Problems in Mathematical Terms 05:39 minutes

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Transcript Problems in Mathematical Terms

Every week, Wanda the Witch forages in the forest for ingredients for her magic spells. Looks like Wanda needs a longer runway for when she wants to land her magic broom, so she finds a golem spell that'll help her conjure a strong and friendly creature to help her clear some trees for her runway. But, in order to figure out how to create her new golem friend, she’ll have to know about Problems in Mathematical Terms. The magic book states that golems have certain characteristics that are constant and some characteristics that can be changed. A constant is a value that stays the same and does not change. Let's take a look at some of the constant characteristics of golems. Every golem has two eyes, four fingers, and a 3-inch top hat. On the other hand, some of the characteristics of golems are not always the same. For example, there are golems of different sizes and strengths. This means that size and strength are variables. A variable is a value that changes, or varies. The size of a golem is based on how many eyes of newt she uses, and the strength depends on how many bloodroots she uses. The number of eyes of newt and bloodroots are also variables, since they can vary as well. In fact, Wanda can choose them! Since Wanda can freely decide what values to use for the number of eyes of newt and bloodroots, these are called independent variables. An independent variable is one which can be freely chosen and doesn't depend on any other variable. On the other hand, size and strength are dependent variables. Dependent variables are determined by what is chosen for the independent variables. Wanda gathers all of her ingredients, dreaming of her new friend, but first she has to decide how big and strong she wants her golem to be. Variables are usually represented by letters. When deciding on which letters to use, we can freely choose them, but it makes sense to choose one that fits the variable's name. Let's call the number of eyes of newt 'n' and the golem's height 'h'. Her magic book tells her that for every eye of newt she adds, the golem will be three feet taller. We can express what we know in an equation by multiplying 3 by 'n', which gives us 'h'. H equals 3 times n. Earlier, Wanda gathered 6 eyes of newt. But how many should she add to her potion? Let's see if we can make a table with what we know. In our table, the independent variable, 'n', can change without being influenced by anything else in the equation, but the dependent variable, 'h', changes everytime there is a change in 'n'. For example, if 'n' equals 1, then 'h' equals 3 times 1, or 3. If 'n' equals 2, then 'h' equals 3 times 2, or 6, and so on. Wanda thinks 12 feet is a good height for a golem, so she drops 4 eyes of newt into her potion. Her golem needs to be strong if he's gonna help her clear trees. It's a good thing she has plenty of bloodroots! For every bloodroot Wanda adds to the potion, the golem will gain the strength of 5 men. It makes sense to use the letter 'b' to represent bloodroot, and 's' to represent strength. That gives us the equation 's' equals 5 times 'b'. Let's see how many bloodroots she should add. Remember, the number of bloodroots in the potion is the independent variable, and the golem's strength is the dependent variable. If Wanda adds just one bloodroot, the golem will have the strength of 5 men, two bloodroots and the golem's strength increases to that of 10 men, and so on. Since Wanda needs the golem to remove trees for her. She adds 5 bloodroots to the potion, which should give the golem the strength of 25 men! Okay, let's review what we've learned. In mathematical equations, there are two types of terms: constants and variables. There is only one type of constant, but there are two types of variables: independent and dependent. The independent variables in our story were the eyes of newt and bloodroots, since they could be chosen freely. The dependent variables were size and strength, since they were determined by the choice of independent variables. Let's get back to Wanda and see what's cookin'. Looks like things didn't go as planned... But she still likes her new friend.

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  • Identifying independent and dependent variables to write and evaluate equations.

    Tipps

    The independent variable is freely chosen and is not determined by any other variable.

    The dependent variable changes every time there is a change to the independent variable.

    For example: If color of a Golem is 4 times the number of pumpkins, the equation can be represented by $c=4p$ where $c$ is the color of the Golem and $p$ is the number of pumpkins.

    Lösung

    Independent variables are chosen freely and can change without the influence of anything else in the equation. The examples of independent variables used here are:

    • eyes of newt
    • bloodroots
    Dependent variables change every time there is a change with the independent variable. This means they depend on the independent variable. The examples of dependent variables used here are:
    • size of the Golem
    • strength of the Golem
    Magic Spell #1: For every eyes of newt Wanda has, the Golem will be 3 feet tall.
    • The equation is $h=3n$
    • If $n=1$ then $h=3$
    • If $n=4$ then $h=12$
    • Using the equation $h=3n$ we substitute $12$ for height and recognize that means Wanda needs $4$ eyes of newt for her potion.
    Magic Spell #2: For every bloodroot Wanda adds to the potion, the Golem will gain the strength of 5 men.
    • The equation is $s=5b$.
    • If $b=1$ then $s=5$.
    • If $b=5$ then $s=25$.
    • Use the equation $s=5b$ and substitute $5$ for bloodroots to give Wanda's Golem the strength of $25$ men.

  • Determine which of the following are independent or dependent variables, or constant terms.

    Tipps

    In mathematics, a constant term is a term in an algebraic expression that has a value that is constant or cannot change. This is because it stands alone, and does not contain any variables.

    Independent variables are freely chosen and do not depend on any other variable.

    Dependent variables are determined by the independent variables.

    Lösung

    1. Constant Term

    • $3$ in $y=2x+3$
    • $3$ is the only term that never changes in the equation and therefore called a constant term. No matter what the values $x$ and $y$ are, 3 will never change.
    • The $2$, in $y=2x+3$, is the coefficient of $x$.
    2. Dependent Variable
    • The $y$ in $y=6x$.
    • The $y$ is the dependent variable because it changes based on $x$.
    • For example, if we choose $x$ to be 2, then $y=6(2)$ which is $12$.
    • If we choose $x$ to be $-3$, then $y=6(-3)$ which is $-18$.
    • Therefore, $y$ is dependent on the independent variable, $x$.
    3. Independent Variable
    • The $x$ in $y=7x+1$.
    • The $x$ is called the independent variable because we can freely choose its value and it is not determined by any other variable.
    • For example, we can choose $x$ to be $1$, $\frac{1}{2}$, $-2$, and so on, to find as many values as we want for $y$.
    4. Variables
    • The $x$ and $y$ in $y=-2x+9$.
    • The $x$ and $y$ are called variables because they can change or vary.

  • Decide which statements about variables and constants are true.

    Tipps

    In the equation, $h=3n$, $n$ is an independent variable because it can be freely chosen.

    In the equation,$ y=x+1$, $1$ is a constant.

    In the equation $h=3n$, $h$ is the dependent variable because it depends on the independent variable, $n$.

    Lösung

    A constant is a value that stays the same and does not change.

    Independent variables can be freely chosen and do not depend on any other variable.

    Dependent variables are determined by independent variables.

    Wanda's magic book states:

    • The eyes, fingers and hat of a Golem never change.
    • A Golem's size is based on the number of eyes of newt.
    • A Golem's strength is based on the number of bloodroots.
    The following statements are true:
    • The size and strength of a Golem are dependent variables because size and strength are determined on the number of eyes of newt and bloodroots that Wanda chooses.
    • The eyes, fingers and hat on a Golem are constants because they never change.
    • Bloodroots and eyes of newt are independent variables because they are freely chosen.
    • There are two types of variables: independent and dependent.
    The following statements are false:
    • Independent variables are determined by dependent variables is false because dependent variables are determined by independent variables.
    • The size of a golem is an independent variable is false because the size of a Golem is a dependent variable.
    • Constants can change is false because constants never change.

  • Set up the linear equation.

    Tipps
    • A constant term is a number that never changes.
    • The independent variable is chosen freely.
    • The dependent variable is determined by the independent variable.

    The cost of $\$ 3$ per shirt plus a $\$ 2$ donation is represented by the equation $c=3s+2$, where $c$ is the cost and $s$ is the number of shirts.

    The cost of $\$ 0.99$ per mango plus a $\$ 2$ discount is represented by the equation $c=0.99m-2$ or $c=0.99m+-2$, where $c$ is the cost and $m$ is the number of mangos.

    Lösung

    1. $c=2d+175$

    • The dependent variable is $c$ because it depends on $d$.
    • The independent variable is $d$ because it can be freely chosen.
    • The coefficient of the independent variable is $2$.
    • The constant term is $175$ because it never changes.

    2. $c=1.25t+10$

    • The dependent variable is $c$ because it depends on $t$.
    • The independent variable is $t$ because it can be freely chosen.
    • The coefficient of the independent variable is $1.25$.
    • The constant term is $10$ because it never changes.

    3. $c=1.50p+-5$

    • The dependent variable is $c$ because it depends on $p$.
    • The independent variable is $p$ because it can be freely chosen.
    • The coefficient of the independent variable is $1.50$.
    • The constant term is $-5$ because it never changes.

  • Identify the dependent and independent variables.

    Tipps

    The independent variable is freely chosen and is not determined by any other variable.

    The dependent variable is determined by the independent variable.

    In the equation $k=2t$, $t$ is the variable that is freely chosen, and $k$ is the variable that depends on $t$.

    Lösung

    1. $d=2t$

    • $d$ is the dependent variable because the donation amount is determined by how many tickets they sell. $t$ is the independent variable because Ella and Jahmal can choose how many tickets they want to sell.

    2. $c=3.50b$

    • $c$ is the dependent variable because the cost depends on how many boxes, $b$, the independent variable, Antonio buys.
    3. $f=4p$
    • $f$ is the is the dependent variable because the number of flowers depends on how many pots, $p$, the independent variable, Maya buys.

  • Find the parts of the story which correspond to different mathematical terms.

    Tipps

    Mi Xuan is a photographer. She charges a one time sitting fee of 30 dollars. Then it costs 15 dollars per photo that you buy. The total cost for photos by Mi Xuan can be found by creating an equation.

    • The independent variable in this situation is the the number of photos you choose. This is represented by per photo in the description.

    Mi Xuan is a photographer. She charges a one time sitting fee of 30 dollars. Then it costs 15 dollars per photo that you buy. The total cost for photos by Mi Xuan can be found by creating an equation.

    • The dependent variable in this situation is the the total cost because it depends how many photos you choose to buy.

    Mi Xuan is a photographer. She charges a one time sitting fee of 30 dollars. Then it costs 15 dollars per photo that you buy. The total cost for photos by Mi Xuan can be found by creating an equation.

    • The constant term in this situation is the the one time sitting fee of 30 dollars because it doesn't change.
    Lösung

    1. Kai is doing a science experiment for school. She starts with water that is $75^{\circ}$F. Every time she heats the water in the microwave the overall temperature. increases 10 degrees per minute.

    • per minute is the independent variable because Kai can choose how many minutes she wants to heat the water in the microwave.
    • overall temperature is the dependent variable because the temperature of the water is determined by how many minutes Kai heats the water.
    • 75 is the constant term because the original temperature of the water never changes.
    2. Jackson wants to purchase a gym membership. He doesn't know the total cost of the membership. However, he does know there is a one time $10$ dollar registration fee, and a monthly fee of $25$ dollars.

    • monthly fee is the independent variable because Jackson can choose the number of months he would like a gym membership.
    • total cost of the membership is the dependent variable because the membership cost is determined by how many months Jackson uses the gym.
    • 10 is the constant term because it is a one-time membership fee that never changes.