SCORE Mathematics

Standards Connections

How Many Different Paths

Grade Levels: 4

Objective: To teach students logic and problem-solving skills.

Process: Have your students view the "Hearts" problem. Let them try to figure out how many paths there are. If they need help, they can read about Pascal and then read the hint to solve the problem. Print out the worksheet for the students, make an overhead, and have the students explain their solutions to the class. Let them share how the other links were used to solve the problem.

With Pascal's Triangle, they are adding the previous two numbers to get the number directly below the others. You can also show the connection with multiplication. Starting with the first 1, this is 2 to the 0 power, the next two 1's are 2 to the 1st power (2) or in other words 1+1 or 2x1, the next row is 2 to the 2nd power (4) or 1+2+1 or 2x2, etc. continuing through Pascal's Triangle. So that when you reach the word "Hearts", there should be 32 different paths, because it is the 5th row (which is 2 to the 5th power. In other words 2x2=4 x2=8 x2=16 x2=32) Better yet, if you went to the 5th row and added the numbers going across, you would get the same answer (1+5=6 +10=16 +10=26 +5=31 +1=32).

Student Lesson: How Many Different Ways

Evaluation: Give the students another problem, such as "Kites", "Shamrock", etc. with more letters than "Hearts." Have them figure out, by looking at Pascal's Triangle, how many paths they find.

Extensions: Have the students make their own problems to try to stump the class. You could also have them take this information home and assign them to do a problem for homework.

Conclusion: Your students might be really excited about this lesson and want to make other connections with Fractals, Sierpinski's Triangle, Factorials, and other Discrete Mathematics information.

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California Mathematics Academic Standards:

Grade 4:
Statistics, Data Analysis and Probability
2.0 Students make predictions for simple probability situations
2.1 represent all possible outcomes for a simple probability situation in an organized way (e.g., tables, grids, tree diagrams)

Mathematical Reasoning
1.0 Students make decisions about how to approach problems

1.2 determine when and how to break a problem into simpler parts

2.0 Students use strategies, skills, and concepts in finding solutions

2.6 make precise calculations and check the validity of the results from the context of the problem

3.0 Students move beyond a particular problem by generalizing to other situations

3.2 note method of deriving the solution and demonstrate conceptual understanding of the derivation by solving similar problems

NCTM K-4:

STANDARD 1: MATHEMATICS AS PROBLEM SOLVING
STANDARD 3: MATHEMATICS AS REASONING
STANDARD 4: MATHEMATICAL CONNECTIONS
STANDARD 11: STATISTICS AND PROBABILITY