Everyday math teacher manual

math kernel library reference manual and math manual for students by students and discrete math solution manual pdf
KomalMittal Profile Pic
KomalMittal,India,Teacher
Published Date:13-07-2017
Your Website URL(Optional)
Comment
Lab Activity Manual MATH 112-113: Math for Elementary Teachers Dr. Jonathan Duncan 2014-2015Chapter 1 Logic and Foundations Logic is the hygiene the mathematician practices to keep his ideas healthy and strong. Hermann Klaus Hugo Weyl Contradiction is not a sign of falsity, nor the lack of contradiction a sign of truth. Blaise Pascal Contents 1.1 What Makes a Good Math Teacher? . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Patterns in Pascal's Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Stacking Cereal Boxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Scoring Darts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 A Logic Puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6 Glicks and Glocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.7 Using Venn Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.8 Counting with Venn Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 11.1 What Makes a Good Math Teacher? In this activity we will examine several aspects of good mathematics teaching. After discussing attitudes and re ecting on previous experiences with math teachers, students' will work as a group to create a job description for their ideal math teacher. Objectives:  To understand how one's own attitude towards mathematics can a ect a classroom.  To determine the characteristics and behaviors that typify e ective math teachers. I. Class Discussion What is your attitude towards math and how did it develop? In what ways did your teachers' attitudes in uence your own? Participate in the class discussion of this question II. Individual Re ection Now take a few minutes to think about your math teachers from elementary or high school. Try to recall at least one particularly good teacher and at least one teacher who in uenced you negatively. Re ect on what these teachers said or did and how it a ected you. III. Group Brainstorming When all members of your group have had adequate time for re ection, share your stories with your peers. Then, work together as a group to compile a list of qualities, skills, and behaviors of both good and poor math teachers. (a) Good Math Teachers: (b) Poor Math Teachers: IV. Group Project: Create a Mathematics Teacher Job Description Reference the lists you made above and class discussions to create a one-page type-written job descrip- tion for the ideal elementary school mathematics teacher. Include the most desirable qualities in a mathematics teacher and exclude those qualities which are least desirable. Do, however, be realistic. Adapted from Frank Lester, Jr. Mathematics for Elementary Teachers via Problem Solving, 5. 21.2 Patterns in Pascal's Triangle The triangle of numbers that we know today as Pascal's Triangle was well known in the ancient world. It can be found in Indian mathematics as early as the 2nd century BC, in Persian writings from the 10th century AD, and in 11th century Chinese manuscripts. However Blaise Pascal's 15th century treatise was one of the most complete study of the patters and properties of this interesting array of numbers. Objectives:  To introduce students to the prevasiveness of patterns.  To identify patterns in large sets of numbers. I. Pascal's Triangle We start this activity by lling in the numbers for the rst thirteen rows of Pascal's triangle. Begin and end each row with a one. Any other entries in the triangle should be lled in with the sum of the two entries directly above it. The rst few numbers are done for you. 1 1 1 1 2 1 3 31.2. Patterns in Pascal's Triangle Chapter 1. Logic and Foundations II. Finding Patterns Locate each of the following sets of numbers in Pascal's triangle. Describe the pattern in which these numbers appear. Some patters may involve adding several numbers together, such as the square numbers 1, 4, 9, 16, . . . which appear in the third diagonal as 1, 1+3, 3+6, 6+10, . . . . (a) Counting Numbers: 1, 2, 3, 4, . . . (b) Triangular Numbers: 1, 3, 6, 10, . . . (c) Fibonacci Numbers: 1, 1, 2, 3, 5, 8, . . . (d) Powers of 2: 2, 4, 8, 16, . . . III. Visual Patterns The arrangement of Pascal's triangles leads to some interesting visual patterns. Use a highlighter or pencil to shade in the cells which would contain the indicated types of numbers. Below these triangles, write a sentence describing the patters you see. Multiples of 2 Multiples of 3 Adapted from Bassarear Mathematics for Elementary Teachers Explorations, 2. 41.3 Stacking Cereal Boxes Inductive reasoning is a useful problem solving tool. In inductive reasoning, you make generalizations and solve complicated problems by looking at smaller examples of the same problem. This activity asks you to make use of inductive reasoning as well as several other problem solving strategies. Objectives:  To use various problem solving strategies possibly including: inductive reasoning, using manipulatives, drawing a picture, making a table, and looking for a pattern.  To solve a real-world problem without direct guidance. I. Stacking A Given Number of Boxes A store clerk was told that she had to stack cereal boxes in a display window. She must create a complete triangle as shown below out of the given number of boxes. Use the provided tiles to help you visualize these problems. Record your answers below, drawing pictures when helpful. (a) How many boxes should the clerk place on the bottom row in order to stack 45 cereal boxes? Musical Charms Musical Musical Charms Charms Musical Musical Musical Charms Charms Charms (b) Suppose that the clerk needs to stack 210 boxes. How many should be put on the bottom row? (c) What if the clerk is given 301 boxes? 51.3. Stacking Cereal Boxes Chapter 1. Logic and Foundations II. Starting with A Certain Number of Boxes Suppose that instead of being given a total number of boxes, the clerk was instead told how many boxes to put on the bottom row. How would she decide how many boxes she would need for the triangle? (a) What if the clerk used 30 boxes in the bottom row? (b) How many boxes are needed if the bottom row is to haven boxes? Your answer should be a formula in terms of n. 61.4 Scoring Darts How do we prove something mathematical? Certainly nding and analyzing patterns is an important part of this process, but how do we convince others that our solution to a problem is in fact correct? In this activity you will work with these concepts in the context of scoring a dart game. Objectives:  To understand the concept of a mathematical proof.  To see a model of the reasoning process used in proofs. I. Finding and Testing Patterns You have four darts and a dart board as shown below. Your score for a game is calculated by adding the values of the regions you hit with each of your four darts. Answer the following questions, making sure to explain your reasoning. (a) Which of the following game scores are possible? 6 10 13 15 20 28 3 5 7 1 (b) Predict the kinds of scores that are possible and not possible. Justify your prediction. (c) Systematically list all of the possible scoring outcomes (e.g. 3, 5, 5, 1). Do the total scores match your prediction? 71.4. Scoring Darts Chapter 1. Logic and Foundations II. Proving an Assertion Many students make use of the following assertion when answering part (b) of the previous section. Prove that this assertion is true. The sum of two odd numbers is always an even number. Hint: You may use algebra or geometry to prove this assertion. Think about how to represent odd and even numbers using one of these methods. 81.5 A Logic Puzzle Many people buy books full of logic puzzles and solve them for fun. In case you are not one of them, your instructor will work through an example logic puzzle with the class to help you become familiar with how to solve these puzzles. As for why we solve them, consider the objectives below. Objectives:  To develop a logical representation of given information.  To use the process of elimination and deductive reasoning. I. A Sample Puzzle Janet Davis, Sally Adams, Collete Eaton, and Je Clark have the following occupations: architect, carpenter, diver, and engineer. Find the occupation of each using the following clues.  The rst letter of each person's last name and occupation are di erent.  Je and the engineer are going sailing together.  Janet lives in the same neighborhood as the carpenter and the engineer. II. Your Group Problem A game warden in charge of a bear population in a certain park is troubled. One of a family of 4 bears is stealing potato salad from picnickers. Based on the warden's observations, can you nd the name and color of the culprit?  The bears are named Wilbur, Bob, Sally, and Jane. They are brown, black, cinnamon, and gray, not necessarily in that order. The bears also have strange dietary habits. One of them likes only sh, another eats only berries and nuts, the third is a vegetarian (no meat or sh), while the last one eats anything.  The warden has seen the black and brown bears by the river eating sh. The gray bear will eat apples out of the warden's hand.  Wilbur and Sally do not like nuts or berries.  The gray and brown bears are female.  Jane eats sh, but it is not her favorite dinner.  A bear who would eat potato salad would have to be willing to eat absolutely anything 91.5. A Logic Puzzle Chapter 1. Logic and Foundations 101.6 Glicks and Glocks In each of the problems below, a statement is given which you may accept as true. Based on that statement, decide if each of the lettered statements is \true", \false", or if the truth value \can not be determined". Work with your group to reach a consensus on each question. Record your answers and a short explanation. Objectives:  To practice deductive reasoning  To learn to make and recognize valid arguments I. All Glick numbers that are greater than 20 are even. (a) No odd number greater than 20 is a Glick. (b) If an even number is greater than 20, then it is a Glick. (c) No odd number is a Glick. (d) No number that is less than 20 can be a Glick. (e) All even numbers are Glicks. (f) 33 is not a Glick. II. If a number is a multiple of 6 or is divisible by 7, then it is a Gluck. (a) 17 is a Gluck. (b) 42 is a Gluck. (c) 38 is not a Gluck. (d) No even number is a Gluck. (e) No positive number less than 5 is a Gluck. (f) All Glucks are divisible by 42. 111.6. Glicks and Glocks Chapter 1. Logic and Foundations III. All odd numbers that are less than 29 and all divisors of 48 are Glacks. (a) All Glack numbers less than 15 are odd. (b) 14 is a Glack. (c) All odd numbers less than 21 are Glacks. (d) 16 is a Glack. (e) No odd number greater than 27 is a Glack. (f) Exactly Nine Glacks are prime. IV. All Glock numbers are divisible by 2 and are multiples of 13. (a) 169 is a Glock. (b) No prime number is a Glock. (c) 52 is a Glock. (d) All Glocks are even. (e) All Glock numbers are divisible by 26. (f) All numbers divisible by 26 are Glocks. Adapted from McGinty, Robert & Van Beynen, John (1985). Deductive and analytic thinking. The Mathematics Teacher, 78(3), 192. 121.7 Using Venn Diagrams Venn diagrams are an extremely useful tool for organizing and visualizing sets. In this exercise, you will use given Venn diagrams to identify sets. Objectives:  To determine if a number is an element of a given set.  To represent set membership using Venn Diagrams. I. Identifying Sets In each of the following Venn diagrams,A andB are sets selected from the following list. Use the given elements to identify the sets A andB in each diagram. Sets may be used more than once or not at all. multiples of 2 odd numbers prime numbers divisors of 24 whole numbers smaller than 10 multiples of 3 whole numbers larger than 10 multiples of 6 multiples of 5 divisors of 15 A B A = 5 B = 21 30 Explanation: A B A = B = 18 4 Explanation: 15 A B A = 12 B = Explanation: 4 15 131.7. Using Venn Diagrams Chapter 1. Logic and Foundations A B A = B = 14 10 Explanation: A B A = 12 B = 4 Explanation: 17 A B A = 7 B = 3 Explanation: 1 A B A = B = 21 20 7 Explanation: A B A = B = 3 15 Explanation: 5 Adapted from McGinty, Robert & Van Beynen, John (1985). Deductive and analytic thinking. The Mathematics Teacher, 78(3), 189. 141.8 Counting with Venn Diagrams Use the Venn Diagram provided below to help you answer the following question. Keep a careful record of your steps as you ll in each region of the Venn Diagram. Provide justi cation for each of those steps. Objectives:  To use visual representations to organize information.  To follow a systematic approach in solving a problem and justify solution steps. I. A class contains 32 students, 25 of whom are freshmen and 18 of whom are girls. Ten of the freshmen girls play sports. In total, 12 girls in the class play sports. Seven freshman boys play sports. All together, 21 students play sports. Finally, ve of the freshman girls do not participate in any sport. How many boys in the class are not freshmen and do not play sports? Justi cation: 151.8. Counting with Venn Diagrams Chapter 1. Logic and Foundations II. Now create your own counting problem. Your problem should use three di erent sets and should give enough information for a student to ll in each region of the Venn diagram below. Work as a group and be creative Problem Statement: 16Chapter 2 Numeration and Operation What are numbers? What is the nature of arithmetical truth? Friedrich Ludwig Gottlob Frege It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the rst rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity. Pierre-Simon Laplace Contents 2.1 Whole Numbers and Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 Modeling Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 Patterns and Circle Clocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Interpreting Remainders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.5 Playing with Operation Sense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.6 Properties of Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.7 Numeration Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.8 Modeling Numerals in Other Bases . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.9 Modeling Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.10 Skeletal Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.11 Mental Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.12 Modeling Multiplication and Division . . . . . . . . . . . . . . . . . . . . . . . . 41 2.13 Multiplication and Division Base Six . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.14 Doubling Algorithms for Multiplication . . . . . . . . . . . . . . . . . . . . . . . 45 2.15 Mental Multiplication and Division Tricks . . . . . . . . . . . . . . . . . . . . . . 47 2.16 String Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 172.1 Whole Numbers and Counting The whole numbers provide a basis for arithmetic and will be the starting point for our exploration of number sets. It is therefore important that we develop a good understanding of the whole numbers. For most, the rst introduction to the whole numbers comes through counting objects. In this activity, you will explore the process of counting and how it relates to the whole numbers. Objectives:  To understand how the whole numbers are used in counting.  To understand the characteristics of counting and set cardinality. I. Connecting Counting and Sets with Cardinality If a child can correctly say the rst ve counting numbers, \one, two, three, four, ve," will the child necessarily be able to determine how many books are in a collection of ve books? To help you think about this question, complete the following tasks. (a) The following pictures describe errors that very young children commonly make when rst learning to count objects. After understanding these errors, write down the characteristics of correctly counting a set of objects. How does this process connect to the cardinality of a set? "3" "1" "2" "3" "4" "1" "2" "5" "6" "4" Child 1 Child 2 (b) Suppose a teacher asks two students to determine how many blocks there are in a set. Both students count as shown below. "1" "2" "3" "4" "5" Then the teacher asks the students \So how many blocks are there?" They respond di erently this time, as shown on the next page. 182.1. Whole Numbers and Counting Chapter 2. Numeration and Operation "1" "2" "3" "4" "5" " Five all together" Child 1 Child 2 Which student understands counting better? What is the di erence? (c) Suppose that the teacher takes the original ve books and asks the student to count that there are ve of them. The teacher then covers these books up and puts one more beside them, as shown below. The teacher asks \Now how many books are there in all?" Here are some possible responses.  Child 1 is unable to answer.  Child 2 says \1,2,3,4,5" while pointing to the covered books, and then says \6" while pointing to the new book. Finally, the child says \there are 6 books."  Child 3 says \5" while pointing to the covered books, then points to the new book and says \6." Finally, the child says \there are 6 books." Compare these di erent responses. Which response shows a better understanding of counting and why? Adapted from Beckmann, Sybilla. Mathematics for Elementary Teachers Activity Manual, 3rd Edition, Addison Wesley, 2011. Pages 2-4. 19