Reasoning About Sums and Factors

COUNTING THINGS YOU CAN'T SEE

This activity is an exercise in numeracy and also in spatial reasoning.



Counting things you can't see

THE 10-BEANS PROBLEM

Students reason about ways to distribute 10 beans, giving them practice with make-10 combinations, divisibility, and evens/odds.




GIRLS PLAYING BALL

This activity was adapted from a puzzle in Boris Kordemsky's The Moscow Puzzles.  It gives young students a visual way to explore factors of 12.




THE TRIANGLE PUZZLE

The Triangle Puzzle is a great exercise in problem-solving for young children.  I like this puzzle especially because there are multiple solutions, and by finding one or two, students will generalize and reason about what other solutions there must be.


SAME-SUM PUZZLES

The "same-sum puzzles":  These are different versions of the triangle puzzle.  Students try to place consecutive numbers so that they get the same sum along each line of the puzzle.  Also like the triangle puzzle, some of the puzzles have multiple, related solutions.



THE DOMINO WINDOW PUZZLE

This puzzle comes from Kordemsky's The Moscow Puzzles.  It can be used even with young students who are not yet fluent with single-digit addition.




DIVIDING TIME

This classic puzzle is good for problem solving and for practicing addition, and it can also be used as a springboard to talk about Gauss' strategy for finding the sum of an arithmetic sequence.




CRYPTARITHMS

These are simple versions of the classic puzzle.  Students reason about which digit each letter represents.


MULTIPLICATION PUZZLES (missing digits)

Reasoning about missing digits, missing factors.



FACTOR PUZZLES

These "Factor Puzzles" give young students an informal introduction to factor trees and have them reasoning and making discoveries about primes before they even learn the definition of a "prime number."  Students should be told simply (1) that every number is the product of the numbers that branch off of it from below, (2) and that they are not permitted to use the number 1 in any of the spaces.

Note:  The puzzles start very easy and get quite hard.  The hardest puzzles are for more advanced students who are comfortable reasoning about prime factors.



THE TWO-BAGS PROBLEM


This puzzle (with a different visual presentation) came from mathpickle.com.  Students are asked to put the numbers 1 through 8 in two "bags" such that no number is the sum of any other two numbers in the same bag.  There is a single solution to this problem.  As an extension, ask students how many numbers they can place in three bags. (Start with 1 and count forward, placing the numbers with the same restriction that no number is the sum of any other two numbers in the same bag).  This problem is much harder, and is a great extension for those who find the solution to the two-bag problem.




NUMBER-SENTENCE PUZZLES
These puzzles are a gentle introduction to problem-solving for young children who don't yet have the stamina for some of the puzzles above.  Students also gain experience with the inverse relationships between addition/subtraction and multiplication/division as they reason about the missing numbers.


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