Hearts and Stars!

If  +  +  +  +  = 18

And  +  = 8

What is the value of one ? What is the value of one?

Correct or Incorrect?

For each of the following, a student evaluated the given expression. Your job is to determine whether the student is correct or not. If you think the student is wrong, explain the mistake and provide the correct answer.

  1. Fiona says that
  2. Malia says that
  3. Kevin says that
  4. Tyler says that
  5. Carlos says that
  6. Jennifer says that
  7. Lillian says that

What’s the Number?

In each equation below, x can be replaced with a number that makes the equation true. Find such a number.


Quotient Rule

In this problem represents a positive number.

The quotient rule for exponents says that if and are positive integers with , then

.

After explaining to yourself why this is true, complete the following exploration of the quotient rule when :

  1. What expression does the quotient rule provide for when ?
  2. If , simplify without using the quotient rule.
  3. What do parts (a) and (b) above suggest is a good definition for ?
  4. What expression does the quotient rule provide for ?
  5. What expression do we get for if we use the value for found in part (c)?
  6. Using parts (d) and (e), propose a definition for the expression .

Delivery Trucks

A company uses two different-sized trucks to deliver sand. The first truck can transport cubic yards, and the second cubic yards. The first truck makes trips to a job site, while the second makes trips. What quantities do the following expressions represent in terms of the problem's context?

Moving Parentheses!

Evaluate the following numerical expressions.

  1. )
  2. )

Can the parentheses in any of these expressions be removed without changing the value of the expression?

What’s the Difference?

Write an expression for the sequence of operations.

  1. Add 3 to , subtract the result from 1, then double what you have.
  1. Add 3 to , double what you have, then subtract 1 from the result.

Tell the Truth!

Decide whether each equation is true or false, and explain how you know.


Amusement Park Tickets

Malia is at an amusement park. She bought 14 tickets, and each ride requires 2 tickets.

  1. Write an expression that gives the number of tickets Malia has left in terms of , the number of rides she has already gone on. Find at least one other expression that is equivalent to it.
  1. represents the number of tickets Malia has left after she has gone on rides. How can each of the following numbers and expressions be interpreted in terms of tickets and rides?

14 −2

  1. also represents the number of tickets Malia has left after she has gone on rides. How can each of the following numbers and expressions be interpreted in terms of tickets and rides?

7 2

Blueprint

Maggie is working from a blueprint in which 3 inches represents 14 feet. Write an equation that represents the relationship between the actual lengths (a), in feet, and the blueprint lengths (b), in inches?