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.
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 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 Mastery Math Method masterclass,
where you can learn how to make your math stick in 10 days.
Go to disability labs.com forward slash calendar to register.
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.