Evaluating Expressions

Evaluating Expressions exercise

• Calculate the length of hair after given number of months.

  Hints

  You can use a linear function to represent the length of Rap-Punzel's hair. The unknown quantity is the number of months.

  Let's look at another example: The amount of money you receive on your birthday increases by $\$5$ each year. You start with $\$15$.

  • After one year, you receive $\$15+\$5=\$20$
  • After two years, you receive $\$20+\$5=\$25$

  You have to multiply the number of months that have passed by the rate her hair grows, then add $50$.

  Solution

  In the beginning, Rap-Punzel's hair is $50$ inches long. Because her hair grows $5$ inches per month, we know that Rap-Punzel's hair grows as follows:

  • After one year: $50+5=55$ inches
  • After two years: $55+5=60$ inches
  • After three years: $60+5=65$ inches
  • After four years: $65+5=70$ inches
  • After five years: $70+5=75$ inches
  • After six years: $75+5=80$ inches

  We can plug the values into a function table:

  $\begin{array}{c|c|c|c|c|c|c} x&1&2&3&4&5&6\\ \hline \text{length}&55&60&65&70&75&80 \end{array}$

• Find an equation that represents Rap-Punzel's hair length after $x$ months.

  Hints

  Rap-Punzel's hair grows $5$ inches per month. He hair has grown:

  • 10 inches after $2$ months
  • 15 inches after $3$ months

  To avoid split ends, Rap-Punzel has to trim her hair. This means she has to cut her hair. Is her hair getting longer or shorter?

  You can simplify expressions and equations by combining like terms. For example: $2$ apples plus $3$ bananas plus $4$ apples results in $6$ apples and $3$ bananas.

  Solution

  Poor Rap-Punzel and poor Prince MC. Rap-Punzel's hair grows $5$ inches per month. But she has split ends. To avoid getting split ends, she has to cut her hair $2$ inches per month.

  How can we write this as a mathematical expression?

  • Let's represent the rate of growth: $5x$
  • Next,write the expression to represent Rap-Punzel trimming her hair: $-2x$
  • Finally, we add $10$

  Together we have $5x-2x+10$.

  We can combine the like terms to get $3x+10$ as our final expression.

• Evaluate how long Rap-Punzel's hair is after one year.

  Hints

  First, determine the expression that represents the length of Rap-Punzel's hair after $x$ months. $x$ can be any number of months: $1$, $2$, $3$, ...

  Rap-Punzel's starting hair length is the constant.

  To write the simplified expression, the coefficient can be found by determining the difference between the growth rate of Rap-Punzel's hair minus the monthly trimming in order to avoid split ends.

  Solution

  First, let's write an expression representing the length of Rap-Punzel's hair after $x$ months:

  • To start, Rap-Punzel's hair is $10$ inches long. This is our constant.
  • The variable is the unknown quantity of months, which we'll call $x$.
  • The coefficient can be found by calculating the net growth per month of Rap-Punzel's hair. In this case, her hair grows $5$ inches per month, but she cuts $2$ inches each month as well. So our coefficient becomes $5-2=3$.

  Combining this information, we have the expression: $3x+10$.

  To determine the length of Rap-Punzel's hair after one year, we can plug in $x=12$.

  $\begin{array}{rcl} 3(12)+10&=&\\ 36+10&=&46 \end{array}$

  So after one year, Rap-Punzel's hair is $46$ inches long.

  The tower is a BIT taller than $46$ inches... Poor Prince MC.

• Determine how long it takes until the magic beanstalk grows to a height of $100$ inches.

  Hints

  The expression is:

  coefficient $\times x +$ constant.

  To isolate the variable $x$, use opposite operations:

  • Multiplication ($\times~\longleftrightarrow~\div$)
  • Division ($\div~\longleftrightarrow~\times$)
  • Addition ($+~\longleftrightarrow~-$)
  • Subtraction ($-~\longleftrightarrow~+$)

  Check your solution by plugging in your value for $x$:

  $0.5x + 20$

  Did the expression simplify to $100$?

  Solution

  Waiting for Rap-Punzel's hair to grow takes too long and learning how to climb is too expensive, so Prince MC decides to buy some magic beans to grow a plant. The beanstalk has:

  • an initial height of 20 inches - this is the constant
  • a growth rate of 0.5 inches per day - this is the coefficient
  • the variable $x$ to the unknown number of days

  So we can write the expression: $0.5x+20$.

  To figure out the number of days the beans need to reach $100$ inches, we must solve the equation for $x$:

  $0.5x+20=100$

  We use Opposite Operations to isolate $x$:

  $\begin{array}{rclcl} 0.5x + 20y &=& 100 \\ \color{#669900}{-20} && \color{#669900}{-20} \\ 0.5x &=& 80 \\ \color{#669900}{\times 2} && \color{#669900}{\times 2} \\ x &=& 160 \end{array}$

  So, after $160$ days, more than $5$ months, the magic beanstalk will be $100$ inches high.

  Is this tall enough to reach Rap-Punzel?

• Label the different parts of the expression.

  Hints

  A constant is independent of the variable.

  The term above represents the length of Rap-Punzel's hair after $x$ months.

  In the beginning, Rap-Punzel's hair is $50$ inches long.

  Solution

  This expression represents the length of Rap-Punzel's hair after $x$ months.

  • The coefficient to the variable is the rate of growth, $5$.
  • $x$ represents the unknown number of months.
  • Finally, our constant is the starting length of her Rap-Punzel's hair in inches, $50$.

• Decide which function table belongs to which equation.

  Hints

  To match to the correct function table, make sure more than $x$-$y$ pair satisfies the equation.

  For each equation on the right, plug in several different $x$s and compare the answers.

  Solution

  A function table is a useful tool used to set up a linear equation.

  This is what it looks like:

  $\begin{array}{c|c|c|c} x& & & \\ \hline f(x)&&& \end{array}$

  • You can plug in different values for the variable $x$ and check the corresponding $f(x)$.

  Let's look at some examples:

  • $f(x) = 3x + 4$

  Try out $x = 3$. Plug in $3$ for $x$, and we get:

  $\begin{array}{rcl} 3(3) + 4 &=& \\ 9 + 4 &=& 13 \end{array}$

  So our function table should look like this:

  $\begin{array}{c|c|c|c|c|c} x& 1&2 &3&4&5 \\ \hline f(x)&7&10&13&16&19 \end{array}$

  $~$

  • $f(x) = 4x + 3$

  $\begin{array}{c|c|c|c|c|c} x& 1&2 &3&4&5 \\ \hline f(x)&7&11&15&19&23 \end{array}$

  $~$

  • $f(x) = 2x + 5$

  $\begin{array}{c|c|c|c|c|c} x& 1&2 &3&4&5 \\ \hline f(x)&7&9&11&13&15 \end{array}$

  $~$

  • $f(x) = 5x + 2$

  $\begin{array}{c|c|c|c|c|c} x& 1&2 &3&4&5 \\ \hline f(x)&7&12&17&22&27 \end{array}$