Everyday Maths: Book 3 - Ages 11-13 by Teacher Superstore

For Ages 11 - 12 r o e t s Bo r e p ok u S

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Everyday Maths Book 3

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Mathematics problems set in a real world . te context. o c . c e her r o st super

Written by Jane Bourke. Illustrated by Rod Jefferson. © Ready-Ed Publications - 1997 Published by Ready-Ed Publications P.O. Box 276 Greenwood WA 6024 Email: info@readyed.com.au Website: www.readyed.com.au COPYRIGHT NOTICE Permission is granted for the purchaser to photocopy sufficient copies for non-commercial educational purposes. However this permission is not transferable and applies only to the purchasing individual or institution.

ISBN 1 86397 169 6

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Contents 4 7 8 9 10

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Teachers’ Notes and Problem Solving Strategies Problem Solving Check These Out! Calculating Circumference Life in the Wheel World Volume and Capacity Archimedes in the Pool Archimedes’ Pool Challenge Problem Solving: Using a Table Hamburger Headache Space: Investigating Positions Building Boom Problem Solving: Number Problems Kilojoule Capers Graphs: Recording and Analysing Data Archimedes’ Diet 1 Archimedes’ Diet 2 Problem Solving: Looking for a Pattern Galileo’s Used Car Yard Problem Solving: Number Problems Transport Trouble! Problem Solving: Solve a Simpler Problem Building a Nest Egg Measurement: Investigating Perimeter Fenced In Volume and Capacity Pythagoras’ Packages 1 Pythagoras’ Packages 2 Measurement: Investigating Area Galileo’s Gazette Problem Solving: Using a Table Sports Stars Problem Solving: Itineraries and 24 Hour Time Archimedes’ Adventures 1 Archimedes’ Adventures 2 Archimedes Phones Home Problem Solving: Percentages Easy Money Graphs: Interpreting Data What’s the Weather Like? 1 What’s the Weather Like? 2 Problem Solving: Area and Perimeter Crazy Car Yards 1 Crazy Car Yards 2 Reading Timetables: Train Trips 1 Train Trips 2 Problem Solving: Distances and Averages Cool Climbing 1 Cool Climbing 2 Problem Solving: Fishing Fever Brainteasers More Brainteasers Even More Brainteasers Answers Record Sheet

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Teachers’ Notes The idea of problem solving activities often conjures up images of numbers and objects that have no direct meaning for students other than teaching the basic problem solving strategies. The activities in this book are designed to present real-life problems in a realistic context so as to provide children with situations in which everyday problem solving and comprehension skills are required. The activities are based around recurring characters who find themselves exposed to a range of problems that need to be solved; the sort of problems that students may one day encounter.

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Problem Solving Strategies

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Most pages include a challenge activity, usually an extension of the main problem, which will further consolidate comprehension skills. Included throughout the book are brainteaser pages which focus on a particular problem solving strategy, highlighted at the foot of the page. These brainteasers can be photocopied and individually glued on to card so as to create a set. Students might like to think up their own brainteasers to add to the set.

There are many strategies for solving everyday maths problems. Some of the main problem solving strategies have been explained below. In some cases examples of problems are given where the particular strategy can be applied. Guess and check: Probably the first strategy children might try and definitely the easiest. By making a guess and checking their answer, children have a point of reference on which to base all other guesses.

© ReadyEdPubl i cat i ons An example: •off o r evi ew p r p ose s182. on l y• I am thinking twor consecutive numbers thatu when multiplied give A guess might be to try 14 x 15 which would give 210. Obviously the next guess would try lower numbers.

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Act it out: Students quite often need to visualise the problem, especially where people or objects are concerned. Counters, coins and students can be used to help solve the problem. Examples: There are 48 players in the darts championships. Each player stays in the competition until they lose a game. How many games must be played to find the club champion?

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A caterpillar crawls 2 m up a tree every day. Every night it slips back 50 cm. The tree trunk is 10.5 m tall. How long will it take for the caterpillar to reach the top of the trunk? Make a model: When problems cannot be acted out the next best thing is to make a model using cubes, matches and so on. Make a drawing, diagram or graph: Graphs and diagrams are particularly useful for trying different combinations or clarifying information. An example: Jack has a rectangular field that has an area of 360 m. What are the possible dimensions of the rectangle?

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Look for a pattern: This strategy can be used in many number and space activities to help simplify the problem. Number patterns: It takes three matches to make a triangle, 5 matches to make 2 triangles. How many matches are needed to make 3 triangles? Spatial Patterns: How many squares are there on a checker board? Construct a table: By organising data in a more meaningful way children can better see relationships, patterns and possibly missing information. This strategy is best used where different information is given about each person or object in the problem. A table can include all the information and eliminate irrelevant information.

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Swimming

Tim

Peter Kelly Max

Tennis

Netball

Hockey

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An example: Tim, Peter, Max, Jane, Tannie and Kelly each play sport over the weekend. They all play a different sport. Match the person to their sport based on the following: Tim doesn’t like swimming but enjoys cricket; Peter likes tennis more than swimming; Kelly enjoys netball; Max won’t play hockey; Jane doesn’t like cricket or diving; Tannie plays the sport that Max doesn’t like. Cricket

Diving

Jane

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The solution can be found through the process of elimination.

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Tannie

Make a list: All possibilities can be listed when using this strategy and again the process of elimination can be used.

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An example: You have three T-shirts: red, blue and yellow; and four pairs of jeans: green, black, navy and light blue. How many different combinations can you wear? Restate the problem: This is best used to make sure students fully comprehend the problem and know what they need to do to find the solution. An example: At the supermarket Sarah bought some groceries. All the items she bought were the same price and she bought as many things as the total number of pence she was charged for each item. If her bill was $6.25 how many things did she buy? This could then be rephrased as: How many things can be bought with $6.25 where they all cost the same amount? Ready-Ed Publications

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Solve a simpler problem: By exploring a simpler problem, an apparently difficult task can be made easier. Students can look for a pattern and then transfer this pattern to the larger problem. An example: There are 20 people at a meeting. Everyone shakes hands with each person once. How many handshakes take place? This could be tried with a group of five and then children can look for a pattern. Account for all possibilities: This strategy can be used in addition to some of the strategies already mentioned such as making a list.

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An example: Emily is playing Monopoly and wants to buy Park Lane at a cost of $400. She has four $100 notes, ten $50 notes, seventeen $20 notes, eight $10 notes, fifteen $5 notes and six $1 notes. How many different combinations of $400 can she hand over to the banker? Use logical reasoning: This strategy involves students using what they already know to solve a problem. A solution can be reached when logical reasoning is used to draw conclusions about mathematics. Strategies involve using models, known facts, properties and relationships to explain thinking. An example: Ann, Brendan, Cathy and Daryl all play an instrument in the school band. They play the tuba, violin, flute and harp. Ann plays the harp and Brendan does not play the violin. If Cathy plays the tuba, what does Daryl play?

Work backwards: This strategy works best when a problem is stated so that the final outcome is clear. In such a case the condition that existed earlier needs to be determined.

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Flight departure Delay of one hour and fifteen minutes Arrived two hours early

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Tips for Students

10.30 10.30 less 1.15 = 9.15 9.15 - 2.00 = 7.15 pm

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Michael arrived at the airport and noticed that he had to wait two hours for his flight. A delay of one hour and fifteen minutes was announced. Michael’s flight eventually departed at 10.30 pm. At what time did Michael actually arrive at the airport?

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(Perhaps these notes could be placed on a wall chart.)

Make sure you understand the problem. Have a go even if you just play around with the problem. Try a variety of strategies. Learn from your mistakes. Keep a record of your working out for the bigger problems so that you can refer back if needed. Check your answers.

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Name: __________________

Check These Out! 1. Archimedes is confused. A barrel full of sand weighs 40 kilograms and the same barrel filled with gold nuggets weighs 60 kilograms. If the gold weighs twice as much as the sand, how much would the empty barrel weigh?

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2. A class of young students were riding past Pythagoras’ window. As they rode past he counted 64 bike wheels. There are 25 students in the class. How many were on tricycles and how many were on bicycles?

Use this table to help you guess and check: Bikes:

Trikes: Total:

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3. At Pythagoras’ farm there is a paddock containing both peacocks and sheep. You counted 30 eyes and 44 feet. Use the table below to help you determine how many of each are in the paddock. Peacocks:

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Total Eyes: Total Feet:

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Sheep:

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Hint: You know that every animal has a certain number of eyes.

Challenge: Zany Zoo Galileo visited Zorba’s Zany Zoo and came across a cage that held a strange mix of jungle animals and creatures. In this cage he counted 11 heads and 20 feet. He also noticed that there were twice as many jungle creatures with four feet as there were with two feet. How many creatures of each kind were in the cage? Hint: Could there be other creatures in the cage?

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Name: __________________

Life in the Wheel World Pythagoras has been cycling around the lake. His bike is fairly old and one brake pad has worn away on the front wheel. A certain part of the wheel always touches the brake pad making a short grinding noise. It only does this once every revolution.

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Pythagoras has been meaning to buy an odometer for his bike so he can measure the distance around the lake. However he has decided to use his brain instead. He knows that he has 48 cm diameter tyres on his bike and thinks he can work out the circumference. He has placed the wheel so that it will grind immediately and is going to count the noises he hears.

π is approximately equal to 3.14

x diameter (D).

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Remember: Circumference of a circle = pi

D = 48 cm

Pythagoras counted 250 grinding noises when he rode once around the lake.

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1. Approximately how far in metres is the distance around the lake? .................................

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How many grinding noises should Pythagoras expect to hear if he rides:

2. Three times around the lake? .........................................................................................

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3. Five times around the lake? ............................................................................................ 4. Exactly half way around the lake? ................................................................................... 5. If he rides for exactly four and a half kilometres?............................................................ 6. If he rides at twice the speed around the lake once? ......................................................

Challenge: A tree doubled its height each year until it reached its maximum height of 9.6 metres. It took ten years to reach this height. How many years did it take for the tree to reach half its maximum height?

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Name: __________________

Archimedes in the Pool Archimedes has a pool in his garden. It has been raining all day and the pool is full to the brim. The rectangular pool measures 4 m long and 2.5 m wide. It is 3.75 m deep in all areas of the pool because Archimedes once dived in the shallow end and created a huge hole! Since then it has been levelled out to 3.75 m.

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1. What is the total capacity of the pool? ............................................................................... Volume, Capacity and Mass are related. Look at the table below:

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Volume (cubic centimetres) 1 cm 3 10 cm 3 100 cm 3 200 cm 3 500 cm 3 1000 cm 3

Mass (grams) 1g 10 g 100 g 200 g 500 g 1 kg

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Capacity (millilitres) 1 mL 10 mL 100 mL 200 mL 500 mL 1 L

2. Archimedes needs to chlorinate the pool every week. He uses a litre of chlorine for every 15 cubic metres of water. How much chlorine does Archimedes need each week?

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$12.99 10 Litres

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Bulk Chlorine Supplies

$79.50 90 Litres

A year’s supply of Chlorine from Chlorine World

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Clean Chlorine

$125.99 130 Litres

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3. How much chlorine does Archimedes need for one year? .............................................. 4. Which offer should Archimedes take if he puts the same amount of chlorine in his pool all year round? ........................................................................................................ 5. Will he have any chlorine left over? ...................................................................................

Challenge: If three hours ago it was as long after one o’clock in the morning as it was before one o’clock in the afternoon, what time would it be now? Ready-Ed Publications

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Name: __________________

Archimedes’ Pool Challenge Archimedes had a measure on the side of his pool to let him know when the water levels rose and fell. One day the water level was right on the line. Archimedes hopped in and noticed that the level rose above the line. Using a marker he drew a line at the new level.

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When he hopped out he noticed the level returned back to the normal mark. Archimedes wanted to know if the level the water rose had anything to do with his weight.

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Using the hose, he filled the pool up to the exact level it reached when he jumped in. He decided to measure the amount of water he had added by removing it in bucket loads and counting the buckets. He knew that each bucket held 6 litres.

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1. Archimedes weighs 138 kilos. How many bucket loads will he need to remove to return the water level to the normal mark?

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2. Archimedes’ family thought he was crazy. He told Mrs Archimedes he could guess how much she weighed if she hopped in the pool and did not splash around too much. Archimedes measured the levels again and used the bucket to work out how much water was displaced. How much does each person weigh if Archimedes removed the following bucket loads?

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The whole family, including Archimedes, then got into the pool and Archimedes marked the levels again. This time Aunty Agnes got in and she weighed 72 kilos. Later on when they were out of the pool, Archimedes looked at the two different levels and said, “I know exactly how many buckets of water are needed to make the water level rise to the mark I made!” 3. How many buckets are needed? ................. Hint: Don’t forget Aunty Agnes!

Challenge: A worm is at the bottom of a twenty metre well. It can crawl upwards at the rate of four metres a day but at night it always slips back three metres. At this rate how long will it take before the worm can crawl out of the well? Page 10

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Name: __________________

Hamburger Headache Every Sunday Galileo and his family eat hamburgers for dinner. Mrs Galileo gets confused as everybody likes different things in their hamburger, so she now phones through an order to the hamburger shop before she gets there. She knows that: Everyone in the family likes meat, lettuce and tomato; She likes gherkin but doesn’t like beetroot or cheese; Galileo likes cheese but not gherkin or egg; Gary Galileo does not like egg or beetroot but loves gherkin; Gina Galileo does not like beetroot or gherkin but likes cheese.

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Mrs Galileo phoned through her order and was given two extra burgers for making things easier. Here is the order she gave:

egg

cheese

beetroot gherkin

cheese

beetroot gherkin

cheese

beetroot gherkin

egg

cheese

beetroot gherkin

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1. Which burger is for which person? Write their names beside their burgers.

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2. Who is most likely to eat the leftover burgers? Write their names beside the burgers.

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Hint: It may help you to draw up a table to solve the problem.

A regular hamburger with meat, lettuce and tomato costs $3.50 with extras costing:

Egg: Beetroot: Cheese: Gherkin:

35c 45c

3. What would the total cost of the order be? .....................................................................

Challenge: How many times can you subtract the number 6 from 36? Ready-Ed Publications

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Name: __________________

Building Boom Galileo has been busy designing new buildings for the town of Mathemia. He is using 2 cm blocks to build models of the buildings and has decided that Mathemia needs at least 7 buildings to make it look like a decent sized town. He has 50 cubes. Create seven building models using all of the cubes and draw them on a piece of grid paper.

r o e t s Bo r e p ok u S Left side

Under each building write the volume and surface area for each building.

Galileo is concerned about what the buildings will look like from other angles. Choose the largest building you have drawn.

Top View

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On the grid below draw what your building will look like from the top, sides and front.

Right side

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Challenge: The Mathemia Hospital that Galileo designed looks like this: Front view Right Side view Left side view

Top view

Draw the building on grid paper. How many cubes are used? Page 12

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Name: __________________

Kilojoule Capers Archimedes hopped into the bath one day and because he was so huge all the water flowed over the edge of the bath. Since Archimedes flooded out the bathroom, Mrs Archimedes has been on at him about his weight. She has suggested he counts the kilojoules in his food and tries to reduce his intake. Archimedes is going to aim for a daily kilojoule intake of 10 000 kj. At this rate and with regular exercise he should lose approximately one kilo a week. According to his height and frame he should weigh 75 kg. Currently Archimedes weighs 138 kg. 1. If he keeps up with regular exercise and follows the diet, when can he expect to reach

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his ideal weight?..............................................................................................................

Kilojoule Counter 1. Two scoops of vanilla ice cream 2. A large hamburger 3. A chocolate fudge sundae 4. A milkshake 5. A cream bun 6. A fried chicken drumstick 7. A slice of pizza

800 2300 1250 1715 950 585 1260

kj kj kj kj kj kj kj

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2. Archimedes thinks that if he starts doing more exercise he might be able to still lose the weight but also enjoy some of his food luxuries. From the table below, what combinations of activities can Archimedes “swap” for these luxury items? Kilojoules used per minute: Jogging Moderate cycling Basketball Swimming Golf Rollerblading Tennis Brisk walking Running up stairs

20 14 12 15 8 12 15 14 25

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vi. .................................................................... vii. ....................................................................

Describe two ways that Archimedes can burn off 2000 kilojoules ......................................... .............................................................................................................................................. What is the slowest way he can use up 1500 kilojoules? .....................................................

Did you know? Celery has hardly any kilojoules in it because it mostly consists of water. When you eat a stick of celery you are burning off more kilojoules through chewing and swallowing than the celery is worth! If you add peanut butter you have another story! Find out how many kilojoules you would be adding!

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Name: __________________

Archimedes’ Diet 1 Archimedes has been on his diet for eight weeks now and has recorded his progress in his notebook. Weight at the end of each week: 1

Week

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Week

132.5 kg

136 kg

Week

131 kg

136.5 kg

Week

132.5 kg

134 kg

Week

130 kg

Draw a graph showing Archimedes’ weight loss below:

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Kg 139 138 137 136

135 ©R eadyEdPubl i cat i ons 134 •f or r evi ew pur posesonl y•

132

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4 5 Weeks

It looks like Archimedes lost eight kgs in the eight weeks as 138 - 130 equals 8. In fact Archimedes lost more than eight kgs. Explain how he did this. 1. What is the total weight loss for the 8 week period? ....................................................... 2. In which week did Archimedes lose the most amount of weight? ................................... 3. On average how much weight has Archimedes been losing each week? ....................... 4. If Archimedes continues at this rate how long will it be before he reaches his target weight of 75 kg? .............................................................................................................. Page 14

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Name: __________________

Archimedes’ Diet 2 Archimedes is going to his daughter Angel’s wedding in eight months time. The wedding date is exactly 40 weeks away. Angel has asked him to give her away at the wedding and Archimedes thinks it would be worth reaching his goal before the wedding so that he can celebrate after.

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1. What weekly weight loss average does he need to maintain if he is to reach his target for Angel’s wedding? .......................................................................................................

2. Archimedes has been on the diet for another

eight weeks. He has lost another 14 kilos and is down to 116 kg. He has discovered that since taking up swimming, he has started to increase his average weekly weight loss by half a kilo.

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© R e a d y E d P u b l i c a t i o n s he reaches his target if he keeps swimming every day? •f orr evi ew pur posesonl y• ........................................................................................................................................ Approximately how many weeks will it be before

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Challenge: Skinny Galileo has been teasing Archimedes about his weight and has been boasting about his own weight loss. He claims to have lost 12 kilos in the last six months. Mrs Galileo just laughs and says “Sorry Archimedes but he is telling the truth!”. Archimedes has known Galileo for many years and has never seen him be anything but a size 12 weighing about 75 kg. How can this be? Ready-Ed Publications

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Name: __________________

Galileo’s Used Car Yard Galileo has six cars in his yard and each one has a different reading on the odometer. He has decided to pose a problem and give away his shiny yellow Porsche to anyone who can solve his puzzle. The puzzle is to work out the exact number of kilometres that each car has driven. Galileo has provided the following clues:

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1. The old green convertible has driven more than 200,000 km and less than 300,000 km. Only two digits appear on the odometer and the numbers in the thousand, ten thousands and hundred thousands place each equal half the value of the digits in the ones, tens and hundreds place. 2. The blue Volkswagen’s odometer shows approximately 1/3 of the amount of the green convertible. The five digits on the odometer form a palindrome. This means that the number reads the same from front or back, such as 53435 or 2332. 3. The orange van has nearly 24,850 more kilometres on it than the VW. The number contains only two digits which form a repeating pattern. There are no 7’s in the number.

4. The purple truck has as many kilometres as the orange van and the VW combined.

© ReadyEdPubl i cat i ons The newest carr on r Galileo’s yard, a black Mustang, has done only 21,642 kilometres. • f o e v i e w p u r p o s e s o n l y • This is exactly half of what the Ferrari has done.

5. The white Ferrari has only 25% of the amount of the purple truck. 6.

Write the odometer readings for each of the cars:

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Convertible:

Volkswagen: Van:

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Truck: Ferrari:

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Mustang:

Challenge: Galileo has another car yard where Smooth the super sales representative is employed. Smooth sold three cars in his first week on the job. The second week saw him sell 9, with sales of 18 in the third week and 30 in the fourth week. If Smooth continues at this rate, for how many number of weeks will he have been working when he sells 84 cars in one week? Page 16

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Name: __________________

Transport Trouble! Pythagoras, Galileo and Galileo’s two children went camping one weekend at Cubic Camp. They decided not to take the car and caught a bus to the train stop at Millimetre Road. From there they caught the Treasure train and rode for two zones until they reached their destination, Mathemia. To get to the Cubic Campground they had to catch a ferry across Lake Litre. From the other side of Lake Litre a Base Bus took them to the office at Cubic Camp.

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Base Buses Adult $1.50 Child 80c

Millimetre Road Buses Adult $1.50 Child 80c

Treasure Trains One Zone Adult: $1.75 Child: $1.15 Two Zones Adult: $2.50 Child: $1.50

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These are the public transport fare rates for the area:

Fast Ferries Adult: One way Return -

$4.00 $7.00

Child: One way Return -

$3.00 $5.00

1. Calculate the total cost the group will pay for transport to Cubic Camp and home

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Mrs Galileo has offered to pick the group up on Sunday night and drive everyone home by taking the bridge over the lake.

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2. By how much will their transport cost be reduced? ......................................................... At camp they come across a large lake with an island in the middle which they wish to visit. They are able to hire a boat at the jetty, however, it can only carry a maximum of 90 kilos at a time. Pythagoras weighs 80 kilos, Galileo weighs 75 kilos, Gary Galileo weighs 45 kilos and Gina Galileo weighs 40 kilos.

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3. What is the least number of trips they can make so that everyone arrives at the island safely? .............................................................................................................................

Challenge: It takes one minute for Archimedes to cut through a wooden log. He wants to cut the log into ten equal pieces. How long will it take? Ready-Ed Publications

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Name: __________________

Building a Nest Egg On Archimedes’ first birthday he was given $1 by his grandparents. For his second birthday they gave him $2 and every year after they doubled the amount that they had given him the year before. One year on his birthday, Archimedes received $16 384 from his grandparents.

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What birthday did Archimedes celebrate? ..............................

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A) Invest the money in a 3 year term deposit at 14% interest.

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Archimedes’ grandparents stopped the birthday gifts at this age and he decided to choose the best investment for his money so that when he was 18 he would have a solid nest egg. He had three options to choose from:

B) Invest the money for 1 year at 16% interest and then in a two year fixed deposit at 12% interest. C) Spend $3000 on shares and place the rest in a three year fixed deposit with 15% interest. The shares may increase their value but will not decrease.

1. Work out how much money each option would make if the shares stayed the same rate. Remember Archimedes is investing $16 384.

100 New total

= $30 = $230

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To calculate interest for one year on $200 at 15% p.a. 200 x 15

A) .....................................................................................................................................

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B) ..................................................................................................................................... C) ....................................................................................................................................

2. Suppose Archimedes decided to invest according to Option C. He bought 6000 shares at 50c each. How much would Archimedes make on his shares if the value of the shares rose by: 12c ........................................................

5c ..............................................................

19c ........................................................

45c ............................................................

3. If the shares rose by 65c, how much money would Archimedes have had altogether on his 18th birthday? ........................................................................................................

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Name: __________________

Fenced In Pythagoras is building a fence around his farm. He has measured the total perimeter and has found the shortest rectangular distance to be 48 km. He needs to build the fence with three layers of wiring and expects that he will need to put a post in the ground at every metre. Pythagoras is going to put a metal gate in one side which will be two metres long.

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1. How much wire will Pythagoras need? ............................................................................

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2. How many posts will he need to dig holes for? ............................................................... Wooden posts cost $10.00 for a bundle of 12 and wire costs $0.35 a metre. The metal gate will cost $25.00. 3. What is the total cost of the materials needed for Pythagoras’ fence? ............................... Galileo says he can construct the fence and organise all the materials for $100 000 using the above prices.

4. If he uses ten workers, how much can he afford to pay each one after he has

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deducted $1000 for his own fee? .....................................................................................

Challenge:

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Pythagoras has seen an advertisement in the paper for an area of land that has become available near his farm. The advertisement states:

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A prime piece of real estate situated in the Chimerical Valley. This large triangular field offers views as far as the eye can see. 123456 123456 123456 123456 123456 123456 123456 123456 123456 123456 123456

320 m

780 m

If the price of the land is calculated by measuring the total area and 1 square metre is equal to $500, what should Pythagoras be paying? Ready-Ed Publications

Page 19

Name: __________________

Pythagoras’ Packages 1 Pythagoras’ Delivery Service is flourishing. He has worked out that exactly 200 deliveries have been made in the last 4 days. His delivery truck container has the following dimensions:

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Length = 5 m, Height = 4 m, Width = 2.5 m.

Volume = L x W x H

1. What is the capacity of the storage area?.......................................................................

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...............................................................................................

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2. On Monday he delivered 20 deluxe fridges. There was no room left to fit in anything else and the fridges were identical. Based on what you know about the capacity of Pythagoras’ truck, calculate the volume of one of the fridges.

3. On Tuesday Pythagoras was asked to help Archimedes move some boxes and furniture to one of Galileo’s new buildings. Archimedes had four large boxes with the following dimensions:

Vc = .......................................... © ReadyEdPubl i at i ons Box 2: Length = 90 cm, Width = 75 cm, Height = 75 cm V = .......................................... •f orr evi ew pur pos esonl y• Box 3: Length = 1.5 m, Width = 1.5 m, Height = 1 m V = .......................................... Box 4: Length = 60 cm, Width = 60 cm, Height = 60 cm Calculate the volume of each box.

V = ..........................................

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Box 1: Length = 2.5 m, Width = 80 cm, Height = 1.5 m

4. How much room was left in Pythagoras’ truck? ...............................................................

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5. Was he able to fit in ten solid jarrah storage trunks that are 2.5 m tall, 3 m wide and

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50 cm long? .....................................................................................................................

Challenge: Taxi Trick?

Pythagoras, Galileo and Archimedes take a taxi to the city. When they reach the Town Hall the meter reads $25. Each man hands over a $10 note. The driver gives back five $1 coins as change. Each man takes one of the dollar coins and they leave the remaining two $1 coins for the driver as a tip. Each man has spent $9 making a total of $27 spent between the three. The driver has two dollars bringing the total up to $29. Explain what happened to the other dollar. Page 20

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Name: __________________

Pythagoras’ Packages 2 6. On Wednesday Pythagoras delivered 80 microwave ovens to Mathemia Microwaves. He had just enough room left in the truck to fit a wardrobe that had a capacity of 2 m3. All the microwaves were the same size. What was the volume of each microwave?

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........................................................................................................................................ 7. How many items must Pythagoras have delivered on Thursday? ...................................

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8. After his busy week, Pythagoras thought about expanding the business to include another driver. He hired D’artagnan who has a dump truck that can transport sand and dirt. The tray capacity of the dump truck is 2000 kilograms or 2 tonnes. Pythagoras has organised for D’artagnan to drop quantities of mulch at six different homes. Each home wants a different quantity:

Home 1

750 kg

Home 2

349 kg

Home 5

1.25 tonnes

Home 6

985 kg

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What is the minimum number of trips that D’artagnan can take to deliver the correct load to each house?

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Challenge: Because the quantities D’artagnan has to deliver are all different how can he be sure each home receives the correct amount of mulch? Ready-Ed Publications

Page 21

Name: __________________

Galileo’s Gazette Galileo runs a small weekly paper and uses his paper to advertise his car yards. He usually just squeezes in his advertisement when he has space left over which he cannot sell to anybody else. This week he has 16 pages and over those pages he has found a total of 6 spare spaces on different pages. The dimensions for each space are:

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page 3 ............. 10 x 12 cm page 4 ............. 14 x 5 cm page 7 ............. 6 x 5 cm

page 9 ............. 8 x 3 cm page 11 ........... 20 x 2 cm page 14 ........... 13 x 3 cm

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1. What is the total amount of space left over? ...................................................................

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2. If Galileo sells advertising at a rate of $1.50 per square centimetre, how much will he

be losing this week? .........................................................................................................

Galileo has decided to offer a special deal to cover his costs. Now an advertiser can get an advertisement for $1.00 per square centimetre plus a bonus advertisement for half the price again. Pythagoras has rung up to advertise his farming supplies business. Because of the cheap rates he wants the biggest sized advertisement possible for this issue. 3.

i) On what page should he ask for his advertisement to be placed? ..........................

ii) How much will this particular advertisement cost? ..................................................

cost of both advertisements? ...................................................................................

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4. How much space is still spare in Galileo’s Gazette?.......................................................

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Galileo wants to make at least $100 on the remaining space. He knows the advertisements will be small and is prepared to offer a very low rate. He will keep the space on Page 7 for his car yard.

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5. Describe two ways in which Galileo could aim to make at least $100 on the remaining space. a)

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Challenge: How many cubic metres of dirt are in a hole measuring 6 metres wide, 4 metres long and 3 metres deep? Page 22

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Name: __________________

Sports Stars Pythagoras, Archimedes and Galileo enjoy their regular game of basketball on the weekend. Their team is called Mathemia Magic and they are tipped to take out the championship.

r o e t s Bo r e p ok u S

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1. Galileo has had a spectacular season and doubled the amount of points he scored every time he played. In the first game he scored 4 points, the second 8, and so on. How many points did Galileo score in the eighth game

of the season? ..............................................................

.....................................................................................

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2. In the Allstars Basketball League there are 13 teams. Mathemia Magic play all the other teams twice and play half the other teams three times. If they have two byes a season, how many games are in the season?

3. Every week after their game Archimedes, the captain, buys everyone in the team an ice cream, a cola or a milkshake. There are 13 players in the team and last week he spent $10. Ice creams 40c Colas $1.00 Milkshakes $1.20

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What did Archimedes buy? .............................................................................................

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4. It is finals time and the top five teams have been announced. Mathemia Magic were included in the five, however Archimedes has lost the list. Five of his players remember certain parts: Pythagoras: Galileo: Newton: Einstein: Edison:

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“I know we were not first or third.” “I know that the Prisms, the Polygons and the Angles were all included.” “I remember that the Polygons were higher than the Prisms and that the Angles were higher than the Perimeters.” “Mathemia Magic were higher than the Angles and the Perimeters.” “I’m fairly sure that the Prisms were fifth.”

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List the order of the teams in the top five. 1. .....................................

2. .....................................

4. .....................................

5. .....................................

3. .......................................

6. Mathemia Magic have a range of uniforms that they mix and match depending on who they are playing. There are three types of singlets, three types of shorts and three different caps. How many different combinations could they wear? ....................................................... Ready-Ed Publications

Page 23

Name: __________________

Archimedes’ Adventures 1 Archimedes is going on a tour around the world. Here is his planned itinerary:

Itinerary

r o e t s Bo r e p ok u S Using 24 hour times

Thursday 22nd May

Depart Sydney on Qantas at 0945 Arrive Honolulu Wednesday 22nd May 2305

Friday 23 May

Depart Honolulu on Qantas at 0030 Arrive Vancouver at 0850

Thursday 29 May

Depart Vancouver on British Airways at 2000 Arrive London on Friday 30 May at 1315

Friday 30 May

Depart London on British Airways at 1740 Arrive Rome at 2100

Tuesday 1 July

Depart London on Thai Airways 1305 Arrive Bangkok Wednesday 2nd July 0650

Friday 4 July

Depart Bangkok on Qantas at 0750 Arrive Perth at 1530

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Depart Perth on Qantas Airways at 0010 Arrive Sydney at 0610

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Wednesday 21 May

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Study Archimedes’ itinerary carefully and answer the questions over the page.

Challenge: Archimedes has calculated that his flight from Vancouver to London will take 17 hours and 15 minutes. He doesn’t think this can be correct yet he has checked the departure and arrival times twice with the travel agent. A friend has told him it only takes about 9 hours and 40 minutes. If his plane is a normal plane and is not stopping anywhere what is the reason for the difference? Page 24

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Name: __________________

Archimedes’ Adventures 2 Use Archimedes’ Itinerary to answer the following: 1. For how many nights will Archimedes be away? .............................................................

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2. What is the total number of flights that Archimedes will take? ........................................ 3. How many different airlines will he be using? ................................................................. 4. Explain why Archimedes appears to fly “back in time” when he leaves Sydney on

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Thursday and arrives in Honolulu on the Wednesday. ................................................... ........................................................................................................................................ ........................................................................................................................................

5. How many days will he spend travelling from Rome to London? .................................... 6. Using 12 hour time, at what time does Archimedes arrive in:

a) Honolulu? ....................................................................................................................

b) London on May 30? ....................................................................................................

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Challenge:

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7. What is the shortest flight Archimedes will take? ............................................................

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Archimedes urgently needs travel insurance. Which deal should he take?

Relax Insurance $200 for 45 days

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Dodge Insurers Special: $50 for 10 days

ZYX Insurance $100 for 35 days

Page 25

Name: __________________

Archimedes Phones Home Archimedes is well into his tour of the world and wants to let everybody know what an exciting time he is having. Use the world times on the map below to help check Archimedes’ calls. (Hint: Use pencil in the circles so that you can change the GMT for different questions). 0°

30°

60°

90°

120°

E 150°

180°

W 150°

120°

90°

60°

30°

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Moscow London

Amsterdam Paris

Rome

Beijing

Bombay

Hanoi

San Francisco

Teipei Hong Kong Manilla

Singapore

Johannesburg

Perth

Sydney Auckland

New York Washington

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Athens

Did you know? The earth is divided into 24 time zones. The base time zone is at 0° longitude and passes through the Royal Greenwich Observatory in England. Each zone spans 15° longitude although some borders are bent to fit in certain regions. The time is set according to Greenwich Mean Time (GMT).

N.B. On the map used on this page zones represent 2 hours.

1. When it is Noon GMT what is the time in: (Use your atlas to locate some cities.) b. Vancouver ....................

c. Rome ...............................

Sydney .........................

e. Bangkok ......................

f. London ..............................

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Honolulu........................

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2. When Archimedes arrived in Sydney at 6.10 am he wanted to ring his mother in Perth. He decided it would be best to wait until it was 8.30 am in Perth.

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What time would it have been in Sydney? .........................................................................

3. The next night when he stopped over in Honolulu, he gave his cousin in London a quick call to let him know he was on his way to Vancouver. He rang his cousin at 11.30 pm. What would the time have been in London? ...................................................... 4. Archimedes arrived in Vancouver at 8.50 am on Friday. He decided to ring his brother in New Zealand. What time and day would it have been? ...................................... 5. When Archimedes arrived in Bangkok at 6.50 am, he remembered it was his friend Newton’s birthday. Newton was holidaying in London so Archimedes rang him straight away. What time was it in London? ....................................................................... 6. What time should Archimedes have waited till before he rang Newton? ............................. Page 26

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Name: __________________

Easy Money Over the years Pythagoras has spent some time purchasing rare items in the hope of making some money on them. He wishes to sell some of them to finance his new house. He has a collection of records, coins and stamps and only wants to sell one of his collections. 1. His coin collection was valued at $650 dollars ten years ago. He has been told it would double its value every five years. How much would it be worth now? Hint: Round all answers to the nearest dollar.

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........................................................................................................................................

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2. He started collecting long playing records twenty years ago and was told each one would double its value every four years. He bought 8 records twenty years ago at $6.50 each. He then bought 5 more records twelve years ago at $10.50 each. Finally he bought 3 records eight years ago at $26.50 each. If each one has doubled its value every four years what is the total value of the record collection? Show your working here:

........................................................................................................................................ ........................................................................................................................................ ........................................................................................................................................

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3. His stamp collection was started thirty years ago with forty rare stamps worth $87 altogether. These stamps increased their values at a rate of 35% every five years. Over the years Pythagoras added more rare stamps to his collection and had them valued at $500 eight years ago. He was told these other stamps would increase by half every two years. What is the total value of the stamp collection? Show your working here:

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4. Pythagoras needs to know which collection will make the most money. He is going to sell the one that will make the least amount of money. He will need to compare the rates of increase over the years. Which collection should he sell and why?

Challenge: What will each collection be worth in forty years time? Did you make the right decision for Pythagoras? Ready-Ed Publications

Page 27

Name: __________________

What’s the Weather like? 1 Pythagoras was wondering what sort of weather his friend Archimedes would be experiencing on his world tour. The climate graphs below show the average annual rainfall and temperature for the cities included in Archimedes’ trip.

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Sydney

Vancouver

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Temperature

Honolulu

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Complete the climate graph for Bangkok using the data: Months Average Rainfall Average Temp. Page 28

300 290 265 190 26°

55 100 175 200 245 275

27° 26° 26° 27° 28° 29° 30°

29° 28° 27° 26° Ready-Ed Publications

Name: __________________

What’s the Weather Like? 2 Use the weather graphs on Page 28 to find answers to these: 1. Archimedes arrives in Sydney on May 21. What is the approximate average temperature in Sydney at this time of the year? .................................................................

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2. Consider the time of year that Archimedes is away. What city will be expecting the highest rainfall when he visits? (You might need to refer to Archimedes’ Itinerary).

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........................................................................................................................................

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3. Which cities will be experiencing temperatures between 5°C and 15°C while Archimedes

is visiting? ........................................................................................................................

........................................................................................................................................

4. Of the six cities, which has the highest yearly rainfall? ....................................................... 5. Which city has the most range in rainfall between the months of April and May?

© ReadyEdPubl i cat i ons Which city has the greatest annual range in temperature? ................................................. •f orr evi ew pur posesonl y• Which city has the greatest annual range in rainfall? .........................................................

........................................................................................................................................

6. 7.

8. Which city has the highest average monthly temperature for the month of July?

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........................................................................................................................................ 9. Which city has the lowest range in rainfall? .......................................................................

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10. Which city has annual temperature and rainfall averages most similar to where you

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live? .................................................................................................................................

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Page 29

Name: __________________

Crazy Car Yards 1 Galileo is looking to buy several new car yards as his current businesses are doing very well. He has seen a number of car yards that he is interested in. He has learnt from experience that a good car yard has a wide range of cars to choose from so he will be looking for a yard that will hold a large number of cars. In the past Galileo has noted that cars placed along the edges of the car yard have sold much better than cars squashed up like sardines in rows.

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1. When choosing his new car yards should Galileo be more concerned about the area

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or perimeter of the yard? Why? ...................................................................................... ........................................................................................................................................

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2. Galileo has seen five car yards for sale. Each has an area of 2400 m2, however they all have different perimeter lengths. List six sets of rectangular dimensions (L x W) that have this area: For example: 120 m x 20 m. ..............................................................

..................................................................

..............................................................

..................................................................

© ReadyEdP ubl i cat i ons .................................................................. Using a scale of 1 square to 100 metres, draw five different polygons that have an •f orr evi ew pur posesonl y• area of 2400 metres . Each car yard plan may have as many sides as you like. ..............................................................

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Ready-Ed Publications

Name: __________________

Crazy Car Yards 2 1. Calculate the largest perimeter possible for a shape that has an area of 2400 m². Draw your shape here giving dimensions for the length of each side.

r o e t s Bo r e p ok u S

50 m

20 m

© ReadyEd.................................................................. Publ i cat i ons .................................................................. •40f o r r e v i e w p u r posesonl y• m Show room

Show your working here:

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2. Galileo can park three cars every ten metres and he usually uses the inside area of a yard for his showroom and office. Assuming he has five sports cars in his showroom, how many cars will Galileo be able to display in each of the car yards below:

..................................................................

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60 m

80 m

Show room

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Show room

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20 m

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Challenge: What is the maximum number of cars Galileo can have on display if the area of the car yard is 2400 m² and he only has cars along the perimeter of the yard? Assume he can only have five cars in the showroom and still fits three cars every ten metres. Don’t forget to leave enough room for the showroom. Compare your answers with other students and check the area equals 2400 m². Ready-Ed Publications

Page 31

Name: __________________

Train Trips 1 Pythagoras lives in Scalene City and will be working in Mathemia for the next two weeks, measuring the angles for Galileo’s new buildings. He has the train timetable in front of him but doesn’t want to waste too much time in travel. It takes Pythagoras five minutes to walk from Mathemia Station to his work at Galileo’s office.

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Trains to Mathemia: (am) Scalene City

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Triangle Rd

Equilateral Lane

Mathemia

5.33 5.48 6.03 6.18 6.33

5.37 5.52 6.07 6.22 6.37

5.52 6.07 6.22 6.33 6.52

7.03 7.18

7.07 7.22

7.22 7.33

6.03. 6.18 6.33 6.48 7.03 7.05 7.33 7.48 7.50

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5.30 5.45 6.00 6.15 6.30 6.45* 7.00 7.15 7.30*

Isosceles Avenue

Trains run every five minutes until 9.15 am. These trips take approximately 45 min. each. 9.15 9.45

9.18 9.48

9.22 9.52

9.33 10.03

9.48 10.18

Trains From Mathemia: (pm) Triangle Rd

Isosceles Avenue

Scalene City

3.00 3.10 3.20

3.11 3.21 3.31

3.26 3.36 3.46

3.30 3.40 3.50

3.33 3.43 3.53

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Trains run every five minutes until 5.00 pm. 5.40 6.20 6.40 7.00

5.51 6.31 6.51 7.11

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Equilateral Lane

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Mathemia

6.06 6.46 7.06 7.26

6.10 6.50 7.10 7.30

6.13 6.53 7.13 7.33

Trains run every hour on the hour until midnight. Use the train timetable to answer the questions on Train Trips 2.

Page 32

Ready-Ed Publications

Name: __________________

Train Trips 2 Use the timetable on Page 32 to answer these. 1. How many trains run from Scalene City to Mathemia after 7.30 am? ................................. 2. How long is a train trip at 6.15 am from Scalene City? ......................................................

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3. What reason might explain the 45 minute trips between 7.30 am and 9.15 am? ................ ........................................................................................................................................

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4. Pythagoras wants to spend at least 8 hours in Mathemia as he has lots of triangles to

time possible waiting for and catching trains?

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measure. What train trips could he take to make sure he spends the least amount of

a) ..................................................................................................................................... b) .....................................................................................................................................

5. On Tuesday Pythagoras needs to drop the kids off at school before he catches the

© ReadyEdPubl i cat i ons expect to arrive back at Scalene City in the afternoon if he works an eight hour day in •f orr evi ew pur posesonl y• Mathemia? ....................................................................................................................... morning train. He cannot get to the station until at least 8.30 am. What time can he

6. On Thursday Pythagoras accidentally slept in. He arrived at Scalene City Station at

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10.05 am. When can he expect to arrive back at the station this evening?......................... 7. On Friday Pythagoras caught the 7.00 am train to Mathemia. On the way the train

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had mechanical problems and did not arrive until 8.15 am. How long must the train

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have stopped for? ............................................................................................................ 8. Pythagoras left on the 6.45 am train on the following Monday. At 12.00 pm he had a dentist appointment, so he left work at 11.10 am. For how long had Pythagoras been at work on that day? .............................................................................................................

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Page 33

Name: __________________

Cool Climbing 1 Galileo, Archimedes and Pythagoras decided to go on a climbing adventure in the mountains of Peru. For the journey they hired a guide who would show them the ropes and prepare the meals. Their guide, Hachu, was a very experienced climber and knew all the shortcuts. Their journey would take two weeks and their goal was to climb to the summit of Mt Pisco, located near Machu Picchu, 3000 metres above sea level, in the Andes.

r o e t s B r e Daily Schedule o p ok u S

They made the following distances each day and set up camp each night.

(1000 m)

Day 2

Reach Lake Camp

(800 m)

Day 3

Trek to Col Camp

(500 m)

Day 4

Rest

Day 5

Reach Crevasse Camp

(200 m)

Day 6

Scale Ice Wall

(100 m)

Day 10

Back to Summit Camp

(32 m)

Day 11

Return Crevasse Camp

(220 m)

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Arrive River Camp

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Day 1

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Return to Col Camp

Day 13

Col Camp to Lake Camp

Day 14

Day 15

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Rest © Rea dyEdPubl i cat i ons Day 8 Reach Summit Camp (120 m) •f orr ev i ew pur poseson l y• Day 9 Reach Pisco Summit (32 m) Day 7

(200 m)

(500 m)

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(800 m)

From River Camp the group hiked back to town which was

1000 m. They were now 3000 m above sea level.

Use the daily schedule to answer the questions on the next page.

Page 34

Ready-Ed Publications

Name: __________________

Cool Climbing 2 Galileo, Archemides and Pythagoras are reflecting on their journey. Use their information to answer the following. (Hint: Round all the answers to the nearest metre.) 1. What was the total distance covered over the two week trek? .......................................

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2. What was the average distance in metres covered each day? ....................................... 3. Over which four day period on the trek was the most distance covered? .......................

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........................................................................................................................................ 4. Suppose the group did not take rest days and made the same distances as above.

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What would the average daily distance have been? .......................................................

5. How far is it from River Camp to Col Camp? ................................................................... 6. How tall is Mt Pisco if River Camp is 4000 m above sea level? ...................................... 7. What was the average daily distance covered during the ascent of Mt Pisco?

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y• ........................................................................................................................................ ........................................................................................................................................

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8. What was the average daily distance covered during the descent of Mt Pisco?

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Challenge: Now that they have mastered Mt Pisco, the group wants to attempt Mt Huascaran, at a height of 6768 metres. Base Camp is located at 3500 metres above sea level and they would like to complete the summit and return down the mountain in twelve days. What distance must they aim to average each day if they are to complete the climb on schedule?

Ready-Ed Publications

Page 35

Name: __________________

Fishing Fever Pythagoras, Galileo and Archimedes went fishing one day at Mathemia Marina. They put all the fish they caught into a large ice bucket, however, when they returned home they realised they had mixed up all the fish! In their bucket they counted seven herring, four mullet, two bream, two cod and a small shark.

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“ I caught at least four herring and two mullet,” said Pythagoras. “ I caught more than five fish and at least one of each kind,” said Archimedes. “ I caught three fish but I didn’t catch any cod or bream,” said Galileo.

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1. What did each man catch?

Archimedes

Galileo

.........................................

........................................

..........................................

.........................................

........................................

..........................................

.........................................

........................................

..........................................

.........................................

........................................

..........................................

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Pythagoras

Later on that day Archimedes and Pythagoras went out in Galileo’s fishing trawler. They decided to put some crab nets in and ended up catching a mixture of lobsters and crabs. Galileo counted 80 heads and noted that there were three times as many lobsters as there were crabs. Archimedes counted 300 legs and he knows that lobsters have 8 legs and crabs have 6 legs.

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Number of crabs

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2. How many crabs and how many lobsters did the fishermen catch? Use the table to help you work out the answer.

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Galileo wants to sell his latest catch at the Mathemia Markets. He knows mullet sells for $6.25 a kilo and red cod sells for $13.50 a kilo. Based on these prices, Galileo has calculated that he will receive exactly $91.00. He has exactly twice the amount of one kind of fish than the other. Kilos of mullet Kilos of red cod Total Profit 3. How many kilos of mullet? ................................................................................................ 4. How many kilos of red cod?.............................................................................................. Page 36

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Name: __________________

Brainteasers Try these extra tricky brainteasers. Some of the answers are not what you think!

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1. Ten coins are arranged as shown. How can you move just one coin so that when added either vertically or horizontally, two rows of six coins will be formed?

2. There are six glasses in a row. The first three are filled with water and the last three are empty.

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1. 2. 3. 4. 5. By moving only one glass, can you arrange them so that the pattern becomes full, empty, full, empty, full, empty?

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3. The power has gone off in Pythagoras’ house but he needs to pack his case for his trip. He wants to be certain his socks match but cannot see the colours at all. In his sock drawer he has 14 blue socks and 16 red socks. What is the smallest number of socks he needs to pack in his suitcase to make sure he has at least two socks of the same colour?

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Page 37

Name: __________________

More Brainteasers

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2. Which of these numbers can be evenly divided into two? 1

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1. Gary Galileo was asked how old he was. He replied, “In two years I will be twice as old as I was five years ago.” How old is Gary?

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costing a dollar more than the other chocolate. What were the individual prices of the chocolates?

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4. Galileo has two coins that add up to 25c. If one of the coins is not a 20c what are the two coins?

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Even More Brainteasers

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1. Archimedes was at the market and noticed that three plums were worth the same amount as two apples. He wants to buy 24 plums. How many apples are they worth?

2. Pythagoras was looking at his favourite object, triangles. He used a protractor to measure the corners. One corner was 13 degrees. He then got out his magnifying glass which magnifies everything by three times. Under the glass, how big would the angle measure?

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3. Pythagoras and Archimedes played backgammon. They played nine games. Each won the same number of games yet there were no draws. How is this possible?

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Answers

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Check These Out! (Page 7) 1. 20 kilograms. 2. 14 on tricycles, 11 on bicycles. 3. 8 peacocks, 7 sheep. Challenge: There are 4 four footed animals, 2 two footed animals and 5 snakes! Life in the Wheel World (Page 8) (Note: These answers are based on the circumference of the wheel being rounded to 150 cm.) 1. 375 metres, 2. 750 noises, 3. 1250 noises, 4. 125 noises, 5. 3000 noises, 6. 250 times; Distance won’t change, only time. Challenge: Nine years. The tree doubled its height each year which means one year before it would have been at half the height it is today. Archimedes in the Pool (Page 9) 1. 37.5 cubic metres. 2. He needs 2.5L of chlorine each week. (37.5 m3 divided by 15). 3. Archimedes needs 130 litres for the whole year. 4. Two possible answers: Chlorine World is the cheapest for a year’s supply. However, if Archimedes buys his chlorine from Bulk Chlorine Supplies, there will be 50 litres left over which he can use the following year. Challenge: It would be 10.00 am. Halfway between the two times would be six hours (7.00 am). Three hours added onto 7.00 am would make it 10.00 am. Archimedes’ Pool Challenge (Page 10) 1. 23, 2a. 90 kg, 2b. 48 kg, 2c. 54 kg. 3. Aunty Agnes takes up 12 buckets making a total of 67 buckets. Challenge: It will take 17 days. Basically the worm moves at the rate of one metre each day. After Day 1 he would be at the one metre mark, Day 2, the two metre mark and so on. On Day 16 he will crawl from fifteen metres to nineteen metres but will then slip back to sixteen metres. On Day 17 he will crawl four metres to the top of the well and then he can crawl out. Hamburger Headache! (Page 11) 1. Galileo: Hamburger with cheese and beetroot Mrs Galileo: Hamburger with egg and gherkin Gary: Hamburger with gherkin and cheese Gina: Hamburger with cheese and egg. 2. Galileo is most likely to eat the hamburger with just beetroot as nobody else likes it. Gary will probably eat the gherkin and cheese hamburger as only he and Mrs Galileo like gherkin. However, Mrs Galileo doesn’t like cheese. 3. The total cost would be $17.55. Remember they were given the two extra hamburgers for free. Challenge: You can only subtract six once because after that the number is no longer 36 and you will be subtracting from 30, 24 and so on. Building Boom (Page 12) Answers will vary. Challenge: 3. Kilojoule Capers (Page 13) 63 weeks or approximately 1 year and two and a half months. 2. Check combinations. Archimedes Diet 1 (Page 14) Archimedes put on some weight while on the diet and so he had to lose this weight as well. After Week 3 he had put on half a kg and after Week 7 he had gained 1.5 kg. This had to be lost as well so he ended up losing 10 kgs over the entire period. 1. 10 kgs, 2. Archimedes lost 2.5 kgs in Weeks 4 and 8, 3. 1.25 kgs, 4. Archimedes now has 55 kgs to lose. At this rate he is losing 1.25 kgs a week. It will take 44 weeks to lose 55 kgs. Archimedes’ Diet 2 (Page 15) 1. Archimedes has 40 weeks to lose 55 kgs. He should aim to lose 1.375 kgs a week or better still 1.4 kgs. This way he will lose 56 kgs! One to spare considering he will probably eat too much at the wedding! 2. At a rate of 1.75 kg weight loss per week, Archimedes will take approximately 23 and a 1/2 weeks to lose 41 kg. Challenge: Galileo is telling the truth. Every fortnight he gains a kg bringing his weight up to 75 kg. Every other week he loses a kg and goes back to 74 kg. This has been going on for six months and in that time he has gained 12 kgs and lost 12 kgs! He never puts on any more than one kg at a time and so his weight changes would not be noticed.

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Galileo’s Used Car Yard (Page 16) Convertible: 222,444 km, Volkswagen: 74,147 km, Van: 98,989 km, Truck: 173,136 km, Ferrari: 43,284 km, Mustang: 21,642 km. Challenge: Smooth will sell 84 cars in his seventh week at work. He is currently increasing his sales at a rate going up by three each week. Week 1 he sold 3 (+3), Week 2 he sold 9 (+6), Week 3 he sold 18 (+9) etc. Transport Trouble (Page 17) 1. Buses: $18.40, Trains: $16.00, Ferries: $24.00. Total: $58.40. 2. $27.20. 3. It will take five trips: Trip One: The two children cross with one of them getting out at the island and the other rowing back to the jetty. Trip two: Galileo rows across and the child waiting on the island rows back to the jetty. Third trip: The two children cross again and one stays behind. Fourth trip: Pythagoras rows across, gets out and the child rows back to the jetty. Fifth trip: The two children row back together. Challenge: Nine minutes because he only needs to make nine cuts to end up with ten pieces. Building a Nest Egg (Page 18) Archimedes turned fifteen. 1. A) This option will see a return of $24,273.62. B) This option will return $23,840.43. C) If the shares stay the same there will be a return of $23,355.39. 2. 12c: $720, 5c: $300, 19c: $1140, 45c: $2700. 3. Archimedes will have $20,355.39 plus $6,900 from the shares; a total of $27,255.39. Fenced In (Page 19) 1. At least 144 km of wire (i.e. exactly 143.994 km). 2. At least 47 999 (minus one for where the gate takes up two metres). 3. Total cost: $90 422.07. 4. Each worker should receive $857.79. Challenge: $62 400 Pythagoras’ Packages 1 (Page 20) 1. 50 cubic metres or 50 m³. 2. 2.5 m³. 3. 1) 3 m³; 2) 0.5 m³; 3) 2.25 m³; 4) 0.22 m³. Total Volume = 5.97 m³. 4. Space left in truck = 44.03 m³. 5. Yes; each trunk has a volume of 3.75 m³ making a total of 37.5 m³. The total amount taken up by the boxes and the trunks would be 43.47 m³. Challenge: Nothing happened to the other dollar. The driver has $25 plus a two dollar tip making a total of $27. The three men each have a dollar: $27 + $3 = $30. Pythagoras’ Packages 2 (Page 21) 6. 0.6 m³. 7. 85. 8. The total amount of mulch to be delivered is 4.984 tonnes or 4984 kg. Dan will need to make at least 3 trips to deliver all the mulch. Challenge: Before Dan leaves the mulch suppliers he should measure exactly how much a wheelbarrow load will hold. He can then measure out the mulch in wheelbarrow loads. If a wheelbarrow can hold 75 kilograms, Dan would need 10 wheel barrow loads to make 750 kilograms. If the total quantity was not divisible by 75, he could measure the remainder using a bucket with a known capacity. Galileo’s Gazette (Page 22) 1. 323 cm², 2. $484.50, 3. i) Page 3, ii) $120, iii) $132, 4. 179 cm². 5. a) Galileo could offer the three ads altogether for the one flat rate of $100. b) If he can’t attract anybody who wants all three advertisement spaces he could sell them individually at a .... reduced rate. Galileo has 179 cm² left in advertising space. If he offers 56c per square centimetre he will make $100.24 on the space. Challenge: A hole is a hole, there would be no dirt. Sports Stars (Page 23) 1. 512 points, 2. 32, 3. 4 colas, 3 milkshakes and 6 ice creams, answers may vary, 5. 1st -Polygons, 2nd - Mathemia Magic, 3rd - Angles, 4th - Perimeters, 5th - Prisms, 6. 27.

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Archimedes’ Adventures 1 (Page 24) Challenge: The trip takes 9 hours and 40 minutes. Archimedes is flying across several time zones. In Vancouver the time will be 5.40 am while in London it will be 1.40 pm. Archimedes’ Adventures 2 (Page 25) 1. 43 nights, 2. Seven flights, 3. Three Airlines, 4. Because he is travelling across the International Date Line, 5.15 days, 6. 11.05 pm, 1.15 pm, 6.50 am, 7. London to Rome. Challenge: Relax Insurance Archimedes Phones Home (Page 26) 1. a) 2 am, b) 4 am, c) 1 pm, d) 10 pm, e) 7 pm, f) Noon, 2. 10.30 am, 3. 9.30 am, 4. 2.50 am on Saturday, 5. It was only 10.50 pm of the previous day. He was early. 6. He needs to wait until it is 8 am in Bangkok. Easy Money (Page 27) 1. $2 600. 2. 8 records at $6.50 = $52.00. These have doubled their value five times coming to a total of $1664.00. 5 records at $10.50 = $52.50. These have doubled their value three times equalling $420.00. 3 records at $16.50 = $79.50. These have doubled their value twice equalling $318.00. The total value equals $2402. 3. The first set of stamps have increased by 35% every five years. 35% of 87 = $30.45. The next increase will be 35% of $117.45 and so on. Over 30 years there will be six increases. Today the stamps are worth $527. The other stamps were worth $500 eight years ago. They increased by 50% every two years so six years ago they would have been worth $750, four years ago - $1125, two years ago - $1687, today - $2531. The total value equals $3058. 4. He should sell the coins as the other items are increasing at a faster rate. He could also sell the first set of stamps as they will not make that much, as they are only increasing by 35% where other items are increasing by 50% and 100%. Challenge: Coins - $665 600, Records - $2 459 648, Stamps - First set = $5814, Second set = $8 416 223, Total value = $8 422 037. What’s the Weather Like? 2 (Page 29) 1. 24°C, 2. Honolulu - average monthly rainfall for May ≅ 118 mm, 3. Vancouver and London, 4. Bangkok, 5. Bangkok, 6. Honolulu, 7. Bangkok, 8. Rome, 9. London. Crazy Car Yards 1 (Page 30) 1. If it is more beneficial to have cars placed along the edges, Galileo should be concerned about the length of the perimeter. A long perimeter will mean more cars can be placed around the edges and less space inside will be wasted. A large area may not necessarily mean a large perimeter. 2. Answers will vary: 240 x 10, 80 x 30, 40 x 60, 50 x 48, 25 x 96, 75 x 32. 3. Answers will vary. Crazy Car Yards 2 (Page 31) 1. Answers will vary. 2. a. 71 cars, b. 65 cars, c. 77 cars Challenge: Galileo could have an L shaped car yard as shown. The total perimeter is 420 m and this would allow 126 cars in the yard. Together with the five cars in the showroom the total number is 131 cars. 5. 6.13 pm, 6. 8.33 pm, 7. 42 minutes, 8. 4 hours.

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20 showroom 30

10 200

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Train Trips 2 (Page 33) 1. 28, 2. 33 minutes, 3. Peak Travel times - more people are getting on and off so the train spends more time at each station, 4. a) Catch the express at 6.45, return on 3.12 train. Travel time = 53 minutes, b) Catch the express at 7.30, return at 4.00. Travel time = 65 minutes. All other trips will take at least an hour and six minutes in travel time. Cool Climbing 2 (Page 35) 1. 5504 m, 2. 367 m, 3. Days 12, 13, 14, 15, 4. 423 m, 5. 1300 m, 6. 5752 m, 7. 306 m, 8. 459 m. Challenge: The summit of Mt Huascaran is 3268 m above the Base Camp. The group needs to average 272 metres a day if they are to complete the climb in 12 days. Fishing Fever (Page 36) 1. Pythagoras - Four herring, two mullet, Archimedes - One herring, one mullet, two bream, two cod, one shark, Galileo - Two herring, one mullet. 2. 10 crabs and 30 lobsters, 3. 7 kilos of mullet, 4. 3.5 kilos of red cod. Brainteasers: (Page 37) 1. Move the bottom coin and place it on top of the coin in the centre of the cross. 2. Pour the water from the second glass into the fifth glass. 3. Only three. If there are only two colours, the third sock he picks will have to match one of the other two. More Brainteasers (Page 38) 4. Gary is 12. Five years ago he was 7 which will make his birthday in two years twice that. 5. All of them. For example 9 can be evenly divided into 4 and a ½. 6. $1.05 and 5c. 7. One of the coins is not a 20c but the other coin is. The coins are a 5c coin and a 20c coin. Even More Brainteasers (Page 39) 8. 16 apples. Two apples equals three plums, this means that apples are worth 2/3 of plums and 2/3 of 24 equals 16. 9. The angle size would not change. 10. Pythagoras and Archimedes were not playing each other! They each played nine games with Galileo and each won six games.

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Student Names

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