- Number lines are an
essential mathematical model.
However, if you are
like me, we didn't grow
up doing math using number lines.
And so if we didn't use it
and we didn't learn that way,
it becomes hard for us to
do that with our students.
I'm Christina Tondevold, the
Recovering Traditionalist,
and today we're gonna take
a look at my top five ways
to use number lines in the math classroom.
I hope you'll stick around as we build
our math minds so that we can build
the math minds of our students.
So I'm gonna give you the quick overview
of the top five and then I'm gonna show
some slides that show you
what it looks like using
a number line for these five things, okay.
So our top five are
rounding, which goes along
with last week's Vlog post.
If you didn't watch that,
there's a link to it
below this video that will take you
to last week's video all about rounding
using a number line.
Addition and subtraction.
We're gonna talk about
multiplication and division
on a number line.
We're also gonna talk about
how number lines build
spatial relationships for our students.
And then the last thing I
want to talk about is using
number lines to help
kids understand percents
and finding percent of type questions.
Okay, so let's dig into
number one, rounding.
Alright, so I'm not gonna
go to in-depth on rounding
because there is the other
video that I'll link to
that will give you more in-depth,
but the whole idea is
getting kids to place
the number we're trying to round
on the number line based
upon the benchmark numbers
that we're trying to think
about does it round up
to this amount or down to this amount.
And if they can place it on a number line,
that visual helps them to see
which amount it's closest to.
And it's a better way to start out
than the way that I learned which was
if it's five or higher, you round up.
So if you want more details,
head back to the Vlog post
all about using number lines to round.
Let's jump into the second one which is
adding and subtracting.
There are lots of different strategies
that kids will use when
they're adding and subtracting.
Way different than when I grew up
or even when I first started teaching.
It was still about just learning
the traditional algorithm.
Now the traditional
algorithm is part of it,
but part of what we're trying to do is let
kids solve addition and subtraction,
multiplication and division,
any way that they can.
As long as it's efficient and accurate,
if it is based in their understanding,
it's an okay way to solve the problem.
So I want to show you
some of those strategies
that kids kids might come up with
and this is not all of
them, but just some of them.
And how the number line can really help
kids to visualize it.
Not just the kids who are doing it
'cause they're seeing how
all of these numbers work,
but the number line can help
the other kids understand
what that kid was doing.
So I'm gonna start off with fractions
because fractions are
not friendly some times.
But one of the things
that's helpful is that
kids will want to make
the fraction friendly.
So if I'm adding two and three-fourths
and I've got to add three
and two-fourths onto that,
one of the things that
kids will do is say man
if I just added a fourth,
I could get to three.
And then once they're at three,
that's one of those kind of
benchmark friendly numbers,
once they're there, it
becomes really easy for them
to just add the remaining piece
of the three and one-fourth
to get to their answer.
So again, as I'm talking
about these strategies,
I don't think I said this explicitly,
but these are not things
you have to teach to kids.
It's things that kids will naturally do
if they understand numbers
and how numbers work.
Kids will do these strategies.
And the number line is
a great way for them
to take what they're doing in their head
and have a way to show it on paper.
The number line is a
notation, it's a model.
The number line itself is not a strategy.
There's nothing that tells me what to do
on the number line.
Kids will use them in different ways.
Alright, let's take a look
at an example of subtraction.
Okay, a lot of times,
kids will want to round
one of those numbers to a friendly number.
If one of the numbers is
near a friendly number,
they will want to round.
The hard part with subtraction is that
once they get here, this
becomes really easy.
I know that the answer to that is 47,
but the hard part is they don't know
what to do next.
They knew that they changed the problem,
but they aren't quite sure what
how to undo what they did, right?
They will say to themselves,
oftentimes they will say,
well I added 11 to get to the 300
so I should subtract 11 because they feel
like it needs to balance out.
However, when we show it on a number line,
and we use that notation
to show their strategy
and we show how they started at 347
and they jumped back 300,
once we show that visual,
a lot of times kids are
like wait, okay wait,
I jumped back too far.
I went farther than I was supposed to
so I actually need to add the 11 back in.
So a lot of times when kids are struggling
and they've got a
strategy part of the way,
like they've got this idea,
but it's not quite working right,
having a visual model of that strategy can
really help them understand it better.
So a number line is
always, just seems to be
a great strategy, a great notation to help
them visualize the strategy.
Okay, let's move into
multiplication and division.
And when it comes to multiplication,
where kids first start out is that
they see it as repeated
addition or a skip-counting.
And that's a fine place
to start, but it is not
the only way we want kids
to view multiplication.
It's not an additive process.
Multiplication is a
multiplicative comparison.
You're multiplying an amount.
So we want to also be
attaching the groups of,
this idea of groups of.
Just changing our wording a little bit,
not just saying it's five
plus five plus five plus five,
but it's one group of five,
then it's two groups of five,
and three groups of five, and so on.
And I'm also a firm believer in it because
number lines are a little
abstract for kiddos.
So when you're first starting out,
don't be surprised if number
lines are not that easy
for kids to use.
And actually number lines are recommended
to start at second grade.
So kindergarten and first
grade should not be using
number lines.
Instead next week I'll talk
about what we should be using
instead of a number line.
So just know as kids in
second and third grade are
starting to use a number line,
it may be a little abstract.
'Cause we're asking kids to
go from seeing visuals of five
to now just having a big hop of five.
They're not seeing five things.
So attaching these visuals can
really help kids understand
and have another visual to tie it to.
Now when it comes to
division, I'm gonna use
a context here to help us out.
But because I like
having stuff in context.
I think it's the best way
to be doing mathematics
in the classroom.
But let's say we have a
story problem like this
where Ms. Kyle has 20 jolly ranchers.
And students earn five Jolly Ranchers
when they complete their AR points.
So how many students can Ms.
Kyle give Jolly Ranchers to
once they complete their points?
Now this idea, what
kids will naturally do,
even before they officially
start doing division,
is they will visualize
those 20 Jolly Ranchers
and say okay five goes to this kid,
five goes to this kid, and
so as kids are visualizing
those five Jolly Ranchers going to one kid
and five Jolly Ranchers
going to another kid,
it's nice to have a model
to notate that thinking.
And so a number line can be a great model
for that thinking.
And again, remember it
is kind of abstract.
So we're starting at 20
and then they take away
one group of five.
Then they've taken away
two groups of five.
And they take away three groups of five
until they get down to
no Jolly Ranchers left.
That's essentially what they do
when they're physically
modeling this problem
with manipulatives.
And then the number line
is just a way to notate
that thinking for kids.
Okay, spatial relationships is a really
huge piece for kids
and their understanding
of how numbers relate to each other.
Spatial relationships is about giving kids
a visual but talking about
how those visuals relate.
And so with small amounts,
it's nice to just have
visual pictures of you
know seven and five,
and talking about what's the same
and what's different.
But as we move forward,
we want to help kids use
the number line as one of those visuals.
So that if they were to place
seven on this number line,
where would it go?
How do you know it goes there?
What does it tell you
about how it relates to 10?
Where would five go?
Where would 12 go?
All of those kinds of things
because it's a nice visual
but again, number lines
are pretty abstract.
So you want to first
start with actual things
where they can see seven things
in comparison to 10 things.
However, as kids start moving
up into larger amounts,
we don't want them to
have to draw 70 things
in order to see how it
compares to 100 things.
So the number line gives us a model
that is extendable, right?
We can use it for all kinds of numbers
that we are working with kids about.
So we want to start with physical things,
move to visual images, but
then the number line can be
a great visual to help
us see relationships
between amounts.
And even when it comes to fractions,
this becomes a huge visual for our kids.
Number lines are a great
visual for fractions
and understanding how
fractions relate to each other.
If I wanted to place three-fourths,
where would I put it on that number line?
And kids might partition,
but then things like
if I wanted to place 13-16ths,
where would I put that on the number line?
Or even 15-16ths?
I may not be asking the kids
to partition it into 16ths,
I want them to know that 15-16ths
is darn close to a whole.
That's the idea is that
they can place things
on this number line
that gives them a visual
to then talk about how
numbers relate to each other.
Same thing as we go into decimals.
We want kids to be able to know.
And if I want to put 32
hundredths on this number line,
where would it go?
Does knowing how it
relates to 50-hundredths
or one-half help me out at all?
So all of those pieces,
it extends into all types
of numbers that kids are gonna work with.
Alright, the last one I'm
gonna talk about is percents
because oftentimes kids
struggle with percents.
And I will say, not even kids, but adults.
I was the adult who was a math teacher
and when it came time to be figuring out
the percent of tip that I need to leave
for a waiter or waitress,
I was pulling out
my cell phone to figure it out
because I didn't have any visuals.
I didn't have any strategies
for figuring out percents.
And a number line is
a great way to do that
and what it essentially becomes
is a double number line.
So we start out with
this idea of zero to 100,
but that 100 is really
100% or the whole amount
we're working with.
So if my bill was $24,
right, I could figure out
lots of things on here.
But not even just in
that scenario of a bill.
What if I just wanted to know 50%?
Would being able to place this
on a number line help me out?
50% should be half, right?
And so what's half of the 24?
Now most of the time, hopefully they know
that fairly quickly, but
then we can use that visual
to help us with other things
like 25% or one-fourth, right.
We want kids to be able to visualize
where that would fall on the number line
and use that to help them out.
And we start with friendly amounts,
but then you can move to things
that maybe aren't as friendly,
and show them on the number line.
Now here's a typical problem
that we do with percents.
Ms. Thompson has 12 Labradors.
This is 20% of all her show dogs.
How many show dogs does she have?
When I was in school,
we were taught to set
this up as an equation
and you know figure out
what goes on one side of the equal sign
and figure out what the unknown is,
and once I learned that I could show this
on a number line, it
changed everything for me.
It helped me so much in
understanding percents of, right.
In this story, I know
that 12 Labradors is 20%
of what she has for show dogs.
And that's what I've
modeled on the number line.
From there, the goal is
to figure out what's 100%.
How many show dogs does she have?
Well the cool part about
once it's here is that
kids will figure it out
in lots of different ways.
And in ways that build their number sense.
They aren't just computing
because remember,
it's about building math minds,
not just creating calculators.
I could plug this into a
calculator and get the answer.
I want to encourage my kids to be thinkers
not just calculators.
Yes, I want them to get the answer.
Yes, I want them to calculate it.
But I want them to build their math mind
while they do that.
So you might see kids from here say
well if 20% is 12, 40%
would be double that,
and then if I doubled that,
that would gets me 80%,
and then when I get 80%,
all I have to do is add
that 20, go back to it, add that 20% on
to figure out what the 100% would be.
You might have some kids
who don't see it that way
and they say man, I can
just halve that 20%,
that would give me 10%.
And once I know 10%, that
could tell me what 100% is.
I mean they're using their number sense,
but the number line is
helping them visualize
this idea, right, to
build their understanding
while they're also computing
and figuring out the answer.
Now I will say that this
is not a complete list.
If you have a favorite way that you love
to use number lines that
I did not talk about,
let us know down in the comments.
Or if you have a favorite activity,
link to it down in the comments.
Help all of us build our math minds
because this is not the only way to use
number lines in the classroom.
Just my top five.
So if you want a short little summary
of each of these in a PDF download,
there's a link below this video
and at the bottom of the Vlog post
where you can request a download of it
that will give you these top five ways
and show some visuals along with it.
Alright, well as always,
I hope this helped you
build your math mind so you can build
the math minds of your students.
Have a great day.
