Square Units and Second Power, then Square Roots STANDARDS Square Units and Second Power, then Square Roots Cubic Units and Cube Numbers In between what whole numbers is the square root? A pattern of Powers of 10’s Scientific Notation Exponent Properties END SHOW PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

GRADE 7: Number Sense GRADE 8: Algebra 2.1 Understand negative whole-number exponents. Multiply and divide expressions involving exponents with a common base. 2.4 Use the inverse relationship between raising to a power and extracting the root of a perfect square; for an integer that is not square, determine without a calculator the two integers between which its square root lies and explain why. 2.1 Entender exponentes enteros negativos. Multiplicar y dividir expresiones que involucran exponentes. 2.4 Usar la relación inversa entre elevar una potencia y sacar su raíz cuadrada perfecta; para un entero que no es cuadrado, determinar sin calculadora los dos enteros entre los cuales se encuentra dicha raíz y explicar porqué. GRADE 8: Algebra 2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, and taking a root. They understand and use the rules of exponents. 2.1 Los estudiantes entienden y usan operaciones como tomar el opuesto, encontrar el reciproco, y sacar la raíz. Ellos entienden y usan las reglas de los exponentes. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

What is the area of the square? STANDARDS 1 1 1 What is the area of the square? 1 2 1x1 = = 1 What is the length of the side? 1 = 1 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

What is the area of the square? STANDARDS 4 3 2 1 2 2 What is the area of the square? 2 2x2 = = 4 What is the length of the side? 4 = 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

What is the area of the square? STANDARDS 9 8 7 6 5 4 3 2 1 3 3 What is the area of the square? 3 2 3x3 = =9 What is the length of the side? 9 = 3 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

What is the area of the square? STANDARDS 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 4 4 What is the area of the square? 4 2 4x4 = = 16 What is the length of the side? 16 = 4 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

What is the area of the square? STANDARDS What is the area of the square? 5 2 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 5x5 = = 25 The SQUARE OF A NUMBER is the total of square units used to form a larger square. 5 What is the length of the side? 5 25 = 5 The SQUARE ROOT OF A NUMBER is the opposite of the square. It is when you find the lenght of the side in a square with a given number of square units. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

THE SQUARE ROOT OF A NUMBER STANDARDS THE SQUARE OF A NUMBER THE SQUARE ROOT OF A NUMBER STANDARDS 3x3 = 3 2 9 8 7 6 5 4 1 =9 = 3 2x2 = 2 4 3 1 = 4 = 2 1 1x1 = 2 = 1 4x4 = 4 2 16 15 14 13 12 11 10 9 8 7 6 5 3 1 = 16 = 4 1x1 = 5 2 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 4 3 1 = 25 = 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

SUMMARIZING: STANDARDS 3 3 2 We say 3 SQUARE or THREE TO THE SECOND POWER. 3x3 = =9 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

SUMMARIZING: STANDARDS 5 We say 5 SQUARE or FIVE TO THE SECOND POWER. 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 4 3 2 1 5 2 We say 5 SQUARE or FIVE TO THE SECOND POWER. 5x5 = = 25 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

What means 7 square? Why? STANDARDS 7x7 = 7 = 49 7 49 square units. 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

What means 13 square? Why? STANDARDS 13x13 = 13 = 169 13 2 Why? 13x13 = = 169 13 169 square units. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

What is the volume for these cubes? 3 1 2 1 1 2 2 3 1 3 3 1x1x1 = = 1 2 3 2x2x2 = = 8 3 3x3x3 = = 27 1 CUBED 2 CUBED 3 CUBED 4 4 3 4x4x4 = =64 4 4 CUBED 4 STANDARDS PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Three to the THIRD POWER What is 3 cubed? 3 3x3x3 = = 27 3 Three to the THIRD POWER 3 OR 3 That is 27 cubic units! STANDARDS PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

What is 4 cubed? 4 4x4x4 = =64 4 4 Four to the THIRD POWER 4 OR 3 4x4x4 = =64 4 4 Four to the THIRD POWER 4 OR That is 64 cubic units! STANDARDS PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Which whole numbers is between? 3 STANDARDS Which whole numbers is between? 3 What is the largest perfect square that can be made with 3 square units? Adding 1 more: Taking out 2: 3 2 1 2 4 3 1 1 There is no possible perfect square with 3 square units. 1 2 2 1x1 = = 1 2x2 = = 4 4 1 = 2 = 1 We either take out 2 or add 1. 1 4 The square root of 3 is between 1 and 2. 4 3 2 1 3 3 1.73 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Which whole numbers is between? 8 STANDARDS Which whole numbers is between? 8 What is the largest perfect square that can be made with 8 square units? 3 Adding 1 more: Taking out 3: 2 There is no possible perfect square with 8 square units. 2 2x2 = = 4 3 2 3x3 = =9 4 = 2 We either take out 3 or add 1. 9 = 3 4 9 The square root of 8 is between 3 and 4. 4 3 2 1 8 8 2.83 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Which whole numbers is between? 12 STANDARDS Which whole numbers is between? 12 What is the largest perfect square that can be made with 12 square units? Adding 4 more: Taking out 3: 4 3 There is no possible perfect square with 12 square units. 3 2 4 2 3x3 = =9 4x4 = We either take out 3 or add 4. = 16 9 = 3 16 = 4 9 16 The square root of 12 is between 3 and 4. 4 3 2 1 12 12 3.46 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Can you continue the pattern? STANDARDS Finding a pattern: 10 4 10x10x10x10 = 10 2 10 3 10x10 = 10x10x10 = 10x1000 100 10x100 10000 1000 10 5 100 square units. 10 10x10x10x10x10 = 10 1000 cubic units. 10x10000 100000 10 6 10x10x10x10x10x10 = 10x100000 1000000 Can you continue the pattern? PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

ONE MILLIONS HUNDRED THOUSANDS TEN THOUSANDS ONE THOUSANDS HUNDREDS TENS ONES 1000000 100000 10000 1000 100 10 1 10x10x10x10x10x10 10x10x10x10x10 10x10x10x10 10x10x10 10x100000 10x10000 10x1000 10x100 10x10 10x1 10 6 10 5 10 4 10 3 10 2 10 1 10 Then: 500 = 5 x 100 8x10 6 =8x1000000 =5x10 2 and =8000000 3x10 5 =3x100000 7000 = 7 x 1000 =7x10 3 =300000 9x10 =9x1 =9 STANDARDS PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Write in scientific notation: 1,750,000 10x10 10 2 100 10x10x10 3 1000 10x10x10x10 4 10000 10x10x10x10x10 5 100000 10x10x10x10x10x10 6 1000000 10x100 10x1000 10x10000 10x100000 10x1 1 ONES TENS HUNDREDS ONE THOUSANDS TEN THOUSANDS HUNDRED THOUSANDS ONE MILLIONS 1 7 5 Then: 10 6 x 1,750,000 = 1.750 millions = 1.750 This is the number in scientific notation! 10 6 x OR 1 ,7 5 0 , 0 0 0. = 1.750 STANDARDS 6 places to the left PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Write in Scientific Notation the following numbers: 1,320 11.45 10 3 x 10 3 x 1,320 = 1.32 11.45 10 3 x 10 4 x 3 places to the left = 1.145 1 place to the left 37,900 237.6 10 5 x 10 4 x 37,900 = 3.79 237.6 10 5 x 10 7 x = 2.376 4 places to the left 2 places to the left 55.91 10 1 x 55.91 = 5.591 1 place to the left STANDARDS PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Write in Standard Notation the following numbers: 2.85 10 3 x = 2.85 1000 8.093 10 1 x = 8.093 10 = 2,851. = 80.93 = 2,851 5.71 10 6 x 27.9 10 2 x = 5.71 1000000 = 27.9 100 = 5,710,000. = 2790. = 5,710,000 = 2,790 STANDARDS PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Observe the following pattern: ONE THOUSANDS TEN THOUSANDTHS TENS HUNDREDTHS HUNDREDS THOUSANDTHS TENTHS ONES 1 10 1 100 1 1000 1 10000 1000 100 10 1 10x 1 10 10x 1 100 10x 1 1000 10x 1 10000 10x100 10x10 10x1 10x 1 100000 10 3 10 2 10 1 10 10 -1 10 -2 10 -3 10 -4 .1 .01 .001 .0001 What pattern do you see emerging in the exponents? They decrease from left to right! STANDARDS What about the decimals? PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

STANDARD 1.2 How many hundredths does the unit have? 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 = PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

How many parts does the tenth have? STANDARD 1.2 How many parts does the tenth have? 10 9 8 7 6 5 4 3 2 1 = PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Penny STANDARD 1.2 1 cent A hundredth of a dollar. 1¢ $1 =$ .01 100 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Dime STANDARD 1.2 10 cents 10¢ A tenth of a dollar $1 =$ .1 10 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Dollar STANDARD 1.2 = Name = 1 dollar Worth = $1.00 Worth = 10 dimes Word = 100 cents Worth = 100 ¢ PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Hundredths Tenths Units or Ones STANDARD 1.2 DECIMAL POINT PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Write in Scientific Notation the following numbers: 1 10000 x 10 -4 x .000345 = 3.45 .000345 = 3.45 OR 4 places to the right 10 -4 x = 3.45 1 1000 x 10 -3 x .00675 = 6.75 .00675 = 6.75 OR 3 places to the right 10 -3 x = 6.75 STANDARDS PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Write in Standard Notation the following numbers: 4.35 10 -4 x 1 10000 x 4.35 10 -4 x = 4.35 OR 000 =.000435 4 places to the left = 4.35 10000 =.000435 7.26 10 -7 x 1 10000000 x = 7.26 7.26 10 -7 x OR 000000 =.000000726 7 places to the left = 7.26 10000000 =.000000726 40.1 10 -6 x 1 1000000 x = 40.1 40.1 10 -6 x OR 0000 =.000000726 6 places to the left = 40.1 1000000 =. 0000401 STANDARDS PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Product of Powers: Do you remember? STANDARDS 10 = 10x10x10x10x10x10 6 = 10x10x10x10x10x10 10x100000 = 1000000 = Let’s look for a pattern: Exponential Form Standard Numbers What is then? 10 1 x 10 2 = 10x10 =100 7 1 x 7 1 Z 1 x Z 1 Z 2 = 7 2 = 10 1 2 x 10 3 = 10x100 7 1 x 7 2 7 3 = Z 1 x Z 2 Z 3 = =1000 10 1 3 x 10 4 = 7 1 x 7 3 7 4 = Z 1 x Z 3 Z 4 = 10x1000 =10000 10 1 4 x 10 5 = 7 1 x 7 4 7 5 = Z 1 x Z 4 Z 5 = 10x10000 =100000 10 6 = 7 1 x 7 5 7 6 = Z 1 x Z 5 Z 6 = 10 1 5 x 10x100000 =1000000 Product of Powers: For any real number a and integers m and n a m n = a m+n Write the expressions as a single power of the base: x x 2 5 = x 2+5 y y y 2 5 7 = y 2+5+7 = x 7 = y 14 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Let’s look for a pattern: then STANDARDS 1 10x10x10x10x10x10 10 1x10x10x10x10x10 1 = If =100000 Let’s look for a pattern: then Exponential Form Standard Numbers What is happening? 1000000 10 =100000 10 6 1 10 5 = 10 6 1 10 6–1 = 10 5 = 100000 10 =10000 10 5 1 10 4 = Quotient of powers: For any real number a, except a=0, and integers m and n 10000 10 =1000 10 4 1 10 3 = 1000 10 =100 10 3 1 10 2 = a m n = a m-n 100 10 =10 10 2 1 10 1 = Simplify the quotients: 10 =1 10 1 10 = x 9 3 y 7 6 = x 9–3 =y 7–6 = x 6 = y PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

STANDARDS Exponential Form Standard Numbers 10 =1 Let’s concentrate in this part: If Exponential Form Standard Numbers 10 =1 10 1 10 1–1 = 10 = then Power to the zero: a = 1 UNDEFINED! (4y) = 1 (-3kp) = 1 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Let’s look for a pattern: then STANDARDS 1 10x10x10x10x10x10 10 1x10x10x10x10x10 1 = 100000 1 = If Let’s look for a pattern: then Exponential Form Standard Numbers Power with Negative Exponents: 1000000 10 100000 1 = 10 1 6 10 1–6 = 10 -5 = For any real number a, and any integer n, where a = 0 100000 10 10000 1 = 10 1 5 10 -4 = 10 1–5 = a = -n n 1 a 10000 10 1000 1 = 10 1 4 10 1–4 = 10 -3 = 1000 10 100 1 = 10 1 3 10 -2 = 10 1–3 = a = -3 3 1 a x = -5 5 1 x 100 10 10 1 = 10 1 2 10 1–2 = 10 -1 = y = -9 9 1 y PRESENTATION CREATED BY SIMON PEREZ. All rights reserved