Mathematics: Order of Operations and Mental Math Techniques for Whole Number Arithmetic - , Study notes of Mathematics

Download Mathematics: Order of Operations and Mental Math Techniques for Whole Number Arithmetic - and more Study notes Mathematics in PDF only on Docsity! 1. Order of Operations: a. PEMDAS: Parenthesis, Exponents, Multiply, Divide, Add, Subtract 2. Mental math techniques a. count on/back: begin with the subtrahend and count up to the minuend, i. find the missing-addend by counting up from the subtrahend. Record the count for each power of ten. The difference is the sum of the values from the count. b. choose compatible numbers: look for numbers that add equally to a number of tens (51 + 19=70) c. break apart numbers: when you break apart numbers to add the 10's and 1's separately. i. 48+36=. 1. First add the tens, 40+30=70. 2. Next add the ones 8+6=14. 3. 48 + 36=84 d. use compensation: changing one number to make it easier (2830, 1110 etc) e. equal additions: For each place-value, an equal amount is added to both the minuend and subtrahend to form a basic fact subtraction. The actual subtraction may be completed by either the take-away or missing-addend approach. i. 67-49 1. 67 (+1) – 49 (+1) = a. 68-50 =18 3. Estimation strategies a. Rounding: round each number to the 100s b. substitute compatible numbers: c. front-end with and without adjustment: use front part of numbers i. 350 + 220= 1. Add the front digits in each number. a. 300 + 200 =500 2. Adjust to account for the remaining digits. a. 50 +20 =70 3. 500 + 70=570 d. Clustering: When all numbers are near each other i. 39+38+43+40= 1. 40 x 40= 160 e. range estimation: estimate a range that addition falls under 4. different algorithms for whole number addition in any base a. standard algorithm i. begin with the right-hand column and proceed to the left one column at a time. The exchange is recorded at the top of the next column. 4 3 7 1 1 1 + 5 2 1 3 6 8 6 9 3 9 5 8 + 4 2 5 + 4 5 8 7 9 3 1 1 5 1 5. Lattice Addition Algorithm a. This is basically the whole-group algorithm recorded in a different format. b. expanded algorithm i.Write the number in expanded notation and then perform the operations. 437 = 400 + 30 + 7 368 = 300 + 60 + 8 693 = 600 + 90 + 3 + 521 = 500 + 20 + 1 + 425 = 400 + 20 + 5 + 458 = 400 + 50 + 8 900 + 50 + 8 700 + 80 + 13 1000 + 140 + 11 = 958 = 700 + 80 + 10 + 3 = 1000 + 100 + 40 + 10 + 1 = 700 + 90 + 3 = 1000 + 100 + 50 + 1 = 793 = 1151 6. different algorithms for whole number subtraction in any base a. standard algorithm: Exchanges completed only on the minuend. The actual subtraction may be completed by either the take-away or missing-addend approach. 7 8 5 11 13 2 14 8 13 – 3 5 3 1 3 17 3 4 9 3 4 3 2 1 2 4 7 – 1 5 2 7 – 7 5 8 1 9 6 6 4 8 9 i. b. expanded algorithm, c. adding the complement algorithm i. In base ten, the complement of a whole number is found by subtracting each digit from 9. Number complement 6 9 3 + 4 5 8 1 1 11 40 1 1 5 1 3 6 8 + 4 2 5 1 3 00 87 7 9 3 4 3 7 + 5 2 1 0 8 00 59 9 5 8 8. different algorithms for whole number division in any base a. standard algorithm--“long division” i. The procedure is repeated subtraction where the largest possible power of ten multiple is subtracted each time and the quotient is written above the dividend. b. expanded algorithm--“scaffolding” i. This is a more efficient version of repeated subtraction. The procedure is to subtract multiples of the divisor. ii. Note that the multiple chosen maybe any number that is less than the dividend. 89  17 1 7 89 – 34 2 55 – 34 2 21 – 17 1 4 5 89  17 = 5 R 4 9. determine whether or not a relation (rule of assignment) is a function a. Definitions: i. A relation is a set of ordered pairs where the first components of the ordered pairs are the input values and the second components are the output values. ii. A function is a relation that assigns to each input number EXACTLY ONE output number. 1. Be careful. Not every relation is a function. A function has to fit the above definition to a tee b. Vertical line test (no 2 points of y can correspond to one point of x) 10. evaluate a function for a particular value of a variable a. To evaluate a function, we insert a given x value, a number in the domain, and see what number we get, which is a number in a range. i. To evaluate f(4) ;f(4) = 2(4) = 8 1. We just evaluated f(x) for the value x = 4. 11. understand the concept of a “function machine” a. allows you to experiment with various functions. 12. solve an equation for one variable in equations that contain two variables a. Use substitution 13. set of natural numbers a. counting numbers (starts with one…) 14. factors and multiples a. Factor: a number that is multiplied to get a product i. 1, 3, 9 are factors of 9 b. Multiple: the product of a quantity by an integer i. 36, 45, 63 are some multiples of 9 15. divisibility rules for the numbers 2, 3, 4, 5, 6, 8, 9, 10, and 11 a. 2: even numbers b. 3: if the sum of all digits is divisible by 3 c. 4: last two digits in your number divisible by 4 d. 5: ends in 5 or 0 e. 6: divisible by 2 and 3 f. 8: the last 3 digits are divisible by 8 g. 10: ends in a 0 h. 11: If you sum every second digit and then subtract all other digits and the answer is 0 or divisible by 11 16. the Factor Test Theorem (use to find and list all distinct factors of a number) a. If f(a) = 0, then (x-a) is a factor of f(x). 17. prime number vs. composite number vs. the number one a. prime: natural number that has exactly two distinct natural number divisors: 1 and itself b. Composite: positive integer which has a positive divisor other than one or itself c. Number 1: neither prime or composite: 18. the Factor Test Theorem (use to tell whether or not a number is prime) a. ?? 19. the Fundamental Theorem of Arithmetic (a.k.a: Unique Factorization Theorem) a. any integer greater than 1 can be written as a unique product (up to ordering of the factors) of prime numbers. i. ii. 20. Different ways to find GCF and LCM a. list out factors and multiples b. use prime factorizations c. d. Euclidean Algorithm to find GCF i. Divide larger number by smaller ii. If remainder is not 0, divide first divisor by remainder 1. 16573/ 2478= 6 R. 1705 a. 2478/1705= 1 R. 773 e. use the GCF-LCM Theorem