Math Monday 3

So Many Ways to Make a Dollar

Age: 5 & up (depending on the challenge you give, even adults could be challenged by these)

Concept/ Skills: Money, Interval Counting, Place Value (foundation), Organizing your Thinking, Flexibility in Thinking, Reasoning & Logic (foundation of proofs), Problem-Solving

Materials: a big pile of various coins

Object: To figure out lots of different ways to make one dollar.

How to play: For a 5-7 year old, simply dump out a bunch of money and ask them, "How many ways do you think you can make a dollar?" Tell him to show each way and leave them all laid out so you can look at them. If the kid pretty good at working on this independently, but you want to up the motivation a bit, you could say, "How many different ways can you make a dollar in 10 minutes?" Then at the end of the time, you can lather on the praise when they have found SO MANY different ways to make a dollar. Or if they answer your original question by saying, "I think I can make a dollar in 12 different ways." Then ask, "how long do you think it'll take you to do that?" "OK, I'll time you to see how long it takes you to come up with 12 ways."

For an older kid who's more familiar with counting money, you can ask the more challenging question, "how can you make a dollar using exactly __ coins?" If you think they really need a challenge, ask them one you don't know the answer to and let them really think through it! Remind them to keep track of what they tried so they'll be able to tell you about the process. Older kids don't even need the actual coins, they can just use paper. Even if they do use coins to figure it out, you might want them to keep track on paper.

How to maximize the math learning going on: Let's say you're doing this with a 5-year-old. You ask the big question, "How many ways do you think you can make a dollar?" and they tell you one way. Xander's first response was the simplest, "A Sacajawea dollar!" Ask, "is that the only way to make a dollar?" He replied, "No, we could do 100 pennies." "OK, let's show all the ways we can make a dollar right here on the floor." He is kind of new to counting money, so I found I had to ask him leading questions like, "How many dimes do you think it would take to make a dollar?" And then, "How can you check to see if that guess is right?" I also needed to guide him to organize the money in rows to make checking your counting easier. We grouped the pennies by 10s, each cluster arranged in a 2x5 array. Doing this together helps him to develop the skill of organizing his thinking and his work, which will help him solve math problems more efficiently. Looking for patterns is a big part of math, and it's hard to see those patterns if you leave everything in a jumbled pile. I also found that while Xander is great at counting by 2s, 5s, and 10s, switching gears in the middle is really challenging for him. It takes practice to actually think about what it means to count by numbers instead of just rattling them off like he memorized. Counting money is really good for this. And learning that concept of one thing (like a dime) being worth more than 1 is great for the developing sense of place value.

Let's say your doing this with a 7-year-old (or maybe older--you'll get the feel for where she is as you discuss it). You gave her 10 minutes to see how many ways she could come up with. When the time is up, ask her to show you her ways and tell you how she came up with them. "Do you think you found every way possible? How do you know? Can you think of a more efficient way to find every single way possible?" Those thought provoking questions really increase the challenge level of the problem. At that point, if they seem ready for it and haven't come up with it on their own, you could suggest creating a table with all of the different coins along one axis and using tally marks to keep track. Point out simple things like, "There is only one way to make a dollar with 4 quarters, so you could mark it like this. But if I try 3 quarters, I could first do the 2 dimes and 1 nickel way. (Mark it.) Then I could mark 3 quarters again (mark it) and use the other coins to make $0.25 in a different way. Is there a different way to do it that still uses 2 dimes?"

Let's say you asked an even older kid to make a dollar using exactly 5 coins. After working on it for a few minutes, he says, "it's not possible." This is one of the best opportunities to stretch his capacity to reason and prepare him for more formal proofs to come. "Can you prove that this is not possible?" If you can prove that it can't be done, then you'll be done with the problem. Make sure you write it down in a way that I can follow what you were thinking." If you ask one that turned out to be too easy for him and he solves it super fast, ask him, "do you think you can come up with a number of coins with which it is not possible to make a dollar?" You can actually do these kind of problems verbally on a road trip or something. If you come up with a challenging math problem for them to think about every time they complain, "Mom, there's nothing to do," they're bound to learn something. (Even if they just learn not to complain to mom about boredom, which is a valuable thing in its own right!)

I just love to get kids thinking! We surely wouldn't want our kids brains to turn to mush, now would we? Or our own, for that matter!