Teaching Fractions in Tier 1: Addressing Misconceptions Using Multiple Representations

Abstract

Many elementary students, particularly those with or at risk for mathematics learning disabilities (MLDs), demonstrate persistent fraction misconceptions, which can impede conceptual understanding and fraction computation. This article describes four common fraction misconceptions and provides targeted instructional practices for addressing each misconception, emphasizing the use of multiple representations through the length, area, and set models of fractions. The article focuses on addressing these misconceptions at Tier 1 level given the widespread difficulties with fraction understanding and common Tier 1 constraints (e.g., curriculum pacing, limited reteaching opportunities, and overemphasis of symbolic computation). By intentionally integrating multiple representations within fraction instruction, using precise mathematical language, teachers can preemptively address students’ whole-number bias and support students in developing a more robust and flexible understanding of fractions. Although situated within Tier 1 instruction, the strategies can be adapted for use across Tiers 2 and 3 settings to support students with or at risk for MLD.

Mathematics instruction in elementary school follows a developmental continuum, beginning with whole number concepts and progressing toward more complex numerical systems, including rational numbers (National Governors Association Center for Best Practices & Council of Chief State School Officers [NGA & CCSSO], 2010). Within this progression, the Common Core State Standards designate formal introduction to foundational fraction concepts in Grade 3, building on students’ early understanding of equal partitioning and fraction magnitude (NGA & CCSSO, 2010; Shin & Bryant, 2015). Thereafter, fractions are revisited and expanded upon across subsequent grade levels, with a sustained focus on numbers, operations, and rational number reasoning. Because fractions serve as a critical bridge between whole-number understanding and advanced mathematical concepts such as algebra (Booth & Newton, 2012), early fraction instruction must establish a solid foundation that prepares students for increasingly complex mathematics. However, national assessments suggest that many students’ early experiences with fractions often fail to prevent misconceptions or to support deep understanding (National Mathematics Advisory Panel, 2008).

Despite the central role of fractions in the mathematics curriculum, many students struggle to develop proficiency in this domain. According to the 2024 National Assessment of Educational Progress (NAEP; National Center for Education Statistics [NCES], 2024), 61% of fourth-grade students scored below the proficient level in mathematics. On the two items related to rational numbers, 48% of fourth graders were unable to correctly plot a mixed number on the number line and 81% were unable to correctly order decimal numbers. Proficiency in fourth grade entails the ability to add and subtract whole numbers, fractions, and decimals in both single- and multi-step problems (NGA & CCSSO, 2010). It also encompasses the capacity to identify, compare, and solve problems involving fractions in real-world contexts. These findings highlight a significant gap in students’ understanding of fractions and grade level expectations. More alarmingly, students with or at risk for learning disabilities experience greater difficulties with fractions, and their performance demonstrate limited growth over time (Barbieri et al., 2021; Gesuelli & Jordan, 2024). Mathematics learning disabilities (MLDs) refer to persistent difficulties in understanding and using mathematical concepts that occur beyond typical instructional gaps or exposure. As a result, students’ misconceptions, particularly in fractions, are often more entrenched and less responsive to brief or informal remediation than those seen in students who struggle with mathematics (Fuchs et al., 2012; Namkung & Fuchs, 2019).

Addressing Fraction Misconceptions Through Tier 1 Instruction

Core instruction, or Tier 1, takes place in the general education classroom setting, where elementary students with or at risk for MLD typically spend the majority of their school day (e.g., Barrio et al., 2021; Poch et al., 2022). Unfortunately, consistent challenges arise in Tier 1 mathematics instruction, including pacing pressures in the curriculum, limited opportunities to revisit foundational concepts, and instructional materials that overemphasize symbolic computations (Afrillia et al., 2022; Yao et al., 2021). Nationally, schools often prioritize reading over mathematics. For example, nearly twice as many public elementary schools have reading specialists (48%) as mathematics specialists (23%), and a similar gap exists in private schools (NCES, 2022). Schools that do provide scheduled intervention blocks often prioritize supplemental reading instruction over targeted mathematics support (Christopher & Nesbitt, 2023; Zhang & Xin, 2024). This underscores the critical importance of effective core mathematics instruction that addresses the needs of students with or at risk for MLD. Because strong Tier 1 instruction can support initial fraction learning, intensifying instruction at the Tier 1 level serves as a cost- and time-efficient way to remediate existing difficulties and reduce later risk for MLD (Barrett & VanDerHeyden, 2020; Björkhammer et al., 2024; VanDerHeyden & Codding, 2015).

Given the known challenges associated with fraction learning and the limitations of relying solely on small-group interventions, it is imperative to invest in whole-class instructional practices. By doing so, educators can provide comprehensive support to a larger number of students (Björkhammer et al., 2024). In the current paper, we provide practical recommendations for general education teachers delivering Tier 1 mathematic instruction to support students, especially those with or at risk for MLD, in understanding fraction concepts. One of the most effective ways to do so is through the purposeful use of multiple mathematical representations.

Using Multiple Representations to Teach Fraction Understanding

Mathematical representations are a primary mechanism for building conceptual understanding of fractions and are especially advantageous for students with or at risk for MLD (Bouck et al., 2023; Bouck & Sprick, 2018; Ennis & Losinski, 2019; Flores et al., 2024). The concrete-representational-abstract (CRA), or integrated concrete-representational-abstract (CRA-I), framework is a research-validated instructional strategy to build deep understanding of mathematical concepts, with demonstrated effectiveness across mathematical content areas (e.g., Bundock et al., 2021; Flores et al., 2024; Flores & Hinton, 2022; Morano et al., 2020). Related frameworks such as virtual-representational-abstract (VRA) further highlight flexible adaptation and utility across representations (i.e., virtual manipulatives; Bouck & Sprick, 2018). The CRA or CRA-I instructional framework can be broken into three phases: concrete, representational, and abstract. In traditional CRA, the phases are introduced sequentially, starting with concrete models and progressing to abstract (e.g., Bouck & Sprick, 2018). In CRA-I, the phases are integrated so that students can model and practice with multiple representations simultaneously (e.g., Flores et al., 2024, 2025). In Tier 1 instruction, CRA-I can be carried out through teacher modeling that links multiple representations during whole-class lessons. For example, teachers can model with virtual manipulatives (e.g., interactive number line) or concrete manipulatives (e.g., fraction tiles via a document camera) while students work with their own sets of manipulatives in pairs or groups. Another example might be when introducing one-fourth as a unit fraction, the teacher might use fraction tiles to identify and model a one-fourth tile above a one-whole tile while drawing and labeling the corresponding position of the fraction on a 0–1 number line. The teacher, then, can ask students to explain how the tiles and the location of one-fourth on the number line represent the same quantity. Researchers indicate that both the CRA and CRA-I reap similar benefits when used to build understanding of fractions (Morano et al., 2020).

The concrete step is most often used when introducing a new mathematical concept or skill and involves modeling the concept using three-dimensional mathematical manipulatives (Miller & Mercer, 1993; Witzel et al., 2003). To build understanding of fractions as numbers, common manipulatives include fraction tiles, counters, Cuisenaire rods, or pattern blocks. Figure 1 shows a variety of examples of how a teacher can illustrate the concept of $\frac{1}{4}$ using CRA or CRA-I, as well as their role in building a part–whole (i.e., area, set) or measurement (i.e., length) understanding of fractions. Representational, or pictorial, models are two-dimensional representations that are often printed or drawn on paper. Examples include a number line (e.g., 0–1 number line with $\frac{1}{4}$ hash marks) or pictorial representations (e.g., a box split into four equal squares with 1 square shaded to represent $\frac{1}{4}$ , a drawing of a pie cut into four pieces with one piece shaded, etc.). The final step, abstract, is the ultimate goal of instruction where students understand the meaning of the numbers and mathematical symbols and use them to solve problems (e.g., seeing $\frac{1}{4}$ and understanding what it represents).

Figure 1.

Using area, length, and set models within CRA/CRA-I to represent identification of “one-fourth.”

Although CRA and CRA-I are powerful frameworks for building understanding of mathematics concepts, the effectiveness of these models depends on how instruction is delivered. Consistently, researchers demonstrate that systematic and explicit instruction is a critical practice of successful mathematics instruction for students with or at risk for MLD, ensuring that students understand not only what the representations show but how and why the mathematics behind them work (Archer & Hughes, 2011; Ennis & Losinski, 2019; Gersten et al., 2009). Within this approach, teachers should explicitly introduce the material name and related vocabulary, then model how to use the material to represent or solve mathematical problems. For example, a teacher may provide modeling and think-aloud using precise mathematical language, then lead guided practice with students through stations (i.e., small groups), followed by independent practice opportunities. In this way, systematic and explicit instruction serves as the mechanism through which CRA and CRA-I support student learning.

One final consideration is to incorporate appropriate mathematical models based on the specific instructional goal or misconception being addressed (Brave et al., 2024). Teachers often have a variety of manipulatives available in their classroom, but some manipulatives may be better suited to illustrate specific mathematical concepts. For example, length models (e.g., number lines, fraction tiles) are particularly effective when introducing fractions as numbers with magnitude or when comparing fraction size, whereas area (e.g., fraction circles, shaded shapes) and set (e.g., counters) models are effective for developing part–whole understanding and equal partitioning and sharing. Although area models are frequently used in classrooms, overreliance on them without integrating length or set models can inadvertently reinforce whole-number bias, where students misapply whole-number properties to fractions (Ni & Zhou, 2005). Because these models emphasize different aspects of fractions, emphasis is well placed on the development of a length and a part–whole understanding as distinct (Morano et al., 2019; Siegler et al., 2011). The following section illustrates how these models can be used to address common misconceptions.

Addressing Common Misconceptions About Fractions Using Mathematical Models

Students with or at risk for MLD often exhibit persistent misunderstandings about fractions (Bottge et al., 2014; Schumacher & Malone, 2017). In this section, we describe five common misconceptions students may have about fractions and offer strategies for addressing them using length, area, and set models.

Misconception #1: Students Treat the Numerator and Denominator as Independent Whole Numbers

The term fraction refers specifically to positive rational numbers represented symbolically as $\frac{a}{b}$ , where a whole is partitioned into b equal parts; both a and b are positive natural numerals (NGA & CCSSO, 2010). When first encountering fractions, such as $\frac{1}{4}$ , $\frac{1}{8}$ , or $\frac{1}{20}$ , some students may interpret the numerator and denominator as two unrelated whole numbers rather than as components of a single numerical value with magnitude (Ni & Zhou, 2005). This misconception can lead to procedural errors, especially in fraction operations (Schumacher & Malone, 2017). Therefore, it is critical that teachers establish early on that a fraction represents one single number and that the numerator and denominator work together to define its value.

To support this understanding, teachers can implement the following instructional sequence:

Step 1. Establish Equal Partitioning Using Unit Fractions

Begin by introducing unit fractions (i.e., fractions with a numerator of 1) to ground students’ understanding of equal partitioning. Using area or set models, the teacher partitions a whole into equal parts (e.g., halves, thirds, fourths) and explicitly labels each part using precise mathematical language (e.g., “one-fourth”; Fuchs et al., 2021). Emphasize that the denominator indicates the number of total equal parts in the whole. Students engage by identifying, sharing, or constructing unit fractions, ensuring that each part is equal in size. This step establishes the foundational meaning of the denominator as defining equal-sized units.

Step 2. Build Proper Fractions Using on Unit Fractions

Once students demonstrate understanding of unit fractions, guide students in composing proper fractions by combining unit fractions (e.g., $\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=\frac{3}{4})$ . Using the same model, explicitly connect the numerator to the number of unit fractions being counted (e.g., “three one-fourths”). Students can then represent and describe fractions by counting equal-sized parts using pictorial or concrete representation. In this step, students see that the numerator and denominator are not independent; rather, the numerator counts how many of the equal parts defined by the denominator are being considered.

Step 3. Represent the Same Fraction Across Multiple Models (CRA-I)

After students can compose fractions using area or set models, the teacher can introduce the length model of fractions (e.g., fraction tiles or number lines) to reinforce that fractions are quantities with magnitude (Lesner et al., 2023; Rojo et al., 2023). First, model how a fraction such as $\frac{3}{4}$ is located on a number line as a point between 0 and 1, emphasizing that it represents a value in length. Students then practice locating common, or benchmark fractions such as $\frac{1}{4}$ and $\frac{1}{2}$ on number lines and compare their locations. Then, prompt students to connect these representations to prior models (i.e., area or set) by asking questions such as “How does the length of $\frac{3}{4}$ match the three shaded fourths?” This step helps shift students’ understanding from discrete parts to continuous magnitude, building their conceptualization of fractions as numbers.

Step 4. Make the Meaning of Numerator and Denominator Explicit

Prompt students to articulate the meaning of the numerator and denominator across representations. For example, by seeing $\frac{3}{4}$ , the teacher may ask, “What does the 4 tell us?” (possible response: the whole is divided into four equal parts) and “What does the 3 tell us?” (possible response: we counted three of those four equal parts). Students are encouraged to explain their reasoning using models and precise vocabulary. Again, reinforce that the numerator and denominator function together to describe a single quantity, rather than as two separate whole numbers.

Instructional Supports

Throughout instruction, model their reasoning think-alouds and consistently connect concrete, pictorial, and abstract representations (CRA-I). For example, when introducing the fraction, $\frac{1}{2}$ , a teacher might simultaneously display a fraction circle, a number line, and the abstract representation while asking students to explain how each representation shows the same quantity. Maintaining consistent mathematical language (e.g., “one-fourth,” “denominator”) across representations supports students in connecting meanings.

Figure 2 illustrates a sample teacher–student dialogue that integrates explicit instruction, multiple representations, and precise mathematical language to build students’ understanding of $\frac{1}{4}$ . To further support teachers in incorporating representations while maintaining consistent mathematical language across CRA-I phases, Figure 3 provides sample teacher prompts aligned with each phase of the CRA-I instruction. These prompts are intended to help teachers reinforce the connections across representations and maintain a focus on fraction meaning.

Figure 2.

An example of teacher–student dialogue on the meaning of one-fourth.

Figure 3.

Examples of teacher prompts within CRA-I.

Misconception #2: Students Believe That the Greater the Denominator, the Greater the Value of the Fraction

Many students hold this misconception because they overgeneralize from their experience with whole numbers (Ni & Zhou, 2005). For example, students may incorrectly conclude that $\frac{1}{8}$ is greater than $\frac{1}{4}$ because 8 is greater than 4. This misconception reflects a limited understanding of fraction magnitude and the inverse relationship between the number of equal parts and the size of each part. Whereas Misconception #1 focuses on recognizing the numerator and denominator as components of a single quantity, this misconception reflects students’ difficulty reasoning about how changes in the denominator affect the fractional value. Building on students’ understanding that fractions represent a single value with magnitude, instruction should extend this concept by emphasizing how the denominator influences the size of that value when the whole remains constant.

To support this understanding, teachers can implement the following instructional sequence:

Step 1. Revisit Equal Partitioning With a Fixed Whole

Briefly revisit equal partitioning to prepare students for comparison. Rather than reintroducing unit fractions, teachers can focus on maintaining a consistent whole using the length, area, or set model of fractions while partitioning it into different numbers of equal parts (e.g., halves, fourths, eighths). Students can decompose these partitions and verify that each set of parts composes the same whole (i.e., three $\frac{1}{3}$ fraction circle pieces cover a one-whole fraction circle; five $\frac{1}{5}$ fraction tile pieces cover a one-whole fraction tile). This helps ground students in understanding a fixed reference (i.e., whole) and how the number of parts affects their size.

Step 2. Compare Unit Fractions With Different Denominators

Present unit fractions with different denominators basing them on the same whole (e.g., $\frac{1}{2}$ , $\frac{1}{4}$ , $\frac{1}{8}$ ), and prompt students to compare their sizes. Using length or area models such as fraction titles or fraction circles, students observe that as the number of equal parts increases, the size of each part decreases. The teacher can ask, “Which piece is larger? How do you know?” and guide students to justify their reasoning based on the structure of the partitioning. At this point, it is beneficial for the teacher to state and reinforce the inverse relationship between the denominator and its size.

Step 3. Apply Understanding to Real-World and Set Contexts

Real-world sharing contexts (e.g., sharing a box of cookies, which represents the whole, among a different number of people) can extend students’ understanding of the relationship between the total number of equal parts (i.e., denominator) and the size of each equal part. For example, the teacher may ask, “If two people share a box of 12 cookies equally, how many cookies does each person get? If six people share the same box of 12 cookies equally, how many cookies does each person get?” Set and contextual models help highlight this relation in meaningful situations and support transfer beyond pictorial representations.

Step 4. Generalize the Concept Using Other Representations

Pattern blocks are another powerful area model for helping students visualize and compare fraction sizes. For example, if a hexagon represents one whole, students can cover it with two trapezoids (i.e., $\frac{1}{2}$ ), three rhombuses (i.e., $\frac{1}{3}$ ), or six triangles (i.e., $\frac{1}{6}$ ). By manipulating and comparing the blocks, students can see that as the number of equal parts increases, each piece becomes smaller. The teacher can then prompt students to explain why $\frac{1}{6}$ is less than $\frac{1}{3}$ , despite 6 being greater than 3.

Instructional Supports

Throughout instruction, teachers should model using think-alouds and precise mathematical language, while demonstrating the concept via CRA-I. Importantly, maintaining a fixed whole across comparisons is essential to ensure that students focus on how the denominator affects the size of each part. Figure 4 provides an example mini-lesson structure illustrating how these steps can be implemented in practice. This format supports teachers in planning and implementing brief, targeted lessons within Tier 1 instruction that integrate multiple representations and explicit instruction. Additional sample activities addressing Misconceptions 1, 3, and 4 are provided in Supplementary A.

Figure 4.

Sample activities to address Misconception #2.

Misconception #3: Students Struggle to Understand and Find Equivalent Fractions

Equivalent fractions are conceptually challenging because they conflict with students’ understanding that different numbers represent different amounts (Ni & Zhou, 2005). For example, a student may view $\frac{1}{2}$ and $\frac{2}{4}$ as numbers representing different values because the numerators and denominators differ. This misconception reflects difficulty coordinating fraction magnitude with multiplicative reasoning, particularly the understanding that a fraction can be renamed by multiplying the numerator and denominator by the same factor.

To address this misconception, teachers can implement the following instructional sequence:

Step 1. Establish Equivalence Through Area Models Using the Same Whole

Using a consistent whole (e.g., a fraction circle, a fraction tile, a 0–1 number line), teachers can guide students to represent fractions with different denominators (e.g., $\frac{1}{2}$ and $\frac{2}{4}$ ) using area models such as rectangles or circles. Students then shade one of two equal parts and two of four equal parts from the same whole to compare shaded regions (see Figure 5; Siegler et al., 2013). The teacher prompts students to describe what they notice about the amount of space covered. This allows students to observe that different partitions of the same whole can result in the same quantity.

Figure 5.

Using length, area, and set models to find equivalent fractions.

Step 2. Use Non-Examples to Clarify What Is Not Equivalent

Teachers can also use non-examples to show students that the shaded area of one out of two equal parts and one out of three equal parts using the same whole are not the same, or equivalent. Students compare and identify differences in the shaded areas and explain why the fractions are not equal. This step helps sharpen students’ understanding by contrasting equivalent and non-equivalent fractions, prompting them to attend to both the number of parts (i.e., the denominator) and their size.

Step 3. Build Equivalence Through Partitioning and Iteration (Paper Folding)

Another fun activity is paper folding. For example, students fold a rectangular or circular paper into halves, shade one half, then fold the paper again to create fourths; they observe that each half is now made up of two-fourths, creating a visual and tactile representation that $\frac{1}{2}$ equals $\frac{2}{4}$ . Teachers can extend this activity by repeatedly folding the paper and recording equivalent fractions (e.g., $\frac{8}{16}$ , $\frac{16}{32}$ , $\frac{32}{64}$ ). This activity helps students see that equivalent fractions result from partitioning the same whole into smaller equal parts, keeping the shaded half constant.

Step 4. Represent Equivalence Using Length Models

Length models, such as fraction tiles or number lines, can further support the conceptual development of equivalency (Rodrigues et al., 2023). For example, students place $\frac{1}{3}$ and $\frac{2}{6}$ on the same number line or align fraction tiles to show that both reach the same point or length (see Figure 5). The teacher then prompts students to explain how different fractions can be placed at the same location on the number line. Whereas area models highlight equal partitioning, length models emphasize that equivalent fractions correspond to the same magnitude on a continuous number line.

Step 5. Connect Representations to Mathematical Strategies

At this step, teachers explicitly connect representations such as fraction tiles to the multiplicative nature of equivalent fractions by highlighting how both the numerator and denominator are multiplied by the same factor (e.g., $\frac{1}{2}$ × $\frac{2}{2}$ = $\frac{2}{4}$ ). As students identify that as two $\frac{1}{4}$ tiles equate one $\frac{1}{2}$ tile, the teacher can show that multiplying the factor (i.e., 2) to both the numerator and denominator preserves the same value because of the identity property (because $\frac{2}{2}$ = 1). The explanation bridges conceptual understanding with procedural fluency (i.e., finding the least common multiple) beyond the use of specific models.

Instructional Supports

As the concept becomes more complex, teachers should ensure they consistently pair concrete or pictorial representations with the abstract to help students see the mathematics. Figure 5 illustrates how the length, area, and set models can be used to generate and compare equivalent fractions. Providing students with multiple representations and ample opportunities for think-alouds and practice can solidify a robust understanding of fraction equivalence.

Misconception #4: Students Combine the Numerals Across When Adding or Subtracting Fractions

Difficulties with fraction addition and subtraction are well-documented, particularly for students with or at risk for MLD (Siegler & Lortie-Forgues, 2017). They often add or subtract fractions by combining the numerators and denominators across (e.g., $\frac{3}{2}+\frac{1}{5}=\frac{4}{7}$ ; $\frac{3}{2}-\frac{1}{5}=\frac{2}{3}$ ), revealing their misapplication of whole number properties and limited understanding of fractional properties (Schumacher & Malone, 2017; Tian & Siegler, 2017). To address this fundamental misconception about fractions, students need a solid understanding of the meaning of denominators and numerators, and that fraction addition or subtraction involves combining or finding the difference between equal-sized parts.

To support this understanding, teachers can implement the following instructional sequence:

Step 1. Reinforce That Fraction Addition Involves Combining Equal-Sized Parts

Begin by revisiting fraction addition with like denominators using area models like fraction circles, length models like number lines, or set models two-color counters. For example, students combine $\frac{1}{4}+\frac{2}{4}$ by physically joining one $\frac{1}{4}$ piece and two $\frac{1}{4}$ pieces to arrive at the sum of three $\frac{1}{4}$ pieces, or $\frac{3}{4}$ . The teacher explicitly states that the size of each part (i.e., denominator) remains the same, while the number of parts (i.e., numerator) changes. Students should be describing the process using precise mathematical language (e.g., “three one-fourths, or three-fourths”). This step establishes that fraction addition is grounded in combining equal units of the same whole, not numerals independently.

Step 2. Use Length Models to Represent Fraction Addition

Next, number lines can show fraction addition as movement along the continuum, helping students see that they are counting equal-sized jumps. For example, students solve $\frac{1}{4}+\frac{2}{4}$ by making successive jumps of one-fourth along the number line. Teachers may incorporate the gesture of chunking (i.e., consistent spacing between the thumb and index finger) to reinforce that each interval represents the same-sized unit (see Rodrigues et al., 2023). Students observe that the denominator defines the unit size through chunking, which remains unchanged, as the numerator counts how many units are combined.

Step 3. Contrast Correct and Incorrect Strategies Using Models

Next, explicitly address the misconception by comparing worked examples and non-examples (i.e., solution steps by combining numerals across). For example, using a length model (i.e., fraction tiles or double number lines), the teacher shows that $\frac{1}{4}+\frac{1}{6}\ne \frac{2}{10}$ by modeling $\frac{2}{10}$ above the combined $\frac{1}{4}$ and $\frac{1}{6}$ sum, pointing out that $\frac{2}{10}$ does not align with the actual combined length of $\frac{1}{4}$ and $\frac{1}{6}$ (see Figure 6). From this, students can have the opportunity to analyze why the method of adding the numerals across fails and recognize that combining denominators across changes the size of the unit, resulting in an inaccurate representation of the sum (Lortie-Forgues et al., 2015).

Figure 6.

Using fraction tiles to add fractions with unlike denominators.

Step 4. Introduce Common Denominators Through Equivalent Fractions

Building on students’ understanding of equivalence (see Misconception 3), teachers can guide students to generate equivalent fractions with a common denominator before combining. Using the area or length model, teachers model fraction pieces such as $\frac{1}{4}$ and $\frac{1}{6}$ , then rename the addends as equivalent fractions with a common denominator by finding pieces that cover each piece perfectly so that both numbers are now expressed in equal-sized parts (i.e., $\frac{3}{12}$ and $\frac{2}{12}$ ). Teachers can then prompt students to explain how the value of each addend remains the same even though the representation changes.

Once students see that both addends have like denominators, they can meaningfully combine the addends to yield the correct sum (i.e., $\frac{5}{12}$ ). In this way, the instruction explicitly connects the generation of equivalent fractions to addition and subtraction of fractions with unlike denominators.

Instructional Supports

Throughout, emphasize the role of the denominator as defining unit size and the necessity of maintaining equal-sized units during operations. Teachers may continue to use multiple representations to model fraction addition and subtraction with unlike denominators, connecting the concept of equivalence to operations.

Supporting Students’ Precision With Fraction Notation

In addition to common misconceptions, students with or at risk for MLD may demonstrate errors in reading, writing, or labeling fractions (e.g., switching the numerals in the numerator and the denominator, reading $\frac{3}{4}$ as “four-thirds,” or writing $\frac{6}{5}$ instead of $\frac{5}{6}$ ). These errors often reflect an incomplete understanding of fraction structure and the distinct roles of the numerator and denominator (see Misconception 1). Supporting students’ precision in fraction language and notation is an important complement to core instruction, as it ensures that students can accurately interpret and articulate fraction quantities.

Teachers can support students’ precision with fractions through the following practices:

• Use consistent and precise mathematical language: Provide vocabulary instruction and display vocabulary terms on a word wall to help students internalize concepts. Provide opportunities for students to practice saying and apply fraction vocabulary terms to support their reasoning, especially during activities involving concrete or pictorial representations.

• Frequently check for understanding: Prompt students to monitor their own understanding by asking questions such as, “Does your model match the problem?” or “What does the denominator tell you in this scenario?”

• Provide brief fluency-building activities every day: Practice matching vocabulary terms (e.g., “unit fraction,” “benchmark fraction,” “proper fraction,” and “improper fraction”) with spoken, written, and pictorial representations. Practice estimating the location of a proper fraction on a 0–1 number line; extend the practice with improper fractions or mixed numbers on a 0–2 number line. Practice generating multiple iterations of a given fraction to find equivalent fractions.

Conclusion

A strong foundation in fraction understanding is essential for students’ long-term success in mathematics. However, persistent misconceptions can impede learning, particularly for students with or at risk for MLD. This article describes five common fraction misconceptions and illustrates how purposeful use of multiple representations can preemptively and proactively address these misconceptions for students with or at risk for MLD within Tier 1 settings.

Rather than relying on a single model or moving too quickly to symbolic procedures, instruction should strategically incorporate length, area, and set models of fractions to highlight both part–whole relations and fraction magnitude, while maintaining consistent use of precise mathematical language. Instruction should also integrate multiple opportunities for student response (e.g., think-alouds) and fluency-building to solidify students’ fraction learning.

Teachers may encounter challenges related to instructional time, planning, behavior management, and coordinating and monitoring the use of multiple representations, particularly when supporting students who may require additional scaffolding. However, these challenges may be mitigated through intentional scope and sequence, clear modeling and explanation, and opportunities for students practice. Although this manuscript emphasizes Tier 1 instruction, these recommended practices can be adapted and intensified for use in Tiers 2 and 3 settings. When implemented consistently, these practices can reduce students’ reliance on rote strategies, strengthen conceptual understanding, and better prepare students for fraction operations and more advanced mathematical reasoning.

Supplemental Material

sj-docx-1-isc-10.1177_10534512261446611 – Supplemental material for Teaching Fractions in Tier 1: Addressing Misconceptions Using Multiple Representations

Supplemental material, sj-docx-1-isc-10.1177_10534512261446611 for Teaching Fractions in Tier 1: Addressing Misconceptions Using Multiple Representations by Jessica Mao, Marah Sutherland and David Fainstein in Intervention in School and Clinic