Standards - Mathematics

Models such as population growth should be recognized as recursively developed models.
The recursion process can be applied to many situations.
A sequence lists the solutions of a set of related problems.

Vocabulary

Difference equation
Recursive process
Recursive formula
Fractals
Population growth models
Sequences
Series

Aligned Learning Resources

Classroom Connection: Benchmark Fractions

Mathematics (2019) Grade(s): 09-12 - Applications of Finite Mathematics

MA19.FM.14
Use mathematical induction to prove statements involving the positive integers.

COS Examples
Examples: Prove that 3 divides $2^{2n}- 1$ for all positive integers n; prove that $1 + 2 + 3 + … + n = n(n + 1)/2$; prove that a given recursive sequence has a closed form expression.

Unpacked Content

Knowledge

Students know:

How to find equivalent expressions.

Skills

Students are able to:

Show that a statement is true for the first case, generally n=1.
Show that a statement is true for n=k+1 if it is assumed that the statement is true for n=k.

Understanding

Students understand that:

Proof by induction is a way of proving statements that includes two steps.

Vocabulary

Proof by mathematical induction

Mathematics (2019) Grade(s): 09-12 - Applications of Finite Mathematics

MA19.FM.15
Use mathematical induction to prove statements involving the positive integers.

COS Examples
Examples: Prove that 3 divides $2^{2n}- 1$ for all positive integers n; prove that $1 + 2 + 3 + … + n = n(n + 1)/2$; prove that a given recursive sequence has a closed form expression.

Unpacked Content

Knowledge

Students know:

How to calculate combinations.

Skills

Students: Use recursive pattern to construct Pascal’s triangle.

Compare combinations to each row of Pascal's triangle to identify each row as the set of all combinations for a given set of objects.

Understanding

Students understand that:

Each row in Pascal’s triangle is the number of combinations of N take r where N is the row of the triangle starting with N=0 and r is the entry in the row from left to right.

Vocabulary

Pascal's Triangle
Recursion
Combinations

Standards - Mathematics

MA19.FM.11 Find patterns in application problems involving series and sequences, and develop recursive and explicit formulas as models to understand and describe sequential change. COS ExamplesExamples: fractals, population growth

COS Examples

Knowledge

Skills

Vocabulary

MA19.FM.12 Determine characteristics of sequences, including the Fibonacci Sequence, the triangular numbers, and pentagonal numbers. COS ExamplesExample: Write a sequence of the first 10 triangular numbers and hypothesize a formula to find the nth triangular number.

COS Examples

Knowledge

Skills

Understanding

Vocabulary

MA19.FM.13 Use the recursive process and difference equations to create fractals, population growth models, sequences, and series.

Knowledge

Skills

Understanding

Vocabulary

MA19.FM.14 Use mathematical induction to prove statements involving the positive integers. COS ExamplesExamples: Prove that 3 divides $2^{2n}- 1$ for all positive integers n; prove that $1 + 2 + 3 + … + n = n(n + 1)/2$; prove that a given recursive sequence has a closed form expression.

COS Examples

Knowledge

Skills

Understanding

Vocabulary

MA19.FM.15 Use mathematical induction to prove statements involving the positive integers. COS ExamplesExamples: Prove that 3 divides $2^{2n}- 1$ for all positive integers n; prove that $1 + 2 + 3 + … + n = n(n + 1)/2$; prove that a given recursive sequence has a closed form expression.

COS Examples

Knowledge

Skills

Understanding

Vocabulary

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Grade(s)

Course(s)

Content Area

MA19.FM.11
Find patterns in application problems involving series and sequences, and develop recursive and explicit formulas as models to understand and describe sequential change.

COS Examples
Examples: fractals, population growth

MA19.FM.12
Determine characteristics of sequences, including the Fibonacci Sequence, the triangular numbers, and pentagonal numbers.

COS Examples
Example: Write a sequence of the first 10 triangular numbers and hypothesize a formula to find the nth triangular number.

MA19.FM.13
Use the recursive process and difference equations to create fractals, population growth models, sequences, and series.

MA19.FM.14
Use mathematical induction to prove statements involving the positive integers.

COS Examples
Examples: Prove that 3 divides $2^{2n}- 1$ for all positive integers n; prove that $1 + 2 + 3 + … + n = n(n + 1)/2$; prove that a given recursive sequence has a closed form expression.

MA19.FM.15
Use mathematical induction to prove statements involving the positive integers.

COS Examples
Examples: Prove that 3 divides $2^{2n}- 1$ for all positive integers n; prove that $1 + 2 + 3 + … + n = n(n + 1)/2$; prove that a given recursive sequence has a closed form expression.