Building a Mental Number Line
Mental math and number sense. These are the building blocks of a strong mathematical thinker. But how do we get kids there?
We must first start with our own mindset of what math is. For far too long, we have thought about math as a content area - knowledge to impart. But math is actually a skill that we are all ready to do. We are great math teachers when we are aiding our kids in using their math skills. It would probably help if instead of looking at our math block as content (what content am I teaching today) we looked and called it as a skill (what skill are students using today?)
Once students have come into contact with our number system (the part of math that was made to match up with our approximate sense of number in our natural brain) (for more about that, join the Natural Number course here!) they are ready to start using those numbers to do math skills.
Addition and subtraction are content. Manipulating numbers is a skill.
Word problems are content. Problem-solving is a skill.
But we have content to teach right? Good news! Students can build content knowledge through math skills (they’re also called processes and they’re in our standards). All you have to do is teach them the social conventions.
“We call this adding.” “We call this area.” “We call his the distributive property.” “We call these numbers fractions” (more on fractions here)
So, once we start realizing that students need to build math skills, we are ready to help them build their mental number line.
The mental number line is a beautiful tool that the human brain is more than ready to build. Through the Natural Number course you can learn more about how the human brain is born with an approximate sense of number (they study babies and it is fascinating!). Because that is in the course, I won’t go over it here.
Because our brains already have this sense of number, we just need to attach our number system to it. This is not laborious for students. It’s actually quite fun! Students love to think, solve, and use reasoning (don’t believe me? Who has argued with a kid…? They love it).
Here are a few activities that take very little time and resource that will help your students with their mental number line.
GROUND-BREAKING: For students to have a healthy mental number line, they must work with number lines.
I know I’m blowing your mind. But I’m not talking about the number lines that we put on students desks and they use as a tool. I’m talking about number lines as a physical visual for the thinking we want to happen in their heads.
It looks like this.
Let’s make a number line. (students draw the line)
Start the number line with 0, end it with 10
How much is between 0 and 10? (10) We can put a big jump and call this 10 (if this is difficult for students, back up and have them draw all the 10 jumps in and stay there)
If the whole thing is 10, what is half? (draw two equal parts/two equal jumps and the line in the middle)
So this part is 5 and this part is 5. The middle is… (5)
Where would 4 go? Put a sticker where 8 would go. Draw a dot where 2 would go. Etc.
Let’s make a number line. (students draw the line)
Start the number line with 0, end it with 10
How much is between 0 and 100? (100) We can put a big jump and call this 100.
If the whole thing is 100, what is half? (draw two equal parts/two equal jumps and the line in the middle)
So this part is 50 and this part is 50. The middle is… (50)
Where would 40 go? Put a sticker where 80 would go. Draw a dot where 20 would go. Etc.
Let’s make a number line. (students draw the line)
Start the number line with 100, end it with 300
How much is between 100 and 300? (200) We can put a big jump and call this 200
If the whole thing is 200, what is half? (draw two equal parts/two equal jumps and the line in the middle) (if students struggle with half, draw 200 with a model and have them physically break it in half)
So this part is 100 and this part is 100. The middle is… (200)
Where would 170 go? Put a sticker where 250 would go. Draw a dot where 210 would go. Etc.
Once the lines are built, it’s time to ask questions like “what is that number close to?” “what is that number far from?” “what hundred is the number closest to?” “what ten if the number closest to?”
All of this work is helping students begin seeing the line in their head.
If students struggle with this type of number line, it would benefit to try a vertical number line. It makes logical sense for the numbers to get bigger as you move up and smaller as you move down and may help students, especially young students who struggle with reading and aren’t “left to right” thinkers yet. It’s also easier to visualize a vertical number line going up infinitely (and negative numbers go below the “ground”) so it’s a good visual all around.
Once students have had experience with visual number lines as a tool for thinking, they are ready to move to mental thinking. One of our favorite class games for this is “close, closer, closest”.
It’s very cool because all you need is a whiteboard & marker (and not even that when your kids get better at it) and like…5 minutes (or you can take up 30 minutes if you go in depth - it’s a beautiful activity).
Write a number on the board. Have students tell you the number.
Ask a student to tell you something that is “close to” that number (or something that number is close to)
Another student must say a number that is closer
Another students must say a number that is closest
That is it!
Story time: I did this activity with my 3rd graders yesterday. The number was 172. A student said 200. Next student said 170. Next student said 173 or 171 (my students all usually stop here and say the one above and below because they know someone else will “steal it” if they don’t say both.
I could have stopped there. But I asked
“Can anyone get closer?” And the sweetest little voice from one of my students who is not traditionally high-functioning math student pipes up. “uhh….171 and a half…” and my students erupted into “oh yeahhhh!! that would work! Oh man!'“ so I wrote it on the board “171 1/2” and they ended up talking themselves into 171 and three-fourths. That was as close as they could get that day.
No prompting, just days of thinking about number and making sense of them with visuals.
If that version of the game is tricky for your students at first, try this one.
Put a number on the board. Ask students to tell you the number.
Say “I’m going to put another number on the board. You say if our number is close to or far from that number”.
Write numbers (always start small and build up) (for 32 you could put 30, 40, 800, 5, etc.)(the cool thing is no student is actually wrong…they could say 5 is close because it certainly is closer than 800!) The point of this activity isn’t to be “right” or “wrong” (that’s math as content thinking). The goal is to build a skill. If students can reason why the number is close or far, that is the goal.
Eventually, when you have many numbers up there, ask a kid to come find the number that is closest to your number (in our example case it would be the 30).
If students struggle, make the number line. Have students put all your numbers on it where they go. Physically see where 32 is and who it is close to.
Those are two simple activities you can start with students to build their mental sense of number.
The absolute best thing you can do for students to help their mental number sense is to retain numerical value at all times. Limit activities and algorithms that reduce numbers to digits. For example, students should always see 78 as seventy-eight (or sixty and eighteen, fifty and twenty-eight), etc.) and NOT as 7 and 8.
7 and 8 is 15…
If we could avoid students splitting their numbers up to calculate or follow algorithms, we will be aiding their mental number line to see the numbers where they truly are. This will help them with mental math, rounding, estimation, and more.
If you have students who have already been trapped in digit-thinking, start by only giving them problems horizontally. Sometimes this little trick will prompt them to solve with value and mental sense instead of reducing numbers to digits and trying to remember rules.
Because again…math isn’t a set of rules. It is a beautiful skill.
and it is natural!
Happy teaching!
