This content advocates for teaching integer addition and subtraction through conceptual understanding and a consistent framework, rather than relying on memorized rules, to foster deeper mastery and confidence in students.
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welcome to how we teach integer addition
and subtraction in this video we will
cover the problem with teaching rules
for integer operations and a consistent
framework for understanding integer
addition and subtraction that we can use
instead integer addition and subtraction
can quickly become frustrating for both
students and
instructors integer operations are often
treated as a set of rules to memorize
but rules are easy to misremember or
misapply which is a recipe for wrong
answers and low
confidence but here at Mathnasium we
teach for conceptual understanding
giving students a consistent framework
for understanding all integer addition
and subtraction problems builds
Mastery let's explore the problem with
rules in this interaction between an
instructor and a
student how much is -3
+8 um netive 11 great how do you
know the signs were the same so I added
them awesome What about3 minus 2
2
uh1 okay how did you get
that the signs were different so I
subtracted them that's the rule for
addition but this is
subtraction oh it's not the same let's
five8 not quite how did you get
that subtracting negative is like adding
so I added
them okay but it's like adding a
positive so what's -3 +
5 it's
ne8 okay but the signs are different so
we subract them but you said we were
stupid when we teach integer addition
and subtraction consistency is key
memorizing rules is frustrating for both
the student and the
instructor rules that are not based on
conceptual understanding are easy to
misremember or
misapply wrong answers lead to low
confidence so we need a framework that
works for all integer addition and subtraction
subtraction
problems let's start with our
prerequisite knowledge
negative numbers are the opposite of
positive numbers and we already know how
operations with positive numbers work
whenever you add a positive number you
make the starting number bigger and when
you subtract a positive number you make
the starting number
smaller since we know how positive
numbers work we can use opposites to
understand how negative numbers
work for example 3 + 5 means start at
three and get bigger by five
which gives an answer of positive
8 since adding a positive number means
get bigger adding a negative number must
do the opposite we get smaller therefore
3 +5 means start at three and get
smaller by 5 we can see on the number
line that we end up at
-2 similarly 3us 5 means start at three
and get smaller by five since
subtracting a posit positive number
means get smaller subtracting a negative
number must do the opposite we get
bigger therefore
3-5 means start at three and get bigger
by five we can see on the number line
that we end up at posi
8 we can reason through all four of
these problems using the same three
questions where are we starting from are
we getting bigger or smaller and where
will we end
up let's try one of the problems from
framework let's try another how much is
58 not quite how did you get
that subtract a negative number is like
adding so I added them let's think about
from3 and are we getting bigger or smaller
bigger that's right
why because subtract negative is the
same as ADD and a positive it's true
that both of those things make a number
bigger let's think about why what
happens every time we subtract a positive
positive
number it gets smaller that's right and
negative numbers are the opposite of
positive numbers so what do you think
would happen if we subtracted a negative
number instead
opposite of smaller is bigger so it gets
bigger exactly so let's go back to our
problem you said we're starting from
negative3 and we'll get bigger by how
much by five so if we start at negative3
and get bigger by five where will we end
up at two that's right how did you
figure it
out well we have to go up three to get
to zero and then we still have to go up
two more great job and hey what would
ne3 plus
5 start AT3 get bigger by five oh it's
the same thing so two exactly
subtracting a negative number and adding
a positive number give the same answer
even though they're not the same problem
bigger in the interaction the instructor
asked all three key questions where are
we starting from are we getting bigger
or smaller and where will we end up but
just as critical Ally the instructor
asked the most important question after
almost every answer the student gave
why as always at Mathnasium it is
essential to check for
metacognition a correct answer alone
can't tell you whether the student used
efficient strategies the only way to
know what's going on in their head is to
ask a student can produce the correct
answer to the problem 3 minus neg5 in
multiple ways a student who has
memorized the rules might think
subtracting a negative
is like adding a positive and 3 + 5 is8
so the answer is eight whereas a student
who has a conceptual understanding of
integers might picture a vertical number
line and think we're starting at three
and getting bigger by five so we end up
at positive 8 which means 3 -5 is
8 a student who correctly answers that3
+ 5 is 2 might be thinking of a vertical
number line where we start at 3 and
count up by five one at a time so we end
up at
pos2 which means that -3 + 5 is 2 but a
student who has learned efficient
strategies will picture the same
vertical number line and think we're
starting atg -3 and getting bigger by
five it takes three to get up to zero
and then we still need to go two more so
we end up at positive2 and -3 + 5 is 2
let's explore how we can use real world
examples to make problems more
relatable horizontal number lines can
seem arbitrary to students why does
going right mean getting bigger and
going left mean getting
smaller vertical number lines which we
have been using throughout this video
clearly communicate the concepts of
getting bigger as going up and getting
smaller is going down because many
things in the real world work this way
at add real world scenarios to your
visual Clues can make this Association
even more powerful let's think about the
problem -2 + -2 using the idea of
digging a hole in the ground we can
think of the ground as representing zero
on a number line so starting at
negative2 means we are two below zero
which is like a hole that is 2T deep
underground adding another -2 is like
digging two more feet down into the
ground which means we reach a depth of 4
feet underground meaning that -2 + -2 is
-4 let's try another -2 minus
-3 starting at -2 again means we are
starting with a hole in the ground that
is 2 feet deep if adding a negative
meant digging further down then
subtracting a negative must mean the
opposite we're going back up by
3et first we must get to zero so we fill
in our twoot deep hole in the ground but
we still need to go up one more so we
can pile up a foot of dirt on the ground
getting us to 1 foot above the ground
which means that -2 - -3 is postive
1 let's take a look at another
example suppose you have $400 in your
bank account which we can represent with
a positive
400 you want to buy a Nintendo switch
which costs $300 so you go to the store
and buy it with your debit card this
adds a charge of $300 to your card and
since you're reducing your bank balance
by spending we can repr ENT this by
adding -300 to our balance so how much
do we have now you are now the owner of
a Nintendo switch and have $100 left in
your bank account so 400 plus -300 is
positive 100 now suppose you decide you
would rather have a different color
switch that you found online so you go
to the store to return the one you just
bought when you return the switch the
store removes the the $300 charge from
your debit card which we can represent
as minus
300 so what do you have now your bank
balance has returned to $400 which means
that 100 minus -300 is positive
400 let's review some key facts to
remember every time you teach adding and subtracting
subtracting
integers don't give students rules to
memorize or allow them to just change
the signs it's easy to forget something
you've memorized but much harder to
forget something you conceptually understand
understand
for every problem do ask three important
questions where are we starting from are
we getting bigger or smaller and where
will we end up and never skip the most
important step of all checking for
why thanks for watching how we teach
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