Fractions - Nevada Mathematics Project

Report
Fractions: Fourth
Grade
What is a Fraction?
•What is a fraction?
•Think of a scenario
to represent 1/3 and
create a model.
Explore Fractions Using
Fraction Bars
• http://www.mathsisfun.com/numbers/fraction-numberline.html
• What fractions add up to one whole?
• Create equivalent Fractions
What patterns would you want your
students to notice?
• Number line
• Area Model
• Set Model
• Develop understanding of fractions as numbers.
• CCSS.Math.Content.3.NF.A.1
Understand a fraction 1/b as the quantity formed by 1 part when a
whole is partitioned into b equal parts; understand a fraction a/b as
the quantity formed by a parts of size 1/b.
• CCSS.Math.Content.3.NF.A.2
Understand a fraction as a number on the number line; represent
fractions on a number line diagram.
• CCSS.Math.Content.3.NF.A.2.a
Represent a fraction 1/b on a number line diagram by defining the
interval from 0 to 1 as the whole and partitioning it into b equal
parts. Recognize that each part has size 1/b and that the endpoint of
the part based at 0 locates the number 1/b on the number line.
• CCSS.Math.Content.3.NF.A.2.b
Represent a fraction a/b on a number line diagram by marking off a
lengths 1/b from 0. Recognize that the resulting interval has size a/b
and that its endpoint locates the number a/b on the number line.
Explain equivalence of fractions in special cases, and compare fractions by
.
reasoning about their size
• CCSS.Math.Content.3.NF.A.3
Explain equivalence of fractions in special cases, and compare fractions by
reasoning about their size.
• CCSS.Math.Content.3.NF.A.3.a
Understand two fractions as equivalent (equal) if they are the same size, or the
same point on a number line.
• CCSS.Math.Content.3.NF.A.3.b
Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3.
Explain why the fractions are equivalent, e.g., by using a visual fraction model.
• CCSS.Math.Content.3.NF.A.3.c
Express whole numbers as fractions, and recognize fractions that are equivalent
to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 =
6; locate 4/4 and 1 at the same point of a number line diagram.
• CCSS.Math.Content.3.NF.A.3.d
Compare two fractions with the same numerator or the same denominator by
reasoning about their size. Recognize that comparisons are valid only when the
two fractions refer to the same whole. Record the results of comparisons with
the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction
model
4.NF.1 Equivalent Fractions
• Explain why a fraction a/b is equivalent to a fraction (n × a)/(n
× b) by using visual fraction models, with attention to how the
number and size of the parts differ even though the two
fractions themselves are the same size. Use this principle to
recognize and generate equivalent fractions.
• How would student explain equivalent fractions?
• What do students need to understand in order to do this?
• 2/4
½
Equivalent Fractions
• Students subdivide the equal parts of a fraction, resulting in a
greater number of smaller parts.
• Students discover that this subdividing has the effect of
multiplying the numerator and denominator by n.
1 x 4 parts and 4 x 4 parts.
4.NF.2 Comparing Fractions
• Compare two fractions with different numerators and
different denominators, e.g., by creating common
denominators or numerators, or by comparing to a benchmark
fraction such as 1/2. Recognize that comparisons are valid only
when the two fractions refer to the same whole. Record the
results of comparisons with symbols >, =, or <, and justify the
conclusions, e.g., by using a visual fraction model.
Comparing Fractions
• Students create common numerators or common
denominators by renaming one fraction or both.
• When comparing fractions with the same numerator, they use
prior knowledge about the relative sizes of fractional parts.
2 and 1
5
3
How would you rename these
fractions to compare them?
Comparing Fractions
• Complete the following comparisons without using equivalent
fractions. Make a note of how you did them…
1 and 8
8
9
7 and 5
8
6
4 and 3
9
4
What are some student misconceptions about comparing
fractions?
4.NF.3 (a-b) Fractions as a Sum of
Unit Fractions
a. Understand addition and subtraction of fractions as joining and
separating parts referring to the same whole.
b. Decompose a fraction into a sum of fractions with the same
denominator in more than one way, recording each
decomposition by an equation. Justify decompositions, e.g., by
using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ;
3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
Describe a common misconception for this problem:
2 + 1
5
5
Fractions as a Sum
of Unit Fractions
• All fractions can be seen as a sum or difference of two other
fractions.
• Think in terms of adding/subtracting copies of 1/b.
“Just like 3 dogs + 9 dogs is 12 dogs, or 3 candies + 9 candies is 12 candies, or
3 children + 9 children is 12 children, 3 fifths + 9 fifths is 12 fifths” (Small,
2014, p. 51).
• This line of thinking enables students to clearly understand
why only the numerators are added or subtracted.
Fractions as a Sum
of Unit Fractions
• Students rename mixed
numbers as improper
fractions and vice versa.
• Students decompose
fractions and mixed
numbers in more than one
way:
mrs-c-classroom.blogspot.com
Before teaching the “shortcut,” allow
students plenty of time to reason with
models.
4.NF.3 (c-d) Adding and Subtracting
Mixed Numbers
c. Add and subtract mixed numbers with like denominators, e.g.,
by replacing each mixed number with an equivalent fraction,
and/or by using properties of operations and the relationship
between addition and subtraction.
d. Solve word problems involving addition and subtraction of
fractions referring to the same whole and having like
denominators, e.g., by using visual fraction models and
equations to represent the problem.
Adding and Subtracting
Mixed Numbers
• Students rename mixed numbers as improper fractions, then
add or subtract.
But…
• They notice that sometimes it is easier to add or subtract the
whole number and fraction separately.
How would you solve each problem? Why?
Adding and Subtracting Mixed
Numbers
• Students can “count up” to find the difference between mixed
numbers (see page 52 in Uncomplicating Fractions).
OR
• They might prefer to regroup part of the whole number in the
greater mixed number.
Explain how to regroup the first
mixed number in this problem.
4.NF.B.4 Multiplying Fractions by a Whole
Number
• CCSS.Math.Content.4.NF.B.4a Understand a fraction a/b as a
multiple of 1/b. For example, use a visual fraction model to represent
5/4 as the product 5 × (1/4), recording the conclusion by the
equation 5/4 = 5 × (1/4).
• CCSS.Math.Content.4.NF.B.4b Understand a multiple of a/b as a
multiple of 1/b, and use this understanding to multiply a fraction by
a whole number. For example, use a visual fraction model to express
3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n ×
(a/b) = (n × a)/b.)
• CCSS.Math.Content.4.NF.B.4c Solve word problems involving
multiplication of a fraction by a whole number, e.g., by using visual
fraction models and equations to represent the problem. For
example, if each person at a party will eat 3/8 of a pound of roast
beef, and there will be 5 people at the party, how many pounds of
roast beef will be needed? Between what two whole numbers does
your answer lie?
Interpreting the meaning of
multiplication
• It is important to let students model and solve these problems
in their own way, using whatever models or drawings they
choose as long as they can explain their reasoning.
• Once students have spent adequate time exploring
multiplication of fractions, they will begin to notice patterns.
• Then, the standard multiplication algorithm will be simple to
develop. Shift from contextual problems to straight
computation.
How can you solve the following
problem? How many different ways
can you solve it?
Kristen ran on a path that was ¾ of a mile in length. She
ran the path 5 times. What is the total distance that
Kristen ran?
Interpreting the meaning of
multiplication
• Adding 4/5 3 times ( 4/5 +4/5 +4/5)
• 4 fifths + 4 fifths + 4 fifths = 12 fifths, or 12/5
• The result of three jumps of 4/5 on a number line, beginning at 0
• The number of fifths of a 2-D shape if 3 groups of 4 fifths are
shaded.
Understand decimal notation for fractions, and
compare decimal fractions
• CCSS.Math.Content.4.NF.C.5 Express a fraction with
denominator 10 as an equivalent fraction with denominator
100, and use this technique to add two fractions with
respective denominators 10 and 100.2 For example, express
3/10 as 30/100, and add 3/10 + 4/100 = 34/100.
• CCSS.Math.Content.4.NF.C.6 Use decimal notation for
fractions with denominators 10 or 100. For example, rewrite
0.62 as 62/100; describe a length as 0.62 meters; locate 0.62
on a number line diagram.
• CCSS.Math.Content.4.NF.C.7 Compare two decimals to
hundredths by reasoning about their size. Recognize that
comparisons are valid only when the two decimals refer to the
same whole. Record the results of comparisons with the
symbols >, =, or <, and justify the conclusions, e.g., by using a
visual model.
Interpreting decimals
• Representing tenths and hundredths and decimals as a sum.
The 10-to-1 relationship continues indefinitely.
What is a common misconceptions students
have about decimal place value?
Students must
understand the
equivalent
relationship
between tenths
and hundredths.
Representing Decimals on a
Number line
One of the best length models for decimal
fractions is a meter stick. Experiences
allow students to compare decimals and
think about scale and place value.
Comparing Decimals
• Reason abstractly and quantitatively. Develop benchmarks; as
with fractions: 0, ½, and 1. For example, is seventy-eight
hundredths closer to 0 or ½, ½ or 1? How do you know?
• Using decimal circle models. Multiple wheels may be used to
conceptualize the amount. Or, cut the tenths and hundredths
and the decimal can be built.
• Why do many students
think .4 < .19?
Activity: The Unusual Baker
• George is a retired
mathematics teacher who
makes cakes. He likes to
cut the cakes differently
each day of the week. On
the order board, George
lists the fraction of the
piece, and next to that, he
has the cost of each piece.
This week he is selling
whole cakes for $1 each.
Determine the fraction and
decimal for each piece.
How much will each piece
cost if the whole cake is
$1.00?
The Unusual Baker
• CCSS 4.NF.A.1 Equivalent fractions
• CCSS 4. NF.A.2 Compare two fractions
• CCSS 4. NF.B.3a Understand addition and subtraction of
fractions
• CCSS4.NF.B.3b Decompose a fraction into a sum of fractions
with the same denominator.
• CCSS4. NF.C.6 Use decimal notation for fractions with
denominators 10 or 100.
• CCSS4.NF.C.7 Compare two decimals to hundredths by
reasoning about their size.

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