Formats for Problems: Larger Tasks

Chapter 9

Formats for problems: Larger tasks

Each year, during the Rotterdam Marathon, students in grade 8 of a school near Rotterdam get this assignment:

"For the marathon, a chalked line is drawn through the streets of exactly

42 kilometers and 195 meters. But how do they now the correct distance? Some kind of wheel is used, with a ticker that counts the number of rotations of the wheel.

Now suppose we want to organize our own marathon in our own city. Somebody chalks, we take a bicycle wheel (diameter 68 centimeters), use a ticker and start walking.

Which number will be at the ticker after 42 kilometers, 195 meters?

Now we walk in the opposite direction, to control our measurements. Alas, during this walk we get a small puncture in the tire and the tire slowly gets emptier and emptier.

As we arrive at the end of the chalked line, is there a larger number on the ticker? A smaller number? Does it make no difference at all? Explain your answer."

(Of course, if a school is not situated near Rotterdam, this context is of less interest for students.)

That is what this chapter is all about. It is meant to show many different opportunities for larger tasks. They assess other skills than can be assessed in a written test. Some examples of these larger tasks are:

investigations that can be given as homework;
assignments to be carried out in small groups in the classroom;
two-stage tasks, making it possible to give feedback during the process;
things a teacher can do when students are all excited about something that happened at school which makes it necessary to think of something special to do instead of the normal schoolwork;
ideas for rehearsing basic skills by the end of the year.

But above all, the tasks shown in this chapter are meant to stimulate your own creativity. Examples of work done by colleague teachers can be used in different forms in your own classroom. The choice depends more on school culture or views of the teacher than on conditions to be fulfilled by the format of the task.

A balanced assessment program will probably contain one or two of these larger tasks for one school year.

Homework assignment

The first example in this chapter was given by a teacher who changed the following problem in her ‘End of unit’-test into a homework assignment for her students. She did not feel this problem was a very realistic one. If you make a frame for a photograph yourself, the area of the glass is not important, the dimensions are. And, given the dimensions of the glass, the dimensions of the wooden frame have to be larger than those of the glass panel.

1.
Jenny has a photo of her dog. She is making a frame to put the photo in. For the frame she needs glass and a strip of wood for the edges. The photo is 10 inches long and 5 inches wide.
a. How much glass does Jenny need for the frame? Show your work
b. How many inches of wooden strip does Jenny need? Show your work.
Answers:
a. Jenny needs 10 x 5 = 50 square inches of glass.
b. Jenny needs 2 x 10 + 2 x 5 = 30 inches of wooden strip.
Age: 12, 13
Level: 1
Content: Geometry, area, perimeter
Context is relevant, situation daily life

The teacher used the content of the problem and changed it into a homework assignment that was far more interesting for her students. The products that were handed in a good week later showed a larger variety in quality and choice of the presentations.

2.
Imagine you get home with your first math test of the year. You have got an “A”!! Somebody suggests: “You should frame that one!” The neighbor, who is just visiting, says:
“That’s a great idea. You should frame it but of course you do not want some ordinary frame. I will make one just for you. Just tell me exactly how you want to have your frame. Of course I will put a sheet of glass in it and a nice wooden edge around it. Write down for me all I need to know: How you want it, what kind of materials I need and how much of it. I will buy everything for you.”
The neighbor does not want to buy too much or not enough materials. Pleas write down all your calculations.
Most students really got into this task. One wanted small lamps on the frame, one wrote underneath the assignment: “And the name of my neighbor is Anderson, he did a wonderful job.”

Two-stage task

A two-stage task can have different forms. For instance, the first part may consist of a simple quizz to assess basic skills that are important for the subject. The teacher discusses the results with the students and makes sure all students master the mathematical skills and knowledge needed to complete the second stage of the task which is an open problem based on the same mathematical content.

Another possibility is to monitor the process of problem solving where students hand in the first results of a large task. They get feed back from their teacher and can adjust their findings and their plans before starting with the escond stage. This may help them not to get entangled in dead ends and help them find the right path through solving the problem.

The next example was used for students in grades 7 and 8.

Doing your own statistics project

Until now you have learnt some statistical techniques to present data; for example a bar graph, a pie graph or a box plot. You also learnt how to summarize facts: by the mean, median, or modus.
Now you are going to carry out your own investigation. This task has two stages. For each stage you get score points.

Stage 1

First you choose something you want to investigate. There are many possibilities.

A few suggestions are:

Compare two newspapers and investigate which one on average has the largest number of photographs/advertisements in it.
Find out how much time teenagers spend watching television each day.
Find out if there is a difference in doing sports between girls and boys.

Maybe you find a quite different subject on your own that is allowed also.

When you have chosen a subject, you have to decide how you are going to investigate it. How will you gather all the information you need (the data)? If you are going to interview people, make also a list of questions, a survey.

Write down a short report about the things you plan to do. At least you have to describe:
the subject you have chosen;
the way you are going to gather the data;
if relevant: a list of questions you are going to ask. If you use a survey, you have to interview at least 25 people)
how you are going to handle your data statistically. You have tow use at least two different diagrams, and if possible, calculate the mean, modus and median of your data.

You will get some score points for this plan.
You have to hand your report at …… (date). I will read it, and write down some comments on your work.
At…. (date) I will discuss your plan with you. If we both agree, you can start with the second stage: carry out your investigation.

Stage 2

You can start carrying out your investigation and think of a way to present your findings.
If you want to, you can make a poster presentation, but you are also allowed to make a short report (1 or 2 pages is enough).
In your presentation you have to describe:

the subject you have chosen;
the way you gathered the data;
a statistical presentation of your data (as described in the plans we discussed)
and, very important: your conclusions.

You have to hand in your report at ……… (date).
For this report you will also get some score points. The total number of score points for both stages will be used to mark this work.

Group work

By “Group work” in this case we do not refer just to students working together in class at problems in their books as many students do at a daily basis. Here a special assignment is given to carry out in a group where all group members are equally responsible for the results, conclusions and reports, and a mark is given either to individual students or to the group as a whole.

When doing mathematics in (small) groups, students are forced to work together with others, share ideas and try to convince the others of the correctness of your views, which makes it a worthy activity to carry out once in a while in class. Many examples can be given, here is one that was done by younger students.

4.
Jessica and David decided to design their own game to be used at Jessica’s birthday party. Winners will get nice prices. This is what their game looks like:
A wooden box which measures about 150 centimeters by 100 centimeters has an opening for David’s rabbit to enter. At each corner of the box there is a carrot.
The rules of the game are:
Each child chooses a corner. The rabbit is then placed at the entrance by David. You win the game if the rabbit chooses to eat part of the carrot in your corner first.
After playing the game a few times, Jessica says: “This is not a fair game!”

Why is the game not fair? Explain your answer.
Now design your own game. Your game should be fair, and if you use an animal, the game should be fun for the animal as well.

Indicate clearly how the game can be made and explain the rules of your game.

Hand in one report for the group as a whole. Put it in a plastic binder and make sure your work is neatly done.

Age: 8 - 12
Level: question 1: Level 2,
question 2: Level 3,
question 3: Level 1.
Content: Uncertainty and chance, geometry
Context is relevant, situation daily life.

Make it up yourself!

1.A mathematics contest

By the end of the school year, teachers sometimes want to rehearse all of the basic skills and knowledge being taught over the now nearly past year. One way to do so is to have the students make their own “math contest”, consisting of short answer questions about all of the subjects in the book. A teacher could ask students to hand in 25 questions (and answers!) and compose a quiz later herself. It is important students recognize their own questions, so these are altered as little as possible. An example of questions for a contest, made by a group of Dutch students age 15, is given below.

5.
Math Contest.
Answer as many questions as possible. Each correct answer is awarded with one score point. The student who gets the highest score is winner of the contest of the year!!

How many erasers are in a dozen?
Which formula “wins” in the long run: 400x4 or 40x5?
What does the graph of a linear relationship look like?
What is the name of the graph that fits a squared relationship?
How many faces has a cube?
How many glasses of lemonade could be poured from a 1 liter bottle of Coca Cola?
In 813 give names for “8” and “13”
Where does the sun go down?
How can one compute the volume of a cylinder?
Write down the first five odd numbers.
How many prime numbers are even?
Where do you find the number ?
What is the result of 12 squared?
How many cars are in a traffic jam length 5 kilometers?
Give the names of three different triangles.
Write down the Pythagorian Theorem
A cube has edges of length 8 centimeters. What is the total area of this cube?
Compute 12 x 2 ÷(4 x 3)
How many different colors do you need for a map as a minimum?
What is the formula for the area of a triangle?
What percent is 4 taken from 25?
Write down a formula of a parabola.
How many faces has a sphere?
For which x is 8x – 22 = 18 right?
How many degrees are the angles in an equilateral triangle?

Some schools are able to make a computer version of the contest and have all students answer the questions on the computer but using paper and pencil is also possible.

One teacher told us she found out that very weak students sometimes limit themselves to a lot of questions about the same subject where other students pose questions that go far beyond the possibilities of the average student in class. Moreover, often important misconceptions became clear while analyzing the questions the students handed in. Of course these needed a discussion in class before the contest “Who knows most about basics?” started.

2.Make your own test

The former was of course just one example of socalled Own Productions. One teacher said he does the following:

“About say once or twice a year, I ask my students, to Make their own test, to go

with the chapter in the book that was just finished. Each student hands in a 20 minute test with problems on different levels. Of course answers should be given as well. Sometimes this work is marked as an assignment itself (and students are told so beforehand) but at another time I promise them to use one or two appropriate problems in the test given to the whole class. So if you find your own problem in the test you have an advantage over other students because you know the right answer! It is surprisingly to see how well able students are to make their own problems after they had some experience in doing so. It is something many of them like doing, and sometimes I even get problems from students that are better than my own!”

3.Make a crib

At the time teachers themselves went to school they never ever used cribs, but all of them know their students do if they get the chance. So why not use it instead of abuse it? Then all of them have at least studied the chapter to some extend. Ask the students to make a crib for the chapter to be assessed in tomorrow's test. Limit the dimensions of the piece of paper they may use in order to give each student equal opportunities. Tell them they may use this crib for the test as an aid. By the way, many students find out they do not have to use it once they made it and implicitly that was the teacher's goal!

The cribs are handed in together with the test results. Some cribs will show that the student is not able to make a good summary of the chapter. When discussing their cribs with them the teacher has an opportunity to teach them how to summarize.

4.Use the internet

If computers are available for students, either at home or at school, they can get an assignment where the internet is useful. They are asked to write a short essay (give an indication of the expected number of pages!). Examples of possible subjects are given, if they want to choose another subject themselves, they first should discuss this with the teacher. An example of the way this was done at one school is given below. This example was meant for 15 year old students but the subjects can easily be changed into suitable ones for younger students.

6.
For this investigation, you work in pairs. Just one essay, mentioning both names, is expected form you.

Choose one of the subjects shown below. If you want to investigate another subject, please contact your teacher before you get started!
Which mathematical questions emerge from the subject? Remember the mathematical content of your essay is what counts most.
Use the internet or books from the library to find information about the subject. Take care however, you are supposed to process this information for your essay and you must be able to explain everything you wrote down. I do not want a bundle of pages printed directly from the internet or copied from a book.
Write an essay about your subject. This should not be at least 1 1/2 pages and not over four pages in print.
Note the time you used for this assignment in a log. Time used should not exceed three hours.
Remember that negative results are also results, you do not always find what you were looking for.
I am interested in the process as well as in the product.

Here are the subjects you may choose from:

The Canadian Yvan Dutil sent a message into deep space Spring 1999.
Does global warming really exist?
Who invented the Pythagorian Theorem?
Why do we write numbers the way we do?
What is the connection between Florence Nightingale and statistics?
How does a doctor know the amount of medicine a patient needs?

Assessment of the essays.
You get score points for the following parts of your essay:

Essay is handed in following the usual instructions 2 score points

(Neatly covered in plastic, names, date, class on it and
on time;

Quality of the mathematical questions; 2
Processing the information that was found; 2
Answers for the questions, content of the essay; 2
Books and/or web sites used, other sources of information; 1
Extra for high performance. 1 +

Total 10 score points
Good luck and lots of success and pleasure with this assignment.

Summary