Mastering Multiplication and Division: Understanding Operations, Properties, and Equations

1. Chapter 1 (1.3,1.4) The End of the Beginning

2. 1.3 Multiply and Divide • Multiplication = Repeated addition of the same number • 2 times 3 = 2 + 2 + 2 • The numbers being multiplied = factors • The answer to a multiplication problem = product

3. Ways to show multiplication Dot Parenthesis (on one of all of the factors) 4(7) or (4)(7) Nothing ( like 3 times B is 3B)

4. Big Problems • 439 (206) • Multiple Multiplication • 4(5)(7)(1)

5. Properties of Multiplication • 1. If you multiply by zero, you get zero • 2. If you multiply a factor by one, you get that factor • 4x=4 x=1 • 3. If you have a problem where everything is being multiplied (i.e. no addition, subtraction, etc.) the order in which you multiply does not matter. • (2*3)4 = 2(3*4)

6. Try a few • Find the product of 8,704 and 93 • 809,472 • Evaluate 8ab when a = 4 and b = 2 • 8*4*2 = 8 * 8 = 64 • What’s 70 * 9,000 (Hint, there is a shortcut) • Take 7 * 9 and add 4 zeros = 630,000 • Is 11 a solution for 11x = 121 • Yes because 11*11 = 21

7. Exponents • That little number at the top right • 23 = 2*2*2 = 8 • In ab, a is the base • Multiple Exponents (Numbers and Variables) • 1432 = 1*1*1*1*3*3 • c2b3 = c*c*b*b*b

8. Division • Important Note • Top = Numerator • Bottom = Denominator • Zero can never be the Denominator • That is undefined

9. Try a few more • What is 24(32) • 144 • What is 80 • 1 • What’s 7694/24 ? • 320r14 or 320.583333333333333333

10. Applications and formulas • Area of a rectangle = length times width • Area of a square = side times side • Rate = distance / time

11. Prime Factorization • Use your method or the Tree method • Prime number  Divisible by 1 and itself. • 12 = 3 * 4 = 3 *2*2 = 3 *22 • Question: What is the exponent on the 3? • Try 56, 28 • Note: Factors are all the different possibilities

12. 1.3 Homework • 1 thru 187 EOO • 10, 16, 34, 48, 56, 78, 82, 98, 124, 132, 158, 178 Giant Idaho Potato

13. 1.4 Equations • Equations have an equals sign! • x+ 3 = 7 • Remember solving algebraic equations is all about opposites. • i.e. do the opposite of the whatever the mathematical operation.

14. Solving Stuff • What you want • The variable to = a constant • Like y=5 • What's the opposite of: • Addition • Subtraction • Multiplication • Division • Exponents • Square Roots

15. First form • X+a=b • X+3=5 • Try to get simplify first (PEMDAS) • Try to isolate the variable • Do the opposite • X+3 =5 -3 -3 • X =2

16. Example • Y+5=9 • -5 -5 • Y = 4 • Check your answer • Sub in what you found for Y into the original equation • Does 4+5 = 9 • You Bet!

17. Things are what they appear • 3=T+1 • It’s the same thing • 3=T+1 • -1 -1 • 2=T • Check your answer

18. The second type • Form ax=b • 2x=6 • What’s the operation between the 2 and the x? • What's the opposite? • Do it! • 2x=6 • 2 2 • x=3

19. Try a few • 37 = a + 12 • a=25 • 3z=36 • z=12 • 13+m=13 • m=0 • 70=14n • n=5

20. Turning Words into Numbers and Letters • Take these piece by piece and they will be much much easier! • 3 Part process • Assign a variable to one of the unknowns • Use that variable to write an expression for any other unknowns • Write the variable expression

21. Equals • Equals • Is • Was • Is equal to • Represents • Is the same as

22. The product of seven and a number Equals twenty-eight 7n = 28 7 7 Examples • The product of seven and a number equals twenty-eight. Find the number. • Unknown: n n = 4

23. Try Some • Twelve added to a number is sixty. Find that number. • 48 • Using the formula A = P + I, where A is the investment value, P is the original investment, and I is the interest, to find the interest earned on an original investment of $18,000 that now has a value $21,060. • $3,060 • The sum of eleven and a number equals fifty-two. Find that number • 41

24. Homework 1.4 • 1 thru 40 EOO • 12, 22, 28, 30, 38, 40