Understanding Expressions
What does it mean to rewrite an expression in a different form?
Rewriting an expression means to express it in an equivalent form, often to make it easier to work with or understand. For example, you might factor an expression or expand it.
What is the result of expanding (4(x + 3))?
The expanded form is 4x + 12, using the distributive property: 4 x (x + 3) = 4x + 12
Solve the equation (2x + 5 = 13). What is (x)?
Subtract 5 from both sides:
2x = 8
Divide by 2
x=4
What are complementary angles? Give an example.
Solve (3x - 5 = 10)
Add 5 to both sides:
3x = 15
Divide by 3:
x = 5
If you have the expression (3(x +2)), what is an equivalent expression?
If you factor (6x + 12), what is the factored form?
The factored form is 6(x+2), since both terms share a common factor of 6.
If (3(x + 4) = 21), what is the value of (x)?
First, expand:
3x + 12 = 21
Subtract 12 from both sides:
3x = 9
Divide by 3
x = 3
If two angles are supplementary and one measures 70 degrees, what is the measuer of the other angle?
Supplementary angles add up to 180 degrees
180 degrees - 70 degrees = 110 degrees
If you have (4.5 + 2x = 12), what is (x)?
Subrtact 4.5 from both sides:
2x = 7.5
Divide by 2:
x = 3.75
How can rewriting (2x + 4) help you understand the relationship between (x) and the total?
Rewriting 2x+4 can help you see that the total is always 4 more than twice the value of x. It helps clarify the slope (2) and the y-intercept (4) in a linear equation.
How do you apply the distributive property to the expression (5(x + 4) - 3)?
Distribute the 5 to (x+4)
5(x + 4) = 5x + 20
5x + 20 - 3 = 5x + 17
Write an equation for a problem where a person has (p) apples and buys (q) more.
The equation could be:
p + q, where p is the number of apples they already have and q is the number they buy.
How can vertical angles help you find an unknown angle in a figure?
Vertical angles are congruent, meaning they have the same measure. If you know one of the vertical angles, the other is the same.
How would you solve the equation (2(x-3) = 8)?
2x - 6 = 8
Add 6 to both sides:
2x = 14
Divide by 2
x = 7
Why is it important to use variables in mathematical expressions?
Variables represent unknown values or quantities that can vary. They allow for generalizations and the ability to model real-world situations.
What is the sum of (2x + 3) and (4x - 5)?
Combine like terms:
(2x + 4x) + (3 - 5) = 6x - 2
How would you compare an algebraic solution to an arithmetic solution for (5x = 30)?
The alegebraic solution invloves solving 5x = 30 by dividing both sides by 5: x = 6
Write and equation to find an unknown angle (x) if one angle is 45 degrees and they are supplementary.
Solve for x:
x - 180 degrees - 45 degrees = 135 degrees
Describe how you would assess the reasonableness of the answer for the problem (5x = 25).
Check by substituting x=5x = 5x=5 back into the equation:
5(5)=25, which is true, so the answer is reasonable.
Give an example of a real-world situation where rewriting an expression clarifies the problem.
If you're buying a phone case for $20 and a screen protector for $10, the total cost is 20+10=30 Rewriting it as 20+10=3020 + 10 = 3020+10=30 clarifies the total price.
How can you use properties of operations to simplify (x + x + 2x)?
Combine like terms:
x + x + 2x = 4x
Create a word problem that leads to the equation (2x + 3 = 11)
A person has twice the number of apples as another person, plus 3 more apples. The total number of apples is 11. Find how many apples the other person has.
Equation: 2x+3=11
If two adjacent angles measure (x) and (2x), how can you find (x)?
Since the two angles are adjacent, they form a straight line, so they are supplementary:
x + 2x = 180 degrees
Solve for x:
3x = 180 degrees
x = 60 degrees
Solve the multi-step problem: A store sells pencils for (0.50) each. If you buy (x) pencils and pay a total of (5), what is the equation, and how would you solve for (x)?
The equation is:
0.50x = 5
Solve for x:
x = 5/0.50=10