Transcript How Many Valentines?

How Many Valentines?
Question…
There are five friends:
Morgan Ryan Mel Hannah Ben
On Valentine's Day, every friend gives a valentine to
each of the other friends.
How many valentines are exchanged?
You have 4 minutes to figure it out on paper…
Time’s
:04up!
:03
:02
:01
minutes
minute left
left
The answer is 20!
•The answer is 20. And if this isn't the answer you
got, figure out what you did wrong.
•For example, if you got 10, you want to be sure to
count each valentine that you gave a friend as well
as the valentine that friend gave you.
•If you got 25, remember that you don't give a
valentine to yourself.
Problem Solving Strategies
•On the next few slides we will show you 11
strategies for solving the problem.
•There is no one right way. They are all correct! But
some strategies work better than others, depending
on the problem you are trying to solve.
•As we explore each one, think about which
strategy most closely matches the strategy you
used…
Method # 1: Make a List
The names of the friends are Morgan, Ryan, Mel,
Hannah & Ben. Let's list the valentines that each friend
gives, starting with Morgan's valentines.
1.
2.
3.
4.
5.
Morgan gives valentines to: Ryan, Mel, Hannah, & Ben
Ryan gives valentines to: Morgan, Mel, Hannah, & Ben
Mel gives valentines to: Morgan, Ryan, Hannah, & Ben
Hannah gives valentines to: Morgan, Ryan, Mel, & Ben
Ben gives valentines to: Morgan, Ryan, Mel, & Hannah
If you count all the people receiving
valentines on the list it totals 20 valentines.
Method # 2: Act It Out
Get four friends. Now there are five of you.
Give valentines to each other. Then collect all
the valentines and count them.
There are 20. Kinesthetic learners like this
best because they “learn by doing.”
Method # 3: Draw a Diagram or Picture
Suppose the names of the friends are A, B, C, D, and E. You
could draw a diagram to represent the exchanging of
valentines. Everyone’s picture might look different.
Visual learners LOVE this strategy!
Method # 4: Guess and Check
or Trial and Error
Use logical reasoning to think about a reasonable answer. If
there are 5 people giving valentines to one another, there
must be at least 5 valentines given out but no more than 25.
You have just narrowed down the answer to between 5-25.
Method # 5: Make a Model
We’ll use A, B, C, D, and E again as the names of the
friends. We make a model of all the friends. Each pink
square is a valentine. The gray squares show that each
person does not send a valentine to himself or herself.
Count the pink squares
(20). There are 20
valentines exchanged.
You could use actual
different colored blocks to
model the situation as well.
Method # 6: Find a Pattern
• What if there was only 1 person? No valentines are
exchanged…
• What if there were only 2 friends instead of 5? There would
be 2 valentines exchanged.
• Three friends, there would be 6 exchanged.
• Four friends, there would be 12.
What's the pattern? 0, 2, 6, 12,...?
Between the 1st and 2nd numbers is a difference of 2. Between
the 2nd and 3rd, a difference of 4. Between the 3rd and 4th a
difference of 6. And so on. If the pattern were to continue,
the next number would be a difference of 8—and 20
valentines would have been exchanged.
Method # 7: Make a Table
We can use the same information on the previous slide, but
organize the information in a table.
Number of
People
1
2
3
4
5
Valentines
Exchanged
0
2
6
12
?
Method # 8: Make a Graph
We can use the same information, but organize it on a graph.
You can see the number of valentines goes up very quickly as
more people are added to the exchange.
Number of
People (x)
Valentines
Exchanged (y)
1
0
2
2
3
6
4
12
5
?
Method # 9: Write an Equation
Let’s do some algebra! You know that if there are n people, each
will send out n - 1 valentines (because you don’t send one to
yourself, right?)
So the total number of valentines v is:
v = n (n - 1)
Because we have 5 friends, n = 5 in this situation.
v = 5 x (5-1)
v=5x4
v = 20
This may seem like a lot of work to solve this problem, but think
how much easier this would be if you had to do the same problem
for 100 friends. Do you really want to list out all the people, or
create a grid, or a star for that? Try using this strategy now to
determine the number of valentines for 100 friends…
Method #9 continued:
Use the Equation
Using our equation: v = n (n - 1)
In the case of 100 friends, n = 100 in this situation. So,
v = 100 x (100-1)
v = 100 x 99
v = 9,900
Now isn’t that easy?
Method 10: Use Simpler Numbers
• What if the original problem didn’t involve only 5 friends,
and instead the entire sixth grade at Lovinggood? This
would be an excellent opportunity to use this strategy. By
using smaller numbers (like 5) you can quickly see the
pattern or come up with an equation to use for larger
numbers.
• This is also an excellent thing to do when you have
problems that include fractions and decimals which can
be harder to work with.
Method 11: Work Backwards
Questions…
•So, which strategy did you use originally?
•Did you use a strategy that was not shown?
•Which strategy would you use if you had to do the
same problem again? What if the problem involved
LARGER numbers?
•Why is it important to learn many ways to solve a
problem?
Problem Solving Plan
1.
2.
3.
4.
5.
6.
Understand the question
Find relevant information
Make a plan
Take action
Look back
Explain