options: Functions: students learn to define, use, and evaluate functions to model relationships between quantities.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.
problem_activity: Activity
The goal of this activity is for students to understand that the way an equation is written to define a function depends on how the domain of the function is interpreted. In the previous activity, starting the sequence
\(C\)
with
\(C(0)\)
made sense since
\(n\)
was defined as the number of cuts while
\(C(n)\)
is the area of a paper square after the cuts. In this activity, students consider two equations written for a sequence represented by a visual pattern. One of the equations assumes the sequence starts at Step 0 and the other assumes the sequence starts at Step 1.
To help all students understand that both equations generate the same sequence and correctly represent the visual pattern, during the activity and whole-class discussion students should be encouraged to use precise language as they explain why an equation is valid (MP6). Confusion is likely to arise unless students are clear about the definition for
\(n\)
each equation is assuming. Some students may find the idea that both equations are equally valid challenging, so it is important to take time during the discussion for these students to understand that part of mathematics is recognizing what assumptions you are making and addressing them appropriately.
Monitor for students with clear definitions to select to share during the whole-class discussion. In particular, students that create different representations, such as tables, to see that the two equations result in the same list of numbers as their recursive definitions should be highlighted.
Making a spreadsheet available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Arrange students in groups of 2.
Student Facing
A Sierpinski triangle can be created by starting with an equilateral triangle, breaking the triangle into 4 congruent equilateral triangles, and then removing the middle triangle. Starting from a single black equilateral triangle:
[IMG]
Let
\(S\)
be the number of black triangles in Step
\(n\)
. Define
\(S(n)\)
recursively.
Andre and Lin are asked to write an equation for
\(S\)
that isn't recursive. Andre writes
\(S(n)=3^n\)
for
\(n\ge0\)
while Lin writes
\(S(n)=3^{n-1}\)
for
\(n\ge 1\)
. Whose equation do you think is correct? Explain or show your reasoning.
Student Response
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Student Facing
Are you ready for more?
Here is a geometric sequence. Find the missing terms.
3, ____, 6, ____, 12, ____, 24
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
Students may assume that at least one person has to be wrong because their equations aren't equivalent. Encourage them to make a table for each equation that starts with the first value for
\(n\)
according to the definitions.
Activity Synthesis
The goal of this discussion is for students to understand why Andre and Lin's equations are both representations of the visual pattern due to the interpretation of
\(n\)
. Display the visual pattern for all to see. Invite previously selected students to share their reasoning about why either Andre or Lin is correct, recording student reasoning for all to see. If students still think starting with 0 is unusual after the share out, give some more time for students to reason about Andre's point of view, and then select students to share. (For example, Andre could be thinking about
\(n\)
as the breaking apart phase, so for the solid black triangle that has happened 0 times.)
Conclude the discussion by telling students that identifying an appropriate domain for a function is partly dependent on the situation and partly dependent on how they see the relationship. There are often many correct equations that represent a function. An important takeaway for students is that when they write an equation to represent a situation, they need to be clear what domain they have identified so other people can correctly interpret what they've done.
Writing, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to prepare students for the whole-class discussion by providing them with multiple opportunities to clarify their explanations through conversation. Before the whole-class discussion begins, give students time to meet with 1–2 other groups to share their response to the question, “Whose equation do you think is correct?” Invite listeners to ask questions, and to press for details and mathematical language.
Design Principle(s): Optimize output (for explanation); Cultivate conversation


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