Transcript Day-2-True-False-and-Open-Sentences

True, False, and Open Sentences
An introduction to algebraic equations,
also called open sentences
5+
= 13
Is this equation true or false?
5+
= 13
Mathematical sentences like this one are called open
sentences. They’re neither true nor false, because
there’s a part of the sentence – the box in my equation
– that isn’t a number. The box is called a variable,
because you can vary what number you put into it or
use to replace it.
5+ 8
= 13
How can we make it true?
7·6=
Is this an open sentence?
-4=3
Is this an open sentence?
-4=3
-4=3
I can use a triangle for my variable.
I can also use a square for my variable.
x –4=3
Is this an open sentence?
-4=3
-4=3
x
–4=3
I can use a triangle for my variable.
I can also use a square for my variable.
I can also use the letter “x” for my variable.
+
=
10
How is this sentence different than the
others?
5 +
5
=
10
When you have the same shape, the
same number must go in each shape.
5
+
5
+
5
x
=
10
=
10
=
10
5
+
x
When you have the same shape, or
letter, they must represent the same
number. What does x equal? X = 5
+
=
10
When you have different shapes, the
numbers can be different BUT they
can also be the same.
0 + 10, 1 + 9, 2 + 8, 3 + 7, 4 + 6, 5 + 5,
6 + 4, 7 + 3, 8 + 2, 9 + 1, 10 + 0
x
+
+
y
=
=
10
10
When you have different letters, the
numbers can be different BUT they
can also be the same.
0 + 10, 1 + 9, 2 + 8, 3 + 7, 4 + 6, 5 + 5,
6 + 4, 7 + 3, 8 + 2, 9 + 1, 10 + 0
(6x
) + 8 = 38
=
5
6x(
+ 8) = 78
=
5
( 6 x 5) + 8 = 38
6 x ( 5 + 8) = 78
Same numbers on the left but the answers on
the right are very different. Parentheses help
you to know the order of operations for a
problem.
Make two open ended sentences for your
partner to solve using a variety of variables and
formats (shapes, letters, parenthesis, etc.)
Pick a Number
Revisiting Open Ended Sentences
Pick a number from 0 – 25 (
Now
Multiply by two
Then add seven
(
· 2) + 7 =
)
(
· 2) + 7 = 57
What is x?
Guess and check
“Undoing”
(
· 2) + 7 = 57
Guess and check
( 22 · 2) + 7 = 51 Too small
Too Small
20
21
22
23
24
25
26
27
28
(
· 2) + 7 = 57
Guess and check
( 26 · 2) + 7 = 59 Too big
Too Big
Too Small
20
21
22
23
24
25
26
27
28
(
· 2) + 7 = 57
Guess and check
( 24 · 2) + 7 = 55 Too Small
Too Big
Too Small
20
21
22
23
24
25
26
27
28
(
· 2) + 7 = 57
Guess and check
( 25 · 2) + 7 = 57 Too Small
Too Big
Too Small
20
21
22
23
24
= 25
25
26
27
28
(
· 2) + 7 = 57
Undo the seven I added to it
(
· 2) = 57 – 7
(
· 2) = 50
Undo the 2 I multiplied it by
= 50 ÷ 2
= 25
• Equations are mathematical sentences with
equal signs.
• In an equation, the expressions on either side
of the equals sign name the same quantity.
•
•
•
•
•
3+4=7
6 x 7 = 42
9 + 7 = 10 + 6
3 x 8 = 32 – 8
10 – 6.5 = 3.5
Inequalities – sentences that do not have an
equal sign
• 3 + 9 ≠ 10
• 7x4>8x3
• 3+4<3+5
More equations
(5 x 2) + 6 = 16
5 x (2 + 6) = 40
The numbers and operations on the left side are
the same, but the quantities on the right side
are different. The parentheses used in math
sentences indicate that you should perform
the operations within the parentheses first.
Order of Operations
If no parentheses are included, then the mathematical
convention called “order of operations” must be
applied. This convention says to first perform all
multiplication and division in order from left to right,
and then to perform all addition and subtraction in
order from left to right.
3 + 2 x 5 = 13
2 + 6 x 6 ÷ 2 = 20
While parentheses aren’t necessary in these
equations for students who understand the
convention for the order of operations, they are
useful to include for clarity:
3 + 2 x 5 = 13
3 + (2 x 5) = 13
2 + 6 x 6 ÷ 2 = 20
2 + (6 x 6 ÷ 2) = 20
But if the parentheses were placed differently in
these equations, the quantities on the right side
of the equal signs would change.
3 + 2 x 5 = 13
2 + 6 x 6 ÷ 2 = 20
3 + (2 x 5) = 13
2 + (6 x 6 ÷ 2) = 20
(3 + 2) x 5 = 25
(2 + 6) x 6 ÷ 2 = 24
Another mathematical convention that is useful
for students to learn is to use a dot to represent
multiplication.
5x3 = 5·3
Using the dot is especially helpful with algebraic
equations to avoid confusing an x used as a
variable with an x used to indicate
multiplication.
A difference between arithmetic equations and
algebraic equations is that arithmetic
equations are either true or false. Algebraic
equations, however, involve variables, and
therefore aren’t true or false.
An equation such as x + 3 = 5, for example,
becomes true if you replace the variable, x,
with 2. Equations like this are also called open
sentences, since they’re open to a decision
about whether they’re true or false until you
decide on the value for the variable.
Are these equations true or false?
8+4=5+7
5=4+1
6·0=6
Are these equations true or false?
7 + 8 x 2 = 15 + 10 + 5
3 ÷ 3 x 4 + 15 = 100 ÷ 4
5 ÷ 10 + 4 = 9 ÷ 2
Write three grade appropriate equations that
are true and three that are false in your journals.