The Skeleton Tower Task

The Skeleton Tower (A rather Halloweeny name, don't you think?) is a tower built of cubes in a pyramid fashion, with 4 stair steps extending from the center of the tower. In a tower that is 6 cubes high, there are 66 cubes altogether.

A tower 3 cubes high contains 15 cubes.

A tower 4 cubes high contains 28 cubes.

A tower 5 cubes high contains 45 cubes. 

The following table helped me in determining the number of cubes for each level. The highlighted portion was completed using mathematical equations only, without actually counting the cubes in a pyramid. 

Height	Cubes in all	Looking for Patterns
10	190	9 x 4 = 36 +1=37             *10x 19 = 190 153+37=190
9	153	8 x 4= 32 +1 = 33           *9 x 17= 153 33+ 120= 153
8	120	7 x 4=28 +1=29              *8 x 15= 120 91+29= 120
7	91	6 x 4=24 +1=25               *7 x 13= 91 66+25= 91
6	66	66-6=60          66-45=21 (the height x 4 +1) 60/4=15
5	45	45-5=40          45-28=17 (the height x 4 +1) 40/4=10
4	28	28-4=24          28-15=13 (the height x 4 +1) 24/4=6
3	15	15-3=12         15-6= 9 (the height x 4 +1) 12/4=3
2	6	6-2=4 4/4=1

The question now is whether or not there can be an equation to find the number of cubes in the pyramid at N cubes high. 

I noticed the pattern with an asterisk next to it in the table above. Obviously, 6 x 11 is 66. Initially, I thought the multiplication was just counting up odd numbers. Then I realized that 6 x 2 = 12, 7 x 2= 14, 9 x 2= 18, etc. So I wondered if I took N, multiplied it by 2, subtracted 1, then multiplied it by itself....would I get a correct answer? 

That would be [(N x 2)-1] x N.

Let's see if it is true if used for N = 4: 

[(4 x 2) - 1] x 4= [(8)-1] x 4= 7 x 4 = 28 (It works here, yay!) 

Now N= 10

[(10 x 2)-1] x 10= [(20)-1] x 10 = 19 x 10= 190 (Yay again!)

Using the table, I was able to identify patterns that would help formulate an equation to solve the problem. Oddly enough, the pattern that helped the most was just a side note when I noticed that all of the numbers in the not highlighted section were multiples of the height in cubes. 

Chicken Nuggets Task...Otherwise known as 'Now I am Hungry'

McDonald's sells chicken nuggets in boxes of 6, 9, or 20. Obviously one could purchase exactly 15 nuggets by buying a box of 6 and a box of 9. Using only combinations of boxes of 6, 9, and/or 20 nuggets:

Could you purchase exactly 17 nuggets?
     No, because two boxes of 6 would result in 12. Two boxes of 9 would result in 18, which is too much. A box of 6 and a box of 9 is already noted at a total of 15 nuggets.

How would you purchase exactly 53 nuggets?
   

What is the largest number for which it is impossible to purchase exactly that number of nuggets? 

     This question is difficult to explain, as numbers are infinite. I think the most important point here is that the number obviously cannot be prime, as that would eliminate the divisibility of it by 6, 9 and/ or 20. I did notice that continuing from 53 all the way to 100 resulted in some form of divisibility by the three numbers combined. Once I realized this, I went backwards to see if maybe that largest number was below 53. 

52: 2 boxes of 20, 2 boxes of 6

51: 1 box of 9, 7 boxes of 6

50: 1 box of 20, 5 boxes of 6

49: 2 boxes of 20, 1 box of 9

48: 2 boxes of 9, 5 boxes of 6

47: 1 box of 20, 3 boxes of 9

46: 2 boxes of 20, 1 box of 6

45: 5 boxes of 9

44: 1 box of 20, 4 boxes of 6

43: Hm. I am stumped here!  Perhaps the answer to this question is 43?

Let's say you could only buy the nuggets in boxes of 7, 11, and 17. What is the largest number for which it is impossible to purchase exactly that number of nuggets? 

   To figure this out, I looked to see if there was a pattern for the above numbers. It is possible that I am not thinking about this 'mathematically' enough, however I can't seem to determine a formula or pattern to finding 43 as the number above. The best I could come up with is using a hundreds chart, I marked out all numbers divisible by 7 (in black). I kept going with the chart, marking out every 11th, 7th, or 17th number from one that was already eliminated. The highest number I have not marked out is 37. 

In closing, there must be a more mathematically sound method of solving this problem that I have not yet discovered. As it stands, I feel fairly confident that I figured out the solution....albeit in a rather elementary fashion. 

Mathematics Curriculum Notes

   After viewing the following video, I felt that I connected to the thoughts presented by M.J. McDermott. As a fifth grade teacher, it is frustrating to see that students cannot multiply or divide fluently. We have recently worked in multiplication and division in my classroom with very scattered results. Many of my students did not know how to set up a traditional multiplication algorithm. Some knew how to use the ‘cluster math’ depicted in the video, others were familiar with the lattice method (although a precious few actually used it successfully on more than one or two problems at a time), and several were familiar with partial products. By not using the traditional algorithm, my students are not able to master any particular form of multiplying. In addition to this, when student are asked if their answer is reasonable, they cannot seem to grasp the concept of determining reasonability because they are more focused on which method they can use to determine reasonability. When we moved into long division, I saw that students looked for an alternative way to do it. One even attempted to create his own ‘division lattice’. It seems all of these ‘tricks’ have made students less willing to persist in traditional mathematics for mastery of skills.

https://www.youtube.com/watch?v=Tr1qee-bTZI#t=437

   Based on Chapter 9 from the Handbook of Research in Mathematics Education, there is a web of curricula that leads to student understanding. A few of the points in this web are the Teachers’ orientations toward curriculum, Teacher beliefs and knowledge, and Teachers’ professional identity. After reading this, I wondered how a state could have a set list of standards; yet have teachers teaching a completely different form of mathematics. I have a recent personal connection to this. My daughter is in 3^rd grade in NC. Her math homework recently included subtraction problems called “Expanded Subtraction”. Below is an illustration of the method she was instructed to use. Where you see a (c) is the place she was instructed to use a calculator.

   My question for this, is exactly what part of NC or Common Core Math standards does this fall under? Additionally, if my daughter was encouraged to use a calculator for two separate parts of this problem, why would she not logically use a calculator to simply solve 284-95 to begin with? Finally, the interpretation of this subtraction skill was different between two teachers in the same building. While my daughter’s teacher did it this way, another teacher taught it the way you see below. This exemplifies the sections we see in Explanations for Transformations from the Handbook of Research in Mathematics.

   As a home-educated student, I used Saxon Mathematics during the course of my education. Saxon Math was very straightforward, teaching the use and mechanics of all mathematics skills I use today. I cannot say that I particularly enjoyed math as a student, due in part to the literary nature of my childhood, but I did not hate it or feel inadequate using the skills acquired through traditional problem solving.  As a teacher for four years, only one of those years have I had an actual math textbook to utilize in teaching. The experience with it was strenuous at best due to a heavy focus on games, investigating math concepts and unrelated content mixed in throughout the book. Students utilized a very large workbook version of the textbook that included 1-2 problems per page. The problems always seemed either very easy or rather convoluted, neither of which was beneficial to my students.

Stein, Mary Kay, Janine Remillard, and Margaret S. Smith. "Chapter 8: How Curriculum Influences Student Learning." Handbook of Research in Mathematics Education. New York, NY: MacMillan Co, 1992. 319-23. Print.

TED 663 Project Post 1

Over the course of the next few months, I will be posting on various Math topics in relation to a graduate class I am taking.

Project Overview
1. Who's classroom? and 2. Describe your relationship with the teacher.

During the Mathematical Leadership project, I will be working with the fifth grade teammate at my school. To maintain her privacy, here I will refer to her as Nichole. This year I have moved to fifth grade in a much smaller school than I am familiar with. The fifth grade team consists of myself and a first year teacher. Due to the smaller team size and the first year status of my teammate, I am the grade level chair this year. This is a new experience for me, but I look forward to using the opportunity to assist Nichole in implementing meaningful mathematics lessons for her students.

3. What instructional problems (math concepts) do their students have?

As it is the beginning of the year, we have not yet ascertained any particular problems or mathematical concept deficits. Nichole has noticed that her students are fluent in utilizing the lattice method for multiplication. She also noted they have a decent grasp on the traditional algorithm and know most of their math facts. There are a few students who will need remediation in this concept, however. Math concepts to be covered in 5th grade include multiplication (lattice and traditional, as well as using powers of 10), order of operations in equations, patterns in numbers (with a coordinate plane), adding and subtracting unlike fractions, and several more.

4. What types of approaches or instructional modifications might help?

Instructional approaches vary based on the topic of the lesson. In the case of the multiplication standards, teacher directed instruction is utilized for at-level instruction. Students who are not performing at grade level receive independent instruction during I.E. time as well as guided math groups. During IE, the focus is on skills the students are lacking in that may be unrelated to current standards. During guided math, the focus is on the strand or standards being currently taught while instruction is broken down more and/ or increased in cognitive expectations if necessary.

5. How will you measure student growth?

For each unit, students take a pre-assessment to determine their knowledge of the standards. The information is recorded and used in grouping students for guided math. During instruction, formative assessments in the form of exit tickets assist in knowing which students are progressing and who needs additional assistance. At the end of the unit, students take a post assessment to determine final growth and to allow teachers to plan for IE time in any standard that is not yet mastered.

My whirlwind life!

This past year has been crazy busy! Have you ever had one of those days where you completely lost track of time? That was this year for me. I began grad school in January, then got married in March to the sweetest old man.

The newest change is in employment...I have moved to a closer school system and will be teaching 5th grade next year. Although it is a bit nerve wracking to change grades, I am very much ready for the new adventure. In preparation, I have been reading up on word study in the upper grades. I hope to post more on that soon!