Tie and Lift Math Problem

Take a long rope; tie it to the bottom of the goal post at one end of a football field.

Then run it across the length of the field (120 yards) to a goal post at the other end. 

Stretch it tightly, and then tie it to the bottom of that goal post, so that it lies flat against the ground.

Now suppose that I add just 1 foot of slack to the rope, so that now I can lift it off the ground at the 50-yard line. 


How high can I lift it up?


(the answer is surprising)

POW: Locker Problem

(actual lockers at my school)


Here is a long time favorite middle school math problem:


Imagine you are at a school that has 100 lockers, all shut.


Suppose the first student goes along the row and opens every locker.


The second student then goes along and shuts every other locker beginning with locker number 2.


The third student changes the state of every third locker beginning with locker number 3. (If the locker is open the student shuts it, and if the locker is closed the student opens it.)


The fourth student changes the state of every fourth locker beginning with number 4.

Imagine that this continues until the 100 students have followed the pattern with the 100 lockers. 


At the end, which lockers will be open and which will be closed? 


Which lockers have been switched the most often? 


How many lockers, and which ones, were touched exactly five times?


(This is a great investigation into the nature of numbers and their factors. It is also easy to make this a hands-ons investigation using a deck of cards and flipping them up and down as needed. Finally, it is a great investigation to show how to deal with a huge problem by working with smaller parts: instead of 100, students work with 10 or 20 and see what patterns arise)

Exemplary POW Essay

The title of this POW is Going Global. In this problem you have to figure out the area of the field of Shakespeare’s Globe Theater. Using this you have to figure out how many people can stand in the field if they each need 4 square feet. The theater is an icosagon, this is a 20 sided figure. You have to figure out the interior angle. To figure out these problems you have to assume that the inner field is a circle and it is 12.5 feet away from the outside. The diameter of the icosagon is 100 feet. And the dimensions of the rectangular stage are 53 feet by 26.5 feet. The center of the front of the stage is the exact center of the building. The key math content in this POW is geometry because the whole POW is about finding areas and angles of shapes.



I solved the problem of how big the interior angle is by first cutting the icosagon into 20 triangles. Each triangle cut each interior angle in half. Then I noticed that there was a strait line going across the icosagon cutting it into two sets of ten triangles. A strait line is 180º, so I divided 180 by 10 = 18. I know that the total angle of a triangle is 180, so I subtracted 18 from 180 = 162. The two remaining angles were the same so I divided 162 by 2. But that is only half of the total interior angle so I multiplied it by 2. This told me the interior angle was 162º. I knew what to do because we spent a lot of time on angles in 6th and 7th grade and you explained a way to solve the problem.



I figured out how many people could stand in the field, by first figuring out how big the circle of the field would be if part of it was not cover by the stage or backstage. First I figured out how big the radius was. The diameter of the icosagon is 100 feet and the inner field is a circle and it is 12.5 feet away from the outside. I multiplied 12.5 by 2 = 25 I did this because there were 12.5 feet on each side. then I subtracted 25 from 100 = 75 to get the diameter. I divided 75 by 2 = 37.5 to get the radius. Then I squared the radius and multiplied it by pi to get the area of the circle (37.5 x 37.5 = 1,406.25 x 3.14 = 4,415.625 square feet). Then I found the area of the stage and subtracted it from the field circle (53 x 26.5 = 1,404.5)(4,415.625 - 1,404.5 = 3,011,125). I could not figure out how to find the part of the stage that was cut off by the backstage. When I went to school Owen helped me figure out that if you continued the draw of the field into the backstage and you drew lines from the edges of the backstage to the front center stage, you would cut the stage in half and the over all circle of the stage into a forth. I then divided the over all circle of the stage in froths(4,415.625/4 =1,103.90625), I divided the stage area in half (1,404.5/2 = 702.25), and then I subtracted the half of the stage by forth of the circle to get the part of backstage that was not part of the field (1,103.9625 - 702.25 = 401.7125). Then I subtracted the area of the field backstage from the area of the field minus the stage plus the area of the field backstage to get the area of the field (3,011.125 - 401.7125 = 2,609.4125). Then I divided the area of the field by 4 to get the total number of people that fit in the field (2,609.4125/4 =about 652 people). About 652 people fit in the field. My resources were my old math notebooks and my classmates.


My answer is the interior angle is 162º and 625 people can fit in the field. My answer is correct because I checked it with my classmates. I did not find a second answer. This POW reminded me of some of the Verdana POW’s were we had to use triangles to figure out areas of a circle and how long different routes from the adult village were.



What interested me about this problem was how logical it was to figure out how big the portion of the field was backstage once you thought about i for a while. And it surprised me how useful triangles were in a POW about circles and icosagons. What challenged me was figuring out how big the portion of field was backstage. I think I managed my time very well because I solved the POW in three days and wrote it up on the forth. But next time I should grapple with my problems a little longer before I ask for help.

7th Grade POW: Captain Future

The title of this week’s POW was “Captain Future.” In this problem, Captain Future, Joan Rundall, Otho and Grag have to get over a bridge in 17 minutes. There has to be 3 trips going across the bridge and 2 trips coming back over the bridge because a metal box must guide 2 or less people across the bridge at a time. Captain Future can cross the bridge in 1 minute, Joan Rundall can cross the bridge in 2 minutes, Otho can cross the bridge in 5 minutes and Grag can cross the bridge in 10 minutes. With this information, I had to figure out a combination or combinations to get all four of the people to the other side of the bridge. This POW was a logic problem.

My first step in this POW was to draw a little diagram and write out some key information so I could use it as a reference whenever I needed it. Next, I started to think of a logical way to organize the four different people. I thought that maybe if I moved the two people who could cross the bridge in the least amount of time first, then I could find the answer. So, I tested this hypothesis and tried many different combinations. After a couple of attempts, I started to notice a pattern I think is key. Captain Future and Joan Rundall have to be the people who move both back and forth across the bridge. Using my new information, I was able to find a combination that was successful in 17 minutes. After finding the first combination, I simply switched around the order a little to see if there were any other possible combinations. I used some objects to help me visualize the combinations.

There are two possible combinations for crossing the bridge in 17 minutes. The first combination is Captain Future and Joan Rundall go over, Captain future comes back, Otho and Grag go over, Joan Rundall comes back and Captain Future and Joan Rundall go over. The second combination is Captain Future and Joan Rundall go over, Joan Rundall comes back, Otho and Grag go over, Captain future comes back and Captian Future and Joan Rundall go over. I know my solution is correct because I used objects to visualize the two combinations, added up the total number of minutes multiple times and checked with other students after I had solved the POW. There are no other possible answers to this problem because any other combination wouldn’t follow at least one of the specific rules for this POW.

This POW was interesting because it was a pretty simple logic problem with a bunch of twists, which made it more complicated than it seemed. The POW challenged me at first because I didn’t know how to go about starting the problem, initially. This POW reminded me of other logic problems we used to do in math class, where several trips had to be made to carry a certain number of objects. I managed my time extremely well this week and planned out the work load very nicely. Next time, I would like to try to use more than one visual reenactment to solve a logic problem.

Sample 8th Grade POW essay

The title of this week's P.O.W is Unraveling. In this problem, the Captain finds a book called Our Historyand in it, she reads about Captain of the Ferdinando, Jurakan, who made seven journeys around the Carribean. On each of those seven journeys, he got a famous treasure which he deafeted an opponent to get. In this problem, you have to find out the order of his voyage and each of the people he deafeated and prizes he acquired. The pertinent information in this problem are that there were seven places that he went on his voyage and each of the 13 clues are the other important information. The main math topic in this problem is solving word problems.

The first step I took to solve this problem, was that I made a list of all the islands, people, and prizes involved in the problem. Then I made a chart with all of the places, people, and prizes in a row on the top and the numbers 1-7 in a column on the side of the top row for all seven voyages Jurukan took (look at previous post for chart.) Then I put x's in the boxes for all of the things that could not be possible and +'s for all of the things that were true in the voyage. To find all of the possible and not possible steps in his trip I used all of the thirteen clues. The first clue I looked at was Clue 11, which said that on Jurukan's 7th and last voyage, he stole the Princess Parlina, who became his wife and that on his first voyage, he sailed to the Goya Island. I filled in those two clues and put x's in all of the other prizes on the 7th voyage and all the places for the first voyage. Then, I mostly looked at the clues that said which things came before others like Clue 1, 5, 7, 8, and 9 because it helped me determine which things could not be on which voyage because it was too early or too late. Once I figured out more things by doing that, I looked at which things couldnt be with something else like in Clue 6 or 10. When I found a new thing that was true, I marked that any other thing in that category could not be possible for that voyage and that eliminated a lot on the chart. I just kept going back and reading the clues throughout the problem to find another detail. When I had filled out the whole table, I read back through all of the clues and made sure they were true in my table and that nothing was repeated in my table. I knew that I should try this strategy becuase when I made lists seperating all of the places, prizes and people, it seemed the easiest to organize my thoughts on a chart because it showed every detail of the problem by marking what things were possible and not possible along the way instead of just having the final answer written down. A problem I dealt with was remembering which step I last took becuase there are so many spaces on my chart and if I messed up and figured out that something did not match up with one of the clues, I would have to delete a lot of things in my table until it made sense again. A resource I used was James because he showed me what kind of chart he used and it helped me figure out how to make my chart to put my answers in.

My solution is that the first voyage Jurukan took was to Goya Island, where he defeated Pirate Guanin for the Black Pearl. Then he went to Lukiyo Island where he defeated Viejo Tomas for the Lamp of the Ditas. From there, he went to Amabala for the Sapphire of Sezu which he got from Zerena. His fourth voyage was to Jutia Island, where he got the Caribe Ruby from General Bure. Then he went to Natia for the Mime Emerald from Macana. His sixth voyage was to Tabata, where he defeated Queen Nasa for the Inriri Diamond. Lastly, he went to Baya where he defeated Chief Fotu for Princess Parlina. I know my solution is correct because I read through all of the clues to make sure that it matched up with them. There are no other possible answers for this problem.

Something that was interesting about this P.O.W is that I thought it would be very easy when we first got the problem because it was just like a puzzle but it turned out being the hardest P.O.W and it took me a VERY long time. This problem challenged me because, to solve it you had to organize the problem really well becuase there were so many clues and that was challenging for me. This problem reminded me of the problems where we had to figure out the ages of people in a problem by using different clues or comparisons between each of the peoples ages. I think I managed my time well and much better than I usually do with other P.O.W's but the problem was so challenging that it still took me a really long time. Next time, I might try to start even earlier so that if the problem is really hard, I can solve a little more of it everyday instead of spending a lot of time on it for one day because it is so hard. 

7th Grade POW: Can't Get No Respect

          Mr. Bott was surprised to see Mr. LeFour at his classroom door. He couldn’t remember the last time they had spoken to each other in the past couple of years. Ever since Mr. LeFour was hired and took over the sport department from Mr. Bott, there had been friction. Mr. Bott had dedicated many years to Willowside Middle School’s sports teams. True, the school rarely had a winning season, but Mr. Bott felt that the real purpose wasn’t to win, but to instill a sense of honor and discipline in his athletes. It wasn’t a very popular position, he agreed, but an important one nonetheless. 

Mr. LeFour, on the other hand, was all about winning. He applied the same discipline he learned in math to the development of a first class sports program.  He carefully watched the students during recess pick up games or after school events. He felt he had a good eye for talent and he went out of his way to cultivate it in select students. His technique, controversial to many on staff, produced one winning season after the next. The medal wall in the cafeteria was completely full and the awards shelf sagged under the weight of 10 winning seasons in a row. Mr. Strehl, the school principal, wasn’t about to risk this by asking Mr. LeFour to change his practices. Mr. Bott resented this.

So, when Mr. LeFour offered him the volunteer soccer coaching position, Mr. Bott had to consider his options before deciding what to do. On the one hand, he was excited to return to coaching. Starting up a soccer team from scratch intrigued him. But it had been a long time since he had taken on such responsibility. He wondered what Mr. LeFour’s expectations would be of him and the team. What if they did not have a winning season? How would the athletes react? But he knew down deep that the challenge of coaching a new soccer team was just what he needed to jolt him out of his lethargic existence. He accepted.

Unfortunately, there was very little time to get ready. He met with the students interested in the soccer team one afternoon. There were only six of them. “Not enough, not enough” he muttered as he looked at them. Justin and Sydney were there, of course. Richard, Kyle and Amanda also came to that first meeting. Richard had been recruited by Mr. LeFour after trying out for (but not making) the baseball team. Kyle and Amanda were there because their parents were pushing them to get more involved in extra curricular activities. Griffin was ostensibly there only as a reporter for the school newspaper, but found himself interested in playing on the team as well. Almost at the end of the meeting, after Mr. Bott had talked about honor and discipline, in walked Doni Einstein. “Is this the soccer meeting?” he asked. Mr. Bott nodded before saying, “Yes, but if coming late is your habit, you might want to reconsider.” Doni looked unsure what to do until Sydney stood up and went over to him.

“We are SO glad you want to play on the team. Now, with seven, we have the minimum number required by the league. Welcome, Doni.” She shook his hand as the other looked on, incredulous.

“Speaking of the league,” interrupted Mr. Bott, “I almost forgot to tell you about the schedule. There are 8 total teams in the league, including us, and all teams need to play each other twice in the season. The problem is that there are only two fields big enough around here to play soccer on and the baseball league dominates them. The only times available were Tuesday and Thursday evenings and early Sunday mornings.” There was a collective groan from the students. Mr. Bott passed out the team’s game schedule. 

“I don’t want to be disrespectful or anything, Mr. Bott,” said Amanda a little cautiously, “but this schedule is horrid. You have us playing most of our games late on Tuesdays and Thursdays, not to mention this crazy 6 am Sunday game. There’s no way my parents will agree to this. Isn’t there anything we can do?”

“Honor and discipline, Amanda, honor and discipline. It’ll be a great character builder,” replied Mr. Bott in a clear, firm voice. Sydney watched Amanda go all quiet and worried that she would drop out of the team. She stood up quickly.

“Mr. Bott, Amanda, Everyone: listen. I propose we look at the schedule and see if we can help the league make it a little more humane for everyone. I mean, after all, there are only 8 teams. It can’t be that hard, right?”

Your Task: You are challenged to create a more reasonable soccer schedule given the limitations imposed on the league by the baseball schedule. Make a clear schedule that indicates when, where and at what hour each team will play for the minimum number of weeks necessary for everyone to have completed the two game requirement. Be realistic with the times of the games.  Assume a soccer game takes one hour to play and that the play-off schedule will be determined post season.

8th Grade POW: Chapter 10: Unravelling

Cynthia lay in her hammock, exhausted yet unable to rest. Nothing on this trip seemed real now. Not even 24 hours earlier she was on the Rim with the Adults. There had been hope of finding a way off this confusing island. Then the rocks came hurling through the sky. She saw Jesse hit in the head and fall, then, in short order, Margie and Jeanie and several Adults around them were hit. Then Miguel. Confusion ensued and Cynthia felt hands grab her over the edge of the Rim and into a series of covered trenches that offered protection from the rocks raining down on them. The Verdania Adults were shouting and running through the trenches. It felt like an old World War II movie. 

Dula ran up to her, “The other in your group, the Captain, is down that way with two of our people. They will evacuate you. You must leave, now!” Dula pushed Cynthia down the trench, “Go!”

Cynthia ran up to the Captain, who was strangely distant and silent.  The trenches led down the side of the Rim and back into the Verdania Caldera. They moved quickly. The walked briskly through the night, along moon lit paths through the forests until finally arriving to a small village.  They left Cynthia and the Captain in a small cabin with several hammocks hanging from posts. The two adults nodded as they left, closing the door behind them.

“What has happened? Oh my god, what about the kids? What are we going to do, Captain?”

The Captain paced along the floor, impatiently. “The kids are fine, at least ours. Dula told me that attacks on the Rim happen only occasionally, but to date, nothing has ever reached the Caldera floor. The children don’t even know about this. It’s all some sort of elaborate scheme to protect them from the Deesors.”

“Who are the Deesors?” asked Cynthia.

“Don’t know. There wasn’t time to find out. I think we are ok here, though. Feel bad about the others on the Rim though.” She looked down for moment. Cynthia wondered if she would cry.

But then the Captain looked up, dry eyed, and started pacing around the cabin again. She stopped at a table in the back corner of the room. There was a small bookcase on the wall behind the desk. The Captain rummaged through the books with idle curiosity. They were ancient, dusty old books, some of them handwritten in fancy calligraphy. Most of them were in Spanish or French, though a few were written in English as well. She picked up one book and saw the title Our History. 

“What a silly title,” she thought to herself. But as she breezed through the first couple pages, she realized that perhaps she could finally come to understand the origins of the people on this crazy island. She read:

Jurakan, Captain of the Ferdinando, made his life's fortune on seven perilous voyages around the Caribbean, each of which took him to a different island where he acquired a famous treasure from his dangerous opponents. From the following notes taken from Jurakan's log, you can find where Jurakan went, what prize he acquired, and whom he outsmarted on each of his seven voyages?

1. Jurakan was on Jutía Island on an earlier voyage than the one when he bested Macana; his meeting with Macana was on an earlier voyage than the one on which he acquired the Inrirí Diamond.
2. The Jurakan's attack against Viejo Tomás wasn't the one in Baya or the one in Goya.
3. Queen Nasa wasn't the foe from whom Jurakan got the Mimé Emerald.
4. Neither Chief Fotu nor the Pirate Guanín was the victim when Jurakán gained possession of the Lamp of  Ditas.
5. Jurakan outfoxed General Buré on the voyage immediately after the one that took him to Amabala and immediately before the one that brought him the Mimé Emerald as a prize.
6. The Caribe Ruby wasn't acquired in either Natia or Tabatá.
7. Immediately after tricking Viejo Tomás on one voyage, Jurakan got the Sapphire of Sezu on his next adventure.
8. Jurakan acquired the Caribe Ruby on a voyage later than the one on which he had to defeat Zerena; the voyage on which he beat her was later than the trip to Lukiyó Island.
9. Jurakan's adventure in Baya immediately followed his journey to the land ruled by Nasa.
10. The encounter with Macana wasn't in fabled Tabatá.
11. On his 7th and last voyage, Jurakan stole the Princess Parlina, who became his wife. On his 1st voyage, Jurakan sailed to the Goya Island.
12. King Fotu wasn't the foe who possessed the Black Pearls of the Puño.
13. The Sapphire of Sezu wasn't the prize Jurakan outwitted the Grand Buré to get.

Exemplary 7th Grade POW Essay

This week’s POW was called “Join The Bacteria Team.” In the problem, Justin was in Mr. Bott’s math class, on Friday afternoon right before spring break, and he notices that the tables are dirty (there were 2 bacteria on the table originally). To solve this problem, I had to figure out how many bacteria would be on the table after spring break was over, if the number of bacteria doubled every hour. I had to make a table to show the growth of the number of bacteria as well as a graph. This problem was about using scientific notation and finding patterns.

My first step in this problem was to make the table. To do this, I selected a time in the afternoon for the growth to start (2 pm on Friday), and end (9 am on Monday). Then, I carefully filled the table out, starting at 2 and doubling each number, (for example 2+2=4, 4+4=8) making sure to use scientific notation when the numbers got larger than a couple thousand. After filling out all of the table, and rechecking my work multiple times, I started making the graph. To make the graph, I split up the data into sections, meaning that I graphed for everyday separately. This allowed me to see a pattern that the data goes up at an even pace very quickly (see sample of graph and table below). I used a calculator to ensure that all my calculations were correct.

The total number of bacteria after spring break was about 1.13 times 1071. I know my solution is correct because all the calculations were check and I compared my results with other students after I had solved the problem. There are other solutions to this problem because some people might have chosen to start at 1 pm or later and chosen to end at 8 pm or earlier. Plus, since this problem will never have an exact answer, there will be many variations.

This problem was very interesting to me because I had never really thought that bacteria grew that quickly till I solved the problem. This problem challenged me because at first I did not realize that the graph would be so difficult to make, especially when there are so many huge numbers. This POW reminded me of counting beans in 1st grade, because there were so many beans and the number just kept getting larger and larger. I managed my time fairly well, however I could have done more work over the week rather than the weekend. Next time, I would like to try and find a rule for the POW which will make it easier to complete the table.

This problem is to figure out the lengths of 3 different routes bisecting the circular Caldera rim and valley. The pertinent information is that the Kid’s village is in the exact center of the circular valley which has a 100km radius. The northern-most point on the rim is Sentry Point 1. Hope Lake is 30km south of there. Sentry Point 2 is due west of Hope Lake. The Adult’s village is 15km south of Sentry Point 2. We need to figure out the direct overland distance (“as the crow flies”) from Sentry Point 1 to the Adult’s village. We also need to figure out the distance on the usual adult route from Sentry Point 1 to the Adult village, going south to Hope Lake, then West to Sentry Point 2 and south to the Adult village. Finally, we also need to calculate the distance if you went on the rim from Sentry Point 1 to Sentry Point 2 and then South to the village. The main math topic imbedded in this problem is geometry.

My first step in solving this problem was to draw out the situation on a diagram/map. Heres what it looks like:

I decided to figure out the easiest thing first which was the usual adult route, since I knew the distance from Sentry Point1 to Hope Lake is 30km and the distance from Sentry Point 2 to Adult’s village is 15km. I needed to figure out the distance from the Hope Lake to Sentry Point 2. I realized that this length represented one side of a right triangle formed by Hope Lake, Kids Village and Sentry Point 2. I knew the distance from Hope Lake to Kids village was 70km. I also knew the distance from Kid’s village to Sentry Point 2 was 100km, because it is the radius of the circle. I used the Pythagorean theorem as follows:

4900 + b2 = 1000
b2 = 5100
b=71

So I added the 3 lengths and got the total usual adult path was 116km.



Next, I decided to figure out “as the crow flies” distance. I realized that this distance was the hypotenuse of yet another triangle formed by Sentry Point 1, Adult’s village and a point 15km due south of Hope Lake. I knew the lengths of two of the sides were 45km and 71km. I used Pythagorean theorem (see above) to figure out that the hypotenuse was 84km.





Finally, I needed to figure out the distance in the around-the-rim route. I found a formula online that was based on angles that was like this:

I needed to figure out the angle of the two lines between Kids village and Sentry Point 1 and Kids village and Sentry Point 2. I decided to use a protractor to measure the angle after measuring my lines carefully. Here’s what my diagram looked like:

With the protractor I measured the angle at 45 degrees. Plugging the numbers into the formula. I got:

200km (the diameter) * 3.14159 * 45/360 = 78.5km or 79

I then added the extra 15km south to the Adult’s Village from Sentry Point 2 to 79 and got a total of 94km. The resources I used to solve the problem were my Mac, keynote (for diagrams), the Internet (for formulas) and my Mom.



My solution is that the longest route is the usual adult route at 116km, the next longest is the rim route at 94km, and the shortest was the “as the crow flies” at 84km. I think my answers are correct. I do not think there are other answers. It makes sense that the shortest route would be the most direct (a line between 2 points).

This problem was hard because it took a while to discover the triangles that could lead to the answer. I also wasn’t sure using the protractor was the only way to get the arc length, but it was the only way I could figure out. Because I used a protractor, my answer might be less precise than if I had just used a formula. I think I managed my time poorly, but it was all I could do as I had a lot to do. 

Willowside POW # 8: Join The Bacterial Team

"Justin decided that the only way to get out of Mr. Bott’s remedial math class and into Mr. LeFour’s algebra class was to join the baseball team. The only problem was that he didn’t have a clue how to play. For his plan to work, Justin would have to impress Mr. LeFour with his skills. Unfortunately it wasn’t Justin’s math skills that Mr. LeFour would notice...."


Installment # 8 of the ongoing mathematical saga of Willowside Middle School.

http://willowsidesaga.blogspot.com/

New Middle School Problem of the Week Saga

The Willowside Saga
is the story of Justin, a new kid at Willowside Middle School in the middle of nowhere. Justin used to live in New York, but for mysterious reasons, his parents moved across the country to a tiny, rural area surrounded by fruit trees and weeds.

Justin is desperate to figure out how to fit in.

There is a wide cast of characters in this saga. Each chapter ends with a significant math problem designed to challenge and extend a middle schoolers mathematical thinking. It is part of a bigger program of problem solving and writing to "make our thinking visible".

The Willowside Saga is the precursor to Verdania: A Mathematical Odyssey.

Problem of the Week: Why Students Need to Write and Think

Why Problem of the Week assignment ?

The POW’s are a central part of a strong math program for middle schoolers. It requires them to tackle problem solving with an investigative or non-routine type of problem.

People too often do math from a procedural approach. Schools too often teach a procedure, with or without justification, and then have the students practice these procedures on very similar problems. While this has a place, it reminds be of learning how to read words and phrases, but never delve into the meaning of a book.

The essay format I teach is designed with specific ideas in mind. 


First, read and understand the problem, and show this understanding by stating the key info, discarding the irrelevant, and identifying what type of problem it is.

The second paragraph explains approach to the problem. What worked and what led astray. It is not always about getting it “right” the first time, but rather, the investigation of methods and the recognition that we often don’t do things in isolation, but search for help.

The third paragraph describes the answer and lays out alternative answers or multiple answers, as many problems are not single answer events.

The last paragraph is a reflection of the learning process with a small eye on the future (behavioral adjustment to improve learning).

The maximum size of the essay is one page, perhaps 500 words or so. Most paragraphs can be 5-7 sentences.

Value process much more than procedure in math. See the importance in the "why" of something we do and understanding it vs. the sole focus how of doing something and perhaps not understanding as well.


Recognize and promote the value of "spaced practice" in all academic endeavors, math included.

Families and students come to appreciate the impact of writing structured essays explaining logical thinking on their everyday approach to other subject areas. The approach is unusual for many  who had a different educational experience.

Give it a try. Make Thinking Visible. 

8th Grade Problem of the Week: Chapter 9: Harvesting

       Teena and Oto, the leaders of a very young tribe of tree dwellers, proved to be excellent hosts. They had saved Jake from some life threatening attack (which he still did not understand) and provided food and shelter for his Aunt and the other shipwrecked passengers once they had calmed down and understood that Jake had not been kidnapped in reality. Justin was instrumental in convincing the Captain, as well has his mother, that they were not in any danger. It wasn’t that he knew this for a fact. It was more of an intuitive feeling. Something in the serene and joyful voice of Teena inspired confidence. He felt safe.

For her part, Teena seemed to be drawn to Justin and Jake as well. She showed them around the village, which was built inside a grove of huge mango trees. The villagers lived in huts built high in the branches and accessed by rope ladders hanging down during the day and drawn up at night. Between the huts was a system of rope bridges that connected them together, resembling a horizontal spider web of sorts. Justin loved walking back and forth across the bridges. The mango laden branches hung down as if offering up their fruit to the villagers. The villagers harvested the mangoes and placed the extra ones in a cart at the edge of the grove.

Although the Captain could sense there was little immediate danger, she could not ignore a nagging feeling of discomfort at the way Teena ignored her, the obvious leader of this group, in favor of Justin and her nephew, Jake. She found it odd that the oldest people in this tribe were in their early 20’s. They all seemed quite competent and even skillful in the way they took care of their village, but it was strange none the less.

The Captain tried to catch Teena’s attention several times to ask where the adults were. She had told Jake that she and Oto were the leaders of the tribe. But where were the parents, the younger kids and the grandparents? She couldn’t get a straight answer from Teena. Finally, she confided in Cynthia, Justin’s mother, about her worries that some terrible information was being withheld from them.

“You know something? I’ve been wondering the same,” said Cynthia quietly one day. “It’s like they are orphans or something. They are polite to me, to be sure, but they rarely talk to me about anything real. It’s almost as if they look straight through me when I am with them. I feel awkward asking them again.”

      “I’ve been thinking, why don’t we ask Justin and Jake? They seem to have an easier time talking with Teena.”

The boys were confused by the Captain’s request when she spoke with them later that day. They hadn’t noticed that Teena and Oto were basically ignoring the adults in favor of them. But now that they thought about it, they also began wondering about the adult villagers. They asked Teena about them.

“Oh, Adults are far, far away from here.”

“Why?”

“Why? Because that is the way of things. Adults live on the Rim and Youngsters live in the Grove. “

“Don’t you miss them?” asked Justin.

“Miss them? Why? We send them mangoes each month.”

Jake and Justin looked at each other.

“Adults are very busy people. We send them mangoes to help out and let them know we are ok.”

“How far away is the Rim?”

“It’s about 100 kilometers away. Our cart hold exactly 100 mangoes. When the Messengers are sent off with the mangoes, they have strict orders not to eat more than one mango per kilometer between the two of them. If they ate any less, though, they would die.” 

       Justin did a quick mental calculation and blurted, “Don’t they end up eating all the mangoes before arriving?”

“Oh, That is why we wait until we have 300 mangoes ready. We’ve developed a system of caches, sort of like holding bins, to leave mangoes along the way.  The Messengers go back and forth between the Grove and the different drop off points. At some point, they have collected all the mangoes they can at a certain drop off point and they set off to the Rim with their final load.”

“I’m confused,” said Jake. “How do the Messengers get back? 

Teena looked at him placidly, “Come back? Why would they do that?”

Your Task:  

Determine the maximum number of mangoes the Youngsters can get to the Adults on the Rim?

Chapter 6: Where are the traffic jams when you need them?

Where are the traffic jams when you need them?

“Here we go again” thought a morose Justin as he stepped out of his mother’s car onto the curb. Winter break was over, his trip back to New York a mere memory and his dread of stepping back into the Willowside social scene was almost too much to bear. He was still in Mr. Bott’s horrible math class (“We think it is best for you right now,” the school counselor had explained distractedly) and he hadn’t quite figured out how to fit in with the flannel shirt and baseball cap crowd that dominated this school. Talking about this to his parents was a lost cause: his father was rarely home anymore and his mother was either working or on the phone as she plopped down the next microwaved dinner for him in the evenings.

He looked around at the cliques of kids milling around the entrance of the school, excitedly jabbering about their winter breaks, their new jackets or other nonsensical mush. Most of them probably just stayed around the area during the break; hanging out at home, playing video games or calling each other on the phone. Few, if any, would have left the area, much less cross the country, as he had. Once again he felt out of place.

Justin saw Tyrone, Juli and Wendy over by Mrs. E’s classroom. He walked over towards them just as the bell rang. They were all in Mrs. E’s English class together. They waited for him to catch up, quickly said hi and then tucked in the door. Mrs. E, in her customary way, was not there yet. She always came in late, with stacks of papers under her arms and a huge, airy smile on her face. She was, in many ways, the opposite to Mr. Bott’s militancy. But that did not necessarily make her class any easier on Justin. He never knew what to expect from such a loosely organized teacher as Mrs. E. But even if she did confuse him, she never offended, which was a major relief.

“How was New York?” asked Juli as they sat down. 

“Great, snowy, but great...” Just then Mrs. E breezed into the room, laid down her stacks of papers, smiled broadly at the class, sighed, and said “Isn’t it great to be back, you guys.” General, good nature groaning filled the room. 

“I’m glad you’re back, Justin,” Juli whispered secretly. Justin smiled shyly.

“Welcome back to a bright new year,” continued Mrs. E. “I am so excited. Every year is a gift, you know. Remember that. Now, we have a new student, in case you hadn’t notice. Her name is Cassidy and she comes to us from New York City, can you believe that?” 

Justin’s felt himself jerk to attention. Did she just say New York City? He looked over to see that, indeed, a new girl was sitting on the other side of the classroom. She had long, shiny black hair and a fancy pair of lavender tinged glasses. She looked calm, poised and above all, sophisticated. 

“I think it is so wonderful to get new students this time of year. It is a gift, you know. Remember that” Everything was a gift to Mrs. E. But this time, Justin agreed with her.

“Let’s do an activity to help us reconnect and bond with our peers, ok? After all, we are social animals. We need our peers and they need us. Cooperation is a gift. Remember that. Let’s work on that. I have this game called Traffic Jam. It’s played first in groups of 4, then groups of 6, then 8 and so on. There are five stepping stone in a row. On the two left-hand stones, facing the center, stand two people. Justin and Juli, why don’t you be those two people?” 

Justin knew Mrs. E would pick him. She always did for some reason. He and Juli walked up to the front of the class.

“On the two right-handed stones stand two other people. Let’s have Doni and Cassidy stand there, ok?” 

Suddenly Justin was much more interested in the outcome of this game. 

“The center stone is not occupied. The point of this game is that everyone on the left side must somehow move to the right side, and vice versa, with the center stone left unoccupied in the end. Here are some basic rules: after each move, each person must be standing on a stone. If you start on the left, you can only move towards the right. You you’re on the right, you can only move towards the left. You can leap frog a person if the stone on the other side is empty. You cannot jump over more than one person at a time. And finally, only one person may move at a time.”

“The winning team will show us how to change positions in the least number of moves. Then we’ll try it with 6 people and 7 stones (always the center stone unoccupied). This will be so much fun! Let’s start!”

Justin wasn’t sure how much fun the game was going to be, but he was sure interested in knowing more about Cassidy.

Your task: what are the minimum moves for 4, 6, 8 and 10 people games? Is there a pattern that helps?