Structural Overview
Section 1: Working with Unit Fractions
Section 2: Equivalence and Comparison of Fractions
Section 3: Operations with Fractions

 

 

About the Structural Overview

 

Fractions have their roots in equipartitioning. A fraction is always defined relative to the whole to which it refers. Young students learn this as they equipartition collections and single wholes. They begin to understand “½” as a description of half of the whole collection or as a description of one-half of a single whole, shared fairly between 2 people. They know that the whole collection or single whole are each 2 times as large as that fair share.

 

As the Structural Overview illustrates, this Learning Trajectory focuses on the transition from unit fractions to fractions in the form of a/b in Grade 3, then on equivalence and comparison of fractions, leading up to operations with fractions in Grades 4-5.

 

CCSS-M Description

Descriptor

Section 1: Working with Unit Fractions

3.NF.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.

Sharing a single whole fairly (as a region, for example, circle or rectangle) is a valuable (and powerful) way to introduce fractional parts to students [3; 5]. Students are introduced to this notion in Standard 3.NF.1 in the Equipartitioning LT and they name one fractional part as “one-nth” of the original whole. If the original whole is equal to 1, this “one-nth” is called a unit fraction.

 

Students know the three criteria for successful and correct equipartitioning:

1) the desired number of parts,

2) equal-sized parts, and

3) exhausting the whole [16].

 

They find it easier to exhaust the whole and create even-valued numbered parts (2, 4, 8) than odd-valued numbered parts (3, 5, 7) [16; 17]. In these instances, shape matters as well. For example, on a circle, students initially often find it easier to create six parts than three parts, as the 2-split (creating two equal-sized parts, or halves) is a basic way for them to start [5; 11; 17]. They soon realize, however, that this cannot be the starting point for an odd split. They can name and write a unit fraction as a description of one part of a single whole (see Standard 3.G.2 of the Equipartitioning LT) [3; 5; 26]. For example, students name the shaded region of the figure below as “one-fifth,” and write this as 1/5:

 

 

From equipartitioning, students also know that having more equal parts in a whole makes each share smaller, hence 1/a > 1/(a+1) [3; 7]. They recognize that the parts do not need to be congruent as long as they are the same size. Students are also flexible in creating fractional parts because they understand the composition of splits—for example, they can create 12 equal parts by splitting a whole into 4 equal parts and then splitting each of those four into 3 equal parts, often using the two dimensions of a rectangle (see Standards 2.EDP.D and 3.NF.1 of the Equipartitioning LT) [4].

 

In Standard 2.MD.6 of the Length, Area, and Volume LT, students work primarily with whole numbers represented on a number line. For this standard, they equipartition the length between 0 and 1 into n equal parts and identify each of those lengths as 1/n long. A common challenge for students in working with the number line is to understand the relationship between the length of the segment and the naming of a point or position on the number line [10]. Students work with strips of a given length (created by folding or equipartitioning) to emphasize length and create number lines from these to emphasize that points are named based on their distance from 0. Thus, they learn that the point 1/3 is located one-third of the distance from 0 to 1. Working with “fraction bars” or “fraction strips,” like those shown below, children avoid the misconception that the points 1/2, 1/3, 1/4, 1/5, … are equally spaced on a number line [10] by aligning them and attempting to measure different fractions using a particular fraction.

 

 

Rulers provide another resource for learning to name equipartitions as fractions [10; 25]. Students use the ruler to identify lengths of 1/2, 1/4, 1/8, and 1/16 of an inch and their positions on the ruler. It is important to stress that the inch is the referent unit in these instances. Other challenges include asking students to 1) identify 1/2, 1/4, 1/3, 1/6, and 1/12 on a ruler using a foot as the referent unit, and 2) use a meter stick to identify 1/10th and 1/100th of a meter.

 

Finally, given a particular length, students should have experiences naming the length as a unit fraction relative to different referent units [26]. For example, suppose that a line segment with length X is given below.

 

 

Students should be asked challenging questions such as:

1.         Write the length of line segment AB in terms of X.

2.         If line segment AB has a length of 1 inch, how would it be named relative to X? What would you name a line segment that has length 5 x AB?

 

3.NF.2.b Represent a fraction a/b on a number line diagram by marking off a lengths of 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.

A critical next step is to model and name non-unit fractions. This process begins with regional representations (pizza slices, sections of oranges, etc.) and is extended to include number lines. The advantage of regional representations is their familiar contexts for students [8; 25].

 

To begin, students name regions of wholes as a/b (where a < b) as a copies of 1/bth and that a/b is a times as long or as large as 1/bth [15].

 

For example, students name the figure below as 2/5.

 

 

Two common errors need to be discussed and addressed.

 

Error 1: Naming the un-shaded region rather than the shaded region. For example, students name the fraction represented by the drawing above as 3/5 because there are 3 un-shaded sections and 5 total sections.

 

Error 2: Naming the shaded sections over (in relation to) the un-shaded sections as a fraction of the whole. For example, students name the fraction represented by the drawing above as 2/3 because they are relating the number of shaded regions (2) to the number of un-shaded regions (3).

 

The first error is a matter of convention: in fraction problems, the shaded part is the focus. The second error is evidence of a stronger misconception: students may fail to understand that the denominator of a fraction refers to the number of parts in the whole [1; 10]. A particularly compelling context for this misconception is a basket with 3 red marbles and 2 white marbles. An answer of 3:2 describes the ratio of red marbles to white marbles (known later as odds) while the fraction 3/5 describes what fraction of the total marbles are red.

 

From equipartitioning, students know that b 1/bths = 1. By equipartitioning single wholes, they have named each share as 1/bth and understand that the whole is b times as large as one share. Hence, from the figures below, b/bths = 1 [3; 5].

 

 

Next, students extend their understanding of fractions to include fractions of the form a/b in which a > b. They name such fractions in two ways: as mixed numbers and as improper fractions. At all times, students identify the referent unit (the value of 1 whole) connected with their naming.

 

Note to teachers: Students identify the referent unit associated with a fraction number as 1 (or b/b), and they name the fraction as a number of equal parts of a given size relative to that unit (e.g., 7/4 is 7 parts of size 1/4 relative to a whole consisting of 4 parts of the same size, 1/4).

 

Students represent fractions greater than 1 using regions and number lines. If regions are used, it is imperative to provide clear information about what the referent unit is. Students’ flexibility with the referent unit is a critical part of understanding fractions [5; 12]. If a quantity is named as a fraction (less than or greater than one), then one must assume that the intended referent unit is “one” of that quantity [5; 15]. For example, when students are asked to compare 1/5 to 1/7, they must assume that these two numeric values are compared relative to the same unit of one [6; 23]. An advantage of representing fractions on the number line is that in elementary grades, number lines are constructed to have a consistent, clearly identified, and repeated unit of one.

 

Students should also be challenged to consider problems in which the referent unit is more than one, and asked to find a fraction of that unit. For example: “Suppose that Marge has one-half dozen eggs. How many eggs does she have?” (Referent unit: dozen eggs = 12 eggs). Likewise, in the example shown below, “How much of the two circles are shaded?” or “Name the shaded parts.”

 

 

Students note that if the referent unit is a single whole, the representation shows 6/5ths. However, if the referent unit is 2 wholes, the representation shows 6/10ths or 3/5ths. Similarly, if the referent unit is equal to one of the resulting fractional parts of an n-split performed on a single whole (1/5th from the 5-splits shown above), then the name of the shaded parts becomes their count (6 in the case above). Students at this level are not expected to convert between referent units, but should be challenged to give names or measurements of the same quantity using more than one referent unit.

 

The number line also provides a particularly effective context to extend beyond proper fractions (less than one) to improper fractions (greater than or equal to one, expressed for example, as 5/4) and mixed numbers, because one can continue to iterate on the unit fraction past 1 [18].

 

For proper fractions between 0 and 1, students compare a/b to c/b (like denominators) based on the sizes of a and c and explain their reasoning. Students also identify the referent unit of 1 relative to a/b for both proper and improper fractions.

 

Combining the fractions less than 1 and those greater than 1 into one set of rational numbers, students name and distinguish between the meanings of numerators and denominators [1; 5].

 

Section 2: Equivalence and Comparison of Fractions

3.NF.3.c Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.

 

Students come to know that any whole number n is equivalent to the fraction n/1 by applying earlier Standard 3.NF.2.b with a referent unit of 1 as 1/1. Conversely, any fraction n/1 is equivalent to the whole number n. Students also recognize from earlier Standard 3.NF.2.b that n/n equals one.

 

They extend this to recognize that 2n/n equals 2, and so on. Students justify this result in multiple ways including:

1) stating that 1 is n times as large as 1/nth;

2) counting unit fractions from 0;

3) iterating lengths on a number line; and

4) identifying a referent unit of 1 and relating to division by considering a fair-sharing collections problem which are typically equipartitioned without remainder (12/3 = 4) [3; 13; 18; 21].

 

Note to teachers: Counting starts from 0 and concludes with the nth count, while iterating has to begin with a given unit and is therefore only repeated n-1 times, as shown in the figures below:

 

 

3.NF.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).

 

Students recognize and generate equivalent fractions using both their knowledge of equipartitioning and by exploring the number line [5; 18]. These contexts give students two models with which to work.

 

When equipartitioning a whole for a certain number of people (see Standards 2.G.3 and 3.G.2 of the Equipartitioning LT), students should be challenged to find multiple ways to share [4; 21]. On both a rectangle and a circle, this may lead to a composition of splits (see Bridging Standard 2.EDP.D of the Equipartitioning LT) [4]. When asked for multiple ways to share, some students use a co-splitting strategy (see earlier Bridging Standards 2.EDP.A and 3.EQP.A of the Equipartitioning LT) when sharing for p people [3; 21]. They split the whole into kp parts of size 1/kp (where k is an integer) and give each person k parts. Generally students start with k = 2 but realize that they can make k any integer. For example, when asked to share a circular cake among 2 people, students split the cake into (2 x 2 = 4) equal parts and give each person 2 of the parts. They may then notice that they could split the cake into (2 x 3 = 6) equal parts and give each person 3 of the parts, as shown below. Indeed, they could go on to progressively split the cake into (2 x k = 2k) parts and give each person k parts.

 

.          

 

The above strategy may be accompanied with a visual justification by students. If not, students should be challenged to justify their covariation strategy [3] with objects, such as paper or play-dough “cakes.”

 

Students have experience sharing circles and rectangles for 2, 3, 4, 5, 6, and 8 people. When applying the above covariation strategy, they may justify it by placing, for instance, 2 halves on top of 4 fourths and noticing that each one-half completely covers exactly two-fourths.

 

Students are also challenged to use a number line to find fractions that are equivalent to 5/3. Using their knowledge of earlier Standard 3.NF.2.b, students first locate 5/3 on the number line and mark each 1/3. They create a representation similar to the one below.

 

 

Students can then equipartition each interval of length 1/3, out of the 5/3, into fourths to create 20 smaller equal intervals. They recognize each of these smaller intervals as having a length of 1/12th units.

 

 

From the figures above, students note that 5/3 and 20/12 lie on the same location on the number line. They conclude that 5/3 is equivalent to 20/12.

 

Another approach to recognizing equivalent fractions is to use “fraction bars” or “fraction strips” [10; 18; 19; 25]. However, the number line approach is powerful because it requires students to equipartition a line into equal parts, which is important in their cognitive development of number and length measurement [10; 25].

 

3.NF.3.a Understand two fractions as equivalent (equal) if they are the same size or the same point on a number line.

 

Students prepare to compare and order fractions with either unlike numerators or denominators, using a variety of strategies including using models, using a number line [1; 8; 10; 11; 18; 19; 22; 24-26], comparing with benchmark fractions, through equipartitioning, and eventually symbolically [9; 20; 26].

 

To facilitate comparing, students understand, describe, and use models (including the number line) to identify and generate equivalent fractions, addressing relations among halves, fourths and eighths, and between thirds and sixths. Fractions are shown to be equivalent using models in three ways:

 

1)      sets

2)      using areas or regions (e.g., rectangles, circles)

3)      locating identical positions on a number line [18; 25]

 

When using regions, students learn that the wholes can be equipartitioned or split into smaller units. Thus, any given shaded region can be named in more than one way. In moving to number lines, students draw lengths using rulers as models for equivalence, learning the conventions that govern the use of different-sized tick marks.

 

However, for most students, it is easier to find equivalent fractions by splitting a region into smaller equipartitions and composing smaller regions into larger units [12; 18]. For example, they can perform a 2-split (creating 2 equal parts) on half of a circle and reason that they have created two “fourths” because the same action could be applied recursively to the other half, yielding an identical result; thus

2/4 = 1/2. This requires them to check that the larger and smaller fractional parts both co-measure (i.e., cover or span equally) the shaded and un-shaded region.

 

Dot drawings can also be useful in student learning of equivalent fractions [2; 25]. These are used to represent different ways of grouping, and facilitate students’ understanding that equivalent fractions can be viewed as a problem of forming various size subgroups. For example, 8/12 can be viewed as two groups of four over three groups of four as in the figure below.

 

 

8/12 can also later be viewed as 2/3 + 2/3 + 2/3 + 2/3 as ratio units when considering “2/3” as a ratio unit (see Standard 6.RP.2 in the Ratio and Proportion and Percents LT) [6].

 

From an equipartitioning standpoint, equivalent fractions can be introduced using sets of items or fair shares. Students learn that equivalent fractions can be represented by different distributions of items and people [5; 18]. For example, consider the problem: “Is 6 pizzas fairly shared among 8 people at one table, equivalent to 3 pizzas fairly shared among 4 people at each of two tables?” At first, students may perform the actual equipartitioning for each table and see that if 6 pizzas are shared among 8 people at one table, each person gets 3/4 of a pizza, the unit ratio of pizzas per person. Likewise, they notice that when sharing 3 pizzas among 4 people, each person also gets 3/4 of a pizza, which is true at both of the two tables with these numbers of pizzas and people. Later, using co-splitting, students notice that a 2-split performed on 6 pizzas being shared among 8 people yields two tables with 3 pizzas being shared among 4 people at each, with the equivalence being ensured by the splitting action.

 

As students become sophisticated in working with fraction equivalence, they should be asked to consider what happens when the referent unit changes. One way to begin to approach this is to ask them if all halves of congruent objects are equal. Consider the figures below that show several different 2-splits on a set of congruent objects:

 

 

Some students may state an equipartitioning argument: If a half is the result of splitting a whole into 2 equal parts, then all halves are equal.

 

Note to teachers: In terms of the importance of referent units, one could ask, for example: “If we were to split a bag of 10 apples into two bags of five apples each, is each half equal?” A direct answer based simply on the quantity of apples as a referent unit is “yes.” However, this type of context could also promote a discussion of the relationship of referent units to quantity, shape, and type of units, and whether the two sets of apples created in the example above, or any other naturally varying (i.e., not necessarily congruent) discrete objects, are exactly equal. In this example, the referent unit is the “bag of apples.” One could measure the bag of apples by number (10), and half would be 5 apples. But what if the apples are different in size or color? One could measure the bag of apples by weight, and half of the bag would be the apples that make up half of the weight of the total bag.

 

The two bags could contain the following apples:

    

 

Some students may consider each of these as half of the bag, as they each contain 5 apples. However, other students may argue that the sets are not equal, since the set on the left has only one small apple and the other set has two (and so forth). It is important to have these discussions with students as they learn the significance of the referent unit and the relationship between the referent unit.

 

4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x 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.

 

Students are challenged to use different representations for a/b and (n x a)/(n x b) to reason about and justify the equivalence for various values of n, and are asked to generate representations for other equivalent fractions.

 

One way to initially approach this problem is to give students a number of equivalent fraction pairs and ask them how fractions with different numbers of parts are located at the same location on the number line. For example, having represented 14/12 and 42/36 as the same location on the number line, students recognize that the denominator and numerator are both increased by a factor of 3. They also recognize that one important consequence of this is that the locations of 14/12 and 42/36 on the number line are made the same because a) there are 3 times as many parts, and b) the parts themselves result from the 3-split equipartitioning of the 1/12th-intervals of 14/12, so for the same referent unit 1, there are 3 times as many parts for 42/36 as for 14/12.

 

Conversely, students could reassemble groups of two 1/12th-intervals of 14/12 into 1/6th-intervals. They recognize that though the denominator and the numerator are both reduced by a factor of 2, but because there are half as many parts from the reassembly of the 1/12th-intervals of 14/12, the number line locations of 7/6 and 14/12 are the same.

 

 

Students also recognize the same pattern from other pairs of equivalent fractions and conclude that two fractions are equivalent if and only if their numerators and denominators are both increased or decreased by the same factor.

 

For example, students recognize that the first pair of fractions are equivalent, but that the second and third pairs of fractions are not equivalent:

1.      , both numerator and denominator increased multiplicatively by the same factor, 2

2.      , numerator and denominator decreased (multiplicatively) by different factors

3.      , numerator and denominator increased additively, and not by multiplicative factors

 

Similar to Standard 3.NF.3.b earlier, students also recognize this idea through equipartitioning and examining shaded regions on a circle or rectangle. The main difference between this standard and Standard 3.NF.3.b earlier is that in the previous standard, students are not expected to generate the mathematical property that underlies the equivalence of fractions.

 

For example, students use a shaded region, such as on a circle, to represent the fraction 12/18. They then equipartition each of the 1/18th parts using a 2-split to create 36 equivalent smaller intervals. They recognize each smaller part as the unit fraction 1/36 and note that while the shaded region remains the same, 12/18 is equivalent to 24/36. Conversely, students reassemble every 4 parts of the region represented by 24/36 into a larger part and that 9 such resulting parts (named as 1/9) can be composed to form the whole circle. They recognize that because the denominator is reduced by a factor of 4, the fraction 6/9 has a region equivalent to 24/36 because it requires one-fourth as many parts as the reassembly of the parts of size 1/36.

                       

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Using sets of items or fair shares, students represent and reason about the equivalence of fractions with different distributions of items and people, using a similar inferential process [5; 18].

 

Students should also gain practice symbolically manipulating fractions, and simplifying them into equivalent fractions, by factoring both the numerator and denominator (see also Standard 4.OA.4 in the Division and Multiplication LT). They learn that when identical factors occur in the numerator and denominator of a fraction, their quotient’s equivalence to 1 permits the removal of those factors based on commutativity and the property of multiplicative identity.

 

For example, .

 

3.NF.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.

 

After understanding how to recognize and generate equivalent fractions, students begin to compare fractions numerically. They reason about the denominators when the numerators are equal, and about the numerators when the denominators are equal [14]. For example, 3/8 is less than 3/4, because 3 items shared among 8 people results in a smaller fair share than 3 items shared among 4 people. Likewise 3/8 is greater than 2/8, because 3 items shared among 8 people results in a larger fair share than 2 items shared among 8 people, or because 3/8 is to the right of 2/8 on the number line. They also learn to code these relationships using symbols (e.g.,

3/8 < 3/4).

 

Note to teachers: Similar comparisons of fractions can be made using part-whole contexts, but this can be confusing to students or lead to misconceptions when dealing with improper fractions in which the numerator is larger than the denominator. Therefore, it is suggested to acknowledge such explanations by students, but to use caution in overgeneralizing and to avoid presenting that reasoning by itself. For example, students may express“3/4 of people have brown eyes” as “3 out of 4 people have brown eyes”. However, the “part out of whole” language cannot be used with improper fractions.

 

Students should also be asked whether their comparisons still make sense if the referent unit changes. For example, they should be reminded of questions like, “Are all halves the same?” and challenged with questions such as, “Suppose John has 12 apples in a bag and that Louise has 18 apples in a bag. Would you rather have half of John’s bag of apples or half of Louise’s bag of apples if you wanted to eat the most apples?”

 

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 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.

Students compare fractions with different numerators and denominators using four strategies that build on prior work by:

 

1) Using visual models (e.g., circle diagrams and fraction bars) to compare two fractions [1; 10; 18; 22; 24-26]. For example, they see that 1/2 > 1/3 (and avoid the misconception that if a > b then

     1/a > 1/b) by drawing on their equipartitioning knowledge of sharing a cake for two people and a cake for three people (including their understanding of qualitative and quantitative compensation), recognizing that a piece of a cake that was shared fairly for two is larger than a piece of a cake that was shared fairly for three.

 

2) Comparing fractions to benchmark fractions (0, ½, ¼, and 1) [1; 8; 19]. For example, students reason that 2/4 is equivalent to the benchmark fraction 1/2 by using a visual model or by reasoning numerically. Then, since 5/8 is greater than the benchmark fraction 1/2, 5/8 is greater than 2/4.

 

3) Comparing the denominators of two fractions whose numerators are equal, or comparing the numerators of two fractions whose denominators are equal [14], (as in Standard 3.NF.3.d earlier in this LT).

 

4) Relating the use of models to numeric strategies. For example, as students become more proficient, when they are asked to compare 5/6 and 9/12, they can change 5/6 into the equivalent fraction 10/12, which permits direct comparison with 9/12, as in Standard 3.NF.3.d earlier in this LT.

 

In employing each of these strategies students should be reminded to clearly state the referent unit, especially if that referent unit is not equal to 1.

 

Section 3: Operations with Fractions

4.NF.3.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.

Students can decompose a fraction into additive parts, writing it multiple ways as a result of composing unit fractions [21]. They learn to do this for all rational number fractions, a/b, where b does not equal 0.

 

Students should be challenged to combine or join single unit fractions (e.g., 1/2 or 1/3 or 1/4) in multiple ways to make other fractions. For example, they should be able to decompose 3/4 into 3/4 = 1/4 + 1/4 + 1/4 and know that since 1/4 + 1/4 = 2/4 = 1/2, then 3/4 can also be decomposed as 3/4 = 1/2 + 1/4.

 

They should justify their decompositions using visual models such as circle diagrams, fraction bars, and equipartitioning.

 

4.NF.3.a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

 

Before students are formally introduced to the mechanics of fraction addition and subtraction, they need to be able to reason about the structure of a fraction a/b [8]. One way for students to understand a/b is to decompose a/b into a parts of size 1/bth (see Standard 3.NF.2.b earlier in this LT). This helps them to further understand that fractions can be “joined” (added) and “separated” (subtracted), if (and only if) the fractions are taken as parts of the same whole.

 

Proceeding in this way, students’ understanding of writing additive number facts with two or more addends can be applied to adding multiple 1/bths to create a/b. Decomposing a/b into a parts of size 1/bth can be used as a basis for transitioning to common denominators in addition. That is, students proceed from this fact to add a/b + c/b to get (a+c)/b. This helps students avoid the misconception that, for example, 1/4 + 1/4 = 2/8 (i.e., the misconception of adding fractions by adding the numerators and the denominators) [18; 25]. Models such as rulers, with which children can see that 1/4 inch + 1/4 inch ≠ 2/8 inch can also help to establish this correct reasoning.

 

Students can add and subtract using fractions with like denominators for all problem types from addition and subtraction, including joining, separating, and comparison problems [18; 25].

 

4.NF.3.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.

 

Students add and subtract fractions with like denominators based on an understanding that the denominator describes the size of the part, and the numerator describes the number of those parts. They apply this knowledge to adding and subtracting mixed numbers and/or improper fractions [1; 10; 18].

 

Based on their understanding of composing and decomposing (in the sense of addition) fractions, students know that the addition of a parts of size 1/bth composes a fraction a/b. They also realize that any fraction a/b can be decomposed to a parts of size 1/bth each. Combining these two pieces of information, they add fractions with like denominators by adding the numerators (e.g., number of parts) and justify this understanding using circle graphs and the number line. To deal with mixed numbers, students then use their understanding of Standard 3.NF.3.c earlier in this LT, and can rewrite 1 as n/n (with n being any whole number) or any whole number k as kn/n. So for example, students know that to add 1 3/4 + 2/4 they can rewrite 1 3/4 as (4/4 + 3/4) to get 7/4, and then 7/4 + 2/4 = 9/4.

 

Fraction addition and subtraction assumes that students have learned, retained, and can apply their knowledge of the referent unit (see Standard 4.NF.3.a earlier in this LT) [5]. Students can be asked to create a counterexample to show what would happen if the referent unit were not in common (e.g., 1/4 foot + 1/2 inch).

 

4.NF.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.

 

Students add and subtract decimals to the hundredths by using place value (e.g., ones to ones, tenths to tenths, hundredths to hundredths; see Standard 5.NBT.7 in the Place Value and Decimals LT), which may involve lining up the decimal points of the numbers. Attention to place value helps students avoid a common misconception of right-justifying decimals with different numbers of decimal places.

 

Students use their knowledge of equivalent fractions and of adding fractions with like denominators to solve word problems and add fractions with denominators of 10 and 100 by writing a/10 as

(a x 10) / 100.

4.NF.3.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.

Students apply their knowledge of adding and subtracting fractions with like denominators by solving word problems [12; 18]. For example, suppose that to prepare dinner, Shayla needs 7/4 cups of milk to make macaroni and cheese, 5/4 cups of milk to make gravy, and 3/4 cup of milk to make a cake. How much milk does she need all together?

 

Macaroni and Cheese

Gravy

Cake

 

Students may add the fractions as 7/4 + 5/4 + 3/4, or they may convert the improper fractions to mixed numbers. Using the model above, they notice that the 3/4 cup from the macaroni and cheese could combine with the 1/4 cup from the gravy to make 1 whole cup, and thus simplify the addition.

 

5.NF.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.

Most students do not yet know how to add fractions with unlike denominators, for example 2/3 + 3/4. They should be presented with examples that lead to explorations of and questions about the addition (or subtraction) of such fractions. They should be asked to provide other examples to show that merely adding the numerators to each other and the denominators to each other does not work. For example, they show that 1/2 + 1/4 does not equal 3/6 because they know that 3/6 is equal to 1/2, and 1/2 + 1/4 is more than 1/2. Another way to demonstrate this idea is with the number line, where students see that 1/3 + 1/3 does not equal 2/6.

 

Another bridging activity is to pose to students the possibility of finding a common or like denominator, for example, in the case of

1/2 + 1/4 where they can use their knowledge of equipartitioning to see that 1/2 = 2/4.

 

Students first learn to add and subtract fractions with like and unlike denominators within specified fraction families based on the understanding that the denominator describes the size of the part, and the numerator describes the number of those parts [1; 10]. Using these facts, students find equivalent fractions with common denominators in two ways:

1)         Students use models, knowledge of equivalent fractions developed in earlier grades, or other relationships among fractions that they have previously developed. For example, to solve 2/3 + 5/12, students recognize that the unit fraction 1/3 is equivalent to 4/12, so 2/3 would be equivalent to 8/12, making 8/12 + 5/12 = 13/12 (or 1 1/12) [2; 18; 25].

 

2)         As students become more proficient, they add and subtract fractions with unlike denominators by finding common denominators and equivalent fractions using least common multiples (LCM’s) or greatest common factors (GCF’s) [25].

 

Students learn to factor in Standard 4.OA.4 of the Division and Multiplication LT. Although LCM’s and GCF’s are not formally introduced until 6th grade (see Standard 6.NS.4 in the Division and Multiplication LT), they can be introduced to students here as a more systematic approach to finding common denominators than the guess and check methods used in some curricula.

 

For example, to add 7/12 + 4/15 students reason that the LCM of 12 and 15 is 60.

 

7/12 x 5/5 = 35/60    4/15 x 4/4 = 16/60

 

Therefore, 7/12 + 4/15 = 35/60 + 16/60 = 51/60.

 

The greatest common factor of 51 and 60 is 3. So 51/60 = 17/20 in lowest terms: 7/12 + 4/15 = 17/20.

 

Students may also use Venn diagrams to determine the LCM or GCF for a pair of numbers and then apply this to finding a common denominator for a pair of fractions in order to adding them [2]. For example, to add 5/18 + 9/24, students determine the GCF of 18 and 24 using a diagram similar to the one shown below, where the factors of each number are placed in the appropriate areas of the diagram: (common) prime factors of both numbers in the intersecting area, additional prime factors of 18 in the left area, and additional prime factors of 24 in the right area.

 

 

Knowing that the GCF is 2 x 3 = 6, students then reason that 5/18 can be multiplied by 4/4, because 4 (= 2x2) is the other factor of 24. Likewise, 9/24 can be multiplied by 3/3. Students then add the fractions by writing and evaluating the expression

5/18 (4/4) + 9/24 (3/3).

 

Note to teachers: Using the same Venn diagram, the LCM of 18 and 24 could be determined by finding the product of every number in the diagram (i.e., 3 x 2 x 3 x 2 x 2 = 72).

 

5.NF.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.

Students recognize that when contextual problems involve the addition or subtraction of fractions referring to the same whole but with unlike denominators, they must find a common denominator in order to solve the problem. This is done using either a visual model or an equation [18; 23; 25].

 

For example, consider the following problem:

Todd’s family took two pies to a dinner party. At the end of the evening, there was 1/6th of one pie left over and 4/9th of the other pie left over. How much pie does Todd’s family have left to take back home?

 

Students may draw pictures like the ones below to show the remaining amounts of each pie.

 

They then perform a composition of splits on each pie to create pieces of the same size (named by a common denominator, 1/18th) by using a 3-split on the piece left from the first pie and 2-splits on the pieces left from the second pie, as shown below:

 

The number of pieces are then added together to determine the total amount of pie that was left over for the two pies.

 

Students may also write an equation representing this process and use skills from Standard 5.NF.1 earlier, to solve the equation:

1/6 + 4/9 = 3/18 + 8/18 = 11/18.

 

NOTE

Standards covering the multiplication and division of fractions are located in the Division and Multiplication LT (see Section 5, Bridging Standards and Standards 4.DVM.B, 4.NF.B, 4.NF.4.a, 4.NF.4.b, 4.NF.4.c, 5.NF.4.a, 5.DVM.A, 5.NF.4.b, 5.NF.5.b, 5.NF.5.a, 5.NF.6, 5.NF.7.a, 5.NF.5.b, 5.NF.5.c, 6.NS.1). There are also a few standards in the Ratio and Proportion, and Percents LT that discuss ratios as operators and as fractions that are relevant to this topic.

 

 

 

References

 

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4          Confrey, J., Maloney, A., & Nguyen, K.H. (in press). Understanding over Time: The cognitive underpinnings of learning trajectories. Journal for Research in Mathematics Education.

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