Download multiplication, division, area, perimeter

Algebraic and
Geometric
Thinking
Quantity/
Magnitude
Proportional
Reasoning
Numeration
Language
Form of a
Number
Equality
Base Ten
The Components of Number Sense
© 2007 Cain/Doggett/Faulkner/Hale/NCDPI
Students don’t see the
connection between X , ÷
and geometry when we
inundate them with formulas
If they understand these
relationships they will become
fluent in area and perimeter
problems
Note the connection between
the linear measurements and
the area measurements.
What we say now:
Okay, what formula are you going to use?
A = l*w
But it’s a square right? So what formula
do we use for a square?
A = s*s
Good! Now plug in what you know and…
Our new habit:
Remember, you all already understand the
relationships between one dimension and
two dimensions.
14
Let’s look at these again:
6
We know from our work
With multiplication that if we know these two
lengths we know how many little one x one
squares are in here, right? These two factors
give us this product.
Arrays: Build the fundamental understanding
of these relationships and watch them connect
and grow!
16 linear units long
7
linea
r
units
long
112 little one by one squares!
Link Mult/Div to Geometry!
This is deepening knowledge
But look, this is Geometry! Our two
factors tell us how long the linear sides
are, right?
And what about our product? That is our
AREA. We measure AREA in 1x1
squares. If we say 1 square unit, that’s
exactly what we mean!
What we do now:
Area is the thing on the inside; rug;
carpet, etc.
Perimeter is the measurement around;
fence;string, etc.
This is good but it is not precise enough
to have clear impact on student thinking
about measurement.
Thinking in dimensions:
the measurement/geometry link
Teachers should always have a piece of
string one unit long, and a square that is 1 x
1 unit long available!
Hold these up whenever talking about area
and perimeter or area versus perimeter
Do we mean 3 of these? Or 3 of these?
How do you know?
Fluency - Practice lots of these!
So if I know this and this
What else can I figure out?
7
3
7
Great, now what if I knew this
21
And this?
I have the Area and one side length.
Can I figure out the other side length?
Yes! Multiplication and Division help us go
back and forth depending on what we need!
Symbolic Level - Make connection
So what we have been doing the past
week is figuring out area and perimeter
because we understand how they are
related.
Mathematicians write this out with symbols,
but they just mean what we already know
how to do!
Area = base * height
Symbolic Level - BIG IDEA
Area = base * height
You will see lots of different ways to write this. You need to
always go back to what you know: it’s just what we do with
our multiplication arrays!
A = l*w
A = s*s
A = b*h
A = 1/2 b*h
Don’t be fooled! It is all the same big idea!
Practice: pairing symbolic with representational (arrays).
Arrays
Distributive property
Base-ten
16 x 7
10
7
10*7
6
6*7
Arrays and
distributive property
(X + 6) 7
(X
7
X*7
+
6)
6*7
Several Warm-ups where
students move from problems in
the form of 7(x + 6) to 7x + 42.
Simplifying.
And also from the form of 7x + 42 back
to the area model and 7 (x +6).
Factoring.
Arrays and
distributive property
(X + 6) 7
(X
7
X*7
+
6)
6*7
We know that we have 7x + 42. Still don’t know
How much the 7x is because we are only given
One dimension (the 7) and not the other (x).
Finally, Problems that have students solve for x given
what they know.
For instance, give them the area of a rectangle, one
dimension and part of the other dimension and solve
for x.
7(x + 6) = 77.
Draw out the rectangle and fill in everything you
know. What does the x have to be in order for the
area to be 77? Figure this through guess and check.
Students will build on this algebraically by the end of
the course.
Finally, more Warm-ups that have students figure this
out using the algebra they learn from Hands-onEquations AND modeling it using geometry (these
rectangles) should be put into Units 5 and 6.
Arrays and
distributive property
(X + 6) 7
(X
7
X*7
+
6)
6*7
But what if we knew the whole area was 77? Could we
Work backwards to figure out what x is?