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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 31
Lesson 31: Problems in Mathematical Terms
Student Outcomes
 Students analyzean equation in two variablesto choose an independent variableand dependent variable.
Students determine whether or not the equation is solved for the second variablein terms of the firstvariable
or viceversa. They then use this information to determine which variableis theindependent variableand
which is the dependent variable.
 Students create a tableby placingthe independent variablein the firstrow or column and the dependent
variablein thesecond row or column. They compute entries in the table by choosingarbitrary values for the
independent variable(no constraints) and then determine what the dependent variablemustbe.
Classwork
Example 1 (10 minutes)
Example 1
Marcusreadsfor 𝟑𝟎minuteseach night. He wantsto determinethetotal number ofminuteshe will read over thecourse
of amonth. He wrote the equation 𝒕 = 𝟑𝟎𝒅to represent thetotalamount oftimethat hehasspent reading, where
𝒕 representsthe total numberofminutesread and 𝒅 representsthe number ofdaysthat he read during themonth.
Determinewhichvariableisindependent and which isdependent. Then createatable to show how many minuteshe has
read in the first seven days.
Number ofDays
(𝒅)
Total Minutes Read
(𝟑𝟎𝒅)
𝟏 𝟑𝟎
𝟐 𝟔𝟎
𝟑 𝟗𝟎
𝟒 𝟏𝟐𝟎
𝟓 𝟏𝟓𝟎
𝟔 𝟏𝟖𝟎
𝟕 𝟐𝟏𝟎
 When setting up a table, we want the independent variablein the firstcolumn and the dependent variablein
the second column.
 What do independent and dependent mean?
 The independent variable changes, and when it does, it affects the dependent variable. So, the
dependent variable “depends” on the independent variable.
 In this example, which would be the independent variableand which would be the dependent variable?
 The dependent variable is the total number of minutes read because it depends on how many days
Marcus reads. The independent variable is the number of days that Marcus reads.
 How could you use the tableof values to determine the equation if it had not been given?
 The number of minutes read shown in the table is always 30 times the number of days. So, the
equation would need to show that the total number of minutes is equal to the number of days times 30.
Independent variable _________________
Dependentvariable _________________
Number ofDays
Total Minutes Read
MP.1

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 31
Example 2 (5 minutes)
Example 2
Kiradesignswebsites. She can create threedifferent websiteseachweek. Kirawantsto createan equation that willgive
her the total number ofwebsitesshe can design giventhenumber ofweeksshe works. Determinetheindependentand
dependentvariables. Create atableto show the number ofwebsitesshe can design overthe first 𝟓weeks. Finally, write
an equation to represent thenumberofwebsitesshe can designwhen given any number ofweeks.
 How did you determine which is the dependent variableand which is theindependent variable?
 Because the number of websites she can make depends on how many weeks she works, I determined
that the number of weeks was the independent variable and the number of websites designed was the
dependent variable.
 Does knowing which one is independent and which one is dependent help you write the equation?
 I can write the equation and solve for the dependent variable by knowing how the independent variable
will affect the dependent variable. In this case, I knew that every week 5 more websites could be
completed, so then I multiplied the number of weeks by 5.
Example 3 (5 minutes)
Example 3
Priyastreamsmoviesthrough acompany that chargeshera $𝟓monthly fee plus $𝟏.𝟓𝟎per movie. Determinethe
independent and dependent variables, write an equationto model thesituation, and createatable to show thetotal cost
per month given that she might stream between 𝟒and 𝟏𝟎moviesin amonth.
 Is the flatfee an independent variable,a dependent variable, or neither?
 The $5 flat fee is neither. It is not causing the change in the dependent value, and it is not changing.
Instead, the $5 flat fee is a constant that is added on each month.
 Why isn’tthe equation 𝑐 = 5𝑚 + 1.50?
 The $5 fee is only paid once a month. 𝑚 is the number of movies watched per month, so it needs to be
multiplied by the price per movie, which is $1.50.
Independent variable_____________________
Dependentvariable______________________
Equation____________________________________________________________
# ofWeeks
Worked (𝒘)
# ofWebsites
Designed (𝒅)
𝟏 𝟓
𝟐 𝟏𝟎
𝟑 𝟏𝟓
𝟒 𝟐𝟎
𝟓 𝟐𝟓
# ofweeks worked
# ofMovies (𝒎) Total Cost Per Month (𝒄)
𝟒 𝟏𝟏
𝟓 𝟏𝟐.𝟓𝟎
𝟔 𝟏𝟒
𝟕 𝟏𝟓.𝟓𝟎
𝟖 𝟏𝟕
𝟗 𝟏𝟖.𝟓𝟎
𝟏𝟎 𝟐𝟎
# ofwebsites designed
𝒅 = 𝟓𝒘, where 𝒘 is thenumber ofweeks worked and 𝒅 is thenumber ofwebsites designed
Independent variable______________________
Dependentvariable_______________________
Equation______________________________
# ofmovies watched per month
Total cost per month
𝒄 = 𝟏. 𝟓𝒎+ 𝟓 or 𝒄 = 𝟏.𝟓𝟎𝒎+ 𝟓
MP.1

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 31
Exercises1–4 (15 minutes)
Students work in pairs or independently.
Exercises1–4
1. Sarah ispurchasing pencilsto share. Each package has 𝟏𝟐pencils. The equation 𝒏 = 𝟏𝟐𝒑, where 𝒏 isthenumber
of pencilsand 𝒑 isthe number ofpackagescan be used to determine thetotalnumber ofpencilsSarah purchased.
Determinewhichvariableisdependent and which isindependent. Then make atable showing thenumber of
pencilspurchasedfor 𝟑–𝟕packages.
𝒑, thenumber ofpackages, is theindependent variable.
𝒏, thetotal number ofpencils, is thedependentvariable.
# ofPackages (𝒑) Total #ofPencils (𝒏 = 𝟏𝟐𝒑)
𝟑 𝟑𝟔
𝟒 𝟒𝟖
𝟓 𝟔𝟎
𝟔 𝟕𝟐
𝟕 𝟖𝟒
2. Charlotte reads 𝟒bookseach week. Let 𝒃bethe number ofbooksshe readseachweek and let 𝒘be thenumber of
weeksthat she reads. Determine whichvariable isdependent and which isindependent. Then writean equation to
model the situation, and make agraph that showsthe numberofbooksread in under 𝟔weeks.
The number ofweeks, 𝒘, is theindependent variable.
The number ofbooks, 𝒃, is thedependent variable.
𝒃 = 𝟒𝒘
# ofWeeks (𝒘) # ofBooks (𝒃 = 𝟒𝒘)
𝟏 𝟒
𝟐 𝟖
𝟑 𝟏𝟐
𝟒 𝟏𝟔
𝟓 𝟐𝟎
3. A miniature golfcoursehasaspecial group rate. You can pay $𝟐𝟎plus $𝟑per personwhen youhave agroup of 𝟓or
more friends. Let 𝒇 be thenumber offriendsand 𝒄be the total cost. Determine which variableisindependentand
which isdependent,and writean equation that modelsthe situation. Thenmake atable to show thecost for 𝟓to
𝟏𝟐friends.
The number offriends, 𝒇, is theindependent variable.
The total cost, 𝒄, is thedependentvariable.
𝒄 = 𝟑𝒇 + 𝟐𝟎
# ofFriends (𝒇) Total Cost (𝒄 = 𝟑𝒇 + 𝟐𝟎)
𝟓 𝟑𝟓
𝟔 𝟑𝟖
𝟕 𝟒𝟏
𝟖 𝟒𝟒
𝟗 𝟒𝟕
𝟏𝟎 𝟓𝟎
𝟏𝟏 𝟓𝟑
𝟏𝟐 𝟓𝟔

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 31
4. Carlosisshopping for school supplies. He boughtapencil box for $𝟑, and he also needsto buy notebooks. Each
notebook is $𝟐. Let 𝒕 = the total costofthe suppliesand 𝒏 be thenumber ofnotebooks. Determine which variable
isindependent and which isdependent, and writean equation that modelsthe situation. Then make atable to
show the cost for 𝟏to 𝟓notebooks.
The total number ofnotebooks, 𝒏, is theindependentvariable. 𝒕 = 𝟐𝒏 + 𝟑
The total cost, 𝒕, is thedependentvariable.
# ofNotebooks (𝒏) Total Cost (𝒕 = 𝟐𝒏 + 𝟑)
𝟏 𝟓
𝟐 𝟕
𝟑 𝟗
𝟒 𝟏𝟏
𝟓 𝟏𝟑
Closing(5 minutes)
 Use this time for partners to sharetheir tables with another set of partners.
 End classwith a quick discussion of how the students determined which variableis independent and which is
dependent.
 The dependent variable is affected by changes in the independent variable.
 I can write a sentence that one variable depends on another. For example, the amount of money
earned depends on the number of hours worked. So, the money earned is dependent and the number
of hours worked is independent.
Exit Ticket (5 minutes)

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 31
Name Date
Lesson 31: Problems in Mathematical Terms
Exit Ticket
For each problem, determine the independent and dependent variables,writean equation to represent the situation,
and then make a tablewith at least 5 values that models the situation.
1. Kyla spends 60 minutes of each day exercising. Let 𝑑 be the number of days,and let 𝑚 represent the total minutes
of exercisein a given time frame. Show the relationship between the number of days and the total minutes of
exercise.
2. A taxi cab servicecharges a flatfee of $8 plus an additional $1.50 per mile. Show the relationship between the total
costand the number of miles driven.
Independent Variable
Dependent Variable
Equation
Independent Variable
Dependent Variable
Equation

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 31
Exit Ticket Sample Solutions
For each problem,determinetheindependent and dependent variables, write an equationto represent thesituation,and
then make atable with at least 𝟓valuesthat modelsthe situation.
1. Kylaspends 𝟔𝟎minutesofeach day exercising. Let 𝒅 be thenumber ofdays, and let 𝒎 represent thetotal minutes
of exercise inagiven timeframe. Show therelationship between the number ofdaysand the total number of
minutes.
Tables may vary.
# ofDays # ofMinutes
𝟎 𝟎
𝟏 𝟔𝟎
𝟐 𝟏𝟐𝟎
𝟑 𝟏𝟖𝟎
𝟒 𝟐𝟒𝟎
2. A taxi cab service chargesaflat fee of $𝟖plusan additional $𝟏.𝟓𝟎 per mile. Show the relationshipbetween the
total cost and the numberofmiles driven.
Tables may vary.
# ofMiles Total Cost
𝟎 𝟖. 𝟎𝟎
𝟏 𝟗. 𝟓𝟎
𝟐 𝟏𝟏.𝟎𝟎
𝟑 𝟏𝟐.𝟓𝟎
𝟒 𝟏𝟒.𝟎𝟎
Problem Set Sample Solutions
1. Jaziyah sells 𝟑houseseach month. To determine thenumber ofhousesshe can sellin any given numberofmonths
she usesthe equation 𝒕 = 𝟑𝒎, where 𝒕isthe total number ofhousessold and 𝒎 isthe numberofmonths. Name
the independent and dependent variables. Then create atableto show how many housesshe sellsin lessthan 𝟔
months.
The independent variableis thenumber ofmonths. Thedependent variableis thetotal number ofhouses sold.
# ofMonths Total Number ofHouses
𝟏 𝟑
𝟐 𝟔
𝟑 𝟗
𝟒 𝟏𝟐
𝟓 𝟏𝟓
Independent Variable
DependentVariable
Equation 𝒄 = 𝟏. 𝟓𝟎𝒎 + 𝟖
Number ofMiles
Total Cost
𝒎 = 𝟔𝟎𝒅
Number ofDays
Total Number ofMinutes
Independent Variable
DependentVariable
Equation

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 31
2. Joshuaspends 𝟐𝟓minutesofeach day reading. Let 𝒅 be thenumber ofdaysthat he reads, and let 𝒎 represent the
total minutesofreading. Determinewhich variableisindependent and which isdependent. Then writean
equation that will modelthe situation. Make atable showing the number ofminutesspentreading over 𝟕days.
The number ofdays (𝒅)is theindependent variable.
The total number ofminutes ofreading (𝒎)is thedependent variable.
𝒎 = 𝟐𝟓𝒅
# ofDays # ofMinutes
𝟏 𝟐𝟓
𝟐 𝟓𝟎
𝟑 𝟕𝟓
𝟒 𝟏𝟎𝟎
𝟓 𝟏𝟐𝟓
𝟔 𝟏𝟓𝟎
𝟕 𝟏𝟕𝟓
3. Each package ofhot dog bunscontains 𝟖buns. Let 𝒑 be the number ofpackagesand 𝒃be the totalnumber ofbuns.
Determinewhichvariableisindependent and which isdependent. Then write an equationthat willmodel the
situation and make atable showing the number ofhot dog bunsin 𝟑to 𝟖packages.
The number ofpackages (𝒑)is theindependent variable.
The total number ofhot dog buns (𝒃)is thedependent variable.
𝒃 = 𝟖𝒑
# ofPackages Total #ofHot Dog Buns
𝟑 𝟐𝟒
𝟒 𝟑𝟐
𝟓 𝟒𝟎
𝟔 𝟒𝟖
𝟕 𝟓𝟔
𝟖 𝟔𝟒
4. Emmawas given 𝟓seashells. Each week she collected 𝟑more. Let 𝒘be the number ofweeksand 𝒔be thenumber
of seashellsshe hastotal. Which variable isindependent and which isdependent? Writean equation to modelthe
relationship, and make atable to show howmany shellsshe hasfrom week 𝟒to week 𝟏𝟎.
The number ofweeks (𝒘) is theindependent variable.
The total number ofsea shells (𝒔)is thedependent variable.
𝑺 = 𝟑𝒘+ 𝟓
# ofWeeks Total #ofSea Shells
𝟒 𝟏𝟕
𝟓 𝟐𝟎
𝟔 𝟐𝟑
𝟕 𝟐𝟔
𝟖 𝟐𝟗
𝟗 𝟑𝟐
𝟏𝟎 𝟑𝟓

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 31
5. Emiliaisshopping for fresh produce at afarmer’smarket. She bought awatermelon for $𝟓, and she also wantsto
buy peppers. Each pepperis $𝟎.𝟕𝟓. Let 𝒕 = the totalcost ofthe produce and 𝒏 be thenumberofpeppersbought.
Determinewhichvariableisindependent and which isdependent, and write an equation that modelsthesituation.
Then make atable to show the cost for 𝟏to 𝟓peppers.
The number of peppers, 𝒏, is theindependent variable. 𝒕 = 𝟎. 𝟕𝟓𝒏 + 𝟓
The total cost, 𝒕, is the dependentvariable.
# ofPeppers (𝒏) Total Cost
𝟏 𝟓. 𝟕𝟓
𝟐 𝟔. 𝟓𝟎
𝟑 𝟕. 𝟐𝟓
𝟒 𝟖. 𝟎𝟎
𝟓 𝟖. 𝟕𝟓
6. A taxi cab service chargesaflat fee of $𝟕plusan additional $𝟏.𝟐𝟓 per mile. Show the relationshipbetween the
total cost and the numberofmiles. Which variableisindependentand whichisdependent? Write an equation to
model the relationship, and make atableto show the cost of 𝟒to 𝟏𝟎miles.
The number ofmiles, 𝒎, is theindependent variable.
The cost, 𝒄, is thedependent variable.
𝒄 = $𝟏.𝟐𝟓𝒎 + 𝟕
# ofMiles Total Cost
𝟒 $𝟏𝟐
𝟓 $𝟏𝟑.𝟐𝟓
𝟔 $𝟏𝟒.𝟓𝟎
𝟕 $𝟏𝟓.𝟕𝟓
𝟖 $𝟏𝟕
𝟗 $𝟏𝟖.𝟐𝟓
𝟏𝟎 $𝟏𝟗.𝟓𝟎