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 ππ(π₯π₯) = 2π₯π₯ + 5 Yes, Dina?

So you said yesterday that ππ β1 stands for an inverse function

This is correct.

1

But we learned that this power (-1) means 1 over, that is 5β1 = , right?

5

Right.

So is this the same symbol or what?

Excerpt from Amandaβs 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: Itβs a circleβ¦

Teacher: Of course, [Teacher draws a β1β beside the circle.] Now what is it?

Student: Ten. The circle is a zero.

Teacher: Very good. [Teacher erases the β1β and writes a big β2β and β5β, using the circle as a decimal

point.] How about now?

Student: Itβs 2.5. The circle is a decimal.

Teacher: Excellent. [Teacher erases the β2β and β5β and β100β to the left of the circle and βCβ 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 itβs on the bottom,

sometimes itβs on the top. With the β-1β, itβs always small and itβs 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 ππ(π₯π₯, π¦π¦) = ππ(π₯π₯, π¦π¦), we get the equation π¦π¦ = 2π₯π₯ + 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 π₯π₯ in π¦π¦ = 2π₯π₯ + 1, I get a only one value for π¦π¦?

T:

But you called π¦π¦ = 2π₯π₯ + 1 an equation.

S:

So is it both or what?

T:

β¦

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

Γlvarez, 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 βlimited conception of what the equal sign

means is one of the major stumbling blocks in learning algebraβ (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

Page 1 of 5

Γlvarez, 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 β 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 β=β

results as

is

results as

is

a. 3 + 5 = _____

computes

2

b. π(π₯) = π₯ + 5

c.

2

π

(π π₯

ππ₯

define

+ 4π₯) = ____

d. π΄ = ππ 2

same e. 2 sin π₯ cos π₯ = sin 2π₯

is

f.

π(π₯) = cos π₯ + π₯

2

same g. π₯ + 3π₯ = π₯ β 1

is

h. π = {π β β:

π

2

β β}

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

Page 2 of 5

Γlvarez, 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 studentsβ 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 Davidβs 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.

π₯ 2 β1

.

π₯β1 π₯β1

c. Ashton is given the following exercise: Find lim

Ashton shows the following work:

Comment upon Ashtonβs 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

Γlvarez, 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 β(π₯) = π₯ 2 β 3 and π(π₯) = 2π₯.

a) What is the meaning of β(π₯) = π(π₯)? Explain. H and R have only two intercepting points in common

b) Discuss why π₯ 2 β 3 = 2π₯ 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 π(π₯, π¦) = π¦ and π(π₯, π¦) = 2π₯ + 1. The graphs of f and g are provided

below for reference.

Graph of π§ = π(π₯, π¦)

Graph of π§ = π(π₯, π¦)

Graphs of π§ = π(π₯, π¦) and π§ = π(π₯, π¦)

a. What is the meaning of π(π₯, π¦) = π(π₯, π¦)? Explain. Means list of points were they intersect

b. Discuss why π¦ = 2π₯ + 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 π¦ = 2π₯ + 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 π¦ = 2π₯ + 1 arises from π(π₯, π¦) = π(π₯, π¦) itβs 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 π = {(π₯, π¦) β β Γ β: π¦ = 2π₯ + 1 } represents the solution set for

Text

π(π₯, π¦) = π(π₯, π¦). Does Olegβs observation

help clarify either Aniβs or Vassoβs

claims? Explain.

Page 4 of 5

Γlvarez, 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. Smithβs 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 π(π₯) = 2βπ₯ + 3 when π₯ = 9.

Given functions π and β, evaluate/simplify/solve π(π₯) = β(π₯).

Evaluate/Simplify/Solve 5π₯ + 2 when π₯ = 2.

Evaluate/Simplify/Solve 2(π₯ 2 + π₯ + 1) β 5(π₯ 3 + π₯) + ππ₯ 2 .

Evaluate/Simplify/Solve 2π₯ 2 + 3π₯ = 5π₯ + 2.

Evaluate/Simplify/Solve (cos 2 π₯)(sin 2π₯) + 2(sin3 π₯)(cos π₯) β cos π₯

Find β(3) by evaluating/simplifying/solving β when π₯ = 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 βevaluateβ is used incorrectly or is

problematic. Explain your reasoning. f(x)= 2 βx +3 would have evaluate used incorrectly

c. Give an example like those above in which the term βsolveβ is used incorrectly or is problematic.

Explain your reasoning.

d. Give an example like those above in which the term βsimplifyβ 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 βsolve,β βevaluate,β or βsimplifyβ 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 Γlvarez; Co-PIs: Dr. Theresa Jorgensen & Dr. Kathryn Rhoads

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