K-8 Mathematics

Base-10 (Decimal)Using the number, 357.35, lets break down what each value represents. Decimal 10's Point 1/100 V V V3 5 7 . 3 5^ ^ ^100's 1's 1/10<–––––––– –––––––––> x10 ÷10Base-10:Ones - 10^0Tens - 10^1Hundreds - 10^2Thousands - 10^3 When in base form, no value can exceed the number of the base. For example, In base-10 there are 10 numeric digits:0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.Therefore, any value in base-form cannot be larger than 9.So in base-5, the digits are 0, 1, 2, 3, and 4.And again but in base-3, 0, 1, and 2.

Expanded Notation:285 = 2 hundreds + 8 tens + 5 ones. = 200 + 80 + 5 = (2 x 100) + (8 x 10) + (5 x 1) = (2 x 10^2) + (8x10^1) + (5 x 10^0)Note: When moving from left to right, the exponent must decrease by -1.(Numbers to the power of 0 always equal 1)For expanded notation, we are going to use the number 1342 with a base of 5. 1342513425 = (1 x 5^3) + (3 x 5^2) + (4 x 5^1) + (2 x 5^0) = (1 x 125) + (3 x 25) + (4 x 5) + (2 x 1) = 125 + 75 + 20 + 2 = 2222123 = (2 x 3^2) + (1 x 3^1) + (2 x 3^0) = (2 x 9) + (1 x 3) + (2 x 1) = 18 + 3 + 2 = 23

Division 8 ÷ 4 -> ÷ Division Sign8/2 -> / Division/ Fraction Bar2 ⟌8 -> ⟌ The rinculum Quotient V 42⟌8 <- Dividend^DivisorStandard Algorithm Place Value Explicit 1 4 6 1 4 64 ⟌5 8 5 4 ⟌5 8 5- 4 | | - 4 0 0------V | -------- 1 8 | 1 8 5- 16 | - 1 6 0 ---------V -------- 2 5 2 5 - 2 4 - 2 4 --------- --------- 1 <-- Remainder 1 <-- RemainderAlternative Algorithm167 Pokemon Cards12 in each booster packHow many Packs? 13 Packs 12 ⟌ 1 6 7 - 1 2 0 --> 10 packs ----------- 47 - 36 --> 3 packs ------------- 11Standard Algorithm 2. Place Value 3. Expanded Notation 2 ^ 19 19 19 x 13 x 13 x 13 ---------- --------- ---------- 3 x 9 = 27 -->20 10 + 9 5 7 3 x 10 = 30 10 + 3 + 1 9 0 10 x 9 = 90 ---------- ---------- + 10 x 10 = 100 110 + 7 2 4 7 --------------------- ----------- 2 4 7 100 + 90 + 0 200 + 40 + 7 ------------------ 2 4 7

Fractions:Meanings:Part-WholeQuotientRatioModels:Surface AreaLengthSets (groups of things)20 Students13 girls7 boysFractions:Girls 13 Boys 7------- = ----- --------- = ----Whole 20 Whole 20Not Fractions:Boys 7 Girls 13-------- = ---- ------- = ----Girls 13 Boys 73/7 > 1/74/5 > 4/93/7 < 5/89/10 > 3/44/ 4 = 1 1.4./ 1.4 = 1x/ x = 1xy/ xy =1

There was ¾ of a pie in the refrigerator. John ate 2/3 of the left over pie. How much pie did he eat?First start by drawing out 3/4 on paper.Then shade 2/3 of whats left of the pizza.It would be 3/6 still shaded.Jim, Ken, Len, and Max have a bag of miniature candy bars from trick-or-treating together. Jim took ¼ of all the bars, and Ken and Len each took 1/3 of all the bars. Max got the remaining 4 bars. How many bars were in the bag originally? How many bars did Jim, Ken, and Len each get?Jim: 1/4 = 3/12 = 12 barsKen: 1/3 = 4/12 = 16 barsLen: 1/3 = 4/12 = 16 barsMax: 4 bars = 1/12Draw a 3 x 4 grid containing 12 squares inside of it.

Adding:1.24 + 1.35 = ? 1.24+ 1.35------------- 2.5921.34 + 90.319 = ? 21.34+ 90.319--------------- 111.659Subtracting:34.45 - 32.23 = ? 34.45- 32.23--------------- 2.2212.338 - 8.23 = ? 12.338- 08.230--------------- 4.108

Multiplication:0.34 x 2 = ? 0.34x 2----------- 0.68 Answer:0.34 x 2 = 0.68 Division:1 ÷ 8 = ? 0.1 2 58 ⟌ 1.0 0 0 -8 | | ---- V | 2 0 | -1 6 | ----- V 4 0 -4 0 ---------- 0Answer:1 ÷ 8 = 0.1250.125 = 12.5%0.125 = 1/8

Decimals:a) 0.128 < 0.234 < 0.45 <0.9b) 0.23 < 0.3 < 0.378 < 0.98c) 0.003 < 0.03 < 0.033 < 0.303 < 0.33 < 3.003Place Value: Hundred HundredThousands Thousands Tens Tenths Thousandths Thousandths V V V V v V 100,000 10,000 1,000 100 10 1 . 1/10 1/100 1/1,000 1/10,000 1/100,000 ^ ^ ^ ^ ^ Ten Hundreds Ones Hundredths Ten Thousands Thousandths**If decimals are repeating, only use the line to represent what is being repeated** __0.21212121 = 0.21 _0.555555 = 0.5 __0.2345454545 = 0.2345

-17 + (10) = -7-10 - (8) = -183 * (-4) = -716 / -4 = -4

Understand the question/ problem:What are you asked to find/ show?Can you restate the problem?Can you draw a picture or diagram to help you solve the problem?Plan how to solve the problem:Draw a picture/ diagramLook for a patternWork backwardsMake the problem simplerImplement the plan to solve the problem:Try different strategiesDo not get discouragedCarrying out the plan is usually easier than devising the planLook back (reflect):Is it a reasonable answer?Did all questions get answered?Is there an easier way?What did you learn?

I have three 5-cent stamps and two 9-cent stamps. Using one of more of these stamps, how many different amounts of postage can I make?There are three 5-cent stamps - ©©© There are also two 9-cent stamps - ©©© ©© ©©© © ©©©©©©©©©©©©©©©©©©©©©©©11 different postage combinationsExplanation: First I started off with laying out the different combinations of the 5-cent and 9-cent stamps, there are three 5-cent (blue) stamps and two 9-cent (red) stamps. Each combination containing both types of stamps are all unique as they have different numbers of both the 5-cent and 9-cent stamps. I first started with different combinations of 5-cent stamps with the 9-cent stamps by adding one, two, or three 5-cent stamps to each combination of 9-cent stamps. For example, when I had one 5-cent stamp I needed to add one 9-cent stamp and then another 9-cent stamp to a different 5-cent stamp so it went up +1 each time until I ran out of 9-cent stamps. I did this with other combinations of 5-cent stamps such as having two or three 5-cent stamps with one or two 9-cent stamps. Adding up all the different combinations of stamps it is a total of 11 different postage combinations.

Meaning:Addition (Put together/ Join)Identity PropertyWhen adding the number 0 to any value, the identity remains the same.a + 0 = a 4 + 0 = 4Commutative PropertyThe order of numbers in an equation will equal the same value.a + b = b + a3 + 4 = 4 + 3Associative PropertyGrouping of numbers does not matter when adding values.(a + b) + c = a + (b +c)Addends3 + 4 = 7 <–––– Sum^ ^AddendsMeanings:Subtraction:Take away (5 - 2 = 3)Comparison**Missing addend** (3 + __ = 7)7 - 3 = 4 <–––– Difference^. ^--- SubtrahendMinuend

`Multiplication ––> Repeated addition3 x 4 = 12 ––> Product^ ^Factors3 x 2 = 3 groups of 2Identity property of multiplicationAny number multiplied by 1, equals one).a x 1= a7 x 1 = 7-3 x 1 = -33/5 x 1 = 3/57 x 0 = 0 (any number multiplied by 0, equals 0).Commutative property of multiplicationThe order of numbers being multiplied does not matter.a x b = b x a7 x 3 = 3 x 7Associative property of multiplicationWhen multiplying numbers together in an equation, the grouping does not matter.(a x b) x c = a x (b x c)(3 x 7) x 2 = 3 x (7 x 2)Distributive property of multiplicationWhen a number is multiplied by a sum, it is the same as multiplying that number by adding the sum and partial products together.a x (b + c) = (a x b) + (a x c)3 x 7 = 3 x (5 + 2) = (3 x 5) + (3 x 2)

Addition AlgorithmsAmerican Standard:1V384+235––––––619Partial Sums:3| 8| 4+2| 3| 5––|––|–––| |91 | 1 |5 | |––––––––––619Partial Sums With Place Value:3| 8| 4+2| 3| 5––|––|–––| |91 | 1 |05 | 0 |0––––––––––619Left - to - Right:384+235––––––500110+ 9––––––619Expanded Notation:100V384 = 300 + 80 + 4+235 = 200 + 30 +5–––––––––––––––––––600 + 10 + 9 = 619

Subtraction AlgorithmsAmerican Standard: 4 13 V V 4 5 3 -2 3 6–––––– 217European/ Mexican: 1V 4 5 3 4V-2 3 6–––––– 217Reverse Indian: 1V 4 5 3-2 3 6––––––2 2 1 7217Left - to - right: 13 V 4 5 3-2 3 6––––––2 0 02 0107–––––217Expanded Notation: 40 13453 = 400 + 50 + 3236 = 200 + 30 + 6 –––––––––––– 200 + 10 + 7217Integer Subtraction: 4 5 3-2 3 6––––– -3 20 200––––217

a is divisible by b if there is a number c that meets the requirement: b * c = aEx. 10 is divisible by 5 because 2 * 5 = 102 * 5 = 10 5 and 2 are factors of 105 * 2 = 10 5 and 2 are divisors of 1010 ÷ 2 = 5 10 is divisible by 2 and 510 ÷ 5 = 2 10 is a multiple of 2 and 5Divisibility Rules:Ending:By 2: 0, 2, 4, 6, 8By 5: 0, 5By 10: 0Sum of Digits:By 3: Sum of digits is divisible by 3By 9: Sum of digits is divisible by 9Last Digits:By 4: Last two digits are divisible by 4By 8: Last three digits are divisible by 8Extras:By 6: If it is divisible by both 2 and 3By 7: Double the last digit then subtract from the new number to see if it is divisible by 7.By 11: "Chop off", add, check if divisible by 11.Examples:*770: 2, 5, 7, 10, 11*136: 2, 4, 8Factors:28: 1, 2, 4, 7, 14, 2860: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 607: 1, 791: 1, 7, 13, 911 –> Identity Multiplication Element0 –> Identity Addition ElementPrime Numbers (0-60)2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59

GCF: Greatest Common FactorGCF (24, 36):1. List Method:24: 1, 2, 3, 4, 6, 8, 12, 2436: 1, 2, 3, 4, 6, 9, 12, 18, 36GCF (24, 36) = 122. Prime Factorization Method:*USE TREE*24: 2 * 2 * 2 * 336: 2 * 2 * 3 * 3GCF (24, 36) = 2 * 2 * 3 = 12 24 / \ 6 4 / \ / \ 3 2 2 2 2 * 2 * 2 * 3 = 24 36 / \ 6 6 / \ / \ 3 2 3 2 2 * 2 * 3 * 3 = 36LCM: Least Common MultipleLCM (24, 36):1.List Method24: 24, 48, 72, 9636: 36, 72, 108LCM (24, 36) = 722. Prime Factorization MethodLCM (24, 36) =GCF * 2 * 3 = ^Unused Factors from the GCF12 * 2 * 3 = 72

Prime Factor Trees 24 / \ 6 4 / \ / \ 3 2 2 2 2 * 2 * 2 * 3 = 24 48 / \ 12 4 / \ / \ 6 2 2 2 / \ 2 3 2 * 2 * 2 * 2 * 3 = 48

Addition:1/4 + 2/4 = 3/43/12 + 2/12 = 5/12Subtraction:5/8 - 4/8 = 1/88/9 - 2/9 = 6/9 = 2/3Improper Fractions:5/6 + 2/3 = V V5/6 + 4/6 = 9/6 = 1 3/6 –> 1 1/2 (Mixed Number)6/10 - 2/5 = V V6/10 - 4/10 = 2/10 = 1/5

Multiplication (Part of a Part):1/2 of 1/21/2 x 1/2 = 1/41/2 x 1/4 = 1/81/3 x 1/8 = 1/24Division:2/3 ÷ 4/5 =(Keep –> Change –> Flip)2/3 x 5/4 = 10/12 = 5/62/3 ÷ 4/5 = 2/3 / 4/5 = 2/3 x 5/4 / 4/5 x 5/4 V V 10/12 20/20 = 110/12 / 1 = 10/12 = 5/6

a) 11% of 45 is what number?0.11 * 45 = 4.95Multiply the decimal and the whole number together and then add.b) 9% of what number is 170.09 * n = 17–––– ––––0.09 0.09n = 188.8 <–– Repeating decimal / 189Divide both sides by the decimal.c) 17% is what % of 2517/25 = 0.68 | V 68%Divide the percentage by the whole number.