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 + = 13 How can we make it true? 8

7 · 6 = Is this an open sentence?

- 4 = 3 Is this an open sentence?

- 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 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?

+=10 When you have the same shape, the same number must go in each shape. 5 5

+=10 x+x=10 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 , 1 + 9, 2 + 8, 3 + 7, 4 + 6, 6 + 4, 7 + 3, 8 + 2, 9 + 1, ,

+=10 x + y = 10 When you have different letters, the numbers can be different BUT they can also be the same , 1 + 9, 2 + 8, 3 + 7, 4 + 6, 6 + 4, 7 + 3, 8 + 2, 9 + 1, ,

( 6 x ) + 8 = 38 = 5

6 x ( + 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

( · 2) + 7 = 57 Guess and check ( 26 · 2) + 7 = 59 Too big Too Small Too Big

( · 2) + 7 = 57 Guess and check ( 24 · 2) + 7 = 55 Too Small Too Small Too Big

( · 2) + 7 = 57 Guess and check ( 25 · 2) + 7 = 57 Too Small Too Small Too Big = 25

( · 2) + 7 = 57 Undo the seven I added to it ( · 2) = 57 – 7 ( · 2) = 50 = 25 Undo the 2 I multiplied it by = 50 ÷ 2

Equations are mathematical sentences with equal signs. In an equation, the expressions on either side of the equals sign name the same quantity = 7 6 x 7 = = x 8 = 32 – 8 10 – 6.5 = 3.5

Inequalities – sentences that do not have an equal sign ≠ 10 7 x 4 > 8 x < 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 x 5 = 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: x 5 = (2 x 5) = x 6 ÷ 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 x 5 = x 6 ÷ 2 = (2 x 5) = (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. 5 x 3 = 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? = = · 0 = 6

Are these equations true or false? x 2 = ÷ 3 x = 100 ÷ 4 5 ÷ = 9 ÷ 2

Write three grade appropriate equations that are true and three that are false in your journals.