Introduction: Developing "Number Sense" With Solar Powered, Recycled, Numeric Devices
Two trains leave the station at the same time on parallel tracks. One travels 560 miles in the same amount of time it takes the second one to travel 630 miles...and why do we care?
Did you spot anything special about the numbers used in this problem (that isn't a problem)? If you have good "number sense" you most likely noticed that 2, 5 and 7 are factors of both numbers. If you didn't catch that, you might want to give this instructable a try.
Students working on multi-step math problems often struggle reaching the correct answer because they have to stop and calculate a small part of the problem and by the time they have that worked out they don't remember what they needed it for. If we can train their brains to solve basic math equations without conscious effort math will get easier for them.
I developed this activity about 10 years ago in my tutoring business. I have used it successfully with students from 5 to 75. The specialized, "Solar Powered, Recycled, Numeric Devices" needed are two decks of ordinary playing cards!
Step 1: Getting Started
First- Get two decks of playing cards
Second- Remove all Kings, Queens, Jacks and Jokers (except for that one kid in the back row. He may learn something here!)
I use the cards as random number generators. I prefer two decks of cards because of the greater possible card sets. I use the aces as 1's. That gives me eight sets of the digits 1-10. This number works very well for three or four students to play together.
It is not necessary to shuffle too much or to deal the cards out one at a time to the students. Students don't even need to hide their cards from each other. Some of the best teaching in this game comes from one student seeing something a neighbor can do with their cards and giving them a clue (i.e. "I see a way you can make it with four cards!")
Step 2: Creating Equations With Single Digit Answers
Fourth- Turn over the next card (or cards) as the "answer card"
Here, you see the hands of four players as well as "the answer" they are seeking. In this case, the "2" is the answer. It is the job of the students to devise equations with that answer using the cards in their hands. We often talk about factors as we begin to analyze the answer card. That becomes more important when working with double digit answers. For now, we will keep it simple.
This works well to introduce various operations to young students because each card has built-in counting points. Today, I had a 5 year old doing division!
The goal: Use as many cards as possible to create as many equations as possible but only use each card one time! That implies that since an equation requires at least two cards, no more than two equations can be created in any one hand. It is possible to create two equations using either four or five cards. For our purposes, the player who used the most cards to create the most equations wins the round and collects the cards of the other players. These are simply piled aside to serve as a score keeping device.
Possible problems in order from best to...not so best:
- Two separate equations using all five cards (one with 2 cards and one with 3 cards)
- Two separate equations using four cards
- One equation using all five cards
- One equation using four cards
- One equation using three cards
- One equation using two cards
Let's look at some of the possible problems we can create with these cards.
Given the hand 10, 6, 3, 2 & 2...
We can't use the other 2 or the 6
One equation using 3 cards.
One equation using 4 cards.
Division of like numbers always results in 1 so, in this case, 2 divided by 2 equals 1
Subtraction: 3 minus the 1 created above equals 2
We have created one equation using all five cards. That is hard to beat but not impossible.
Addition of the above results: 1+1=2
Same results as the previous try. One equation using all five cards.
Depending on the level of students, you may want to have a conversation about "order of operations". Since our tables have glass tops, I often give a student a dry-erase marker and have them lay out their cards as a proper equation with parenthesis and all.
Step 3: A Few More Examples With Single Digit Answers
Let's look at two more example hands.
Given the hand 10, 9. 7, 7, 6 there are two possible equations
9-7=2 and 6/(10-7)=2
This hand is unbeatable with two equations using all five cards.
The same hand can produce more options.
9-7=2 and 6/(10-7)=2
Again, two equations using all five cards can not be beaten.
Our last hand does not fair too well.
8/4=2 and another 8/4=2 is pretty good but they can not use their 5.
It is hard to explain how exciting it is to watch a child who has little, or no, number sense as they start catching on to the relationships of numbers using this simple card game. Today, I was working with a new student who is in the 8th grade. She struggles with math but, today, there was a spark in her eye as she played. A boy in her tutoring session is in the same school. The girl asked which lunch period he had lunch. When she learned that they shared the same lunch time she said, "We should play this at lunch. We could teach it to our friends!" Who needs to get paid money when students make comments like this?
Step 4: Expand the Equations Via Double Digit Answers
Expanding the "Answers" to double digits introduces a new set of concepts and challenges. We determine if the number is prime or not and if it can be the product of just two of our cards. If it is prime or requires a number larger than 10, we know that there is no way to make more than one equation this round. Remember, we can only use each card one time. It may be possible to make more than one equation but we must choose only one to play that round.
Our target "Answer" this round is 49.
With the hand of: 10, 5, 4, 3, 1
A perfect hand.
With the hand of: 9, 6, 6, 2, 1
Another perfect play using multiples to get close to the answer.
The hand: 10, 8, 3, 1, 1
Factoring sure helps.
Step 5: You Can Play It by Yourself Too!
The Solitaire Game-
If you want to practice your math facts to build your math sense but no one will play with you, here's the way to do it.
Deal just 4 cards, face up. These are the cards you use to build your equation. Flip over the top card on the deck. This is your "answer card". When you find the solution, remove only the cards you use for the equation including the answer card. Turn over new cards until you have 4 cards on the table again. Flip the next card for the answer and start again. You "win" the game when you remove every card from the deck and the table with your final equation.
When teaching the solitaire game to my students, I show them the way to play and guide them through a few rounds. Once I think they have the idea, I turn the 4 cards for the equation but do not turn an answer card. Then, we check their understanding by asking if they can find a way to make each digit (1-10) from the 4 cards in play. Here's what that looks like:
If the equation cards are 1, 4, 5 and 7 what can you make? Remember, you must use at least 2 cards for each equation. The results:
We made it! It is not always that easy.
Try this set yourself:
4, 6, 7, 10
I don't know about you but finding an equation for 6 was tough.
One more for you:
8, 8, 9, 10
I can't figure out how to make a 4 or a 5, but the rest of them are there.
Step 6: In Closing...Bring on the Smiles!
Number sense makes sense. I have played this game with hundreds of students and their parents. Many of these students took a timed test on the times tables (up to 12x12) on the same day I taught them the MathCards game. The students who played the game often and practiced their times tables also, often were able to complete the tables in less than half the time required on their first try. It is so much fun to see a student "get it".
With this game, you can prove that "having fun can teach you something!" Trent believes it. Just look at the pride on his face when his Math grades started climbing.
We have a be nice policy.
Please be positive and constructive.