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Section 1

Introducing the Language of Algebra

Fill in the blanks.A _____ is the result of an addition. A _____ of a subtraction. A _____ is the result of a multiplication. A ____ is the result of a division.

Fill in the blanks._____ are letters (or symbols) that stand for numbers.

Fill in the blanks.A number, such as $8,$ is called a ______ because it does not change.

Fill in the blanks.Variables and numbers can be combined with the operations of addition, subtraction, multiplication, and division to create algebraic ______

Fill in the blanks.An ______ is a mathematical sentence that contains an $=$ symbol. An algebraic ______ does not.

Fill in the blanks.An equation such as $c=10 \mathrm{m},$ which expresses a relationship between two or more variables, is called a ______

Fill in the blanks.The _____ axis of a graph extends left and right and the vertical axis extends up and down.

Fill in the blanks.The word _____ comes from the title of a book written by an Arabian mathematician around A.D. 800 .

Classify each item as an algebraic expression or an equation.a. $m+18=23 \quad$ b. $m+18$

Classify each item as an algebraic expression or an equation.a. $30 x \quad$ b. $30 x=600$

Classify each item as an algebraic expression or an equation.a. $\frac{c-7}{5} \quad$ b. $\frac{c-7}{5}=7 c$

Classify each item as an algebraic expression or an equation.a. $r=\frac{2}{3} \quad$ b. $\frac{2}{3} r$

What arithmetic operations does the expression $\frac{12+9 t}{25}$ contain? What variable does it contain?

What arithmetic operations does the equation $4 y-14=5(6)$ contain? What variable does it contain?

Construct a line graph using the data in the following table.$\begin{array}{|c|c|}\hline \text { Hours } & {\text { Pay }} \\ \hline \text { worked } & {\text { (dollars) }} \\ \hline 1 & {20} \\ \hline 2 & {40} \\ \hline 3 & {60} \\ \hline 4 & {80} \\ \hline 5 & {100} \\ \hline\end{array}$(GRAPH CAN'T COPY)

Use the data in the graph to complete the table.$\begin{array}{|c|c|}\hline & {\text { Depth }} \\ \hline \text { Minutes } & {\text { (feet) }} \\ \hline 0 & {} \\ \hline 5 & {} \\ \hline 10 & {} \\ \hline 15 & {} \\ \hline 20 & {} \\ \hline\end{array}$(GRAPH CAN'T COPY)

Fill in the blanks. The symbol $\neq$ means ______ ______ ______ ______.

Fill in the blanks. The symbols ( ) are called ______

Fill in the blanks. Write the multiplication $5 \times 6$ using a raised dot and then using parentheses.

Fill in the blanks. Give four verbs that can be represented by an equal symbol $=$

Write each expression without using a multiplication symbol or parentheses.$$4 \cdot x$$

Write each expression without using a multiplication symbol or parentheses.$$P \cdot r \cdot t$$

Write each expression without using a multiplication symbol or parentheses.$$2(w)$$

Write each expression without using a multiplication symbol or parentheses.$$(x)(y)$$

Write each division using a fraction bar.$$32 \div x$$

Write each division using a fraction bar.$$3 0 \sqrt{ 9 0 }$$

Write each division using a fraction bar.$$5 \sqrt{ 5 5 }$$

Write each division using a fraction bar.$$h \div 15$$

Use the given line graphs to answer the following questions. Explain what the dashed lines in the graph below help us find.

Use the given line graphs to answer the following questions. What is the value of 35 -year-old machinery?

Use the given line graphs to answer the following questions. Business. Refer to the graph below. Find the income received from 30 customers.

Use the given line graphs to answer the following questions. Business. Refer to the graph below. Find the income received from 70 customers.(GRAPH CAN'T COPY)

Express each statement using one of the words sum product, difference, or quotient.$$8(2)=16$$

Express each statement using one of the words sum product, difference, or quotient.$$45 \cdot 12=540$$

Express each statement using one of the words sum product, difference, or quotient.$$11-9=2$$

Express each statement using one of the words sum product, difference, or quotient.$$65+89=154$$

Express each statement using one of the words sum product, difference, or quotient.$$x+2=10$$

Express each statement using one of the words sum product, difference, or quotient.$$16-t=4$$

Express each statement using one of the words sum product, difference, or quotient.$$\frac{66}{11}=6$$

Express each statement using one of the words sum product, difference, or quotient.$$12 \div 3=4$$

Translate each verbal model into an equation. (Answers may vary, depending on the variables chosen.)(VERBAL MODEL CAN'T COPY)

The amount of sand that should be used is the product of 3 and the amount of cement used.

The number of waiters needed is the quotient of the number of customers and $10 .$

The weight of the truck is the sum of the weight of the engine and $1,200 .$

The number of classes still open is the difference of 150 and the number of classes that are closed.

The profit is the difference of the revenue and $600 .$

The distance is the product of the rate and 3 .

The quotient of the number of laps run and 4 gives the number of miles run.

The sum of the tax and 35 gives the total cost.

Use the formula to complete each table.$$d=360+L$$$$\begin{array}{|c|c|}\hline \text { Lunch time } & {\text { School day }} \\ {\text { (minutes) }} & {\text { (minutes) }} \\ \hline L & {d} \\ \hline 30 & {} \\ \hline 40 & {} \\ \hline 45 & {} \\ \hline\end{array}$$

Use the formula to complete each table.$$b=1,024 k$$$$\begin{array}{|c|c|}\hline \text { Kilobytes } & {\text { Bytes }} \\ \hline k & {b} \\ \hline 1 & {} \\ \hline 5 & {} \\ \hline 10 & {} \\ \hline\end{array}$$

Use the formula to complete each table.$$t=1,500-d$$$$\begin{array}{|c|c|}\hline & {\text { Take-home }} \\ \hline \text { Deductions } & {\text { pay }} \\ \hline d & {} \\ \hline 200 & {} \\ \hline 300 & {} \\ \hline 400 & {} \\ \hline\end{array}$$

Use the formula to complete each table.$$w=\frac{s}{12}$$$$\begin{array}{|c|c|}\hline \text { Inches of } & {\text { Inches of }} \\ {\text { snow }} & {\text { water }} \\ \hline s & {w} \\ \hline 12 & {} \\ \hline 24 & {} \\ \hline 72 & {} \\ \hline\end{array}$$

Use the data in the table to complete the formula.$$d=\frac{e}{ }$$$$\begin{array}{|c|c|}\hline \text { Eggs } & {\text { Dozens }} \\ \hline e & {d} \\ \hline 24 & {2} \\ \hline 36 & {3} \\ \hline 48 & {4} \\ \hline\end{array}$$

Use the data in the table to complete the formula.p=____ c $$\begin{array}{|c|c|}\hline \text { Canoes } & {\text { Paddles }} \\ \hline c & {p} \\ \hline 6 & {12} \\ \hline 7 & {14} \\ \hline 8 & {16} \\ \hline\end{array}$$

Use the data in the table to complete the formula.$$I=c$$$$\begin{array}{|c|c|}\hline \text { Couples } & {\text { Individuals }} \\ \hline c & {I} \\ \hline 20 & {40} \\ \hline 100 & {200} \\ \hline 200 & {400} \\ \hline\end{array}$$

Use the data in the table to complete the formula.$$t=\frac{p}{}$$$$\begin{array}{|c|c|}\hline \text { Players } & {\text { Teams }} \\ \hline p & {t} \\ \hline 5 & {1} \\ \hline 10 & {2} \\ \hline 15 & {3} \\ \hline\end{array}$$

The number of calories that a 125-pound adult burns doing general house cleaning chores is three times the number of minutes spent cleaning.a. Write a verbal model using the word product that describes the relationship between calories burned and minutes cleaning.b. Write a formula using the variables $c$ and $m$ that describes the relationship between calories burned and minutes cleaning.c. Use your answer to part b to complete the following table.$\begin{array}{|c|c|c|c|c|c|c|}\hline m & {10} & {20} & {30} & {40} & {50} & {60} \\ \hline c & {} & {} & {} & {} & {} & {} \\ \hline\end{array}$d. Use the data from the table to construct a line graph. Scale the horizontal axis in units of 10 minutes. Scale the vertical axis in units of 30 calories.

Traffic Safety. As the railroad crossing guard drops, the measure of angle 1 (written $\angle 1$ ) increases while the measure of $\angle 2$ decreases. At any instant the sum of the measures of the two angles is $90^{\circ} .$ Complete the table. Then use the data to construct a line graph. Scale each axis in units of $15^{\circ} .$$\begin{array}{|c|c|}\hline \text { Angle 1} & {\text { Angle } 2} \\ {\text { (degrees) }} & {\text { (degrees) }} \\ \hline 0 & {} \\ \hline 15 & {} \\ \hline 30 & {} \\ \hline 45 & {} \\ \hline 45 & {} \\ \hline 60 & {} \\ \hline 75 \\ \hline 90 \\ \hline\end{array}$

Many students misuse the word equation when discussing mathematics. What is an equation? Give an example.

Explain the difference between an algebraic expression and an equation. Give an example of each.

In this section, four methods for describing numerical relationships were discussed: tables, verbal models (words), graphs, and equations. Which method do you think is the most useful? Explain why.

In your own words, define horizontal and vertical.

Complete the formula. t= ___ s+___$\begin{array}{|c|c|}\hline s & {t} \\ \hline 18 & {55} \\ \hline 33 & {100} \\ \hline 47 & {142} \\ \hline\end{array}$

Suppose $h=4 n$ and $n=2 g .$ Complete the following formula: h= ___ g