Reimagining Fractions with the Linear Abacus®: From Parts to Counts
Linear Abacus
The Games Book Series · Stage 7 · Fractions
Skip Counting Fractions
Blog 8 of 12
Reimagining Fractions with the Linear Abacus®:From Parts to Counts
When children count fractions the same way they count whole numbers, the number system stays whole.
Have you noticed your child’s confidence suddenly falter when fractions enter the mathematics curriculum? You’re not alone. While many children develop solid understanding of whole numbers, fractions often mark a troubling shift where mathematics can begin to feel disconnected and abstract. This struggle isn’t just about academic performance — it can shape a child’s mathematical identity and influence their relationship with mathematics for years to come.
The challenge with traditional fraction instruction is that it often presents fractions primarily as “parts of a whole,” using circular pies or pizzas and other area models that emphasise division of objects and shapes. While this approach has value, it creates a conceptual gap between whole numbers and fractions, making fractions seem like an entirely new mathematical system with different rules.
The “Skip Counting Fractions” game from the Linear Abacus® Games Book offers a revolutionary approach by treating fractions as countable quantities. Rather than seeing 1/3 merely as “one part out of three equal parts,” children discover that fractions can be counted just like whole numbers. They experience firsthand how three 1/3s make one whole, how six 1/3s make two wholes, and how this pattern continues predictably. This approach builds a seamless connection between whole numbers and fractions, maintaining the coherence of the number system.
When children use the Linear Abacus® to skip count by unit fractions, they’re not just learning procedures — they’re developing a fundamental understanding of how our number system extends beyond whole numbers while maintaining its essential properties. This embodied experience establishes fractions as “names of places in an order that answers ‘how many?'” — just like whole numbers, only with a different unit being counted.
The Mathematics
The Mathematics Behind the Game
Fractions represent one of the most challenging transitions in elementary mathematics. What Wu and Lamon independently observed about students’ difficulties with fractions — that they stem from conceptual rather than procedural issues, specifically from seeing fractions as disconnected from the whole number system children have already mastered — confirms what the Linear Abacus® makes visible from the first count: fractions are not a new system, they are the same system counted in smaller units.
Traditional approaches typically introduce fractions as parts of a whole, using area models like pies or rectangles. While these models help visualise a single fraction, they have limitations:
- They don’t naturally show how fractions are ordered on a number line
- They make comparing fractions difficult, requiring common denominators
- They don’t extend easily to improper fractions or mixed numbers
- They treat fractions as fundamentally different from whole numbers
The innovative approach embodied in the Linear Abacus® reconceptualises fractions by emphasising their connection to whole numbers through the concept of unit fractions:
- A unit fraction (like 1/3) is the basic unit being counted
- Multiple copies of the unit fraction can be counted (2/3, 3/3, 4/3, etc.)
- Fractions naturally extend the whole number system rather than representing a new system
- Skip counting by unit fractions parallels skip counting by whole numbers
For example, when children understand that 2/3 means “two one-thirds,” they can understand why 2/3 + 1/3 = 3/3 in the same way they understand why 2 + 1 = 3. This powerful conceptual connection helps children see fractions not as mysterious entities with arbitrary rules, but as natural extensions of counting.
The Game
Skip Counting by Fractions
The Framework
During “Skip Counting Fractions,” every position on the string has two names — its fraction name and its whole number name when it reaches a whole. The child who notices that 3/3 and 1 are the same position, without being told, has understood something about fractions that procedural instruction rarely achieves.
Talk: Probing Questions
- “If we’re counting by thirds, how many thirds make a whole?”
- “How would you name this position on the abacus? Can you name it in different ways?”
- “What pattern do you notice as we skip count by one-thirds?”
- “How is skip counting by one-thirds similar to or different from skip counting by ones?”
- “What happens when we continue counting past one whole?”
“We’re going to skip count by thirds on the Linear Abacus®. Before we start, how many one-thirds do you think would make one whole?”
“Three one-thirds make a whole.”
“Great! So if we count 1/3, 2/3, 3/3… what comes next?”
“I’m not sure… maybe we start over?”
“That’s a good question. Let’s keep counting and see what happens. If 3/3 is the same as 1, what would come after that?”
“Maybe… 4/3?”
“Let’s try it on the Linear Abacus® and see if that makes sense!”
This dialogue helps children understand fraction sequences as natural extensions of whole number counting, with 3/3, 6/3, 9/3, etc., corresponding to the whole numbers 1, 2, 3, etc. This is a first introduction to improper fractions — which is approached through counting unit fractions.
Do: Mathematical Gestures
The physical manipulation in “Skip Counting Fractions” makes the ordering of fractions tangible:
This physical representation helps children visualise fractions as positions on a number line and understand how improper fractions naturally extend beyond one whole. The colour patterns on the Linear Abacus® can also help children recognise patterns in the fraction sequence.
A grade 3/4 student counting by fifths. He is connecting this sequence to a bag of marbles. He notices that five fifths are equivalent to one.
What would the counting sequence be if this student made the second bag? Write this out and discuss this with your child.
Write: Mathematical Symbolism
Recording the fraction sequence helps children connect physical positions to mathematical notation:
1. Create a table showing the fraction counting sequence:
Position 2: 2/3
Position 3: 3/3 (or 1)
Position 4: 4/3 (or 1 1/3)
Position 5: 5/3 (or 1 2/3)
Position 6: 6/3 (or 2)
2. Note equivalent ways to express the same position:
6/3 = 2
9/3 = 3
3. Create a visual number line showing the fraction sequence and how it aligns with whole numbers:
For more advanced students, introduce skip counting by different unit fractions (1/2, 1/4, 1/5) and compare the patterns. This helps children see how the denominator determines the “unit” being counted.
The Deeper Idea
The Revolutionary Insight: Fractions as Counts
The “Skip Counting Fractions” game embodies a profound insight about our number system: fractions represent countable quantities, not just parts of a whole. This perspective transforms how children understand fractions in several powerful ways.
Perhaps most importantly, this approach makes fraction arithmetic coherent. Addition of fractions with the same denominator becomes analogous to addition of whole numbers — we are simply counting how many unit fractions we have in total. This alignment reduces cognitive load and helps children see mathematics as a unified system rather than a collection of disconnected procedures.
Beyond the Game
Extending Learning Beyond the Game
These fraction concepts can be reinforced through everyday activities:
- Cooking: “This recipe needs 1/3 cup of sugar. If we’re making 2 batches, how many third-cups do we need?” (2/3 cup) “What if we make 3 batches?” (3/3 or 1 cup)
- Time: Use a clock to show fractions of an hour. “Each 5-minute mark represents 1/12 of an hour. If we count by twelfths, how many minutes is 5/12 of an hour?” (25 minutes)
- Measurement: Use a measuring tape with centimetres. “If 1/4 of a metre is 25 centimetres, how long is 3/4 of a metre?” (75 centimetres)
- Money: Use pennies and quarters. “If 25 pennies make 1 quarter, how many pennies would 3/4 of a dollar be?” (75 pennies)
Connect these activities to other Linear Abacus® games like “First to 3,” which builds on the understanding of fractions as counts by having children add fractions to reach a target.
What to Look For
Parent Reflection Guide
To assess your child’s understanding of fractions through this game, look for:
Common misconceptions to watch for include: thinking that fractions must always be less than 1; not recognising that the same position can be named in different ways; struggling to connect the physical representation to symbolic notation; viewing the fraction sequence as disconnected from whole numbers; and confusion about the relationship between the denominator and the size of the unit fraction.
Looking Forward
Fractions as a Bridge to Algebraic Thinking
While fractions often represent a stumbling block in mathematical development, when approached through the Linear Abacus® counting model, they can actually serve as a powerful bridge to algebraic thinking. This approach develops several key capabilities:
By treating fractions as an extension of counting rather than a separate mathematical domain, the Linear Abacus® approach maintains the coherence of children’s mathematical understanding as they move beyond whole numbers, setting the stage for successful engagement with algebra and higher mathematics.
What Independent Research Confirms
What Lamon and Carpenter independently observed — that children who understand fractions as measurable quantities perform better on fraction tasks than those who only understand fractions as parts of a whole — confirms what Skip Counting Fractions makes visible from the first bead: a fraction is a position you can count to, not a shape you cut up.
What Wu and Ma independently observed about the “measurement interpretation” of fractions — that it better supports children’s understanding of fraction magnitude, comparison, and arithmetic — confirms what the Linear Abacus® makes structurally explicit: 2/3 means two one-thirds, and adding another one-third gives you three one-thirds, which is the same position as 1. The arithmetic follows from the counting.
What neuroscience researchers independently observed about successful mathematics learners — that they develop an integrated understanding of number magnitude that includes fractions — confirms what the Linear Abacus® number line makes possible from the beginning: fractions and whole numbers occupy the same string, in the same order, counted by the same logic.
Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children’s mathematics: Cognitively guided instruction. Heinemann.
Dehaene, S. (2011). The number sense: How the mind creates mathematics (revised and updated edition). Oxford University Press.
Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Reidel.
Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books.
Lamon, S. J. (2012). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers (3rd ed.). Routledge.
Ma, L. (2010). Knowing and teaching elementary mathematics (Anniversary ed.). Routledge.
Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62(4), 273–296.
Wu, H. (2011). Understanding numbers in elementary school mathematics. American Mathematical Society.
Skip Counting Fractions: Fractions as Countable Quantities
Use the Linear Abacus® to explore how fractions can be counted in sequences, just like whole numbers.
Game at a Glance- Skip Counting Fractions — Establish a unit fraction on the string (e.g., each bead = 1/3), then count along the string naming each position: 1/3, 2/3, 3/3, 4/3… Use pegs to mark whole number positions and discover that the counting sequence never stops — it just passes through familiar landmarks.
- Unit fraction — A fraction with numerator 1 (e.g., 1/3, 1/4) — the basic unit being counted.
- Skip counting — Counting by equal intervals (e.g., counting by 1/3: 1/3, 2/3, 3/3…)
- Equivalent expressions — Different ways to name the same position (e.g., 3/3 = 1, 4/3 = 1 1/3)
- Improper fraction — A fraction where the numerator is greater than the denominator (e.g., 4/3, 7/4) — simply a position past one whole.
- Mixed number — A whole number and a fraction combined (e.g., 1 1/3, 2 3/4)
“How many one-thirds make a whole?” · “What comes after 3/3 in the counting sequence?” · “How can we name this position in different ways?” · “How is skip counting by one-thirds similar to skip counting by ones?”
Ready to Play?
Use this blog alongside the Games Book for the full set of probing questions, illustrated student reasoning, and diagrams that bring each game to life.
The Games Book contains thirty games, complete instructions, probing questions, student reasoning examples, and diagrams — everything you need to sit down and play with confidence.
