Transcript La-STEM Math Academies

La-STEM Math Academies for ENFA and LA Educators
“Transforming Numbers & Operations
and Algebra Instruction in Grades 3-5”
DAY 2 of 8
“My heart is singing for joy this morning. A
miracle has happened! The light of
understanding has shone upon my little pupil’s
mind, and behold, all things are changed.”
Anne Sullivan (A teacher of the blind and deaf)
GOALS:
•Understand fractions and decimals represented as parts
of a unit or area, parts of a collection of objects, and
locations on a number line
•Use concrete materials to represent fractions and
decimals as parts of a unit or area, parts of a collection of
objects, and locations on a number line
•Name fractional and decimal parts of a unit or area, of a
collection of objects, and of a number line
Fair Shares
• Children must be aware of two
components of fractions:
1. The number of parts
2. The equality of the parts
(equal size, but not necessarily shape)
Rational Numbers
• A rational number is a number that can be
written in the form a/b where a and b are
integers and b≠0.
• In the set of rational numbers, a is called the
numerator (Latin word meaning number) and
b is called the denominator (Latin word
meaning namer), of the fraction.
• QUESTION: Is every fraction a rational number
and is every decimal a rational number?
5 Main Interpretations of Fractions
1. Fractions as parts of wholes or parts of sets
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A unit is partitioned equally into equivalent parts.
2. Fractions as the result of dividing two numbers
▫
The quotient- meaning results when a number of objects are
shared by a set number.
3. Fractions as the ratio of two quantities
▫
Compares a part to a whole.
4. Fractions as operators
▫
A fraction acts on another number by stretching or shrinking it.
5. Fractions as measures
▫
The length marked on a number line or subunits.
NOTE: equal parts and equal size pieces (but not necessarily
identical shapes) are ESSENTIAL when dealing with fractions.
Different ways to express: “I have $3
I’d like to share with 4 of you.”
…..with 4 of you.
3 4
Or
Or
4 3
3
4
Research indicates:
• Using models is critical in understanding
fractions.
• Younger grades are better at this than
later grades.
• Models help clarify what is being written
symbolically.
• Sometimes it helps to do the same
activity with different models.
1. Area or Region Model
• This is the place we usually begin for MOST
students but students have to understand what
we mean by AREA.
• Area models involve sharing something that can
be cut into smaller parts.
• Circular models are good about emphasizing the
amount that remains but not very good when the
fractions move beyond ½, 1/3, ¼, 1/5 or when
we have to operate with fractions.
Area Model
An AREA model using pattern blocks
Use your Pattern Blocks: Activity 1
2. Measurement or Length Model
• Length is the critical factor in this model-instead of the area of the unit
• The number line is significantly more
sophisticated that most other models
• Each number represents a distance to the
labeled point from zero
Length Model
Navigating Through Number and Operations
Cuisenaire Rods-- a LINEAR model
More practice with a linear model…
Number Line-- Activity 2
1. Find the Cuisenaire Rod that allows you to
partition the second number line into halves. Mark
½ on the number line. What is the rod color?
2. Find the Cuisenaire Rod that allows you to
partition the third number line into thirds. Mark
1/3 and 2/3 on the number line. The rod color is?
3. Continue in this manner until you have partitioned
and labeled the final three number lines into
fourths, sixths, and twelfths.
Number Line Fraction Division
Helping Students Reason About
Fractions as Numbers
1. What number is halfway between zero and
one?
2. What number is halfway between zero and
one-half?
3. What other numbers are the same as onehalf?
4. What number is ¼ more than ½?
5. What number is 1/6 more than ½?
6. What number is 1/6 less than one?
cont.
More questions for understanding…
7. What number is 1/3 more than one?
8. What number is halfway between 1/12
and 3/12?
9. What number is closest to zero?
10.What number is closest to one?
11. What would you call a number halfway
between zero and 1/12?
3. Set or Discrete Model
• The whole is understood to be a set of objects and
subsets make the fractional parts
• Good at establishing real world uses of fractions and
ratio concepts
• Most common manipulative is two-color counters
• We also use set models in showing data.
[1/4 of boys are blond]
Set Model
Navigating Through Number and Operations
Lynne Tullos, LDOE
2010
Draw a small square on your paper.
• If the square = 2/5, draw 1 whole
Lynne Tullos, LDOE
2010
Look at your Pattern Pieces
• If this piece = 3/5 unit
• How much of a unit is this piece =?
• Draw the unit piece.
Using Fraction Circles, Strips, or Fraction Bars
Lynne Tullos, LDOE
2010
Understanding Size with Fraction Bars
▫ What do you notice about fraction bars?
▫ Compare fraction bars to circle fractions.
Lynne Tullos, LDOE
2010
Understanding Size with Fraction Bars
▫ What do you notice about these bars?
▫ Get out the 3rd and 4th bars: without finding a
common denominator, which is larger, 1/3 or
1/4?
▫ Which is larger, 1/5 or 1/8? How do you
know?
Lynne Tullos, LDOE
2010
Understanding Size with Fraction Bars
▫ Which is larger 7/8 or 8/9?
▫ Which is larger 2/3 or 11/12?
▫ Which is larger 5/8 or 5/7?
▫ NOTE: if the numerators are the same, you must
look at the size pieces!
Lynne Tullos, LDOE
2010
REMEMBER To Ask Children
• How can you tell which fraction is larger?
• What must you consider?
• What strategies can you use?
Modeling Decimals
• Notations for money are the first thing to come
to mind.
• $127.95 means 127 dollars and some part of
another dollar.
• The decimal point separates the dollars from the
parts of a dollar.
Modeling $127.95
• Because 95¢ is 95/100 of a dollar, we have
$127.95 = 127 + 95/100 dollars
• Because 95¢ is 9 dimes and 5 cents; one dime
=1/10 of a dollar and 1 cent = 1/100 of a dollar,
95¢ is 9 x 1/10 + 5 x 1/100
Consequently:
$127.95 =
1 x 102 + 2 x 101 + 7 x 100 + 9 x 1/10 + 5 x 1/102
GOALS:
•Compare fractions and decimals by using concrete
models
•Compare fractions and decimals by using
benchmarks
•Order fractions and decimals by using concrete
models, benchmarks, and parallel number lines
Lynne Tullos, LDOE
2010
Which Comes First?
Lynne Tullos, LDOE
2010
2/47
4/5
1/14
7/8
4/9
10/13
6/14
1/35
5/9
3/100
6/7
5/12
Sort these 12 fractions into
the categories below and
write what you notice:
• Close to 0:
• Close to ½:
• Close to 1:
Lynne Tullos, LDOE
2010
Sort these 12 fractions into the categories
below and write what you notice:
2/47, 4/5, 1/14, 7/8, 4/9, 10/13, 6/14, 1/35,
5/9, 3/100, 6/7, 5/12
▫ Close to 0: 2/47, 1/14, 1/35, 3/100
 (numerator is small compared to denominator)
▫ Close to ½: 4/9, 6/14, 5/9, 5/12
 (numerator is close to ½ of the denominator)
▫ Close to 1: 4/5, 7/8, 6/7, 10/13
 (numerator and denominator are about the same size)
Lynne Tullos, LDOE
2010
Looking for Fractions:
• Find a fraction between 3/7 and 4/7;
Lynne Tullos, LDOE
2010
Looking for Fractions:
• Find a fraction between 3/7 and 4/7;
• Find another fraction between 3/7 and 4/7
that isn’t ½.
Lynne Tullos, LDOE
2010
Denseness Property of Rational
Numbers:
• Between every 2 rational
numbers, there is another
rational number!!
More/Less Fraction Tree
Lynne Tullos, LDOE
2010
• Choose 1 fraction
• Choose a fraction smaller (left) and larger (right) than the first fraction
• Continue taking turns until someone misses their turn with an incorrect
fraction.
GOALS:
•Represent fractions and terminating decimals by using
visual models
•Determine fraction and decimal equivalents by using
tenths and hundredths grids, parallel number lines,
clocks, money, and calculators
•Understand relationships among fractions and
decimals
Taking an Hour for Clock Fractions
• Engage students by
asking questions such as:
▫ How many minutes in
an hour?
▫ How many minutes
after the hour is it when
the minute hand is
pointing to the 6?
Clock Fractions
▫ What are some ways you can use to find this?
 This leads to discussing 30 minutes out of 60 is ½ of
an hour
▫ How many minutes after the hour is it when the
minute hand is pointing to the 3?
 Since we were ½ way around the clock showed 30
min., we must be ½ of 30, or 15 minutes.
 The clock hand divides the clock into 4 parts so 15
minutes must be ¼ of the clock.
Decimals Make Cents!
Ten Tenths Make A Whole
• The definition of equivalent
is an equal amount.
• Of course, we usually see a
zero (0) in the hundredths
place to show we are dealing
with money.
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.1 = 10 cents
.2 = 20 cents
.3 = 30 cents
.4 = 40 cents
.5 = 50 cents
.6 = 60 cents
.7 = 70 cents
.8 = 80 cents
.9 = 90 cents
.10 = 10 cents
.20 = 20 cents
.30= 30 cents
.40 = 40 cents
.50 = 50 cents
.60 = 60 cents
.70 = 70 cents
.80 = 80 cents
.90 = 90 cents
The Hundredths Place
• Hundredths place is the
last digit to the right and
represents coins smaller
than a dime.
• .01 = 1 cent
• .02 = 2 cents
• .03 = 3 cents
• .04 = 4 cents
• .05 = 5 cents
• .06 = 6 cents
• .07 = 7 cents
• .08 = 8 cents
• .09 = 9 cents
• Look at the following
numbers. Can you
identify the number in
the hundredths place?
• What coins can be used?
• $2.12
• $1.47
• $0.94
• $5.55
• $0.83
• $20.06
Fraction and Decimal Equivalents
Simplified
Fraction
½
Number of Shaded
Numerator Written Fraction Written
Squares (Numerator) W/Denominator of as a Decimal Number
100
50
50/100
0.50 (or 0.5)
Contact Information
• STEM: Science, Technology, Engineering, and
Mathematics Goal Office
• Lynne Tullos
▫ [email protected]
• Jenny Foot
▫ [email protected]
• Dr. Guillermo Ferreyra
▫ [email protected]