missliebschteaches.weebly.com · web viewstandard 2: knowledge of reasoning and proof. candidates...

Math PortfolioStandardsProcess Standards (Standards 1-7)The process standards are based on the belief that mathematics must be approached as a unified whole. Its concepts, procedures, and intellectual processes are so interrelated that, in a significant sense, its “whole is greater than the sum of the parts.” This approach would best be addressed by involvement of the mathematics content, mathematics education, education, and field experience faculty working together in developing the candidates’ experiences.Likewise, the response to the disposition standard will require total faculty input. This standard addresses the candidate’s nature and temperament relative to being a mathematician, an instructor, a facilitator of learning, a planner of lessons, a member of a professional community, and a communicator with learners and their families..Standard 2: Knowledge of Reasoning and ProofCandidates reason, construct, and evaluate mathematical arguments and develop an appreciation for mathematical rigor and inquiry.Indicators2.1 Recognize reasoning and proof as fundamental aspects of mathematics..

Rationale: I enjoy problems that ask students to reason why formulas work or why math concepts exist, because the more sense a student can put into their own language and learning the more they will develop. It isn’t just rote memorization then, but actual critical thinking that will advance the students learning. Here they are even asked to make a conjecture about an idea that have not been taught but can be easily seen in the relationships of the work they are doing. Even if a student is wrong, questions like this is a great way for a teacher to understand how much of the concept they understand and how they look at questions like this one.

NCTM Standards (2003) – Elementary Mathematics Specialists 22.2 Make and investigate mathematical conjectures.16.1 11. Find the missing sizes of the angles marked for each question and make an educated guess (a conjecture) about the sum of the sizes of all the exterior angles of every quadrilateral, using just one exterior angles at each vertex and give a justification that your conjecture will always be correct

This problem first applies reasoning that the students will have learned in class that every straight line is 180 degrees, so when looking at the exterior angles if the known angle on the interior then a simple equation of taking 180 minus the interior angle will give you the exterior angle. However it also asks students to find interior angles and make the connection between the sum of the interior angles and their relationship to exterior angles.

2.3 Develop and evaluate mathematical arguments and proofs. 4. When asked to find the sum of the interior angles in a hexagon a student writes the following. Comment on whether the student’s mathematical reasoning is correct or incorrect. If it is correct, explain how you know. If it is incorrect, explain what was incorrect about the student’s thinking and what he/she would have to do to correct the error.

A: While the student is on the right track, she is missing a few steps. She was smart to break down the polygon into triangles, and now that she has done that, it is actually easier to explain her mistake. If we simply times 180 by 6 that means we are accounting for the ALL of the angles in each triangle she drew but they aren’t part of the exterior angle. So to get rid of those interior angles we do not we subtract 2 from the number of the side of polygon (effectively we are also subtraction the two sides of the triangles that make up the shape that we are not using). Therefore, the formula she should have used was really very, close to what is shown, except she forgot to subtract 2 from the polygon side shape.2.4 Select and use various types of reasoning and methods of proof.

1. Is it possible for the number of edges, the number of faces, and the number of vertices of a polyhedron to all be even? If so, give an example. If not, explain why not.

A: This problem directly relates to Euler’s Formula which states for any polyhedral the number of vertices plus the number of faces will equal the number of edge plus 2, or V+F=E+2. Using Euler’s formula a table can be made to begin to see if the all numbers could be even, so for this table only even vertices and faces will be used.

Vertices Faces Edges 2 2 2 4 4 6

It is a small start, so I then took some of the polyhedron manipulatives made in math class to test my hypothesis. Using shape A, which is a pyramid, I counted the faces (4) and vertices (4) and edges (6). For shape E, which is a cube, I counted the faces (6) and the vertices (8) and used Euler’s Formula to figure out the edges; 8+6=14-2, 8+6=12. All even numbers.

This problem is great for showing students why double checking work is important. In order to feel like a valid answer, it should be shown at least two ways how it is possible or not possible.

Standard 4: Knowledge of Mathematical ConnectionsCandidates recognize, use, and make connections between and among mathematical ideas and in contexts outside mathematics to build mathematical understanding.

Indicators4.1 Recognize and use connections among mathematical ideas.5. Find the measure of angle A of the triangle. Explain each step/connection to get to angle A. (NOTE: Angle A, B and C are the INTERIOR angles of the triangles at those given points.)

A: This Problem requires students to solve the angles in a particular order before being able to solve for angle A, if they don’t make this connection the problem will seem almost impossible to solve. Students must also think of multiple ways to solve for angles as this problem requires 2 different ways and one is dependent on the correct application of the other. It’s a great problem to cover multiple steps and understanding student thinking about angles. 4.2 Recognize and apply mathematics in contexts outside of mathematics.4.3 Demonstrate how mathematical ideas interconnect and build on one another toproduce a coherent whole.

This problem helps show how different shapes can cover a space. What I love about tessellations is that in completely involves math but most of the instances where tessellations are found are in architecture or art. It completely involves the students to think of math in a whole different way and apply it to the world around them.

Standard 6: Knowledge of TechnologyCandidates embrace technology as an essential tool for teaching and learning mathematics.Indicator6.1 Use knowledge of mathematics to select and use appropriate technological tools, such as but not limited to, spreadsheets, dynamic graphing tools, computer algebra systems, dynamic statistical packages, graphing calculators, data-collection devices, and presentation software.11.2 Build and manipulate representations of two- and three-dimensional objects using concrete models, drawings, and dynamic geometry software.

A: Using this program was a great experience and even though I am not usually a fan of computers in early childhood classrooms this is a program I can stand behind because students still need to understand the basics of building a shape and how translations, rotations, etc work in order to work the program.

Standard 8: Knowledge of Mathematics PedagogyCandidates possess a deep understanding of how students learn mathematics and of the pedagogical knowledge specific to mathematics teaching and learning.Indicators

8.1 Selects, uses, and determines suitability of the wide variety of available mathematics curricula and teaching materials for all students including those with special needs such as the gifted, challenged and speakers of other languages.NCTM Standards (2003) – Elementary Mathematics Specialists 48.4 Plans lessons, units and courses that address appropriate learning goals, including those that address local, state, and national mathematics standards and legislative mandates.8.7 Uses knowledge of different types of instructional strategies in planning mathematics lessons.

MAT

HTessellation

Intro: Game and PowerPoint, LOOK FORS:

student background

knowledge in discussions

between what is and isn’t a

tessellation, and math and art.

Tessellations and Math:Using manipulatives have students figures out what

shapes can make tessellations, study the relationships in small

groups-do a problem talk to figure out if they can

express a formula to test whether a shape can

tessellate or notLOOK FORS: observe and

note take what groups and individuals are saying

and their thought process, prompt with

questions where necessary

CCSS.Math.Content.4.OA.A.1

Interpret a multiplication equation as a comparison,

e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative

comparisons as multiplication equations

Make it Tessellate!

Student will create their own

tessellations using either

graph or plain paper. Students

may cut out shapes or draw

them out. On the board

will be a several other drawn shapes that

students can identify if the tessellate for

extra discovery work for those who work fast and partially

started graphs sheet will be available to

accommodate students

Look FORS: students to

relate what they studied about

shapes yesterday to

understand how to create one

today

Math in Real Life:Brainstorm: Where do we

see tessellations in real life?

5-10 min discussionDiscovery Field Trip:

Take the students on a walk around the school, possibly outside to take

notice of any tessellations they see, I will take

pictures and have them printed

When students return have a recap discussion of what the noticed and why they think tessellations might

be relevant in the real world.

CCSS.Math.Content.4.MD.A.3

Apply the area and perimeter formulas for rectangles in

real world and mathematical problems.

Classify two-dimensional figures based on the

presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a

specified size. Recognize right triangles as a category, and identify right triangles.

Final Project Study:

As a large end of year

assessment students will

create their own concept pages

for items learned to be

part of a reflection book. Today students will work on the

initial draft of their

tessellation concept page, pictures from the Discovery Walk will be

available as well as foam pieces, magazines, and

graph paper. Creativity is

encouraged but the concept

must be expressed

completely and correctly on the

page. Summative

Assessment:Possible items students can include, must

be at least two: Name of

concept, rules that govern

what is and isn’t a tessellation, examples and non-examples of objects that

tessellate, formula for if an

object can tessellate,

examples of real

http://www.corestandards.org/Math/Content/4/MD/A/3/

http://www.corestandards.org/Math/Content/4/MD/A/3/

http://www.corestandards.org/Math/Content/4/OA/A/1/

http://www.corestandards.org/Math/Content/4/OA/A/1/

Monday Tuesday WednesdayThursday Friday

Lesson Plan FormatNAME: Ellie Liebsch

Lesson Title: Tessellations 1 Grade level4 Total Time: 30 min

# Students: 20

Learning Goal:(Content Standard/Common Core)

CCSS.Math.Content.4.OA.C.5Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itselfClassify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.

Target Goal or Skill: Have student begin a concept of what a tessellation is

Essential Question(s): How is a tessellation described?

Topical question(s): What is a tessellation

Instructional Objective(s):

Students will be able to define what makes a tessellation

Assessment (Criteria / Look Fors/ Performance Tasks)

Formative Assessment: SlideshowTo understand what the students background knowledge might be, take observational notes on answers as students view each picture, use small whiteboards or sheet protectors to have students answer

Summative Assessment:

Disabilities/Diverse Needs Represented Student Accommodations and/or Modifications

Provide printouts for students whom notetaking may be hard, prompt with questions for pictures on the slides that may confuse students

Instructional Procedures(including specific times)

Introduction:(including motivational hook where applicable)

Learning Activities:

Closure:

Introduction: Start students off on tessellation game, let them play for about 10 minutes before prompting them to look at the game board, and how it the shapes are used in point play. Allow the students to play for about 5 more minutes before having a 5 minute grand discussion about what they noticed in the game. (20 min total)Game link: http://calculationnation.nctm.org/

Using PowerPoint from NCTM Illuminations page http://illuminations.nctm.org/Lesson.aspx?id=5903 Walk students through pictures of examples and non-examples of tessellations. (10 min)

Close: Remind students what factors must be met for it to be defined as a tessellation.

Language Demands: Function Vocabulary Syntax Discourse

Vocabulary Tessellation

Grand Discussion:

5 Questions (Bloom’s or DOK)

Curriculum (APA)e.g. Investigations in Number, Data, and Space. (2012). Pearson.

Materials ProjectorLaptop(s) if students are playing on own instead of as a class

Notes This lesson is mostly about the discussion of what a tessellation is and is not. This day is to give students a sturdy foundation.

http://illuminations.nctm.org/Lesson.aspx?id=5903

http://calculationnation.nctm.org/

Lesson #2 Lesson Plan Format

NAME: Ellie Liebsch

Lesson Title: Tessellations 3 Grade level: 4 Total Time: 30 min

# Students: 20


CCSS.Math.Content.4.OA.C.5Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself

Target Goal or Skill: Students understand the concepts and rules of tessellation well enough to build their own

Essential Question(s): What components make up a tessellation?

Topical question(s): How are tessellations created?


Students will create their own tessellation pictures


Formative Assessment: ObservationListen and observe the conversations students are having as they build their tessellations, prompt students who are struggling to remember either the formula for tessellating objects or rules for what a tessellation is. Student record math thinking while creating tessellation for teacher review

Summative Assessment:


Provide graph paper, cut out shapes, leveled pre-made tessellations to accommodate students

Extra shapes to test tessellations for students who work quickly




Closure:

Introduction: Review slideshow from Monday along with pages from http://www.tessellations.org/tess-escher1.shtml

Today you will be creating your own tessellations! You may use plain or graph paper, cut shapes or drawn. Be creative with your shapes, keep any math thinking on a separate piece of paper to be turned in for feedback.

Observe students as they work, talk with as many students one on one about what their thinking is, encourage discussion with other students about possible places other than are, one may find a tessellation.



AnglesRepetition



Materials Plain paper, gridded paper, foam cut out shapes, rulers, extra pencils, markers, colored pencils, crayons, scissors, glue

Notes

http://www.tessellations.org/tess-escher1.shtml

Lesson #3 Lesson Plan Format

NAME: Ellie Liebsch

Lesson Title: Tessellations 5 Grade level: 4 Total Time: 30 min

# Students: 20


CCSS.Math.Content.4.OA.C.5Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself

Target Goal or Skill: Students shown a full and sturdy understanding of tessellations

Essential Question(s): How can we teach and learn about tessellations?

Topical question(s): What do tessellations mean to you?


As part of a year-long project students, students will create a first draft to their tessellation concept page


Formative Assessment:

Summative Assessment: This page is a summative assessment; students are encouraged to be creative and free in how the page is displayed and presented but must make sense of the concept and include at least 2 of the following: Name of concept, tessellation formula, rules for tessellations, explanation of angles in tessellation, examples and non-examples of tessellations, real life examples of tessellations.


Provide graph paper, cut out shapes, leveled pre-made tessellations, typed words, pictures from Discovery Walk, magazines to accommodate students




Closure:

Introduction: student review session, students and teacher all participate in discussion and Q&A sessions for any questions about concepts, pictures are up and available to students. After students feel comfortable they are set to get to work on their first drafts of concept page

Constant communication and check in with students to see how they are doing is needed, this time instead of readily prompting the student, as them first what they think they need to do in order to answer their own question.



AnglesRepetition



Materials Plain paper, gridded paper, foam cut out shapes, rulers, extra pencils, markers, colored pencils, crayons, scissors, glue

Notes

Rationale: These lessons are developed to allow students team work and individual work, it brings real life into math as well as art. The Problem Talk is a great way for students to feel express themselves in math without fear of being wrong in front of the whole class, students can scaffold off of each other as they walk through what each person knows. The groups all have a facilitator to make sure everyone gets heard and for the teacher, the student’s thinking is displayed right in front of them. The hands on experiences also allow plenty of accommodation for exceptional learners without sacrificing practice or content.

Content (Standards 9-13)Candidates’ comfort with, and confidence in, their knowledge of mathematics affects both what they teach and how they teach it. Knowing mathematics includes understanding specific concepts and procedures as well as the process of doing mathematics. That knowledge is the subject of the following standards.Standard 11: Knowledge of GeometriesCandidates use spatial visualization and geometric modeling to explore and analyze shapes, structures, and their properties.Indicators11.1 Use visualization, the properties of two- and three-dimensional shapes, and geometric modeling.Section 17.2 Question 1Which of the shapes A-K meet the criterion given in each statement?

A: All the faces are parallelogramsB, E, F

B. A base is pentagonal regionG

C. All the triangular faces are equilateralA, C, H

D. None of the triangular faces is equilateralD, G, I, J

E. Two faces are parallel and congruent, but the shape is not a prismH, J, K

F. More than one face is an isosceles trapezoid regionK

G. All the edges are congruentA, E, H

H. No two edges are parallelA, G

Using cutouts labeled A-K students can physically manipulate the objects to help them answer questions that would otherwise be quite difficult to solve. Especially when beginning students off in area of geometry, a teacher can’t expect students to be able to visualize 3D shapes in their head without having seen them before. Even simple shapes like cubes can be hard for students to visualize and manipulate in the heads. Here the question asks to sort a variety of 3D objects based on a feature they have. To do this even with 2D drawn pictures of flats would be hard for many students.

11.2 Build and manipulate representations of two- and three-dimensional objects using concrete models, drawings, and dynamic geometry software.A: Using this program was a great experience and even though I am not usually a fan of computers in early childhood classrooms this is a program I can stand behind because students still need to understand the basics of building a shape and how translations, rotations, etc work in order to work the program. 11.3 Specify locations and describe spatial relationships using coordinate geometry.11.4 Apply transformations and use symmetry, congruence, and similarity.

What I love most about this problem is that it not only asks student to mathematically manipulate an object, by estimates none the less, but it also asks students to describe what they find difficult. It starts a conversation about math and gives the students a chance to express what they find the hardest without embarrassment. This problem can be a formative assessment to see how well students understand an idea and what information may be the most difficult and thus re-explained in a new way.

Standard 12: Knowledge of Data Analysis, Statistics, and ProbabilityCandidates demonstrate an understanding of concepts and practices related to data analysis, statistics, and probability.Indicators12.1 Design investigations that can be addressed by creating data sets and collecting, organizing, and displaying relevant data.13.1 Select and use appropriate measurement units, techniques, and tools.

Rationale: The reason why I chose two standards for this problem was that it invariably needs accurate measuring and knowledge of how to measure in proper units as well as being a multi-tiered problem that gives the students one key piece and then asks them to collect the variables and figure out the best way to solve the problem. This is a word problem that forces students to critically think and use more than just one type of math knowledge. In order to find the distance to Beantown, which is not pictured on the scale map the students have to find the distances between San Carlos and River City and then figure out the scale factor. As they work through the problem, it becomes clearer what each next step to be as it becomes the only logical step and a shape between the three cities starts to form; and now the problem becomes less about map distances and more about geometry and similarities. However, since the answer does require a distance the context of the problem is never really lost. It’s a great example of bringing real life into math. On a road trip there may be places pictured on one map that is not on another but the maps don’t have the same scale so figuring out the distance requires one to create a similar object.

12.2 Use appropriate statistical methods and technological tools to analyze data and describe shape, spread, and center.

Rationale: This is a great question for students. Not only does it deal with numbers but it has students explore what information can and cannot be extrapolated from the information. Then it also asks students to infer data from the graphs as well.

12.3 Apply the basic concepts of probability.

Rationale: These problems show different ways to answer probability questions. Probability is a difficult concept for students to understand so it’s important they have multiple strategies when tackling these problems.Standard 13: Knowledge of MeasurementCandidates apply and use measurement concepts and tools.Indicators13.2 Recognize and apply measurable attributes of objects and the units, systems, and processes of measurement.13.3 Employ estimation as a way of understanding measurement units and processes.

Rationale: This problem works well for these indicators as students need to apply situations, not just numbers, to units-which also applies meaning to the units. Students may not think of time as a measurement until they are given a situation in which time need to be measured. Since this are categorizing questions students also get a chance to understand we can apply estimations of units to a category without having to expressly show the number.

missliebschteaches.weebly.com · web viewstandard 2: knowledge of reasoning and proof. candidates...

Documents