Algebra Math Y8 By Valéria Kumoto Topic Summary Exponentiation Indicator : Calculate the exponentiation of Integer Numbers ATL’s Ability to manage information • Present information in various formats. Valéria Kumoto Social Research • Social: Use appropriate strategies for organizing complex information. • Collect, record and verify data. 3 Goals Main Goal Topic Goals In the final of the class the students will be able to: The student will be able to recognize - Express numbers in expanded form by using exponents; exponential patterns in problems and solve them in familiar situations in different contexts. - Name the terms of an exponent and read it properly; - Calculate an exponent using a scientific calculator. . Valéria Kumoto 4 Vocabulary Speak it! Frames Key Words - When a number is raised to a power, the number that is used as a factor is the ………, - Base - Exponent - ………is the number that indicates how many times the base is used as a factor, - Power - Square - Cube - ……… is a number produced by raising a base to an exponent, - A ……… number is a number raised to the second power, - A ……… number is a number raised to the third power. Valéria Kumoto Agenda – Topic 2 March, 23th March,23 rd March, 23th March, 23th March th March, nd March, nd March, nd Learning Cycle Warm up Warm up Investigate Reflect Investigate Create Knowledge Create Knowledge Go Further Discuss Practice Practice Problems Discuss Discussion about the class Go Further Reflect Thinking Routines Warm up Warm Up Warm up Legend of rice on chessboard You can find the complete Legend in the book: The man who counted/Malba Tahan - /Chapter 16 /pg 46. Instructions: Watch the video bellow regarding the Legend of rice on the chessboard and discuss the legend. https://www.youtube.com/watch?v=CM6bLzvstYk https://www.youtube.com/watch?v=A7k3XDopD2U - To work in pairs/ in groups (of three, of four, etc), - It is necessary a scientif calculator, or a laptop or a notebook. Warm up Legend of rice on chessboard Warm up The King’s Chessboard Problem: In The King’s Chessboard, the wise man requests as his reward grains of rice following the rule: 1 grain of rice for the first square on the chessboard, two grains of rice for the second square, four grains of rice for the next square, then eight grains of rice, and so on, for all 64 squares on the chessboard. Exponents There is an Mathematical operation, named Exponent, that helps up to solve problems like this, when we have a number being doubling or tripling or quadrupling and so on consecutively. The increase is quick 8 minutes Investigate Task Statement : Legend of rice on chessboard Investigate Legend of rice on chessboard 1) How would you determine the total amount of grains of rice on the last square? (64th square) 2) Can you think of another way you might have figured out the solution in question 1? Describe it. 3) How would you determine the total amount of grains of rice for all 64 squares? Warm up Legend of rice on chessboard Warm up 4) Complete the chart bellow and calculate the number of grains of rice on the 64th square? You can use the scientific calculator. Squares in the board Total grains placed Repetead Multiplication (using the same number) 8 2x2x2 1st square 2st square 3st square 4st square 5st square 6st square 7st square 8st square 9st square ... 64st square 5) Were you surprised at how the numbers grew? Why or why not? Another notation Create Knowledge Create Knowledge Inquiry questions 1) What is an exponent? 2) How can we represent an exponent and what are the names of its terms? 3) How can we calculate an Exponent? 4) How can we read them? 5) Is there an exponential symbol? If yes, what is this symbol? 6) When is any number raised to the number one, what can we conclude? 7) When the base of an exponent is equal one, what can we conclude? 8) When the base of an exponent is equal zero, what can we conclude? 9) Why is any non-zero number raised to the power of zero equal 1? And what happens when we raise zero to zero? 10) Does the exponentiation have the commutative property? 1) What is an exponent? It’s an Arithmetic Operation just like the four main Arithmetic Operation that we use in Math, with are: + : Addition - : Subtraction × : Multiplication ÷ : Division 1) What is an exponent? Multiplication Examples: a) 2+2+2+2+2+2+2+2 = 8×2 (or 2×8) � The number 2 is being adding by itself eight times. � Instead of adding the number 2 eight times, it is possible to write 8×2 b) 3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3 = 16×3 (or 3×16) � The number 3 is being adding by itself sixteen times. � Instead of adding the number 3 sixteen times it is possible to write 16×3 Multiplications are a shorthand way to write repeated addition. 1) What is an exponent? If I have a number being multiplying by itself many times, how can we write it in a shorthand way? For example: a) 2×2×2×2×2×2×2×2= ➔ The number 2 (factor) is being multiplying by itself eight times, ➔ Instead of multiplying the number 2 eight times, it is possible to use exponents to write it in a simpler way: b) 3×3×3×3×3×3×3×3×3×3×3×3×3×3×3×3 = ➔ The number 3 (factor) is being multiplying by itself sixteen times, ➔ Instead of multiplying the number 3 sixteen times, it is possible to use exponents to write it in a simpler way Exponents are a shorthand way to write repeated multiplications. 2) How to represent and what are the name of the terms of an exponent? Base Exponent The small number written above and to the right of a number is called exponent and the underneath the exponent is called base. • Base is the number that is multiplied by itself. • Exponent indicate how many times the base is multiplied by itself. 3) How to calculate an Exponent? a) 62 c) 35 62 = 6 × 6 = 36 Multiply the base 6 2 times. b) 33 35 = 3 ×3 × 3 × 3 ×3 = 243 Multiply the base 3 5 times. d) 45 33 = 3 × 3 × 3 = 27 Multiply the base 3 3 times. 45 = 4 × 4 × 4 × 4 × 4 = 1024 Multiply the base 4 5 times. 4) How to read exponents? Example: Ordinal numbers Two to the fourth (4th) power Two raised to the power 4 Two to the fourth Two to the power 4 4) How to read exponents? Example: Ten to the sixth (6th) power Ten raised to the power 6 Ten to the sixth Ten to the power 6 4) How to read exponents? When the base is raised to the second power we can use the word squared to mean “to the second power.”. Example: Why is squared? Three to the second power Three raised to the power 2 Three to the second Three to the power 2 Three squared To square a number, we need to multiply it by itself. Example: 3 squared = 32 = 3 • 3 = 9, that is the area of a square with the side 3. 3 3 Area: 3 • 3 = 32 = 9 4) How to read exponents? When the base is raised to the third power we can use the word cubed to mean “to the third power.”. Example: Why Cubed? Four to the third power Four raised to the power 3 Four to the third Four to the power 3 To cubed a number, we need to multiply the number by itself three times. Example: 3 cubed = 33 = 3• 3 •3 = 27, that is the volume of a cube with the side 3. Volume: 3 • 3 • 3 = 33 = 27 3 Four cubed 3 3 5) Is there an exponential symbol? If yes, what is the symbol? If expoents are Math operation, don’t we need a operation symbol ? Operation Symbol Symbol Name Addition + Plus sign Subtraction - Minus sign Multiplication × Times sign Division ÷ Division sign Exponent ? ? ➔ We don’t need a special operation symbol to represent an exponent because of the way the numbers are written. ➔ We can recognize this operation because the exponent is written smaller and up at the top of the base: 5) Is there an exponential symbol? If yes, what is the symbol? We don’t need to use any symbol to represent an exponent but there are sometimes, for example when you use the calculater or a computer to calculate an exponent or in a computer programming, we need to use a symbol. Caret 3 ^2 123 ^ 1233 2 ^ 60 5) Is there an exponential symbol? If yes, what is the symbol? 3 ^9 = 19,683 3^9 19683. 3×3×3×3×3×3×3×3×3 ^ Observation: We can find other symbols in the scientific calculator: 5) Is there an exponential symbol? If yes, what is the symbol? Let’s Practice with Calculators 3^4 = 81 5^3 = 125 10^4 = 10000 7^7 = 823543 6) When is any number raised to the number one, what can we conclude? itself Case 1: Any number raised to the power 1 is equal to ______ 12 7) When the base of an exponent is equal one, what can we conclude? 1 Case 2: The number 1 raised to the any power is equal to ___ The number 1 is being multiplying by itself 23 times 8) When the base of an exponent is equal zero, what can we conclude? 0 Case 3: The number 0 (with the exception of itself) raised to the any power is equal to ___ The number 0 is being multiplying by itself 23 times 9) Why is any non-zero number raised to the power of zero equal one? 1 Case 4: Any number (with the exception of 0) raised to the power 0 is equal to ___ Let’s Follow a Pattern –1 = 16 ÷2 ÷2 –1 –1 =9 ÷2 –1 ÷2 –1 =3 =2 =1 –1 =1 = 10000 ÷10 = 1000 ÷3 –1 =4 –1 ÷3 = 27 =8 –1 –1 = 81 ÷10 = 100 ÷3 –1 ÷3 –1 = 10 =1 ÷10 ÷10 9) And what happens when we raise zero to the zero power? Is it still one? undefined Case 4: (The exception): Zero to the Zero Power is _______________ Case 4: Any number to the power 0 = 1. 0 =1 Case 3: Zero to any power = 0. 0 0 =? 0=0 When mathematicians have two perfectly good rules that give different answers for some problem like 00, they say the answer is undefined. 10) Does the exponentiation have the commutative property? Addition Multiplication 2+3 =3+2 =5 2×3 = 3×2 6= 6 Subtraction 9-3 ≠ 3-9 6 ≠ -6 Commutative Division Exponentiation 2÷1 ≠ 1÷2 2 ≠ 0.5 ≠ 8 ≠ 9 Not Commutative Example: Exponentiation So, Exponentiation doesn’t have commutative property. Common Mistake Practice Legend of rice on chessboard -Correction- Practice Legend of rice on chessboard The King’s Chessboard Problem: In The King’s Chessboard, the wise man requests as his reward grains of rice following the rule: 1 grain of rice for the first square on the chessboard, two grains of rice for the second square, four grains of rice for the next square, then eight grains of rice, and so on, for all 64 squares on the chessboard. 1 2 4 8 16 32 64 128 Squares in the board Total grains placed 1st square 1 2 4 8 256 512 2st square 3st square 4st square 5st square 6st square 7st square 8st square 9st square 10st square 16 32 64 128 256 512 Repetead Multiplication (using the same number) 2x2 2x2x2 2x2x2x2 2x2x2x2x2 2x2x2x2x2x2 2x2x2x2x2x2x2 2x2x2x2x2x2x2x2 2x2x2x2x2x2x2x2x2 Other Notation Legend of rice on chessboard 1) How would you determine the total amount of grains of rice on the 64th square? Analysing the pattern and multiplying the number 2 sixty three times. 2) Can you think of another way you might have figured out the solution in question 1? Describe another method. It’s possible to use the concepts of exponentiation. 3) How would you determine the total amount of grains of rice for all 64 squares? Use the exponential to calculate the amount of rice in each square and then, adding the amount of grains on the 64 squares. Legend of rice on chessboard 4) Complete the chart bellow to help you to answer the following question. Calculate the number of grains of rice on the 64th square? You can use the calculator. Using a scientific calculator, we have: 2^63= 9,223,372,036,854,775,808 Legend of rice on chessboard 5) Were you surprised at how quickly the numbers grew? Why or why not? Exponential growth will become fast, even if it’s start slow. https://www.youtube.com/watch?v=k1s1Jg3C6Q&list=RDCMUC8w7ynzLYXpRbPrfpDbMHlw &start_radio=1&t=6 How many grains of rice did the man ask the king? Add each of the 64 squares together we have the total of: 18,466,744,073,709,551,615 grains of rice. Discuss Discuss Discuss In what other situations can we use exponentiation concepts to help us to solve problems? Discuss Go Further Go Further Problem Problem https://thecollegeinvestor.com/17145/would-you-rather-have-a-penny-that-doubles-each-day-for-a-month-or-1million/#:~:text=Now%20that%20you've%20read,30%2C%20you%20would%20have%20%245%2C368%2C709.12. (395) Double a Penny Everyday for 30 Days - Learn The Power of Compound Interest - YouTube Reflect Reflect Thinking routines Fill the blanks in a Mental Mind provided by the teacher. This Mental Mind included all the 10 inquiry questions. Reflect Kahoot Practice Set by playing Kahoot! 1. Enter in the site: https://kahoot.it Practice in the link: 2. Write the pin number: 3. Play! 49 Class - Goals Express numbers in expanded form by using exponents; Name the terms of an exponent and read it properly; Calculate an exponent using a scientific calculator. END Valéria Kumoto MATH 51 Attachment In what other situations can we use exponentiation concepts to help us to solve problems? Slide 56: Viral growth, Slide 57: Grow your wealth. In what other situations can we use exponentiation concepts to help us to solve problems? Viral growth A virus, if unchecked, tends to spread through a population in an exponential manner. I'm sure, if you've been watching as the current coronavirus pandemic has begun its spread across the globe, you'll have heard the term "R-naught" or R0 being used. This refers to how infectious a virus is, and is a measure of how many other people one person may infect. An R0 of 2 means that, on average, one person will infect two others. This growth and spread from one person to two more will take place over a certain length of time, maybe several days. Which is why containment, via isolation, is so important. So back to the chess board example. If one person infects two others, those two will infect two more each, resulting in 4 more new infections. There are now 7 infections in total, the original one, two more, then four more... Over each period of days numbers of newly infected double and re-double, just like the grains of rice on a chess board. After 10 periods of doubling, from one initial infection, there are new 512 people infected. The 11th doubling infects 1,024 more, and the total number infected is now 2,047. Many of the original "generations" of infected will by this time have had an "outcome". They have either recovered or died. CFR (case fatality rates) can begin to be estimated from these figures. The 21st period of doubling produces over 1 million new cases. In what other situations can we use exponentiation concepts to help us to solve problems? Grow your wealth The value of an investment, as interest paid on it compounds over the years, can be thrilling as one approaches retirement. A sum of $1,000, invested at the age of 20, compounded at 13% growth per year, over a term of 45 years will grow to an incredible hoard of wealth. At the end of Year 1 the investment is worth $1,130, and when re-invested for the second year, will gain a further 13%. This results in a sum of $1,277 at the end of Year 2. The maths is simple. You just take the original 1,000 and multiply it by 1.13. The answer is multiplied by 1.13 again to give the result for the end of the second period. Continue this multiplication 45 times, and you have the resulting cash lump sum payout at age 65, from your initial investment of $1,000 at age 20.The amazing result is $216,497. If you'd invested just $5k at age 20, and for 45 years didn't add a single extra cent to your retirement fund, you'd retire a millionaire. $5,000 compounded at 13% over 45 years, comes out at $1,082,484. OK, so I know what you're thinking: "But Ian, you can't earn 13% on your capital. Also, this doesn't take in to account interest rate variables, or the devaluation of your currency due to inflation."You're right, of course. This is just a simple example to further illustrate how compound interest, or exponential growth, works over time to create huge numbers. I've used the optimistic growth rate of 13% for a reason, as you'll soon see. Let's look at another example, perhaps of a more relevant nature right now.
