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How Many Valentines?

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**:02 :01 :03 :04 Time’s up! Question… minutes left minute left**

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… :02 minutes left :01 minute left :03 minutes left :04 minutes left Time’s up!

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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.

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**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…

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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. 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.

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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.”

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**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!

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**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.

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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.

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**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.

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**Method # 7: Make a Table Number of People Valentines Exchanged 1 2 3 6**

We can use the same information on the previous slide, but organize the information in a table. Number of People Valentines Exchanged 1 2 3 6 4 12 5 ?

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**Valentines Exchanged (y)**

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 2 3 6 4 12 5 ?

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**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 = 5 x 4 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…

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**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?

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**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.

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**Method 11: Work Backwards**

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**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?

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**Problem Solving Plan Understand the question Find relevant information**

Make a plan Take action Look back Explain

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