Download Sequences - Mater Academy Lakes High School

CARE
Curriculum Assessment Remediation Enrichment
Algebra 2
Mathematics CARE Package #9
Domain Functions: Building Functions
Cluster Build a function that models a relationship between two quantities
Standards MAFS.912.F-BF.1.1a
Write a function that describes a relationship between two quantities.
a. Determine an explicit expression, a recursive process, or steps for
calculation from a context
MAFS.912.F-BF.1.2
Standards
Write arithmetic and geometric sequences both recursively and with an explicit
formula, use them to model situations, and translate between the two forms.
Domain Algebra: Seeing Structure in Expressions
Cluster Write expressions in equivalent forms to solve problems
Standards MAFS.912.A-SSE.2.4
Derive the formula for the sum of a finite geometric series (when the common ratio
is not 1), and use the formula to solve problems. For example, calculate mortgage
payments.
CURRICULUM
Performance Task
In a video game called Snake, a player moves a snake through a square region in the plane, trying to eat the white
pellets that appear.
If we imagine the playing field as a 32-by-32 grid of pixels, then the snake starts as a 4-by-1 rectangle of pixels, and
grows in length as it eats the pellets:
 After the first pellet, it grows in length by one pixel.
 After the second pellet, it further grows in length by two pixels.
 After the third pellet, it further grows in length by three pixels.
 etc.
Let L(n) denote the length of the snake after eating n pellets. For example, L(3)=10.
1. How long is the snake after eating 4 pellets? After 5 pellets? After 6 pellets?
2. Find a recursive description of the function L(n).
3. Find a non-recursive expression for L(100), and evaluate that expression to compute L(100).
4. What is the largest number of pellets a snake could eat before he could no longer fit in the playing field? That
is, how long is a perfect game of snake?
ASSESSMENT
The Mini-MAF includes standards MAFS.912.F-BF.1.1a, MAFS.912.F-BF.1.2, and MAFS.912.ASSE.2.4. Use the following table to assist in remediation efforts.
Questions
1-6
7-9
10-12
Standards
MAFS.912.F-BF.1.2
MAFS.912.F-BF.1.1a
MAFS.912.A-SSE.2.4
Lesson
9-2 Series and Summation Notation
9-3 Arithmetic Sequences and Series
9-4 Geometric Sequences and Series
9-1 Introduction to Sequences
9-5 Mathematical Induction and Infinite Geometric
Series
REMEDIATION / RETEACH
Key Vocabulary – Sequences and Series
Sequence
Terms of a sequence
Series
Summation nation
Sigma notation
Arithmetic sequence
Common difference
Arithmetic series
Explicit rule
Recursive rule
Iteration
PERFORMANCE TASK Considerations & Solution Guide
This task has students approach a function via both a recursive and an algebraic definition, in the context of a famous
game of antiquity that they may have encountered in a more modern form. The content underlying the algebra is the
sum of the first n natural numbers (also known as the nth triangular number):
1+2+3+⋯+n=n(n+1)2.
In addition to the general notions of recursive and algebraic functions, the task could be used either
as a motivation for, or an application of, these types of sums. As an introduction to triangular
numbers that might proceed this task, see http://www.illustrativemathematics.org/tasks/1830.
Students may need help in understanding the play of the game, especially with regard to the turning
and growing mechanism of the snake. It may be helpful to provide visual aids for this -- for
example, the motivation for this task was the animated gif http://i.imgur.com/dAtcCfH.gif showing
the perfect game addressed in the question (shows that indeed the maximum length can be
achieved!).
1. Since L(3)=10, we can most easily compute L(4) by adding 4 to L(3), to get
L(4)=L(3)+4=10+4=14, rather than starting from scratch, that is by calculating
L(4)=4+1+2+3+4=14. Similarly, we find L(5)=L(4)+5=14+5=19, and L(6)=L(5)+6=19+6=25.
2. Generalizing the previous part, to compute the length after n pellets, we take the length
after n−1 pellets, and add on n more pixels. Algebraically, this reads
L(n)=L(n−1)+n.
To complete the specification of a recursive function, we also need to include a starting
value, which we are given as L(0)=4.
3. The recursive definition gives L(100)=L(99)+100, but this doesn't particularly help for
computing this value. Instead, the "start from scratch" method gives
L(100)=4+1+2+3+⋯+98+99+100.
One particularly neat method for evaluating this proceeds as follows:
1+2+3+⋯+98+99+100=(1+100)+(2+99)+(3+98)+⋯+(50+51)=50⋅ 101=5050.
We conclude that L(100)=4+5050=5054. After 100 pellets, the snake will be 5054 pixels
long.
4. We begin by noting that the 32-by-32 grid of pixels contains 322=1024 pixels, so by the last
part the snake definitely runs out of room by the time it eats 100 pellets. We are looking for
the largest number n such that L(n)<1024, which algebraically takes the form
4+1+2+3+⋯+n<1024.
Here we can either repeatedly apply the trick of the last part of this problem to solve for n
by trial-and-error, or generalize the previous part of the problem to arrive at the formula
1+2+3+⋯+n=n(n+1)2:
the method from part (c) applies directly if n is even while if n is odd we find n−12 groups of
n+1 along with the middle number n+12 and, adding these up, we check that the formula still
holds. Substituting this formula into L(n)<1024, we need to solve
4+n(n+1)2<1024,
or n(n+1)<2040. Again, we have a variety of methods for finding the largest such n, ranging
from educated trial-and-error to solving the quadratic equation
n2+n−2040=0
which gives n≈44.7. With either method, we conclude that the snake can eat 44 pellets
before running out of room.
Remediation/Reteaching Resources - Sequences and Series
Resource Links
Introduction to Sequences
Description
Reteaching and ELL instruction and exercises for
reaching all learners around MAFS.912.F-BF.1.1a.
Identify Relationships
Finding terms of a sequence by using a
recursive formula
Video instruction and
MAFS.912.F-BF.1.1a.
examples
to
support
Finding terms of a sequence by using a
explicit formula
Writing rules for sequences
Iteration of fractals
Arithmetic Sequences and Series
Reteaching and ELL instruction and exercises for
reaching all learners around MAFS.912.F-BF.1.2.
Use Patterns
Geometric Sequences and Series
Use a Graphic Organizer
Identifying arithmetic sequences
Finding the nth Term given an arithmetic
sequence
Finding missing terms
Finding the nth term given two terms
Finding the sum of an arithmetic series
Identifying geometric sequences
Fining the nth term given a geometric
sequence
Finding the nth term given two terms
Finding geometric mean
Finding the sum of a geometric series
Video instruction and
MAFS.912.F-BF.1.2.
examples
to
support
ENRICHMENT
Extra for Experts(SPS Online)
1. How many cubes are needed to build this tower? Show your calculations.
2. How many cubes are needed to build a tower like this, but 12 cubes high? Explain how you figure out your answer.
3. How would you calculate the number of cubes needed for a tower (n) cubes high?