Have my students share their strategies, as there are multiple ways to solve one problem. Students' strategies could work for others who are struggling to understand the strategy I am teaching with. POLYA's PROBLEM SOLVING 1. Understand the problem 2. Make a plan 3. Carry out the plan 4. Look back, reflect Have my students repeat the problem in their own words to hear if they understand it or not Example problem: 7 people in a room, everyone shakes hands one time, how many handshakes? To solve each person is assigned a letter 1=A 2=B 3=C 4=D 5=E 6=F 7=G 6 handshakes AB AC AD AE AF AG 5 handshakes BC BD BE BF BG (BA is not used because AB already shook hands) 4 handshakes CD CE CF CG (CA and CB is not used because AC and BC already shook hands) 3 handshakes DE DF DG (the pattern continues) 2 handshakes EF EG 1 handshake FG Add together all of the handshakes 6+5+4+3+2+1 = 21 handshakes total Example #2: 4 3cent stamps (x) 3 7cent stamps (o) Any combination, how many combinations? x xx xxx xxxx o oo ooo (7 combinations) + xo xxo xxxo xxxxo (4 combinations) + xoo xxoo xxxoo xxxxoo (4 combinations) + xooo xxooo xxxooo xxxxooo (4 combinations) = 19 OR 7 + 3x4 = 19 Reminder: With 3 items, each having their own total and different possibilities of combinations, use the Vinn Diagram

Developed as a consistent way to record quantity System to record quantities in a consistent manner

This is the base we use. Base 10 is a positional system. It matters where you put the numbers and you need zeros to hold places. Such as 804 means something different than 84. Consistent 1:10 1's 10's 100's 1000's etc... 10 1's = 10 10 10's = 100 10 100's = 1000 and so on. OR 1x10^0 = ones 1x10^1 = tens 1x10^2 = one hundreds 1x10^3 = thousands DECIMALS/FRACTIONS are negative exponents ex 1/10 = 1x10^-1 or 10 x 1/10 10

BASE 2 1's = 1x2^0 2's = 1x2^1 4's = 1x2^2 8's = 1x2^3 16's = 1x2^4 32's = 1x2^5 64's = 1x2^6 etc... EXAMPLE: 101001 base 2 = (1x2^5) + (0x2^4) + (1x2^3) + (0x2^2) + (0X2^1) + (1x2^0)= (32) + (0) + (8) + (0) + (0) + (1) = 41 What is 17 in base 2 = (1x2^4) + (0x2^3) + (0x2^2) + (0x2^1) + (1x2^0) = 1 0 0 0 1 = 10001 base 2 BASE 3 1's =1x3^0 3's =1x3^1 9's =1x3^2 27's =1x3^3 81"s =1x3^4 BASE 4 1's =1x4^0 4's =1x4^1 16's =1x4^2 64's =1x4^3 BASE 5 1's =1x5^0 5's =1x5^1 25's =1x5^2 125's =1x5^3 BASE 6 1'S =1X6^0 6'S =1X6^1 36'S =1X6^2 216'S =1X6^3 1296'S =1X6^4 etc...

Addition: 3 (addent) + 4 (addent) -------------- 7 (sum) ----------------------------------------------------------------------------------- 1. Standard American Algorithm>Right > Left 576 + 279 855 No reference to place value 2. Partial Sums 5 7 6 + 2 7 9 -------------- 1 5 1 4 7 -------------- 5 5 8 No explicit refernece to place value 3. Partial Sums w/ emphasis on place value 5 7 6 + 2 7 9 ---------------- 1 5 1 4 0 7 0 0 --------------- 8 5 5 4. Left to Right 5 7 6 + 2 7 9 --------------- 7 0 0 1 4 0 1 5 ---------------- 8 5 5 5. Expanded Notation>Place Value explicit 100 10 576 = (500) + (70) + (6) + 279 = (200) + (70) + (9) ---------------------------------------- 800 + 50 + 5 = 855 6. Lattice Algorithm or Latice Method 5 7 6 + 2 7 9 -------------- 0 1 1 7 4 5

Subtraction Top Number - minuend 34 Bottom Number - subtrahend - 12 Answer - difference 22 Take Away - I have 7 cookies, I ate 4, how many do I have left? Comparison - I have 4, Jordan has 3, How many more do I have than Jordan? Missing Addent - I have 4, Jordan gave me some more, now I have 7. How many did Jordan give me? Students must understand place value before they can understand how to "borrow" or carry over. 1. American Standard Right to Left>no emphasis on place value *Should be the last step learned 4 16 1 5 7 6 - 2 8 9 ------------- 2 8 7 2. European - Mexican right to left no emphasis on place value add one to subtrahend and add one (of its place value) to minuend 17 16 5 7 6 - 32 98 9 --------------- 2 8 7 3. Reverse Indian left to right no emphasis on place value borrowing from answer working left to right 1 1 5 7 6 - 2 8 9 --------------- 3 2 9 2 8 7 4. Left To Right Place value is emphasized 170 16 5 7 6 - 2 80 9 ----------------- 3 0 0 2 0 0 9 0 8 0 7 ---------------- 2 8 7 5. Expanded Notation begin with blocks, then to expanded notation emphasis on place value 400 160 16 5 7 6 = (500) + (70) + (6) - - 2 8 9 = (200) + (80) + (9) ------------------------------------------- 200 + 80 + 7 = 287 6. Integer Subtraction 5 7 6 - 2 8 9 ----------------- -3 -10 300 300-10 =290 290-3= 287

Multiplication 3 factor x 2 factor ------- 6 product Array: xxxxxxxxxx xxxxxxxxxx xxxxxxxxxx <----0----1----2----4----5----6----7----8----9----> Number Line Make a number line on the floor and have students actually jump Use students as representation of numbers, groups, etc... Multiplication is repeated addition 2+2+2+2 = 8 4x2=8 4 groups of 2 = 8 = 4x2 [NOT 2x4] 2, 4 times = 4x2 (4 of 2's) Clock minutes count by 5 or multiply by 5 ex 7x5+35 minutes

1. Commutative Property The order does NOT matter 8 x 3 = 3 x 8 The answer is the same A x B = B x A x y = y x Use arrays to help with understanding of this (how we looked at the arrays being turned on the overhead projector) 2. Identity Property of Multiplication When I multiply any number by 1 (one), the numbers identity does not change. a x 1= a 93 x 1= 93 3. Zero Property Any number multiplied by 0 (zero), the answer is always 0 (zero) a x 0 = 0 93 x 0 = 0 4. Associated Property Grouping The way I group my numbers does NOT matter (AxB) x C = A x (BxC) 5. Distributive Property A x (B + C) = (A x B) + (A x C) 3 x 7 = (3 x 5) + (3 x 2) = 3 x (5+2) 3 x 9 = (3 x 5) + (3 x 4) = 3 x (5 + 4) Break the array down in to smaller numbers Instead of pushing to memorize, help to understand the process

1. Standard American Algorithm 23 x 14 -------- 92 + 230 --------- 322 2. Expanded Notation 10 2 3 23 = 20 + 3 x x 1 4 14 = 10 + 4 --------- ------------------------ 9 2 100 90 + 2 2 3 0 200 + 30 + 0 --------- ------------------------ 3 2 2 300 + 20 + 2 = 322 3. Place Value 2 3 x 1 4 ----------- 4 x 3 = 12 4x 20 = 80 10 x 3 = 30 10 x 20 = 200 12 + 80 + 30 + 200 = 322 4. Lattice