Understanding Open Number Lines in Mathematics
An open number line is a flexible visual tool in mathematics that helps students solve addition, subtraction, multiplication, and division problems. Unlike a traditional number line, which comes pre-marked, an open number line starts blank, allowing students to add numbers and labels as they work through a problem.
To use an open number line for addition or subtraction, students mark the starting number and use “jumps” to represent adding or subtracting values until they reach the final answer. For example, to solve 37 + 25, students might start at 37, jump 20 to reach 57, and then jump 5 more to land on 62.
For subtraction, they might jump backward in steps, such as starting at 42 and jumping back 10 to reach 32, then back 5 to reach 27.
When applied to multiplication or division, open number lines can represent repeated addition or subtraction. For example, solving 4 × 6 might involve making four jumps of 6 on the number line, while solving 24 ÷ 6 could involve starting at 24 and making equal backward jumps of 6 to determine how many jumps it takes to reach 0.
Open number lines are especially useful for fostering understanding of key concepts like place value, the distributive property, and number flexibility. Allowing students to decompose and compose numbers helps build a strong foundation for conceptual understanding, preparing students to transition to more abstract methods and algorithms.
Why Are Open Number Lines Important?
In the early grades, open number lines are used to introduce basic concepts like counting on, counting back, and simple addition or subtraction. For instance, when learning to subtract, students can use an open number line to find the difference by jumping backward from a starting number. This visual approach makes subtraction more intuitive and helps students understand it as “finding the distance between numbers.”
As students progress, open number lines become a powerful tool for representing more complex ideas, such as adding or subtracting larger numbers, understanding fractions, or working with decimals. When students label their own increments on an open number line, it helps them take ownership of their strategies and strengthen their problem-solving skills.
Using Open Number Lines for Computation
Using an Open Number Line to Add
Open number lines are an incredibly useful tool, especially as students begin working with multi-digit numbers. When using an open number line for addition, a student starts at one of the addends and counts forward by the other addend, breaking it into manageable chunks that make sense to them. The beauty of open number lines lies in their flexibility, allowing students to approach problems in ways that align with their individual thinking.
For example, let’s explore two different ways a student might use an open number line to solve 125 + 347.
The student below began at 125 and added 347 by first adding the hundreds, then the tens, and finally counting up by the 7 ones:
The next student approached the problem differently. They also started at 125 and counted forward by 347, but first added a small part of the addend to reach the next friendly number, in this case, 130. This strategy can help students avoid errors by working with numbers that are easier to manage. After adding 300 and then 40, they were left with just 2 more to add, simplifying the process:
These are just two examples of how this problem can be solved. A student might choose to break apart 347 in a different way, or they might decide to start at 347 and count forward by 125 instead. Regardless of the approach, as long as the calculations are accurate, all strategies will arrive at the same sum.
Using an Open Number Line to Subtract
When using an open number line for subtraction, students can explore multiple meanings of subtraction. Initially, they might use the number line to jump backward, reinforcing the idea of subtraction as “taking away.” However, as numbers become more complex, counting backward can be challenging for many children. This is where understanding subtraction as finding the difference, or the distance between two numbers, can be particularly useful.
Let’s explore this concept by examining how two students might use an open number line to solve 462−284.
The student below started at 462 and subtracted 284 by counting backward. While they broke 284 into smaller chunks to subtract, such as taking away 80 from 262 and then 4 from 182, these steps can be challenging for some students, particularly when working with more complex numbers or regrouping:
Another way to approach the problem is by placing 284 and 462 on the open number line. Instead of subtracting, students can focus on finding the distance between the two numbers. This distance represents the difference and provides the solution to the problem:
Some students might find the following approach easier because it allows them to count forward on the number line rather than backward:
This student began with small jumps to reach friendly numbers that are easier to work with, while other students might prefer to start with larger jumps to get as close to the ending number as possible before making smaller, precise jumps to reach the exact location. Regardless of the approach, as long as the calculations are accurate, all strategies will lead to the same answer.
Using an Open Number Line to Multiply
Students can use an open number line as a tool to track skip counting when multiplying. For example, to solve 6 × 4, a student can skip count by fours six times on the number line. The point where they land after completing all the jumps represents the product.
Using an Open Number Line to Divide
Using an open number line to divide is helpful because it gives students a clear, hands-on way to see how division and multiplication are connected. It works a lot like using a number line for multiplication since the size of each jump is already known. The difference is that in division, the ending number is given, and students figure out how many jumps it takes to get there.
For example, when solving 20 ÷ 4, students use the number line to find out how many jumps of 4 are needed to reach 20.
Since it took five jumps of 4 to reach 20, the student can see that 20 ÷ 4 is 5.
Developing Mental Representations
As students gain experience, they can practice solving problems mentally using open number lines. Instead of drawing or writing out every step, they visualize the line and mentally track their jumps. For example, when solving 45 + 32, a student might think, “Start at 45, add 30 to get 75, then add 2 more to get 77.” This progression from physical materials to drawings and eventually to mental strategies ensures students develop a flexible, efficient approach to mathematical problem-solving.
Encouraging Mathematical Communication with Open Number Lines
Open number lines naturally invite discussion and sharing of strategies. Encourage students to compare their number lines and explain their choices:
- “What made you choose that number as your starting point?”
- “Why did you break apart the number that way?”
- “What made you choose to count forward or backward?”
- “Was there another way you could have solved this problem using the number line?”
- “How does your strategy compare to someone else’s? What’s similar or different?”
- “If you solved the problem again, would you do anything differently? Why or why not?”
- “Could you have reached the same answer using fewer jumps?”
These conversations help students see that there are many ways to solve a problem and that their reasoning matters. Building this confidence in mathematical communication not only strengthens their skills with open number lines but also supports their overall growth as mathematicians.
