The NUMBER LINE – A Hidden Treasure in Plain Sight!

Just about every elementary math classroom has one - I've worked in three different school districts in my career and having a number line displayed on the wall has been the standard in each of those districts. Let's pause for a minute and think about that phrase "displayed on the wall"....

WALL  ART

or

LEARNING  CHART

A number line display? Is that really what we want? A display rather than a tool. I want my number line to be a tool – a number line is a powerful learning aid when it's used. Now think about where most teachers hang their number lines... in just about every classroom, the number line is hung on the wall wa-a-y-y up high in a location where access is challenging to our young learners who benefit from using the number line as a hands-on learning tool. 

Why do we hang this amazing math tool in a place where access is difficult? Lack of wall space is certainly one of the reasons, perhaps it's just become a classroom tradition, and maybe we just don't understand the full value and the range of applications that support instruction when using a number line. 

Many teachers have already discovered one of the solutions: Keep your wall art for reference BUT make sure students have hands-on access to other number lines (perhaps one is taped to their desk or maybe various number lines have been placed in their hands-on math kits). Okay, so now that we've brought the number line down to students, let's talk about a few of the many amazing (and perhaps surprising!) ways to use a number line to support learning. 

AMAZING (and perhaps surprising!)
WAYS to USE a NUMBER LINE to SUPPORT LEARNING

 

Did you know that the CCSS Standards which have been adopted by many states, specifically names "Number Line" as as strategy/model 30 times (!). In addition to those 30 callouts, the number line can be an effective modeling tool for many other learning moments, as well. 

Many classrooms already use the number line to model addition and subtraction, one more and one less, counting and before and after concepts, but what about multiplication? division? elapsed time? unit conversions? and rounding? Can a number line be used effectively to solve for these types of problems? 

Yes, yes, yes, yes, and yes! 

MULTIPLICATION

If we know 4 x 20, how could we use the number line as a tool to visualize 4 x 19? Those 4 jumps of 20 would be 4 jumps of 19, so we'd need to compensate for the 4 big jumps by backing up 4 (1 for each 20 that was really a 19). What if we were calculating 4 x 18? How could we use our visual model of 4 x 20 to solve that equation? Are you thinking we could do 4 x 20 then back up 8 (compensating for the 2 extra spaces with each jump of 20)? What if the original problem had been 5 x 19 instead of 4 x 19? How might the number line model be the same? How might it be different? 

DIVISION

I love using the number line model for division. You'll likely notice that this is just a partial quotient method that utilizes the number line. Many students (and teachers!) find this number line model easier to understand and manage with division equations than using the box method or area model. 

Notice how we take advantage of multiplication facts that we know to get to the quotient. Take a look for yourself, what do you think? 

ELAPSED TIME

A number line is a simple way to straighten out our traditional analog clock.  We can use a number line model by creating friendly jumps to make calculating elapsed time more efficient. 

Check out the two models below that help students (adults, too!) calculate elapsed time. Because I use the number line model so much, I can now visualize a number line in my head when trying to figure out what time I will arrive at my destination 🚘.

How are these two models the same? How are they different?

Now we simply add the values of the  friendly jumps to calculate that the workout was 38 minutes long (that's not a bad workout on a busy day, right?)

 

ROUNDING

This is a procedural TRICK!
This is NOT a good model

I Googled rounding strategies the other day and found mostly a bunch of rounding tricks (mnemonic devices to remember and various rhymes - some that were so complex that I couldn't see how that possibly made it easier!). Anyway, we don't need tricks – tricks are just that, ways of tricking people into believing that they know or understand a concept. We need to teach strategies that actually help students to conceptually understand the mathematical concept; after all, we want them to understand the relationship of numbers and then generalize their understandings in other applications, right?

The number line model can help us give relational context to a number. Imagine we wanted to round the population of Palm Springs to the nearest thousand. I can see that there are 44 thousand (plus a few more) people who currently reside in Palm Springs. The question is whether the "plus some more" puts us closer to the next thousand???

The Critical Importance of Benchmark Numbers 

Benchmark numbers are the KEY to UNDERSTANDING rounding numbers. 

In the model below, I know 44 thousand (plus) live in Palm Springs. So the value is more than 44 thousand but less than the next higher thousand (45 thousand). These values become my bookends. The benchmark value that falls exactly halfway between the two values is 44,500. 

Next I consider if my number is approaching 44,500 (making it closer in value to 44,000 rather than 45,000), or has my value of 44,612 already passed my halfway benchmark of 44,500 and is now closer to 45,000?

Think about how this method builds a conceptual understanding rather than relying on procedural memory, or worse, a catchy rhyme that requires no understanding of the math that drives the math. 

CONVERSIONS

I love using a double number line for conversions. I first started using this model when I learned that 24K jewelry is made of 100% gold, and being the mathcurious person I am, I wanted to figure out the percentage of gold in my 14K gold wedding ring. 

Recently, a group of learners used the number line model for a conversion related to a Washington DC landmark: The Washington Monument. 

Let's begin with a bit of background for the problem: The Washington Monument was built in two phases. The first phase was 152 feet tall, but they ran out of money to complete it, so the monument stayed incomplete for two decades until Congress decided to pass a joint resolution in 1876 to complete the project. The completed monument is 555 feet tall - at the time, it was the tallest structure in the world. 

Back to the math.... As you study the progression of the model, you will notice they reflect the progression of thinking. 

How does this number line model for unit conversions scaffold thinking better than the algorithm method of solving the problem? 

How can we use 1 foot = 12 inches to determine how many inches are in 100 feet?

 

How does knowing 100 feet, help us quickly know the number of inches in 50 feet? 

We have the conversion for 100 and for 50, that's 150 of the 152 feet. 

We just need 2 more feet.

How did we determine that 152 x 12 = 1824 without actually multiplying 152 and 12?

What math concepts can you teach using the Number Line Model to build students' conceptual understanding of the relationship that numbers share?