Developing Number Relationships with the Linear Abacus®: Comparison, Strategy, and Equivalence
Linear Abacus
The Games Book Series · Stage 3 · Number Relationships
What’s the Difference? · Reach 100 Beads · Make It Balance
Blog 4 of 12
Developing Number Relationships with the Linear Abacus®:Comparison, Strategy, and Equivalence
When children can see how numbers relate to each other, everything else in mathematics becomes possible.
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
The Mathematics Behind the Games
Understanding how numbers relate to each other is essential for developing algebraic thinking and flexible problem-solving strategies. What Marilyn Burns independently observed in her research on children who develop number relationships — that they think more flexibly and solve problems more capably than those who only memorise procedures — confirms what the Linear Abacus® makes visible from the very beginning of play.
Three key relationship concepts are explored in these games:
- Difference — Understanding that subtraction can be viewed as finding the distance between two numbers
- Strategic Number Selection — Making purposeful choices about which numbers to use based on their relationships to target values
- 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. 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 span 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.
Game One
What’s the Difference?
The Framework
Talk: Mathematical Communication
When playing “What’s the Difference?”, effective questioning helps children articulate their understanding of the comparison relationship:
Talk: Probing Questions
- “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 ___?”
“You’ve created the numbers 67 and 38. How can we find the difference between these numbers?”
“I can count from 38 up to 67.”
“That’s one way! And what would that counting tell us?”
“How far apart the numbers are.”
“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:
Level 3/4 student using place value to name the difference
Write: Mathematical Symbolism
Recording the comparison process helps solidify understanding of the difference concept:
“Let’s write down how you found the difference between 67 and 38.”
Writes “67 – 38 = 29”
“Now, can you show how you counted on from 38 to reach 67?”
Writes “38 + 2 = 40, 40 + 20 = 60, 60 + 7 = 67, so the difference is 2 + 20 + 7 = 29”
“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 span to the mathematical concept of difference.
Game Two
Reach 100 Beads
The Framework
When playing “Reach 100 Beads”, thoughtful questioning helps children develop strategic thinking about number relationships — because every choice involves reasoning about where a number will land on the string before a single bead is moved.
Talk: Probing Questions
- “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?”
“You’ve rolled a 3 and a 9. You could make 39 or 93. Which would you choose and why?”
“I’m already at 31, so if I add 93, that would be way over 100.”
“How do you know that without adding the exact numbers?”
“Well, 31 plus 90 would already be more than 100. I know this because 10 plus 90 is 100.”
“Great estimation! So what’s your choice?”
“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:
Write: Mathematical Symbolism
Recording the game progress helps children track their thinking and strategies:
| Roll | Number Created | Running Total | Reasoning |
|---|---|---|---|
| Start | — | 0 | — |
| 3, 1 | 31 | 31 | Either order works; keep going |
| 3, 9 | 39 | 70 | 93 would exceed 100 |
| 1, 2 | 21 | 91 | Getting close — one more careful roll |
| 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).
Game Three
Make It Balance
The Framework
When playing “Make It Balance”, focused questioning helps children develop their understanding of equivalence — that the equal sign is not an instruction to calculate, but a statement about two expressions landing on the same position on the beads.
Talk: Probing Questions
- “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?”
“You’ve rolled 5, 1, 9, and 2. How might you arrange these with operations to make both sides equal?”
“I could do 5 × 2 = 9 + 1.”
“That’s an interesting idea! How can you check if both sides are equal?”
“5 times 2 is 10, and 9 plus 1 is also 10.”
“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:
Level 1/2 student showing how two additions land on the same number (6 + 5 = 7 + 4)
Write: Mathematical Symbolism
Recording equivalence through multiple representations deepens understanding:
“Now that you’ve shown this on the abacus, can you think of other ways to reach the number 10 using different operations?”
Writes “20 ÷ 2 = 10” and “15 – 5 = 10”
“That shows great flexible thinking! All of these different expressions are equivalent because they all equal 10.”
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.
Beyond the Game
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.
What to Look For
Parent Reflection Guide
To assess your child’s understanding of number relationships through these games, look for:
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.
What Independent Research Confirms
What Carpenter, Franke, and Levi independently observed in their research on the equal sign — that children who develop a relational understanding of it are better prepared for success in algebra — confirms what Make It Balance makes visible from the first game: that the equal sign is a statement about two expressions landing on the same position, not an instruction to produce an answer.
What Fosnot and Dolk independently observed about models that reveal mathematical relationships — that multiple representations deepen children’s understanding — confirms what TALK DO WRITE makes possible from the beginning: the same structure expressed through language, gesture, and symbol simultaneously.
What Blair and Raver independently observed about executive function skills — that cognitive flexibility, inhibitory control, and planning are strongly associated with mathematical achievement — confirms what these three games build naturally: the child who pauses before sliding beads to ask “where will this land?” is exercising precisely the skills that matter.
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.
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.
Games at a Glance- What’s the Difference? — Span the distance between two numbers on the string and name it using towers and wall.
- Reach 100 Beads — Choose how to arrange your rolled digits to advance strategically toward 100 without going over.
- Make It Balance — Arrange four rolled digits with operations so that both sides of the equal sign land on the same bead.
- Difference — The distance between two numbers, found by subtraction and seen as a span on the string.
- Strategic thinking — Making choices based on understanding where a number will land before moving a bead.
- Equivalence — Different expressions that land on the same position on the Linear Abacus®.
- Equal sign — A symbol showing that expressions on both sides name the same position.
- Balance — When both sides of an equation have the same value.
“How can you find the difference efficiently?” · “What strategy will get you closest to the target?” · “How can you prove both sides are equal?”
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.
