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1 C oloring Discrete Structures Provided by TryEngineering - L e s s o n F o c u s Is it true or false that Discrete Structures and Discrete Mathematics are the same thing? This is the kind of question that is asked in this field or both fields if they are indeed different. Most Middle School students see a mix of discrete and continuous math without ever noticing the difference. This lesson introduces them to areas of mathematics that computer scientists use to do computational problems. Search techniques through discrete structures are illustrated through graph traversal and graph coloring. A g e L e v e l s Intended for (US Middle School grades 6-8) Can be used in lower High School (e.g. 9 th grade) O b j e c t i v e s Introduce students to: the relationship between Discrete Structures and Discrete Mathematics. the difference between discrete and continuous phenomenon. how to answer a question with sets. how a discrete problem is solved through search: specifically graph coloring. A n t i c i p a t e d L e a r n e r O u t c o m e s Students will be able to: explain the difference between continuous and discrete structures. discuss the difference in perspective of discrete math and discrete structures. form a problem statement as a logical proposition, a Venn diagram, and an adjacency graph. color a map with the least number of colors. A l i g n m e n t t o C u r r i c u l u m F r a m e w o r k s See attached curriculum alignment sheet. I n t e r n e t C o n n e c t i o n s (Continuous vs. Discrete) (Map coloring) R e c o m m e n d e d R e a d i n g O p t i o n a l W r i t i n g A c t i v i t y Is it true or false that Discrete Structures and Discrete Mathematics are the same thing? Defend your answer. Coloring Discrete Structures Page 1 of 10

4 b. If there were differences, call up two students who had different solutions, and ask them to stand at opposite sides of the room. c. Ask the rest of the class to join one of those students if they have that student s name on their sheet. d. If anyone remains seated, call up one of them and have them stand apart from the others. Ask the remaining seated students to join the new solution if that student s name is on their sheet. Do this until everyone is standing. Clearly there is more than one solution. 7. There might have been a flaw in the procedure in step 6. It s possible that students got it wrong that two solutions that appeared to be the same were different, or that two different, were the same. Ask everyone to check whether they have the name of someone in a different group on their sheet. If they do, and if you have time, check their answers and regroup. If you don t have time, simply use this as an object lesson, that a proof is only as good as the information coming in. But if this happens to end on a good note for this to happen. You still know that there is more than one solution, but you still don t know if the solution is unique. These are the kinds of problems solved with discrete structures! Time Needed 2 sessions, at most 1 hour each. You should be able to do the first two activities in a single session, and the coloring problem in the third. If you move more quickly, you can always have them create their own drawing and challenge a friend to color it in. Coloring Discrete Structures Page 4 of 10

5 Some classic four color examples Coloring Discrete Structures Page 5 of 10

6 C oloring Discrete Structures S t u d e n t R e s o u r c e : Discrete Mathematics Topics ( ematics) Theoretical computer science Information theory Logic Set theory Combinatorics Graph theory Probability Number theory Algebra Calculus of finite differences Geometry Topology Operations research Game theory, decision theory Discrete Structures Topics ( Discrete_structures) Sets Logic Boolean Algebra Proof Techniques Counting Graphs and trees Discrete Probability Boolean algebra Set theory From Meriam-Webster.com: Continuous: continuing without stopping; happening or exiting without a break or interruption Discrete: Map: Set (noun, def 2): Subset: separate and different from each other a picture or chart that shows the rivers, mountains, streets, etc. in a particular area; a picture or chart that shows the different parts of something a group of things of the same kind that belong or are used together a group of things, people, etc. that is part of a larger group From Wikipedia.org Set: In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three. Set intersection: (paraphrased) In mathematics, the intersection of two sets A and B is the set that contains all of the elements A that also belong in B, but no others. In mathematics, the four color theorem, or the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color.. Coloring Discrete Structures Page 6 of 10

7 C oloring Discrete Structures S t u d e n t W o r k s h e e t 1 : Is it true or false that Discrete Structures and Discrete Mathematics are the same thing? The question above poses a question in a way that we might be able to answer it with logic. It is a very simple research statement. To answer it definitively, you would have to do some research on the Internet to see whether reliable sources make a distinction or not, and who refers to structures and who refers to mathematics. A place to start to just address the question is to look at the topics considered in each. Using set theory, if the two sets of topics are equal, then they are the same. With a partner, use the lists of topics from the Student Resource Sheet to decide whether a topic is part of Discrete Mathematics and which are part of Discrete Structures. A Venn Diagram gives you a way to view the elements in a set. Use the Venn diagram below and the definitions on the Student Resource Sheet to place topics in the correct parts of the diagram. Using the definitions on the Student Resource Sheet: Are all topics in the intersection (are the sets equivalent)? Is one set a subset of the other? Math Structures Both Math and Structures Coloring Discrete Structures Page 7 of 10

8 C oloring Discrete Structures S t u d e n t W o r k s h e e t 2 : Can this picture be colored in with just for colors? The rule is that the same color can touch at a corner but it can t be used in two adjacent areas. Is there more than one way to color this picture with just four colors? You will answer this question as a class! Note to readers of the draft. The general outline show here will be used, however the graphic needs to be edited so that the tiny areas are not shown. This image can be four colored in more than one unique way. Proof by two pictures Coloring Discrete Structures Page 8 of 10

9 C oloring D iscrete Structures F o r T e a c h e r s : A l i g n m e n t t o C u r r i c u l u m F r a m e w o r k s Note: All lesson plans in this series are aligned to the Computer Science Teachers Association K-12 Computer Science Standards, and if applicable also the U.S. Common Core State Standards for Mathematics, the U. S. National Council of Teachers of Mathematics' Principles and Standards for School Mathematics, the International Technology Education Association's Standards for Technological Literacy, the U.S. National Science Education Standards and the U.S. Next Generation Science Standards. National Science Education Standards Grades 5-8 (ages 10-14) CONTENT STANDARD E: Science and Technology As a result of activities, all students should develop Understandings about science and technology National Science Education Standards Grades 9-12 (ages 14-18) CONTENT STANDARD E: Science and Technology As a result of activities, all students should develop Understandings about science and technology Next Generation Science Standards & Practices Gr.5-8 (ages 10-14) Practice 5: Using Mathematics and Computational Thinking Use mathematical representations to describe and/or support scientific conclusions and design solutions. Principles and Standards for School Mathematics (all ages) Problem Solving Standards Solve problems that arise in mathematics and other contexts Connections Recognize and apply mathematics in contexts outside of mathematics Representations Use representations to model and interpret physical, social, and mathematical phenomena Common Core State Practices & Standards for School Mathematics (all ages) CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them. CCSS.MATH.PRACTICE.MP4 Model with mathematics. CCSS.MATH.PRACTICE.MP7 Look for and make use of structure. Standards for Technological Literacy - All Ages Nature of Technology Standard 2: Students will develop an understanding of the core concepts of technology Coloring Discrete Structures Page 9 of 10

10 Coloring Discrete Structures F o r T e a c h e r s : A l i g n m e n t t o C u r r i c u l u m F r a m e w o r k s CSTA K-12 Computer Science Standards Grades 6-9 (ages 11-14) 2 Level 2: Computer Science and Community (L2) Computational Thinking: (CT) 8. Use visual representations of problem states, structures, and data (e.g., graphs, charts, network diagrams, flowcharts). 14. Examine connections between elements of mathematics and computer science including binary numbers, logic, sets and functions. CSTA K-12 Computer Science Standards Grades 9-12 (ages 14-18) 5.3 Level 3: Applying Concepts and Creating Real-World Solutions (L3) 5.3.A Computer Science in the Modern World (MW) Computing Practice and Programming (CPP) 12. Describe how mathematical and statistical functions, sets, and logic are used in computation. Coloring Discrete Structures Page 10 of 10

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Teacher Candidate: Grade Level/Subject Unit Title Lesson Title Duration Lesson Outcomes Chiara Shah 9 th /Algebra I Unit 4: Solving and Graphing Inequalities 6.1 Solving One-Step Linear Inequalities 45

### Problem of the Month: Fair Games

Problem of the Month: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards:

2009 Leaders Notes Grade 8 Module 3 page 1 General Materials and Supplies: Handouts 10 and 11 8 ½ x 11 paper white boards markers chalk tape tape measure ruler yard stick meter stick tennis balls in the

E XPLORING QUADRILATERALS E 1 Geometry State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space. Statement of Purpose: The activities in this

### Introduction to learning with Sphero

Introduction to learning with Sphero Hello there, and thanks for taking a look at Sphero and Education! The lessons in the SPRK program teach math, physics, and computer science concepts using hands-on,

Provided by TryEngineering - Lesson Focus Lesson focuses on how software engineers design computer games and other software. Student teams work together to develop a simple computer program using free