MATH 1332 TUTA Math Equation and The Function Questions

Description

Math 1332
Journal 1: Functions and Equations
In this assignment, you will be completing a scripting task. You will be given the beginning of an
interaction between a two students, and your task will be to continue this imaginary conversation.
Below is an example of a scripting task assigned to a student, Amanda. Her scripting task involves
interaction between a teacher and student. This is an excerpt of the script Amanda wrote:
You are given the beginning of an interaction between a teacher and a student and your task is to
extend this imaginary interaction in a form of a dialogue between a teacher and a student (or
several students). You may also wish to explain the setting, that is, the circumstances in which the
particular interaction takes place.
T:
S:
T:
S:
T:
S:
T:
So today we will continue our exploration of how to find an inverse function for a given
function. Consider for example 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥 + 5 Yes, Dina?
So you said yesterday that 𝑓𝑓 −1 stands for an inverse function
This is correct.
1
But we learned that this power (-1) means 1 over, that is 5−1 = , right?
5
Right.
So is this the same symbol or what?
Excerpt from Amanda’s script:
Teacher: Some symbols in math mean different things in different scenarios. [Teacher draws a small
circle on the board.] What does this symbol represent?
Student: It’s a circle…
Teacher: Of course, [Teacher draws a ‘1’ beside the circle.] Now what is it?
Student: Ten. The circle is a zero.
Teacher: Very good. [Teacher erases the ‘1’ and writes a big ‘2’ and ‘5’, using the circle as a decimal
point.] How about now?
Student: It’s 2.5. The circle is a decimal.
Teacher: Excellent. [Teacher erases the ‘2’ and ‘5’ and ‘100’ to the left of the circle and ‘C’ to the
right.] What is the circle now?
Student: 100 degrees. The circle looks like a degree symbol.
Teacher: Correct. So the circle can mean a lot of different things. It just depends on what else is
around it that determines how it is used.
Student: But in all of these examples, the circle is moving around. Sometimes it’s on the bottom,
sometimes it’s on the top. With the ‘-1’, it’s always small and it’s always above a number or letter.
So how can the exact same symbol in the exact same place mean two different things?
…(continues)
Task and excerpt from: Zazkis, R., Kontorovich, I. (2016). A curious case of superscript (-1): Prospective secondary
mathematics teachers explain. The Journal of Mathematical Behavior, 43, 98-110.
1|P a g e
Math 1332
Journal 1: Functions and Equations
Part 1
10 points
You are given the beginning of an interaction between a teacher and a student pertaining to Functions
and Equations Exploration. Your task is to extend this imaginary interaction in a form of a dialogue
between a teacher and Isaac.
T:
S:
When we set 𝑓𝑓(𝑥𝑥, 𝑦𝑦) = 𝑔𝑔(𝑥𝑥, 𝑦𝑦), we get the equation 𝑦𝑦 = 2𝑥𝑥 + 1. Yes, Isaac?
T:
This is correct.
S:
So, you said yesterday that a function has a unique output for each input. So,
when I input a value for 𝑥𝑥 in 𝑦𝑦 = 2𝑥𝑥 + 1, I get a only one value for 𝑦𝑦?
T:
But you called 𝑦𝑦 = 2𝑥𝑥 + 1 an equation.
S:
So is it both or what?
T:
…
Right.
Part 2
5 points
Explain your choice of approach, that is, why did you choose a particular example, what student
difficulties did you foresee, why did you find a particular explanation appropriate, etc.
When you wrote this dialogue, what classroom setting did you have in mind? What did you assume the
students already knew?
2|P a g e
Álvarez, J.A.M., Jorgensen, T., & Rhoads, K. (2018) Enhancing Explorations in Functions for Preservice
Secondary Mathematics Teachers Project, The University of Texas at Arlington.
Lesson 6:
Functions and Equations
In school mathematics teaching, using vocabulary in mathematically precise ways is important because
there are ambiguities in or common uses of certain terms that can cause confusion when students are
confronted with the implied or informal meanings in a mathematical situation. This lesson focuses on
the meaning of the term equation and the different constructed meanings associated with the use of
the equal sign. Carpenter, Franke, & Levi (2003) assert that a “limited conception of what the equal sign
means is one of the major stumbling blocks in learning algebra” (p. 22).
Exploration 6.1: What is an equation?
1. When you hear the word equation, write down what immediately comes to your mind.
a formula to solve a type of math problem
a. Each group member should write down, three examples of equations.
b. Share your examples with your group members. Create a group list of examples for which
there is consensus (note that you may throw some examples out after considering
duplicates or there may be examples offered that not everyone agrees are equations).
4+8=12
y=2x+8
y=mx+b
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Álvarez, J.A.M., Jorgensen, T., & Rhoads, K. (2018) Enhancing Explorations in Functions for Preservice
Secondary Mathematics Teachers Project, The University of Texas at Arlington.
2. Consider the following definition of equation: An equation is a mathematical statement that
asserts the equivalence between two quantities.
a. Which examples from the group list generated in part 1(b) above would be considered
equations according to this definition? Explain you reasoning.
4+8=12, y=2x+8, y=mx+b becuase they are a mathematical statement that has two quantities and declares something
b. According to the definition provided, is 1 + 3 = 4 an equation? Explain why or why not.
yes because it meets three out of three in the definition
c. According to the definition provided, is 1 + 3 = 4 = 11 − 7 an equation? Explain why or
why not. no because it doesnt meet the definition of two quantities
Exploration 6.2: Constructed meanings of the equal sign
1. The use of the equal sign evokes several constructed meanings that can cause some confusion, in
particular in K-12 mathematics (e.g. Knuth, Stephens, McNeil, & Alibali, 2006; Kieran, 1981). Identify
and discuss the meanings that the following uses convey:
Meaning of “=”
results as
is
results as
is
a. 3 + 5 = _____
computes
2
b. 𝑓(𝑥) = 𝑥 + 5
c.
2
𝑑
(𝑒 𝑥
𝑑𝑥
define
+ 4𝑥) = ____
d. 𝐴 = 𝜋𝑟 2
same e. 2 sin 𝑥 cos 𝑥 = sin 2𝑥
is
f.
𝑔(𝑥) = cos 𝑥 + 𝑥
2
same g. 𝑥 + 3𝑥 = 𝑥 − 1
is
h. 𝑆 = {𝑛 ∈ ℕ:
𝑛
2
∈ ℕ}
computes
Give another example conveying the same meaning.
7+9=
k(x) = x^2
8x(a-3b) =
equivalent D=2r
equivalent a^2+b^2=c^2
define
s(x)=sin x +2
equivalent
define
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Álvarez, J.A.M., Jorgensen, T., & Rhoads, K. (2018) Enhancing Explorations in Functions for Preservice
Secondary Mathematics Teachers Project, The University of Texas at Arlington.
2. Consider the following situations arising from students’ work.
a. David has no problem with computations such as 3 + 5 = ____, but has trouble with
3 + 5 = 2 + ___. David may have a limited understanding of the use of the equal sign. Which
meaning may David be missing? How would you know?
David might be missing the equivalent meaning. When we did the
chart the equations with the equal sign in it in that context was
decided to be called equivalent because in order to solve it you have
to find out the number that makes the equation have the same
numbers on both sides.
b. David (from part a) is given the following word problem.
Jim has 2 apples and Mary has 3 apples. How many apples do Jim and Mary have all
together? Kim comes along with 6 oranges, how many pieces of fruit do Mary, Jim,
and Kim have all together?
David writes:
Discuss David’s work and any connections to possible issues regarding the meaning of the equal
sign he displayed in part (a).
While David got the right answer he should not write it like that because it can mess
with how he sees the equal in equations. he should have written two separate
equations 2+3=5 and 5+6=11.
𝑥 2 −1
.
𝑥→1 𝑥−1
c. Ashton is given the following exercise: Find lim
Ashton shows the following work:
Comment upon Ashton’s work shown. References to the definition of equation and the
various meanings of the equal sign should be included in your commentary.
Page 3 of 5
Álvarez, J.A.M., Jorgensen, T., & Rhoads, K. (2018) Enhancing Explorations in Functions for Preservice
Secondary Mathematics Teachers Project, The University of Texas at Arlington.
Exploration 6.3: Function or equation?
1. Consider the functions ℎ(𝑥) = 𝑥 2 − 3 and 𝑟(𝑥) = 2𝑥.
a) What is the meaning of ℎ(𝑥) = 𝑟(𝑥)? Explain. H and R have only two intercepting points in common
b) Discuss why 𝑥 2 − 3 = 2𝑥 is an equation. Is it also a function?
it is a an equation because it fits the description, not a function because it has a collection not two sets
2. Now consider the functions 𝑓(𝑥, 𝑦) = 𝑦 and 𝑔(𝑥, 𝑦) = 2𝑥 + 1. The graphs of f and g are provided
below for reference.
Graph of 𝑧 = 𝑓(𝑥, 𝑦)
Graph of 𝑧 = 𝑔(𝑥, 𝑦)
Graphs of 𝑧 = 𝑓(𝑥, 𝑦) and 𝑧 = 𝑔(𝑥, 𝑦)
a. What is the meaning of 𝑓(𝑥, 𝑦) = 𝑔(𝑥, 𝑦)? Explain. Means list of points were they intersect
b. Discuss why 𝑦 = 2𝑥 + 1 is an equation (part of your discussion should involve the definition
of equation). It’s an equation because it fits the description but it’s not a function
because it’s just quantities without relation
i. Ani claims that 𝑦 = 2𝑥 + 1 is also a function. What assumptions is Ani making?
Explain. She’s assuming that there is no Z axis and that it’s like y=mx+b
ii. Vasso claims that since 𝑦 = 2𝑥 + 1 arises from 𝑓(𝑥, 𝑦) = 𝑔(𝑥, 𝑦) it’s not a function.
Explain. He has the full context of the equation which is why he knows it’s not a function
iii. Oleg says that 𝑆 = {(𝑥, 𝑦) ∈ ℝ × ℝ: 𝑦 = 2𝑥 + 1 } represents the solution set for
Text
𝑓(𝑥, 𝑦) = 𝑔(𝑥, 𝑦). Does Oleg’s observation
help clarify either Ani’s or Vasso’s
claims? Explain.
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Álvarez, J.A.M., Jorgensen, T., & Rhoads, K. (2018) Enhancing Explorations in Functions for Preservice
Secondary Mathematics Teachers Project, The University of Texas at Arlington.
Exploration 6.4: Applying understanding of functions and equations to teaching
Mr. Smith is teaching a high school mathematics class and is writing a few homework and assessment
questions as he plans the school year. Review Mr. Smith’s questions and circle the most appropriate
word(s) he should use in the question.
Consider the following:
i.
ii.
iii.
iv.
v.
vi.
vii.
Evaluate/Simplify/Solve 𝑓(𝑥) = 2√𝑥 + 3 when 𝑥 = 9.
Given functions 𝑔 and ℎ, evaluate/simplify/solve 𝑔(𝑥) = ℎ(𝑥).
Evaluate/Simplify/Solve 5𝑥 + 2 when 𝑥 = 2.
Evaluate/Simplify/Solve 2(𝑥 2 + 𝑥 + 1) − 5(𝑥 3 + 𝑥) + 𝜋𝑥 2 .
Evaluate/Simplify/Solve 2𝑥 2 + 3𝑥 = 5𝑥 + 2.
Evaluate/Simplify/Solve (cos 2 𝑥)(sin 2𝑥) + 2(sin3 𝑥)(cos 𝑥) − cos 𝑥
Find ℎ(3) by evaluating/simplifying/solving ℎ when 𝑥 = 3.
a. For which (i-vii), did your group decide there was more than one appropriate instruction?
Explain. None look like there is more to me
b. Give an example like those above in which the term “evaluate” is used incorrectly or is
problematic. Explain your reasoning. f(x)= 2 √x +3 would have evaluate used incorrectly
c. Give an example like those above in which the term “solve” is used incorrectly or is problematic.
Explain your reasoning.
d. Give an example like those above in which the term “simplify” is used incorrectly or is
problematic. Explain your reasoning.
e. Create guidelines that Mr. Smith can use for determining when it is appropriate to use the
instructions “solve,” “evaluate,” or “simplify” on his homework assignments and assessments.
Your guidelines should address exercises or tasks involving functions and equations.
solve should have both quantities before the equal sign, evaluate should have two quantities with an
equal sign in between, simplify should have numbers without an equal sign present.
References
Carpenter, T. P., Franke, M. L ., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in
the elementary school. Portsmouth, NH. Heinemann.
Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12, 317-326.
Knuth, E.J., Stephens, A.C., McNeil, N.M., & Alibali, M.W. (2006). Does Understanding the Equals Sign Matter?
Evidence from Solving Equations. Journal for Research in Mathematics Education, 37(4), 297-312.
Acknowledgement
This material is partially based upon work supported by the National Science Foundation Improving Undergraduate
STEM Education (IUSE) program under Grant No. DUE #1612380. Any opinions, findings, and conclusions or
recommendations expressed in this material are those of the author and do not necessarily reflect the views of the
NSF. Thanks to Janessa Beach for research assistance.
Principal Investigator: Dr. James A. Mendoza Álvarez; Co-PIs: Dr. Theresa Jorgensen & Dr. Kathryn Rhoads
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