Digital Fabrication and Aunt Sarah: Enabling Quadratic Explorations via Technology. Michael L. Connell University of Houston  Downtown


 Samson Miles
 4 years ago
 Views:
Transcription
1 Digital Fabrication and Aunt Sarah: Enabling Quadratic Explorations via Technology Michael L. Connell University of Houston  Downtown Sergei Abramovich State University of New York at Potsdam Introduction When viewed as a content area, mathematics has a split personality. To use an example from language, there are parts of mathematics that function very much like a noun (the concepts of mathematics), while others function more like a verb (procedures, which many think of as actually doing math ). Such ideas form the basis for later, more formalized procedures. The role technology can play in visualizing these ideas for learners should not be overlooked. The following graph, for example, was created using a computerbased spreadsheet but could just as easily have been done using a smartphone app, an online graphics program or a handheld calculator. Figure 1. Graph of Y = 2X + 3. Things get a little complicated when the mathematics described has both noun and verblike features (i.e., requiring understanding of both content and process components). For example, the number 2 can be a noun describing a position in a sequence or how many of something one might have. In this case, we are clearly using the nounlike features. In a different context, however, 2 can describe: how many times something appears as in the case of filling a bowl of cereal 2 times; a base used by computers to represent other numbers (this is also called Binary); or the power to which a quantity is raised as shown in X For students to develop meaningful mathematical understandings, they should have many rich experiences in mathematics from these two markedly different perspectives. So, as we select appropriate technology we need to allow them to experience mathematical structures containing both concepts to think about the nounlike content features, and processes to think with the verblike procedural features. Once a teacher can see this dualism about mathematics, it has major impacts on potential roles of technology in the mathematic classroom. This can be shown very clearly when considering multiplication strategies. Multiplication is used to compute area, and area can be used to illustrate multiplication so both the concept and procedure can be illustrated at once. The Algebra Tiles application from the National Library of Virtual Manipulatives (found at ml ) was used in Figure 2 to model (X+1)(Y+2).
2 Figure 2. (X+1)(Y+2) In Figure 2 we see a rectangle being formed from placing representative tiles along two dimensions: X+1 in the vertical direction, and Y+2 in the horizontal direction. The resulting algebraic product is shown by the area itself. To fill this rectangle the student needs to use an XY piece, two X pieces, one Y piece, and two single squares. When this is written out in standard form it shows that (X+1)(Y+2) = XY + 2X + Y +2. In order to get to this point, however, students need to be able to utilize both the conceptual and procedural aspects of the representation created through interaction with this application. Digital Fabrication. One of the major tasks in the high school environment lies in developing the ability to work with abstract concepts. With its ability to provide an interface between functions and graphs technology can be an invaluable tool in this effort. The figures shown in this section were created with the Graphing Calculator (version 4.0) produced by Pacific Tech (Avitzur, 2011). There are a number of alternatives which could be used to do traditional graphing (Figure 3); yet, there exist very few tools (e.g., Wolfram Alpha) capable of graphing segments or their borders (Figures 4 and 5). It is a tribute to the power of technology that once the math is understood there are a variety of tools which might be used appropriately. The opposite is also true, however. If the math is not understood it does not matter what kind of tools you have access to. Consider the question of constructing the graphs of the functions y = x and y = x 2 in a single drawing. This construction is shown in Figure 3 and leads to the question of constructing just the parabolic segment and its reflection on the line X=1 as shown in Figure 4. Figure 3. Y=X and Y=X 2 Figure 4. A parabolic segment and its reflection in the line X=1. In order to construct the parabolic segment, one has to describe the points inside it in the form of inequalities. First, an xcoordinate of any point (x, y) that belongs to the parabolic segment satisfies the inequalities 0 < x <1, where x = 0 and x =1 are the points of intersection of the graphs y = x and y = x 2. Second, its ycoordinate satisfies the inequalities f (x) < y < g(x) where f (x) = x 2 and g(x) = x. These properties of the points that belong to the parabolic segment can be expressed in the form of simultaneous inequalities x  y > 0, y  x 2 > 0,x > 0,x <1.
3 In addition, the reflection of the parabolic segment in the line x = 1 can be expressed through another set of inequalities by substituting 2 x for x (2  x)  y > 0, y  (2  x) 2 > 0,(2  x) > 0,(2  x) <1. Figure 5. Digital fabrication of e thick borders of the parabolic segment and its reflection. Likewise, the set of points that belong to the border of the parabolic segment can be described through inequalities. First, the graph of the upper border (a part of the line y = x) can be described as a set of points (x, y) for which the values of the coordinates x and y are e  close to each other; that is, y  x < e. Second, the graph of the lower border (a part of the parabola y = x 2 ) can be described as a set of points (x, y) for which the values of y are e  close to the values of x 2. Finally, once again, the inequalities 0 < x <1 characterize the points that belong to the border. In the context of the Graphing Calculator these properties of the points that belong to the border of the parabolic segment can be expressed in the form of the union of simultaneous inequalities é y  x < e, x > 0, x <1; ê ë y  x 2 < e, x > 0, x < 1. Adding another union of simultaneous inequalities é y  (2  x) < e, 2  x > 0, 2  x < 1; ê ë y  (2  x) 2 < e, 2  x > 0, 2  x <1 yields the righthand side of the digital fabrication shown in Figure 5. Note that in Figures 4 and 5 e = Using technology to enable students to construct graphs of areas in the plane and their borders by using twovariable inequalities illustrates the way in which software can embody a mathematical definition (Conference Board of the Mathematical Sciences, 2001, p. 132). Aunt Sarah and the Farm. Aunt Sarah wants to help her nephew Jack. However, she does not want to simply give him money. Instead she will provide him with a 10 dkm x 10 dkm plot of land provided he keeps it fenced. At the end of the year she will reduce the width by 1 dkm and increase the length by 1 dkm so that in the second year he will have an 11dkm x 9 dkm plot. This will be done each year until there is nothing left but a fence (i.e., 20 dkm x 0 dkm). This way it will be up to Jack to work hard and make the most of this opportunity. Help Jack explore what to expect over the next 10 years. As a start, for each year find: 1. How much land will Jack lose from the preceding year?
4 2. How much land will Jack lose from the first year? 3. How will the shape of his farm change over time? 4. How many feet of fencing will it take to fence it in? The mathematics which underpins the Aunt Sarah problem allows for multiple competencies, both on the part of the teacher and that of the student, to be addressed. Without the use of supporting technology it typically takes several days of tedious calculations for sufficient data to be generated to get to the richer underlying mathematics. Thanks to the spreadsheet, the explorations of Aunt Sarah s farm allow more time to be spent on building connections between deeper levels of mathematical content than was previously possible including a powerful link forward from prealgebra into limits and precalculus. It is recommended that initial problem setting and procedure choices be done prior to the introduction of the spreadsheet. A good teacher question to ask at this point is: Is there a way to predict what happens in the 5th year? The 6th? With the addition of spreadsheet an excellent bridging question is, How can we organize our work to make prediction easier? This last question quite commonly leads to a row and column layout which can be directly translated into a spreadsheet later. In exploring Aunt Sarah and the Farm problem, prior to the introduction of the spreadsheet, some fascinating mathematics can be shown. If you draw out what the farm would look like each year on a single figure, one will get the following: Figure 6. The changing shape of the land From here, the possibilities for exploration open up. For example, to show the land lost for any given year relative to the beginning year (the fourth year is shown), take the rectangle of land gained for that year (A), rotate it (B and C), and place it inside the original figure to show the total amount lost (D). Figure 7. Where the land is going from Year One It can quickly be shown that the land lost for each year that this is done will be a perfect square which certainly hints as some interesting patterns to come! To show the land lost for any given year relative to the preceding year (the difference between the fourth and fifth year is shown) take the rectangle of land gained for that year (A), rotate it (B and C), and place it inside the preceding figure to show the total amount lost (D).
5 Figure 8. Where the land is going from the preceding year These sketches both represent specific processes and the solution to problems. This preliminary exploration provides a context for the following spreadsheet explorations as well as providing important clues for exploration. The following screenshot from shows one possible way of representing the problem situation. Figure 9. Data and calculations confirmed using a spreadsheet When relationships between cells are observed and can be generalized the formula bar can be used to create many of the cells, for example cell B12 was defined as being: Figure 10. The Formula (Function) bar. This makes it easy to copy cell B12, together with its attributes, and easily copy these filling in the respective columns. This ability to copy relationships between cells, including functional relationships, helps in the students understanding and exploration of the mathematical situation. In a like fashion each of the following cells can be defined using the formula bar as being: Figure 11. Using the Formula (Function) bar. The increment in year was then defined in cell A13 as being: Figure 12. Last step in creating the spreadsheet
6 Once relationships are recognized and their underlying functions identified, it becomes easy to create meaningful tables of values. In this example, we can see this by copying cell A13 into cells A14 through A22. In a like fashion it is possible to copy cells B12, C12 and D12 into cells B13 through B22, C13 through C22 and D13 through D22. This is a bit different than the typical use of data tables serving as the basis for function identification. In this case, the function is created first and used to create a table of data for exploration. An examination of the formula bar for Column B (fx=10+a12), Column C (fx=10a12), and Column D (fx=b12*c12) provides a possible avenue to explore the concept of difference of squares (i.e., the length (10+A12) and width (10A12)) being used in the area calculation. In this case, Column D s function is equivalent to B12*C12 which in turn is equivalent to (10+A2)*(10A12). Depending upon the classroom, this may not be followed up, but it does provide an important clue which could be utilized in further exploration into the mathematics underlying the Aunt Sarah and the Farm problem. By making explicit the relationships between cells the formula bar can often be used in this fashion to gain hints as to potential mathematical underpinnings. The mathematics which may be found beneath the rules can then be made available for student explorations. The remaining columns look at some of the other interesting interactions immediately springing from the problem situation. Each cell in Column E, E13 for example, was computed using the following convention: Figure 13. Differences from the preceding year When this is done the sequence of odd numbers is generated, leading to questions concerning where this shows up in the graphical and functional representations generated in the group activity. Column F was generated using the following: Figure 14. Differences from the first year The $ sign preceding the D and the 12 indicates that this location will be locked in and used as the reference for each of the cells generated by copying it. This ensures that each subsequent years difference will be computed taking the first year as the comparison. Now a sequence of squares is generated, once again leading to questions concerning where this shows up in the graphical and functional representations generated. Using these columns the following graphs were generated: Figure 15. Data to function to graph to story! It is now up to the students to describe which series gives rise to each graph and why. They should also be able to link their graphic representation created prior to the use of the spreadsheet (typically, done using graph paper) to these graphs.
7 An important conclusion that one can draw from this investigation is that given perimeter of rectangle, no smallest area exists whereas square (that is, rectangle with congruent adjacent sides) has the largest area. However, as noted by Kline (1985), A farmer who seeks the rectangle of maximum area with given perimeter might, after finding the answer to his question, turn to gardening, but a mathematician who obtains such a neat result would not stop there (p. 133). This note motivates extending Aunt Sarah and the Farm problem using the computational power of a spreadsheet. Technology enabled extensions. Of course, technically, in order for a line graph to be properly used a case must be made that there will not be any changes in the line as the difference between sampling times becomes infinitely small. This provides an easy link to the calculus which may be made via the spreadsheet. This can be shown by first changing the spreadsheet so that the change point occurs every month instead of every year. This action effectively changes the difference between points on the line graphs by 1/12. This is easily done by changing cell A13 to be: Figure 16. Links to advanced math Now we can reconstruct the earlier graphs using this more finely tuned set of measurements. When this is done the graphs created look like the following: Figure 17. Identical curves. This is the identical shapes as shown in the earlier set of graphs. The underlying equivalency can be better shown by changing the chart type to not plot the locations of the individual data points. In a like fashion we can narrow the limit to the day, the hour, the minute to any degree we might choose in each case since the underlying functions are the same the graphs will maintain the same shape! Technology has enabled us to develop in a very intuitive fashion the notions of limit which underpin calculus. Without technology this amazing development is not possible. Summary If we take the studentcentered and meaning driven approach to mathematics education advocated in this paper, the question becomes what tools and abilities are necessary for success and how can educational technology be used as a tool in acquiring these? This is a crucial question as the nature of the "tools" which are provided to students to "thinkwith" come to significantly shape their performance and cognitive styles (Connell, 2001). For example, twodigit division may constitute a legitimate problem when paper and pencil are the only tools available
8 for the student to use but are no longer a problem when calculators are available. When a computer is available for the students use, the situation shifts again. A legitimate problem with a computer might involve the identification and selection of what data to include in the problem, identification of the problem goals, and selection of appropriate procedures and control statements to obtain and verify the desired results. Let us be careful not to transfer a misplaced belief that mathematics education is solely about developing speed of process over to our thinking about technology uses. Modern technology is capable of blinding speeds of process so this cannot be viewed as our end goal. If a student is to internalize and construct meanings from experiences, there must be time to reflect upon the nature of the experiences and how they connect with the students' existing mathematical knowledge (Abramovich & Connell, 2014). Great care must be taken to allow students to construct their own knowledge and representations and then establish the linkages with other (also student constructed) tools, representations, and concepts many of which are technology dependent. References Abramovich, S., & Connell, M. L. (2014). Using technology in elementary teacher education: A sociocultural perspective. ISRN (International Scholarly Research Network) Education, Article ID , 9 pages, doi: /2014/ Avitzur, R. (2011). Graphing calculator (Version 4.0). Berkeley, CA: Pacific Tech. Conference Board of the Mathematical Sciences. (2001). The mathematical education of teachers. Washington, DC: The Mathematical Association of America. Connell, M. L. (2001). Actions upon objects: A metaphor for technology enhanced mathematics instruction. In D. Tooke & N. Henderson (Eds.), Using information technology in mathematics education (pp ). Binghamton, NY: Haworth Press. Kline, M. (1985). Mathematics for the nonmathematician. New York: Dover.
AGS THE GREAT REVIEW GAME FOR PREALGEBRA (CD) CORRELATED TO CALIFORNIA CONTENT STANDARDS
AGS THE GREAT REVIEW GAME FOR PREALGEBRA (CD) CORRELATED TO CALIFORNIA CONTENT STANDARDS 1 CALIFORNIA CONTENT STANDARDS: Chapter 1 ALGEBRA AND WHOLE NUMBERS Algebra and Functions 1.4 Students use algebraic
More informationDublin City Schools Mathematics Graded Course of Study GRADE 4
I. Content Standard: Number, Number Sense and Operations Standard Students demonstrate number sense, including an understanding of number systems and reasonable estimates using paper and pencil, technologysupported
More informationClassroom Connections Examining the Intersection of the Standards for Mathematical Content and the Standards for Mathematical Practice
Classroom Connections Examining the Intersection of the Standards for Mathematical Content and the Standards for Mathematical Practice Title: Considering Coordinate Geometry Common Core State Standards
More informationGUIDE TO THE CUNY ASSESSMENT TESTS
GUIDE TO THE CUNY ASSESSMENT TESTS IN MATHEMATICS Rev. 117.016110 Contents Welcome... 1 Contact Information...1 Programs Administered by the Office of Testing and Evaluation... 1 CUNY Skills Assessment:...1
More informationMath Grade 3 Assessment Anchors and Eligible Content
Math Grade 3 Assessment Anchors and Eligible Content www.pde.state.pa.us 2007 M3.A Numbers and Operations M3.A.1 Demonstrate an understanding of numbers, ways of representing numbers, relationships among
More informationStatewide Framework Document for:
Statewide Framework Document for: 270301 Standards may be added to this document prior to submission, but may not be removed from the framework to meet state credit equivalency requirements. Performance
More informationTabletClass Math Geometry Course Guidebook
TabletClass Math Geometry Course Guidebook Includes Final Exam/Key, Course Grade Calculation Worksheet and Course Certificate Student Name Parent Name School Name Date Started Course Date Completed Course
More informationMontana Content Standards for Mathematics Grade 3. Montana Content Standards for Mathematical Practices and Mathematics Content Adopted November 2011
Montana Content Standards for Mathematics Grade 3 Montana Content Standards for Mathematical Practices and Mathematics Content Adopted November 2011 Contents Standards for Mathematical Practice: Grade
More informationUNIT ONE Tools of Algebra
UNIT ONE Tools of Algebra Subject: Algebra 1 Grade: 9 th 10 th Standards and Benchmarks: 1 a, b,e; 3 a, b; 4 a, b; Overview My Lessons are following the first unit from Prentice Hall Algebra 1 1. Students
More informationExploring Derivative Functions using HP Prime
Exploring Derivative Functions using HP Prime Betty Voon Wan Niu betty@uniten.edu.my College of Engineering Universiti Tenaga Nasional Malaysia Wong Ling Shing Faculty of Health and Life Sciences, INTI
More informationPaper 2. Mathematics test. Calculator allowed. First name. Last name. School KEY STAGE TIER
259574_P2 57_KS3_Ma.qxd 1/4/04 4:14 PM Page 1 Ma KEY STAGE 3 TIER 5 7 2004 Mathematics test Paper 2 Calculator allowed Please read this page, but do not open your booklet until your teacher tells you
More informationMathematics Success Level E
T403 [OBJECTIVE] The student will generate two patterns given two rules and identify the relationship between corresponding terms, generate ordered pairs, and graph the ordered pairs on a coordinate plane.
More informationMathematics subject curriculum
Mathematics subject curriculum Dette er ei omsetjing av den fastsette læreplanteksten. Læreplanen er fastsett på Nynorsk Established as a Regulation by the Ministry of Education and Research on 24 June
More informationJanine Williams, Mary Rose Landon
TInspire Activity Janine Williams, Mary Rose Landon Course Level: Advanced Algebra, Precalculus Time Frame: 23 regular (45 min.) class sessions Objectives: Students will... 1. Explore the Unit Circle,
More informationAre You Ready? Simplify Fractions
SKILL 10 Simplify Fractions Teaching Skill 10 Objective Write a fraction in simplest form. Review the definition of simplest form with students. Ask: Is 3 written in simplest form? Why 7 or why not? (Yes,
More informationMathUSee Correlation with the Common Core State Standards for Mathematical Content for Third Grade
MathUSee Correlation with the Common Core State Standards for Mathematical Content for Third Grade The third grade standards primarily address multiplication and division, which are covered in MathUSee
More informationIf we want to measure the amount of cereal inside the box, what tool would we use: string, square tiles, or cubes?
String, Tiles and Cubes: A HandsOn Approach to Understanding Perimeter, Area, and Volume Teaching Notes Teacherled discussion: 1. PreAssessment: Show students the equipment that you have to measure
More informationStandard 1: Number and Computation
Standard 1: Number and Computation Standard 1: Number and Computation The student uses numerical and computational concepts and procedures in a variety of situations. Benchmark 1: Number Sense The student
More informationExtending Place Value with Whole Numbers to 1,000,000
Grade 4 Mathematics, Quarter 1, Unit 1.1 Extending Place Value with Whole Numbers to 1,000,000 Overview Number of Instructional Days: 10 (1 day = 45 minutes) Content to Be Learned Recognize that a digit
More informationCharacteristics of Functions
Characteristics of Functions Unit: 01 Lesson: 01 Suggested Duration: 10 days Lesson Synopsis Students will collect and organize data using various representations. They will identify the characteristics
More informationMathematics Scoring Guide for Sample Test 2005
Mathematics Scoring Guide for Sample Test 2005 Grade 4 Contents Strand and Performance Indicator Map with Answer Key...................... 2 Holistic Rubrics.......................................................
More informationNCSC Alternate Assessments and Instructional Materials Based on Common Core State Standards
NCSC Alternate Assessments and Instructional Materials Based on Common Core State Standards Ricki Sabia, JD NCSC Parent Training and Technical Assistance Specialist ricki.sabia@uky.edu Background Alternate
More informationDIDACTIC MODEL BRIDGING A CONCEPT WITH PHENOMENA
DIDACTIC MODEL BRIDGING A CONCEPT WITH PHENOMENA Beba Shternberg, Center for Educational Technology, Israel Michal Yerushalmy University of Haifa, Israel The article focuses on a specific method of constructing
More informationGCSE Mathematics B (Linear) Mark Scheme for November Component J567/04: Mathematics Paper 4 (Higher) General Certificate of Secondary Education
GCSE Mathematics B (Linear) Component J567/04: Mathematics Paper 4 (Higher) General Certificate of Secondary Education Mark Scheme for November 2014 Oxford Cambridge and RSA Examinations OCR (Oxford Cambridge
More informationOPTIMIZATINON OF TRAINING SETS FOR HEBBIANLEARNING BASED CLASSIFIERS
OPTIMIZATINON OF TRAINING SETS FOR HEBBIANLEARNING BASED CLASSIFIERS Václav Kocian, Eva Volná, Michal Janošek, Martin Kotyrba University of Ostrava Department of Informatics and Computers Dvořákova 7,
More informationRelationships Between Motivation And Student Performance In A TechnologyRich Classroom Environment
Relationships Between Motivation And Student Performance In A TechnologyRich Classroom Environment John Tapper & Sara Dalton Arden Brookstein, Derek Beaton, Stephen Hegedus jtapper@donahue.umassp.edu,
More informationMath 96: Intermediate Algebra in Context
: Intermediate Algebra in Context Syllabus Spring Quarter 2016 Daily, 9:20 10:30am Instructor: Lauri Lindberg Office Hours@ tutoring: Tutoring Center (CAS504) 8 9am & 1 2pm daily STEM (Math) Center (RAI338)
More informationFlorida Mathematics Standards for Geometry Honors (CPalms # )
A Correlation of Florida Geometry Honors 2011 to the for Geometry Honors (CPalms #1206320) Geometry Honors (#1206320) Course Standards MAFS.912.GCO.1.1: Know precise definitions of angle, circle, perpendicular
More informationMissouri Mathematics GradeLevel Expectations
A Correlation of to the Grades K  6 G/M223 Introduction This document demonstrates the high degree of success students will achieve when using Scott Foresman Addison Wesley Mathematics in meeting the
More informationGrade 2: Using a Number Line to Order and Compare Numbers Place Value Horizontal Content Strand
Grade 2: Using a Number Line to Order and Compare Numbers Place Value Horizontal Content Strand Texas Essential Knowledge and Skills (TEKS): (2.1) Number, operation, and quantitative reasoning. The student
More informationFull text of O L O W Science As Inquiry conference. Science as Inquiry
Page 1 of 5 Full text of O L O W Science As Inquiry conference Reception Meeting Room Resources Oceanside Unifying Concepts and Processes Science As Inquiry Physical Science Life Science Earth & Space
More information1 35 = Subtraction  a binary operation
High School StuDEnts ConcEPtions of the Minus Sign Lisa L. Lamb, Jessica Pierson Bishop, and Randolph A. Philipp, Bonnie P Schappelle, Ian Whitacre, and Mindy Lewis  describe their research with students
More informationChapter 4  Fractions
. Fractions Chapter  Fractions 0 Michelle Manes, University of Hawaii Department of Mathematics These materials are intended for use with the University of Hawaii Department of Mathematics Math course
More informationCAAP. Content Analysis Report. Sample College. Institution Code: 9011 Institution Type: 4Year Subgroup: none Test Date: Spring 2011
CAAP Content Analysis Report Institution Code: 911 Institution Type: 4Year Normative Group: 4year Colleges Introduction This report provides information intended to help postsecondary institutions better
More informationDeveloping a concretepictorialabstract model for negative number arithmetic
Developing a concretepictorialabstract model for negative number arithmetic Jai Sharma and Doreen Connor Nottingham Trent University Research findings and assessment results persistently identify negative
More informationEdexcel GCSE. Statistics 1389 Paper 1H. June Mark Scheme. Statistics Edexcel GCSE
Edexcel GCSE Statistics 1389 Paper 1H June 2007 Mark Scheme Edexcel GCSE Statistics 1389 NOTES ON MARKING PRINCIPLES 1 Types of mark M marks: method marks A marks: accuracy marks B marks: unconditional
More informationAlgebra 2 Semester 2 Review
Name Block Date Algebra 2 Semester 2 Review NonCalculator 5.4 1. Consider the function f x 1 x 2. a) Describe the transformation of the graph of y 1 x. b) Identify the asymptotes. c) What is the domain
More informationINTERMEDIATE ALGEBRA PRODUCT GUIDE
Welcome Thank you for choosing Intermediate Algebra. This adaptive digital curriculum provides students with instruction and practice in advanced algebraic concepts, including rational, radical, and logarithmic
More informationProbability and Statistics Curriculum Pacing Guide
Unit 1 Terms PS.SPMJ.3 PS.SPMJ.5 Plan and conduct a survey to answer a statistical question. Recognize how the plan addresses sampling technique, randomization, measurement of experimental error and methods
More informationSpreadsheet software UBU104 F/502/4625 VRQ. Learner name: Learner number:
Spreadsheet software UBU104 F/502/4625 Learner name: VRQ Learner number: VTCT is the specialist awarding organisation for the Hairdressing, Beauty Therapy, Complementary Therapy, Hospitality and Catering
More informationDiagnostic Test. Middle School Mathematics
Diagnostic Test Middle School Mathematics Copyright 2010 XAMonline, Inc. All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by
More informationAlignment of Australian Curriculum Year Levels to the Scope and Sequence of MathUSee Program
Alignment of s to the Scope and Sequence of MathUSee Program This table provides guidance to educators when aligning levels/resources to the Australian Curriculum (AC). The MathUSee levels do not address
More informationMathematics process categories
Mathematics process categories All of the UK curricula define multiple categories of mathematical proficiency that require students to be able to use and apply mathematics, beyond simple recall of facts
More informationTIM: Table of Summary Descriptors This table contains the summary descriptors for each cell of the Technology Integration Matrix (TIM).
TIM: Table of Summary Descriptors This table contains the summary descriptors for each cell of the Technology Integration Matrix (TIM). The Technology Integration Matrix (TIM) provides a framework for
More informationSelf Study Report Computer Science
Computer Science undergraduate students have access to undergraduate teaching, and general computing facilities in three buildings. Two large classrooms are housed in the Davis Centre, which hold about
More informationRIGHTSTART MATHEMATICS
Activities for Learning, Inc. RIGHTSTART MATHEMATICS by Joan A. Cotter, Ph.D. LEVEL B LESSONS FOR HOME EDUCATORS FIRST EDITION Copyright 2001 Special thanks to Sharalyn Colvin, who converted RightStart
More informationMinitab Tutorial (Version 17+)
Minitab Tutorial (Version 17+) Basic Commands and Data Entry Graphical Tools Descriptive Statistics Outline Minitab Basics Basic Commands, Data Entry, and Organization Minitab Project Files (*.MPJ) vs.
More informationWord Stress and Intonation: Introduction
Word Stress and Intonation: Introduction WORD STRESS One or more syllables of a polysyllabic word have greater prominence than the others. Such syllables are said to be accented or stressed. Word stress
More informationGrade 6: Correlated to AGS Basic Math Skills
Grade 6: Correlated to AGS Basic Math Skills Grade 6: Standard 1 Number Sense Students compare and order positive and negative integers, decimals, fractions, and mixed numbers. They find multiples and
More informationExcel Intermediate
Instructor s Excel 2013  Intermediate Multiple Worksheets Excel 2013  Intermediate (103124) Multiple Worksheets Quick Links Manipulating Sheets Pages EX5 Pages EX37 EX38 Grouping Worksheets Pages EX304
More informationLLD MATH. Student Eligibility: Grades 68. Credit Value: Date Approved: 8/24/15
PUBLIC SCHOOLS OF EDISON TOWNSHIP DIVISION OF CURRICULUM AND INSTRUCTION LLD MATH Length of Course: Elective/Required: School: Full Year Required Middle Schools Student Eligibility: Grades 68 Credit Value:
More informationTOPICS LEARNING OUTCOMES ACTIVITES ASSESSMENT Numbers and the number system
Curriculum Overview Mathematics 1 st term 5º grade  2010 TOPICS LEARNING OUTCOMES ACTIVITES ASSESSMENT Numbers and the number system Multiplies and divides decimals by 10 or 100. Multiplies and divide
More informationHardhatting in a GeoWorld
Hardhatting in a GeoWorld TM Developed and Published by AIMS Education Foundation This book contains materials developed by the AIMS Education Foundation. AIMS (Activities Integrating Mathematics and
More informationWorkshop Guide Tutorials and Sample Activities. Dynamic Dataa Software
VERSION Dynamic Dataa Software Workshop Guide Tutorials and Sample Activities You have permission to make copies of this document for your classroom use only. You may not distribute, copy or otherwise
More informationNumeracy Medium term plan: Summer Term Level 2C/2B Year 2 Level 2A/3C
Numeracy Medium term plan: Summer Term Level 2C/2B Year 2 Level 2A/3C Using and applying mathematics objectives (Problem solving, Communicating and Reasoning) Select the maths to use in some classroom
More informationAn Introduction to the Minimalist Program
An Introduction to the Minimalist Program Luke Smith University of Arizona Summer 2016 Some findings of traditional syntax Human languages vary greatly, but digging deeper, they all have distinct commonalities:
More informationLearning Disability Functional Capacity Evaluation. Dear Doctor,
Dear Doctor, I have been asked to formulate a vocational opinion regarding NAME s employability in light of his/her learning disability. To assist me with this evaluation I would appreciate if you can
More informationNotetaking Directions
Porter Notetaking Directions 1 Notetaking Directions Simplified CornellBullet System Research indicates that hand writing notes is more beneficial to students learning than typing notes, unless there
More informationIntroduction to the Practice of Statistics
Chapter 1: Looking at Data Distributions Introduction to the Practice of Statistics Sixth Edition David S. Moore George P. McCabe Bruce A. Craig Statistics is the science of collecting, organizing and
More informationPreAP Geometry Course Syllabus Page 1
PreAP Geometry Course Syllabus 20152016 Welcome to my PreAP Geometry class. I hope you find this course to be a positive experience and I am certain that you will learn a great deal during the next
More informationOhio s Learning StandardsClear Learning Targets
Ohio s Learning StandardsClear Learning Targets Math Grade 1 Use addition and subtraction within 20 to solve word problems involving situations of 1.OA.1 adding to, taking from, putting together, taking
More informationField Experience Management 2011 Training Guides
Field Experience Management 2011 Training Guides Page 1 of 40 Contents Introduction... 3 Helpful Resources Available on the LiveText Conference Visitors Pass... 3 Overview... 5 Development Model for FEM...
More informationTCC Jim Bolen Math Competition Rules and Facts. Rules:
TCC Jim Bolen Math Competition Rules and Facts Rules: The Jim Bolen Math Competition is composed of two one hour multiple choice precalculus tests. The first test is scheduled on Friday, November 8, 2013
More informationTASK 2: INSTRUCTION COMMENTARY
TASK 2: INSTRUCTION COMMENTARY Respond to the prompts below (no more than 7 singlespaced pages, including prompts) by typing your responses within the brackets following each prompt. Do not delete or
More informationMaximizing Learning Through Course Alignment and Experience with Different Types of Knowledge
Innov High Educ (2009) 34:93 103 DOI 10.1007/s1075500990952 Maximizing Learning Through Course Alignment and Experience with Different Types of Knowledge Phyllis Blumberg Published online: 3 February
More informationIntroductory thoughts on numeracy
Report from Summer Institute 2002 Introductory thoughts on numeracy by Dave Tout, Language Australia A brief history of the word A quick look into the history of the word numeracy will tell you that the
More informationUnit 3: Lesson 1 Decimals as Equal Divisions
Unit 3: Lesson 1 Strategy Problem: Each photograph in a series has different dimensions that follow a pattern. The 1 st photo has a length that is half its width and an area of 8 in². The 2 nd is a square
More informationAbout How Good is Estimation? Assessment Materials Page 1 of 12
About How Good is Estimation? Assessment Name: Multiple Choice. 1 point each. 1. Which unit of measure is most appropriate for the area of a small rug? a) feet b) yards c) square feet d) square yards 2.
More informationPage 1 of 11. Curriculum Map: Grade 4 Math Course: Math 4 Subtopic: General. Grade(s): None specified
Curriculum Map: Grade 4 Math Course: Math 4 Subtopic: General Grade(s): None specified Unit: Creating a Community of Mathematical Thinkers Timeline: Week 1 The purpose of the Establishing a Community
More informationWhat is PDE? Research Report. Paul Nichols
What is PDE? Research Report Paul Nichols December 2013 WHAT IS PDE? 1 About Pearson Everything we do at Pearson grows out of a clear mission: to help people make progress in their lives through personalized
More informationFourth Grade. Reporting Student Progress. Libertyville School District 70. Fourth Grade
Fourth Grade Libertyville School District 70 Reporting Student Progress Fourth Grade A Message to Parents/Guardians: Libertyville Elementary District 70 teachers of students in kindergarten5 utilize a
More informationMTH 141 Calculus 1 Syllabus Spring 2017
Instructor: Section/Meets Office Hrs: Textbook: Calculus: Single Variable, by HughesHallet et al, 6th ed., Wiley. Also needed: access code to WileyPlus (included in new books) Calculator: Not required,
More informationThe lab is designed to remind you how to work with scientific data (including dealing with uncertainty) and to review experimental design.
Name: Partner(s): Lab #1 The Scientific Method Due 6/25 Objective The lab is designed to remind you how to work with scientific data (including dealing with uncertainty) and to review experimental design.
More informationIntroducing the New Iowa Assessments Language Arts Levels 15 17/18
Introducing the New Iowa Assessments Language Arts Levels 15 17/18 ITP Assessment Tools Math Interim Assessments: Grades 3 8 Administered online Constructed Response Supplements Reading, Language Arts,
More informationSouth Carolina College and CareerReady Standards for Mathematics. Standards Unpacking Documents Grade 5
South Carolina College and CareerReady Standards for Mathematics Standards Unpacking Documents Grade 5 South Carolina College and CareerReady Standards for Mathematics Standards Unpacking Documents
More informationWiggleWorks Software Manual PDF0049 (PDF) Houghton Mifflin Harcourt Publishing Company
WiggleWorks Software Manual PDF0049 (PDF) Houghton Mifflin Harcourt Publishing Company Table of Contents Welcome to WiggleWorks... 3 Program Materials... 3 WiggleWorks Teacher Software... 4 Logging In...
More informationAP Calculus AB. Nevada Academic Standards that are assessable at the local level only.
Calculus AB Priority Keys Aligned with Nevada Standards MA I MI L S MA represents a Major content area. Any concept labeled MA is something of central importance to the entire class/curriculum; it is a
More informationAnswers: Year 4 Textbook 3 Pages 4 10
Answers: Year 4 Textbook Pages 4 Page 4 1. 729 2. 8947. 6502 4. 2067 5. 480 6. 7521 > 860 7. 85 > 699 8. 9442< 9852 9. 4725 > 4572. 8244 < 9241 11. 026 < 211 12. A number between 20 and 4800 1. A number
More informationUsing Virtual Manipulatives to Support Teaching and Learning Mathematics
Using Virtual Manipulatives to Support Teaching and Learning Mathematics Joel Duffin Abstract The National Library of Virtual Manipulatives (NLVM) is a free website containing over 110 interactive online
More informationDesigning a Rubric to Assess the Modelling Phase of Student Design Projects in Upper Year Engineering Courses
Designing a Rubric to Assess the Modelling Phase of Student Design Projects in Upper Year Engineering Courses Thomas F.C. Woodhall Masters Candidate in Civil Engineering Queen s University at Kingston,
More information*Lesson will begin on Friday; Stations will begin on the following Wednesday*
UDL Lesson Plan Template Instructor: Josh Karr Learning Domain: Algebra II/Geometry Grade: 10 th Lesson Objective/s: Students will learn to apply the concepts of transformations to an algebraic context
More informationGetting Started with TINspire High School Science
Getting Started with TINspire High School Science 2012 Texas Instruments Incorporated Materials for Institute Participant * *This material is for the personal use of T3 instructors in delivering a T3
More informationThe Singapore Copyright Act applies to the use of this document.
Title Mathematical problem solving in Singapore schools Author(s) Berinderjeet Kaur Source Teaching and Learning, 19(1), 6778 Published by Institute of Education (Singapore) This document may be used
More informationWelcome to ACT Brain Boot Camp
Welcome to ACT Brain Boot Camp 9:30 am  9:45 am Basics (in every room) 9:45 am  10:15 am Breakout Session #1 ACT Math: Adame ACT Science: Moreno ACT Reading: Campbell ACT English: Lee 10:20 am  10:50
More informationSAT MATH PREP:
SAT MATH PREP: 20152016 NOTE: The College Board has redesigned the SAT Test. This new test will start in March of 2016. Also, the PSAT test given in October of 2015 will have the new format. Therefore
More informationCal s Dinner Card Deals
Cal s Dinner Card Deals Overview: In this lesson students compare three linear functions in the context of Dinner Card Deals. Students are required to interpret a graph for each Dinner Card Deal to help
More informationBRAZOSPORT COLLEGE LAKE JACKSON, TEXAS SYLLABUS. POFI 1301: COMPUTER APPLICATIONS I (File Management/PowerPoint/Word/Excel)
BRAZOSPORT COLLEGE LAKE JACKSON, TEXAS SYLLABUS POFI 1301: COMPUTER APPLICATIONS I (File Management/PowerPoint/Word/Excel) COMPUTER TECHNOLOGY & OFFICE ADMINISTRATION DEPARTMENT CATALOG DESCRIPTION POFI
More informationOCR for Arabic using SIFT Descriptors With Online Failure Prediction
OCR for Arabic using SIFT Descriptors With Online Failure Prediction Andrey Stolyarenko, Nachum Dershowitz The Blavatnik School of Computer Science Tel Aviv University Tel Aviv, Israel Email: stloyare@tau.ac.il,
More informationEGRHS Course Fair. Science & Math AP & IB Courses
EGRHS Course Fair Science & Math AP & IB Courses Science Courses: AP Physics IB Physics SL IB Physics HL AP Biology IB Biology HL AP Physics Course Description Course Description AP Physics C (Mechanics)
More informationPHYSICS 40S  COURSE OUTLINE AND REQUIREMENTS Welcome to Physics 40S for !! Mr. Bryan Doiron
PHYSICS 40S  COURSE OUTLINE AND REQUIREMENTS Welcome to Physics 40S for 20162017!! Mr. Bryan Doiron The course covers the following topics (time permitting): Unit 1 Kinematics: Special Equations, Relative
More informationFoothill College Summer 2016
Foothill College Summer 2016 Intermediate Algebra Math 105.04W CRN# 10135 5.0 units Instructor: Yvette Butterworth Text: None; Beoga.net material used Hours: Online Except Final Thurs, 8/4 3:30pm Phone:
More informationThis table contains the extended descriptors for Active Learning on the Technology Integration Matrix (TIM).
TIM: Active Learning This table contains the extended descriptors for Active Learning on the Technology Integration Matrix (TIM). The Active attribute makes the distinction between lessons in which students
More informationFunctional Skills Mathematics Level 2 assessment
Functional Skills Mathematics Level 2 assessment www.cityandguilds.com September 2015 Version 1.0 Marking scheme ONLINE V2 Level 2 Sample Paper 4 Mark Represent Analyse Interpret Open Fixed S1Q1 3 3 0
More informationRobot manipulations and development of spatial imagery
Robot manipulations and development of spatial imagery Author: Igor M. Verner, Technion Israel Institute of Technology, Haifa, 32000, ISRAEL ttrigor@tx.technion.ac.il Abstract This paper considers spatial
More informationUsing Calculators for Students in Grades 912: Geometry. Republished with permission from American Institutes for Research
Using Calculators for Students in Grades 912: Geometry Republished with permission from American Institutes for Research Using Calculators for Students in Grades 912: Geometry By: Center for Implementing
More informationMathematics Assessment Plan
Mathematics Assessment Plan Mission Statement for Academic Unit: Georgia Perimeter College transforms the lives of our students to thrive in a global society. As a diverse, multi campus two year college,
More informationCopyright Corwin 2015
2 Defining Essential Learnings How do I find clarity in a sea of standards? For students truly to be able to take responsibility for their learning, both teacher and students need to be very clear about
More informationBroward County Public Schools G rade 6 FSA WarmUps
Day 1 1. A florist has 40 tulips, 32 roses, 60 daises, and 50 petunias. Draw a line from each comparison to match it to the correct ratio. A. tulips to roses B. daises to petunias C. roses to tulips D.
More informationSpinners at the School Carnival (Unequal Sections)
Spinners at the School Carnival (Unequal Sections) Maryann E. Huey Drake University maryann.huey@drake.edu Published: February 2012 Overview of the Lesson Students are asked to predict the outcomes of
More informationFactoring  Grouping
6.2 Factoring  Grouping Objective: Factor polynomials with four terms using grouping. The first thing we will always do when factoring is try to factor out a GCF. This GCF is often a monomial like in
More information