1. science
2. mathematics
3. algebra

4th Grade Unit 2
Focus Content Standards


4.NF.1 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.
4.NF.2 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 . 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.
4.OA.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a
statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal
statements of multiplicative comparisons as multiplication equations.
4.OA.4 Find all factor pairs for a whole number in the range 1–100. Recognize that a whole
number is a multiple of each of its factors. Determine whether a given whole number in the
range 1–100 is a multiple of a given one-digit number. Determine whether a given
whole number in the range 1–100 is prime or composite.
Rationale
This unit focuses on two very important content standards – 1) making sense of equivalent
fractions using visual models to make connections to the symbolic notation of these numbers
AND 2) comparing and reasoning about fractions using the size of unit fractions and other
relationships between the numerator and the denominator. Students must be able to
articulate how the number of copies of a unit fraction helps determine the size of a number.
(4.OA.1) Ex. one eighth seven times or 1/8 +1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 or almost “a whole”,
or one-eighth from a whole (1- 1/8).
The 4th grade standards build upon the 3rd grade standards which focus on the concept of a
unit fraction and simple equivalent fractions using visual models (ex. Folding a piece of paper
into halves, then fourths, then eights, 16ths, etc. seeing what makes a half for different units.)
This is pre-requisite knowledge your students may or may not have this year; therefore, several
scaffolding lessons are available to begin this fourth grade unit. These beginning lessons will
insure that students know how to create and use linear tools for reasoning about the size of
various fractions in the fourth grade lessons.
As students gain more understanding of equivalent fractions using paper-folding and
reasoning, have them look for the factors in this structure. How do they relate? Where are the
factors in the model? How can you prove it? Have students connect the numerical symbols
with the models. See the standard 4.NF.1 and the connection to the first part of 4.OA.4. (Explain
why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models) This
understanding will prepare them for 5th grade content when they will need to work with these
ideas abstractly.
Focus practice standards include: Looking for and making use of structure (ex. the unit fraction
and its copies and the decomposition of fractions into unit fractions and also looking at them
geometrically through paper folding or on a number line plus other visual models) Justify your
thinking and critique the reasoning of others (ex. comparing fractions mentally and proving
and reasoning why fractions are equivalent)
ESSENTIAL UNDERSTANDING - the unit fraction.
Understanding geometrically and numerically the size of the unit fraction and the proportional
relationship between the numerator and denominator and the size of the number is essential
knowledge. Students can and should use area models and number line diagrams (that THEY
create!) to reason about equivalency. Grade 4 students use their understanding of unit
fractions to compare fractions with different numerators and different denominators. The first
step is the ability to mentally reason about the size of numbers by focusing on the size of the
unit fractions (no paper and pencil or conversions; however, paper-folding is
encouraged)…..Before students can learn to compare fractions mentally, they must have a
deep understanding of the size of the unit fraction in relationship to the whole.
Background Knowledge and Strategies
Comparing Fractions Mentally (reasoning with unit fractions)
Important strategy: Have strips of paper of equal length constantly available for students to fold or cut
and then use as a tool to help them communicate and reason about fractions. Focus on the relationship
between the size of different unit fractions in relation to the whole. This tool aids them in their ability to
justify their reasoning. It enables them to connect the visual model to the various equivalent, symbolic
fractional numbers. For example: connect a “half” to the numerical symbols of all the numbers that
equal a half. Why are 2/4, 6/12, 3/6 etc. all one-half? Have students reason using the models.
Here are the different types of fractions that need to be compared mentally from easiest reasoning to
more complex. ( This can be done during number talks.)
1. Compare unit fractions - ex. one-ninth and one-fourth; or one-twelfth and one-eighth
2. Compare fractional numbers with the same denominators, for example and ; five ninths
is composed of 5 (
) or
+
+
+
+
and seven-ninths is composed of 7 (
). In other
words, five of a unit is less than seven of the same unit, in this case, “ninths”.
3. Compare fractions with the same numerator. For example,
and . The unit “eighths”
is larger than the unit “ninths” so seven “eighths” is greater than seven “ninths”.
4. Compare fractional numbers using the benchmark of 1. For example they see that <
because
is less than 1 (and therefore to the left of 1 on the number line) but
than 1 (and is therefore to the right of 1 on the number line).
5. Students can also use the benchmark of one-half as in the example of
and
is greater
. Three-
sevenths is a little less than half (three and a half “sevenths” would be half of seven
“sevenths”) and five ninths is a little more than half (four and half “ninths” would be half of
nine “ninths”) of five-ninths is greater than three-sevenths.
Misconceptions and Challenges

Students’ over-reliance on the pie or pizza model of fractions – they think this is what
fractions are!! In other words, students don’t see fractions as numbers but as pies or
rectangles that are divided and shaded or unshaded.

Inadequate development of the idea that a fraction is a number.

Students view the numerator and the denominator as separate whole numbers which leads
to false conclusions such as 2/5 is greater than ½ because 2 is greater than 1 and 5 is
greater than 2.

They are focused on the number of parts and don’t attend the size of the parts and whether
or not they are equal. For example, they call an area divided into three unequal parts,
“thirds”.
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Understanding the size of the whole affects the size of the parts – ex. ½ of a small pizza is
different than ½ of a large pizza

Understanding that when communicating about an area using a fractional number, the
area does not have to be contiguous. The area of shape can be rearranged and it will still
be the same amount. (conservation of area)

Understanding I can’t compare fractions unless I am talking about the same “whole”.
Example: Dividing nontraditional shapes into thirds, such as triangles, is the same as dividing
a rectangle into thirds. If they are only used to dividing traditional shapes – circles, squares,
and rectangles – they begin to think that all shapes are divided similarly.

Children often do not recognize groups of objects as a whole unit. Instead they will
incorrectly identify the objects. For example, there may be 2 cars and 4 trucks in a set
of 6 vehicles. The student may mistakenly confuse the set of cars as 2/4 of the unit instead of
2/6 or 1/3 (Bamberger, Oberdorf, & Schultz-Ferrell, 2010).
More Strategies

Give students equal sharing tasks to make sense of fractions. Meaning needs to precede
symbols. Sharing tasks that help with understanding of equivalent fractions in which students
will get equivalent answers:
12 children want to share 9 candy bars so that everyone gets the same amount. How
much candy bar will each child get?
There are 6 candy bars for 8 children? If each child gets the same amount, how much
would each child get?
20 children want to share 8 candy bars equally. How much should each child get?

Use the number line to help students make sense of fractional numbers – to actually see
fractions as numbers. After students determine a solution for a sharing task; have them try to
prove it is correct by using a number line. They must determine what each number means
and why (in the case above) it adds up to 5 complete candy bars.

Have each student fold and cut fraction strips – halves, fourths, eighths, sixteenths, and also
thirds, sixths, and ninths. Have these up in the room like a number line so students can
constantly be referring to this visual when they are reasoning about fractions as a number.
Also, use this model for having students count by fractions; for example, count by halves,
sixths, etc. Later, begin counting in different places, for example, count by halves starting at
three-fourths. The count would be three-fourths, five-fourths or one and one-fourth, seven
fourths or one and three-fourths, etc. Have a student point on the fraction strip number line
as the students count.

Make sure students understand WHY we have fractions. For example, for more exactness in
measuring: I don’t need 2 or 3 of something; I need 2 ½. Fractions are necessary numbers
for accurate measurement and communication of ideas. Have students brainstorm all the
times fractions are a MUST!! They use them every day, but when and where?

Write fractions with the unit written in words For example
is written 2 thirds. This format
promotes students to think about the number of times they have a unit fraction. It also help
them see that you would add 2 thirds and 3 fourths together by just adding denominators.
(adjective/noun relationship)

Provide opportunities for students to work with irregularly partitioned, and unpartitioned
areas, lengths, and number lines. Give them meaningful tasks that involve measuring for
purpose – rulers are especially helpful with gaining a deeper meaning of fractions.
Correspond the ruler to the number line. Engineering design tasks or any science task that
has students measuring with rulers, scales, liquid, etc. Help students see how these
measurements are alike although what they are measuring and the unit is different.

Provide opportunities for students to develop their understanding of the importance of
context in fraction comparison tasks: Fractions are representation of quantities, and these
quantities are measured in relation to a unit (or a whole). The meaning of
determined in
part by the size of the whole.

When presenting fractional numbers use the horizontal line as opposed to the diagonal line
when separating the numerator and denominator. It is easier to help students make
observations about the relationship between the numerator and the denominator and
students are less likely to think they are whole numbers…. 2/3 as opposed to .