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Workshop Goals
What should students know and be able to do?
What common difficulties do students have with fractions?
What instructional practices can help students understand fractions?
Understanding the meaning of fractions
Too much school instruction on fractions is with computation. When instruction focuses on the meaning of fractions, it is often too brief and superficial. As a result, students are forced to learn rules and procedures for computations without a sound understanding of what they’re operating on.
What are appropriate goals for fraction instruction? Name fractional parts of regions and sets Find fractions on a number line Represent fractional parts using standard
notation (proper and improper fractions, mixed numbers) and also with concrete and pictorial representations
Understand equivalence Compare and order fractions Make reasonable estimates with fractions Compute with fractions Solve problems with fractions
MLR goals for students 1997 MLR Grade 3
Read, write model, and compare simple fractions with denominators of 2, 3, 4
2007 MLR Grade 3 Students recognize, name, illustrate, and use
simple fractions Recognize, name, and illustrate fractions with
denominators from two to ten Recognize, name, and illustrate parts of a whole Compare and order fractions with like
numerators or with like denominators
MLR goals for students 1997 MLR Grade 4
Read, compare, order, classify, and explain simple fractions through tenths
Solve real-life problems involving addition and subtraction of simple fractions
2007 MLR Grade 4 Students understand, name, illustrate, combine, and use
fractions Add and subtract fractions with like denominators and use
repeated addition to multiply a unit fraction by a whole number List equivalent fractions Represent fractions greater than one as mixed numbers and
mixed numbers as fractions Connect equivalent decimals and fractions for tenths, fourths,
and halves in meaningful contexts
MLR goals for students 1997 MLR Grade 5
Read, compare, order, use, and represent simple fractions (halves, thirds, fourths, fifths, and tenths with all numerators)
Compute and model addition and subtraction with simple fractions with common denominators
Create, solve, and justify the solution for multi-step, real-life problems involving addition and subtraction with simple fractions with common denominators
MLR goals for students 2007 MLR Grade 5
Students understand, name, compare, illustrate, compute with, and use fractions Add and subtract fractions with like and
unlike denominators Multiply a fraction by a whole number Develop the concept of a fraction as
division through expressing fractions with denominators of two, four, five, and 10 as decimals and decimals as fractions
MLR goals for students 1997 MLR Grade 6
Read, compare, order, use and represent fractions (halves, thirds, fourths, fifths, sixths, eighths and tenths with all numerators)
Compute and model all four operations with common fractions
Create, solve, and justify the solution for multi-step, real-life problems with common fractions
MLR goals for students 2007 MLR Grade 6
Students add, subtract, multiply, and divide numbers expressed as fractions, including mixed numbers
Students understand how to express relative quantities as percentages and as decimals and fractions Use ratios to describe relationships between
quantities Use decimals, fractions, and percentages to
express relative quantities Interpret relative quantities expressed as
decimals, fractions, and percentages
Developing the meaning of fractions Fractions have four basic interpretations
Measure Part of a region Part of a set Location on a number line
Quotient 1/3 is what you get when you divide 1 pizza
between three people Ratio
The ration of pizzas to people is 1 to 3, 1:3, 1/3 Operator
There are 1/3 as many pizzas as people
Developing the meaning of fractions
Students should understand all 4 interpretations and how they are interrelated. Present fractions using all four interpretations.
Using manipulatives is important Understanding fractions as parts of
regions may be easiest
Fractions as parts of regions
NCTM lesson: Fun with Fractions
http://illuminations.nctm.org/LessonDetail.aspx?id=U113
Set model—fractions as parts of sets
Get six green triangles Put them into two equal groups
Put them into three equal groups
Fraction Line-Up
Number Line model—fractions as locations on a number line
Find the number line master in your packet
Fraction Representations Using a variety of representations and
asking students to switch between them enhances understanding. Real objects Manipulatives
Fraction circles Fraction rectangles Pattern blocks
Drawings Words Symbols
Fraction Representations
Ask students to make connections
“How are these different representations alike?”
Fraction Representations
Ask students to consider negative examples.
“Why is this not 1/3?”
“Why is this not ¼?”
Fraction Equivalence
Students should have strong conceptual understanding of equivalent fractions based on lots of experience. They can then relate that understanding to numerical methods for generating equivalent fractions. Simplifying fractions Generating fractions with common
denominators
Comparing and Ordering Fractions
Strong concepts of the relative size of fractions, based on experience with physical objects and drawings, supports students’ number sense.
Estimating with fractions depends on ideas about the relative size of fractions.
Without sufficient experience with physical objects, students make errors, often using whole-number thinking when working with fractions.
Comparing and Ordering Fractions
Students should consider these cases: Same denominator Same numerator Fractions with different numerators and
denominators Students might relate fractions to a
benchmark like ½.
Fractions as Quotients Try sharing 12 cookies between 4 people. Try sharing 3 cookies between 4 people.
Use the cookie masters in your materials. Students have more success making thirds,
fifths, etc. if they use toothpics. Try sharing 7 cookies between 3 people.
Help students connect mixed numbers and improper fractions.
Lesson Ideas for mixed numbers
Ask students to share cookies between different numbers of people. Use cookie cutouts and glue. Don’t use a numbers that are multiples. Do several examples.
Share between three, four, and six people.
Operating on Fractions
NCTM recommends using simple denominators that can be visualized concretely or pictorially and are apt to occur in real-world settings.
Emphasis in instruction must shift from learning rules for operations to understanding fraction concepts. Begin by asking students to use fraction
pieces to add fractions.
Adding Fractions
Pick 2 Make a train with two pieces on your
whole strip that are not the same color. Build another train the same length
using pieces that are all the same color. Record. Try to build other one-color trains the
same length. For each, record.
Adding Fractions
Help students make the connection between the procedures for adding fractions and their experience with manipulatives. They might even invent rules for adding fractions!
Multiplying Fractions Use physical objects and drawings to
develop meaning. Use whole number meaning for
multiplication. Repeated addition Use the commutative property and “of”
Use rectangles Help students develop the rules for
themselves
Dividing Fractions
Relate dividing fractions to dividing whole numbers 6 ÷ 2 = 3
“How many times can I subtract 2 from 6?” “How many 2’s are in 6?” Check by multiplying ÷ means “into groups of.”
6 ÷ ½ =