We're Teaching Subtraction Wrong - Try this instead!

Jonily: distance cannot be negative.

Distance is not negative, but
direction can be negative.

So the result of subtraction is a number.

And a direction.

Cheri: Welcome to Tier One Interventions
podcast, where we help your, help

you strengthen your core classroom
instruction so every student has the

chance to thrive, even those students
that need additional instruction.

I'm Cheri Dotterer, and today Jenna Lee
and I are diving into the second of three.

Math rules that never expire.

Subtraction.

This is one of my favorite rules,
not just about the math, it's

about how the brain thinks.

And you'll hear Jonily use,

you'll hear Jona Lee use subtraction
across all grade levels and

jump into tips on how to make
learning stick with these brain.

Brain-based strategies, movement,
and multisensory engagement.

Whether you teach kindergarten or high
school, whether you're an occupational

therapist, let's learn how we can
change the way we instruct mathematics

and help kids make subtraction stick.

Jonily: Hello and welcome to Tier One
Interventions podcast, where we talk

about strengthening your core classroom.

That is I'm your host, Jay-Z.

Jonily Zupancic, your main
Math Ander Math specialist.

Math coach.

Math leader.

And at Tier One Interventions
podcast, we share.

Tips, tricks, techniques, and
strategies to enhance your tier

one core general classroom.

This reduces the number of students
that need tier two and tier

three pullout with a strong core,
tier two, tier three, no more.

It's just a little joke.

Not that we wanna eliminate those
small group pullouts, but we want

our tier one core regular classroom.

Specifically for mathematics to be
accessible for every student, no

matter the ability or disability.

We, she, we teach instructional
and facilitation strategies that

are going to increase learning
memory and retention of content.

Specifically with mathematics.

We want to share strategies with you
to improve, number, sense thinking,

reasoning, and sense making.

Today's focus is going to be
on rule number two of three

math rules that never expire.

These rules are transferable from
preschool through high school

no matter what the number is.

Hey Cheri here.

When we teach a concept that works
at every level from preschool

through high school, we're giving
students the brain pattern that

doesn't have to be unlearned later.

That consistency frees up working memory
so we can focus on reasoning and sense

making and remembering at any stage
and building rather than tearing apart.

Jonily: Some math rules that expire
are things like when subtracting

you can't get a smaller number.

That's only true with certain numbers.

We need a rule that we can teach
consistently from progression.

From preschool through high school
that the rule and the strategy is not

gonna change these specific rules.

And there are three big ones are the rules
that never expire and make high impact.

Powerful results in improving
number sense for our students.

Rule number one is division is counting.

Rule number two is
subtraction is distance.

Rule number three is
multiplication, is rectangle.

No matter what the numbers are.

These rules and processes never expire.

They are transferred from
grade level to grade level.

As we evolve and create
more complexity in numbers.

Today we're gonna focus
on rule number two.

Subtraction is distance.

In the previous episode of Subtraction
is Distance, which was aired on February

18th, 2025, called Hidden Math Patterns
that Make Subtraction easy for everyone.

The introductory version of the
three Math Rules that Never Expire

specifically with Subtraction was
released on that podcast episode.

So look that up.

If you wanna start at the introduction
today, we are gonna do an extension

of that introduction and we are still
going to follow the subtraction.

Is distance cycles A through Z?

But today particularly, we're gonna
focus on two of the extended cycles.

These cycles, and these numbers are
appropriate for any and all grade levels.

And the thoughts and ideas and strategies
that I share are absolutely essential for

improving number sense in your school.

Remember, number sense is an
innate, intuitive understanding

of the size of number.

Another word for size is magnitude.

Understanding the value of number.

For subtraction.

The question is how far
apart are these two numbers?

Subtraction is distance, subtraction
is range, and yes, subtraction are

other things, but some of those other
things are not transferable, especially

when we get to negative numbers.

So if we want to build consistency and
progressions of improving number sense

from preschool through high school.

This rule number two, subtraction
is distance is the strategy

and technique to do just that.

We are gonna focus on two
subtraction cycles today, f and

p in the previous podcast titled.

Hidden math patterns that
make subtraction easy.

From February 18th, 2025, we
focused on cycle A, which is the

introduction to subtraction cycles.

In the show notes, you will get
a copy of this notes version that

will accompany today's podcast.

If you are listening to this
podcast, audio only, I'm gonna be

as clear and explicit as possible.

Telling you about what's on the screen.

However, to catch a glimpse of what I'm
writing at the same time that I'm talking,

you can visit us on YouTube at tier one
intervention podcasts, subtraction cycles,

f and p are as follows, cycle F. These are
the four problems that we are going to.

Model today, eight minus three,
three minus eight 1012 minus 995

and negative three minus negative
eight cycle P 71 minus 38.

38 minus 71.

Seven decimal two five minus
three, decimal five and negative

13 minus negative seven.

As we look through some of these examples,
you're going to see how this transferable

strategy can be used with any number.

This gives access to negative
numbers earlier in the grades.

With this one strategy, we don't have
to change our strategy or technique when

we subtract, when we use this method,
again, this isn't the only method.

There are other methods to use for
different numbers that are procedural.

Procedural focuses on answer, getting.

And efficiency and accuracy improving.

Number sense is negated when we
focus on efficiency and accuracy.

However, improving number sense is
necessary for accuracy and fluency.

Efficiency to be improved.

That all sounds counterintuitive,
but all of them are necessary

for overall math achievement.

So what does it mean to be to
have subtraction as distance?

Subtraction as distance means?

In a problem.

The second number is my starting point
on a number line, and I'm gonna find the

distance between that starting number
and the first number in the subtraction.

Technically this subtraction symbol,
if I put arrows on either end of the

actual subtraction symbol that shows
that I'm looking at a number line.

And the subtraction symbol

in the depth of its meaning is how
far apart are these two numbers?

If I place these two numbers on the number
line, and I go from the second number to

the first number, what is the distance?

What is the range?

How far apart?

Now, as you can see here, my second
number is three, so I'm going

to put three on the number line.

My first number is eight.

I'm gonna put eight on the number
line, so I'm going to start at three.

And I'm gonna go to eight
and I have to figure out how

far apart those numbers are.

Now for these numbers and for some
students, I may have to put tick marks

to say, okay, here's four or 5, 6, 7.

Not drawn to scale, obviously, so that
then we can circle or mark the spaces.

Kids like to count the tick marks,
but we really need to count the

number of jumps or the number of
spaces from three to eight as we are

teaching distance and measurement.

This is a good intervention
for that as well, but also

enhancing work on a number line.

As I've talked in previous
sessions, one of our.

Interventions for number line is to
create that rectangle bar model over

top of the number line and extend the
tick marks so kids can see these spaces.

That's a little less abstract and
much more visual representational

than the abstract number line itself.

So the distance from three
to eight is five units, and

because we went to the right.

We have a result of positive five
if we follow the same strategy.

For this second problem, now our starting
number is eight on the number line.

As a matter of fact, I could have
used the same number line, but

I'm gonna draw a different one.

My starting number is eight, so I'm
gonna put eight on the number line.

And I'm going to move to three.

I'll mark three on the number line.

Now, keep in mind, already I know
that I'm moving in the left direction.

My result's gonna be negative.

Distance cannot be negative.

Distance is not negative, but
direction can be negative.

So the result of subtraction is a number.

And a direction.

Hey, Cheri.

Cutting in here again.

Breaking distances into chugs
is actually how the brain

naturally processes magnitude.

It reduces cognitive load and letting
the prefrontal cortex that says part

right up front here hold smaller
and manageable pieces, making mental

math faster and less stressful.

Jonily: So if I start at eight and
go to three now what's super cool

is subtraction is not commutative.

Meaning if I change the order, I don't
get the same exact answer, but I do get

an answer that shares a relationship.

So if the distance between three and
eight is five, the distance between

eight and three is still five.

The distance doesn't
change, but the direction.

Changes, and that's why my
overall result is negative five.

When we explore subtraction as distance,
we are actually teaching many more

standards than just subtraction.

That's why this improves
overall math achievement because

we're looking at number line.

Measurement distance range.

We are also looking at counting, which
improves number, sense estimation.

Is the distance gonna be
a lot or a little like?

Let's look at this next one.

Grab a different color here.

The next one is 100 1012 minus 9 95.

9 95 is our starting number.

So if I sketch a number line and
I mark 9 95 on my number line.

The number that I'm going to is 1012.

Now using this strategy and technique
and this thinking exercise, and really

what this is a thinking exercise.

It's not a procedure for
answer getting and accuracy.

That is important, but
that's a different exercise.

This exercise is a thinking exercise.

It's a minds on exercise.

It shapes the brain differently
than working and mimicking

a procedure step by step.

This actually increases problem solving
and critical thinking skills because

students are constantly figuring out.

It also improves number sense because
I'm actually physically seeing

values of number through distance.

Hey everybody, it's Cheri again.

If you're loving these strategies
and you want to join our Mastery

Math Method Masterclass, we hold
this hour long event once a month

and we'd love to have you join.

What we talk about there is how to
improve your math skills in 10 days.

Can you imagine 10 days, whether you
are a math teacher or an occupational

therapist, you can see exactly how to
shift teaching so that every student,

no matter their starting point, can
master core math concepts and reserve.

So reserve your seat today and go
over to disability labs.com/calendar

to check out the next event.

Jonily: I'm practicing my
counting techniques, which

increases rate and function.

So all of the standards depend on this
strategy and technique for subtraction.

So if I typically look at this.

In typical traditional procedure,
I would stack and subtract

1012, I'd put 9 95 underneath.

I'd do a lot of borrowing,
blah, blah, blah.

In that procedure, there still is a
lot of room for error and it doesn't

improve conceptual cognitive function.

So with this process, not only.

Is there a little bit of room for error,
but I actually can estimate better and

improve my conceptual cognitive function.

So I'm really working the
brain and exercising the brain.

There's brain growth that is happening
through just thinking about this

through this distance technique.

So if I stack and subtract, there
is room for error, but also in

looking at those two numbers on
top of each other, it still doesn't

trigger to me distance how far apart.

So I can ignore the value of
those numbers, which means I'm not

practicing a good estimate estimation.

Prior to solving the problem here,
where I'm placing the numbers, forces

me to think about the value of the
number, where it's gonna be placed.

And then as I place this,
I'm oh, you know what?

I think 1000 might be there because
I'm thinking a thousand closer to 9

95, a little further away to 1,012.

But really, in all honesty, neither of
those numbers are too far from a thousand.

So that thought process rather
than a procedure is what is

exercising and shaping and
growing the mind to have a better

understanding of number for students.

So when we look at this, we can
see subtraction is distance.

I start at 9 95, I need to go
to 1012, which is to the right.

So my direction is positive.

Distance is always positive, but I
can break this distance into we're

five away here and we're 12 away here.

So the total difference, the
total distance difference is

a combination of five and 12.

If I relate this to a thousand.

Lots of ways to think about that.

We can use manipulatives, we
can use bar models, concrete.

Paper strips, lots of instructional
strategies to use, but the point

is for students to engage in this
experience of figuring out the distance.

How far apart are these two numbers?

So five and 12.

17. The distance is 17.

If I switch this around 9 95
minus 1012, I start at 1012.

I go to 9 95.

The distance is still 17.

The result is negative because
I moved to the left direction.

Now you might be saying again, why don't
we just teach procedures for these?

The issue is the procedures for
single digit numbers, multi-digit

numbers, and then negative numbers
are all different types of thinking.

We teach different
strategies, and that's fine.

We don't have to stop doing that.

But the one strategy that works
for every single number is looking

at subtraction as distance.

Let's look at this example.

So in this example, my second
number is negative eight, and

I'm gonna go to negative three.

Now, typically what happens in some
of these problems is if I have an

example, like negative three, minus
positive eight, kids will frequently

ask, is that second number, is it
an eight or is it a negative eight?

And typically the teacher response is.

What's both?

Technically, let's look at this.

Technically.

Technically.

Remember when we make
that subtraction symbol?

The representation of a number line,
so I'm gonna put arrows on the left

and right end of the subtraction symbol
technically right here as this appears.

This is positive eight.

So I have to mark this at
positive eight on the number line.

So if this is positive
eight on the number line.

Because it's positive eight right now,
but I know what you mean, that that

subtraction can change to addition,
which makes it a negative eight.

But right now it's
subtracting a positive eight.

So that means I'm starting at eight
and then I'm going to negative three.

I'm starting at eight and I'm
gonna land on negative three.

So right away I'm moving in.

The left direction, and I can use these
number lines vertically also up and

down, and I interchange that constantly.

So if I'm starting an eight and
going to negative three, I'm

moving in the left direction.

So direction is gonna be represented by
negative distance is always positive.

So now if I go from
eight, let's say to zero.

That's a distance of eight.

And then we have from zero to
negative three, a distance of three.

There's a distance of 11, the distance
is 11, the direction is negative.

So when we look at this in that way, the
same strategy, technique and thinking

that we used with the previous examples.

We, we end up doing the same exact
thing no matter what the number is.

We're not changing the strategy
or technique because now we have

different types of numbers and we
can negate that misconception that

this is both a negative eight and a
positive eight as it stands there.

It is a positive eight.

So we oftentimes, with our language
and how we're responding to students

are giving them misconceptions,
and that is what's causing them a

number of errors and inefficiencies
in calculation and computation.

So now let's look at the one
that was actually the example

that was actually here.

Which is negative three, minus negative
eight, and again, that subtraction symbol.

If I put arrows on each side,
we can clearly see that my

second number is negative eight,

and I'm going to go to my first
number, which is negative three,

and then I can mark zero just for.

Conceptual, just framework really.

So now I'm going from negative
eight to negative three.

So my result, my direction is positive
distance is gonna be positive.

So now negative seven,
negative six, negative five,

negative four, negative three.

How many hops, how many spaces?

And again, an intervention for students
to have better access to this is to create

the bar model over top of the number line.

And.

Instead of tick marks, create dividing
up a bar so that we can clearly

see there's 1, 2, 3, 4, 5 spaces.

As we look at cycle P. Same thing with all
of these, but the one thing I wanna point

out is this example in some of the cycles.

We add in decimal numbers
and fraction numbers as well.

That's why we're giving you
cycle F and cycle P today.

There's importance in larger
numbers, multi-digit numbers.

Also importance in decimal
numbers and fraction numbers.

But let's see how this works.

If I have a number line.

And my second number is, oh, if I
put my number line here, there's

not gonna be any confusion.

That's a positive three decimal five.

So if I put my three decimal five here

and my Landon number, my distance
from three decimal, five to seven,

decimal two five, and you can see
this gets a little more complicated.

But again, I'm not looking
for efficiency and accuracy.

I'm looking for a thinking exercise.

That's going to contextualize
and conceptualize.

Subtraction being distance so that I
can improve my estimation, my counting

skills, which estimation and counting
are fundamental for number sense.

So if you have students that have
weak number sense, yes, they're

gonna struggle with these thinking
exercises, but the more exercises we

have them do with this, the better
their sense of numbers is going to be.

So here, this is gonna be a little tricky
because it's okay, three and a half.

Where's three?

Where's four?

Maybe this is three.

Maybe this is four.

So there's a little
playing around here with.

How much this is.

Oh, but wait a minute.

I know seven is here.

Actually seven is gonna be a
little closer to that line.

I didn't draw it to scale.

We can have those discussions.

We can actually do draft two with three
decimal five and seven decimal two, five.

So seven is gonna be closer to
seven, decimal two, five, but four

is gonna be further like we can start
to scale it a little bit better.

And modeling this for students and then
having them work independently on it is

going to be the key to engaging them in
deep thinking, reasoning and sense making.

It is not gonna be easy.

It's not about accuracy and efficiency,
it's about playing with it so that they

can figure it out and they can start
to understand value, size, magnitude.

Magnitude meaning how large and
small numbers are and where numbers

are in the number line, and in this
case, how far apart numbers are.

So once we have decimals
and fractions in the mix.

The thinking exercise extends.

We wanna give kids lots of time to think
about this, lots of time to play with it.

And then what I might do is
I might say, you know what?

We're not gonna finish this today.

We're not gonna finish this today.

We're gonna come back to this another day.

Because thinking exercises, if we finish
thinking exercises and we confirm.

Did you know that letting things linger
actually improves thinking skills?

By letting things linger, you're letting
the brain process what you've just taught.

Rather than give it the answer
and let it stop thinking.

Don't give them the answer, let it think.

You'd be surprised how many kids
will come back to you two or three

days later and say, Hey, Mrs.

Or Mr. I think I figured out
the answer to that thing that

we were talking about on Monday.

And you'll be going, why?

How do they, how are they thinking
about this two or three days later?

Their brain's been processing it
ever since you left it linger.

Jonily: Or deny accuracy and efficiency.

Then the brain stops thinking
about the thinking exercise.

However, if we don't confirm or deny
accuracies, we gather lots of answers.

We let the conclusion linger and come
back to it maybe three days later

or a week later, or two weeks later.

The subconscious mind.

Automatically continues to think about it
because there's this cognitive dissonance.

There's not been a
completion or a closure.

What we want in mathematics, we
want the brain to work over time

too often in our math classes.

We have this, and Dan Meyer said it
beautifully a number of years ago.

We have this sitcom approach to
mathematics where, when you watch this

show for 30 or 40 minutes and we expect
everything to be wrapped up at the end

of that time, and we're rushing to get
through and we're not allowing students

to process and have cognitive subconscious
thinking that they're not even aware of.

So we're limiting that.

And so the brain stops working because
there's no dissonance there's no conflict

because everything has been wrapped up.

The only conflict is some of our
students that have slower processing

or didn't get enough time to think.

They stop thinking during math
classes because they know other people

are gonna think for them and other
people are gonna finish for them.

So there's an implication to
the way that we facilitate.

So oftentimes I will let the answer.

The accurate answer, linger on
this and come back to it two or

three times, and that's one of
the reasons for the cycle as well.

These are thinking exercises
to improve number sense.

We use these subtraction cycles and by
using those subtraction cycles, we can

improve number sense and also increase.

Motivation, focus, engagement,
reasoning, and sensemaking.

And those are the culture of math
classrooms that we're trying to

create to produce amazing math.

Iners today was the extended
version of a previous podcast

that was released on February.

18th, 2025 titled Hidden Math Patterns
That Make Subtraction Easy for Everyone.

That podcast episode was the intro
version to today's extended version,

the part two version, but this is also
as part of a series of podcasts that

are three math rules that never expire.

Rule number one division is
counting rule number two.

Subtraction is distance.

And rule number three,
multiplication is rectangle.

I was your host today, miss Jay-Z.

Jonily Zupancic here with Tier
One Interventions podcast.

Hey, thanks for joining us today for
this episode of Tier One Interventions.

If you found today's strategy is helpful,
imagine what your students could achieve.

If every math lesson clicked this
way, make sure you subscribe to us and

set this a comment in the show notes.

The.

Send us a comment and let us know
what you thought of this episode.

Don't forget to grab your spot at
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Go to disability labs.com forward
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Until the next time, go be awesome.

Go be brilliant.

You were put here for such a time as this.

Make tier one strong so that
every student can thrive.