Student s booklet. Bills, Ladders and other Number Diagrams. Meeting 23 Student s Booklet. Contents. May 18, UCI. Town Bills 2 Big and Small

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1 Meeting Student s Booklet Bills, Ladders and other Number Diagrams May 18, UCI Contents 1 Town Bills Big and Small Student s booklet Quarnies Exponent Ladders UC IRVINE MATH CEO

2 1 TOWN BILLS 1 Town Bills (i) Meeting (MAY 18, 016) = 1 + $1.00 (ii) In a town there are three types of bill: Triangle, Pentagon and Circle bills. A visitor has already figured out the following: A Triangle bill is worth $. A Pentagon bill is worth $5. a For each product, determine a combination of bills that will allow you to buy it without receiving any change. An example is shown. Challenge: Can you find all possible combinations of bills? = + = + = + = + $9.00 (iii) $18.00 (iv) $ (v) $101.00

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4 1 b TOWN BILLS For each product(s), determine a combination of bills that will allow you to buy it without receiving any change. Recall that: A Triangle bill is worth $. A Pentagon bill is worth $5. (i) Meeting (MAY 18, 016) = + = + = + = + $.00 (ii) $19.00 (iii) $6.00 Challenge: How do the answers to (iii) and (iv) follow from the previous two answers? (iv) $.00

5 1 TOWN BILLS The Circle Bill c Meeting (MAY 18, 016) Clue 1 + Recall that: A triangle bill is worth $ A pentagon bill is worth $5 With A new bill has come out: the Circle bill. Based on the three clues presented, can you determine the value of a single Circle bill? Note: assume that the value of the circle bill is a positive integer. Also, you are allowed to get change back. 1 You can buy $1.5 Clue With + You cannot buy $1.99 EXAMPLE: With a triangle bill and a pentagon bill (total of $8) you can buy an ice cream cone that costs $.75 and have $5.5 in change, $9.99 $.75 but you do not have enough money to buy a soccer ball that costs $9.99. Clue 1 + With + 5 You can buy $.99

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7 BIG AND SMALL Big and small a Meeting (MAY 18, 016) Complete following diagrams: the We will consider diagrams connecting two different values: a big number on the right, and a small number on the left The number on the line connecting the two numbers indicates the difference between the values (that is, their distance on the number line) ( tennis balls) (1 tennis ball) The tennis ball represents a value. Can you figure out this value? b -1 1 =

8 c (i) BIG AND SMALL Each of the letters x, y, z, w represents a value which you need to figure out based on the diagram. Find the value of the letter and verify it by writing an equation. The first exercise has been done for you. x x = Meeting (MAY 18, 016) (iii) - z z = 0 because 11 7 because + 7 = 11 (iv) This is a tricky one! (ii) y 8 y = 15 because w w w = 6 because

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10 d (i) BIG AND SMALL Meeting (MAY 18, 016) Turn each equation into a diagram, and find the value of the variable. An example is shown. Given the equation x + = 10: x = (iii) x 10 (ii) Given the equation 0 + x = 7x: Given the equation 1 + x = 7: x = (iv) Given the equation 8 + x = 5: x = x =

11 BIG AND SMALL Meeting (MAY 18, 016) BOOKLET CHALLENGE Find the values of x, y and z. / 1/ z 1/ 1 x y 1

12 BIG AND SMALL Meeting (MAY 18, 016) CHALLENGE Find the values of x and y. y x 15 8 x + y

13 QUARNIES Meeting (MAY 18, 016) Quarnies A Quarny is a square divided into four squares (top left, top right, bottom left and bottom right), with some of them shaded (sometimes none of them, sometimes all of them). Here are some examples: A B a Draw every quarny that has its top left square shaded. b Draw every quarny that has its top left square unshaded. The empty quarny The full quarny A one-fourth quarny A one-half quarny Another one-half quarny DISCUSS Orientation matters when considering quarnies. The following quarnies, A and B, are not equal: Why did you obtain the same number of quarnies in part a and in part b? Explain.

14 QUARNIES Meeting (MAY 18, 016) Joining Quarnies Rotating Quarnies We can rotate a quarny one quarter turn clockwise. We indicate this action by the following symbol: R EXAMPLES We can join two or more quarnies, which means that we create a new quarny incorporating all shaded regions of the quarnies that were joined. We use the symbol: R EXAMPLES R J R J R J

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16 c a QUARNIES Meeting (MAY 18, 016) Compute the following operations by shading the quarnies. R R J R J Join them and then rotate. or rotate each and then join them? d a Suppose that we have a collection of three quarnies. We could either: (i) First join all of them and then rotate the quarny, or (ii) Rotate separately each quarny and then join them. Do you think that procedures (i) an (ii) yield the same quarny in the end? Try this with several examples before answering.

17 e a QUARNIES Meeting (MAY 18, 016) In the following problems you start with a collection of quarnies. Try to rotate them and join them until you obtain the full quarny (with all squares shaded). We include an example. If it is not possible to do so, indicate why. EXAMPLE Rotate B Join A, D and C We can choose to rotate B first, then join that A and C A B A C (i) A B A B A B (ii) (iii) C D D C

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19 QUARNIES CHALLENGE We introduce a new operation called mirror (M) or horizontal axis reflection. Here are some examples: M Meeting (MAY 18, 016) (i) Out of all 16 quarnies, how many quarnies do not change when applying the transformation M? Draw them all. (ii) We introduce a transformation X. This transformation permutes the squares in the following way: 1 X 1 Ex: M M So transformation X switches the top right () and bottom right () squares and leaves the other squares fixed. M Pick any two half-full quarnies (each having shaded squares). Can you transform one into the other by performing one or several transformations of type R and X? Try this several times.

20 QUARNIES booklet CHALLENGE We have a circle of quarnies. In each square a different amount of area is shaded. The operation next to the arrow describes how the amount of shaded area changes going from one quarny to the next. Meeting (MAY 18, 016) The circle of life x (/)?? x Complete the picture, filling in the missing operations.??

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22 EXPONENT LADDERS Meeting (MAY 18, 016) Exponent ladders Each of the diagrams shown is called an exponent ladder. We have the base in the bottom and then we go up the ladder by multiplying by the base each time. The length of the ladder (number of values in the ladder) is called the exponent. The top of the ladder is the final result of the repeated multiplication by the base. BASE RAISED TO THE EXPONENT 8 x x 65 x5 x5 5 x5 5 x x6 x x x x x x x x 10 6 x x BASE 7996 x6 EXPONENT 15 x6 6 9 x

23 c EXPONENT LADDERS Meeting (MAY 18, 016) Complete the following exponent ladders /9 10,000

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25 d EXPONENT LADDERS Meeting (MAY 18, 016) Complete the following exponent ladders ,

26 e EXPONENT LADDERS Meeting (MAY 18, 016) Multiply the following two ladders of the same length by multiplying their corresponding values in each position. 6 = 6 6 = 79 ( )6 = 5 = 5 = ( )5 = = 16 = 81 ( ) = ( ) = = 8 x = 7 = = = 9 ( ) = 6 1 = 1 = ( )1 = 6 DISCUSS Is it true that when we multiply two exponent ladders we also obtain an exponent ladder? Explain.

27 EXPONENT LADDERS CHALLENGE A Consider the following ladder of length 9 and base, for the value 9 = 51: 9th 9 = 51 8th 8 = 56 7th 7 = 18 6th 6 = 6 5th 5 = th = 16 rd =8 nd = 1st 1 = Create a new ladder of length from the previous ladder by eliminating positions 1,,, 5, 7 and 8. In other words, keep only positions that are multiples of. We know that a ladder of base b and length looks like this: b b b Ladder B 9 Ladder B What is the base b and the length n of this ladder? b = ( ) n = Ladder A Meeting (MAY 18, 016) Using the fact that Ladders A and B have the same top value, what equality can we conclude? ( ) = ( )

28 EXPONENT LADDERS CHALLENGE B Consider the following ladder of length 8 and base 5, for the value 58 = 9065: Create a new ladder from the previous ladder by only keeping positions that are multiples of. Meeting (MAY 18, 016) We know that a ladder of base b and length looks like this: b b b 8th 8 5 = 9065 b 7 7th 5 = th 56 = th 55 = 15 th 5 = 65 rd 5 = 15 nd 5 = 5 1st 51 = 5 Ladder A Ladder B Using the fact that Ladders A and B have the same top value, what equality can we conclude? 8 Ladder B ( 5 ) = ( 5 ) What is the base b and the length n of this ladder? b = ( ) n =

29 EXPONENT LADDERS CHALLENGE C Consider the following ladder of length 8 and base 5, for the value 58 = 9065: 8th 58 = th 57 = th 56 = th 55 = 15 th 5 = 65 rd 5 = 15 nd 5 = 5 1st 51 = 5 Ladder A Create a new ladder from the previous ladder by only keeping positions that are multiples of. Meeting (MAY 18, 016) We know that a ladder of base b and length looks like this: b b Ladder B Using the fact that Ladders A and B have the same top value, what equality can we conclude? 8 Ladder B ( 5 ) = ( 5 ) What is the base b and the length n of this ladder? b = ( ) n =

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