Page 1
AMT #1
Name
Date
Meeting Day
Please save file as first name last initial AMT1 (ex. CalSAMT1.doc)
Part 1
This is the first Analysis of Mathematical Thinking (AMT) of the semester. These
activities and assignments will focus on important mathematical ideas that underlie the
elementary school mathematics curriculum. This one will focus on the concept of
division.
For Part 1 of this AMT, please respond in writing to the following questions.
(a) How do you divide two numbers? Use specific examples to show algorithms1 for
dividing whole numbers, fractions, and decimals.
Why would you divide two numbers? That is, what does it mean to divide two numbers?
What does the answer to this division tell you? In what circumstances might you want to
divide two numbers?
1
An algorithm is a step-by-step procedure, or recipe, for accomplishing a specific task. The word
“algorithm” comes from the name of one of the most important mathematicians, Abu Ja'far
Muhammad ibn Musa al-Khwarizmi, who lived in what is now Iraq around 800 C.E. The title of
one of his most important books, Hisab al-jabr w'al-muqabala, is the source of our word for the
topic of that book, “algebra.”
AMT #1
Page 2
Name
Date
Meeting Day
Please save file as first name last initial AMT1 (ex. CalSAMT1.doc)
Part 2
In Part 1 of this AMT, you were asked to consider how you would divide two numbers,
and why you would perform this operation. In class, we have watched a video tape2 on
which 9 different students share their strategies for finding the answer to this wholenumber division problem: 159 ÷ 13. In Part 2, you will be analyzing some of these
strategies.
(a) Two of the strategies we watched on the tape are shown below. Explain how each
of these strategies works, and why each works.
Strategy #2: 12 r 3
Strategy #1:
10 10 10 10 10 10 10 10 10 10 10 10 10
1 1 1 1 1 1 1 1 1 1 1 1 1
1
1
1
1
1
1
1
1
1
1
1
1
1
130
13
13
156
12 r 3
2
Schifter, D., Bastable, V. & Russell, S. J. (1999). Developing Mathematical Ideas, Part I:
Building a System of Tens (Segment 6). Parsippany, NJ: Dale Seymour Publications.
AMT #1
Page 3
Name
Date
Meeting Day
Please save file as first name last initial AMT1 (ex. CalSAMT1.doc)
(b) Two more students’ strategies are shown below. Explain how each of these
strategies works, and why each works.
Strategy #3:
159
– 13
146
– 13
133
– 13
120
– 13
107
– 13
94
– 13
81
Strategy #4:
81
– 13
68
– 13
55
– 13
42
– 13
29
– 13
16
– 13
3
12 r 3
13
26
39
52
12 r 3
104
156
(c) Use as many of these 4 student strategies as you can to find the answer to the
3 1
following division problem: 1 
4 2