Transcript The Four Operations & Diagrams SUBTRACTION

Foundations of Algebra
Unit 1 Lesson 1
Math Basics &
Diagrams
Lesson Objectives:
• Have students understand BASIC rules of mathematics
through the use of simple diagrams such as number
lines, basics shapes and plus & minus signs.
Have students reason and make logical sense of
simple diagrams that model basic mathematical rules.
Have students use diagrams that represent their way of
thinking (how they see the problem), as well as to
understand how other students think (how others see
the problem).
AF 1.3
Introduction
How would you show the math
expression 3+2 using basic shapes?
Depending on how your brain is wired, here are two
possible ways of seeing this problem using circles:
+
and
+
Both diagrams show the sum of 3 and 2.
Terminology
DIAGRAM
A diagram is a pictorial representation of some kind of
idea, concept, or math problem. The diagram describes
the math problem.
EXPRESSION
An expression is a math problem that is written using
numbers, operations, letters and symbols. There are two
kinds of expressions:
• Numeric (numerical) expression: A math problem that
contains only numbers and operations.
• Algebraic expression: A math problem that contains
numbers, operations, letters and/or symbols.
Terminology (Continued)
TERM
A term is a value in a math problem separated by an
operation. For example, in 3+2, the number 3 is a term
and the number 2 is also a term.
INTEGERS
Integers are all numbers & their opposites. The number
–6, for example, is an integer because it has an opposite
value, 6.
The Four Operations & Diagrams
ADDITION
To add means to combine terms that are alike. For
example 6 + 5 means we are going to combine 6 of
something with 5 of the same kind of something:
+
Is the answer different if the numbers were –6 + –5?
The answer would be negative, but the diagram would be the
same because we are combining two like-terms, only negative.
The Four Operations & Diagrams
SUBTRACTION
To subtract means to combine OPPOSITE terms. For
example, 5 – 3 can be shown like this:
+
+
–
One blue circle is the opposite of one red circle, so when you
combine them, they cancel each other out.
After all opposites have canceled out, the answer remains.
The answer to 5 – 3 is 2.
The Four Operations & Diagrams
MORE ON SUBTRACTION
You can rewrite a subtraction problem using a plus sign. For
example, 5 – 3 can be rewritten as 5 + – 3. Make sure that
the negative sign travels with its number, though!
+
–
–
–
+
+
Notice in both diagrams the opposite values cancel each other
out even though we have two different diagrams.
The idea here is to remember that opposite values cancel
each other out and 5 pluses added to 3 negatives equals 2.
Double Signs
It’s not a good to have consecutive signs, or double signs,
because they can confuse you.
For example, find the value of 4 – – 5.
If you are lost, then use a diagram:
The answer you get is –1, but
THAT IS NOT CORRECT
because you didn’t remove
the double negatives.
+
+
–
–
–
+
Consecutive negatives cancel each other to become a
positive.
This means that the red circles actually become blue and they
are added to the other blue circles.
The final answer is 9, NOT –1.
Remove Double Signs
To remove double
signs, use the Alien
Face (two negatives
turn into a positive).
The Alien Face is also
used when multiplying
or dividing numbers
with different signs.
▬
▬

The Four Operations & Diagrams
MULTIPLICATION
To multiply means to take a value and add it to itself as many
times as the other number indicates.
Let’s make a diagram that shows the expression 3 x 2. This
can be shown two ways:
x
=
+
2 groups
of 3
or
+
+
The answer is 6
3 groups of 2
The Four Operations & Diagrams
DIVISION
To multiply means to take a value and make a specified
number of groups containing the indicated amount of values.
Let’s make a diagram that shows the expression 12 ÷ 4.
In other words, how many groups of 4 can we make with 12?
=
According to our diagram,
we can make three groups
of 4 if we have 12.
Making Diagrams
To make a mathematical diagram, you can use circles,
stars, or any shape you wish.
The purpose of this unit is to present you with a familiar
topic in a way that may help you make better sense of it.
Example: Create a simple diagram showing 3 + 5:
 + 
or
 + 
Remember, we are combining like-terms, so the shapes
are the same.