6th Grade Math CRCT Week March Madness

Least Common Multiple What is a "Multiple" ? A multiple is the result of multiplying two numbers. To list the multiples we simply list the multiplication facts for a number. The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, etc ... The multiples of 12 are: 12, 24, 36, 48, 60, 72, etc...

Least Common Multiple (Cont.) Example: Find the least common multiple for 3 and 5: The multiples of 3 are 3, 6, 9, 12, 15, ..., and the multiples of 5 are 5, 10, 15, 20, . As you can see on this number line, the first time the multiples match up is 15. Answer: 15

Example: Find the least common multiple for 4, 6, and 8 Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, ... Multiples of 6 are: 6, 12, 18, 24, 30, 36, ... Multiples of 8 are: 8, 16, 24, 32, 40, .... So, 24 is the least common multiple (I can't find a smaller one !)

Least Common Multiple Try it! Find the LCM of 12 and 16

Answers Find the LCM of 12 and 16 48 Find the LCM of 10 and 15 30

Factors Factors are numbers that divide evenly in a number. Factors of the number 12 You can evenly divide 12 by 1, 2, 3, 4, 6 and 12. Therefore, we can say that 1,2,3,4,6 and 12 are factors of 12. We can also say that the greatest or largest factor of 12 is 12.c

The greatest common factor of two or more numbers is the largest number that can divide evenly into each of the numbers. Factors of 12 and 6 You can evenly divide 12 by 1, 2, 3, 4, 6 and 12. You can evenly divide 6 by 1, 2, 3 and 6. Now look at both sets of numbers. What is the largest factor of 12 and 6? 6 is the largest or greatest factor for 12 and 6.

Greatest Common Factor (Cont) Factors of 8 and 32 You can evenly divide 8 by 1, 2, 4 and 8. You can evenly divide 32 by 1, 2, 4, 8, 16 and 32. Therefore the largest common factor of both numbers is 8.

Answers Try It On your On! What is the GCF of 10 and 18? 2

Adding and Subtracting Decimals To add decimals, follow these steps: Write down the numbers, one under the other, with the decimal points lined up Put in zeros so the numbers have the same length Then add normally, remembering to put the decimal point in the answer Try it on your on! 1) 23.56 + 9.81 = 2) 121.01 + 36.222 = 3) 89.8 – 22.11 = 4) 623. 4 – 1.02 =

Try it on your on! 1) 23.56 + 9.81 = 33.97 2) 121.01 + 36.222 = 157.232 3) 89.8 – 22.11 = 67.69 4) 623. 4 – 1.02 = 622.38

Rate(Unit Rate) Example 1: A rate is a ratio that compares quantities in different units. Rates are commonly found in everyday life. The prices in grocery stores and department stores are rates. Rates are also used in pricing gasoline, tickets to a movie or sporting event, in paying hourly wages and monthly fees. A unit rate is a rate where the second quantity is one unit, such as $34 per pound, 25 miles per hour, 15 Indian Rupees per Brazilian Real, etc. Example 1: A motorcycle travels 230 miles on 4 gallons of gasoline. Find the average mileage per gallon. For this problem, simply divide 230miles by 4gallons. The motorcycle gets 57.5 miles per gallon.

Rate (Unit Rate) Example 2: A copy machine makes 45 copies in 25 second. Find the unit rate of copies per second. Step 1: Divide 45 by 25 to find the unit rate. Answer: The copy machine makes 1.8 copies per second. Try it on your On! The train that Earnest was on traveled 540 miles in 5 hours. What was the average speed of the train? What is the price of a jar of Todd’s Own Salsa if its unit price is 40 cents per oz. and the bottle weighs 8 oz.? At Bob’s school tickets for a dance were sold at the rate of 40 per half-hour. How many tickets will be sold in 7 hours?

Answers Try it on your On! The train that Earnest was on traveled 540 miles in 5 hours. What was the average speed of the train? 108 What is the price of a jar of Todd’s Own Salsa if its unit price is 40 cents per oz. and the bottle weighs 8 oz.? 3.20 At Bob’s school tickets for a dance were sold at the rate of 40 per half-hour. How many tickets will be sold in 7 hours? 560

How to Solve an Equation An equation is a statement that two quantities are equivalent. To solve linear equations, you add, subtract, multiply and divide both sides of the equation by numbers and variables, so that you end up with a single variable on one side and a single number on the other side. As long as you always do the same thing to BOTH sides of the equation, and do the operations in the correct order, you will get to the solution.

Solving Equations For this example, we only need to subtract 1 from both sides of the equation in order to isolate 'x' and solve the equation: x + 1 - 1 = 4 - 1 Now simplifying both sides we have: x + 0 = 3 So: x = 3

Try It Out Problems to try 1. X + 7 = 40 2. w – 11 = 106 3. 2w = 12 4. ½ x = 3

Answers Problems to try 1. X + 7 = 40 x = 33 2. w – 11 = 106 w =117 3. 2w = 12 w= 6 4. ½ x = 3 x = 6

Orders of Operations Operations" are math computations to include addition, subtraction, multiplication, division, and using exponents. If it isn't a number it is probably an operation. When you have an expression like ... 7 + (6 × 52 + 3) ... what operation is the first to compute? Step 1: Complete operations in the parentheses first 6 × (5 + 3)=6 × 8=48 Step 2: Evaluate Exponents 5 × 22 =5 × 4= 20 Step 3: Multiply and/or Divide from left to right then add or subtract 2 + 5 × 3 = 2 + 15=17 Step 4: If the problem only has multiplication and division or addition and subtraction, compute operations from left to right 30 ÷ 5 × 3 =6 × 3=18

Orders of Operations Try it On your on! Solve Each 3 x 15 ÷ 5 = 4 + 8 ÷ ( 4 ÷ 2) = 20 – 5 +10 = 5 (10 +17) = 72 ÷ 9 - 5 + 7 =

Answers 3 x 15 ÷ 5 = 9 4 + 8 ÷ ( 4 ÷ 2) = 8 20 – 5 +10 = 25 5 (10 +17) = 135 72 ÷ 9 - 5 + 7 = 10

Expressions and Terms Expression- Consists of numbers and variables. It involves one or more operations. Examples- 2x+1, 3a+8 Term- If an expression is connected only by multiplication, or if it stands by itself. Examples- 5x -8a ½ y x 9

Translating Verbal Expressions Examples for Translating Sentences 1. The sum of twice a number and three 2x + 3 2. Twice the sum of a number and three 2(x + 3) 3. The difference of four times a number and six 4x - 6 4. Four times the difference of a number and six 4(x – 6) 5. Twice the product of three and a number 2(3x) 6. The product of four times a number and negative two is five 4x(-2) = 5

Translating Verbal Expressions the sum of a number and three a number decreased by seven three-fourths of a number the product of x and y

the sum of a number and three x + 3 a number decreased by seven x - 7 three-fourths of a number 3/4x the product of x and y xy

Integer Rules Adding Rules: Positive + Positive = Positive: 5 + 4 = 9 Negative + Negative = Negative: (- 7) + (- 2) = - 9 Sum of a negative and a positive number: Use the sign of the larger number and find the difference. (- 7) + 4 = -3 6 + (-9) = - 3 (- 3) + 7 = 4 Same sign add and keep! Different sign subtract! Take the sign of the larger and then you’ll be exact!

Integer Rules Try It Out! 4 + (-99) = (-8) + 54 = (-43) + (-32) = 109 + (-56) + (-21) =

Answers 4 + (-99) = -95 (-8) + 54 = 46 (-43) + (-32) = -75 109 + (-56) + (-21) = 32

Keep, Change, Change Subtracting Rules: -5+(-3)=-8 Same sign add and keep! 5 - (-3) = Keep5 Change+(Change 3) = 8 5+3=8 Different sign subtract! Take the sign of the larger and then you’ll be exact! (-5) - (-3) = ( -5) + 3 = -2 (-3) - ( -5) = (-3) + 5 = 2

Integer Rules Try It Out! (-9) – 17 = 6-34 = (-5) – (-19) = 98 – (-124) =

Answers (-9) – 17 = -26 6 - 34 = -28 (-5) – (-19) = 14 98 – (-124) = 222

Integer Rules Multiplying Rules: Positive x Positive = Positive: 3 x 2 = 6 Negative x Negative = Positive: (-2) x (-8) = 16 Negative x Positive = Negative: (-3) x 4 = -12 Positive x Negative = Negative: 3 x (-4) = -12 Dividing Rules: Positive ÷ Positive = Positive: 12 ÷ 3 = 4 Negative ÷ Negative = Positive: (-12) ÷ (-3) = 4 Negative ÷ Positive = Negative: (-12) ÷ 3 = -4 Positive ÷ Negative = Negative: 12 ÷ (-3) = -4

Integer Rules (-25) ÷ 5 + (-9)(-3) = (-100) ÷ (-20) = 3 x 15 =

Answers (-25) ÷ 5 = -5 (-9)(-3) = 27 (-100) ÷ (-20) = 5 3 x 15 = 45

Area

Area Try It Out! Find the length of a rectangle whose area is 36 square inches and whose width is 4 inches. Find the area of a parallelogram with a base of 6 centimeters and a height of 8 centimeters. The side of a square is 7in.

Answers Find the length of a rectangle whose area is 36 square inches and whose width is 4 inches. 9 Find the area of a parallelogram with a base of 6 centimeters and a height of 8 centimeters. 48 The side of a square is 7in. 49

Volume

Volume How many cubic feet of water are needed to fill a swimming pool 24 feet long and 15 feet wide to a depth of 5 feet? What is the volume of a cube whose edges are 5 inches long? What is the volume of a dice(number cube) that has edges 3 in long?

Answers How many cubic feet of water are needed to fill a swimming pool 24 feet long and 15 feet wide to a depth of 5 feet? 1800 ft What is the volume of a cube whose edges are 5 inches long? 125 in What is the volume of a dice(number cube) that has edges 3 inches long? 27 in

Review and Practice Evaluate (6 – 3) · 2. Evaluate c + 5 if c = 3. Solve 5 + y = 11. Solve 5x = 55. Multiply 0.25 × 7 Express as ⅛ a decimal.

Answers Evaluate (6 – 3) · 2 6 Evaluate c + 5 if c = 3 8 Solve 5 + y = 11 6 Solve 5x = 55 11 Multiply 0.25 × 7 1.75 Express as ⅛ a decimal .125

Review and Practice 7. What are the prime factors of 120? 8. What is the greatest common factor of 36 and 45? 9. What is the least common multiple of 4, 5, and 10? 10. Express 0.37 as a fraction. 11. Solve 6(-7) 12. Find the value of 1 + 12 × 3 ÷ 6 + 2.

Answers What are the prime factors of 120? 2 x 2 x 2 x 3 x 5 8. What is the greatest common factor of 36 and 45? 9 9. What is the least common multiple of 4, 5, and 10? 20 10. Express 0.37 as a fraction. 37/100 11. Solve 6(-7) -42 12. Find the value of 1 + 12 × 3 ÷ 6 + 2. 9

Review and Practice 13. Evaluate j ÷ k if j = 45 and k = 9. 14. Write 6 · 6 · 6 · 6 · 6 using exponents. 15. Solve 2x + 2 = 8. 16. Find the mode of the set of data. 24, 25, 30, 31, 31, 33, 34, 38, 41, 42, 44, 48, 49, 67 17. Two new pennies weigh 3.15 grams and 3.128 grams, respectively. Find the difference of their weights.

Answers 13. Evaluate j ÷ k if j = 45 and k = 9. 5 14. Write 6 · 6 · 6 · 6 · 6 using exponents. 65 15. Solve 2x + 2 = 8. 3 16. Find the mode of the set of data. 24, 25, 30, 31, 31, 33, 34, 38, 41, 42, 44, 48, 49, 67 31 17. Two new pennies weigh 3.15 grams and 3.128 grams, respectively. Find the difference of their weights. .022

Review and Practice 18. A full tank of gasoline costs $41.40. If the car has a 12-gallon tank, approximately how much does a gallon of gasoline cost? 19. Find the prime factorization of 24. 20. A number cube is marked with 1, 2, 3, 4, 5, and 6 on its faces. If you roll the cube one time, what is the probability that you roll a 5? 21. A coyote can be up to 3 ⅙ feet long. Express this number as a mixed number. 22. What is 3/7+4/7 In simplest form?

Answers 18. A full tank of gasoline costs $41.40. If the car has a 12-gallon tank, approximately how much does a gallon of gasoline cost? 3.45 19. Find the prime factorization of 24. 2 x 2 x 2 x 3 20. A number cube is marked with 1, 2, 3, 4, 5, and 6 on its faces. If you roll the cube one time, what is the probability that you roll a 5? 1/6 21. A coyote can be up to 3 ⅙ feet long. Express this number as a mixed number. 19/ 6 22. What is 3/7+4/7 in simplest form? 7/7 = 1

Review and Practice 23. What is the volume of a shoebox that measures 14 inches by 8 inches by 8 inches? 24. Find the GCF of 32 and 40. 25. Solve y – 8 = 3. 26. Solve (-13) + 5 = 27. What is the LCM of 5 and 4? 28. Translate the phrase '‘nine less than a number'' into an algebraic expression

Answers 23. What is the volume of a shoebox that measures 14 inches by 8 inches by 8 inches? 896 24. Find the GCF of 32 and 40. 8 25. Solve y – 8 = 3. 11 26. Solve (-13) + 5 = -8 27. What is the LCM of 5 and 4? 20 28. Translate the phrase '‘nine less than a number” into an algebraic expression. 9 - y

Review and Practice 29. Find the greatest common factor of 16 and 28. 30. Find the prime factorization of 32. 31. How much change should you receive if you give the cashier $30 for a purchase that costs $24.32? 32. If you work 7.5 hours a day for $6.74 per hour, how much do you make each day? 33. Find the area of a rectangle with length 6 units and width 2 units.

Answers 29. Find the greatest common factor of 16 and 28. 4 30. Find the prime factorization of 32. 2 x 2 x 2 x 2 x 2 31. How much change should you receive if you give the cashier $30 for a purchase that costs $24.32? 5.68 32. If you work 7.5 hours a day for $6.74 per hour, how much do you make each day? 50.55 33. Find the area of a rectangle with length 6 units and width 2 units. 12

Review and Practice 34. Express 4 tickets for $35 as a unit rate. 35. The radius of a circle measures 6 feet. What is the measure of its circumference? Round to the nearest tenth. 36. Solve 3g = -33 37. Evaluate 9.85 + 2.1556 = 38. Solve ⅓ + ⅙ = 39. Solve 1 ½ x ¾ =

Answers 34. Express 4 tickets for $35 as a unit rate. 8.75 35. The radius of a circle measures 6 feet. What is the measure of its circumference? Round to the nearest tenth. 37.68 36. Solve 3g = -33 -11 37. Evaluate 9.85 + 2.1556 = 12.0056 38. Solve ⅓ + ⅙ = ½ 39. Solve 1 ½ x ¾ = 1 ⅛