Fourth Grade
Fourth Grade
Alicia Klaich and Deanna LeBlanc
Progression
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.
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.
Equivalent Fractions
•
Students learn that they can also divide the
numerator and denominator equally to generate
equivalent fractions.
•
Using a model, they accomplish this by equally
combining smaller parts to create larger 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…
What are some student misconceptions about
comparing fractions?
1
and
8
8 9
7
and
5
8 6
4
and
3
9 4
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
.
•
This line of thinking enables students to clearly
understand why only the numerators are added or
subtracted.
“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).
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.
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?
How can you solve the following
problem? How many different
ways can you solve it?
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.
Assessment/Resources
•
Howard County Wikispaces
•
https://grade4commoncoremath.wikispaces.hcpss.org/
•
Learning Trajectories versus Proficency
How visual fraction models aid in explaining equivalent fractions, subdividing parts, comparing fractions, and addressing student misconceptions. Discover effective strategies for understanding and generating equivalent fractions.
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Presentation Transcript
Fourth Grade Alicia Klaich and Deanna LeBlanc
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.
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.
Equivalent Fractions Students learn that they can also divide the numerator and denominator equally to generate equivalent fractions. Using a model, they accomplish this by equally combining smaller parts to create larger 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 decompose fractions and mixed numbers in more than one way: Students rename mixed numbers as improper fractions and vice versa. 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 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 100.2For 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, , and 1. For example, is seventy-eight hundredths closer to 0 or , 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.
Assessment/Resources Howard County Wikispaces https://grade4commoncoremath.wikispaces.hcpss.org/ Learning Trajectories versus Proficency
