Download Math done in the session

Teaching Math
Conceptually
• Students who cannot easily memorize
rules tend to not do as well in math.
• Students who can memorize rules tend to
do better in math, but do not always
understand HOW or Why the math works.
• What are some rules that we teach
students?
• Intergers – add the opposite, two negative
make a positive (but not always)
• Dividing Fractions – Keep Change Flip
• Proportions – Cross Multiplying
• Converting Mixed Numbers and Improper
Fractions
How and Why
•Eventually, they will
probably use the “rule”
because it is faster, but they
will make fewer mistakes if
they truly understand how
the math is working.
•Turn to your neighbor
and discuss how you
usually teach adding,
subtracting, multiplying
and dividing integers.
• When I start teaching integers, I always tell
my students that after we start learning this
there will be some students that miss 3+5
and 9-7 because they are so concerned
about the rules instead of THINKING about
how the numbers make SENSE. They think
they will never miss such easy questions,
but it happens every year!!
Integers with a Number
Line or Thermometer
• + number walk forward (to the right)
• - numbers walk backward (to the left)
• “Plus” go in same direction
• “minus” turn around
One Way to Add Integers Is With
a Number Line
When the number is positive, count
to the right.
When the number is negative, count
to the left.
+
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Examples
We can use a number line to help us add positive and
negative integers.
–2 + 5 == 3
-2
3
Remember:
To add a positive integer we move forwards up
the number line.
Examples
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Examples
–3 + (–4) = –7
-7
-3
Remember:
To add a negative integer we move backwards
down the number line.
is the same as
Examples
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
•This works with Subtraction also.
Students know that – means
opposite.
•Examples
•-5-3 =
8-10=
3-(-3)=
Integers Using Algebra
tiles
• This represents a negative one.
• This represents a positive one.
• A positive one and a negative one cancel
each other and make a zero pair.
2 + 4=6
Positives plus some MORE positives equal
lots of POSITIVES.
-2 + -4=-6
Negatives plus some MORE Negatives equal
lots of Negatives.
2 + -4 = 2
Cross out the zero
pairs and whatever
left is the answer.
Students quickly
figure out that if you
are adding a positive
and a negative it is
actually a
SUBTRACTION
problem
Multiplying and Dividing
2 x 3……….This problem means “two groups
of three” =6
•
•
2 x - 3……..This problem means “two groups of
negative 3” = -6
•
•
Multiplying and Dividing
- 2 x 3……..This problem means “the
opposite of two groups of three”
=
.) - 2 x - 3…….This problem means “the
opposite of two groups of negative three”
Subtracting
•Subtracting with the tiles is a little more
difficult, but it is a good way for kids to
visually see why subtracting involves
adding sometimes.
•We will be saying “take away” for the
subtraction sign and will be using zero
pairs.
5–2=3
• Positive 5 take away positive 2
• That leaves us with positive 3
-5 – (- 2) = -3
• Negative 5 take away negative 2
• That leaves us with negative 3
5 – (-2) = 7???
I don’t have any negatives to take away. I can add as
many zeros as I want without changing the number.
Now I have two negatives to take away and I am left with
7 positives.
-5 – 2 = -7
I don’t have any positives to take away. I can add as
many zeros as I want without changing the number.
Now I have two positives to take away and I am left with
7 negatives.
Multiplying Fractions
In Primary school students associate
multiplying with numbers getting larger and
dividing with numbers getting smaller.
Multiplying proper fractions is probably the first
time the answer to a multiplying question is
LESS than what they started with.
Why bother with pictures?
• • Because we won’t help make mathematics
work for a much larger proportion of the
student body until and unless we recognize
that abstractions like 3⁄4 or rules for dividing
fractions work fine for some students but
must be grounded in pictures and models
for others.
• (Leinwand, 2009. pp. 20-21)
1/2 x 1/3
•How would you
model this?
1/2 x 1/3
•You had a birthday party and had
½ of a cake left over.
1/2 x 1/3
•Your brother ate 1/3 of the left
over cake.
1/2 x 1/3
• How much of the whole cake did your brother eat?
• 1/6 of the whole cake is shaded twice.
3/5 x 3/4
3/5 x 3/4
•9 out of 20 are double colored
•9/20
Try it #1
Try it #2
How would you model this?
Try it #1
Use the blocks (or draw) to show
Try it #2
Try it #3
Use the blocks (or draw) to show
Try it #4
How would you model this?
This is a student’s example
Try it #1
Try it #2
Converting Mixed Numbers
and Improper Fractions
What does this mean?
• The denominator is 4 which means our object (candy
bars) is cut into fourths. We have 2 whole candy bars
and ¼ of another. How many total fourths do we
have?
•
The denominator is 5 which means our object (candy
bars) is cut into fifths. We have 1 whole candy bar and
2/5 of another. How many total fifths do we have?
•
•
•
How many Wholes and how many thirds will this
•
make? 2 whole candy bars and 2/3 of
•
another.
•
•
The site to create models
• This Glencoe site allows the user to explain their
thinking using manipulatives.