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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?