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Greg Hargreaves Welcome. THE INJECTION PROBLEM

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Greg Hargreaves



You live near Center City, which has a population of 45,000. Four years ago a mysterious disease struck Center City. At first a few people became sick, running a high fever with sores all over their bodies. After a week many people of all ages became sick in the same way. One month later the disease disappeared suddenly. A total of 4,865 men, women, and children caught the disease and 805 people died. All of those who survived were left with scars from the sores. Some scars were very bad on individual's arms, legs and faces. A year later the disease broke out again in Center City. When it was over, 5,246 people had caught the disease and 750 people died.

Dr. Martha Stanton an epidemiologist (one who studies who gets diseases), realized that no one who caught the disease the first time and survived, caught the disease the second time. She could tell because no one who had scars from the first outbreak of the disease caught it the second time. She reported the information to the medical doctors in the city. A few doctors and other public health officials proposed the idea that once a person survived the disease they were immune and could not get it again. Dr. Walters, a children's doctor, noticed that in homes where children got the disease first, many family members caught the disease afterwards but very few died. Many of these people had cuts or other open wounds when the first family member became sick. Dr. Walters suggested that people can catch the disease by contact with the material in the infected sores. These people, usually, had a mild case of the disease.

Five doctors in Center City decided if another epidemic of the disease broke out, they would offer to give people the disease as an experiment to test Dr. Walters idea. They explained that this meant taking a bit of the material from the sores of someone who had a light case of the disease and injecting it under the skin of the healthy person. It was expected that almost all people so treated would get the disease and that some might even die. Their idea or hypothesis was that very few people would die and all others would then be immune.

In the next three months the disease broke out in two other countries, 12,500 people caught it, and 1594 died.

A few months later people began getting sick again in Center City and the doctors decided they would give the disease to volunteers by injection. There were 869 people who volunteered to be injected. Of these volunteers, 863 caught the disease and 22 died. At the same time. 4,328 other people In Center City caught the disease and 652 died. Again, no one who previously had the disease caught it again. There were many arguments in Center City as to whether it was right to inject people. This year there was another outbreak of the disease in Center City. None of the injected people who survived caught the disease. Dr. Walters said this helped prove his idea that injection was a good thing to do.

Last week, 15 people where you live caught the disease and one is in your neighborhood. The health department announced yesterday they expect a serious outbreak of the disease. Three doctors in the community are offering to inject anyone who wants to be injected. They explained injected people will probably get the disease and some may die. They said it is an experiment and everyone should think about the information from Center City before making up their minds. They think the injection will cause fewer deaths and those who survive will be immune.

Would you take the injection or recommend the injection to others?

If you had another 24 hours to research, what data would you gather that might help you make your decision? (remember that there is little, if any, medical or scientific research that can be done in 24 hours)


Quality Statement 1

Gathering Data:

School leaders and faculty consistentlygather data and use it to understand what each student knows and is able to do and to monitor and facilitate the student's progress over time.


Quality Statement 2

Plan and Set Goals:

School leaders and faculty consistently useavailable data to understand each student's next learning step. Through collaborative planning and student and parent engagement, they set high goals for improving teaching practice and accelerating each student's learning.


Item Analysis questions

  • Is this the distribution of results you expected?
  • Which questions had the most number of correct response? Why?
  • Which questions had the least number of correct responses? Why?
  • Is there a ceiling/floor effect- that is, are all scores close to the top/bottom?
  • Are there students who are scoring low marks compared to the majority of the students? Can you determine why they are not doing as well as most other students?
  • Can you provide any extra assistance? Is there a group of students who are well ahead of the other students? Is it possible to extend these students or use them to help the others?

Data analysis questions

  • Does the data confirm or oppose my intuitions/gut feelings? Why? Why not?
  • Are there any obvious relationships (possibly causal or otherwise) between different sets of data (e.g. school absences and test results)
  • What does the data mean: for my students? For me? For my programs?
  • What are the personal planning / management implications of the data?
  • What is the practical significance of the data?

Next Steps

On the basis of the data and the interpretations, how can I

1) Improve my responsiveness to my students?

2) Improve the quality of what I do for my students?

3) Be more innovative?

4) How will we know if I have been successful?


Teachers have a difficult role to perform in schools and we – Region and schools - need to support you in the execution of your job


We need to look for ways to say “yes” to students.

We must look for ways of providing students with opportunities for success


We want teachers to be able to build trust with their classes.

To be able to build classrooms that reflect respect and the exchange of ideas.


We want students to be comfortable to explore, to think and be willing to share in a safe environment.

We don’t want to produce robots.


Results from the Regents would suggest that Parts II – IV are poorly done. Students demonstrate difficulty or an unwillingness to solve problems.

We must teach students survival skills in Math. Teach them ways of solving problems and teach them thinking strategies. These are skills that are transferable and will benefit them in their lives – set them up for life-long learning.


What is a Problem?

  • A problem is a question that motivates you to search for a solution.
  • This implies first that you want or need to solve the problem and second that you have to search for a way to find a solution.
  • Whether a question is a problem or an exercise depends on the prior knowledge of the problem solver.

The main reason for learning all about math is to become better problem solvers in all aspects of life.

  • Many problems are multi step and require some type of systematic approach. Most of all, there are a couple of things you need to do when solving problems.
  • Ask yourself exactly what type of information is being asked for.
  • Then determine all the information that is being given to you in the question i.e. recognize the important information
  • When you clearly understand the answers to those two questions, you are then ready to devise your plan

Understand the problem

  • Recognize important information
  • Select a strategy
  • Solve and check
  • (Kaplan)

Problem Solving Strategies

MAKE SURE YOU UNDERSTAND THE PROBLEM.This may seem obvious but it is easy to jump straight into solving a problem before you really understand it.So, sure, have a bit of a play around with it at first, if you like, but then read the problem carefully two or three times if necessary.

MAKING MISTAKES!!It's true. Good problem solvers make plenty of mistakes.You have heard the expression: learn from your mistakes.Well this statement is true. Try things out, make mistakes, then try some other way of attacking the problem.


Problem solvers must be organized and willing to make mistakes and learn from their mistakes.


KEEP A RECORD.If you do not keep a record of what you have done (that is all your rough working & notes), you might end up repeating some of your earlier work without realizing it.This is particularly true if you are going to leave your problem for a while before coming back to it. Make sure you write down exactly where you are up to, so it will be easy to get back into at that later date.

MAKE A LIST, then LOOK FOR A PATTERN.Often in mathematical problems, there are patterns to be looked for that will help in their solution. (Consider the lifetime work of Leonardo Fibonacci)


As teachers, we must be flexible. There are often many ways to solve a problem.

As teachers we must be willing to consider anything the students comes up with – there are often little gems waiting for us.


START WITH THE EASIER PARTS OF THE PROBLEM or MAKE THE PROBLEM SIMPLER.Understanding a simple version of a problem often is the first step to understanding a whole lot more of it.So don't be scared of looking at a simple version of a problem, then gradually extending your investigation to the more complicated parts.

CHECK YOUR ANSWERS.You might think you've got it but you'd better check, just in case.This is the vital and final (hopefully) part of solving a problem.And if it is not the final part, it is just as well you checked - isn't it?So always check your answers.


Do your answers make sense? After all, anything is possible:

If a = b

Then a2 = ab

So a2 – b2 = ab – b2

And (a+b)(a-b) = b(a-b)

Eliminating (a-b) from both sides leaves a+b = b

But a =b

So 2b = b

Dividing by ‘b’ leaves us with 2 = 1


Is the answer important? Can you estimate the answer?

This is such an important skill.

How often do we see students come up with answers that are obviously wrong – but that is what the calculator tells them!


Aunt Alice and the Silver Dollars

Aunt Alice gave each of her three nieces a number of silver dollars equal to their ages.

The youngest felt that this was unfair so they agreed to redistribute the money as follows:

The youngest would split half of her silver dollars evenly with the other two sisters.

The middle sister would then give each of the others four silver dollars.

Finally the oldest sister splits half of her silver dollars evenly between the two younger sisters.

After exchanging the coins, each girl had sixteen silver dollars.

How old are the sisters?


Aunt Alice Solution:

Let x = initial number of dollars for the oldest sister

Let y = initial number of dollars for the middle sister

Let z = initial the number of dollars for the youngest sister

x + y + z = 48

Aunt Alice Solution:

Let x = initial number of dollars for the oldest sister

Let y = initial number of dollars for the middle sister

Let z = initial the number of dollars for the youngest sister

x + y + z = 48

Youngest sister

Middle sister

Oldest sister

Youngest splits half her dollars


y + z/4

x + z/4

Middle sister gives each sister 4 dollars

z/2 + 4

y + z/4 -8

x + z/4 + 4

Oldest sister splits half her dollars with her sisters

¼(x + z/4 + 4) + z/2 + 4

¼(x + z/4 + 4) + y + z/4 - 8

½ (x + z/4 + 4)

Number of coins each has at finish




a) So ½ (x + z/4 + 4) = 16 and (x + z/4 + 4) = 32

b) ¼(x + z/4 + 4) + z/2 + 4 = 16

¼ (32) + z/2 + 4 = 16

8 + z/2 + 4 = 16 this means that z/2 = 4 and so z = 8

c) ¼(x + z/4 + 4) + y + z/4 – 8 = 16

¼(32) + y + 8/4 – 8 = 16

8 + y + 2 – 8 = 16 this means that y + 2 = 16 and so y = 14

d) ½ (x + z/4 + 4) = 16

½ (x + 8/4 + 4) = 16

½ (x + 6) = 16 this means that x + 6 = 32 so x = 26

  • ½ (x + z/4 + 4) = 16
  • ½ (x + 8/4 + 4) = 16
  • ½ (x + 6) = 16
  • x + 6 = 32 so x = 26

At the end: 16 16 16

gave half her coins away

8 8 32

gave 4 to each sister

4 16 28

gave half evenly

to each sister

8 14 26


Sliced Watermelon

A 100 pound watermelon is 99% water.

After being sliced and left uncovered it is 98% water.

What weight of water has been lost?


Watermelon problem solution:

  • Watermelon problem solution:
  • 100lb watermelon contains 99lb of water and hence 1lb of flesh
  • After cutting and drying it is 98% water and hence 2% flesh
  • Now, we know that only the water has gone and the amount of flesh at the start and the finish is the same
  • Therefore 1lb is equivalent to 2% of the final weight.
  • This means the final weight is 50lb (1lb is 2%, how much is 100%)
  • So the weight loss is 100lb – 50lb = 50lb!

Weight of watermelon

Weight of water

Weight of flesh

% water





99 . 100 = 99%


After drying

100 - x

99 - x


(99 – x) . 100 = 98%

(100 – x)


Let x equal the amount of water lost:

(99 – x) . 100 = 98

(100 – x)

9900 – 100x = 9800 –98x

100 = 2x

and so x = 50lb equals the amount of water lost