Solving Word Problems The strategy for solving word problems, - - PowerPoint PPT Presentation

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Solving Word Problems The strategy for solving word problems, - - PowerPoint PPT Presentation

Solving Word Problems The strategy for solving word problems, presented in written form, may be summarized in three words: Alan H. SteinUniversity of Connecticut Solving Word Problems The strategy for solving word problems, presented in written


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SLIDE 1

Solving Word Problems

The strategy for solving word problems, presented in written form, may be summarized in three words:

Alan H. SteinUniversity of Connecticut

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SLIDE 2

Solving Word Problems

The strategy for solving word problems, presented in written form, may be summarized in three words:

Read the Question!

Alan H. SteinUniversity of Connecticut

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SLIDE 3

Solving Word Problems

The strategy for solving word problems, presented in written form, may be summarized in three words:

Read the Question!

We’ll expand on how to read the question.

Alan H. SteinUniversity of Connecticut

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SLIDE 4

Strategy

Alan H. SteinUniversity of Connecticut

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SLIDE 5

Strategy

◮ Look for variables and unknowns

Alan H. SteinUniversity of Connecticut

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SLIDE 6

Strategy

◮ Look for variables and unknowns – represent them by symbols

(letters)

Alan H. SteinUniversity of Connecticut

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SLIDE 7

Strategy

◮ Look for variables and unknowns – represent them by symbols

(letters)

◮ Look for connections or relationships among the variables and

unknowns

Alan H. SteinUniversity of Connecticut

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SLIDE 8

Strategy

◮ Look for variables and unknowns – represent them by symbols

(letters)

◮ Look for connections or relationships among the variables and

unknowns – these are usually expressed through equations or formulas

Alan H. SteinUniversity of Connecticut

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SLIDE 9

Strategy

◮ Look for variables and unknowns – represent them by symbols

(letters)

◮ Look for connections or relationships among the variables and

unknowns – these are usually expressed through equations or formulas

◮ Do something with the equations and formulas

Alan H. SteinUniversity of Connecticut

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SLIDE 10

Strategy

◮ Look for variables and unknowns – represent them by symbols

(letters)

◮ Look for connections or relationships among the variables and

unknowns – these are usually expressed through equations or formulas

◮ Do something with the equations and formulas – solve an

equation when you can; simplify what you can; manipulate what you can

Alan H. SteinUniversity of Connecticut

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SLIDE 11

Strategy

◮ Look for variables and unknowns – represent them by symbols

(letters)

◮ Look for connections or relationships among the variables and

unknowns – these are usually expressed through equations or formulas

◮ Do something with the equations and formulas – solve an

equation when you can; simplify what you can; manipulate what you can Bear in mind – every piece of information should be written mathematically!

Alan H. SteinUniversity of Connecticut

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SLIDE 12

Strategy

◮ Look for variables and unknowns – represent them by symbols

(letters)

◮ Look for connections or relationships among the variables and

unknowns – these are usually expressed through equations or formulas

◮ Do something with the equations and formulas – solve an

equation when you can; simplify what you can; manipulate what you can Bear in mind – every piece of information should be written mathematically! Also remember these steps do not have to be performed in order; rather, one goes back and forth between the steps, generally repeating each step several times.

Alan H. SteinUniversity of Connecticut

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SLIDE 13

Example

Consider the following word problem: Develop a formula for the total cost, including tax, of an item based on its price and the sales tax rate.

Alan H. SteinUniversity of Connecticut

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SLIDE 14

Example

Consider the following word problem: Develop a formula for the total cost, including tax, of an item based on its price and the sales tax rate. We begin by finding all the variables and unknowns:

Alan H. SteinUniversity of Connecticut

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SLIDE 15

Example

Consider the following word problem: Develop a formula for the total cost, including tax, of an item based on its price and the sales tax rate. We begin by finding all the variables and unknowns:

◮ sales tax rate = r

Alan H. SteinUniversity of Connecticut

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SLIDE 16

Example

Consider the following word problem: Develop a formula for the total cost, including tax, of an item based on its price and the sales tax rate. We begin by finding all the variables and unknowns:

◮ sales tax rate = r ◮ total cost = C

Alan H. SteinUniversity of Connecticut

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SLIDE 17

Example

Consider the following word problem: Develop a formula for the total cost, including tax, of an item based on its price and the sales tax rate. We begin by finding all the variables and unknowns:

◮ sales tax rate = r ◮ total cost = C ◮ sales price of the item = p

Alan H. SteinUniversity of Connecticut

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SLIDE 18

Example

Consider the following word problem: Develop a formula for the total cost, including tax, of an item based on its price and the sales tax rate. We begin by finding all the variables and unknowns:

◮ sales tax rate = r ◮ total cost = C ◮ sales price of the item = p ◮ amount of sales tax = T

Alan H. SteinUniversity of Connecticut

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SLIDE 19

Finding Relationships

One way to find connections and relationships is to scan the list of variables and think of whether one of them can be calculated if you know the others.

Alan H. SteinUniversity of Connecticut

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SLIDE 20

Finding Relationships

One way to find connections and relationships is to scan the list of variables and think of whether one of them can be calculated if you know the others. Looking through the list of variables and unknowns, it appears that

  • ne should be able to find the total cost if one knows the price and

the tax.

Alan H. SteinUniversity of Connecticut

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SLIDE 21

Finding Relationships

One way to find connections and relationships is to scan the list of variables and think of whether one of them can be calculated if you know the others. Looking through the list of variables and unknowns, it appears that

  • ne should be able to find the total cost if one knows the price and

the tax. In fact, clearly, the total cost is equal to the sales price plus the tax.

Alan H. SteinUniversity of Connecticut

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SLIDE 22

Finding Relationships

One way to find connections and relationships is to scan the list of variables and think of whether one of them can be calculated if you know the others. Looking through the list of variables and unknowns, it appears that

  • ne should be able to find the total cost if one knows the price and

the tax. In fact, clearly, the total cost is equal to the sales price plus the tax. This can be directly translated into an equation,

Alan H. SteinUniversity of Connecticut

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SLIDE 23

Finding Relationships

One way to find connections and relationships is to scan the list of variables and think of whether one of them can be calculated if you know the others. Looking through the list of variables and unknowns, it appears that

  • ne should be able to find the total cost if one knows the price and

the tax. In fact, clearly, the total cost is equal to the sales price plus the tax. This can be directly translated into an equation, replacing total cost by C,

Alan H. SteinUniversity of Connecticut

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SLIDE 24

Finding Relationships

One way to find connections and relationships is to scan the list of variables and think of whether one of them can be calculated if you know the others. Looking through the list of variables and unknowns, it appears that

  • ne should be able to find the total cost if one knows the price and

the tax. In fact, clearly, the total cost is equal to the sales price plus the tax. This can be directly translated into an equation, replacing total cost by C, is equal to by an = sign,

Alan H. SteinUniversity of Connecticut

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SLIDE 25

Finding Relationships

One way to find connections and relationships is to scan the list of variables and think of whether one of them can be calculated if you know the others. Looking through the list of variables and unknowns, it appears that

  • ne should be able to find the total cost if one knows the price and

the tax. In fact, clearly, the total cost is equal to the sales price plus the tax. This can be directly translated into an equation, replacing total cost by C, is equal to by an = sign, the price by p,

Alan H. SteinUniversity of Connecticut

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SLIDE 26

Finding Relationships

One way to find connections and relationships is to scan the list of variables and think of whether one of them can be calculated if you know the others. Looking through the list of variables and unknowns, it appears that

  • ne should be able to find the total cost if one knows the price and

the tax. In fact, clearly, the total cost is equal to the sales price plus the tax. This can be directly translated into an equation, replacing total cost by C, is equal to by an = sign, the price by p, plus by a + sign,

Alan H. SteinUniversity of Connecticut

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SLIDE 27

Finding Relationships

One way to find connections and relationships is to scan the list of variables and think of whether one of them can be calculated if you know the others. Looking through the list of variables and unknowns, it appears that

  • ne should be able to find the total cost if one knows the price and

the tax. In fact, clearly, the total cost is equal to the sales price plus the tax. This can be directly translated into an equation, replacing total cost by C, is equal to by an = sign, the price by p, plus by a + sign, and the tax by T.

Alan H. SteinUniversity of Connecticut

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SLIDE 28

Finding Relationships

One way to find connections and relationships is to scan the list of variables and think of whether one of them can be calculated if you know the others. Looking through the list of variables and unknowns, it appears that

  • ne should be able to find the total cost if one knows the price and

the tax. In fact, clearly, the total cost is equal to the sales price plus the tax. This can be directly translated into an equation, replacing total cost by C, is equal to by an = sign, the price by p, plus by a + sign, and the tax by T. We thus get: C = p + T.

Alan H. SteinUniversity of Connecticut

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SLIDE 29

Another Equation or Formula

Looking through the variables, it’s also clear there should be a way

  • f calculating the sales tax if we know the tax rate and the sales

price.

Alan H. SteinUniversity of Connecticut

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SLIDE 30

Another Equation or Formula

Looking through the variables, it’s also clear there should be a way

  • f calculating the sales tax if we know the tax rate and the sales
  • price. In fact, we know the tax will equal the tax rate times the

sales price.

Alan H. SteinUniversity of Connecticut

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SLIDE 31

Another Equation or Formula

Looking through the variables, it’s also clear there should be a way

  • f calculating the sales tax if we know the tax rate and the sales
  • price. In fact, we know the tax will equal the tax rate times the

sales price. We can directly translate that into an equation or formula,

Alan H. SteinUniversity of Connecticut

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SLIDE 32

Another Equation or Formula

Looking through the variables, it’s also clear there should be a way

  • f calculating the sales tax if we know the tax rate and the sales
  • price. In fact, we know the tax will equal the tax rate times the

sales price. We can directly translate that into an equation or formula, replacing the tax by T,

Alan H. SteinUniversity of Connecticut

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SLIDE 33

Another Equation or Formula

Looking through the variables, it’s also clear there should be a way

  • f calculating the sales tax if we know the tax rate and the sales
  • price. In fact, we know the tax will equal the tax rate times the

sales price. We can directly translate that into an equation or formula, replacing the tax by T, replacing will equal by an = sign,

Alan H. SteinUniversity of Connecticut

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SLIDE 34

Another Equation or Formula

Looking through the variables, it’s also clear there should be a way

  • f calculating the sales tax if we know the tax rate and the sales
  • price. In fact, we know the tax will equal the tax rate times the

sales price. We can directly translate that into an equation or formula, replacing the tax by T, replacing will equal by an = sign, replacing the tax rate by r,

Alan H. SteinUniversity of Connecticut

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SLIDE 35

Another Equation or Formula

Looking through the variables, it’s also clear there should be a way

  • f calculating the sales tax if we know the tax rate and the sales
  • price. In fact, we know the tax will equal the tax rate times the

sales price. We can directly translate that into an equation or formula, replacing the tax by T, replacing will equal by an = sign, replacing the tax rate by r, replacing times by the multiplication

  • perator ·

Alan H. SteinUniversity of Connecticut

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SLIDE 36

Another Equation or Formula

Looking through the variables, it’s also clear there should be a way

  • f calculating the sales tax if we know the tax rate and the sales
  • price. In fact, we know the tax will equal the tax rate times the

sales price. We can directly translate that into an equation or formula, replacing the tax by T, replacing will equal by an = sign, replacing the tax rate by r, replacing times by the multiplication

  • perator · and replacing the sales price by p.

Alan H. SteinUniversity of Connecticut

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SLIDE 37

Another Equation or Formula

Looking through the variables, it’s also clear there should be a way

  • f calculating the sales tax if we know the tax rate and the sales
  • price. In fact, we know the tax will equal the tax rate times the

sales price. We can directly translate that into an equation or formula, replacing the tax by T, replacing will equal by an = sign, replacing the tax rate by r, replacing times by the multiplication

  • perator · and replacing the sales price by p.

We thus get:

Alan H. SteinUniversity of Connecticut

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SLIDE 38

Another Equation or Formula

Looking through the variables, it’s also clear there should be a way

  • f calculating the sales tax if we know the tax rate and the sales
  • price. In fact, we know the tax will equal the tax rate times the

sales price. We can directly translate that into an equation or formula, replacing the tax by T, replacing will equal by an = sign, replacing the tax rate by r, replacing times by the multiplication

  • perator · and replacing the sales price by p.

We thus get: T = r · p.

Alan H. SteinUniversity of Connecticut

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SLIDE 39

Doing Something with the Formulas

We thus have

◮ C = p + T ◮ T = rp

Alan H. SteinUniversity of Connecticut

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SLIDE 40

Doing Something with the Formulas

We thus have

◮ C = p + T ◮ T = rp

It’s now time to manipulate.

Alan H. SteinUniversity of Connecticut

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SLIDE 41

Doing Something with the Formulas

We thus have

◮ C = p + T ◮ T = rp

It’s now time to manipulate. Looking at the formulas, we hopefully notice that the second gives a formula for one of the variables in the first,

Alan H. SteinUniversity of Connecticut

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SLIDE 42

Doing Something with the Formulas

We thus have

◮ C = p + T ◮ T = rp

It’s now time to manipulate. Looking at the formulas, we hopefully notice that the second gives a formula for one of the variables in the first, so we substitute for rp for T in the first to get

Alan H. SteinUniversity of Connecticut

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SLIDE 43

Doing Something with the Formulas

We thus have

◮ C = p + T ◮ T = rp

It’s now time to manipulate. Looking at the formulas, we hopefully notice that the second gives a formula for one of the variables in the first, so we substitute for rp for T in the first to get C = p + rp.

Alan H. SteinUniversity of Connecticut

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SLIDE 44

Doing Something with the Formulas

We thus have

◮ C = p + T ◮ T = rp

It’s now time to manipulate. Looking at the formulas, we hopefully notice that the second gives a formula for one of the variables in the first, so we substitute for rp for T in the first to get C = p + rp. We may notice we can simplify this to obtain C = (1 + r)p,

Alan H. SteinUniversity of Connecticut

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SLIDE 45

Doing Something with the Formulas

We thus have

◮ C = p + T ◮ T = rp

It’s now time to manipulate. Looking at the formulas, we hopefully notice that the second gives a formula for one of the variables in the first, so we substitute for rp for T in the first to get C = p + rp. We may notice we can simplify this to obtain C = (1 + r)p, but this isn’t strictly necessary.

Alan H. SteinUniversity of Connecticut

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SLIDE 46

Thoughts to Ponder

◮ Most of the work did not involve what most people think of as

math.

Alan H. SteinUniversity of Connecticut

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SLIDE 47

Thoughts to Ponder

◮ Most of the work did not involve what most people think of as

math.

◮ We never really paid attention to what was being asked for,

Alan H. SteinUniversity of Connecticut

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SLIDE 48

Thoughts to Ponder

◮ Most of the work did not involve what most people think of as

math.

◮ We never really paid attention to what was being asked for,

  • ther than noticing at the end that we had found what we

wanted.

Alan H. SteinUniversity of Connecticut

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SLIDE 49

Thoughts to Ponder

◮ Most of the work did not involve what most people think of as

math.

◮ We never really paid attention to what was being asked for,

  • ther than noticing at the end that we had found what we

wanted.

◮ The process, in a way, was mostly information gathering.

Alan H. SteinUniversity of Connecticut