EDCP 342A Unit planning: Rationale and
overview for planning a 3 to 4-week unit of work in secondary school
mathematics
Your name: Niloofar Razzaghi
School, grade & course: R.E. Mountain
Secondary School
Topic of unit: Chapter 5:
Radical Expressions & Equations
(1)
Why do we teach this unit to
secondary school students? Research and talk
about the following: Why is this topic included in the curriculum? Why is it
important that students learn it? What learning do you hope they will take
with them from this? What is intrinsically interesting, useful, beautiful
about this topic? (150 words)
The goal is to show students how to solve equations with
rational exponents and/or radicals. And to extend students' knowledge of
solving to where an equation has either two radicals or two expressions with
rational exponents. This lesson is taught to the grade 11 students and this
topic is included in the curriculum because:
1) To show students what solving radical
equations means.
2)To Give students the expression Isolate-Eliminate-Solve,
to
help them remember what they need to do
first in solving.
3)To Give students practice solving
equations with radicals
and equations with rational exponents.
4)To show students how to solve
equations that have more
than one radical or expression with a
rational exponent.
5)Students get some practice solving
with multiple radicals
and/or rational exponents.
After learning Radical Expressions and Equations, students would be able to simplify radical expressions and perform simple operations such as
adding, subtracting, multiplying and dividing these expressions. With this learning, they would be able to create and solve attractive
and beautiful real-life examples: One of the simplest formulas in
electrical engineering is for voltage, V = √PR, where P is the power in watts
and R is the resistance in ohms.
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(2) A mathematics project connected to this unit: Plan and describe a student mathematics project that will form
part of this unit. Describe the topic, aims, process and timing, and what the
students will be asked to produce, and how you will assess the project. (250
words)
Time:
Due on January 15,2018
Title:
Radical Expressions
and Equations
Instructions: Students
will have to submit a creative project with the following requirements and
they may choose a real-life application of Radical Expressions and Equations
from the following list:
Financial Planning, Radioactive half-life, Geology/Meteorology/Oceanography
Electrical Engineering, medicine,
Physics/Engineering, Supply Chain Management/Business
Goal: The
main goal and the big idea of having this project for the students is to show
them that there are large connections between the real-life Experiences and
math and particularly with Radical
Expressions and equation. Rational expressions are used in many
fields such as economics, data science, medicine, engineering, music
acoustics and many others.
A
After choosing one of the topics, answer
the following questions,
a) What
is the application of Radical Expressions you researched?
b) Describe
its connection to the real world.
c) What
is the impact of your area of research on science and/or
mathematics?
d) How
would your area of research find it more difficult to
work/study if Radical Expressions was not used?
e) Why
is your area of research important/significant to your life,
your
community, and the world?
Formats:
Painting/Drawing, song, short movie, story, PowerPoint, poem, brochure, or
others
NOTE
you can work individually or with partner.
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(3) Assessment
and evaluation: How will you build a
fair and well-rounded assessment and evaluation plan for this unit? Include
formative and summative, informal/ observational and more formal assessment
modes. (100 words)
Formative assessment
As formative
assessment we can use Kahoot.it, quizzes, worksheets, example, unit test,
and IXL. Students as group or
individual will have Kahoot quizzes as formative assessment, and the detailed
solution will be given to check their answers. Also, IXL, which is online
learning/ homework, will be used to check student’s progress in each section.
in class participation, inquiry based questions also are part of formative
assessment.
Summative assessment
As Summative
assessment, I will use quizzes, unit test, homework assignments, project
works which are used to evaluate students’ unit understanding and to determine whether they have learned the material are being taught in the calss.
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Elements of your unit plan:
a) Give a numbered list of the topics of the
10-12 lessons in this unit in the order you would teach them.
Lesson
|
Topic
|
1
|
Rational expression and Equation -
Introduction
|
2
|
Simplifying Rational Expression
|
3
|
Multiplying and Dividing rational
expression
|
4
|
Adding and Subtracting rational expression
(same denominator)
|
5
|
Adding and Subtracting rational expression
(different denominator)
|
6
|
Rational equations
|
7
|
Solve rational equations
|
8
|
Introduction to work problem
|
9
|
Use rational equation to solve the problem
|
10
|
Unit Review
|
(11)
|
Practice Test
|
(12)
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Unit Test
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b)
Write a detailed lesson plan for three
of the lessons which will not be in a
traditional lecture/ exercise/ homework format. These three lessons should include at least
three of the following six elements related to your mathematical topic. (And of
course, you could include more than three!)
These elements should be thoroughly
integrated into the lessons (i.e. not an add-on that the teacher just tells!)
a) history of this mathematics
b) social/environmental justice
c) Indigenous perspectives and cultures
d) Arts and mathematics
e) Open-ended problem solving in groups at
vertical erasable surfaces (“thinking classroom”)
f) Telling only what is arbitrary, and
having students work on what is logically ‘necessary’
Be sure to include your pedagogical goals,
topic of the lesson, preparation and materials, approximate timings, an account
of what the students and teacher will be doing throughout the lesson, and ways
that you will assess students’ background knowledge, student learning and the
overall effectiveness of the lesson. Please use a template that you find
helpful, and that includes all these elements.
Your unit plan is due Monday December 11,
with a possible extension if needed to Friday Dec. 15.
Lesson Plan (1)
Subject: Pre-Cal
|
Grade: 11
|
Date: ------
|
Duration: 70 min
|
Lesson Overview
|
Multiplying and Dividing rational
expression
|
||
Class Profile
|
·
30 students
·
1/3 ELL
|
||
Prior Knowledge
|
Simplifying Rational Expression
|
||
Objectives (Pedagogical
Goal)
|
Multiplication and division of Rational expressions is in
some ways
easier than addition and subtraction of rational
expressions
because a
common denominator does not have to be found. A lot of times in math students
must use past concepts to be able to work all the way through the unfamiliar
problems. In this lesson students will have to remember how to factor,
simplify rational expressions and
multiply
polynomials to be able to complete the multiplication or division problems.
These would be my pedagogical goals for all my
classes including this one:
·
To
improve understanding of some of the previous concepts which would be related
to the new ones
·
To improve understanding
of the nature of the subjects: what is important, how it is practiced….
·
To improve understanding
of the historical development of selected topics.
·
To develop an
unobstructed vision of mathematics.
·
And….to to improve
teaching strategies and have better understanding of students’ strengths and
weaknesses.
|
||
Materials and Equipment Needed for this Lesson
|
·
White board
·
Tablet
·
Markers
·
Computers
|
Lesson Stages
|
Learning Activities
|
Time Allotted
|
Warm-up
(Open-ended problem solving in groups)
|
·
First, I need to see if the students
do not remember the multiplication of fractions
·
Then I will have them discuss it
with their neighbor. If necessary, use an example like 1/ 2 ⋅ 1 /2 to see if they can scale this to the problem
presented.
·
If students are comfortable with multiplying
rational numbers, omit the area model, and ask them to determine the
following products
|
10
min
|
Presentation
|
The rules for multiplying and
dividing rational expressions are the same as the rules for multiplying and
dividing rational numbers, I will start by reviewing multiplication and
division of fractions. When we multiply two fractions we multiply the
numerators and denominators separately:
ab⋅cd=a⋅cb⋅d
When we divide two fractions, we
replace the second fraction with its reciprocal and multiply, since that’s
mathematically the same operation:
ab÷cd=ab⋅dc=a⋅db⋅c
|
10
min
|
Practice and Production
|
Multiplying Rational
Expressions
Q and S do not equal 0.
Step 1: Factor both
the numerator and the denominator.
Step 2: Write as one fraction.
Step 3: Simplify the
rational expression.
Step 4: Multiply any remaining factors in the numerator
and/or denominator.
Show an example using multiplication and go through step
by step to make sure the students understand the process.
Example One: Multiply
Step 1: Factor both
the numerator and the denominator.
Since our problem’s denominators are already factored, we
can move on.
Step 2: Write as one fraction.
Step 3: Simplify the
rational expression.
*Exclude values of original data making denom. = 0.
There
are no remaining factors in the numerator or denominator, so we are done.
|
40
min
|
Closing
(Arts and Mathematics)
|
Ask students to summarize the important parts of the
lesson using writing, or drawing/painting with a partner, individually, or as
a class. Use this as an opportunity to informally assess understanding of the
lesson. Ask students to describe the processes for multiplying and dividing
rational expressions and simplifying complex rational expressions either
verbally or symbolically or using any type or art. I can give them more time
till the next class, if they need to express the lesson using any arts.
|
10 min
|
Lesson Plan (2)
Subject: Pre-Cal
|
Grade: 11
|
Date: ------
|
Duration: 70 min
|
Lesson Overview
|
Adding
and Subtracting rational expression
(same denominator)
|
||
Class Profile
|
·
30 students
·
1/3 ELL
|
||
Prior Knowledge
|
Simplifying Rational
Expression, and multiplying …the
Previous lesson
|
||
Objectives (Pedagogical
Goal)
|
I will remind students how to add fractions with
the same denominator. I will let them work through the following sum
individually. Calculate the following sum: 𝟑 /𝟏𝟎 +
𝟔/ 𝟏𝟎
The solution should be available to the class either by the teacher or by a
student because the process of adding fractions will be extended to the new
process of adding rational expressions.
These would be my pedagogical goals for all my
classes including this one:
·
To
improve understanding of some of the previous concepts which would be related
to the new ones
·
To improve understanding
of the nature of the subjects: what is important, how it is practiced….
·
To improve understanding
of the historical development of selected topics.
·
To develop an
unobstructed vision of mathematics.
·
And….to to improve
teaching strategies and have better understanding of students’ strengths and
weaknesses.
|
||
Materials and Equipment Needed for this Lesson
|
·
White board
·
Tablet
·
Markers
·
Computers
|
Lesson Stages
|
Learning
Activities
|
Time Allotted
|
Warm-up
(Open-ended problem solving in groups)
|
Ask students to work in groups to write what they have learned in their
notebooks or journals in previous lesson. Check in to assess their understanding.
Then I will have students work in pairs to quickly work through some review
exercises. Allow them to think about how to approach those problems, which
involves adding few rational expressions.
|
10
min
|
Presentation
|
Ask students for help in stating the rule for adding and
subtracting rational numbers with the same denominator.
The result below is valid for real numbers 𝑎, 𝑏, and 𝑐. 𝑎/ 𝑏 + 𝑐/ 𝑏 = 𝑎 + 𝑐/ 𝑏 and 𝑎 /𝑏 – 𝑐/ 𝑏 = 𝑎 – 𝑐/ 𝑏. They could generalize the process for two rational
expressions, rearrange terms using the commutative property to combine the
terms with the same denominator, and then add using the above process, or
they could group the addends using the associative property and perform addition
twice.
|
10
min
|
Practice and Production
|
To
add/subtract rational expressions with the same denominator
1.
Add/subtract the numerators. Write this sum/difference as the numerator over
the common denominator.
2. Reduce to lowest terms.
Example
1
Simplify
the following:
Solution
These
fractions already have a common denominator
1: Write this sum as the numerator over the
common denominator:
2: Reduce to lowest terms:
|
40
min
|
Closing
(Arts and Mathematics)
|
Ask students to summarize the important parts of the
lesson using writing, or drawing/painting with a partner, individually, or as
a class. Use this as an opportunity to informally assess understanding of the
lesson. Ask students to describe the processes for multiplying and dividing
rational expressions and simplifying complex rational expressions either
verbally or symbolically or using any type or art. I can give them more time
till the next class, if they need to express the lesson using any arts.
|
10 min
|
Lesson Plan (3)
Subject: Pre-Cal
|
Grade: 11
|
Date: ------
|
Duration: 70 min
|
Lesson Overview
|
Solve rational equations
|
||
Class Profile
|
·
30 students
·
1/3 ELL
|
||
Prior Knowledge
|
Simplifying Rational Expression
, and multiplying …
|
||
Objectives (Pedagogical
Goal)
|
Students will be
able to solve rational equations,
monitoring for the creation of extraneous solutions.
In the previous lessons, students learned to add, subtract, multiply, and
divide rational expressions. in this lesson they can solve equations
involving rational expressions There is more than one method to solve a
rational equation, and in this section, I will two such methods. The first
method is to multiply both sides by the common denominator to clear
fractions. The second method is to find equivalent forms of all expressions
with a common denominator, set the numerators equal, and solve the resulting
equation.
These would be my pedagogical goals for all my
classes including this one:
·
To
improve understanding of some of the previous concepts which would be related
to the new ones
·
To improve understanding
of the nature of the subjects: what is important, how it is practiced….
·
To improve understanding
of the historical development of selected topics.
·
To develop an
unobstructed vision of mathematics.
·
And….to to improve
teaching strategies and have better understanding of students’ strengths and
weaknesses.
|
||
Materials and Equipment Needed for this Lesson
|
|||
·
White board
·
Tablet
·
Markers
·
Computers
|
|||
Thanks for this unit outline, Niloo.
ReplyDeleteQ: What grade is this for? (Grade 9?) And ‘Chapter 5’ doesn’t tell me very much. Remember that the textbook is not the curriculum — and as time goes on and curriculum evolves, textbooks drift farther away from the curriculum.
Rationale: Mostly, you’ve said that we teach this unit so that students will know how to solve these particular types of equations — and that is probably the main rationale for this unit (to develop procedural know-how and fluency). It’s one of the more difficult units in the secondary curriculum to justify for intrinsic reasons. You do find the voltage formula, which uses a single radical expression. Perhaps you could find some other places in science, engineering or other areas where radical and rational expressions are used in equations. A very quick Google search found this, and I’m sure you could find more:
http://www.montereyinstitute.org/courses/Algebra1/COURSE_TEXT_RESOURCE/U07_L3_T3_text_container.html
Project: (Why the Jan. 15, 2018 project deadline, as you won’t be starting your long practicum till later in January?)
There are some good starting points for this project idea, but as it is, I don’t think that high school students could actually complete the project. The instructions are far too vague; these young students have NO experience or knowledge of “Financial Planning, Radioactive half-life, Geology/Meteorology/Oceanography Electrical Engineering, medicine, Physics/Engineering, Supply Chain Management/Business”. They have no way to select a topic, and would probably not understand specialized writing and equations from these disciplines if they tried to read them.
As a teacher, you need to take more responsibility in designing and assigning a project. These young people need some guidance, as well as some degree of choice and freedom. You need to do some research yourself, before devising the project, so that you can put some helpful constraints on the project. Similarly, saying that the format can be... practically anything ("painting/ drawing, song, short movie, story, PowerPoint, poem, brochure or others") is entirely unhelpful. Students will have no idea what you expect or wish them to do. It just looks like you didn't want to think this through and make a decision -- and that' s not workable . It would not be possible for students to actually compl•ete a project described this way!
Assessment plan: Looks OK!
Unit elements: Looks fine
Lesson plans: #1: Warm up -- I suggest using a different example. Rather than 1/2 (1/2), try something like 7/10 (3/4). With 1/2 (1/2) = 1/4, some students might think that the denominator is (2+2)...
ReplyDeleteYou talk about omitting the area model -- what area model? You hadn't mentioned this earlier. It seems to me that the area model of multiplication is a bit difficulty to understand with fractions/ rational numbers, don't you think?
There are typos in your unit plan. For example, "ab•cd" ought to be 'a/c • b/d'.
Be prepared for some of your students NOT to understand division by fractions. Students may not really understand why multiplication by the reciprocal of the divisor is equivalent to dividing by the original divisor. How will you help them understand this if they didn't already get it in earlier grades, but only memorized a procedure?
How does this lesson take students beyond simply working with numerical fractions and into working with rational expressions and equation using variables? I don't see anything beyond about a Grade 6 level in this lesson. Where is the Math 9 content?
This lesson plan is completely teacher-led and procedural. I have no idea what the students might be doing during this lesson.
As a student, I would not be able to using writing, drawing or painting in 10 minutes to summarize this lesson! What is it you expect them to do -- and why? Students will have no idea what you are asking them to do, and it is a trivialization of the arts to sat that they can use art in this situation.
Also, with 1/3 of the class of 30 being English Language Learners, I do not see any accommodation being made to help these ELL students understand the topic. If it is all about you lecturing, how will ELL students make sense of it?
#2: Presentation: Students will not know what is meant by the associative and commutative properties.
ReplyDeleteAgain, I see nothing of rational expressions and equations using variables in this lesson. For students who 'got' fractions years ago, this would seem very babyish. You haven't addressed the Math 9 curricular topics here, and that is a very serious problem.
Also, your format is once again completely teacher-led and lecture-based. There is far too little content, and no consideration of what students might be doing to be learning in an active way. Once again, you finish with 10 minutes for students to summarize the lesson, supposedly using the arts, in a way that would not honour the arts or the students' learning.
#3: Presentation: You have allotted only 10 minutes for students to solve a few example by each of two methods, AND for you to give step-by-step examples of each method. This is not possible -- far too little time!
Practice: Then the following 40 minutes is devoted to solving a single equation -- why?
How will you help students understand why division by zero is problematic? They will need to know this in order to understand why the denominators cannot be zero. Otherwise, you are just teaching meaningless procedures. This is a procedural unit, so it is important to emphasize the meaningful, understanding aspects of it.
Once again, this lesson plan is inadequate: completely teacher-focused, not adequate in terms of content and examples, no sense of active learning for students, no accommodation for ELL students, and a meaningless mention of the arts in the final 10 minutes of class.
Overall: Niloo, this unit plan is not yet adequate to pass our EDCP 342A class. I need to get a revised version of your unit plan by next Sunday, January 7 with revised:
•Project plan
*3 lesson plans
Please read the detailed notes here carefully, and revise accordingly to create a meaningful project that students can actually accomplish, and lesson plans that:
•involve students in active learning,
•minimize teacher-centre lectures,
•make accommodations for ELL students,
•cover the important curriculum for this grade level,
•teach for understanding, not just procedural fluency,
•have variety and interest, and
•incorporate the criteria given for these lesson plans :
"These elements should be thoroughly integrated into the lessons (i.e. not an add-on that the teacher just tells!)
a) history of this mathematics
b) social/environmental justice
c) Indigenous perspectives and cultures
d) Arts and mathematics
e) Open-ended problem solving in groups at vertical erasable surfaces (“thinking classroom”)
f) Telling only what is arbitrary, and having students work on what is logically ‘necessary’"
I will look forward to reading your revised unit plan soon! Thanks Niloo.