Chapter Three: Translations & Word Problems Index: A: Literal Equations B: Algebraic Translations C: Consecutive Word Problems D: Linear Word Problems
Name: Date: Period: Algebra I Literal Equations 3A At this point we should feel very competent in solving linear equations. In many situations, we might even solve equations when there are no actual numbers given. Equations with several variables (letters) are called literal equations. Your job, usually, will be to solve the equation for one of the variables. The letters that do not represent your desired variable move to the other side of the equal sign so that the one variable you are solving for stands alone. Even though there are more letters in these equations, the methods used to solve these equations are the same as the methods you use to solve all equations. Let's take a look at what we means in Exercise 1. Exercise 1: Solve each of the following problems for the value of. In (b), write your answer in terms of the unspecified constants and. (a) (b) The rules for solving linear equations (all equations) don't depend on whether the constants in the problem are specified or not. The biggest difference in #1 between (a) and (b) is that in (b) you have to leave the results of the intermediate calculation undone. Exercise 2: Use your knowledge of algebra to solve each of the following. (a) Solve for B. (b) Solve for (c) Solve for.
Exercise 3: For a rectangle, the perimeter, P, can be found if the two dimensions of length, L, and width, W, are known. (a) Write a formula for the perimeter, P, in terms of L and W. (b) If a rectangle has a length of 12 inches and a width (c) Rearrange this formula so that it "solves" of 5 inches, what is the value of its perimeter? for the length, L. Determine the value of L when P=20 and W=4. There is one last complication we need to look at that is often challenging for students at all levels. Let's take a look at this in the next problem. Exercise 4: When is solved for in terms of and, its solution is which of the following? Show the algebraic manipulations you used to get your answer. Exercise 5: Which of the following solves the equation for in terms of and. Show the manipulations to find your answer.
Name: Date: Period: Algebra I Literal Equations 3A HW 1) If, then what is the value of a in terms of b and r be expressed as? 2) The members of the senior class are planning a dance. They use the equation to determine the total receipts. What is n expressed in terms of r and p? 3) If, then what is the value of v in terms of d? 4) Solve the following equations for. It may help to make up an equation with numbers and solve it to the side to make sure you are not making any mistakes. (a) (b)
5) If, then what is the value of h in terms of a? 6) When traveling abroad, many of the units used are different. One of the most common is the unit of temperature namely Fahrenheit verses Celsius. The conversion between the 2 temperatures is as follows. (a) Using the formula above, convert to Celsius. (b) This conversion formula is very useful if you are given Fahrenheit, but less useful if you know a Celsius temperature. Solve the above equation for Fahrenheit,, and then convert into Fahrenheit. Is there a large difference in Fahrenheit and Celsius? Review Section: 7) When is subtracted from 8) Solve the following equation for x and CHECK! what is the result?
Name: Date: Period: Algebra Literal Equations 3A HW 1.) 2.) Homework Answers 3.) 4.) a.) b.) 5.) 6.) a.) b.) 7.) 8.)
Name: Date: Period: Algebra I Algebraic Translations 3B There will be many instances when we have to translate phrases from English into mathematical expressions. This is a skill that take a lot of practice and time to master. In this lesson, we will begin to build this fluency. Exercise 1: It is important to be able to recognize addition and subtraction in phrases. First, let's begin with some numerical work and then transition to expressions that only contain variables. (a) Write a calculation and a result that represents a (b) Write a calculation and a result that represents a number that is 5 greater than 3. number that is 2 less than 9. (c) Write a calculation and a result that represents the (d) Write a calculation and a result that represents sum of and 8. the difference of 20 and 12. (e)if x represents a number, write an expression that (f) If n represents a number, write an expression that represents a number 10 greater than x. represents a number that is 5 less than n. (g) If y represents a number, write an expression that represents the sum of y and a number one greater than y. (h) If n represents a number, write an expression that represents the difference between a number one larger than n and one smaller than n. Be Careful!!
We also need to be able to translate multiplication and division. Multiplication is typically easier to spot and translate. Let's try some! Exercise 2: Translate each verbal statement into an expression and evaluate the expression if it is numerical. (a) Write an expression for a number that is five (b) If n represents a number, then write an times greater than 2. expression for a number that is twice n. (c) Write an expression for the quotient (or ratio) of (d) If x represents a number, write an expression for 12 and 3. the ratio of x to 5. Now we want to be able to put operations together to create more complex expressions. These can be tricky! It is always important to read them carefully, think about your order of operations, and check them with a real number. Exercise 3: Translate each of the following statements into an algebraic expression. (a) If x represents a number, then write an expression (b) If n represents a number, then write an for a number that is three more than twice the value of x. expression for two less than one fourth of n. (c) If s represents Sally s age and her father is 4 years (d) If x represents a number, then write an less than five times her age, then write an expression expression for three times the sum of x and 10. for her father s age in terms of the variable s. (e) If n represents a number, then write an expression (f) If x represents a number, then write an expression for 7 less than four times the difference of n and 5. for the ration of 3 less than x to 2 more than x. (g) If x represents a number, then write an expression (h) If n represents a number, then write an for the sum of twice x which twice a number one expression for the quotient of twice n and three less larger than x. than n.
(i) If y represents a number, then write an expression (j) If x represents a number, then write an for three-quarters of the difference of y and 8. expression for one half the sum of x and 4. Exercise 4: Neat patterns can occur repeatedly when you play around with numbers. A fairly easy one occurs when you add a number to one less and one more than that number. Do this for a few numbers, x, and record the results. Then, prove a general pattern by writing an expression for the sum of a number with a number one less and a number one more than it.
Name: Date: Period: Algebra I Algebraic Translations 3B HW 1) 2) Which algebraic expression represents 15 less than 3) Which expression represents 5 less than the divided by 9? product of 7 and? (1) (1) (2) (2) (3) (3) (4) (4) 4) Which verbal expression is represented by? (1) twice the sum of a number and four (2) the sum of two times a number and four (3) two times the difference of a number and four (4) twice the product of a number and four
5) The Miller family made mathematical statements out of their ages as follows. Tom is four less than twice Gary s age. Rebecca is the youngest and she is two less than half of Gary s age after it was increased by three. Sam s age is the ratio of seven more than Gary s age to eight less than Gary s age. (a) Translate each of the Miller family members ages into algebraic expressions in terms of Gary s age,. Tom s Age: Rebecca s Age: Sam s Age: (b) If Gary is 11 years old, how old are each of the family members? (c) Using Gary s age, come up with an expression that represents your age in terms of. Be creative! For example, if Mr. Weiler is 43 years old, then his age would be. 6) Our future work in this course will necessitate that we work with what are known as consecutive integers. Integers are the set of positive and negative whole numbers (as well as zero). Integers: { -4,-3,-2,-1,0,1,2,3,4 } Consecutive integers are lists of integers that increase by one unit between them. (a) Fill in the pattern with consecutive integers: 2, 3,,5,,, 8 (b) We can also talk about consecutive even integers and consecutive odd integers. Fill in the patterns. 5, 7, 9,, 13,, -10, -8,,, -2, (c) Regardless of whether we have consecutive even integers or consecutive odd integers, to get from one to another you add what number? If represents the first in a list of consecutive even (or odd) integers, write out the next three terms. What do we add to each term?,,
Review Section: 7) Fill in the blanks: When multiplying with exponents, we must the coefficients and the exponents. When dividing with exponents, we must the coefficients and the exponents. Try these: 8) The length of a rectangle is two less than three times a number, and the width is five morethan that same number. (a) Draw a diagram that represents the rectangle. Be sure to label the sides in terms of the unknown,. (b) Using your diagram, find what the perimeter of the rectangle is in terms of. Write your answer as a simplified binomial. (c) What is the area of the rectangle in terms of? Write your answer as a trinomial. Remember the formula for area of a rectangle is
Name: Date: Period: Algebra Algebraic Translations 3B HW 1.) Puzzle 2.) (1) 3.) (2) 4.) (1) 5.) a.) Tom = Rebecca = Sam = b.) Tom is 18, Rebecca is 5 and Sam is 6. c.) 14 years old: 6.) Fill in examples 7.) Fill in examples. 8.) a.) diagram b.) c.) A = Homework Answers
Name: Date: Period: Algebra I Consecutive Word Problems 3C Although word problems can often be some of the most challenging for students, they give us great opportunities to refine our understanding of the relationships between quantities and how to manipulate expressions to solve equations. When you solve any real world problems in mathematics you are modeling a physical situation with mathematical tools, such as equations, diagrams, tables, as well as many others. As we work through these problems, try to make sure to always do the following: MODELING AND SOLVING LINEAR WORD PROBLEMS 1. Clearly define the quantities involved with common sense variables and let statements. 2. Use your let statements to write out expression for quantities that you are interested in. 3. Carefully translate the information you are told into an equation. 4. Solve the equation remember to mentally note the justification for each step. 5. Check the reasonableness of your answer! This could be the most important, and neglected, step in the modeling/problem solving method. One of the ways we can practice our ability to work with algebraic expressions and equations is to play around with problems that involve consecutive integers. Make sure you know what the integers are: THE INTEGERS AND CONSECUTIVE INTEGERS The integers are the subset of the real numbers (so positive and negative whole numbers). Consecutive integers are any list of integers (however long) that are separated by only 1 unit. Such as: 1, 2, 3 or 5, 6, 7, 8 or or Consecutive Even Integers or Consecutive Odd Integers or Rules to follow for word problems: 1) Unknown starting point means that the first number is always equal to x. 2) CONSECUTIVE integers increase by (+1) 3) EVEN integers increase by (+2) ODD integers increase by (+2) ** This means that the let column will look the same for both even and odd consecutive integers!!!** Let Statements: Consecutive: Consecutive Even: Consecutive Odd: n n n n+1 n+2 n+2 n+2 n+4 n+4 n+3 n+6 n+6
Let's try one! Exercise 1: The sum of four consecutive integers is. What are the four integers? Exercise 2: The sum of three consecutive integers is. What are the three integers? Exercise 3: I'm thinking of three consecutive odd integers. When I add the larger two the result is nine less than three times the smallest of them. What are the three consecutive odd integers?
Exercise 4: Three consecutive even integers have the property that when the difference between the first and twice the second is found, the result is eight more than the third. Find the three consecutive even integers. Exercise 5: Find three consecutive integers such that three times the largest increased by two is equal to five times the smallest increased by three times the middle integer.
Name: Date: Period: Algebra I Consecutive Word Problems 3C HW Direction: Set up let statements for appropriate expressions and using these expressions, set up an equation that allows you to find each number described. Be sure to find EACH integer you are looking for. 1.) Find 4 consecutive even integers such that the sum of the 2 nd and 4 th is -132. 2.) Find two consecutive integers such that ten more than twice the smaller is seven less than three times the larger. 3.) Find three consecutive odd integers such that the sum of the smaller two is three times the largest increased by seven.
4.) In an opera theater, sections of seating consisting of three rows are being laid out. It is planned so each row will be two more seats than the one before it and 90 people must be seated in each section. How many people will be in the third row? Review Section: 5) What is the value of, if and? 6) Solve algebraically for : 7) Expand the following:
Homework Answers Name: Date: Period: Algebra Consecutive Word Problems 3C HW 1.) The four consecutive even integers are and 2.) The two consecutive integers are 14 and 15. 3.) The three consecutive odd integers are and 4.) There will be 32 people in the third row. 5.) 14 6.) x = 4 7.)
Name: Date: Period: Algebra I Linear Word Problems 3D RECALL: Although word problems can often be some of the most challenging for students, they give us great opportunities to refine our understanding of the relationships between quantities and how to manipulate expressions to solve equations. When you solve any real world problems in mathematics you are modeling a physical situation with mathematical tools, such as equations, diagrams, tables, as well as many others. As we work through these problems, try to make sure to always do the following: MODELING AND SOLVING LINEAR WORD PROBLEMS 1. Clearly define the quantities involved with common sense variables and let statements. 2. Use your let statements to write out expression for quantities that you are interested in. 3. Carefully translate the information you are told into an equation. 4. Solve the equation remember to mentally note the justification for each step. 5. Check the reasonableness of your answer! This could be the most important, and neglected, step in the modeling/problem solving method. Exercise 1: The sum of a number and five more than the number is 17. What is the number? Let's carefully set up let statements and an equation that relates the quantities of interest. Then we can solve the equation for the number. Exercise 2: The difference between twice a number and a number that is 5 more than it is 3. Which of the following equations could be used to find the value of the number, n? Explain how you arrived at your answer? (1) (3) (2) (4)
Let's try a harder one: Exercise 3: Three numbers have the sum of 99. The 2 nd number is 3 more than double the first. The 3 rd number is 3 more than the second. Find all three numbers. Exercise 4: Sara has three sisters. Lea is 4 less than 3 times the age of Sarah. Rachel is 3 years less than one-half Sarah s age. Ruth is 1 year older than twice the age of Sarah. If the sum of the ages of the four sisters is 46 years, how old is each sister?
Exercise 5: The difference of 2 numbers is 25. The smaller is 5 more than half the larger. Find both numbers.
Name: Date: Period: Algebra I Linear Word Problems 3D HW 1) The sum of three times a number and 2 less than 4 times that same number is 15. Which of the following equations could be used to find the value of the number,? Explain how you arrived at your choice. Create let statements for the following examples. Be sure to carefully read the question and figure out exactly what you are looking for. Then, set up an equation that summarizes the information in the problem and solve the equation and check for reasonableness. 2.) Tom is 4 more than twice Andrew s age. Sara is 8 less than 5 times Andrews age. If Tom and Sara are twins, how old is Andrew? *Think: What does it mean to be twins in regards to your ages?) 3.) A wireless phone plan costs Eric $35 for a month of service during which he sent 450 text messages. If he was charged a fixed fee of $12.50, how much did he pay per text?
In some cases, the answers you will get won t make physical sense or need a bit of interpreting. Look at the next example and be careful when you interpret your final solution. 4.) Tanisha and Rebecca are signing up for new cellphone plans that only charge for the number of minutes and everything else is included in a monthly fee. Their plans are as follows: (a) Figure out how many minutes the two plans will charge the same amount. (b) Interpret your answer. It may help to read their two plans again and think about which one you would rather pay. Review Section: 5.) Express the product of and in standard form. 6.) When solving the equation, Emily wrote as her first step. Which property justifies Emily s first step? (1) addition property of equality (2) commutative property of addition (3) multiplication property of equality (4) distributive property of multiplication over addition.
Homework Answers Name: Date: Period: Algebra Linear Word Problems 3D HW 1.) (1) 2.) Andrew is 4 years old 3.) Each text costs $0.05 4.) 5.) 6.) (1)