Chapter 3: Thinking Computationally


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1 Chapter 3: Thinking Computationally (Completion Time: 3 weeks) Topics: Linear Search Linear search is the most basic algorithm for searching that students will encounter in the course. This topic should be quite short, just enough to familiarize students with the concept of linear search! Bubble Sort Consider a basic approach to sorting that narrows the scope of our problem to focusing on ordering just two elements at a time, instead of an entire array at a time. Selection Sort Another approach to sorting whereby one minimizes the number of swaps required (relative to bubble sort), but substantially increases the amount of comparing required in order to sort a single element. Insertion Sort The last of the sorting algorithms that doesn't require us to iterate across the array multiple times (as selection and bubble sort do), but this benefit, of course, comes with its own additional costs. Binary Search Students discover an algorithm whose speed can be leaps and bounds better than linear search, but not without a costthe data must be sorted first. Time Complexity Begin to discuss the way in which computer scientists measure the theoretical worstcase (O) and bestcase (W) scenarios when running programs.
2 Unsolvable Problems Computers are amazing, and it seems like there must be nothing a computer cannot do. But as it turns out, computers can't do everything, and never actually will be able to. Spend some time considering some unsolvable problems in computer science, including one of the most famous problems in that category: the halting problem. Simulation Computer programs can be used to create models and simulations, to test hypotheses and generate new understanding and knowledge. Students will explore simulations of existing knowledge and use simulation software to test models. Central Focus: Students will analyze algorithms to compare correctness, run times, efficiencies and inefficiencies. Students will also explore computational problems and explain the difference between solvable and unsolvable problems. Assignments: Programming Problem: RNG To be completed with Linear Search LO Create a new computational artifact by combining or modifying existing artifacts. LO Explain how binary sequences are used to represent digital data. LO Evaluate algorithms analytically and empirically for efficiency, correctness, and clarity. In this program, you ll be implementing a program that allows the user to specify how many numbers they would like generated, each of which is capped at some maximum value, with the user optionally able to seed the generator with a seed of their choosing, otherwise relying on some other seed that is always changing, the canonical example being the current time. The user will be providing all of this information to you at the command line.
3 Programming Problem: Fifteen (Parts 13) (Part 3  HACKER) To be completed with Selection Sort LO Apply a creative development process when creating computational artifacts. LO Create a new computational artifact by combining or modifying existing artifacts. LO Collaborate in the creation of computational artifacts. LO Develop an abstraction when writing a program or creating other computational artifacts. LO Use multiple levels of abstraction to write programs. LO Identify multiple levels of abstractions that are used when writing problems. LO Develop an algorithm for implementation in a program. LO Express an algorithm in a language. LO Collaborate to develop a program. LO Explain how programs implement algorithms. LO Use abstraction to manage complexity in programs. LO Evaluate the correctness of a program LO Employ appropriate mathematical and logical concepts in programming. Part 1 s task is to implement two functions, init and draw. The first requires students to generate the numbers in the board to the reverse order (15, 14, 13, etc.). The latter asks students to fill in the game board with the numbers initiated in init. Part 2 s task is to implement another two functions, move and won, whereby move will simulate the action of sliding tiles around the board and won determines if the board is in a winning state. Part 3 is the hacker edition, which requires students to implement god mode. This mode will automatically solve any solvable configuration of the board. Programming Problem: Sort Race To be completed with Insertion Sort LO Find patterns and test hypotheses about digitally processed information to gain insight and knowledge.
4 LO Explain the insight and knowledge gained from digitally processed data by using appropriate visualizations, notations, and precise language. LO Determine how large data sets impact the use of computational processes to discover information and knowledge. LO Express an algorithm in a language. LO Explain the difference between algorithms that run in a reasonable time and those that do not run in a reasonable time. LO Develop a correct program to solve problems. LO Collaborate to develop a program. LO Explain how programs implement algorithms. LO Use abstraction to manage complexity in programs. LO Evaluate the correctness of a program. LO Employ appropriate mathematical and logical concepts in programming. In this program students will be racing three sorting algorithms: selection sort, insertion sort, and bubble sort. They will tested under three conditions: arrays that are almost sorted (only two elements out of place), arrays in reverse order, arrays in completely random order and arrays that are already sorted. Writing Problem: Analyze This To be completed with Insertion Sort LO Collaborate when processing information to gain insight and knowledge. LO Explain how computing innovations affect communication, interaction, and cognition. In 5001,000 words, prepare an essay on your experience, expectations, growths, triumphs, and struggles up through this point in the course. You can spend as much time as you feel is appropriate on each of these questions, but over the course of your writing you should touch on at least the following topics: Discuss your initial expectations of this course, and about how the course has either aligned with or deviated from these expectations. Write about at least one programming problem and at least one writing problem that you enjoyed or did not enjoy. What specifically about that problem did you enjoy or not enjoy, and why do you feel that way?
5 Explain how your habits as a computer scientist have evolved. How do you do things differently at this point in the course from how you did the same things in Unit 1? For instance, has your debugging strategy changed? Do you find yourself more intuitively counting from 0 instead of 1? Do you find that some of what you ve learned in this course has extended to other domains of interest to you? How? Do you feel you learn the most from watching videos, reading excerpts, reviewing slides, hearing audio, or some amalgam of these? Why do you think that is? Explain how you expect your habits as a computer scientist and programmer to evolve further as you proceed through the course. Touch on an "aha!" moment that you encountered in the course; a moment when suddenly a concept that you struggled with made sense. If you have yet to encounter or experience such a moment, discuss why. Talk about each of these with a classmate, and compare notes. How have your experiences differed and how have they aligned? Programming Problem: Seek To be completed with Binary Search LO Create a computational artifact for creative expression. LO Create a computational artifact using computing tools and techniques to solve a problem. LO Analyze the correctness, usability, functionality, and suitability of computational artifacts. LO Develop an algorithm for implementation in a program. LO Express an algorithm in a language. LO Explain the difference between algorithms that run in a reasonable time and those that do not run in a reasonable time. LO Evaluate algorithms analytically and empirically for efficiency, correctness, and clarity. LO Develop a program for creative expression, to satisfy personal curiosity, or to create new knowledge. LO Develop a correct program to solve problems. Students will be implementing two functions in header.c. They will be creating a sort and a search function. In sort, students must implement one of the sorting algorithms of O(n 2 ), which will sort the elements of the array from smallest to
6 largest. The latter requires them to implement a searching algorithm to find a match to the integer the user provided. Programming Problem: Scramble (Part 1) To be completed with Time Complexity LO Apply a creative development process when creating computational artifacts. LO Create a computational artifact for creative expression. LO Develop an abstraction when writing a program or creating other computational artifacts. LO Use multiple levels of abstraction to write programs. LO Identify multiple levels of abstractions that are used when writing programs. LO Develop an algorithm for implementation in a program. LO Express an algorithm in a language. LO Develop a program for creative expression, to satisfy personal curiosity, or to create new knowledge. LO Develop a correct program to solve problems. LO Use abstraction to manage complexity in programs. LO Evaluate correctness of a program. LO Employ appropriate mathematical and logical concepts in programming. In this problem students will complete the implementation of the popular game scramble, which is similar in spirit to boggle. The object of the game is to identify as many words as possible in a given time constraint. Programming Problem: Breakout (HACKER) To be completed with Unsolvable Problems LO Apply a creative development process when creating computational artifacts. LO Create a computational artifact for creative expression. LO Develop an abstraction when writing a program or creating other computational artifacts. LO Use multiple levels of abstraction to write programs.
7 LO Identify multiple levels of abstractions that are used when writing programs. LO Develop an algorithm for implementation in a program. LO Express an algorithm in a language. Students will build their own implementation of the game breakout in the CS50 Appliance (the appliance needs to be downloaded to the students device). Unlike previous problems whereby users interacted with a program at the command line, this problem will have a graphical user interface (GUI) similar to scratch. Learning Outcomes: I can describe different algorithms for sorting data. I can compare and contrast the efficiencies and inefficiencies of each sorting algorithm. I can define big O and big W time complexities. I can describe scenarios where each sorting method would be the most useful. I can design a program that uses the appropriate sorting algorithm for the situation. I can analyze the results from test cases to measure sorting efficiency. I can describe two different algorithms for searching through data. I can implement a searching algorithm in a program. I can define unsolvable and undecidable problems. Essential Questions: Why is sorting data useful? Why does it matter what method we use to sort our data? What makes bubble sort an effective sort? What makes it ineffective? Why might we decide to keep track of how many swaps we have made on each pass through an array? What advantages does selection sort have over bubble sort? What disadvantages are there? Under what circumstances, if any, is selection sort preferable to bubble sort? What advantages does insertion sort have over either selection sort or bubble sort?
8 What disadvantages are there? Under what circumstances, if any, is insertion sort preferable to either of the previous sorts we ve seen so far? What makes insertion sort fundamentally different from the previous sorts in its approach? When is it advantageous to use linear search? When is it not so advantageous? Under which conditions is it more efficient to use binary search versus linear search on a set of data? Under which conditions is it more efficient to use linear search versus binary search? Why is binary search an O (log n) algorithm? How many steps, maximally, does it take to run binary search on a (sorted) data set of size 64? 4,096? 4,294,967,296? In what ways can we measure the resources that our programs consume? Is it always better to choose the algorithm that runs in O (n) over one that runs in O (n 2 )? Why do you think that we analyze algorithms from a theoretical standpoint using asymptotic notation, instead of just counting run time in seconds or the like? In what ways does this adherence to asymptotic notation (disregarding constants and lower order terms) hinder our ability to speak about algorithms in the real world? How might we use time complexity analysis to our benefit as programmers before we even write any code? Are there other questions that a computer would never be able to answer? Ways to Launch the Lesson: Linear Search Refer back to the phone book demo Have students try the following game with linear search. What limitations do you face? Does it matter what side you start on?
9 Bubble Sort Have a student sort a stack of papers (alphabetically or numerically) then have class describe the algorithm used to sort the array. Discuss why one might want a sorted array vs. an unsorted one. Are there benefits to such an array? Bubble Sort, Selection Sort, and Insertion Sort Have a group of students hold an integer. Sort the array using the various sorting algorithms. Write out pseudocode for the algorithms and try to formalize into C as a class or in small groups. Binary Search Go back to the same page ( and look at the example containing 300 numbers. How is it possible to guess the computers answer in 9 guesses? How might we modify our algorithm leveraging Unit 0 concepts? Time Complexity Compare bubble sort, insertion sort and selection sort for different arrays at (number of data points and types of data are customizable). Which is fastest? Do they vary based on how sorted the data is? How do we assign meaning to these times? Given a large set of data which one would sort the fastest (trick question since they all have big O(n 2 ))? Unsolvable Problems Have students discuss a problem that computers cannot solve in pairs or small groups. Challenge students to think of a way that a computer could in fact solve that problem. Show students and to discuss problems in the realm of CS that are in fact unsolvable.
10 Simulation In small groups, have students discuss what scenarios one might use a simulation to solve or simplify a problem. What benefits are afforded to us when we use simulations? Demos and Activities: Where s the 50? The "Where's the 50?" game that David plays in lecture is usually a riot, either because it goes spectacularly well or spectacularly poorly. Have students flip papers lined up in an array to find the number 50. Materials: Papers to cover numbers written on a chalkboard/whiteboard Ping Pong Search Also fairly dramatic is to do binary search with numbered pingpong balls sitting on plastic cups, smacking away various portions of the "array" as you go through the search. Materials: Ping pong balls Plastic cups *NOTE: The majority of the demos for this chapter are located in Ways to Launch the Lesson section. Visual representations of the sorting algorithms help scaffold students understanding of the differences and similarities between sorts.
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