I. Applications with Variables and Mathematical Sentences

Class Exercise 11/11/2011 I. Applications with variables and mathematical sentences Q1) Alan’s sister is 4 years older than Alan. So we have: Alan is 1 year old => his sister is (1+4) = 5 years old Alan is 2 years old => his sister is (2+4) = 6 years old Alan is 3 years old => his sister is (3+4) = 7 years old A more general way to represent this relation: Alan is years old, his sister is years old.

Q2) Alan’s sister is 4 years older than Alan. If Alan’s age and his sister’s age added together equals 22, how old is Alan? According to Q1, we know that if Alan’s age is , his sister’s age is . The sum of their ages will be: . So we can write the equation: After solving this equation, we can find out .

Q3) Emily reads 8 pages every day. How many pages will she read in days? What equation can you write to find out how many days Emily will spend on reading a 94 pages book?

Q4) A bicycle club bought 500 bicycles. Total money spent is . What mathematical sentence can you write to find out the price for each bicycle? If the cost of each bicycle is reduced by $20, write the mathematical sentence to show how much money will be saved overall.

Q5) Ms. Melissa’s class needs to make 540 paper flowers for a school event. In first 10 days, the class made 32 flowers each day in average. If they must finish the rest of the flowers in 5 days, write an equation to find out how many flowers they must make each day.

Q6) The town library bought many new books. If each shelf holds 300 books, it will need 40 shelves. Write an equation to show how many shelves are needed if each shelf holds 250 books.

II. Algebra Terminologies (2) Like Terms – Terms that consist only of numbers are like terms. Example, 5, 3, 0.4, , etc. Terms that use the same variable to the same degree are like terms. Example, 3, , , etc. Unlike Terms – a) A number and a variable are unlike terms. Example, 5, . b) Terms that use different variables are unlike terms. Example, , , . c) Terms that are to the different power of a variable are unlike terms. Example, , etc.

Addition/Subtraction Rule – To add/subtract like terms, add/subtract the coefficients of the terms. Example: . You can NOT add/subtract unlike terms.

Multiplication Rule – a) Multiply a number and a variable. [Step1: Multipy the coefficients; Step2: Attach the variable at the end] b) Multiply two like terms or unlike terms. [Step1: Multiply the coefficients; Step2: Multiply the variables; Step3: Multiply the answers from the first two steps]

Division Rule – Divide two like terms or unlike terms.[Step1: Divide the coefficients; Step2: Divide the variables; Step3: Multiply the answers from the first two steps]

Practice: Add or subtract each of the following algebraic expressions 1)

Practice: Solve the following multiplication and division problems Class Exercise 11/11/2011 III. Pattern recognition 1) Find the pattern from the following mathematical sentences, then fill in the blanks. 1 x 5 + 4 = 9 = 3 x 3 2 x 6 + 4 = 16 = 4 x 4 3 x 7 + 4 = 25 = 5 x 5 4 x 8 + 4 = 36 = 6 x 6 10 x ______+ 4 = ______= ______x ______; ______x ______+ 4 = ______= ______x 102

2) Solve the first three questions, then follow the same pattern, fill in the answers of the rest 6 questions directly. a) 1 x 8 + 1 = ______(f) 123456 x 8 + 6 = ______b) 12 x 8 + 2 = ______(g) 1234567 x 8 + 7 = ______c) 123 x 8 + 3 = ______(h) 12345678 x 8 + 8 = ______d) 1234 x 8 + 4 = ______(i) 123456789 x 8 + 8 = ______e) 12345 x 8 + 5 = ______

IV. Monks and steamed buns (和尚与馒头) 一百个和尚一百个馒头，大和尚一个人吃三个，小和尚三个人吃一个，问有几个大和尚，几个小和尚？ (100 monks share 100 steamed buns. Each senior monk eats 3 buns, every three junior monks eat 1 bun. How many senior monks are there? How many junior monks are there?) Arithmetic Method-1: Step1: Assume there are only junior monks. 100 buns can feed 300 junior monks. The difference between the assumption and the actual number of monks is: 300 – 100 = 200 (junior monks) Step2: Where do the 200 extra junior monks come from? Think about the connection between number of senior monks and number of junior monks. Based on the number of buns a senior monk eats and a junior monk eats, we know: 1 senior monk eats as many buns as 9 junior monks eat. Therefore, each senior monk accounts for (9 – 1) = 8 extra junior monks. Step3: Based on Step1 & Step2, we can find: senior monk count = 200 (step1) ÷ 8 (step2) = 25 ; junior monk count = 100 – 25 = 75

Arithmetic Method-2: Group 1 senior monk with 3 junior monks, each such group will have 4 monks and eat 4 buns. 100 monks with 100 buns can form 25 such groups. Therefore, there are 25 senior monks and 25x3=75 junior monks.

Algebra Method (Demo only. You don’t need to know how to do it YET): Based on the questions asked, define ‘S’ for number of senior monks, ‘J’ for number of junior monks. There are 2 clues given in the question: 1) Total number of monks: S + J = 100 2) Total number of buns eaten by all the monks: S x 3 + J x () = 100 Solve the above equations: - From 1), we can get: J = 100 – S; - Substitute ‘J’ with ‘100 – S’ in 2): S x 3 + (100 – S) x (1/3) = 100 - Multiply both sides of above equation by ‘3’: 3 x [3S + (100 – S) x (1/3)] = 3 x 100 - Use distributive property to remove ‘[]’ on the left side of the equation: 9S + (100 –S) = 300 - Remove the ‘( )’ on the left side of the equation: 9S + 100 – S = 300 - Combine like terms on the left side of the equation: 8S + 100 = 300 - Subtract ‘100’ from both sides of the equation: 8S = 200 - Divide both sides by 8: S = 200/8 = 25 - Find ‘J’ based on ‘S’: J = 100 – S = 75