In a previous post, we explored four straightforward strategies for comparing fractions with different denominators. These methods are effective in various scenarios and relatively easy for students to grasp, encouraging deeper thinking about fraction concepts during comparisons. Here, in part 2, we delve into utilizing these four strategies with number lines. Integrating number lines allows students to visualize these strategies through concrete representations, fostering a more profound understanding of why these comparison techniques work.
These strategies are perfectly aligned with Common Core State Standards (CCSS) for fraction work on number lines in 3rd grade, and for using fraction models in 4th and 5th grades. You can see the specific 3rd-grade standard here: http://www.corestandards.org/Math/Content/3/NF/A/2/
Why Number Lines are Essential for Fraction Understanding
Often, students are primarily introduced to fractions using area models (like pizza slices or shaded rectangles). While area models are helpful, number lines, though sometimes overlooked, provide crucial exposure to fractions as measurements of length. It’s vital for students to internalize and articulate that a fraction’s position on a number line represents the distance from 0, mirroring how whole numbers function. This understanding is foundational for their broader mathematical development.
A Classroom Story: The Ruler Revelation
I had a significant “aha!” moment during a measurement activity I used across many classrooms. In both fifth and sixth grade, I assumed students were comfortable with fractions and inch measurements. However, I was surprised by how many struggled to measure lengths smaller than ¼ inch on a ruler!
They recognized the ¼ inch markings, but the eighth and sixteenth inch marks were a mystery. Crucially, they didn’t know to count the lines to determine if they were eighths or sixteenths. Hoping to guide them, I asked how they knew the ¼ inch marks represented ¼ increments. Most replied they “just knew” or had memorized that the slightly shorter tick marks than the ½ inch marks were ¼ inch. They hadn’t considered that an inch is divided into 4 equal parts, or lengths! This revealed why they didn’t think to count the marks between 0 and 1 to identify eighths or sixteenths.
It became clear that this connection was often missed because measurement is frequently taught separately from fractions in textbooks. This link is essential for students’ proficiency in measuring length, whether in inches or miles. Number lines serve as the bridge to close this gap.
Strategy 1: Equivalent Denominators – Comparing Same Size Pieces on a Number Line
This strategy is generally the easiest for students and often the first they encounter. However, students who lack a solid understanding of denominators might mistakenly add them when working with fractions that have like denominators. To enhance comprehension, it’s beneficial to refer to “like denominators” as “same size pieces.” This phrasing helps students visualize why it’s logical to compare the number of pieces (numerators) when the pieces are identical in size. It also clarifies why adding denominators when adding fractions is incorrect. Students may not intuitively grasp why simply comparing fractions with different denominators isn’t straightforward.
Using a Number Line:
Presenting various problems comparing fractions with like denominators encourages students to recognize that the whole is consistently divided into the same number of equal-sized pieces.
Note: It’s helpful to present problems using both a single number line to plot both fractions and two aligned number lines for each fraction. This can visually reinforce that with like denominators, the whole length is divided into the same number of equivalent length segments.
Example: Which fraction is greater: ¼ or ¾ ?
Alt text: Number line from 0 to 2 showing 1/4 and 3/4 plotted, illustrating fraction comparison on a number line.
Or
Alt text: Two aligned number lines from 0 to 2, the top line plotting 1/4, and the bottom line plotting 3/4, demonstrating comparing fractions on separate number lines.
Guiding Questions for Students:
- Which fraction is farther from 0?
- Which fraction represents a longer length?
- Which fraction represents a larger number?
- How do you know your answers to the above questions are correct?
This last question is crucial! When students reason and explain why ¼ is smaller than ¾, they demonstrate their understanding that the whole is divided into 4 equal parts and ¾ represents three of these equal lengths.
Note: I recommend using number lines that extend from at least 0 to 2. If students only see number lines from 0 to 1, they might be unprepared for plotting mixed numbers. It also reinforces that any fraction with a numerator smaller than its denominator falls between 0 and 1 on any number line. This concept was surprisingly less obvious to my 5th graders than I anticipated. I noticed this when they encountered number lines extending beyond 1 and became less confident about fraction placement.
Strategy 2 : Same Numerator – Comparing Number of Pieces on a Number Line
Often overlooked, this strategy is quite accessible for students and significantly enhances their understanding of denominators.
Example Task: Which fraction is greater? 3/4 or 3/6?
Alt text: Number lines from 0 to 2 illustrating comparing fractions 3/4 and 3/6, highlighting same numerator fraction comparison on a number line.
Guiding Questions for Students:
- Which fraction is farther from 0?
- Which fraction represents a longer length?
- Which fraction represents a larger number?
- How do you know your answers to the above questions are correct?
By dividing the length of one whole into fourths and sixths, students should visually grasp the difference in segment lengths. They should be able to explain that because fourths are longer than sixths, three longer segments amount to a greater total length.
Strategy 3: Compare to Benchmark Fractions – ½ and 1 on a Number Line
Students can often recognize if a fraction is close to, greater than, or less than benchmark fractions like ½ or 1. When comparing two fractions against ½, they might find one is less than ½ and the other is greater. This single piece of information is enough to conclude that the fraction exceeding ½ is the larger one. Older students can extend this to other benchmark numbers as well.
Comparing fractions using benchmark fractions is a 4th-grade CCSS standard: CCSS.MATH.CONTENT.4.NF.A.2
Example Tasks:
Ask students to list as many fractions as they can that are close to ½ but slightly less than ½ and plot them on the number line.
Ask students to list fractions close to ½, but slightly greater than or equal to ½ and plot them on the number line.
Ask students to write inequalities using the fractions plotted.
Alt text: Number line demonstrating benchmark fractions 1/2 and 1, showing fractions clustered around these benchmarks for comparison.
Repeat the same tasks using one whole as the benchmark instead of ½.
Guiding Questions for Students:
- Explain your method for finding fractions slightly less (or more) than ½ (or one whole).
- How do you determine if a fraction is close to ½?
- How do you determine if a fraction is close to one?
These questions elicit diverse and insightful responses. Students often employ various strategies!
Strategy 4: Missing Pieces – Visualizing the Gap to a Whole on a Number Line
Students analyze two fractions by considering how much each is “missing” to reach ½ or one whole.
This strategy can be more challenging, especially with number lines, until students become comfortable with it. Explicit teaching is often necessary.
Example Task:
Ask students to use number lines to show which fraction represents a longer length between 7/8 and 3/4.
Alt text: Number line comparing fractions 3/4 and 7/8 using the missing pieces strategy, visualized on a number line from 0 to 2.
Guiding Questions for Students:
- What similarities do you notice between 3/4 and 7/8? (Both are one segment away from 1.)
- What fraction represents the missing piece needed to make each fraction equal to 1 whole? (They are each missing one unit fraction – 1/4 and 1/8 respectively.)
- Which fraction has a smaller “missing piece,” making it closer to 1 whole?
Providing practice with various fractions that are “missing” one unit fraction can help students recognize and remember this strategy.
How to Practice Comparing Fractions on a Number Line
Number Talks: Incorporate number talks that require students to use a number line to justify their comparisons of two fractions.
Fraction Card Placement:
This can be simple: write fractions on index cards and stretch a string across a whiteboard. Mark 0 and 1 at each end of the string. Students then use paper clips or clothespins to hang their fraction cards on the string, estimating their placements. Students can place two fractions and explain their reasoning for their positions on the number line.
Fraction Sorts and Games: Design fraction sorts or games focused on comparing fractions using number lines. Students can draw number lines on paper and place small fraction cards on their number lines to compare and sort.
Word Problem Resources: These word problem packets are specifically designed for comparing fractions.
Alt text: Resource cover image for comparing fractions word problems, showcasing differentiation levels and problem types.
Three levels of differentiation including open response, multi-step, and multiple-choice problems. Available in print and digital formats.
Alt text: Resource cover image for 3rd-grade fractions on a number line packet, highlighting differentiation and real-world word problems.
This packet offers real-world word problems requiring students to place fractions on a number line. Available in print and digital formats.
Alt text: Resource cover image featuring fraction sort activity and differentiated problem levels for comparing fractions.
Three differentiation levels with open response, multiplication-step, and multiple-choice problems. Includes a fraction sort activity. The fraction sort cards can also be used for comparing fractions. Available in print and digital formats.
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