Some Solutions to Problems ABC

I have not completed the tabulation of peer evaluation for all groups. Do not panic. They will come in after I am done going through all your problem-solving journal which is more urgent as I need to return these to you by the end of this week.

The journals were graded as well kept (all problems are solved with some attempts at reflection), fairly well kept (all problems are solved but the element of reflection is lacking), adequate (many problems are solved but the documentation but communication of methods is wanting) and needing improvement (either incomplete or many solutions are only partial). A top grade journal would be like the examplar I gave during the course, for most of the problems. Journaling forming part of the participation and attendance score. Attendance and contributing significant ideas form the rest of this score.

Problems Modified for K

Vasandy has a suggestion on how to modify the apples problem for younger learners. Giving kids apples arrange in an AB pattern e.g. alternating rows of 3 and four apples and asking them to tell the number of a tray can be a platform to repeated addition which forms the basis for multiplication in later grade levels. Vicky also gave suggestions on how this problem can be made accessible to kindergarten kids.

The course has ended but I hope some of you will continue to share teaching insights on your respective blogs.

Lesson on Friday and Saturday

We solve two problems - the toothpick problem and the brick wall problem.

On Saturday we discussed the lesson study style lesson plan based on the hotdog problem. We also did Monkey Puzzle before we all setlled down to complete the final problem-solving test.

The Conversation Time on Saturday revolves the cotroversial issue of nurture and nature - Do children take to problem solcing like fish to water? and Do children need help in completing the problem-solving task?

The peer rating for Group 11 is 25 and for Group 12 is 27. The score is derived by removing the highest and lowest rating such that 10 scores remain. These are added up to give a score. A higher rating indicates your peers valued your insights more.

The ratings for Group 10 = 20. Group 8 = 24. Group 7 = 28.

Preliminary Assigment is graded based on Peer Evaluation + Quality of Slides + Individual Notes.

Lesson on Thursday

Listen to the Song which is based on this 1825 nursery rhyme by Sesame Street. There was an earlier version from 1730 but that was a nine version, not seven. It dates back to 1650 B.C. when the Egyptians ruled that part of the world.

Solve the problem to find how many cats and kittens were there?

A video solution is available. My brother posted it on my Facebook.

We also solved the logic problem - three containers contain all yellow cubes, all green cubes abd a mixture of green and yellow cubes. The thing is the labels on the three containers were all mixed up and each container is labelled wrongly. The container labelled Yellow does not contain only Yellow cubes. Find the minimum number of cubes that need to be drawn to ascertain which containers contain what.

After the break we had three conversations and solved a problem on newspaper.

Lesson on Wednesday

Why things float in water (or any fluid,for that matter)? You see when things are in water, the things experience a push. This push is upwards. It is called upthrust, a kind of force.

The size of this push depends on how much water is displaced by the thing. That is why a marble sinks and a heavier soccer ball floats.

We also solve the problem of looking for names of animal that has the same number of vowel as there are consonants.

We had two groups chatting about assessment things during Conversation Time.

I also posed a problem - how to compare the performance of the groups based on the ratings that you give the groups.

For example - Group 3 (Liana's) scored 3,3,3,3,3,3,3,3,3,3,3,2,3,3,3,3,x,3 (x means comment without rating) and Group 4 (Noorlinda's) scored 3,2,3,3,1,2,x,3,2,2,2,2,2,2,x,x,2). How can we determine which team receives a higher rating. Come up with thee different approaches. Do all approaches result in the same outcome?

Test @ Home

This is an optional task. This is the take-home test which will be taken into consideration in the event that you did not do well on the Final Problem Solving Test. Each task is valued at 10 points.

Your submission is graded as Exceeding Expectations (good solutions in all tasks), Meeting Expectations (ggod solution in at least one task) or Approaching Expectations (some attempt at the problem s but without success).

Problem A

Jon has a set of cards, numbered 1 to 100.
He removed some cards with consecutive numbers and found that the total of these numbers are 50.

What are these cards?

Write an explanation to show how you arrive at the solution.
Show that you engage in the look back stage of Polya's model.

Problem B

Find the largest rectangular box (by capacity) that can be made using a sheet of A5 paper.

Explain the steps that you take to make sure this is the largest capacity possible i.e. how do you know this is the largest possible value?

Provide an explanation if a container with a larger capacity floats better.

Problem C
The Door Bell Rang
Ma makes some freshly baked chocolate chip cookies, and her two kids sit down to eat the when ding dong! the doorbell rings! More kids arrive to share the cookies, but just when they sit down, ding dong! Finally, when there is only one cookie for each child, the doorbell rings again. Who is it? Grandma with a new tray of fresh baked cookies! And no one bakes cookies as good as Grandma's! Hutchins sneaks a bit of math into this funny tale.

In the original story, Ma baked 12 cookies. This is my version. Needs to have basic literacy in Singlish.

Cookies, cookies, there's chocolate chip and durian bits too.
One cookie left when put in twos.
Cookies, cookies, enough for you, you, you and me.
One cookie left when put in threes.
Cookies, cookies, good to eat with Milo and Kopi-O.
One cookie left when put in fours.
Cookies, cookies, come, come, die, die must try.
One cookie left when put in fives.
Cookies, cookies, baked from nenek's secret mix.
One cookie left when put in six.
Cookies, cookies, more sedap than the ones you bought in London.
No cookie left when put in sevens.

Give one, or better, two, method(s) to find the number of possible cookies in the given story. Originality is valued.

Footnotes (in case you are not literate in Singlish and local dialects): durian = a type of fruit, Milo = brand of a chocolate drink, Kopi-O = black coffee with sugar, die, die must try = definitely must give it a try, nenek = grandmother, sedap = delicious.

Note for Absentees

Students who are absent for any session or significant part of the session (more than an hour) need to complete make-up work and submit these with your Final Assignment. Indicate clearly on your Final Assignment that you are submitting the make-up work to earn credits for the hours you missed from the course. Letters and other documents to support your absebce must be submitted to Wendy (if not already submitted to the professor).

For Monday, the work should include the Solutions to the two problems as well as a summary of the definition of a problem, Polya's problem-solving model, and five examples of problem solving heuristics.

For Tuesday, the work should include the solutions to the problems. Make short notes about the important of affective factors in successful problem solving. Make a list of strengths and difficulties children face when solving the tangram problems - based on the Conservation Time held this evening.

For Wednesday, we had two groups that chat about assessment - what amazes them when they observe the kids and what the observation say about the child. Please find out the main points of this discusssion and make a summary. Your other task is to make notes on (1) the format of lesson-study style lesson plan and (2) explain why things float.

For Thursday and Friday, the make-up task is to solve the problems done in class.
For Saturday, the make-up is to complete an alternate test and taks based on Conversation Time.

You are also expected to complete tasks (short notes) based on the Conversation Time topics that you missed. You can do this by discussing it with a friend who was present, to read up relevant sections from the book, and to base it on your own experience.

Thursday: (1) Give five examples of tasks you expect K2 children to be able to solve? Use different subject areas. (2) Share two surprising things that you saw during the tangram task.

Friday: (1) Creativity & Brilliance in Problem Solving (2)A parent complained that you are wasting time doing the tamghram puxzzle in class. Write an e-mail to justify why you did the task in class. (3)Difficulties Students Have with Problem Solving.

Saturday: (1) Is problem solving taught or caught? Are some people just born to be good at this? (2) What Help Teacher Can Give Children During Problem Solving?

Lesson on Monday

Problem: Use a given number of sticks (12, 10, 9 and 6) to make four triangles. A triangle is made using three sticks with the vertices (corners) formed using the dough.

How many different figures comprising of four triangles can be formed using 9 sticks?

In this lesson, we discuss what makes a task a problem. We also learn to analyse a problem if it is suitable for a given age group.

The class was introduced to Polya's problem-solving model.

Problem: How many apples are there on each tray? See previous post.

In discussing this problem, the idea of mathematical modelling was introduced. The term problem-solving heuristics were also explained with several examples.

MAT150 BSc05 Wheelock College / SEED Institute

The course on problem solving has begun. Please prepare an exercise book which will be your problem-solving journal - you can refer to this during the test, so take good notes. You will also submit this to be graded for the Participation component of the course. Bring your own crayons, glue, scissors etc if you are the kind who like to make your journal interesting. At least one of you is using your iPad to take notes - in this case you can submit your 'notebook' in the soft copy form. Each problem will be assessed for (a) the quality of solution (b) the quality of presentation of the solution.

Please complete your pre-course task. You will be given time on Monday to form your Conversation Time Groups and have a short meeting. All Groups must be ready to talk about the pre-course work from Tuesday onwards - Groups and Questions will be drawn randomly.

This course is on problem solving in numeracy with extension to other domains.

The required textbook is the same as EDU330. Please be familiar with the chapters on problem solving.

To whet your appetite, here is one based on the photograph of a grocery store in New York City. Look at the green apples. Problem: Explaining your method and stating your assumptions, find the number of apples on one tray. Suppose the tray is a square tray.

Also think about how you will help K2 children solve this problem. What can a teacher do to help the children arrive at a solution? What content areas are required in solving this problem?