The number of Kings that lie in this tomb is, and the number of slaves that lie in this tomb is, where

Transcription

1

2

3 Eve, a mathematician from Australia, was touring the still largely intact Great Pyramid of Giza. It is one of the Seven Wonders of the World and the largest and most ancient of the three pyramids in Egypt s Giza Necropolis. Through a caved in section in the wall of the Queen s Chamber, she spied a previously undiscovered room. Q1 The number of Kings that lie in this tomb is, and the number of slaves that lie in this tomb is, where 8 = 2 2 x 2 = a) How many kings and slaves lie in the tomb? Only one of the following sets of numbers satisfies both equations 2 kings, 12 slaves 3 kings, 24 slaves 4 kings, 12 slaves 5 kings, 20 slaves Nearly b) Roughly how many tonnes were installed daily? Only one of the following numbers satisfies both equations c) Therefore = = Written on the wall were several mysterious messages in hieroglyphics and sets of mathematical equations. Can you help Eve crack the code to decipher the secret messages? = tonnes of stone was installed in the pyramid each day, where = 10 + = 1100

4 Q1 Engraved in the wall of the previously undiscovered room is a series of hieroglyphics in big letters! Crack the algebraic code to read what they say. Each symbol relates to a letter of the alphabet. 2 = = = = = = 100 = = 7 2 = 1 = nd letter of the alphabet = = rd letter of the alphabet = = st letter of the alphabet = Remember 4 = 4 x

5 Q1 Inside the room was a massive gold sarcophagus in the shape of a resting King. An inscription was tattooed across the chest. Decipher the code to read the inscription. 3-3 = 30 2 = 64 3 = = 12 = 3 4 = = = 110 ( + 1) x 5 = 100 = st letter of the alphabet = = nd letter of the alphabet =

6 Q2 A scripture was found in the gold sarcophagus detailing the final events that occurred before the King died. Read through the inscription and answer the questions that follow. a) How many tonnes of vegetables were harvested? a) How many tonnes of barley was harvested? a) How many tonnes of flax was harvested? Hint You will have to create algebraic functions from the information in the scripture to calculate the answers to the questions. b) How many tonnes of wheat was harvested?

7

8

9 Firstly, thank you for your support of Mighty Minds and our resources. We endeavour to create highquality resources that are both educational and engaging, and results have shown that this approach works. To assist you in using this resource, we have compiled some brief tips and reminders below. About this resource This Mighty Minds Fundamentals Lesson focusses on one subtopic from the NAPLAN Tests and presents this skill through a theme from the Australian Curriculum (History, Science or Geography). This lesson is also targeted at a certain skill level, to ensure that your students are completing work that is suited to them. How to use this resource Our Fundamentals Lessons are split into two main sections, each of which contain different types of resources. The student workbook contains The main title page; and The blank student activities for students to complete. The teacher resources section contains This set of instructions; The Teacher s Guide, which offers information that may be needed to teach the lesson; The Item Description, which gives a brief overview of the lesson and its aims, as well as extension ideas; The student answer sheets, which show model responses on the student activities to ensure that answers to the questions are clear; The teacher s answer sheets, which provide a more detailed explanation of the model responses or answers; and Finally, the end of lesson marker. We suggest that you print the student workbook (the first set of pages) for the students. If students are completing this lesson for homework, you may also like to provide them with the student answer pages. Feedback and contacting us We love feedback. Our policy is that if you us with suggested changes to any lesson, we will complete those changes and send you the revised lesson free of charge. Just send your feedback to resources@mightyminds.com.au and we ll get back to you as soon as we can.

10 In arithmetic, an expression only contains numbers and operations (such as +, -, x or ). In algebra, in addition to numbers and operations, pronumerals (x, y, etc.) are used to represent unknown values. e.g. Arithmetic: 15 5 = 3 Algebra: 15 a = 3 To solve a simple algebraic expression, a rule of thumb is to change the equation so that the pronumeral (in this case a ) is by itself on one side of the equals sign, with all numbers on the other. This is called balancing. To get numbers from one side of an equation to the other, you must perform the reverse operation (e.g. subtraction is the reverse operation of addition) to both sides. Remember that an operation always accompanies the number it precedes. Ensure that when moving values from one side of an equation to the other, they are moved according to inverse order of operations (i.e. + and first; x and second; brackets third). See the examples below. Addition In addition problems, subtraction is used to isolate the pronumeral: e.g. w + 8 = 23 Subtract 8 from both sides: w = 23 8 (note: this step is often missed out) Simplify: w = 23 8 Solve: w = 15 Subtraction Subtraction is most easily solved by first converting the subtraction equation to an addition one. e.g. 56 g = 32 convert to addition 56 = 32 + g Subtract 32 from both sides: = 32 + g 32 Simplify: = g Solve: 24 = g Multiplication The reverse operation of multiplication is division. Note that often the multiplication sign is left out of the equation thus 7t is the same as 7 x t. e.g. 7t = 28 Divide both sides by 7: 7t 7 = 28 7 Simplify: t = 28 7 Solve: t = 4 This teaching guide is continued on the next page...

11 ...This teaching guide is continued from the previous page. Division Division problems are most easily solved by first converting to a equivalent multiplication equation. e.g. 54 p = 6 Convert to multiplication: p x 6 = 54 Divide both sides by 6: p x 6 6 = 54 6 Simplify: p = 54 6 Solve: p = 9 21 Note that division equations are often written as a fraction. Thus 7 is the same as 21 7 Squares and Square Roots Square root is the reverse operation of squaring, and vice versa e.g. d 2 = 25 Square root both sides: d = 25 Solve: d = 5 Like Terms A term in algebra is the simplest unit in an expression or equation. It can contain numbers, pronumerals (representing an unknown value and usually expressed as a letter of the alphabet) and indices. Like terms are terms that have the same pronumeral or pronumerals and powers, but can have differing coefficients (a number that comes before a pronumeral e.g. with 3x, 3 is the coefficient and x is the pronumeral). Therefore, 3xy 3 and 8xy 3 are like terms, but 9x 2 z and 2xy 5 are not. Like terms can be added and subtracted from each other, but unlike terms cannot. However, both like and unlike terms can be multiplied and divided. Example: 4de + 4d de d 2 + 3d = 48de + 3d 2 + 3d Distributive Law The distributive law states that a (b + c) = ab + ac The law is applied to expand brackets in order to solve equations. The opposite process is called factorisation, i.e. ab + ac a (b+c) Example: 4(g 2 + 2h) = 4g 2 + 8h Index Laws There are six index laws which dictate how to perform calculations with numbers/pronumerals in index form (i.e. numbers with a power) This teaching guide is continued on the next page...

12 ...This teaching guide is continued from the previous page. 1. Multiplication: when multiplying powers with the same base, add the indices. Rule: a x x a y = ab x + y Example: d 5 x d 8 = d Division: when dividing powers with the same base, subtract the indices. Rule: a x a y = a x y Example: t 17 t 4 = t 13 t 17 Note: division sums are often written as a fraction. Thus, is the same as t 17 t Power of zero: any number, except 0, raised to the power of 0 is equal to 1. Rule: a 0 = 1, when a 0 4. Index of an index: when an index form is raised to another power, multiply the indices. Rule: (a x ) y = a xy Example: (k 4 ) 3 = k Powers of products: when a product is raised to a power, every factor of the product is raised to the power. Rule: (ab) x = a x x b x 6. Powers of quotients: when a quotient is raised to a power, both the numerator and denominator are raised to the power. a Rule: = a ( ) x x b b x t 4

13 Item Description Please note: any activity that is not completed during class time may be set for homework or undertaken at a later date. Pyramid Problems, Hieroglyphics Help and Encoded Inscription Activity Description: By solving simple algebraic functions, students will be able to break a number of codes to reveal facts about the Great Pyramid of Giza and secret messages. They will be required to solve equations to determine the value of unknown symbols as well as substitute given values into equations The first activity Pyramid Problems uses hieroglyphics to solve sets of mathematical questions The second activity Hieroglyphics Help requires the student to rearrange mathematical functions to identify the value of certain hieroglyphics The third activity Encoded Inscription uses hieroglyphic symbols to formulate mathematical equations. The students are to decipher the codes to yield the correct answer Purpose of Lesson: This lesson builds students basic algebra skills in a fun context, using hieroglyphics instead of the standard pronumerals. KLAs: Mathematics, History CCEs: Recognising letters, words and other symbols (α1) Interpreting the meaning of words or other symbols (α4) Synthesising (θ44) Translating from one form to another (α7) Calculating with or without calculators (Ф16) Substituting in formulae (Ф19) Applying a progression of steps to achieve the required answer (Ф37) Suggested Time Allocation: This lesson is designed to be completed in an hour twenty minutes per activity This Item Description is continued on the next page...

14 Item Description continued This Item Description is continued from the previous page. Pyramid Problems, Hieroglyphics Help and Encoded Inscription Teaching Notes: The first activity Pyramid Problems, allows students to gain a fundamental knowledge of substitution in algebra Use simple examples to explain pronumerals (x & y) before beginning this activity A simple example used can be if Tom s age is unknown, we could say that he is x years old. If Mary is three times Tom s age, we could say that Mary is 3x. If Mary is 30, how old is Tom? If 3x = 30, then x = 30/3. The second activity Hieroglyphics Help, requires students to rearrange the symbols in order to solve the mathematical functions It would be useful to get the students to replace the symbol with a pronumeral to make the function clearer Go through the answers as a class after the completion of this activity before moving onto activity three The third activity Encoded Inscription, enables students to use their knowledge of rearranging algebraic functions to solve more difficult questions. Encourage the students to double check each question as they go through the activity Students are encouraged to share their answers and discuss their thinking process Follow Up/ Class Discussion Questions: What other cultures use hieroglyphics or symbols as part of their language? In the second activity, what are some ways to check that the answers calculated were correct?

15 Eve, a mathematician from Australia, was touring the still largely intact Great Pyramid of Giza. It is one of the Seven Wonders of the World and the largest and most ancient of the three pyramids in Egypt s Giza Necropolis. Through a caved in section in the wall of the Queen s Chamber, she spied a previously undiscovered room. Q1 The number of Kings that lie in this tomb is, and the number of slaves that lie in this tomb is, where 8 = 2 2 x 2 = a) How many kings and slaves lie in the tomb? Only one of the following sets of numbers satisfies both equations 2 kings, 12 slaves 3 kings, 24 slaves 4 kings, 12 slaves 5 kings, 20 slaves Nearly b) Roughly how many tonnes were installed daily? Only one of the following numbers satisfies both equations c) Therefore = 1 Written on the wall were several mysterious messages in hieroglyphics and sets of mathematical equations. Can you help Eve crack the code to decipher the secret messages? = 10 tonnes of stone was installed in the pyramid each day, where = 10 + = 1100 = 100

16 Pyramid Problems Question One: Students should have answered the questions by solving the given basic algebraic problems. a) The only set of values of the given four that satisfies both equations is 3 and 24. This is found by substituting each set of possible values to determine if the answer they give is correct: 8 = 24 8 = 3 This gives a correct expression. 2 2 x 2 = 4 x 2 x 3 = 24 This also gives a correct expression Therefore 3 kings and 24 slaves lie in the tomb. b) The correct answer is 1000 as it satisfies both equations such that is the same number. This is found by substituting each possible answer into both equations to determine. If = 1000, then 1000 = 10. Convert to a multiplication equation 1000 = 10 x Divide 10 from both sides and simplify 100 = Solve = 100 If = 1000, then = 1100 Subtract 1000 from both sides & simplify = Solve = 100 c) Therefore = 1, = 10 and = 100

17 Q1 Engraved in the wall of the previously undiscovered room is a series of hieroglyphics in big letters! Crack the algebraic code to read what they say. Each symbol relates to a letter of the alphabet. 2 = = = = = = 100 = = 7 2 = 1 L O O T E R S = 12 th letter of the alphabet = L = 15 th letter of the alphabet = O 20 T = 5 th letter of the alphabet = E = 18 th letter of the alphabet = R = 19 th letter of the alphabet = S = 2 nd letter of the alphabet = B = 23 rd letter of the alphabet = W = 1 st letter of the alphabet =A Remember 4 = 4 x B E W A R E

18 Hieroglyphics Help Question One: Students should have solved the algebraic equations by isolating each variable, then worked out which letter of the alphabet each symbol represented. They then should have substituted these letters in to the code to work out the secret message. 2 = 144 = 144 = 12 Square root both sides, simplify & solve 12 th letter of the alphabet = L = 50 3 = 45 Subtract 5 from both sides Divide both sides by 3 = th letter of the alphabet = O 100 = = 5 x Multiply both sides by = Divide both sides by 5 20 = 20 th letter of the alphabet = T 25 + = = 10 Simplify ( 25 = 5) = 10 5 = 5 Subtract 5 from both sides and solve 5 th letter of the alphabet = E 20 - = = 2 Simplify ( 4 = 2) 20 = 2 + Add to both sides 20-2 = = 18 Subtract 2 from both sides and simplify 18 th letter of the alphabet = R + 1 = = = 19 Substitute for th letter of the alphabet = S 100 = 200 = = 2 Divide both sides by 100 and simplify 2 nd letter of the alphabet = B 161 = = 7 x Multiply both sides by = = 23 Divide both sides by 7 and solve 23 rd letter of the alphabet = W 2 = 1 = 1 = 1 Square root both sides, simplify & solve 1 st letter of the alphabet = A Substitute in all letters for the corresponding hieroglyphic: L O O T E R S B E W A R E

19 Q1 Inside the room was a massive gold sarcophagus in the shape of a resting King. An inscription was tattooed across the chest. Decipher the code to read the inscription. 3-3 = 30 2 = 64 3 = = 12 = 3 4 = = = 110 ( + 1) x 5 = = st letter of the alphabet = = nd letter of the alphabet = K H U F U K H U F I L V E S L I V E S

20 Q1 A scripture was found in the gold sarcophagus detailing the final events that occurred before the King died. Read through the inscription and answer the questions that follow. a) How many tonnes of vegetables were harvested? 2 tonnes a) How many tonnes of barley was harvested? 2 tonnes a) How many tonnes of flax was harvested? 1 tonne Hint You will have to create algebraic functions from the information in the scripture to calculate the answers to the questions. b) How many tonnes of wheat was harvested? 3 tonnes

21 Encoded Inscription Question One: Students should have solved the algebraic equations by isolating each variable, then worked out which letter of the alphabet each symbol represented. They then should have substituted these letters in to the code to work out the secret message. 3-3 = 30 3 = 33 Add 3 to both sides = 11 Divide both sides by 3 11 th letter of the alphabet = K 2 = 64 = 64 = 8 Square root both sides, simplify & solve 8 th letter of the alphabet = H 3 = 7 = 21 Multiply both sides by 3 21 st letter of the alphabet = U 24 2 = = Add 2 to both sides 12 = 2 Subtract 12 from both sides 6 = Divide both sides by 2 and solve 6 th letter of the alphabet = F = 3 = 3 2 = 9 Square both sides, simplify and solve 9 th letter of the alphabet = I 4 = 48 = 12 Divide both sides by 4 and solve 12 th letter of the alphabet = L + 1 = 22 = Substitute for nd letter of the alphabet = V = = 100 Subtract 10 from both sides Divide both sides by 20 = 5 5 th letter of the alphabet = E ( + 1) x 5 = 100 ( + 1) = 20 Divide both ides by 5 = 19 Minus 1 from both sides 19 th letter of the alphabet = S Substitute in all letters for the corresponding hieroglyphic: K H U F U L I V E S

22 Encoded Inscription Question Two: Students should have answered the questions by identifying the algebraic functions within the given text. They then should have substituted the known information into the functions to work out the answers to the questions. From the information provided it can be deduced that the functions required to answer the questions can be made from the text highlighted red. 1. We harvested 3 times more wheat compared to flax 3 x Flax = Wheat 2. The farmers yielded the same amount of vegetables as they did barley Barley = Vegetables 3. The amount of flax harvested was half the amount of barley harvested Barley 2= Flax In the text it gives the information that 2 tonnes of vegetables were harvested. Using this information, the students can work out the answers to the functions established from the text. 1. Barley = Vegetables 2 tonnes of vegetables = 2 tonnes of barley 2. Barley 2= Flax 2 tonnes of barley 2 = 1 tonne of flax 3. 3 x Flax = Wheat 3 x 1 tonne of flax = 3 tonnes of wheat The students are now able to answer the questions with the answers gathered from the functions. a) 2 tonnes b) 2 tonnes c) 1 tonne d) 3 tonnes

23