In Exercises 16, write the following expressions using only positive integer powers.d -2

Mat 105 end of chapter 2 notes Characteristics of our Base-10 system 1. It has 10 symbols which can be used to represent any amount 2. The value of each place is 10 times that of the previous place 3. Each place contains only one symbol 4. The value of the numeral depends on its place in the number 5.The value of a number is determined by multiplying each numeral by its place value and then adding the sum 6. Zero represents both an empty place and an actual amount Mat 105 chapter 3 notes Chapter 3 The Four Fundamental Operations of Arithmetic Key idea: Numbers can be composed and decomposed and research shows that the ability to compose and decompose numbers is critical in elementary school mathematics. Section 3.1 Understanding Addition Words for addition: combine, join, put together Addition of Roman numerals: xxxviii + xxvi Four example problems: 1. Andy has 3 marbles and his older sister Bella gives him 5 more. How many does he have now 2. Keesha and Jose each drank 6 ounces of orange juice. How much juice did they drink in all 3. Linnea has 4 feet of yellow ribbon and 3 feet of red ribbon. How many feet of ribbon does she have 4. Josh has 4 red trucks and 2 blue trucks. How many trucks does he have altogether Discrete and Continuous amounts Model or drawings to represent each and general or “bar”model (a+b=c) Part Part Whole This general model can represent each of the problems above and all of the 4 operations of arithmetic. Some stages for addition: x x x x x x x x a) combine all and count, b) count on 3and45678, c) know that 3 + 5 = 8 Note that problem 1. above can be modeled by the union of the elements of two sets. Addition: If A and B are disjoints sets, containing a and b elements, respectively, then a + b = n (A union B). Number line model is often problematic 100 addition facts: Table 0 to 9 horizontal and 0 to 9 vertical. Using properties can benefit students Identity property of addition Commutative property of addition Associative property of addition Closure property of addition-unique and existence Regroup when adding and subtracting e.g. 36 + 28 = and 64 - 36 = In addition 6+8 > 9 in subtraction 6 > 4 but in both an amount is moved from one place to another and the amount always represents a 10-for-1 exchange ( 10 for 1 or 1 for 10) Trading, exchanging or regrouping rather than carry or borrow. Investigation 3.1b Mental Addition 1. 39 + 57 2. 68 + 35 3. 66 + 19 4. 545 + 228 5. 186 + 125 Leading digit or front end: 68+35 = 60+30+8+5 Compensation: 68+35 = 70+33 Break and bridge: 68+35 = 68+30+5 Compatible numbers: 68+35=65+35+3 Children’s strategies for addition: 48 + 26 a) add up to 10:48 58 68 68 + 6 b) Break second number apart: 48 + 20 = 68, 68 + 6 = 74 c) Use partial sums: 48 26 60 14 74 Partial Sums: 48+26 number line Base 10 blocks More difficult three-digit addition with Base-10 blocks Numbers Base-10 blocks Words 267 2 hundreds, 6 tens, 7 ones 133 1 hundred, 3 tens, 3 ones An algorithm is a single, clearly described method that works in all cases. There are many efficient algorithms used around the world not just the ones you learned. Investigation 3.1d Alternate Algorithm - Lattice Algorithm for addition 6 4 5 +7 2 8 Estimation; The NCTM standards note that students need to develop, analyze and explain procedures for computation and estimation. Addition in base-5: 43+24 Number and operation sense. 1. To take numbers apart and put them back together 2. To move fluently among different representations 3. To recognize when one representation is more useful than another 4. To perform mental computation and estimation flexibly 5. To determine whether an answer is reasonable 6. To understand the effect of different operations on numbers Section 3.2 Understanding Subtraction Five example subtraction problems: 1. Joe had 7 marbles. He lost 2 in a game. How many does he have left 2. Billy has 2 marbles and Yaka has 7. How many more does Yaka have 3. At the beginning of the week, the hospital had 7 ounces of insulin. During the week, 2 ounces of insulin were used. How much insulin did the hospital have at the end of the week 4. You need $7 to go to the movie and you only have $2. How much more do you need 5. Tom has 7 feet of bubblegum rope and Meg has 2 feet. How much more does Tom have Which are discrete and continuous (measurement) Some are “Take-away” and some are “Comparison” Set model and linear (measurement) model. Also missing addend. x x x x x x x x x x x x x x minuend - subtrahend = difference C – B = A Whole C Part A Part B Number line Whole number Properties: no identity, not commutative or associative, not closed Investigation 3.2a Mental Subtraction 1. 65 – 28 65 – 30 =35, 35+2 = 37 compensation 2. 62 – 29 29 + 10 = 39, 39 + 10 = 49, 49 + 10 = 59, 59 + 3 = 62 adding up 33 3. 184 – 125 125 + 60 = 185 60 – 1 = 59 compensation 4. 132 – 36 64 + 36 = 100 so 100 - 36 = 64 64 + 32 = 96 compatible numbers 5. 1000 – 648 648 + 300 = 948 300 + 52 = 352 adding up 64 – 27 Base-10 number line 64 - (20 + 7) 64 -7 -20 57 – 20 37 Base-10 blocks Subtraction 300 – 148 = Investigation 3.13 Alternate Algorithm – European Subtraction Algorithm used other countries 9 8 4 _ 3 6 8 ______________ You can’t take 8 from 4 cross out the 6 and put 7 now subtract 14 – 8 is 6, 8 – 7 is 1 and 9 – 3 = 6 6 2 3 - 1 5 8 . _______________ You can’t take 8 from 3 so cross out 5 and make 6, 13 – 8 = 5. You can’t take 6 from 2 so cross out 1 and make it 2 now 12 – 6 = 6 and 2 from 6 = 4 Words “Borrow” and “Carry” would better describe the process as Exchange, Regroup or Trade. Subtraction in Base-5 321-142