Discussion the summative and formative assessment of 5-11 grade of assessment of CCSS-Math.
At first glance, what would you expect the limit of the function to be as x approaches 2? Find the limit, if it exists. Why is predicting the values of this function at 2 difficult? What would the graph look like at this point?
An assessment commentary and the solution is given in the attached document.
High School Geometry: Polygons and Parallelograms
This learning progression will be taught at Ellensburg High School in a Geometry course. The students in the class this learning progression is being taught in are all good students. By that I mean each student is a diligent and hard worker. The course text is Geometry: Reasoning, Measuring, Applying by McDougal Littell. The learning progression covers sections 6.1-6.3, after which will be a summative assessment. Section 6.1 covers the basics of Polygons; what they are, definitions related to them, procedures for determining a figure is a polygon, and the different types of polygons. Section 6.2 covers the properties of parallelograms; related theorems and proofs. Section 6.3 covers the theorems and proof of parallelograms being quadrilaterals. I have planned to spend a single lesson on each section with another lesson afterward designated to a quiz. Below are the prior CCSS which have been covered that are related this learning progression:
Experiment with transformations in the plane
CCSS.MATH.CONTENT.HSG.CO.A.1
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
Prove geometric theorems
CCSS.MATH.CONTENT.HSG.CO.C.9
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
Make geometric constructions
CCSS.MATH.CONTENT.HSG.CO.D.12
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
The students have mastered these concepts and CCSS through activities and tasks assigned in earlier learning progressions. Thus, the students are prepared and have the appropriate academic knowledge to begin learning CCSS.MATH.CONTENT.HSG.CO.C.11 which covers proving theorems about parallelograms. As I explain the tasks within the learning progression I will explain how the students’ prior academic knowledge was necessary; as well as, how the new concepts build off of the students’ prior knowledge will help them gain a deeper understanding of the new concepts. However, before beginning the explanation of the tasks and assessments I would first like to discuss my instructional strategies.
When teaching the learning progression I will use direct instruction and the use multi-media to communicate the learning targets, the concepts, and the directions for the tasks. The initial discussion will not provide examples. I want the students to struggle and make their own inquiries about how to solve the posed problems. I will then make use of grouping to have students work together on the tasks and make inquiries as a group or as an entire class. I will also allow students who have figured something out to travel to another group and share what they know after they first share with their own group (not a requirement though). This will work effectively because I know that my students are diligent workers and that they have proven they work well in groups. I also know that it will promote them to work together and share procedures and reasoning if I give them the opportunity to do so. Lastly, this learning progression will only include formative assessment except for the last quiz since all work will be done in groups and I want to assess discussion and inquiry not mathematical correctness. For the most part answers will be gone over as a whole class so students that I ca make sure all students have the correct solutions, procedures and reasoning to be studying.
Lesson 1: Polygons
The first lesson does not touch a CCSS but teaches the students the necessary prior-academic knowledge needed to learn HSG.CO.C.11. As stated above my lessons do not begin with lectures or examples of how to solve the related problems within the lesson’s tasks. I will begin this first lesson with some light-hearted dialogue with the students to ease them into a learning-mindset. Afterwards, I will hand out a worksheet which I created. I devised the worksheet so that once finished it can be used as a resource tool for studying and review. The worksheet has this prompt, “[d]irections: For the figures 1-13 state which figures are polygons and why. If it is a polygon name it based upon its number of sides and state whether it is concave or convex. Lastly, state whether or not the polygon is regular or not.” Attached to the worksheet is a sheet with 13 figures; some are polygons and some are not, some are concave polygons and some are convex polygons, and some are regular polygons.
How this activity will work is once I have handed it out students will be directed to tackle the first part of the prompt in their groups without help from me. Students have been working with polygons for quite a while but without me explicitly telling them that have been. Thus, it is my hope that through inquiry some of the students will be able to determine which figures are polygons and which are not. Then those students can share their reasoning with the class. If after about 2 minutes not students have made any progress I will pick a figure and state whether or not it is a polygon. Students will then have an example to go off of. By 4 minutes, whether students are finished or not, I will ask the class to quiet down and for any volunteers to share which they chose to be polygons or not to be polygons. Once all of the students had shared I will go over the definition of a polygon and then go through which are and are not on the worksheet. Just like how we as a class went through this part of the prompt so too will we with the other parts. By the end students will have helped each other create a resource tool with visual representations connected to descriptive details. Students will have examples to study and use as reference on homework problems. Which leads me to the last part of this lesson; a homework assignment which has students answering 5-10 questions picked from the section 6.1 of the text.
Lesson 2: Properties of Parallel Lines
As stated above students have gone over parallel lines and have also constructed parallel lines. Thus, I have decided to begin this lesson by having student construct a parallelogram without actually telling them that. I will direct them to construct two parallel lines on a sheet of paper and extend the lines across the whole sheet. Then I will direct students to create two more parallel lines that are not parallel to the other lines and also extend the lines across the whole sheet. Lastly, I will ask the students if any of them know what they had just created. Hopefully a student might know and share but if not I will explain what they created and how it relates to what we are in the lesson.
The next step in the lesson would be to direct students to discover any properties about the parallelogram they just created; for instance any congruent angles or sides, any supplementary angles, or the fact that their diagonals bisect each other. I will designate 10 minutes to this part of the lesson and while students are discovery the properties I will be walking from group to group posing specific questions to promote the right inquiries. Once those 10 minutes are over I will present the 4 related theorems and ask the students how many of the 4 theorems they discovered and make a tally up on the board (a little competition never hurt). I will then go over two examples of how to use these properties to solve proofs relating parallelograms. The end of the lesson would be designated to their homework assignment which would incorporate basic problems from section 6.2 and 2 hand-made, two-column proofs. Students would only have in-class time for their homework if as a class we went through the example proofs in the time allotted for it.
Lesson 3: Proving Quadrilaterals are Parallelograms
This lesson will be set up exactly the same as the second but with a different introduction. Since the students have already constructed a parallelogram there is no sense in repeating that. Instead, I will open the lesson with this question, “[w]hat are the properties of a quadrilateral?” I will give students some time to discuss this. After a while, 2-3 minutes or so, I will ask students to volunteer what their group concluded the properties of a quadrilateral are. After all of the students have shared what they wanted to I would then go over the properties of a quadrilateral and reference to the properties of a parallelogram we went over the day before. Students should be starting to make the connections in their head that these properties are the same or share similarities. To finish the introductory I would ask the students if a parallelogram is a quadrilateral. Hopefully I would get a volunteer to share their reasoning thus creating a discussion. If not I would lead a discussion as to why it is a quadrilateral. Then as I said before the rest of the lesson would follow the same structure as the one before.
Quiz
All of the assignments, tasks, and activities in the lesson were solely graded upon participation and completion. This is because I always gave the students the answers to the in-class activities at the end which meant I graded only the students’ participation in discussions and group work. The homework assignments were only graded on completion and not correctness because I wanted to promote students to at least provide work for a problem even if they thought they could not do it. A student who does not think they can solve a problem won’t even start it but if completion is the only grade then they will at the least attempt it. Therefore, I have decided to give a quiz half way through chapter 6 so that the students and I have an accurate measure of the conceptual and procedural understanding and reasoning skill that should have been learned in section 6.1-6.3. This assessment will help me determine whether or not the students are ready to move on and it will help the students self-reflect on what they have learned and plan out what they still need to.
High School Algebra: Creating Equations
This learning progression will be taught at Ellensburg High School in an Algebra 1 course. The students in the class this lesson progression is being taught in are all good students. By that I mean each student is a diligent and hard worker. Knowing this I have decided to deviate from the usual required text and use tasks from CCSS Problem-Based Tasks for Mathematics I. These tasks are tied directly to a CCSS and incorporate real-world applications. Knowing that my students are capable I am positive that they will transition well from their regular work from the required text book to these problem-based tasks I have found for them to do.
In between the time this learning progression starts and when the school year starter the students have learned the following:
Reason quantitatively and use units to solve problems.
Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
Define appropriate quantities for the purpose of descriptive modeling.
Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Interpret the structure of expressions.
Interpret expressions that represent a quantity in terms of its context.*
Interpret parts of an expression, such as terms, factors, and coefficients.
Interpret complicated expressions by viewing one or more of their pa rts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.
The students have mastered these concepts and CCSS through activities and tasks out for their required text. Thus, the students are prepared and have the appropriate academic knowledge to begin learning concepts tied the CCSS HSA-CED.1, 3 & 4. Note the reason this learning progression does not cover HSA-CED.2 is because that has already been covered in previous lessons when teaching linear equations. As I explain the tasks for each of the CCSS 1, 3 & 4 I will explain how the students’ prior academic knowledge was necessary; as well as, how the new concepts build off of the students’ prior knowledge will help them gain a deeper understanding of the new concepts. However, before beginning the explanation of the tasks and assessments I would first like to discuss my instructional strategies.
When teaching the learning progression I will use direct instruction and the use multi-media to communicate the learning targets, the concepts, and the directions for the tasks. The initial discussion will not provide examples. I want the students to struggle and make their own inquiries about how to solve the posed problems. I will then make use of grouping to have students work together on the tasks and make inquiries as a group. Just like how these tasks are real-world applications I want the students to also learn that in real life you made need to ask someone for help and work together to solve problems. Because I know that my students are diligent workers and that they have worked well in groups before, I know that by having them tackle real-world, problem-based tasks together will yield more positive results. Lastly, this learning progression will only include formative assessment since all work will be done in groups and I want to assess discussion and inquiry not mathematical correctness. Though, before moving on there will be a section of the lessons where I will help students through effective questioning reach the correct answers almost like how the Socratic Method does it.
Lesson 1: Problem-Based Task: Free Checking Accounts Coaching
The figure to the left is of the first t
ask of the learning progression. As shown in the figure the task is tied directly to the CCSS, A-CED.1 (HS is left off because it is understood this text is for high school mathematics 1). This text has tasks in order of what is required to learn the following concept not just in order of CCSS. Thus, this text incorporates scaffolding to effectively organize tasks and CCSS. The figure above describes a word-problem but it isn’t simply just a word-problem; it is a word-problem that describes a real-world scenario that students know they may one day be faced with creating an innate interest within them to learn how to solve the problem. On the next page of the text shown in the figure there are several questions posed related to the above word-problem:
When beginning this first lesson I will go over the CCSS and break it down for the students to better understand what is expected of the to learn from this task. I will then allow students to start to work through the given problems I have listed above as I walk around and use inquiry-based learning and the Socratic method to invoke deep, and critical thinking. I know that my students are capable of deriving the solution to the posed problems with minimal hints from me because I have taught them the necessary problem-solving skills and mathematical knowledge required to; however, like I said prior I did not provide any examples of procedures. Thus, students will struggle with ideas and procedures but as I walk around the room I may hint or pose effective questions to a group to help them solve a problem.
This task requires students derive an inequality and know what that inequality says about the posed problem; however, the students have not learned anything about inequalities but what they have learned is the structure of one variable equations and how to derive them from given information. The hint I will have to present to the students once I have noticed they have all gathered the information and are discussing how to express it as an equation is that an inequality is written just like an equation and would then present the appropriate syntax and notation to them. The point of this is that the students have been in deep discussion and thought about how their group can express the information after struggle to find an answer it will click for them with the information I present. Because the students know the procedure for deriving a one-variable equation from given information, I know that they are capable of deriving a one-variable inequality once I have given them that hint. But if any of the students have not derived the inequality or did not understand how their group did, I would in the last five minutes of class go over the procedure for deriving the inequality and what the correct inequality is. Their homework would then be to answer that final question (e) and be prepared to discuss it the following lesson. This would allow the students to continue to either find the solutions on their own or struggle with the concept and be confused the following lesson. Why is it good for a student to be confused the following lesson and be unable to finish his/her homework, because I know that they are on the precipice of understanding the concept and with a little nudge they will get it.
Lesson 2: Problem-Based Tasks: Skate Constraints
As shown in the figure the next lesson covers A-CED.3. This lesson is quite a bit more extensive and may require two days to complete. The reason the previous lesson was less extensive is because the students will be continuing to derive inequalities. The first lesson was meant to have students struggle with the idea of whether or not the procedure of deriving a one-variable equation 
could be applied to deriving a one-variable inequality and what an inequality represented in terms of the given information. Now the students will continue to struggle towards complete mastery of those concepts while they learn more and more about inequalities till they have all of the learning blocks necessary to have mastered inequalities. Thus, the students will be using the previous lessons knowledge in this lesson.
I will be using the same lesson outline but with a few minor changes. Instead of letting the students struggle through the posed problems immediately after going over the CCSS and directions I will go over some definitions found in the CCSS: constraints, viable and nonviable. I will need to provide an example of what a constraint is and explain why they are necessary. I would use the previous lessons inequality to do and explain how it would not make sense to have negative deposits hence there must be a constraint stating that “x” constraints must be greater than or equal to zero. I would not tie this example to the students’ posed problems in this lesson because again I want them to become confused and struggle to find the solutions. Once I had finished going over any necessary information I felt like would make the task impossible to do without I would let the students loose to begin working in their groups. The text provides these following posed problems:
This task requires that students create two-variable inequalities which is very new because the students have not worked with two-variable equations yet so they are bound to struggle with how to express the information given as an equation; however, they have the necessary knowledge and problem-solving skills so with I am hoping that with no hints the students will derive the equations themselves. As I did in the previous lesson though, if no students have managed to derive the correct two-variable inequalities I will go over the right way to solve the first inequality maybe 30 minutes into the class so they will have to struggle with the constraints and what the inequalities represent. Then, in the following lesson I would we would go over as a class all of the problems and together answer questions (k).
Lesson 3: Problem-Based Tasks: Bricklayers
This is the last lesson of the learning progression. Now this lesson if important and though it does not have to do with inequalities at first glance it actually does; it has to do with all equations and expressions.
This learning progression is designed for a 10th grade Geometry Class. In this unit students learn about transformations in the plane such as translation, reflection, rotation and glade reflections. The CCSS are
CCSS.MATH.CONTENT.HSG.CO.A.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). CCSS.MATH.CONTENT.HSG.CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. CCSS.MATH.CONTENT.HSG.CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. CCSS.MATH.CONTENT.HSG.CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another
To teach this material I use different teaching methods as well as different activities to elicit students learning. For example, I use modeling to work detail examples and give students a visual aid in the board. At the beginning of the lesson, I use a short warm up that helps students review the material learned in the previous lesson. During and after instruction, students also have the opportunity to work with others and ask questions. To assess students learning, I use their responses to class discussions, questions, and answers to problems in different activities For summative assessment, in this learning progression students will show their knowledge by preforming the different transformations to a shape they select and writing explanations of the changes applied to the figure.
Complete learning progression here High School Geometry
Addressed CCSS for Mathematics:
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.
Addressed Mathematical Practices:
Learning Progression Overview:
The lesson will open up with a brief recap to refresh the students on the prerequisite knowledge that is required. After this, the instructor is to present the concept of an imaginary number. The presentation follows the same format as described in the textbook, “Algebra 2”. To put imaginary numbers into context, the instructor will provide a quadratic equation, such that an imaginary number is produced, for the students to solve. Upon deriving the complex number, the instructor will introduce the properties of the imaginary number, i and complex numbers. The students will then transition into their first entry task of performing operations in complex numbers, beginning with addition and subtraction. Succeeding this, the students will learn proceed to their second entry task of multiplying complex numbers, employing their knowledge of the distributive property. The third entry task will center on the utilization of the conjugate to simplify instances of a number being divided by a complex number. Throughout, the instructor is to prompt the class with short “checks” in the form of verbal questions to assess the progress of the class. Finally, the students will be administered an assessment to gauge their overall mastery of the material. The learning progression can viewed at the link below.
The above attachment is a short assessment of HSG.GMD.B.4. It has students interpret the 3-dimensional objects created by two cross-sections which are circles. Then it has them solve a problem using what they know about the 3-dimensional objects created by rotating those cross-sections.
A-REI Solving Systems of Equations
Alignment 1:
A-REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
A-REI.C.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x2 + y2 = 3.
Solve the following systems of equations. First, find an approximate answer (using a graph) and then algebraically by the given method.
a. x-2y=10
3x+y=9
Graphically: Algebraically using Substitution:
b. 6x+3y=18
3x-3y=9
Graphically: Algebraically using Elimination:
c. y=x2+3x-4
y=x-1
Graphically: Algebraically (student’s choice):
d. y=2x-2
x2+y2=4
Graphically: Algebraically (student’s choice):
Assessment Commentary and Solutions are attached here: im-assessment-solving-systems-of-equations
Alignment 1: HSA.SSE.B.3.A
Factor a quadratic expression to reveal the zeros of the function it defines.
Solve the quadratic equation for x by factoring. If your answer is an improper fraction, state it as a mixed number.
For more information and solutions: illustrative-math