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Dividing Money in Bags – Pattern Recognition

Kids are taught to identify patterns from a very young age. For example, activities like determining “what number comes next in the sequence 1, 3, 5,7”, helps kid notice the similarities and differences in identifying repeating patterns. The same approach can be applied by breaking down complex problems into patterns to eventually solve the larger problem. And, the best way to explain pattern recognition to young kids is by creating a visual representation. We used the same problem solving strategy to solve this critical thinking “Dividing Money in Bags ” math problem by identifying the pattern in the solution.

The “Dividing Money in Bags” Math Problem

A shopkeeper needs to divide 15 pennies into 4 bags. He has to divide the coins over the four bags, so that he can pay any sum of money from 1p to 15p without opening any bag. He can label each bag with the number of pennies inside it.

Can you find out how many pennies he needs to put in each bag?

How to solve it

Give this problem a try and leave your answer in the comments.

Show Solution
 

To visualize the problem, we cut out 4 circles out of paper to represent the bags and gathered 15 pennies.

A bag with 1 penny was easy, since 1 has to be added to zero in order to make itself, we placed one penny on the first circle aka bag.

To make 2 pennies, we had two choices – either place 1 penny on second circle to make two using two ones or make the second bag with two pennies in it. We decided to go with the second choice because it stopped us from using up our bags too quickly and hence, would not give us problem latter.

3 pennies can be made by using bag one and two ( 1 penny + 2 pennies = 3 pennies ).

We can not make 4 pennies with bags 1 and 2, therefore we will need to make bag three with four pennies in it.

dividing money in bags combinations

Furthermore, we can graph the result by continuing on this pattern.

dividing money in bags graph

Since, we cannot make 8 using the previous 3 bags, we need to place 8 pennies in the 4th bag. Therefore, using the 4 combinations of 1, 2, 4 and 8 we can make all the possible combinations from 1 to 15.

money bag final

By visualizing the problem and documenting the solution, helped my kids quickly recognize the pattern and come up with the final solution.

Extension

Finally, can you work on your problem solving skills and find a pattern to solve the “Dividing money in bags” math problem ? How can the shopkeeper divide 7 pennies, 31 pennies, 63 pennies and more ? How many minimum number of bags he requires to make a combination of 7 pennies or 31 pennies and so on ?

We can try making combinations for 7 pennies and see the pattern. As a result, we can easily generalize it. So, for any number of ‘n’ bags, we can make 2^n – 1 combinations. For example, for 3 bags we can make 7 combinations, for 5 bags we can have 31 combinations and so on ).

Another way of looking at the solution is to express the numbers in binary format (i.e. base 2), where a 1 means include the bag with 2^n pennies, and 0 means don’t. For example,

The number 11 can be represented as 1011 ( base 2 ) in binary format.

1011 = 2^3 + 0^2 + 2^1 + 2^0 = 8 + 0 + 2 + 1

Therefore, it implies that we can get 11 pennies from Bag 4, 2 and 1.

Similarly, you can write all the numbers from 0 to 15 with 4 binary digits.

Further Readings

Moreover, if you are looking for ideas to teach your kids complex math concepts like Fibonacci, Symmetry and more in an easy and interesting way, then check some of them here.

  • Fibonacci Series
  • Symmetry – Fold and Hold Sequence
  • Geometry – Sum of all angles of a Triangle
  • Problem Solving – A Bag of Marbles

And while you are here, try out some of the brain teasers and matchstick puzzles.

If you are looking for books on logical thinking and problem solving then these are some great picks.

If you would like to stay connected then please do follow me on my facebook page My World Their Way or join my facebook group Moms Who Inspire of like minded moms who instill a love of learning in kids through fun activities.

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