Cover-Page for the Amazing Race Summative

This summative task is intended to be used for the grade ten applied mathematics course,

MFM2P. It will take the students around the world on an “Amazing Race” type of adventure. They will

visit countries such as Germany, France, China and Tanzania solving problems along the way to allow

them to proceed. Each student will have a passport which consists of a duo tang of questions they must

complete in each country, designated by flags around the classroom. All calculations will give answers

which are factually valid, to teach more than the mathematics. Some of the questions are pencil and

paper calculations, whereas others will require them to climb, dig or measure their way to the answer.

In having the students look at the tilt of the Leaning Tower of Pisa and the shadow cast by the

Eiffel Tower, we cover the concepts of similar triangles, primary trigonometric ratios and Pythagorean

Theorem with regards to right triangles. Minimizing the surface area and maximizing the volume of

three-dimensional figures helps the students to bring back some tea from China while covering big ideas

in the Measurement and Trigonometry strand. In the Modeling Linear Relations strand, we have a pre-

departure quiz which checks their ability to manipulate and solve algebraic equations. Also in this

section, we use graphs of lines and systems of equations to solve problems on maps, to prevent

arguments among airport security, and efficiently count animals on the safari. Canadian James Naismith

and his invention of basketball, as well as the Basilique du Sacré Coeur in Paris, help the students to

reason about quadratic relations and solve problems using graphs, technology and identifying

characteristics. Before starting the first day, the students must complete a pre-departure quiz testing

their abilities to factor and expand quadratic expression.

Over three days of this activity, students can see applications of the concepts they have learned

in class while becoming global citizens and learning about different cultures. There are some questions

which really require them to think and communicate their ideas, while others quickly test a concept

through demonstration of knowledge. This summative is a mathematics adventure to remember.

Materials:

Enough question sheets for every student and two extra copies of each

Enough Pre-Departure Quizzes (one for day one and one for day two) for each student

Enough duo tangs for each student. With a cover page, similar to that of a passport.

Tracking Sheet

16 Large Envelopes

Flags for the following countries: Canada, Australia, Dubai, China, Germany, Ireland, France,

Italy, and Tanzania

Three small Mirrors

Three buckets of rice with numbers 1, 5, 10 and 25 in them.

3 measuring tapes

Class set of Algebra tiles

Enough calculators for each pair to have one to use throughout the class

Stop Watch

Two computers with GeoGebra installed or internet to access the web version of GeoGebra

Instructions:

Prior to the Summative Task:

A week before the summative is to take place, tell the students about the summative task so that they

can prepare and enforce the importance of them being present all three days. As the students will be

required to go outside for one of the tasks and in the hallway to avoid the classroom being too crowded,

tell your principal about your summative task and let them know that students will be moving around

the school a bit during your class for the duration of the three day period.

Print enough question sheets for every student (with two extra of each question) two days before the

summative task is to start. For questions that are three pages in length, staple the question pages

together. Print enough Pre-Departure Quizzes for each student. Print out all the flags as listed above.

Randomly select partners for the summative task. Assign each pair to a starting location by spreading

them out across the countries, trying to space them such that you avoid a lot of groups in a single

location at one time. The spacing will depend on the size of your class. Label the envelopes with their

country. For France, China, Canada and Germany, label two envelopes with 1 or 2 so that the students

know there are two questions associated with that country. Put one question in envelope 1 and the

second question in envelope 2. Put the remaining question sheets in their corresponding country

envelope. Make sure you get enough calculators to give one to each pair of students and a class set of

algebra tiles. Arrange two computers with GeoGebra, whether laptops set up in your classroom or

computers in the library. Make sure you have the rest of the materials ready and put everything in a

single location.

Day 1 of Summative Task:

Before class begins, put up the flags and envelopes for Canada and Germany in the hall, and the rest

around the classroom. At the beginning of class on the first day, give the students their duo tangs, a

passport sheet and a tracking sheet. Explain that although the students are traveling around in partners,

it is important that they do their own work. They both need fill out all the question sheets and all pre-

departure quizzes in order to receive marks. Also encourage students to look at the marking scheme for

each question and to be aware that an additional 15 marks will be awarded for communication on the

overall task. Explain that the tracking sheet is there to help them stay on track with finishing the

summative task within the three days and that they need to keep that timing in mind as they make their

way through their Amazing Race.

Tell each pair of students their starting position. Explain that they need to come to the teacher to get

the materials they need for Canada when they get to that country and where the computers are

available to use for Dubai. This is dependent on what you have arranged (whether laptops in the

classroom or computers in the library). As well, let them know that there are algebra tiles available in

the classroom if they choose to use them.

Day one will begin with the Quadratic Relations Pre-departure Quiz. The pairs must finish these

questions in order to get on the plane to their first location. The teacher will be located at Tanzania, the

safari, to be the timer, given there is no student helper in the classroom or a student teacher. If you

have a helper, you can assign them to this station, and float around the room. If there is only the

teacher, the safari will be located by the classroom door so that you can easily keep your eye on the

students in the class and in the hallway. At the end of the class, collect the students’ passport books.

Take down the flags and envelopes from the hall.

Day 2 of Summative Task:

Before class begins, put up the flags and envelopes for Canada and Germany in the hall, and the rest

around the classroom. Hand out the students passport books. Again, in order to continue on their

travels, they must complete the Solving Equations 1 Pre-departure Quiz. Remind the students to pay

attention to timing of each question. Once they complete this, they can continue on their journey. The

teacher will be located at Tanzania, the safari, to be the timer given there is no student helper in the

classroom or a student teacher. If you have a helper, you can assign them to this station, and float

around the room. If there is only the teacher, the safari will be located by the classroom door so that

you can easily keep your eye on the students in the class and in the hallway. At the end of the class,

collect the students’ passport books. Take down the flags and envelopes from the hall.

Day 3 of Summative Task:

Before class begins, put up the flags and envelopes for Canada and Germany in the hall, and the rest

around the classroom. Hand out the students passport books. Again, in order to continue on their

travels, they must complete the Solving Equations 2 Pre-departure Quiz. Once they complete this, they

can continue on their journey. Remind the students to pay attention to timing of each question. The

teacher will be located at Tanzania, the safari, to be the timer given there is no student helper in the

classroom or a student teacher. If you have a helper, you can assign them to this station, and float

around the room. If there is only the teacher, the safari will be located by the classroom door so that

you can easily keep your eye on the students in the class and in the hallway. At the end of the class,

collect the students’ passport books. The summative task is over. Take down the flags and envelopes

from the hall.

Surname:

First Name:

Student Number:

Tracking Sheet:

I have been too…

Country Suggested Time Arrival Time Completed Time

Canada 35 minutes

Australia 15 minutes

Dubai 12 minutes

China 35 minutes

Germany 23 minutes

Belgium 7 minutes

Ireland 5 minutes

France 25 minutes

Italy 3 minutes

Tanzania 20 minutes

Quadratic Relations Fast Minute

15 minutes

Solving Equations Fast Minute

15 minutes

Before departing Canada you must complete a question about the native sport of basketball. The game of basketball was created in December of 1891 by Canadian Dr. James Naismith. The phys. ed professor, created the game on a rainy afternoon to keep his class active despite the weather. After writing the rules he nailed a peach basket to a 10 foot track and the game of basketball was created. It wasn’t until 1906 that metal hoops replaced the peach baskets. In basketball, free throws are taken by a player who had a foul committed against them by the opposing team. The free throw must be taken at least 15 feet from the basket. The arc a basketball travels along during a free throw can be represented by the following quadratic equation. ‘Granny’ shot: throwing the ball under hand (with two hands) between your legs starting with the ball touching the ground. 1. Consider you are taking a ‘granny’ shot at the free throw line. Suppose the maximum

height the ball reaches is 13 feet when the ball is a horizontal distance of 5 feet from the net. What three points do you currently know on the parabola? What do these points represent?

2. Create a sketch to represent the path of the basketball.

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3. What horizontal distance is covered when the ball is higher than 10 feet high in the air? Explain your reasoning.

4. If the ball were to continue on the parabolic path instead of hitting the net, where would

the ball hit the ground? Explain your reasoning.

You have now arrived in Australia. You are looking at the famous Princess Bridge. You are told that the original bridge was quite different from the one you are now standing in front of. Below is a picture of the original Princess Bridge. Your task is to pretend you painting a new bridge similar to the one in the picture.

You are told the length of the bridge is 140m and are given the following dimensions:

This diagram provides you with a side view of the bridge. 1. If you wanted to paint both sides of the bridge, how many m2, rounded to one decimal, of

paint will you need?

10 m

30 m

2. If you were making this bridge out of pure concrete, and you were told the bridge must be

30 m wide, how much concrete would you need?

Welcome to Dubai! You are at the world’s tallest building, the Burj Khalifa. The building is an astonishing 2723 ft. tall. Your next task is to design a rocket launcher to launch the next clue to your partner standing on the other side. The only rule is that you must stand 500 feet away from the building when launching the clue. It is your job to prove that your rocket launcher would send the clue over the building to your partner. 1. With a partner, measure the height of one person in the pair to the nearest decimal.

Height = _________ft. 2. Assume your partner is the same height as you. Based on your knowledge of symmetry in

quadratics, how far from your current position must your partner stand to catch the clue? Explain.

3. Using the information you currently have make a sketch to represent the path of the clue.

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h

4. Using GeoGebra, input the 3 points into a spreadsheet and do a two variable regression analysis to get the equation of the parabola going through these points.

Show the teacher the end result displaying your graph.

y=

5. How high is the clue 200 feet after you launched it? 400 feet? Round your answers to the

nearest hundredth. Show all your steps.

You have arrived in China and have found that rice is a staple food source. In each of the three buckets of rice there are numbers which you must first dig out using the chopsticks provided. Notice the labels A, B, and C on each of the buckets.

1. Dig once in each of the buckets to get the coefficients for the first equation, then dig again to fill each in the second equation. Solve the system of equations you have created.

Eqn1) Ax + By = C Your equation 1): Eqn2) Ax + By = C Your equation 2):

2. Repeat step one for the equations below and solve the system so you can continue on your travels.

Eqn1) Ax + By + C = 0 Your equation 1): Eqn2) Ax + By – C = 0 Your equation 2):

Tea is the national drink of China. Your task is to bring home a box of tea. You come across a store that sells packages of individual flavours. Each package contains two tea bags in small cylinder containers. The containers have a radius of 5 cm and a height of 2 cm. You are required to bring home 24 different flavours.

1. Create a box out of the provided piece of paper by cutting four equal sized squares out of each corner and folding up the sides. The box must contain 24 tea packages stacked flat on top of each other. You must try to minimize unwanted space. Use this space to brainstorm.

2. What is your mathematical reasoning for the dimensions you chose?

3. If you wanted to cover the box you made in colourful paper, how much paper would you need?

4. How much wasted space you have once you put all the tea packages inside your box. Round to the nearest thousandths.

Welcome to Germany! The two-man German bobsleigh team, André Lange and Kevin Kuske, received gold at the 2010 Olympics. The angle of the start ramp of a race track is usually 30o. The duo has been using different tracks to train, each of which have a different start ramp degree. Their coach has been keeping track of how long it takes the team to reach the bottom of the start ramp according to each angle. Below is the coach’s record of the angle of the ramp and the time it took for them to reach the bottom of the start ramp.

1. Complete the above table by filling in the first and second differences.

2. What conclusions can you make from the first or second differences? 3. Plot the angle of the ramp to time on the graph paper provided. Draw the curve of best fit.

Does this align with your conclusions from question 2?

Angle of Ramp

Time (seconds) First Difference

Second Difference

25o 36

30o 26

35o 18

40o 12

45o 8

50o 6

30o

You are to embark on a journey across Germany and will use the Autobahn for part of your travels. Autobahn is the German word for highway and often does not have a speed limit.

1. You are left Oberhausen at 8:30am and, taking the Autobahn, arrive in Castrop-Rauxel at 8:42am. It is a 38.3km journey.

a. What is the rate of change over this period of time rounded to two decimals? Convert this to a rate in km/h.

b. What does the rate of change represent?

c. Suppose you are driving at a constant speed over the twelve minutes and continue at this speed past Castrop-Rauxel. Draw a line to represent the situation. Be sure to label all important features we have talked about. Use a scale of 6 minutes or 1/10th of an hour for the x-axis.

d. What is the equation of the line?

e. What do each of the co-ordinates of any point on the line represent? x: y:

After your adventures in Germany, you move west to Belgium stopping in Aachen, Germany with co-ordinates (6.5oE, 50oN) before crossing the border. You are travelling to Antwerp, Belgium (4oE, 51oN). You have hired a private plane to fly you from Aachen to Antwerp directly (in a straight line). Your competitors are leaving Duisburg (7.5oE, 51.5oN) and have also hired a private to fly them to Namur (5.75oE, 49.25oN). Use the step below to determine where will your planes meet? Each square on the grid map is 0.25o.

1. Draw a line representing your path and that of your competitors on the grid. 2. What is the slope of the line representing your path from Aachen to Antwerp?

3. Use the slope and one point to determine the equation of the line representing your path.

4. Using the two lines and the diagram, determine where your paths would cross (you are travelling at different altitudes so you would not crash).

Antwerp

Duisburg

Aachen

Namur

You have just arrived in Ireland at the Cliffs of Moher. You are standing at the top looking down enjoying the view. Here you are told that your task while in Ireland is to determine the height of the cliffs of Moher. You are given the following picture from a local with some information they collected from their boat to help you. What is the height of the cliff in meters, rounded to the nearest thousandths of a meter?

ft

Welcome to the historic city of Paris, France! Your math teacher mentioned she wanted you to bring back a postcard from your travels. Being a keen student, you are on a search to find a postcard that incorporates your favourite topic, quadratic relations! Below is a postcard you found in one of the shops you were in. It is a picture of the Roman Catholic church, Basilique Du Sacré Zoeur.

Assume that each box represents 1 unit. According to the above picture:

1. What are the x-intercepts?

2. What is the y-intercept?

3. What is the axis of symmetry?

4. What is the maximum value? What is the minimum value?

y

X

Before leaving Paris you head to the Eiffel Tower. In order to prove you where there, you must take a picture in front of the tower and determine the height of the Eiffel Tower. You ask a lady to take the picture for you. She tells you she is 6 ft. tall. In order to determine the height of the Eiffel Tower before you take the picture, you measure the angle of elevation from the tip of the shadow of the Eiffel Tower to the top of the Eiffel Tower to be 83°. The lady stands 129.783 ft. away when she takes the picture and her shadow meets the tip of the Eiffel Tower’s shadow.

1. Draw a sketch of this situation, filling in the information

you know.

2. Simplify the diagram by transferring the situation into a triangular representation. 3. How long is the shadow of the Eiffel Tower? Round your answer to three decimal places.

4. Using properties of similar triangles, how tall is the Eiffel Tower is rounded to the nearest

thousandth of a meter? Explain the mathematical reasoning for you answer.

Welcome to Italy! You just ate some great Italian pizza and you are now visiting the Leaning Tower of Pisa. Your task is to determine the vertical height of the leaning side. You determine that standing directly beneath the lower side you are 4.89m from the base of the tower. You measure the angle of elevation from the ground to the base of the tower to be 85°. Using the information provided, what is the vertical height of the lower side of the Leaning Tower of Pisa, rounded to two decimal places.

You are trying to get a flight into Tanzania, but the airport security will not let you pass until you answer the following question as it has caused a fight among the staff.

1. The staff at the airport have purchased two orders from the local coffee shop. The first order was $30 and had 8 cups of coffee and 9 cups of tea. The second was $54 and had 12 cups of coffee and 18 cups of tea. One employee says that a single cup of coffee is $1.75 and another says it is $1.50. Who is correct? Show all your work.

You have flown to Tanzania and are required to go into the safari for a challenge. The challenge is to count the number of elephants and ostriches. What makes it challenging is that you are only given some information. You know that elephants have 4 legs and ostriches have 2 legs. You are told the number of legs before you get there and only have a restricted time to count.

2. Come up with a plan which will allow you determine the number of each animal in the most efficient way. Write your plan below. Communicate your ideas clearly. Once you have shown the plan to the safari local you will be shown a picture of the safari for you to implement your plan with.

There are 176 legs in this safari. You have ten seconds to look at this picture and determine the number of each animal. (Hint: 10 seconds is probably not enough time to count all the animals)

You made it back to the school! Now you want to let the groups know that you got there before them. You decide to do this by changing the message on the school sign. Your task is to determine what length of ladder you will need to climb up to change the sign. 1. First you need to determine the height of the bottom of the sign to the ground.

Do this by laying the small mirror provided on the ground exactly 1 meter and 20 cm in front of the base of the school sign (the edge of the sign is 20 cm from the base of the sign pole).

Slowing walk backwards (without the mirror) until you can just see the bottom of the sign in the mirror.

Determine your distance from the mirror. Distance = _____ meters

Measure the distance from the ground to your eye level. Height = _____ meters

Draw similar triangles to represent this situation. Label the diagram and indicate all known measurements with units.

Using your diagram, calculate the height from the ground to the bottom of the school sign.

2. Now that you know the height, you can find the length you need for the ladder. The ladder must be leaning 1 m 70cm from the base of the sign and must lean 20 cm above the bottom of the sign. How long must your ladder be?

Solving Equations Pre-departure Quiz

1. 26 8 v

2. 104 8x 3. 15 0x

4. 87

v

5. 6 54

x

6. 9 13m

7. 0 45

n

8. 2 24

v

9. 2( 5) 2n

10. 5

116

n

11. 20 4 6x x 12. 8 ( 4)x

13. 18 6 6(1 3 )k k

14. 24 24 4(1 )a a

Quadratic Relations Pre-departure Quiz

1.

2.

3.

4.

.

1.

2.

3.

4.