This is a professional development blog. We'll be discussing Challenge Math by Ed Zaccaro. Our discussions will be focused on gifted children.

Sunday, June 6, 2010

Session 1 – Question 3

Complete any one of the Einstein Level Problems using the author’s information and share your experience. Site the page number and the question number in your answer.

1. Einstein Level Question #7, page 18: I chose this question b/c I could see using this as a challenge question in my classroom next year. This isn’t really a difficult question, but there are several steps to complete to find the final answer. These are often the types of questions some students would rather give up on since it is not a quick answer. Applying the information that 1 parsec is equivalent to 3.26 light years, you then take the 50 years and multiply them by 3.26 light years to find a product of 163 light years. Then, you must subtract the year 1998 and the 163 light years to find a difference of 1835 which is the year the explosion would have occurred.

2. I chose the Expert Level Einstein problem #6 on page 17. I chose this problem because it goes perfectly with a book that I use in teaching mathematics to my students. The book is called The Librarian Who Measured the Earth, by Kathryn Lasky. The book deals with Eratosthenes, and in very clear language, and beautiful illustrations, demonstrates how he found the circumference of the earth, and how geography and mathematics go hand in hand. The Einstein problem in the Challenge Math book is straight from this children's book, and it is a multi-step process which includes division and multiplication. Once a students figures that out, the rest of the problem is not too difficult. In any case, I strongly recommend using the book, The Librarian Who Measured the Earth, to introduce the Einstein problem. It's a great marriage of the two, and the students respond well.

3. I choose Einstein level question number 1 on page 53. I choose this one because it uses a strategy that our children have the hardest time using- guess and check. Using my prior knowledge of what I first guessed I realized that it was too low and so I increased my number. This is often what the students will see. Also, I enjoyed that they never gave any of the ages to start off with so you had to read and understand in the application level of math to understand the ages of each person.

4. Pg 51 #3: Find Jim’s age if D=3J, J is half the age of Bill, and the average is 64. This question was a good test of my understanding of averages and interpretation of the problem. The algebraic manipulation of the equations was just plain fun. It took me several tries to make sense of it the equations because it was a little cumbersome. I think that students who enjoy solving algebraic problems would love this section.

5. I chose problem #1 on p.82 to solve. I did try some other problems that I thought would be fun and definitely made me intellectually frustrated, but due to the time of night I wasn't bale to comprehend them, so I am bound and determined to go back and solve them b/c they were guess and check type problems. So instead I chose a problem that I had to work backwards to find the answer, which is a strategy that I find students don't enjoy doing. I don't mind it. On the 5th day Emily would need to make \$371.05. in order to have an average of \$248.95.

6. Session 1-Question 3
I worked on page #51, Chapter 3:Algebra, Einstein Level Question #6. I think this is a great problem to challenge my 6th grade GT students next year. Wording is a little bit confusing when you read it first which I think is a good example of intellectual frustration. It promotes deeper analysis of the problem. Since the total of children’s ages is 46 and they were each given an amount of money equal to the product of half of their age and their grandfather’s age; it really does not matter what their ages are, when you add, total amount of money is 23g where g is the grandfather’s age. I think this could be an aha moment for students who use guess and check method and then analyze the given information deeper. Knowing that 23g=1656, g is 72.

7. I chose problem #5 on page 31 because my students have the hardest time given a large set of data and aren't sure how to organize it. I think students get overwhelmed at first and forget to just break it down piece by piece. I was impressed how easy it was to sort the information into different parts of overlapping venn diagrams. The total number of students in the group is 57. I like all the thinking and reasoning that you had to go through- this problem would generate a lot of discussion and thinking- can't wait to turn them loose with it.

8. Fun , fun, fun! I love these types of questions. Since both Bill and Dave's ages were expressed relationally to Jim's, I let Jim=J. Since Jim is half of Bill's age, then Bill=2J. Dave is 3 times older than Jim, or 3J. If the average of the three ages is 64, then the sum of the three ages would be 64 x 3 (number of ages being added) which equals 192. So, J + 2J + 3J=192. Solving for J gives you 32, then Bill would be 64 and Dave would be 96. Even though I wouldn't necessarily expect my students to solve this way, we do many questions like this one and I always model writing the equations to show the relationships within the problem. By the end of the year I do have many students that are able to show mathematical relationships using these algebraic number sentences, even if their solution process wouldn't perhaps be as algebraic as mine!

9. I chose problem # 5 on p. 51 because there is a similar problem in Exemplars that I have used. We have used Guess and Check in class but I wanted to try the strategy provided in the book.
The 23 cents was confusing so I set that to the side since the amount of pennies was not part of the algebraic equation and that did turn out to be the total number of pennies.
Then I converted all the coins to n, 2n and 6n and made another list of their value as 5 n, 20 n and 150 n.
I divided the \$7.00 that was left by the 175 total value and it gives you 4 as the value of n or the total of 4 nickels. Figuring out the rest was easy mathematically.
It helped to have a sample to go by. I picked a problem where I could input different numbers and values that matched the sample.
I'm sure this is a strategy students would try before venturing into
more difficult problems.
It helps if you are successful with a problem before trying a more difficult one.
I will introduce this prior to the Exemplars problem this year as a choice in problem solving. It will be interesting to see how many will try this avenue.

10. I chose the Expert Level Einstein problem #1 on page 82. The first step was to draw a Rectangular box separated into 5th, each 5th representing a week. The entire rectangle should equal 1214.75 but the 5th week is missing so we have to work backward to determine the missing number. Weeks 1-4 equal 873.70 which means we still need 371.05!!

11. In response to SusanM

Expert Level Einstein problem #3 on page 51 is a blast--I often present these as mystery number problems and put them in workstations for the students to problem solve together. The academic conversations that follow between the students are amazing!!

12. In response to nlopez--It may be an easy problem but it is really big on the interest scale. Almost all students would find this problem exciting because they are dealing with PARSECS!!! The kids love anything related to space- especially things that their parents don't already know!!

13. I chose problem #1, on page 25 (Think 1) and all I can say is I love this strategy and I wish I knew it when I was in school!!!! I am not going to lie, I read about how to solve this way, but once I saw one example, it all became so clear. Oh and it would take 6 days with two added people.

14. In response to Katie K, "Think 1" is a highly effective strategy. I think students do not make use of it enough during elementary school. Making a problem simpler is a good strategy to use when the problem seems too complicated to comprehend. Zaccaro has a good way of explaining it too.

15. Session 1 – Question 3
Complete any one of the Einstein Level Problems using the author’s information and share your experience. Site the page number and the question number in your answer.

Einstein Level – page 82, problem 1:

Emily wanted her average salary for five weeks to be \$248.95. She has already been paid the following amounts: Week 1: \$182.40; Week 2: \$621.00; Week 3: \$52.10; Week 4: \$18.20; How much money must Emily earn during her 5th week of work to average \$248.95 for her five weeks of work?

To earn an average of \$248.95 for five weeks of work, Emily must earn a total of \$1244.75. \$1244.75 ÷ 5 = \$248.95. Emily must earn \$307.80 for her 5th week of work.

16. Oops! The amount needed is \$371.05.

17. Problem #4 on page 51. I chose this problem because you can show students how to substitute variables. It is a nice, challenging problem. I was new to teaching 4th grade math this year (I previously taught 1st grade) so I did not use variables a lot. I plan on using them a lot more next year because they just make sense to a lot of students. It will definitely fall into my “differentiation toolbox.” This problem was a great example of the three steps on pg. 43- collecting, isolating n, and just one n. I started with (N+S+L+R)/4=\$150, then I used the correct values from the problem (3S+S+4S+120)/4=\$150. Nick has \$180, Stacey \$60, Lindsey \$240, and Rick \$120. I actually really enjoy solving equations.

18. In response to Katie K. and A Dunlap-
I love the Think 1 strategy. I plan on using that more next year with all students. I feel that it will allow students to break down even the most complex problem to the simplest form. It kind of reminds me of the old first grade benchmark question about how many eyes do 10 kids have and the kiddos were drawing faces with multiple eyes on them instead of Thinking about 1!

19. I chose Chapter 1: Einstein Level problem #2 on page 17 about thunder and lightning. I was able to calculate that the storm was 1.5 miles away from the north and then 2.5 miles away from the south. I solved the problem incorrectly at first because I did not convert the answer to miles per hour. After dividing the total miles by .2 of an hour I was able to come up with an answer of 20 miles per hour. I know my GT students would love this type of problem because it deals with something that is real to them.

20. In response to PKassir posted on June 7. I would love to use the book, the Librarian Who Measured the Earth. Using the book would be a great way to integrate math, science, and literature. My fifth graders seem to really love the astronomy unit that we teach in science and the astronomy problems from this book fit right in.

21. I solved Einstein problem #1 on page 103. First I calculated the area of Shannon’s plywood by multiplying the length (8 ¼ feet) times the width (4 ½ feet). I decided to convert the mixed-numbers into improper fractions for this calculation, but I double-checked my work by multiplying the dimensions in decimal their decimal form, too. The plywood’s area is 297/8 ft² or 37.125 ft². Next I found the area of the circle that Shannon cut out of her plywood. Since the diameter is 3 ft, I knew the circle’s area was 7.065 ft². I decided to work the circle’s area in decimal form. I had to square half of the diameter, or 1.5 ft, and then multiply it by Π, or 3.14 ft. I subtracted the circle’s area from the plywood’s area and found a difference of 30.06 ft². Since Shannon was selling this left over piece for \$1.75 ft², to complete the problem I multiplied 1.75 times 30.06 and got the product 52.605, which I rounded to \$52.61. There were many steps to this problem. I had convert fractions into decimals, multiply decimals, know the area of a rectangle (without it being referred to as a rectangle), know the area of a circle and how to square a radius, know that I must divide the diameter by two to get the radius, know to subtract the areas, multiply two numbers with decimals to the thousandths, and round! This would be extremely challenging for most of my children because so many of the skills haven’t been learned yet, but all steps to its solution could absolutely be completed once explained. Some of my children would love to solve this problem! I loved solving this problem!

22. In response to NDeans post on June 10th, that thunder and lightning problem is great. I agree that real life problems spark interest especially those related to science, space, nature. I can see my GT students having good group discussion while working on that problem.

23. In response to ReneeR on June 10th, I loved the way you solved this problem: Singapore Math type illustration of the whole and its parts. We were at a training for Singapore Math and it is amazing how you can illustrate and solve even complex algebra problems easily. Author also used Singapore Math method on page 21 and 22 for a fraction problem.

24. In response to Katie K, I m with you!! I wish I would of also known this strategy of Think1 when I was at school. It indeed makes very confusing problems much easieer to solve. Love it!

25. I solved Einstein problem #8 of page 23 because I can present it to my GT students after teaching fractions. The strategy of page 21 & 22, using a rectangle and marking/labeling it with the fractional parts used, makes the problem much easier to solve.

26. I found the questions interesting and didn't want to pick just one:

pg 17 (#2): I tried to work it the way I would as a student and found myself drawing a picture to organize the information. I liked the fact that this was a multi-step problem. Students had to convert the times into distances, calculate the total distances, and then convert this into miles per hour.

pg. 82 (#1) It's hard for me to imagine doing this any other way than a straightforward equation. I like the way the problem incorporated multiplication, addition, and subtraction of decimals. It's a standard algebra problem, but I would be curious to see how younger students would attack this one.

pg. 82 (#6): This was my favorite problem. It requires quite a bit of prerequisite knowledge (understanding of isosceles triangles, how to label the true height of the triangle, definition of line of symmetry, etc. I think this would be a great question where a student would have to go look up some information in order to solve it--never hurts to have some independent research. :)

27. In Chapter 2 on page 36 I choose number 10 the total number of squares of any size. I saw this as a patter problem. Knowing that there is only1 8x8 I then saw that they would be 4 7x7 and 9 6x6 I realized I was adding the next odd number so when I got to the 1x1 yes I had 64 of them .
So adding up the number of squares I added 1+ 4 + 9 + 16 + 25 + 36 + 49 + 64 = 204