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Week 5: Algebra - Patterns


Gretel's Goldfish (Printable PDF)

Hansel has goldfish that quadruple, or become four times as many, every month. Gretel has goldfish that increase by 20 every month. Right now, Hansel has 4 goldfish and Gretel has 128 goldfish. In how many months will they have the same number of goldfish? Show how you arrived at your answer.
 

GPS

As seen in Problem Exploration

M6A2: Students will consider relationships between varying quantities.

a. Analyze and describe patterns arising from mathematical rules, tables, and graphs.
 

Using a spreadsheet, the number of goldfish for Hansel and Gretel can be determined. There is a distinct difference between Gretel's function that increases arithmetically and Hansel's function that increases geometrically.

 

M6A3: Students will evaluate algebraic expressions, including those with exponents, and solve simple one-step equations using each of the four basic operations.

Both Hansel's and Gretel's patterns can be written as one-step equations, in which the number of goldfish is a function of the month.

 

M7A1: Students will represent and evaluate quantities using algebraic expressions.

a. Translate verbal phrases to algebraic expressions.

b. Simplify and evaluate algebraic expressions, using commutative, associative, and distributive properties as appropriate.
 

The number of both Hansel's and Gretel's goldfish can be written in the form of equations. These patterns can be written as algebraic expressions or equations and then evaluated.

 

M7A2: Students will understand and apply linear equations with one variable.

c. Use the addition and multiplication properties of equality to solve one- and two-step linear equations.
 

The number of goldfish that Gretel has is an arithmetic pattern, which can be written as a linear equation, G = 128 + 20t, where G is the number of goldfish and t is the month.

 

M7A3: Students will understand relationships between two variables.

a. Plot points on a coordinate plane.

b. Represent, describe and analyze relations from tables, graphs, and formulas.

c. Describe how change in one variable affects the other variable.

d. Describe patterns in the graphs of proportional relationships, both direct (y=kx) and inverse (y=k/x).
 

Gretel's goldfish can be expressed as an arithmetic function and Hansel's goldfish can be written as an exponential function. Both functions can be written as equations. Gretel: G = 128 + 20t, where G is the number of goldfish and t is the month. Hansel: H = 4*(4^t), where H is the number of goldfish and t is the month. These equations can be evaluated and graphed.

 

M8N1: Students will understand different representations of numbers including square roots, exponents, and scientific notation.

d. Use appropriate technologies to solve problems involving square roots, exponents, and scientific notation.
 

The number of Hansel's goldfish is a geometric function, H = 4*(4^t), where H is the number of goldfish and t is the month. This can be evaluated using a calculator or spreadsheet.

M8A1: Students will use algebra to represent, analyze, and solve problems.

b. Simplify and evaluate algebraic expressions.

c. Solve algebraic equations in one variable, including equations involving absolute values.
 

Both patterns are functions, which can be written as equations, graphed and evaluated during this investigation.

M8A3: Students will understand relations and linear functions.

i. Identify relations and functions as linear or nonlinear.
 

The number of Gretel's goldfish is an arithmetic pattern, which can be written as a linear equation. Hansel's goldfish is a geometric pattern and can be written as an exponential (nonlinear) equation.

 

M8A4: Students will graph and analyze graphs of linear equations.

a. Interpret slope as a rate of change.

b. Determine the meaning of the slope and y-intercept in a given situation.

c. Graph equations of the form y = mx + b.
One of the patterns in the investigation is an arithmetic pattern, which can be written as a linear equation, G = 128 + 20t. This can also be written as y= 20x+128. The y-intercept is 128, which is the number of goldfish that Gretel started with. The slope is 20, which is the number of goldfish that Gretel gains when t increases by 1. In essence, each month Gretel gets 20 more goldfish.