Chapter 6 – Technology

We are living in a time when technology is changing almost faster than we can adapt to it or afford it. New products and upgrades are marketed at a rapid pace. The level of complexity associated with new technology, including artificial intelligence (AI) applications like ChatGPT is so great that an application is not fully implemented and absorbed before its next generation is on the market. The third version of the iPad was out while people were still purchasing the iPad 2. The diversity and depth of technology available to us are increasing at a seemingly exponential rate. This is particularly true in mathematics. In the late nineties and early part of this century, many new programmable graphing calculators (Casio, Hewlett Packard, Sharp, Texas Instruments) were introduced. Texas Instruments introduced the TI-89 and TI-84 Plus graphing calculators as well as the TI Navigator, which allows students to contribute work to the learning environment by projecting calculator work in real time for the entire class. These tools were powerful additions in the mathematics classroom. The irony is that most secondary mathematics classrooms did not fully utilize the power of the new handheld tools available to students. Why? Laptops and tablets are readily available in many classrooms and for home use. The COVID-19 pandemic pushed the envelope to support the use of powerful teaching and learning for both teacher and student. With the recent addition of AI applications, how will the mathematics classroom use technology to advance teaching and learning?DOWNLOAD

Exercises

Exercise 6.1

Define “calculator” in today’s society. Give four different examples of calculators. Compare your list to a peer’s. Were you both thinking of calculators in the same way?
Summarize and react to one article dealing with the use of calculators in a secondary mathematics classroom. Include all bibliographic information.

Exercise 6.2

Name a mathematical concept that you feel would be hindered by the use of calculators and one that would be benefited by the use of calculators. Rationalize your position in both instances.
How would you convince a student that memorization of basic fact tables is a convenience?

Exercise 6.3

Devise a set of problems that could be used as a basis to teach the order of operations for addition, subtraction, multiplication, and division on the set of counting numbers. What is the impact of considering the whole numbers, integers, and reals on your discussion?
Create a set of problems that could be used as a basis to teach the role of exponents and parentheses in the order of operations on the set of rational numbers.
Is there any value to using larger numbers for the entries in the problem sets given to students as they are discovering the aspects of order of operations with their calculators or calculator applications? Why or why not?
List the different sub-problems or readiness skills needed to explore
3 + (4 + 5[6 + 7 × 8] + 9) + 10 × 11 + 12.
In part 4 of this exercise set only addition and multiplication were used. Discuss the advantages or disadvantages of avoiding subtraction and division. When should those operations be inserted into the developmental sequence and why? Where do exponents and roots enter the picture and why?

Exercise 6.4

Discuss the advantages or disadvantages of selecting a sequence of exposures that lead students from excess in division being expressed as remainders, then fractions, and finally decimals.

Exercise 6.5

How does a calculator with a y^x key deal with 3⁷ ÷ 3⁵? Describe how this information can be used to help students learn the laws of exponents. Does the calculator deal with 3⁷ ÷ 3⁵ differently from $\frac{3^{7}}{3^{5}}$ ? If it does, what is the impact on its use as an instructional tool when having your students learn about the laws of exponents?
In part 1 of this exercise, describe your response to a student asking about the difference between 4⁷ ÷ 4⁵ and where the base is not a prime number. Is the answer in terms of 4 or 2?
Computers and calculators typically do not have commas or spaces between periods of numbers as we use in “standard” writing. Does dealing with values such as 2,417,851,639,229,258,349,412,352 without the commas create new problems for students? Why or why not?

Exercise 6.6

Suppose you decide to provide an entry in the “missing factor” column of Table 6.1 in the text. Can the entries in the “number” and “sum of digits” columns be uniquely determined? Why or why not?
Suppose you decide to provide an entry in the “sum of digits” column of Table 6.1 in the text. Can the entries in the “number” and “missing factor” column be uniquely determined? Why or why not?
Devise a calculator routine that can be used for discovering the divisibility rule for 2. Establish a table similar to Table 6.1 in the text.
To assist students in discovering the divisibility rule for 4, the third column of Table 6.1 needs to be changed to show the last two digits of the number being considered as divisible by 4. A similar table would be used for 8 except that the last column would show the last three digits of the number being considered. Devise a calculator routine that can be used for discovering the divisibility rule for 4 or for 8. Establish a table similar to Table 6.1 in the text for the one you selected.
Describe a calculator routine you would use to help students “discover” a divisibility rule for 6.

Exercise 6.7

Create a lesson that uses the calculator to discover the Pythagorean theorem.
Describe two unrelated situations in which a calculator would be a useful tool in assisting a student to learn a concept from geometry.

Exercise 6.8

Summarize the contribution to gambling theory made by a mathematician or technology from some decade.
Report on the role mathematicians played in the development of a topic. For example, how did the needs of surveyors impact the development of trigonometry?

Exercise 6.9

Create a lesson dependent on a graphing calculator. Do it on two different graphing calculators. Describe the advantages and disadvantages of each of the selected calculators. Does your bias show in the description?
Create a lesson dependent on dynamic geometry capabilities. Do it on a Desmos, Classpad.net or Geometer’s Sketchpad. Describe your reactions and feelings.
How do you enter a circle centered at the origin with a radius of 2 into your graphing calculator in order to graph it? Describe the limitations, special considerations, and added knowledge the student would need to possess to do this problem.

Exercise 6.10

Summarize and react to one article dealing with the use of computers in a secondary mathematics classroom. Include all bibliographic information.
Summarize and react to one article dealing with the use of technology other than calculators or computers in a secondary mathematics classroom. Include all bibliographic information.
Identify five different software packages for symbolic manipulation/function plotting on a computer. Investigate at least two of these applications. Summarize the capabilities of each piece. Describe any unique features of each product. Compare and contrast the software, selecting the one you feel would be most beneficial in the classroom.
Investigate at least two different selections of dynamic geometry software available today. Summarize the capabilities of each piece. Describe any unique features of each product. Compare and contrast the software, selecting the one you feel would be most beneficial in the classroom.
Use a symbolic manipulator/function plotting software package to find the value of $2^{2^{2}}, 3^{3^{3}}, 4^{4^{4}}, 5^{5^{5}},$ etc. How large a number will the software allow before the answer cannot be determined? Select another piece of software and repeat the activity. Did you get the same results?

Exercise 6.11

Identify five applications for the mathematics classroom that can be installed on a smartphone such as an iPhone or Android device. Rank order these applications for student learning value in the secondary mathematics curriculum. Justify your ranking.

Exercise 6.12

Do you think purchasing a notebook computer as a college requirement is a viable solution to getting computers into the mathematics classroom? Why or why not?
Do you think purchasing tablet computers such as iPads as a college requirement is a viable solution to getting technology into the mathematics classroom? Why or why not?
Should students be required to use the same software (calculator) that is used for demonstration purposes? Why or why not?
Which of the software (calculators) you have used can be used in the mathematics classroom? Are there other pieces of software you should investigate as a potential tool in the classroom? Why or why not?

Exercise 6.13

E-mail a challenging problem to a fellow educator or teacher-to-be. Make sure you request a response. Judge the answer and send back a reply explaining your reasoning.

Exercise 6.14

Find an Internet site that can be used for secondary mathematics. Develop a lesson plan using that site.
Locate the Math Contest at https://themathcontest.com/ . Try the current Problem of the Week. Your goal is to get your name added to the site. What number on the list did you achieve?

Exercise 6.15

Create a 5 – 8 minute video to teach a specific math concept. Show the video to a peer and ask for feedback. Show the video to a student and ask for additional feedback. Use the feedback from your peer and the student to remake the video. Share the video on a site for the world to see. How do you feel about your product?

Problem Solving Challenges

Computational Madness

How many retangles of any size are on an 8 x 8 chessboard?

Figure 6.12

Hint: What about the large outside square?

Answer

Answer/solution: 1296

On a 1 x 1 chessboard, there is only 1 rectangle. On a 2 x 2 chessboard, there are 9 rectangles (4 small 1 x 1 squares, 1 large 2 x 2 square and 4 1 x 2 rectangles)

On a 3 x 3 chessboard, there are 36 rectangles (9 small 1 x 1 squares, 1 large 3 x 3 square, 4 2 x 2 squares, 6 1 x 3 rectangles, 12 1 x 2 rectangles, 4 2 x 3 rectangles).

Notice that the values are perfect squares of triangular numbers.

1 x 1 1st triangular number 1^2 or 1 rectangle

2 x 2 2nd triangular number 3^2 or 9 rectangles

3 x 3 3rd triangular number 6^2 or 36 rectangles

4 x 4 4th triangular number 10^2 or 100 rectangles

5 x 5 5th triangular number 15^2 or 225 rectangles

6 x 6 6th triangular number 21^2 or 441 rectangles

7 x 7 7th triangular number 28^2 or 784 rectangles

8 x 8 8th triangular number 36^2 or 1296 rectangles

If it takes 10 sheep ten minutes to jump over a fence one at a time, how many sheep could jump over the fence in one hour?

Hint: If 2 it takes 2 minutes for 2 sheep to jump over a fence one at a time, it is not one minute per sheep! The first sheep jumps over at 0 minutes and the second at the 2 minute mark.