbers
 Advanced Topics
from Math Session
 Field of Rational Numbers
Multiplicative Inverse
Transition from Number to Algebra |
| Investigation 7: Dividing Fractions
Connections
with advanced topics
Concept behind dividing fractions
(Prob.
7.1, Part 2, pg 154; 7.2, Part 2, pg 155;
and 7.3 Fol low -Up, Part 1, 3, pg 157)
Application – division in context
(Applications 26, pg 159)
Inverse Operation – reciprocal
(Extensions,
Prob. 45, 46, pg 161) |
Research PSSM Companion
Ch. 7: Fractions and Multiplicative Reasoning
Misconceptions:
a/b denotes only a
part-whole relationship (3/7 vs. 7/3)
Fraction versus
rational number
Understanding –
concept image versus
concept definition
Conceptual Scheme of operations that
constitutes a person’s understanding |
|
Measurement
Schemes
Ratio is at heart of measurement
Multiplicative comparison where
measuring unit
imagined apart from thing measured
Magnitude vs. Measure – magnitude
independent of unit of measure
Numerical equations vs . quantity
equations
Proportional reasoning central to
understanding
measurement – single vs. multiple proportions |
Multiplication
Scheme
Repeated addition conception
supports
evaluation of products but not
conceptualization
Thinking of 1/n
as one out of n parts
Leads to sense
of inclusion
Conceptualized Multiplication
Entailing
multiple proportion
Multiplicative
reasoning |
Division Scheme
Two settings
Sharing or
partitioning – distributing
something among participants
Segmenting or
measuring – putting an
amount into parts
Operative imagery
Envision result
of acting prior to acting
Suppress
attention to the process |
|
Fraction Scheme
Collection of Objects misconception
Comparison vs. Inclusion
Invert and Multiply conception
Rule a/b x c/d = ac/bd
Generalizations
must be rooted in meaning -making
activities, not algorithms
Symbolic rules
should not be provided in
place of generalization from conceptualization |
| Teacher Resources
Fraction
activity-coloring parts of shapes and
ordering: Fraction Sorter
Fraction
activity-using geometric shapes to
introduce the concept of fractions
Fraction
game-card game “Make One” |
|
Adding
Fractions activity
Ordering and
comparing fractions activity
Discussion and
example problems on ordering
fractions |
|
Fraction
Multiplication Activity: From Musser website
Fraction
Division Activity: From Musser website
Lesson Plan -
students break words into fractions to
decode a message: Fraction Hunt |
|
Lesson Plan
linking fractions with jazz rhythms
Lesson Plan on
fraction sense, equivalent fractions ,
comparing fractions
Short Lesson on
Dividing fractions by whole
numbers—includes short definition of reciprocal |
