Teaching Math with Technology
Intro: Today we are going to explore how one concept is built upon through the high
school grades, what the standards are, and how technology can be used as part of a
lesson to enhance learning and engage students.
First, we need to travel back in time to…HIGH SCHOOL. We begin in ninth grade as
students in Algebra 1.
Aside: Did you know that the math SOLs were just revamped? In February of this
year, the new standards were released, but they do not have to be implemented in
the classroom until the 2011-2012 school year.
9th grade:
Read the SOL
Students brainstorm what they know about functions (pairs)
Class creates concept map on board (whole group)
Make sure to note these properties of functions:
 Domain
 Range (Codomain)
 a mathematical relation such that each element of a given set (the
domain of the function) is associated with an element of another set
(the range of the function)
 Representations: algebraic, graphical, tables
 Vertical line test
Function Machine activity (pairs or whole group)
Can you figure out the pattern?
How could you state the function in terms of x?
10th grade:
Read SOL
Combine two functions by substituting one function's formula in place of each x in the
other function's formula. The composition of functions f and g is written f ° g, and is
read aloud "f composed with g." The formula for f ° g is written (f ° g)(x). This is read
aloud "f composed with g of x."
Note: Composition is not commutative. That is, (f ° g)(x) is usually different from (g °
f)(x). The example below illustrates this.
Example:
(f ° g)(x)
f(x) = 3x^2 + 12x – 1 and g(x) = 4x + 1
= 3(4x + 1)^2 + 12(4x + 1) – 1
= 3(16x^2 + 8x + 1) + 48x + 12 – 1
= 48x2 + 72x + 14
(g ° f)(x)
= 4(3x2 + 12x – 1) + 1
= 12x2 + 48x – 4 + 1
= 12x2 + 48x – 3
11th grade:
Read the SOL
You make a purchase at a local hardware store, but what you've bought is too big to
take home in your car. For a small fee, you arrange to have the hardware store
deliver your purchase for you. You pay for your purchase, plus the sales taxes, plus
the fee. The taxes are 7.5% and the fee is $20.
(i) Write a function t(x) for the total, after taxes, on the purchase amount x. Write
another function f(x) for the total, including the delivery fee, on the purchase amount
x.
(ii) Calculate and interpret ( f o t)(x) and (t o f )(x). Which results in a lower cost to
you?
(iii) Suppose taxes, by law, are not to be charged on delivery fees. Which composite
function must then be used? Graph this on your calculator.
(iv) Use this function to find out how much you would pay for a $47.50 order.
This sort of calculation actually comes up in "real life", and is used for
programming the cash registers. And this is why there is a separate button on
the register for delivery fees and why they're not rung up as just another
purchase.
(i) The taxes are 7.5%, so the tax function is given by t(x) = 1.075x
The delivery fee is fixed, so the purchase amount is irrelevant. The fee
function is given by f (x) = x + 20
(ii) Composing, I get this:
( f o t)(x) = f (t(x)) = f (1.075x) = 1.075x + 20
(t o f )(x) = t( f (x)) = t(x + 20) = 1.075(x + 20)
= 1.075x + 21.50
Then I would pay more using (t o f )(x), because I would be paying taxes (the
t(x) formula) on the delivery fee (the "+20" in the f (x) formula). I would prefer
that the delivery fee be tacked on after the taxes, because ( f o t)(x) results in
a lower cost to me.
(iii) If the state is not allowed to collect taxes on delivery fees, then: The
function to use is ( f o t)(x).
(iv) ( f o t)(47.50) = f (t(47.50)) = f (1.075(47.50)) = 1.075(47.50) + 20 = $71.06
12th grade:
Read the SOL (this is still the 2001 version)
When a function is inside another function, use the CHAIN RULE to find the
derivative. So, the chain rule can be used to take the derivative of a composite
function.
As we watch the first three segments of the video tutorial, copy examples and
definitions into your notes.
What are the three steps for the chain rule?
Formal Definition: (f o g)’ = f’(g(x))g’(x)
Universal Design for Learning (UDL)
Brain Networks and Learning Types
• Recognition - Basic knowledge-being able to recall facts; basic
understanding of curriculum content and the ability to recognize symbols
or ideas.
• Strategic - Taking information and then either analyzing or evaluating it.
More analysis, procedural, decision making and problem solving
processes rather than just recalling information.
• Affective - Trying to draw students in to the subject and making them
care about what they are learning and become engaged with the
material
Discussion of other Resources
Math Movies
This is an online module which I created for one of my education classes. Movies can
be used in many different ways in the classroom, from teacher-created resources to
student-created presentations. Feel free to explore the many resources and sites
linked off of the module.
Virtual Math Manipulatives
The Virtual Math Manipulatives are online math activities for each grade level from
the National Library of Virtual Manipulatives. Teachers can click on a topic or on a
grade level and get a list of activities appropriate for their class. There is a complex
index of activities on the site, as well as individual pages for each grade and area of
mathematics. The site covers activities for number and operations, algebra,
geometry, measurement, and data analysis and probability. Grade levels are broken
down into Pre-K-2, 3-5, 6-8, and 9-12.
Teachers can use the Virtual Math Manipulatives as enrichment activities, a fun and
educational way to take a break, something to link to from their home page, or
research tools. The site includes classic math and logic games such as the Tower of
Hanoi, as well as ways to visualize geometric and algebraic concepts. Many activities
are suitable for multiple grade levels or for combination with other classroom
instruction. The site lends itself well as a tool for math laboratory exercises.
WebQuests
A WebQuest is an inquiry-oriented lesson format in which most or all the information
that learners work with comes from the web. The model was developed by Bernie
Dodge at San Diego State University in February, 1995 and has been embraced by
thousands of educators from around the world who create and share their own
WebQuests. While individual WebQuests vary widely, they typically include the
following sections:
The Introduction - this section sets the stage for the activity - often tying to a
real-world situation
The Task - includes the specific purpose and objectives for the WebQuest
The Process - step-by-step process to achieve the task - often divided into
groups where students have specific roles in helping the group achieve the task
The Resources - a collection of carefully selected, and often categorized, links
to Web-based resources that provide students with the necessary information/data to
complete the task
The Conclusion - typically some type of performance-based activity that
challenges students to synthesize what they've learned to demonstrate their
understanding of the objectives outlined in the task
The WebQuest model effectively scaffolds inquiry-oriented activities for learners of
various ages and in various content areas. They tend to focus on critical thinking,
synthesis and analysis, and often challenge students to work collaboratively,
leveraging social constructivist principles. The quality, structure, and emphasis of
WebQuests found online can vary quite a bit. WebQuests range from focused activities
that may be completed in one or two class periods to large-scale projects that may
take several weeks to complete. To use a WebQuest "as-is" in the classroom may be
fairly straight-forward and quick to prepare for. Most often, however, teachers
modify the activity to meet the needs of their students and the constraints of their
classroom. Sometimes you can modify an existing WebQuest simply by adjusting the
procedures and our resources in a handout for your students. More often, though,
teachers will develop a modified WebQuest build on one developed by another
teacher (citing the original creator of course) and post their own version on the Web
(perhaps using a tool like Wikispaces). Teachers may also choose to create their own
original WebQuest from scratch. While this approach will certainly take more time,
this is the most direct way to ensure that the content and process skills align with
your curriculum.
Discussion of other classroom technologies
Document Cameras
Interwrite Tablets
Smartboards
Conclusion: 21st century students are diverse learners who require differentiated
instruction. We should use a variety of instructional methods to benefit the most
students, while still teaching the necessary state standards.