Open number lines have some serious staying power in mathematics! I’m pretty biased, but this has got to be one of the most versatile representations that we can use in class. Our students can use it to develop conceptual understanding about so many topics—the four operations, fraction magnitude, decimals, rounding, place value, elapsed time, and money! Just to name a few!
So, if you’re looking for ideas on how to use open number lines more often in class, keep reading this post and bookmark it for later!
What is an empty number line?
An empty number line is another term for an open number line. Essentially, it is a number line model that has not been labeled with a unit.
With this model, students get to choose where they want to start and what unit they want to work with. Open number lines are great ways to record students’ mental math strategies by the students themselves or by their teacher.
Why are open number lines important?
An open number line is an important representation to use in math class especially during elementary school. As students become more familiar with different types of numbers, such as fractions, decimals, and whole number, this is one tool that can support their number sense.
This visual model can help students make important discoveries about mathematics like these:
1. Numbers never end!
We can use number lines to show that numbers are infinite and go on forever in both directions. We can also show that there is always a number that can fit between two numbers on a number line—whether we label them or not. To help students understand this, a good question to ask at the beginning of the year would be: How many numbers are between 0 and 1? or 0 and 2?
Before learning about fractions and decimals, they may think the answer to the first question is obvious: NONE! Although they may not think numbers are between 0 and 1 at first, highlighting situations in real life that require us to have numbers between 0 and 1 could challenge their thinking (like sharing a cookie with a friend).
As they learn about different kinds of numbers, it could be interesting to see how their answers change throughout the school year. A follow-up question when they understand that numbers between 0 and 1 DO EXIST, might be:
Can we name all of the numbers between 0 and 1?
2. Numbers can be decomposed and recomposed in strategic ways.
When kids look at open number lines created by their classmates, they may notice that everyone used different jumps to get to the same answer. Discussing these models may help some students see a connection between place value and standard algorithms or introduce them to new mental math strategies. For example, two students may have solved 5,568 – 4,423 in different ways:
Exploring these open number lines can help students develop a few standards for mathematical practice, including model with mathematics and construct viable arguments and critique the reasoning of others.
How do you teach open number lines?
I think of open number lines as a strategy so I tend to introduce them during number talks. As students share their mental math strategies, I record their thinking using an open number line.
Another way to introduce your students to an open number line is by using an anchor chart. Students can help you make sense of this new strategy by examining a worked problem displayed on a chart. I encourage my students to ask questions about parts they don’t understand yet, and then we do a think aloud. I present another problem, and we try to use our new strategy and think about how we could jump along the number line.
Finally, I love to use open number lines when solving story problems! You can show various number lines that go along with word problems and ask students which number line would be most helpful when solving a specific word problem. During this discussion, you could point out that each number line starts in different parts of our number system, and ask students: Is it fair for us not to start with 0? Could we build our own number line for a new problem?
How do you add using open number lines?
First, you would place one of your addends on the lefthand side of the number line. It’s more efficient to choose your largest addend, but it may take students time to jump to this conclusion! 🙂
After locating the first addend on the number line, use the second addend to determine how far you need to jump to the right. Remember, you can always break up the distance and jump in strategic intervals. The number you land on is the sum of your two numbers.
For multi-digit addition, students eventually realize that they can jump using the place values in the second addend like in the video below:
Using an open number line to add clearly shows that addition is all about combining or joining two amounts and finding a total, especially when you represent a story problem using this model:
Chris has 240 toy cars. His mom gave him 12 more. How many toy cars does Chris have now?
In the example above, I started with the larger addend, 240. I decomposed 12 into 10 + 2, because it made it easier to mentally add them. Some of my students would have done one jump of 12 while others would have made 12 jumps of one, which are all valid strategies!
How do you subtract using open number lines?
You can start with either number in the subtraction number sentence of word problem. A lot of people say that you need to start with the larger number and then count backwards, but this is only one strategy!
When a student understands that subtraction is all about finding the difference between two numbers, then they can use a counting up strategy as well.
For instance, when solving 276 – 142, you could start with 276 and count backwards. Students can find 100 less than 276 on the number line, which is 176. Then jump to 40 less than 176, which is 136. Finally, students can find 2 less than 136, or 134, which is the final answer. In this example, 142 was the difference and the result of removing 142 from 276 is 134.
Perhaps, another student wants to find the difference between 142 and 276 by counting backwards or subtracting an unknown amount. One hundred can be removed from 276, leaving 176. Removing 30 from 176 will result in 146, and jumping 4 to the left will leave you with 142. The difference between 142 and 276 is 134 yet again!
Another strategy would be counting up to find the distance between 142 and 276. Jumping 100 more than 142 would get you to 242. Jumping 30 more than 242 would get you to 272, and you would need a final jump of 4 to land on 276. The total difference would be 100 + 30 + 4, or 134.
Each strategy is acceptable, but the third strategy illustrates the inverse relationship between addition and subtraction quite nicely and can lead to a wonderful discussion!
The next example shows how an open number line can be used to illustrate a subtraction story problem. I think kids need experiences using open number lines in contextualized and decontextualized problems. I try to do a healthy mix in my classroom.
Chris has 240 toy cars. He lost 12 of them. How many toy cars does Chris have now?
How do you multiply using open number lines?
First, examine the number sentence or story problem and figure out how big your jumps will be. You will also need to know how many jumps you should be making. Then you would start from 0 and repeatedly jump the same amount. When you’ve completed all your necessary jumps, the number you land on is your answer.
For example, you might think of 3 x 12 as 3 jumps of 12. Starting from 0, you would jump 12 and land on 12. A second jump of 12 will get you to 24, and the third jump of 12 would get you to the answer, 36.
When students look at this number line, you could ask them what other operation this model might represent. You might even ask them to write 3 number sentences that go along with the open number line.
Students may automatically notice that 12 + 12 + 12 is another number sentence that goes along with the model. Depending on the goals of the lesson, you could take this opportunity to introduce or reinforce the connection between multiplication and addition.
If you asked students what might happen if we did 12 jumps of 3, some students might think this would lead to a different answer. Another conversation you might have could focus on the commutative property of multiplication.
Using a story context could also lead to a deeper discussion about the commutative property of multiplication and efficiency. Consider the following problem:
There are 12 toy cars in a gift set. Chris’s dad buys 240 gift sets. How many toy cars did Chris’s dad buy?
Technically, if we follow the story problem in the literal sense, we should do 240 jumps of 12. A student who understands the commutative property of multiplication could make the argument that 12 jumps of 240 would still lead to a correct answer and be easier to show on an open number line. Now, that’s number sense in action!
The distributive property could be highlighted on this number line as well. Perhaps, I don’t want to add so many groups of 240. Instead, I may break up my 12 groups so that I can find numbers that are easier to multiply in my head. For me, I would choose 10 groups of 240 and then add on the remaining 2 groups of 240.
How do you divide using open number lines?
When solving a division number sentence or word problem, you may either start from 0 and use a skip counting strategy to count up or start with the dividend and count backwards until you cannot make any more complete jumps. Every jump you make should be the size of the divisor, and you will know you are finished if the last jump is less than or equal to the divisor.
Counting upwards would allow students to see the connection between division, multiplication, and addition. Counting backwards, however, highlights that division is related to repeated subtraction.
Open number lines can be a great visual for solving measurement division word problems like:
Chris has 240 toy cars. While cleaning his room, he puts 12 toy cars in each of his toy bins. How many toy bins did he put toy cars in?
Ultimately, the question we are asking is how many groups of 12 are in 240?
When creating open number lines, students can make the choice to make jumps of 12 until they land on 240. To find the answer, students would count how many times they jumped 12 units on the number line and find that 240 divided by 12 is 20.
Other students may use one jump to mark 10 groups of 12 on their number line, and then mark a second jump with 10 groups of 12 to land on 240. Either way, these strategies lead to a valid solution of 20 groups of 12 fit into 240.
Check out this video if you would like to see a few more examples of modeling division with open number lines:
How else might you use an open number line?
Fractions and Decimals
Each of the examples above can be changed to include one or more fractions or decimals, and the open number line would be just as helpful for finding a solution.
For fractions, it is especially useful to model situations with open number lines, because it helps students understand fractions as numbers. Some kids think that a fraction is two whole numbers–one on top of the other, but placing fractions on a number line can help get past that misconception.
Elapsed Time
Students can find elapsed time, start time, and end times using open number lines. Placing these times on a number line helps reinforce certain concepts related to time. For example, when representing minutes and hours on the number line, kids have to remember that there are 60 minutes between each hour. Also, when counting past 12:00, students need to realize the next hour would be 1:00. Students can decompose numbers to help them keep track of time, especially when counting across the next hour as well. Using the number line helps students think about how each unit of time is related to another.
Rounding and Making Comparison
Judging whether a solution is reasonable or not is a huge part of doing mathematics! So, it makes sense that we give kids experience with rounding and estimating whole numbers, fractions, and decimals.
Instead of showing them the rounding rule right away, we can use open number lines to lay a strong foundation for it. For instance, if we wanted to round the number 4,563 to the nearest ten, kids would have to use what they know about our number system to find the tens this number is between (4,560 and 4,570).
Placing all three numbers on a number line would quickly reveal that 4,563 is closer to 4,560. So, for this rounding problem, we would be “rounding down” to 4,560
Locating numbers on a number line requires students to understand their magnitude and compare numbers to each other. Understanding that numbers increase as you move to the right and vice versa is a central idea that students develop while interacting with this model.
Measurement conversion
A double number line is also very helpful when teaching students how to convert one measurement into another. Although this model looks very different from the open number lines we’ve used up until this point, I still consider them to be open number lines since it’s made of two empty number lines.
If I wanted to know how many feet are in 5 yards, I can create a double number line like this:
This model shows how one yard is related to feet and allows students to count on until they reach the measurement they want to convert. My students love using these when we work on measurement conversions!
Made4Math Recap:
Open number lines are my favorite tool to use during math class, because they can grow with my students! These representations can help students record their thinking and make sense of new mental math strategies. These models also help students develop conceptual understanding for each numbers and operations, and as students learn about fractions and decimals, open number lines help them understand where they all fit in!
I’d love to hear from you! Please leave a comment below!
When’s the last time you used an open number line during a math lesson? How did it help your students develop number sense?
