Transcript tenths - Catalyst

FRACTIONS, DECIMALS, PERCENTS
Real Numbers
π
Irrational Numbers
Rational Numbers
1/2
Integers
.85
Whole Numbers
0
1
-6
Natural Numbers
52
.333
3
34
-14
489
100
30
2
Activity - Rational Numbers

What’s the definition of a fraction?

What is a decimal (how is it related to fractions)?

What is a percent (how is it related to fractions
and decimals)?
Teaching Rational Numbers
Fractions  Decimals  Percent



Fraction instruction begins at grade 1 with
conceptual understanding proceeds to fraction
computation and equivalency in grades 3-6.
Decimal concepts are introduced in grade 4 and
continues with computation through 6.
Percent concepts are introduced in grade 5 and
continues with computation and problem solving
through grade 6.
Fractions


Write a fraction in the box:
Write 5 examples to teach students to read fraction
numbers:

4
How do you reduce
?
24

How do you divide these fractions?
3 2
4  3 
Fractions are conceptually complex


Unfamiliar units: Which is the largest?
1
16

1
9
1
2
1
3
Equivalence
12
  3
16 4
 5  5
1
5
1
5
New vocabulary

Terms
 Numerator
 Denominator
 Proper
 Improper
 Mixed
number

1 5
2 
2 2
Equal partitioning



Children’s background knowledge and experience
of half is dividing into 2 parts
Fractional units require that the parts are equal in
size
Fractional units require determining what the whole
refers to
1
cookie or package of 12 cookies
Conceptual understanding - fair
sharing - solve the following….


Share 5 sandwiches equally among 3 children. How
much can each child have?
Share 4 pizzas with 6 children. How much can each
child have?
Developing conceptual understanding



Fractions = relationship
Fractions ≠ specific amount
Defining the whole - use a variety of examples
 Continuous
quantities - single unit is divided into parts
 Discrete quantities, collections - sets divided into parts

Vary examples and provide explicit instruction
Developing conceptual understanding

One whole; one unit
Continuous quantities

Discrete quantities - collections - sets

Importance of the unit
Activity
Comparing Magnitudes

Comparing whole numbers
2 and 5
1
1
and
2
5

Comparing fractions

Only compare same whole unit
 1/5
of a cake is larger than 1/2 of a cupcake.

Conceptual Understanding of Fractions

Determine the unit - oral practice
 1/2
of the students in class v. 1/2 of a pizza v. 1/2
glass of milk

Manipulatives
 Paper
folding
 Colored circles
Paper strips
Rulers
Fraction tiles
Equivalent
Fractions
Operations are conceptually complex

Range of operations are procedurally different than
whole numbers
3 2
 
4 3
3 2
 
4 3
1 2 3
  

5 5 5
3 2
 
4 3
1 2 2
 
5 5 25
Adding and Subtracting Fractions with
Like Denominators



Show conceptually (with pictures) why you can’t
add different size units
Present procedural strategy for adding and
subtracting fractions with like denominators
Provide examples with like and unlike
denominators
 Students
work problems with like denominators
 Students cross out problems with unlike denominators
See DI Format 12.14
Activity
New Vocabulary and Strategies


Adding and Subtracting Fractions with Unlike
Denominators
New Preskill: Lowest (least) common denominator or
lowest common multiple
3 2
 
4 3

Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40
  Multiples of 3:
New Vocabulary and Strategies


Adding and Subtracting Fractions with Unlike
Denominators
New Preskill: Lowest (least) common denominator or
lowest common multiple
3 2
 
4 3

Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40
  Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30
New Vocabulary and Strategies



Adding and Subtracting Fractions with Unlike Denominators
New Preskill: Lowest (least) common denominator or lowest
common multiple
Rewrite the fraction with like denominators (12)
3 3 9
 
4 3 12

2 4  8
 
3 4  12
Relies on knowing the identity element of multiplication is 1
- any number times 1 equals that number and fractions
equivalent to 1.

New Vocabulary and Strategies





Adding and Subtracting Fractions with Unlike
Denominators
New Preskill: Lowest (least) common denominator or
lowest common multiple
Rewrite the fraction with like denominators (12)
Add
9 8 17
 
12 12 12
5
1
12
17
 Change improper fraction to a mixed
 12 17
12
number
New Vocabulary and Strategies


Reducing Fractions
New Preskill: Greatest Common Factor
24

36

Factors of 24: 1x24, 2x12, 3x8, 4x6
 Factors of 36:
= 1, 2, 3, 4, 6, 8, 12, 24
New Vocabulary and Strategies


Reducing Fractions
New Preskill: Greatest Common Factor
24

36

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
 Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

New Vocabulary and Strategies


Reducing Fractions
New Preskill: Greatest Common Factor
24 12 2
  
36 12 3

Relies on knowing the identity element of
multiplication is 1 - any number times 1 equals that
number and fractions equivalent to 1.
Other Reducing Strategies

What would you do with:
144
432
Repeated reducing with common factors
10
5
4  3
2
Conceptually complex

When you multiply fractions, is the answer bigger or
smaller?
3 2 6 1
  
4 3 12 2

When you divide fractions, is the answer bigger or
smaller?

3 2
 
4 3
3 3
 
4 2
9
1
1
8
8
Conceptual understanding of dividing
fractions

Devin has 3 1/2 cookies. She will give 1/4 of a
cookie to each student as they enter class. How
many people will get a cookie?
 Will
the answer get bigger or smaller?
Conceptual understanding of dividing
fractions

Devin has 3 1/2 cookies. She will give 1/4 of a
cookie to each student as they enter class. How
many students will get cookies?
 Will
4
the answer get bigger or smaller?
+
4
+
4
+
2
= 14
Multiplying Fractions

Procedural Strategy is fairly easy
1 2 2 1
  
2 3 6 3

Critical features:
Differentiating strategy from addition and subtracting
fractions

X sign means “of”: “What is one half of two thirds?”

New Vocabulary and Strategies

3 2

4 3
Dividing Fractions
3 3 2 3 9 6
9
9
      1 
4 2  3 2  8 6
8
8


Conceptual Understanding
 Equivalent
fractions
 Reciprocals (1/2 x 2/1 = 1)
 Identity property of division ( any number divided by 1
equals that number)



Decimals
Understanding of fractions is critical!
 Another way of expressing fraction in base
ten number system
 Start with what’s familiar - money
 Decimal concepts

 Understanding
decimal values and magnitudes
 Decimal and fraction equivalency
 Reading and writing decimal numbers
Conceptual Understanding of Decimals (whole
units v. parts)
= 1.4
= 2.5
= 1.25
Conceptual Understanding of Decimals (whole
units v. parts)
= 1.4
= 2.5
= 1.25
Conceptual Understanding of Decimals
Equivalent Units
.2 = .20
.5 = .50
Decimals on a number line
Reading & Writing Decimal Numbers
3/10
3/100
. = decimal point
One digit after the decimal point tells about tenths
Two digits after the decimal point tells about hundredths
.9
.09 .3 .03 .48
.60
Using place value prompt
Hundreds
Tens
Ones
.
Tenths
Hundredths
Reading and Writing Decimals
Critical Features”

Use minimally different examples
.800
.080 .008
4/100 = .4 .40 .04
.8 = 8/10 8/100 1/8
Operations with Decimals

Conceptual Understanding
 Using

objects / pictures
Procedural Strategies
 Given
problems set up
 Setting up own problems
 Involves
place value concepts
Percent
Understanding of fractions and
decimals is critical!
 Start with what’s familiar

Percent
used for grades
Percent for shopping discounts
Sales tax or tipping
Conceptual Understanding
Percent on the Number Line
Conceptual Understanding of Percent

Circle Graphs
20%
15%
15%
50%
Procedural Strategies for Percent

Changing percents to fractions and decimals

Changing fractions and decimals to percent
Problem Solving with Percent

Calculate % of whole or decimal value
 What
is 10% of 40?
 What is 1% of 200?

Calculating discounts with %
 Find
off.
total cost of a $85.00 coat that is on sale for 40%
Rational Numbers
Beware of misrules
 Use concrete and pictorial representations
to develop a solid conceptual
understanding of rational numbers.
 Make connections between fractions,
decimals, and percent explicit!
