Fractions 1. Part-to-Whole  2. Quotient  3. Ration: Part-to-part  Models 1. Area addressing fractions of surface area  2. Length number lines  3. Set (groups of things) 3 groups with 3 people 1 set of group  1 Whole  Discovery 1: when numerator and denominator are the same it's equal to 1 whole Discovery 2: fractional parts are equivalent parts 1/2 > 1/8 Discovery 3: when students realize that 3/4 > 3/9 because the lower denominator means less pieces of the whole number - if wholes are different then fractions will be different such as if the halfs aren't equal sizes - wholes are different sizes

Example of folding a piece of paper  not a helpful method of using cancellation method  do a method that helps students make connections 

1. If a number is not divisible by 5, can it be divisible by 10? If a number is not divisible by 5 then it means the number doesn't end with 0 or 5 2. If a number is not divisible by 10, can it be divisible by 5? Yes, it can be divisible by 5 because numbers ending in 5 or 0 3. Can two numbers have a greatest common multiple? No, because you can continue finding common multiples of a number so it never ends. Meaning you can only find the least common multiple because there is an infinite amount of numbers meaning there is also infinite common multiples. 4. Mary says that her factor tree for 72 begins with 3 and 24, so her prime factors will be different from Tom's because he is going to start with 8 and 9. What do you say to Mary?  I will tell Mary that it doesn't matter what factors you use or start with because factors of a number never change. Meaning when you look for prime numbers it will get the same result no matter how you begin the factor tree. There is only a set of numbers that multiply for a number, for example, 72's factors are 1,2,3,4,6,8,9,12,18,24,72 so you can only use those numbers to find prime numbers. 5. The radio station gave away a discount coupon for every twelfth and thirteenth caller. Every twentieth caller received a free concert ticket. Which caller was first to get both a coupon and a concert ticket? Show work. Explain thinking process. 20,40,60,80,100,120 12,24,36,48,60,72,84 13,26,39,52,65,78,91,104,.... 60th caller gets coupon and concert ticket. I began by listing the factors of 12 and 13 last because 20 is greater so it will be easier to find a common number by looking at the greater number to compare. Then I listed common multiples of 12 and 13 and looked for the same multiple to be present in 12 or 13 because eiether are giving both a coupon and we are looking for the person getting 1 ticket and 1 coupon not 1 ticket and 2 coupons

Types of numbers Divisibility Rules: divisibility are numbers that divide other numbers with no remainders Factors: mulitiples 10 is divisible by 2 10 is divisible by 5 5 is a factor of 10 2 is a factor of 10 10 is a multiple of 2 10 is a multiple of 5 5 is a divisor of 10 2 is a divisor of 10 Example: a (10) is a divisible by b (5) if there is a number c (2) that meets the requirement: cxb=a 2x5 =10

*Ending by 2: 0,2,4,6,8 by 5: 0,5 - a number is divisible by 5 if it ends with 0 or 5 ex: 25 or 20 * Sum of Digits by 3: sum of digits is divisible by 3 by 9: if sum of digits is devisible by 9 Ex: 378 3+7=10+8=18 18/3=6 or 18/9=2 *Last Digits by 4: if last 2 digits of a number is divisible by 4 by 8: last 3 digits Ex: 316 16 / 4 =4 * By 6: a number is divisible by 6 if it's divisble by 2 & 3, if it's divisible by 2 but not 3 then it's not divisible by 6 and vice versa Ex: 378 - divisible by 2 - divisible by 3 Then it's divisible by 6 as well * By 7: 826 1. Look at the last digit & double it 2x6=12 2. take remaining number after doubling the last digit 3. subtract doubled number from the remaining number 82-12=70 4. 70 is divisible by 7 70/10=7 5. if outcome is divisible by 7 the whole number is divisible by 7 Ex: 3,718 8x2=16 371 -16 = 355 5 x 2= 10 35-10 = 25 25 / 7 = not dividible by 7 * By 11: "chop off method" 29,194 1. chop off the last 2 digits (94) 2. go to remaining number (291) 3. add what was chopped off 291+94= 385 4. repeat process 3 +85 =88 5. check if divisible by 11 88 /11 =8

Practice Examples

Homework #5

Factors: different numbers that multiply together to get the same number 28: 1, 2, 4, 7, 14, 28 1x28 =28 2x14=28 4x7=28 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 42: 1, 2, 3, 6, 7 60: 1, 2, 3, 4, 6, 10, 12, 15, 20, 60 91: 1, 7, 13, 91 Prime: a number with only 2 factors, 1 & itself 3: 1, 3 2: 1, 2 13: 1, 13 11: 1, 11 1:1 NOT A PRIME NUMBER SINCE IT ONLY HAS 1 FACTOR 0: NOT A PRIME OR COMPOSITE NUMBER (multiplicative id- element) - additive identity element Prime Numbers 1- 60 2,3,5,7,11,13,17,23,29,31,37,41,43,47,53,59 * 51 is NOT Prime: 1, 3,17,51 Example: 3x7=21 3 times 7 are factors of 21 21 is divisble by 7 21 is a multiple of 3 & 7 3 is a divisor of 21

Prime Factorization Tree

LCM & GCF

1. American-Standard (not really the best)  R ---> L No mention of place value By natural instict students will think or want to start from Left to Right 2. Partial Sums 3. with emphasis on Place Value   Asking students to do the steps need to talk about place value for student to understand One of the best algorithms 4. Left to Right  More natural for the student to start from left to right mentions place value 5. Expanded Notation  we want to strive for students to understand each step, place value, and why instead of just memorizing the addition and just needing to carry a number 6. Lattice 

 no reference of place value Right to left Last step after learning other methods 2. European-Mexican  instead of 1 less from the value take from the top it changes to 1 more for the bottom value 3. Reverse Indian  Borrow from the bottom instead 4. Left to Right 

1. American Standard  2. Place Value  3. Expanded Notation  4. Lattice  when carrying the value leftover put it in the next channel

1. Standard Long Division  2. Alternate Algorithm 

Base 2: 0,1 Ones=2^0 Two's= 2^1 Four's= 2^2 Eight's= 2^3 Examples: 101011 base 2 = (1x2^5)+(0x2^4)+(1x2^3)+(0x2^2)+(1x2^1)+(1x2^0) = 32+0+8+0+2+1 = 43 101011.11 base 2= 43+(1x1/2)+(1x1/4) = 43+1/2+1/4 = 43+ 2/4+1/4 = 43 3/4

Base 3 & 5 Notes Drawn Example

1. Give the Base-10 numeral for each given number. Use expanded notation to explain your answer: 10^0=1 10^1= 10 10^2=100 10^3=1000 10^4=10,000 10^5= 100,000 a) 41.58= (4x8^1)+(1x8^0)+(5x1/8) = (4x8)+(1x1)+(5/8) = 32+1+5/8 = 33 5/8 b) 13415= (1x5^3)+(3x5^2)+(4x5^1)+(1x5^0) = (1x125)+(3x25)+(4x5)+(1x1) = 125+75+20+1 =221 2. Write the number 12 in each given base: a) Base 9  b) Base 8  c) Base 7 

Homework #3

Addition putting things together Identity Order Property a+0=a 4+0=4 -2+0=0 0.25+0=0.25 3/4+0=3/4 Commutative Property a+b=b+a Associative Grouping Property (a+b)+c= a+ (b+c) (3+4)+1=3+(4+1) Subtraction - different meanings Take away (no difficulty remembering) (makes most sense to students) Comparison? - Katie has 2 markers, Marks has 5 markers, How many more does Mark have? Missing addends - no properties; subtracting 3 is like adding -3 Multiplication "Groups of..." & repeated addition Identity Property ax1=a 7x1=7 (identity doesn't change) Commutative Property axb=bxa Associative Property (axb)xc= ax(bxc) Zero Property ax0=0 (anything times 0 = 0 always)

Ones= 10^0 Tens= 10^1 Hundreds= 10^2 Thousands 10^3 375= 300 +70+5 = (3x100)+(7x10)+(5x1) = (3x10^2)+(7x10^1)+(5x10^0) Base 10= 0,1,2,3,4,5,6,7,8,9 Base 5= 0, 1,2,3,4 Ones= 5^0 Fives= 5^1 Twenty-fives= 5^2 One-Hundred Twenty Fives= 5^3 121 base 5 = (1x5^2)+(2x5^1)+(1x5^0) = (1x25)+(2x5)+(1x1) = 25+ 10+ 1 =36 think about values of the digits depending on what base your in 375.32 = (3x10^2)+(7x10^1)+(5x10^)+(3x1/10)+(2x1/100) 121.5 base 5 = (1x5^2)+(2x5^1)+(1x5^0)+(3x1/5) = (1x25)+(2x5)+(1x1)+(3/5) = 25+10+1+3/5 = 36 3/5

(a) 1,075.31 = 1,000+70+5+3/10+1/100 = (1x1,000)+(7x10)+(5x1)+(3x1/10)+(1x1/100) = (1x10^3)+(7x10^1)+(5x10^0)+(3x1/10)+(1x1/100) (b) 79.003= 70+9+3/1000 = (7x10)+(9x1)+(0x1/10)+(0x1/100)+(3x1/1000) = (7x10^1)+(9x10^0)+(3x1/1000) (c) 1212 base 5= (1x5^3)+(2x5^2)+(1x5^1)+(2x5^0) = (1x125)+(2x25)+(1x5)+(2x1) = 125+50+5+2 (d) 32.12 base 5= (3x5^1)+(2x5^0)+(1x1/5)+(2x1/25) = (3x5)+(2x1)+(1/5)+(2/25) =17 7/25 (e) 2123.34 base 5= (2x5^3)+(1x5^2)+(2x5^1)+(3x5^0)+(3x1/5)+(4x1/25) = (2x125)+(1x25)+(2x5)+(3x1)+(3/5)+(4/25) =250+25+10+3+3/5+4/25 =275+13+15/25+4/25 =288 19/25

Base 3: 0,1,2 Ones=3^0 Threes+3^1 Nines= 3^2 Twenty sevens= 3^3 2122.12 base 3= (2x3^3)+(1x3^2)+(2x3^1)+(2x3^0)+(1x1/3)+(2x1/9) = (2x27)+(1x9)+(2x3)+(2x1)+(1/3)+(2/9) =60+11+3/9+2/9 = 71 5/9 Base 8: 0,1,2,3,4,5,6,7 Ones= 8^0 Eights=8^1 Sixty-fours= 8^2 512s= 8^3

 Using pants and shirts as a demonstration you can see the different combinations that you can do with the orange and purple pants to the white, red, and blue shirts. Each pant can be paired with one of the different shirts meaning you have 3 different combos for each pair of pants. You can draw out each combo or use math. 3 orange pant combos + 3 purple pant combos = 6 combinations or 2 pants x 3 shirts = 6 combinations Multiplication can be done since it shows how much of each pant or shirt you have and multiplying 3 and 2 together you are trying to get the amount of times each clothing piece can be combined to make a different combinations.

 Base 10 is the values of the 10s in different place values. No matter the place the 10 base each digit in one of the different place values shows a different quanitity Ones = 10 of the one's single unit Tens= a unit of 10 units together Hundreds= 10 units of 10 ten's Thousands= 1000 units of 10 of the 100's

Number Base 10 Example

Decimals

Examples

4 Steps to Solving Problems Understand the problem Know what is being asked from you to do or show Can you say what the problem is asking in your own words? 2. Devise a plan plan what strategy you are going to use in order to solve the problem 3. Carry out the plan Trial and Error, if one strategy doesn't work try another one Charts, diagrams, drawing, tables, etc. 4. Reflect/ Look Back Does the answer make sense to the question? What did you learn from doing the problem? Are all the questions answered?

Student Methods

Writing Out

Drawing Pictures

Acting Out

1. There are 12 basketball teams in a league. If each of the teams plays each of the other teams once and only once, how many games take place?  2. I have four 3-cent stamps and three 7-cent stamps. Using one or more of these stamps, how many different amounts of postage can I make? 

1. Jim, Ken, Len, and Max have a bag of miniature candy bars from trick-or-treating together. Jim took 1/4 of all the bars, and Ken and Len each took 1/3 of all the bars. Max got the remaining 4 bars. How many brs were in the bag originally? How many bars did Jim, Ken, and Len each get?  2. Jim, Ken, Len, and Max have a bag of miniature candy bars from trick-or-treating together. Jim took 1/4 of the bars. Then Ken took 1/3 of the remaining bars. Next, Len took 1/3 of the remaining bars, and Max took the remaining 8 bars. How many bars were in the bag originally? How many bars did Jim, ken, and Len each get? How is the problem (with regards to fractions) different from problem 1?  go from the end to beginning to solve Jim = 6 bars , Ken=6 bars , Len= 4 bars

3. There was 3/4 of a pie in the refrigerator. John ate 2/3 of the left over pie. How much pie did he eat?  4. Three-fourths of the class are girls. Two-thirds of the girls have black hair. What fraction of the class if female and dark haired?  1/2 of the class if female and black hair

1. A set of marbles can be divided in equal shares among 2,3,4,5, or 6 children with no marbles left over. What is the least number of marbles this set could have? 2,4,6,8,10,......60 3,9,12,15,.......60 4,8,12,16,......60 5,10,15,20,25,30,35,40,45,50,55,60 6,12,18,24,30,36,42,48,54,60 Least number of marbles for a set is 60 marbles 2. Mary spent two thirds (2/3) of her money. She lost two thirds (2/3) of the remaining amount and then she had $8 left. How much money did she start with?  Started with $72

1. Janice is preparing a recipe that calls for 3/4 of a cup of oil per serving. If Janice needs to prepare 2 and 2/3 servings, how many cups of oil will she need?  2. Marc opened a pizza box. Inside there was 3/4 of a pizza. Marc ate 1/2 of what was in the box. How much pizza did Marc eat?  Marc ate 3/8 of the pizza 3. If **** represent 2/7 of the whole, draw what the whole looks like. 

1. If 1/3 cup sugar is needed to make two loaves of bread, how many cups of sugar are needed for three loaves?  Need 1/2 cup for 3 loaves of bread 2. When the least common denominator is used in adding or subtracting fractions, is the result always a fraction in its simplest form? Explain by giving examples. No, its not in the simplest form because you can still simplify results. Examples: 

Divisibility Rules Prime-composite numbers multiple-factors GCF & LCM Fractions

Decimals fall into 2 categories terminating: no remainders repeating: repeating remainders  Example:   Division demonstration:   Multiplication demonstration"  Visual Model: 

65,534.321 decimals start talking about parts of a whole number and go smaller by 1/10 increases from right to left by 10x A view of decimals   900 hundreths+700 hundredths=16 hundredths  demonstrate students you don't just bring down the numbers (ex. 37)

Examples

   create denominator fraction with any power of 10  carry out long division answer will either be terminating or a repeating decimal  

Percent Problems

1. Change the following fractions to repeating or terminating decimals:  2. Express the following fractions as a decimal and a percent. If your response indicates a repeating decimal, use the bar to clearly indicate your answer.  

1. In the United States 13 out of every 20 cans are recycled. What percent of cans are recycled?  (Different way to solve)  2. 2% of Hamilton Middle School have red hair. If there are 700 students at Hamiltom Middle School, how many students have red hair?  Different way to solve  3. There are 25 students in Ms. Johnson's second grade class. In the class election 4 students voted for Ben, 12 voted Sahil, and 9 voted Maria. What percentage voted for Maria?  Different way to solve 

More % Problems

1. A student takes a test with 45 questions and gets 37 questions right. What is their percent on the test?  2. A factory makes sandals. If they produce 829 sandals in a day and 32% of them are blue, how many sandals are blue?  

11% of 45 is what #?  9% of what # is 17?  17 is what % of 25? 

Integers: Positive & Negative Numbers  don't think place value because then students will think positive and negative numbers are on the same level "Chip Method"  

Addition   Subtraction  

Practice

Multiplication   Multiplication-Division Positive & Negative are inverse operations   

Practice

Using _ for a positive chip and - for a negative chip, explain how you would solve the following problems: a. b.  Write and solve the equations that match the given scenaria: a. The stock of a company dropped 17 points and the following day gained 10 points. What was the next change of the stock's worth?  b. The temperature was -10 degrees Celsius and then it rose by 8 degrees Celsius. What is the new temperature?  c.If I lost 4 pounds a week for 3 weeks, what is my change of weight?