Developing Number Relationships with the Linear Abacus®: Comparison, Strategy, and Equivalence
- Genovieve Grouios
- 20 hours ago
- 11 min read
Have you noticed your child can add and subtract numbers but struggles to understand how numbers relate to each other? Perhaps they have difficulty comparing quantities, making strategic decisions about which numbers to use, or grasping that different expressions can represent the same value. These challenges are common but critical to address, as understanding number relationships forms the foundation for algebraic thinking and higher mathematics.
In our previous blogs, we explored how the Linear Abacus® helps children develop basic counting and operations (Blog 1), place value understanding (Blog 2), and mental strategies for addition and subtraction (Blog 3). Now we're ready to take the next step in mathematical development: helping children understand the relationships between numbers through comparison, strategic decision-making, and equivalence.
The Stage 3 games—"What's the Difference?", "Reach 100 Beads", and "Make It Balance"—build on foundational concepts by engaging children in meaningful comparisons and explorations of number relationships. Unlike traditional worksheets that often treat these concepts procedurally, the Linear Abacus® makes these relationships tangible and visible, positioning numbers as "names of places in an order" that children can physically explore, compare, and manipulate.
The Mathematics Behind the Games
Understanding how numbers relate to each other is essential for developing algebraic thinking and flexible problem-solving strategies. Research by mathematics education experts like Marilyn Burns has shown that children who understand number relationships can think more flexibly about mathematics and develop stronger problem-solving skills than those who only memorise procedures.
Three key relationship concepts are explored in these games:
1. Difference - Understanding that subtraction can be viewed as finding the distance between two numbers
2. Strategic Number Selection - Making purposeful choices about which numbers to use based on their relationships to target values
3. Equivalence - Recognising that different expressions can represent the same quantity
Many children struggle with these concepts because traditional approaches often emphasise computation over relationships. For example, children might be able to subtract 38 from 67 but not understand that the difference represents the distance between these numbers on a number line. Similarly, many children interpret the equal sign (=) as simply "the answer comes next" rather than as a statement of equivalence between expressions.
The Linear Abacus® makes these abstract relationships concrete by allowing children to physically "PINCH" the difference between numbers, strategically select positions to reach a target, and visually confirm that different pathways can lead to the same position on the beads.
What's the Difference Game
Talk: Mathematical Communication
When playing "What's the Difference?", effective questioning helps children articulate their understanding of the comparison relationship:
"How can you use place value to help you name the difference?"
"What method are you using to model the difference on the Abacus String?"
"Are you looking beyond counting by one to find the difference?"
"How can you use number bonds to find the difference more efficiently?"
"How many different pairs can you find on the Linear Abacus® that have a difference of ___?"
A productive conversation might sound like:
Parent: "You've created the numbers 67 and 38. How can we find the difference between these numbers?"
Child: "I can count from 38 up to 67."
Parent: "That's one way! And what would that counting tell us?"
Child: "How far apart the numbers are."
Parent: "Exactly! Can you think of a more efficient way to find that difference using what we know about tens and ones?"
This dialogue encourages the child to think about difference as a distance and to use place value to calculate more efficiently.
Do: Mathematical Gestures
The physical manipulation of beads in "What's the Difference?" makes the concept of difference tangible:
1. After creating two numbers (e.g., 67 and 38), the child places markers at these positions on the Linear Abacus®.
2. Using a "pinching" gesture, the child physically spans the space between the two numbers with their fingers.
3. To quantify this difference, they can count the beads between the two numbers, ideally using groups of ten and remaining ones.
The color coding of the Linear Abacus® (alternating every 10 beads) supports efficient counting in groups. For example, recognising that there are 2 complete sets of 10 beads between the numbers, plus 9 individual beads, helps children see the difference as 29 without counting each bead.
If your child struggles with the pinching concept, start with smaller numbers closer together and gradually increase the gap. You might also try using a ribbon or string to physically connect the two points, making the "span" between them more concrete.
Write: Mathematical Symbolism
Recording the comparison process helps solidify understanding of the difference concept:
1. Encourage your child to write the difference equation: 67 - 38 = 29
2. Then, help them record how they used place value to find the difference:
- Counting on: 38 + 2 = 40, 40 + 20 = 60, 60 + 7 = 67, so 2 + 20 + 7 = 29
- OR using place value: 67 - 38
= (60 + 7) - (30 + 8)
= (60 - 30) + (7 - 8)
= 30 - 1
= 29
A supportive dialogue might include:
Parent: "Let's write down how you found the difference between 67 and 38."
Child: Writes "67 - 38 = 29"
Parent: "Now, can you show how you counted on from 38 to reach 67?"
Child: Writes "38 + 2 = 40, 40 + 20 = 60, 60 + 7 = 67, so the difference is 2 + 20 + 7 = 29"
Parent: "That's a great way to see the difference as the distance between the numbers!"
This progression helps children connect the physical experience of "spanning" between numbers with the mathematical concept of difference. Notice that they also use the build to the next ten facts.
This reasoning demonstrates:
- Understanding difference as the distance between numbers
- Using benchmarks of 10 to count efficiently
- Breaking the counting into strategic parts
- Connecting the physical "pinch" to the mathematical concept of difference
Reach 100 Beads Game
Talk: Mathematical Communication
When playing "Reach 100 Beads", thoughtful questioning helps children develop strategic thinking:
"How do you know this is the bead which represents your number?"
"Can you explain what each digit in your numeral represents?"
"How will you slide beads across without counting by ones?"
"Why did you choose to arrange the digits this way?"
"What's your strategy for getting close to 100 without going over?"
A strategic conversation might sound like:
Parent: "You've rolled a 3 and a 9. You could make 39 or 93. Which would you choose and why?"
Child: "I'm already at 31, so if I add 93, that would be way over 100."
Parent: "How do you know that without adding the exact numbers?"
Child: "Well, 31 plus 90 would already be more than 100, I know this because 10 plus 90 is 100"
Parent: "Great estimation! So what's your choice?"
Child: "I'll choose 39. That will get me to 70, which is closer to 100 but still gives me room for another turn."
This dialogue encourages strategic thinking, estimation, and consideration of number relationships.
Do: Mathematical Gestures
The physical manipulation in "Reach 100 Beads" connects place value to strategic decision-making:
1. After choosing how to arrange their two digits (e.g., deciding on 39 rather than 93), the child slides beads to represent this quantity.
2. Rather than counting each bead individually, they can use place value to move 3 sets of ten beads and then 9 individual beads. Note that depending on where they are sitting, the sets of 10 may not always be a full row of colour. Making ten from different starting points will help them review number bonds.
3. They can use pegs or markers to track their progress toward 100.
The color patterns on the Linear Abacus® help children visualise their progress toward 100, supporting both calculation and strategic planning. Children can see how close they are to the target and make decisions about whether to continue rolling or stop.
If your child struggles with strategic decisions, encourage them to estimate by rounding to the nearest ten. For example, "You're at 70 now. If you roll again, what's the smallest 2-digit number you could make? What's the largest? Would either put you over 100?"
Write: Mathematical Symbolism
Recording the game progress helps children track their thinking and strategies:
1. Create a simple table with columns for the roll, the number created, and the running total.
2. For each turn, record both the digits rolled and the strategic choice made.
3. Note the reasoning behind each decision.
A sample recording might look like:
Starting position: 0
Roll: 3, 1 → Choose 31 → New position: 31
Roll: 3, 9 → Choose 39 → New position: 70
Roll: 1, 2 → Choose 21 → New position: 91
Decision: Stop (to avoid going over 100)
This recording process helps children reflect on their strategy and see the consequences of their choices, reinforcing the relationship between numbers and the importance of strategic thinking.
This student demonstrates:
- Strategic thinking to avoid exceeding 100
- Understanding place value in two-digit numbers
- Planning ahead to anticipate outcomes
- Using efficient calculation strategies (10-folding)
Make It Balance Game
Talk: Mathematical Communication
When playing "Make It Balance", focused questioning helps children develop their understanding of equivalence:
"How do you know both sides are balanced?"
"How can you prove this with the Linear Abacus®?"
"What facts are you using to help you find answers?"
"Can you explain the role of each numeral in your number sentence?"
"Could you place the numerals in a different order and still make the sides balance?"
A concept-building conversation might sound like:
Parent: "You've rolled 5, 1, 9, and 2. How might you arrange these with operations to make both sides equal?"
Child: "I could do 5 × 2 = 9 + 1."
Parent: "That's an interesting idea! How can you check if both sides are equal?"
Child: "5 times 2 is 10, and 9 plus 1 is also 10."
Parent: "Excellent! Can you show me on the Linear Abacus® how both of these calculations land on the same bead?"
This dialogue helps children understand that the equal sign represents equivalence rather than simply "the answer comes next."
Do: Mathematical Gestures
The physical manipulation in "Make It Balance" makes equivalence visible:
1. For one side of the equation (e.g., 5 × 2), the child demonstrates the calculation on the Linear Abacus®, perhaps by showing 2 skips of 5 or creating a 5×2 array with the beads.
2. For the other side (e.g., 9 + 1), they show a different path that ultimately leads to the same position on the abacus string.
3. They physically verify that both calculations end at the same position, confirming the equivalence.
This concrete demonstration helps children understand that equivalence means "same value" rather than "same appearance or give the answer." They can see that different pathways can lead to the same destination on the Linear Abacus®.
If your child struggles with the concept, start with simple equivalences like 5 + 5 = 10 and 10 = 5 + 5 to reinforce that the equal sign works in both directions. Then gradually introduce more complex equivalences.
Write: Mathematical Symbolism
Recording equivalence through multiple representations deepens understanding:
1. Write the balanced equation:
5 × 2 = 9 + 1
10 = 10
2. Draw a number line showing how both sides reach the same position:
3. Create an abacus diagram that shows both calculations leading to the same bead
A supportive extension might include:
Parent: "Now that you've shown this on the abacus and drawn it on a number line, can you think of other ways to reach the number 10 using different operations?"
Child: Writes "20 ÷ 2 = 10" and "15 - 5 = 10"
Parent: "That shows great flexible thinking! All of these different expressions are equivalent because they all equal 10."
This multi-representational approach helps children develop a robust understanding of equivalence that will serve as a foundation for algebraic thinking.
This student shows:
- Understanding of equivalence as different paths to the same value
- Connecting concrete actions to symbolic representation
- Flexibility in thinking about operations
- Verification through the physical model
Extending Learning Beyond the Game
These concepts can be reinforced through everyday activities:
Shopping Comparison: "This item costs $67 and this one costs $38. How much more expensive is the first item?" (difference)
Score Keeping: In card games or board games, discuss strategic decisions about when to take risks versus play conservatively to reach a target score. (strategic number selection)
Cooking Equivalence: "The recipe calls for 1/2 cup of sugar, but we only have a 1/4 cup measure. How many 1/4 cups equal 1/2 cup?" (equivalence)
Temperature Changes: "Yesterday it was 57 degrees and today it's 74. How much warmer is it today?" (difference)
Related Linear Abacus® games that develop these concepts further include "Double, Halve or Keep" and "Add Up Beads" for exploring number relationships from different perspectives.
As your child becomes more confident with these relationship concepts, look for opportunities to extend to decimal numbers or fractions in everyday situations.
Parent Reflection Guide
To assess your child's understanding of number relationships through these games, look for:
Comparison Efficiency: Do they use efficient strategies to find differences, or do they count every step?
Strategic Thinking: Can they anticipate the consequences of their number choices?
Understanding of Equivalence: Do they recognise that the equal sign means "same value" rather than "the answer comes next"?
Flexible Operation Use: Can they use different operations to create equivalent expressions?
Common misconceptions to watch for include:
Seeing the equal sign as an operation ("get the answer") rather than a statement of equivalence
Viewing subtraction only as "take away" rather than also as finding a difference
Making random number choices without considering their relationship to the target
Counting every step rather than using place value to find differences efficiently
Your child is ready for more challenging activities when they consistently demonstrate strategic thinking about number relationships, use efficient comparison strategies, and understand that different expressions can represent the same value.
Research Connection
The approach used in the Linear Abacus® aligns with research on developing algebraic thinking through attention to number relationships. Studies by researchers like Carpenter, Franke, and Levi (2003) have shown that children who develop a relational understanding of the equal sign are better prepared for success in algebra than those who view it procedurally.
The emphasis on multiple representations (physical manipulation, number sentences, and number lines) implements recommendations from researchers like Catherine Fosnot and Maarten Dolk, who advocate for the use of models that reveal mathematical relationships. This multi-representational approach helps children build what education researcher Richard Skemp called "relational understanding"—knowing both what to do and why—rather than merely "instrumental understanding" of following procedures.
Beyond mathematical content, these games develop executive function skills such as cognitive flexibility (considering multiple ways to arrange numbers and operations), inhibitory control (avoiding impulsive choices to optimise strategic decisions), and planning (thinking several steps ahead). Research by Blair and Raver (2014) has shown that these executive function skills are strongly associated with mathematical achievement.
Quick Reference Box
Comparison, Strategy & Equivalence: Key Concepts
Compare two-digit numbers by finding the difference, make strategic decisions to approach a target number, and create balanced equations using multiple operations. Key questions: "How can you find the difference efficiently?", "What strategy will get you closest to the target?", "How can you prove both sides are equal?"
Mathematical Language:
- Difference: The distance between two numbers, found by subtraction
- Strategic thinking: Making choices based on understanding number relationships
- Equivalence: Different expressions that represent the same value
- Equal sign: A symbol showing that expressions on both sides have the same value
- Balance: When both sides of an equation have the same value
References
Blair, C., & Raver, C. C. (2014). Closing the achievement gap through modification of neurocognitive and neuroendocrine function: Results from a cluster randomized controlled trial of an innovative approach to the education of children in kindergarten. PloS One, 9(11), e112393.
Burns, M. (2007). About teaching mathematics: A K-8 resource (3rd ed.). Math Solutions.
Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Heinemann.
Fosnot, C. T., & Dolk, M. (2001). Young mathematicians at work: Constructing number sense, addition, and subtraction. Heinemann.
Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65-97). Macmillan.
Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77(1), 20-26.
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