My learning progression is on converting percents, decimals, and fractions. It will lead to comparing ratios and using fractions and percents for ratio lengths. I will cover how to convert from percent to decimal to fraction and vise versa. I will start off with a hinge question then go in with my lesson. I plan to have the students in groups of four and they will be handed a worksheet of conversion.
learning progression percents edtpa-12sg8wn
This learning progression was designed primarily for a high school Precalculus course. The three Common Core State Standards that this learning progression will be satisfying are from the cluster titled “Extend the domain of trigonometric functions using the unit circle,” these are HSF.TF.A.1, HSF.TF.A.3, and HSF.TF.A.4. In this course, students are focusing on mastering the Common Core State standards for Functions. Throughout this learning progression, students will focus on three mathematical practices which are MP5, MP7, and MP8.
One of the instructional tasks that is included in the Learning Progression is creating a unit circle.
In order to construct the unit circle students first have to form the two special right triangles. Through this activity they will from connections between the unit circle and the special right triangles which will strengthen their understanding of the concepts of the lesson.
The assessment used in this learning progression is the green sheet quiz. On this quiz students are given three angles and then they have to draw the angle and evaluate the 6 trigonometric functions for it. The tasks in the learning progression prepare students for this assessment.
Learning Progression for edTPA – Extended Mathematics
Standards:
Task Summary: Students will be given an inquiry prompt to answer as students work through activity sheets that will have students finding values of numbers in ancient systems (e.g. Egyptian, Babylonian, Mayan), different bases, and in modular expressions. Students will also find solutions for addition and subtraction problems in different bases and with modular arithmetic. Students will also discuss their ideas, findings, and questions using mathematical thinking and reasoning. These tasks are designed to develop students’ mathematical thinking and reasoning.
Assessment Task Summary: Students will be assessed on their mathematical thinking and reasoning by their written work or mathematical discourse. Scoring of the assessment will be done via a rubric based on the standards above and learning targets for each task.
This learning progression is about polynomial division. The common core state standards for this learning progression:
Write expressions in equivalent forms to solve problems.
HSA.SSE.B.3
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
CCSS.MATH.CONTENT.HSA.SSE.B.3.A
Factor a quadratic expression to reveal the zeros of the function it defines.
CCSS.MATH.CONTENT.HSF.IF.C.7.C
Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
Formative Assessment
The formative assessment game used for part of this will be giving the students two cards, one of which is green and the other is red for them to choose whether they agree or not that a certain factor works for a polynomial. There will be examples up on the board and they will be given a factor. They must do the division then hold up the green card if it worked out nicely and the red if it doesn’t work.
High School: Algebra 2
Complex Numbers
This learning progression will be taught to a class that consists of sophomores and juniors in high school who are currently taking Algebra 2. The Common Core State Standards that will be addressed come from two different domains. The first domain is from High School: Algebra-Arithmetic with Polynomials and Rational Expressions. The CCSS Math cluster that will be addressed is “Understand the relationship between zeros and factors of polynomials.” The second domain is High School: Number and Quantity-The Complex Number System. The CCSS Math clusters that will be addressed are “Perform arithmetic operations with complex numbers,” “Represent complex numbers and their operations on the complex plane,” and “Use complex numbers in polynomial identities and equations.” The third domain is from High School: Number and Quantity-The Real Number System. The cluster that will be addressed is “Extend the properties of exponents to rational exponents.” Students will also meet Mathematical Practices 1, 2, 3, 7, and 8.
The textbook that I will use as a resource is Glencoe’s Algebra 2: Integration, Applications, and Connections. The lesson will be taught from sections eight through ten of chapter five of this book. These sections transition students from simplifying expressions including radicals and rational exponents to simplifying expressions containing numbers that are a part of the complex plane.
The central focus of this learning progression is an introduction to complex numbers and the complex plane. The progression begins with the strategies that are used in simplifying expressions involving radicals. These strategies will help student s understand how to use complex numbers and how to simplify expressions that contain complex numbers. Students will be first introduced to what a complex number is and will then learn how to graph them in the complex plane. The purpose of this learning progression is for students to gain a better conceptual understanding of the complex plane and will lead into solving quadratic equations that do not have real solutions. This progression is set up so that the entry tasks from each section review a concept or ask students to think critically about a problem that will help them understand the new information that will be taught during the lesson. How students do on this introductory information will influence where each lesson begins. This will then influence how far we get in the planned lesson and so the next day’s lesson will also be affected. Each lesson has been set up to be flexible and to run off of the previous lesson. Beginning the class with questions that lead students to recall information that they have previously learned and to explore a new way of thinking will help students be more successful during the remainder of the class period and will help students become more interested in what they are learning. Kubiszyn and Borich state in the book Educational Testing and Measurement that by imbedding a formative assessment into each lesson, “well-constructed performance test can serve as a reaching activity as well as an assessment. This type of assessment provides immediate feedback on how learners are performing, reinforces hands-on teaching and learning…it moves the instruction toward higher order behavior.”
Common Core State Standards
Extend the properties of exponents to rational exponents.
CCSS.MATH.CONTENT.HSN.RN.A.1
Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.
CCSS.MATH.CONTENT.HSN.RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Perform arithmetic operations with complex numbers.
CCSS.MATH.CONTENT.HSN.CN.A.1
Know there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real.
CCSS.MATH.CONTENT.HSN.CN.A.2
Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
CCSS.MATH.CONTENT.HSN.CN.A.3
(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
Mathematical Practices
CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.
CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.
Learning Progression
Learning Progression-Complex Numbers-164vuer
High School: Geometry I
Proving Geometric Theorems
By Emily Ivie
This learning progression was designed primarily for students who are in a traditional classroom
setting and have similar mathematical abilities. This class consists of 28 students, mainly
sophomores, who are studying geometry as part of their graduation requirement. The geometry
class has been designed to cover topics at a pace that meets the state requirements of content
topics in high school geometry. Since the class is set up on a semester system, the majority of the
students in this class have been a part of this class since the start of the school year and have had
the same exposure and background knowledge about geometry. Many students enjoy talking and
taking part in discussions during class, that is why I am going to teach this learning progression
in a student-lead conversation.
Standards:
The Common Core State Standards that will be satisfied are from the High School Geometry:
Congruence: Prove Geometric Theorems cluster. In the Congruence domain we will cover
CCSS.MATH.CONTENT.HSG.CO.C.9. Prove theorems about lines and angles and we will also
prove theorems about triangles and parallelograms with standards
CCSS.MATH.CONTENT.HSG.CO.C.10 and CCSS.MATH.CONTENT.HSG.CO.C.11. In this
course, students focus on mastering basic geometry knowledge that is required by the state, while
integrating in common core standards and mathematical practices. In this learning progression
the students will use four mathematical practices including: MP4, MP5, and MP7.
The curriculum these students are learning is based off the McDougal Littell: Geometry by
Larson, Boswell, and Stiff. This textbook is a resource used to design lectures, find worksheets,
and create practice problems through. The start to each class day will begin with a warm up
problem, followed with an introduction to a new topic with notes, classroom activity, and then
discussion. The notes are put up on the projector at the beginning of class so that students who
finish their warm up activity early can start writing the notes.
Accommodations:
Throughout the learning progression accommodations are made for students with IEPs and ELL
students. For the students who need a longer amount of time to write down their notes, they are
able to get the notes from me a day early so that they do not fall behind during lecture and are
able to participate in the discussion during note. Another accommodation made is that I let my 2
ELL students sit next to each other because they feel most comfortable working in partners this
way. In addition to these accommodations, the learning progression designed has activities that
are accessible to all learners. They do not require internet or any other tools beside from the
classroom whiteboards, pen, and paper.
Central Focus:
The central focus of this learning progression is for students to understand how specific theorems
about lines, triangles, and parallelograms were proved and how you can apply them. Students
will also have an understanding as to why it is important to know these theorems when applying
them to everyday life situations.
Lesson 1: Lines
We will start with an introduction to lines: parallel, transversal, and perpendicular. Since many of
the students have already had an introduction to this topic, it will be much easier to engage in a
class in student lead discussion when I ask them, “What does it mean for two lines to be parallel/
transversal/perpendicular?” This discussion of defining certain types of lines will begin to build
their mathematical thinking and conceptual understanding that they will use again during their
partner tasks. During the entry task students will demonstrate MP 7 looking for a way to identify
structure. After reviewing the entry task and understanding these definitions, I will give students
different geometric pictures and we will play a game of “I Spy”, students will be given colored
pencils and required to make a key. We will go around the class and identify specific types of
lines and angles. Once each student has identified and color marked one type of each line and
angle, I will show the students how to prove theorems about line angles. My example I will show
in class will be proving how vertical angle are congruent. I will ask students specific questions to
guide their learning such as: “What do we know about the lines and angles in this diagram? Are
their any linear pairs? What about supplementary?” These questions will formatively assess my
students knowledge about how well they conceptually understand how to identify lines and
angles. When students answer these questions, I will be looking for them to make the connection
between the next steps such as, “since we have angles that are linear pairs, we can use the linear
pair postulate.” This assessment will show students understanding of
CCSS.MATH.CONTENT.HSG.CO.C.9. proving theorems about lines and angles. After students
have worked through the vertical angle theorem, I will ask them to prove that alternate interior
angles are congruent. They will turn in their proof as part of their summative assessment. Once
looking at their proof, I will give feedback based on their reasoning and mathematical thinking.
Lesson 2: Triangles
My next activity will start with reminding my students about the properties of triangle. We will
be expanding our proof knowledge of triangles building off of the prior lesson where students
learned about proving lines and angles and we will try to prove properties about triangles. I will
work through one property about triangles and hold a discussion. After this, we will break up into
groups and I will give each group one theorem about triangles to solve. Then once enough time
has passed, each group will go up to the front board and give a presentation about the theorem
they proved. This lesson aligns with the following standard
CCSS.MATH.CONTENT.HSG.CO.C.10 proving theorems about triangles. Finally once groups
have put their proof up on the whiteboard, I will ask questions to assess their understanding such
as “What does this theorem tell us?”, “How can we apply this postulate to our problem?”, and
“Where do you start when proving this theorem?” These questions are important to ask students
to make sure that they are using tools (such as theorems) appropriately MP5. These
presentations will be a formative and summative assessment to make sure that students can
properly use the new information we just learned as well as explain their answers using old
vocabulary.
Lesson 3: Parallelograms
We will be expanding students understanding of proving line angles and triangles by introducing
parallelogram theorems. Ideally this lesson should be a fun activity that helps students with their
understanding of parallelogram proofs. Students will begin the lesson with a warm up in which
we will cover material learned in the previous day. Students will find a partner and share their
proof completed from the homework the night before. After we finish the entry task I plan to go
over the learning outcomes for the day, which is, students will be able to use their learned
understanding of parallelogram proofs. Shortly after that we will have a class discussion about
what a parallelogram is and I will encourage the use of specific vocabulary words like length,
adjacent, and angle. During our discussion I will hand out 4 parallelogram figures made from
construction paper to each student. We will go through each theorem about parallelograms:
labeling, folding, and drawing on each figure to show understanding for each theorem. Then I
will have the students take the time to try about write up their proofs of these theorems. While
the students working on folding there diagrams I will be walking around the classroom asking
questions about the theorems and how they proved the theorem. During this activity students will
achieve their learning target of CCSS.MATH.CONTENT.HSG.CO.C.11 and MP 4: “Model with
Mathematics” because students will use their parallelogram cuts outs to model their proofs.
This learning progression is made to take place in a high school Algebra classroom of 25 students. This class consists of freshman and sophomores. The desks in the classroom are arranged in groups of four. The lesson is based off the Algebra I textbook. The students have prior knowledge of binomials and what they look like and prior knowledge on the GCF.
learning progression-tt6ir7
This learning progression was designed primarily for a 9th grade algebra course. The three Common Core State Standards that this learning progression will be satisfying are from the cluster titled “Reason quantitatively and use units to solve problems,” these are HSN-Q.A.1, HSN-Q.A.2 and HSN-Q.A.3. In this course, students are focusing on mastering the Common Core State standards for Algebra. Throughout this learning progression, students will focus on three mathematical practices which are MP2, MP4, and MP6.
There are three instructional tasks/activities included in this learning progression. Each one was provided by Illustrative Mathematics.
The following is an excerpt from the learning progression regarding the task “Fuel Efficiency”
Learning Target:
I can use unit conversions and proportions to determine fuel efficiency.
Task: Fuel Efficiency
Sadie has a cousin Nanette in Germany. Both families recently bought new cars and the two girls are comparing how fuel efficient the two cars are. Sadie tells Nanette that her family’s car is getting 42 miles per gallon. Nanette has no idea how that compares to her family’s car because in Germany mileage is measured differently. She tells Sadie that her family’s car uses 6 liters per 100 km. Which car is more fuel efficient?
Guiding questions:
Are the two mileages given in the same form? (no, our task says they aren’t)
When we say 42 miles per gallon this is an example of a (rate)?
How should we set up the problem? (use proportions)
____________________________________________________
At the end of the unit the students will do a project for the summative assessment. The students will be told to look through newspapers and magazines and collect 2 examples of situations to be expressed algebraically. They are then supposed to come up with the algebraic expression that accompanies each. They must also write up what each variable represents, what quantities are involved and what units are being used.
Learning Progression
The CCSS-Math cluster I used for this learning progression is
The mathematical practices used are as follows:
The tasks for this LP are as follows:
I will elicit evidence from the following: