Lecture 1 (3) - Properties of Integers, MAE 121 PDF

SCHOOL OF SCIENCE AND MATHEMATICS EDUCATION Email: [email protected] MAE 121 Properties of Integers Understanding the Basics Mr. W Mangcengeza Lecture 1  MAE 121 Name Mr W Mangcengeza Room and building...

SCHOOL OF SCIENCE AND MATHEMATICS EDUCATION Email: [email protected] MAE 121 Properties of Integers Understanding the Basics Mr. W Mangcengeza Lecture 1  MAE 121 Name Mr W Mangcengeza Room and building Education Faculty / First Floor / Office 113 Phone number 021 959 3363 Email [email protected] Consultation hours Tuesday: 10h30 – 12h00 Friday: 10h30 – 12h00 Otherwise, email me for an appointment outside these hours. Class times Time(s): Wednesday Per 1 (N7) 08h30 – 09h15 Thursday Per 3 (L20) 10h20 – 11h05 Tutorial – Starting next week Time: Tuesday (Per 8) and Wednesday (Per 6)  RULES Attendance of lectures are compulsory. Communicate timeously. MAE 121 is about Geometry. A prospective teacher cannot say s/he does not like a topic in Mathematics. You will be qualified to teach Mathematics – meaning ALL the topics in the Phase. Do not come with excuses that you have a ML background. You enrolled to be a Mathematics teacher. So, learn and learn hard. Practice, practice, practice – This is with a pen and paper. Stay up to date with the sections as they build on each other. Module schedule is on iKamva. Changes can be made depending on how we make progress.  Weighting of assessments Type (for example) Weighting 1. Test 1 33,33% 2. Test 2 33,33% 3. Test 3 33,34% Final Weighting Continuous Assessments 50% Final examination 50% Total 100% National Senior Certificate (NSC) (“Matric”) In 2008 1,2 mil. children started Grade 1 in South Africa. 12 years later (in 2019) this cohort wrote the NSC Please try to estimate (guess): How many (what %) of the 1,2 mil wrote the NSC? How many (what %) of the 1,2 mil wrote mathematics? How many (what %) of the 1,2 mil achieved an “A” (≥ 80%) for mathematics? National Senior Certificate (NSC) (“Matric”) In 2008 1,2 mil. children started Grade 1 in South Africa. 12 years later (in 2019) this cohort wrote the NSC 504 303 (42%) Please try to estimate (guess): How many (what %) of the 1,2 mil wrote the NSC? How many (what %) of the 1,2 mil wrote mathematics? How many (what %) of the 1,2 mil achieved an “A” (≥ 80%) for mathematics? National Senior Certificate (NSC) (“Matric”) 58% did In 2008 1,2 mil. children notstarted "make Grade 1 in South Africa. 12 years later (in 2019) it” this cohort wrote the NSC 504 303 (42%) Please try to estimate (guess): How many (what %) of the 1,2 mil wrote the NSC? How many (what %) of the 1,2 mil wrote mathematics? How many (what %) of the 1,2 mil achieved an “A” (≥ 80%) for mathematics? National Senior Certificate (NSC) (“Matric”) 58% did In 2008 1,2 mil. children notstarted "make Grade 1 in South Africa. 12 years later (in 2019) it” this cohort wrote the NSC 504 303 (42%) Please try to estimate (guess): 222 034 How many (what %) of the (19%) 1,2 mil wrote the NSC? How many (what %) of the 1,2 mil wrote mathematics? How many (what %) of the 1,2 mil achieved an “A” (≥ 80%) for mathematics? National Senior Certificate (NSC) (“Matric”) 58% did In 2008 1,2 mil. children notstarted "make Grade 1 in South Africa. 12 years later (in 2019) it” this>80% cohort did wrote not the NSC 504 303 write mathematics (42%) Please try to estimate (guess): 222 034 How many (what %) of the (19%) 1,2 mil wrote the NSC? How many (what %) of the 1,2 mil wrote mathematics? How many (what %) of the 1,2 mil achieved an “A” (≥ 80%) for mathematics? National Senior Certificate (NSC) (“Matric”) 58% did In 2008 1,2 mil. children notstarted "make Grade 1 in South Africa. 12 years later (in 2019) it” this>80% cohort did wrote not the NSC 504 303 write mathematics (42%) Please try to estimate (guess): 222 034 How many (what %) of the (19%) 1,2 mil wrote the NSC? How many (what %) of the 1,2 mil wrote 4 415 (0,37%) Mathematics? How many (what %) of the 1,2 mil achieved an “A” (≥ 80%) for mathematics? National Senior Certificate (NSC) (“Matric”) 58% did In 2008 1,2 mil. children notstarted "make Grade 1 in South Africa. 12 years later (in 2019) it” this>80% cohort did wrote not the NSC 504 303 write mathematics (42%) Less than Please try to estimate (guess): 222 034 half of a How many (what %) of the (19%) 1,2 mil wrote the NSC? percent scored an “A” How many (what %) of the 1,2 mil wrote 4 415 (0,37%) Mathematics? How many (what %) of the 1,2 mil achieved an “A” (≥ 80%) for mathematics? Children who are taught mathematics as rules to be remembered very soon suspend any attempt at sense-making It’s the questions we can’t answer that teach us the most. They teach us how to think. If you give a person an answer, all they gain is a fact. But give them a question and they’ll look for their own answers. Patrick Rothfuss Mathematics is: Mathematics is: The memorisation of A meaningful, sense- facts, rules, formulas making, problem and procedures solving activity. needed to determine the answers to questions.  Introduction to Integers Definition: Integers are a set of numbers that include all positive whole numbers, negative whole numbers, and zero. Notation: ℤ = {..., -3, -2, -1, 0, 1, 2, 3,... }  Types of Integers Positive Integers: { 1, 2, 3,... } Negative Integers: {..., -3, -2, -1 } Zero: 0  Properties of Integers Closure Property: For any two integers a and b, a + b and a × b are also integers. Commutative Property: For any two integers a and b, a + b = b + a and a × b = b × a. Associative Property: For any three integers a, b, and c, (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c). Distributive Property: For any three integers a, b, and c, a × (b + c) = (a × b) + (a × c).  Additive Identity Definition: The additive identity for integers is 0, because a + 0 = a for any integer a.  Additive Inverse Definition: For every integer a, there exists an integer -a such that a + (-a) = 0.  Multiplicative Identity Definition: The multiplicative identity for integers is 1, because a × 1 = a for any integer a.  Examples and Practice Problems Example 1: Verify the commutative property for 3 and -5. Example 2: Show the associative property for 2, 4, and -1. Practice Problem 1: Find the additive inverse of 7. Practice Problem 2: Verify the distributive property for -3, 2, and 5.  Applications of Integers Examples in Real Life: Temperature: Negative and positive temperatures. Banking: Debits (negative) and credits (positive). Sports: Gaining and losing points.  Summary Key Points Recap: Closure, commutative, associative, distributive properties, additive identity and inverse, and multiplicative identity. Importance of Understanding: Integers are fundamental in various mathematical and real-life applications.  References Books and Articles: Elementary Number Theory by David M. Burton Introduction to the Theory of Numbers by Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery