Congratulations Anne!

The Center for Mathematics Achievement would like to offer our congratulations to our director, Dr. Anne Collins, on being awarded the Association of Mathematics Teacher Educators (AMTE) 2013 Nadine Bezuk Award for Excellence in Leadership and Service in Mathematics Teacher Education!

From left to right: Dr. Nadine Bezuk, Dr. Anne Collins, President of AMTE Dr. Marilyn Strutchens, and Awards Committee Chair Dr. Doug Corey.

We will be posting a video of Anne's presentation soon.  Congratulations Anne!

Developing Number Sense

As our students enter middle school and beyond, too often they lose opportunities to continue developing their number sense. Students at this age tend to appreciate more and more the beauty behind numbers, our number system, and the patterns and relationships among them. For this month’s blog I am resurrecting some fun number questions from the Mathematical Digest, Term 1, 1994, Number 105. This mathematical digest has a wealth of information and challenges to be solved. Enjoy!

Anne

Match the clues with the numbers in the box.

Clues

1. An odd cube
2. The first prime 
3. The fourth triangular number
4. Srinivasa Ramanujan* said that this number was equal to (9²  + 19² ÷ 22)²⁵
5. The second perfect number
6. The smallest odd abundant number
7. 6! + 5! + 4! + 3! + 2! + 1!
8. The first number after 1 to be both a square and a triangular number
9. The ninth highly composite number
10. A three digit palindromic square number
11. G. H. Hardy’s taxi cab’s number 

There are twelve numbers in the box for the eleven clues.  Which number does not have a clue written for it?  This number is featured in a very well-known book written in 1726. What is the name of the book? 

*Srinivasa Ramanujan (1887-1920) has been described as the greatest mathematician India has produced in the last 1000 years. His work has only just started to be appreciated and understood. His formulae are being used in areas such as polymer chemistry, statistical mechanics, computers and even cancer research.

Happy New Year!

Lesley's Center for Mathematics Achievement has some exciting offerings in the upcoming year.  These include monthly Saturday workshops, a Dine and Discuss focusing on the CCSS and PARCC, graduate level math courses in Brockton, Quincy, and Springfield, collaborations with UEI and MoS, and a Summer Institute.  We are excited for the upcoming new year and continuing our work with mathematics teachers and education.  We hope that you can join us for some of these events.  If you want more information, you can find all of it at: CMA Homepage!

And to start of the new year...
How many factors does 2013 have?  How many of the factors are prime factors?

Understanding what it means to understand the structure of mathematics (use the language of the mathematical practices)

Implementing the recently adopted Common Core State Standards requires looking at the vertical progressions of important concepts across the grades. One of these concepts is how arithmetic is generalized in algebra. The algorithms we emphasize in arithmetic may ease the transition to algebraic understanding, or thwart that transition. The Common Core talks about the "traditional" algorithm but not until students have opportunities and experiences to deeply understand the foundational concepts.

Think, for instance, about the addition of the following two-digit numbers:
27 + 33

The way in which the computation is presented may make a difference to students. Some students will see this computation and round 27 up to 30 and 33 down to 30 to get the sum of 60. Others will see this computation written horizontally and need to rewrite it vertically before they can compute:

27

+ 33

If they understand place value they may show partial addends illustrating the following:

27

+ 33

10

+ 50

60

Or show the regrouping method:

1
2	7
+ 3	3
6	0

Of these methods, which do you think best relates to addition in algebra?

Let's think about using place value representation to compute:
 2 tens + 7 + 3 tens + 3.

Written this way it is clear that tens will be combined with tens, and units with units, for a sum of 5 tens + 10. Notice there is no temptation to mix the tens and the units.

Written vertically we get

2 tens	+ 7
+ 3 tens	+ 3
5 tens	+ 10

It makes perfect sense to combine like terms.  This can also be written as
2t + 7 + 3t + 3, which is a standard algebraic representation and would total 5t + 10.

The partial sum best models the foundational method for addition in algebra. Now, this is not to say students should always use the partial sums, but they do help solidify what values are being combined. Once this understanding is mastered then we expect our students to use a more efficient method, especially if we are adding three or four columns and rows.

The same rationalization can be used with multiplication which may also include additional links or connections to the structure of algebra. Consider the same two values but this time think of them as factors:

27 x 33.

Using partial products we get

	2	7		20	+ 7
x	3	3		x 30	+ 3
	2	1		7 x 3 =	21
	6	0		20 x 3 =	60
2	1	0		7 x 30 =	210
6	0	0		20 x 30 =	600
8	9	1			891

Or we could think about decomposing 27 into 30 – 3, and 33 into 30 + 3, which would give partial products of 900 + 90 - 90 – 9, which equals 891.

30	- 3
x 30	+ 3
-3 x 3 =	-9
30 x 3 =	90
-3 x 30 =	-90
30 x 30 =	900
	891

The mathematical structure for this computation is the difference of two squares, a² – b². This is an arithmetic way of showing 27 x 33 = (30 – 3) x (30 + 3) = 30² – 3², which equals 891.

I propose that when we are teaching arithmetic we keep in mind the fact that algebra is the generalization of arithmetic.  What do think?

I look forward to your comments,

Anne

Problem of the Week

A number of children are standing equally spaced around the edge of a large circle and are numbered consecutively.  Abby is 14^th and Emma is 32^nd and are standing directly across from each other.  How many children are there?

In a week, we will post solutions for this problem.  At that time we will also choose one commenter who gave the correct answer to get a copy of Zeroing in on Number and Operations (the commenter who wins can choose from PreK-K, 1-2, 3-4, 5-6, or 7-8).  If you want to be entered into the drawing please comment with your solution, and leave your first name last initial.

We look forward to seeing your responses!

Katie