Polya's Four Steps in Problem Solving (1.3)
Summary
TLDRIn this video, Professor Monty explains Polya's four-step problem-solving method: understanding the problem, devising a plan, carrying out the plan, and reviewing the solution. He demonstrates this method using a practical example of comparing the cost-effectiveness of two products—a jar of honey and a showerhead—by calculating their cost per ounce. He highlights the importance of reviewing the solution to ensure it makes sense and concludes by discussing how smaller quantities can sometimes offer a better deal. The video encourages viewers to explore more lessons in various math topics.
Takeaways
- 🧠 Understand the problem: The first step is to comprehend what the problem is asking, often by rereading it and focusing on the last sentence.
- 📝 Devise a plan: Identify patterns or necessary information and decide how to approach the problem.
- 🔧 Carry out the plan: Solve the problem based on the devised method.
- 🔍 Review your solution: After solving, check the answer to ensure it makes sense and correct any errors if necessary.
- 🍯 Example problem: Compare two products—a jar of honey and a showerhead—to determine the better deal based on cost per ounce.
- 📊 Method of calculation: The plan involves calculating cost per ounce by dividing the price by the number of ounces for each product.
- 💸 Honey calculation: The 8-ounce jar of honey costing $2.19 results in about 27 cents per ounce.
- 🚿 Showerhead calculation: The 12-ounce showerhead costing $3.75 results in about 31 cents per ounce.
- ✅ Conclusion: The 8-ounce jar of honey is the better deal, being cheaper per ounce.
- 🤔 Surprising result: Although larger quantities usually cost less per unit, in this case, the smaller jar offers the better value.
Q & A
What are Polya's four steps in problem solving?
-Polya's four steps in problem solving are: 1) Understand the problem, 2) Devise a plan, 3) Carry out the plan, and 4) Look back and check the solution.
What should you do if you don’t understand the problem at first?
-If you don't understand the problem at first, it's important to reread the problem, identify the key question, and focus on understanding what is being asked.
How can you devise a plan to solve a problem according to Polya’s method?
-To devise a plan, you should identify relevant patterns, understand what information is needed, and determine the steps or approach you will use to solve the problem.
What is the example problem used in the script?
-The example problem in the script is determining which is a better deal between an 8-ounce jar of honey for $2.19 and a 12-ounce jar for $3.75.
How does the speaker devise a plan to solve the example problem?
-The speaker's plan is to calculate the cost per ounce for both options by dividing the price by the number of ounces, and then compare the two values.
What is the calculated cost per ounce for the 8-ounce jar of honey?
-The cost per ounce for the 8-ounce jar of honey is approximately $0.27 or 27 cents per ounce.
What is the calculated cost per ounce for the 12-ounce jar of honey?
-The cost per ounce for the 12-ounce jar of honey is approximately $0.31 or 31 cents per ounce.
Which jar of honey is the better deal, and why?
-The 8-ounce jar of honey is the better deal because it costs 27 cents per ounce, which is cheaper than the 12-ounce jar, which costs 31 cents per ounce.
Why might the result of the problem be surprising according to the speaker?
-The result might be surprising because people generally expect a volume discount, meaning the larger jar should be cheaper per ounce. However, in this case, the smaller jar is actually cheaper.
What does the speaker recommend doing after solving the problem?
-The speaker recommends looking back at the solution to ensure it makes sense, checking for any errors, and confirming whether the answer is reasonable.
Outlines
📚 Problem Solving with Polya's Four Steps
Professor Monty introduces Polya's four-step method for problem-solving, emphasizing the importance of understanding the problem first. He explains that often the key to the problem lies in the last sentence of the problem statement. The second step is to devise a plan, which could involve looking for patterns or determining what information is necessary or irrelevant. The third step is to execute the plan to solve the problem, and the fourth step is to review the solution to ensure it makes sense. To illustrate these steps, he presents a problem comparing the value of an 8-ounce jar of honey for $2.19 versus a 12-ounce jar for $3.75. He suggests calculating the cost per ounce to determine which is the better deal.
🔍 Analyzing the Better Deal
Continuing from the previous paragraph, Professor Monty calculates the cost per ounce for both options. For the 8-ounce jar, he finds it to be approximately 27 cents per ounce, and for the 12-ounce jar, it's about 31 cents per ounce. He concludes that the 8-ounce jar offers a better deal. He then reflects on the result, noting that it's a bit surprising since typically buying in bulk should offer a lower cost per unit. However, given the closeness of the numbers, he finds the outcome reasonable. He encourages viewers to like and subscribe for more educational content, mentioning that he covers a variety of subjects including statistics, algebra, calculus, and liberal arts.
Mindmap
Keywords
💡Polya's Four Steps
💡Understand the Problem
💡Devise a Plan
💡Carry Out the Plan
💡Review the Solution
💡Cost per Ounce
💡Volume Discount
💡Mathematical Reasoning
💡Rereading the Problem
💡Surprising Results
Highlights
Introduction to Polya's four-step problem-solving process
Step 1: Understand the problem by reading and rereading
Emphasizing the importance of understanding the problem's core question
Step 2: Devise a plan to approach the problem
Focusing on identifying patterns and necessary information
Step 3: Carry out the plan to solve the problem
Step 4: Review the solution to ensure it makes sense
Example problem: Comparing the cost-effectiveness of two products
Understanding the problem: Determining which product offers a better deal
Planning to calculate the cost per ounce for each product
Calculating the cost per ounce for the 8-ounce jar of honey
Calculating the cost per ounce for the 12-ounce showerhead
Comparing the cost per ounce to determine the better deal
Surprising result: The smaller jar of honey is cheaper per ounce
Reviewing the calculation to ensure accuracy and logical consistency
Conclusion: The 8-ounce jar of honey is the cheaper option
Encouragement to like and subscribe for more educational content
Invitation to engage with a variety of educational topics on the channel
Transcripts
hi everyone it's professor monty here
and let's talk about paulia's four steps
in problem solving
so the first step as you can see i
understand the problem so we need to
read the problem figure out what's going
on you probably know as well as i do a
lot of times you read a problem you're
like i don't even know what they're
asking so obviously we may need to
reread the problem and really understand
what the problem is asking us to do a
lot of times it's the very last sentence
in a paragraph is what the real question
is
devise a plan so whether this is looking
for a pattern understanding what's
needed and what's not needed
all the different things that we need to
look at to try to figure out okay how am
i going to
approach and figure out this problem
that was stated
and then three we have to carry out that
plan and actually solve the problem
and then four once we've solved the
problem let's look back check our answer
make sure it makes sense
so that we can see if he does
did i do this right was there something
i was looking at wrong if it doesn't
make sense we need to look back and see
if maybe we did something wrong okay i'm
going to erase this let's actually do a
problem where we do these steps
okay so i've got a problem set up for us
which is a better deal
okay that's the question our options are
an 8 ounce
jar of honey
for 219
or
a 12 ounce showerhead
375.
put a question mark right there there's
our question
okay so that's the entire question and
so we've got to figure out okay
first of all let's understand what
they're asking this one's pretty
straightforward so
understanding isn't going to be that big
of a deal okay i see what they're saying
which is a better deal which leads us
into okay we need to devise a plan for
how we're going to figure this out
well let's see how much maybe
the honey cost per ounce
and so that's my plan so we're going to
say let's figure out the cost
per ounce
and mathematically that's going to be
cost
divided by ounces and let's just compare
the two
so the first one the cost is 219.
the ounces was eight ounces
and then we can calculate that we'll
compare that to the other one which is
375
for 12 ounces
okay i've got a calculator right here
let's go ahead and figure that out so i
just go 219 divided by 8
and i get
0.27375
now this is in cents i'll just round to
the nearest sense it's about we'll
change this to approximately
it's 0.27 so about 27 well if we're in
dollars still it's about 0.27 per ounce
and notice mathematically dollars is on
top dollars on top ounces on the bottom
ounces are on the bottom so that's 27
cents per ounce
a lot of times we write this this way
it's 27
cents per ounce if we want to do that i
got rid of the decimal i could change it
to from dollars to cents or you could
just say it's
0.27 dollars per ounce whichever way you
want to look at
okay let's look at the other one
3.75
divided by 12.
and that's 0.3125 so about 31
about
31 cents per ounce
well if we're looking for the better
deal obviously we want to want to do
and so we say oh this is the better deal
the eight ounce jar
for 219. now let's see if that makes
sense
we expect usually that
when they charge us to buy something if
we buy more of something or be cheaper
per ounce or per whatever they're
selling
well these aren't that far apart so it
seems reasonable
is it surprising maybe it is a little
surprising we expect what they call
volume discounts where we don't get one
this time this time the
fewer ounces the smaller jar is actually
cheaper per ounce
if these were widely
different numbers then i think maybe i
did something wrong
i don't think i did anything wrong here
so we would say oh using the method i've
looked back it maybe it's a little
surprising but it's reasonable since
these numbers are pretty close together
and so in this case the eight ounce
charcoaling was the cheaper one
so i hope you like this little example
of
polya's
four-step
procedure go ahead and like this if you
like this so other people can
can see it can find it if you want to
subscribe to my channel that would be
great i've got a lot of videos out there
not only in this class but in statistics
algebra calculus
the variety of classes this is a matter
of liberal arts
hopefully that helps you out hopefully
we can get a good connection and i'll
talk to you next time
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