Tom wants to find two real numbers, a and b, that have a sum of 10 and have a product of 10. He makes this table.


 Aubrie Newman
 5 years ago
 Views:
Transcription
1 Sum and Product This problem gives you the chance to: use arithmetic and algebra to represent and analyze a mathematical situation solve a quadratic equation by trial and improvement Tom wants to find two real numbers, a and b, that have a sum of 0 and have a product of 0. He makes this table. Sum of a and b a b Product of a and b a. How can Tom find b if he knows a? b. Draw a ring around any of these algebraic statements that express the relationships between a and b correctly. a + b =0 b = a +0 b =0 " a a =0 " b a =0b b =0a ab =0 b = 0 a! 2. Explain what happens to the product of a and b (in the last column of the table) as a increases from 0 to0.! Copyright 2007 by Mathematics Assessment Page 36 Sum and Product Test 0
2 3. Tom looks at the table and says, If the product of a and b is 0, a must be somewhere between and 2 or between 8 and 9. a. Explain how the table shows this. b. Tom tries to find the value between and 2. He decides to try a =.5 Complete this table to show his calculations. Sum of a and b a b Product of a and b 0.5 c. Which of these two statements is correct? Explain how you decided. a must be between and.5 a must be between.5 and 2 d. Find the values of a and b correct to two decimal places. Show your calculations. 8 Copyright 2007 by Mathematics Assessment Page 37 Sum and Product Test 0
3 Sum and Product Rubric The core elements of performance required by this task are: use arithmetic and algebra to represent and analyze a mathematical situation solve a quadratic equation by trial and improvement Based on these, credit for specific aspects of performance should be assigned as follows. a. Gives correct answer: Subtract a from 0 or divide 0 by a. points section points b. Draws rings around! a + b =0 b = a +0 b =0 " a a =0 " b a =0b b =0a ab =0 b = 0 a 2 2. Gives correct! explanation such as: The product starts at zero, increases up to a maximum of 25, then decreases back to zero. 3.a b c d Gives correct explanation such as: The product is less than 0 when a is and more than 0 when a is 2, so a must be between and 2. Similarly for 8 and 9. Gives correct answer: 8.5, 2.75 Gives correct explanation such as: The product is less than 0 when a is and more than 0 when a is.5, so a must be between and.5 Gives correct answers:.3 and 8.87 Shows correct work such as:.3 x 8.87 = x 8.88 = Total Points 8 5 Copyright 2007 by Mathematics Assessment Page 38 Sum and Product Test 0
4 Sum and Product Work the task and look at the rubric. What are the key mathematical ideas being assessed in this task? What do you think students might struggle with? Look at student work for b. How many of your students recognized and circled the multiplication and division relationships for a and b? Why do you think so many students struggled with this? Now look at the explanations in 2, 3a, and 3b. What was missing from students explanations? Did they focus on a and b instead of the product? Did they forget to quantify and describe how the pattern changes? Did their explanations make sense or did they describe something irrelevant? Did they describe a pattern that didn t fit all the data? What types of opportunities do students have in your class to make sense of a table of data and describe trends? How do you help students understand the qualities and expectations about mathematical explanations or descriptions? What kinds of class discussions do students engage in to help them see differences in qualities of explanations and learn the logic of a mathematically convincing argument? Look at work for part 3d. How many of your students did not attempt this part of the task? How many of your students picked values for a and b that did not add to 0? How many of your students picked negative values for a or b? How many of your students chose numbers with only decimal place? How many of your students only tried or showed evidence of one set of values, rather than testing for closer combination or to check that their solution was optimal? How many of your students used their response from the table in 3b? How many of your students tried to use algebra to solve for a and b, but got stuck because they couldn t factor the expression or didn t know what to do after they made an equation? What are the implications for instruction that you see from looking at this work? What are the big ideas that students are missing? Geometry
5 Looking at Student Work on Sum and Product Student A is able to write very specific explanations. The student is also able to use algebra to solve for a and b in part 3d. Student A Geometry
6 Many students attempted to use algebra to solve for a and b, but struggled because they did not remember the quadratic formula. See the work of Student B. Student B Geometry
7 Student C had difficulty working with decimals. The student subtracted incorrectly in 3b. The student incorrectly uses a negative sign in the quadratic formula. Student C does not check to see if the answer makes sense. A brief glance should show that the values do not at to 0 in his solution to 3d. Student C Geometry
8 Student D notices a pattern in the differences between the products, rather than trying to describe the products themselves. Do student sin your class get enough opportunities to make descriptions about trends? Student D Many students struggled with the demands in part 3d. Student E only tried one set of numbers and doesn t test to see if it can be improved. Student F finds the 2 closest sets of values, but does not understand decimal place value enough to pick the best choice. Student E Student F Geometry
9 Geometry Task 3 Sum and Product Student Task Core Idea 3 Algebraic Properties and Representations Use arithmetic and algebra to represent and analyze a mathematical situation and solve a quadratic equation by trial and improvement. Represent and analyze mathematical situations and structures using algebraic symbols. Solve equations involving radicals and exponents in contextualized problems. Mathematics of the task: Ability to read and interpret constraints and make sense of a table of values Ability to make mathematical descriptions of information in a table and quantify relationships Understand and use guess and check and/or use the quadratic formula to find an optimal solution to match the constraints of the task Recognize multiple representations of expressions for two sets of constraints to describe the relationship between two variables Based on teacher observations, this is what geometry students knew and were able to do: Solving for b and finding the product of a and b in part 3b of the task. Describing how to find b in part a of the task Describing why a must be located in a particular place on the number line based on values in the table in 3a Areas of difficulty for geometry students: Calculating and comparing with decimals Identifying algebraic expressions to show the relationship between a and b if their product is 0 Describing the trends in the product column of the table Explaining why a must be between and.5 Finding the values for a and b to two decimal points Solving a quadratic expression that doesn t factor Geometry
10 The maximum score available for this task is 8 points. The minimum score for a level 3 response, meeting standards, is 4 points. Most students 94% could fill out the guess and test table for a and the product of a and b in part 3. More than half the students, 62%, could find use division or subtraction to find a in part, describe how the product changes as the value of a increases from0 to 0, explain why the value of b falls between two values on the table, and fill in a guess and test table. 7% of the students met all the demands of the task including identifying 5 ways to express the relationship of a and b using both constraints of the problem and calculate values of a and b to two decimal places to come closest to a product of % of the students scored no points on this task. 75% of these students attempted the task. Geometry
11 Sum and Product Points Understandings Misunderstandings 0 75% of the students with this score attempted the task. Students had difficulty using the guess and test table in 3b. They couldn t compute Students could subtract.5 from 0 to solve for b and multiply with decimals to complete part 3b of the task. 4 Students with this score could usually describe the product trend in 2, explain why a is between certain values in 3a, fill in the table in 3b, and describe why the value of a is located between and.5. 5 Students could find the value of b using algebra, describe the trend in products, fill in the table, and describe where a was located. accurately with decimals for find b. Students struggled with describing the pattern in the products in the table. Students did not quantify information, giving examples such as a and b cycles, so the product also cycles. Some students gave inaccurate trends, such as the product is always 0 or the product always increases. 9% of the students did not attempt this part of the task. Many students struggled with describing how to find a in part. 9% said to use a formula. Some said it s a pattern. Others could not express the mathematics using descriptions reversing the operation, such as subtract 0 from a, or using the wrong variable, subtract b from 0. Some students used both variables in their rules, like subtract b from a or multiply a and b. Students struggled with finding all the equivalent expressions in b and solving for a and b to two decimal places. 37% did not attempt to solve for a and b in 3d. 0% attempted to use algebra, but could not complete the process using the quadratic formula. 7 Students could not find the multiplication and division expression that met the constraints for a and b in part b of the task. 70% of all students omitted these options. 8 Students could use a guess and test table to think about the relationships between two variables with different constraints. They could describe trends and reason about the value of a that would meet all the constraints. They could find the value of a and b to two decimal places. Geometry
12 Implications for Instruction Students at this grade level should be comfortable doing calculations with decimals. Many students struggled with subtracting decimal values from 0 to find b. Are there structures in place in your school geometry program to allow students to fill in holes or misconceptions in their arithmetic skills? Can students drop in and out of remedial or ramp up: classes to work on specific topics like decimal calculations, percents, or expressions and equations? Students at this grade level should understand operations, knowing the inverse relationship between addition and subtraction and the inverse relationship between multiplication and division. Students should be able to express these relations in words and symbols. Students should routinely have opportunities to solve rich tasks which incorporate a variety of current and past knowledge to solve nonroutine problems. Part of the problemsolving repertoire should involve identifying all the constraints needed in the solution. Another part of the problemsolving repertoire should include asking themselves questions, such as is this the optimal solution?, can I get closer?, how do I know? Helping students gain this skill of selftalk is important. Good students seem to have this idea, but making the ideas of selftalk explicit, helps all students to pick up this strategy. A great deal of time is devoted in traditional American algebra textbooks to factoring quadratic expressions to find solutions to the problem. In many European texts, more time is devoted to going straight to using the quadratic formula, which will work for all cases. This would be a good topic for discussion in planning for an Algebra Course or to consider when evaluating Algebra texts. How would this allow more time for other important algebraic topics? What topics would you like to see developed in more detail? Geometry
Circles in Triangles. This problem gives you the chance to: use algebra to explore a geometric situation
Circles in Triangles This problem gives you the chance to: use algebra to explore a geometric situation A This diagram shows a circle that just touches the sides of a right triangle whose sides are 3 units,
More informationGrade Level Year Total Points Core Points % At Standard 9 2003 10 5 7 %
Performance Assessment Task Number Towers Grade 9 The task challenges a student to demonstrate understanding of the concepts of algebraic properties and representations. A student must make sense of the
More informationBalanced Assessment Test Algebra 2008
Balanced Assessment Test Algebra 2008 Core Idea Task Score Representations Expressions This task asks students find algebraic expressions for area and perimeter of parallelograms and trapezoids. Successful
More information0.75 75% ! 3 40% 0.65 65% Percent Cards. This problem gives you the chance to: relate fractions, decimals and percents
Percent Cards This problem gives you the chance to: relate fractions, decimals and percents Mrs. Lopez makes sets of cards for her math class. All the cards in a set have the same value. Set A 3 4 0.75
More informationPerformance Assessment Task Bikes and Trikes Grade 4. Common Core State Standards Math  Content Standards
Performance Assessment Task Bikes and Trikes Grade 4 The task challenges a student to demonstrate understanding of concepts involved in multiplication. A student must make sense of equal sized groups of
More informationNumber Factors. Number Factors Number of factors 1 1 1 16 1, 2, 4, 8, 16 5 2 1, 2 2 17 1, 17 2 3 1, 3 2 18 1, 2, 3, 6, 9, 18 6 4 1, 2, 4 3 19 1, 19 2
Factors This problem gives you the chance to: work with factors of numbers up to 30 A factor of a number divides into the number exactly. This table shows all the factors of most of the numbers up to 30.
More informationHow Old Are They? This problem gives you the chance to: form expressions form and solve an equation to solve an age problem. Will is w years old.
How Old Are They? This problem gives you the chance to: form expressions form and solve an equation to solve an age problem Will is w years old. Ben is 3 years older. 1. Write an expression, in terms of
More informationHigh School Algebra Reasoning with Equations and Inequalities Solve equations and inequalities in one variable.
Performance Assessment Task Quadratic (2009) Grade 9 The task challenges a student to demonstrate an understanding of quadratic functions in various forms. A student must make sense of the meaning of relations
More information1. I have 4 sides. My opposite sides are equal. I have 4 right angles. Which shape am I?
Which Shape? This problem gives you the chance to: identify and describe shapes use clues to solve riddles Use shapes A, B, or C to solve the riddles. A B C 1. I have 4 sides. My opposite sides are equal.
More informationGCSE MATHEMATICS. 43602H Unit 2: Number and Algebra (Higher) Report on the Examination. Specification 4360 November 2014. Version: 1.
GCSE MATHEMATICS 43602H Unit 2: Number and Algebra (Higher) Report on the Examination Specification 4360 November 2014 Version: 1.0 Further copies of this Report are available from aqa.org.uk Copyright
More informationInteger Operations. Overview. Grade 7 Mathematics, Quarter 1, Unit 1.1. Number of Instructional Days: 15 (1 day = 45 minutes) Essential Questions
Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Describe situations in which opposites combine to make zero.
More informationHigh School Algebra Reasoning with Equations and Inequalities Solve systems of equations.
Performance Assessment Task Graphs (2006) Grade 9 This task challenges a student to use knowledge of graphs and their significant features to identify the linear equations for various lines. A student
More information1.6 The Order of Operations
1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative
More informationProperties of Real Numbers
16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should
More informationSession 7 Fractions and Decimals
Key Terms in This Session Session 7 Fractions and Decimals Previously Introduced prime number rational numbers New in This Session period repeating decimal terminating decimal Introduction In this session,
More informationG C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
Performance Assessment Task Circle and Squares Grade 10 This task challenges a student to analyze characteristics of 2 dimensional shapes to develop mathematical arguments about geometric relationships.
More informationN Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Performance Assessment Task Swimming Pool Grade 9 The task challenges a student to demonstrate understanding of the concept of quantities. A student must understand the attributes of trapezoids, how to
More informationThe Deadly Sins of Algebra
The Deadly Sins of Algebra There are some algebraic misconceptions that are so damaging to your quantitative and formal reasoning ability, you might as well be said not to have any such reasoning ability.
More informationHigh School Functions Interpreting Functions Understand the concept of a function and use function notation.
Performance Assessment Task Printing Tickets Grade 9 The task challenges a student to demonstrate understanding of the concepts representing and analyzing mathematical situations and structures using algebra.
More informationSolving Rational Equations
Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,
More informationUsing games to support. WinWin Math Games. by Marilyn Burns
4 WinWin Math Games by Marilyn Burns photos: bob adler Games can motivate students, capture their interest, and are a great way to get in that paperandpencil practice. Using games to support students
More informationTime needed. Before the lesson Assessment task:
Formative Assessment Lesson Materials Alpha Version Beads Under the Cloud Mathematical goals This lesson unit is intended to help you assess how well students are able to identify patterns (both linear
More informationSequences. A sequence is a list of numbers, or a pattern, which obeys a rule.
Sequences A sequence is a list of numbers, or a pattern, which obeys a rule. Each number in a sequence is called a term. ie the fourth term of the sequence 2, 4, 6, 8, 10, 12... is 8, because it is the
More informationMathematics Assessment Collaborative Tool Kits for Teachers Looking and Learning from Student Work 2009 Grade Eight
Mathematics Assessment Collaborative Tool Kits for Teachers Looking and Learning from Student Work 2009 Grade Eight Contents by Grade Level: Overview of Exam Grade Level Results Cut Score and Grade History
More informationF.IF.7b: Graph Root, Piecewise, Step, & Absolute Value Functions
F.IF.7b: Graph Root, Piecewise, Step, & Absolute Value Functions F.IF.7b: Graph Root, Piecewise, Step, & Absolute Value Functions Analyze functions using different representations. 7. Graph functions expressed
More informationLies My Calculator and Computer Told Me
Lies My Calculator and Computer Told Me 2 LIES MY CALCULATOR AND COMPUTER TOLD ME Lies My Calculator and Computer Told Me See Section.4 for a discussion of graphing calculators and computers with graphing
More information6 th grade Task 2 Gym
experiences understanding what the mean reflects about the data and how changes in data will affect the average. The purpose of statistics is to give a picture about the data. Students need to be able
More informationMathematics. What to expect Resources Study Strategies Helpful Preparation Tips Problem Solving Strategies and Hints Test taking strategies
Mathematics Before reading this section, make sure you have read the appropriate description of the mathematics section test (computerized or paper) to understand what is expected of you in the mathematics
More information3.1. RATIONAL EXPRESSIONS
3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers
More informationKenken For Teachers. Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles June 27, 2010. Abstract
Kenken For Teachers Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles June 7, 00 Abstract Kenken is a puzzle whose solution requires a combination of logic and simple arithmetic skills.
More informationUnit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.
Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L34) is a summary BLM for the material
More informationPerformance Assessment Task Baseball Players Grade 6. Common Core State Standards Math  Content Standards
Performance Assessment Task Baseball Players Grade 6 The task challenges a student to demonstrate understanding of the measures of center the mean, median and range. A student must be able to use the measures
More informationLesson Plan. N.RN.3: Use properties of rational and irrational numbers.
N.RN.3: Use properties of rational irrational numbers. N.RN.3: Use Properties of Rational Irrational Numbers Use properties of rational irrational numbers. 3. Explain why the sum or product of two rational
More information0.8 Rational Expressions and Equations
96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions  that is, algebraic fractions  and equations which contain them. The reader is encouraged to
More informationOrder of Operations More Essential Practice
Order of Operations More Essential Practice We will be simplifying expressions using the order of operations in this section. Automatic Skill: Order of operations needs to become an automatic skill. Failure
More informationPlaying with Numbers
PLAYING WITH NUMBERS 249 Playing with Numbers CHAPTER 16 16.1 Introduction You have studied various types of numbers such as natural numbers, whole numbers, integers and rational numbers. You have also
More informationGeometry Solve real life and mathematical problems involving angle measure, area, surface area and volume.
Performance Assessment Task Pizza Crusts Grade 7 This task challenges a student to calculate area and perimeters of squares and rectangles and find circumference and area of a circle. Students must find
More informationPartial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:
Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than
More informationMATH 10034 Fundamental Mathematics IV
MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.
More informationObjective. Materials. TI73 Calculator
0. Objective To explore subtraction of integers using a number line. Activity 2 To develop strategies for subtracting integers. Materials TI73 Calculator Integer Subtraction What s the Difference? Teacher
More informationMath Released Set 2015. Algebra 1 PBA Item #13 Two Real Numbers Defined M44105
Math Released Set 2015 Algebra 1 PBA Item #13 Two Real Numbers Defined M44105 Prompt Rubric Task is worth a total of 3 points. M44105 Rubric Score Description 3 Student response includes the following
More informationClick on the links below to jump directly to the relevant section
Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationMATH 60 NOTEBOOK CERTIFICATIONS
MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5
More informationProblem of the Month: Perfect Pair
Problem of the Month: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards:
More informationMathematical goals. Starting points. Materials required. Time needed
Level N of challenge: B N Mathematical goals Starting points Materials required Time needed Ordering fractions and decimals To help learners to: interpret decimals and fractions using scales and areas;
More informationBinary Adders: Half Adders and Full Adders
Binary Adders: Half Adders and Full Adders In this set of slides, we present the two basic types of adders: 1. Half adders, and 2. Full adders. Each type of adder functions to add two binary bits. In order
More informationTo Evaluate an Algebraic Expression
1.5 Evaluating Algebraic Expressions 1.5 OBJECTIVES 1. Evaluate algebraic expressions given any signed number value for the variables 2. Use a calculator to evaluate algebraic expressions 3. Find the sum
More informationVerbal Phrases to Algebraic Expressions
Student Name: Date: Contact Person Name: Phone Number: Lesson 13 Verbal Phrases to s Objectives Translate verbal phrases into algebraic expressions Solve word problems by translating sentences into equations
More informationWeek 13 Trigonometric Form of Complex Numbers
Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working
More informationCurrent California Math Standards Balanced Equations
Balanced Equations Current California Math Standards Balanced Equations Grade Three Number Sense 1.0 Students understand the place value of whole numbers: 1.1 Count, read, and write whole numbers to 10,000.
More informationCORE Assessment Module Module Overview
CORE Assessment Module Module Overview Content Area Mathematics Title Speedy Texting Grade Level Grade 7 Problem Type Performance Task Learning Goal Students will solve reallife and mathematical problems
More informationMultiplying and Dividing Signed Numbers. Finding the Product of Two Signed Numbers. (a) (3)( 4) ( 4) ( 4) ( 4) 12 (b) (4)( 5) ( 5) ( 5) ( 5) ( 5) 20
SECTION.4 Multiplying and Dividing Signed Numbers.4 OBJECTIVES 1. Multiply signed numbers 2. Use the commutative property of multiplication 3. Use the associative property of multiplication 4. Divide signed
More informationMathematical goals. Starting points. Materials required. Time needed
Level A3 of challenge: C A3 Creating and solving harder equations equations Mathematical goals Starting points Materials required Time needed To enable learners to: create and solve equations, where the
More informationPrentice Hall. California Edition of Algebra 1  Classics Edition (Smith/Charles) 2008. Grade 8
Prentice Hall Grade 8 California Edition of Algebra 1  Classics Edition (Smith/Charles) 2008 C O R R E L A T E D T O California s Map for a Basic Grade Level Program Grade 8 PROGRAM DESCRIPTION Prentice
More informationNCTM Curriculum Focal Points for Grade 5. Everyday Mathematics, Grade 5
NCTM Curriculum Focal Points and, Grade 5 NCTM Curriculum Focal Points for Grade 5 Number and Operations and Algebra: Developing an understanding of and fluency with division of whole numbers Students
More informationThis unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.
Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course
More informationExponents and Radicals
Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of
More informationHow To Factor Quadratic Trinomials
Factoring Quadratic Trinomials Student Probe Factor Answer: Lesson Description This lesson uses the area model of multiplication to factor quadratic trinomials Part 1 of the lesson consists of circle puzzles
More information2.5 Zeros of a Polynomial Functions
.5 Zeros of a Polynomial Functions Section.5 Notes Page 1 The first rule we will talk about is Descartes Rule of Signs, which can be used to determine the possible times a graph crosses the xaxis and
More informationConnections Across Strands Provides a sampling of connections that can be made across strands, using the theme (fractions) as an organizer
Overview Context Connections Positions fractions in a larger context and shows connections to everyday situations, careers, and tasks Identifies relevant manipulatives, technology, and webbased resources
More informationSection 4.1 Rules of Exponents
Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells
More informationSample Fraction Addition and Subtraction Concepts Activities 1 3
Sample Fraction Addition and Subtraction Concepts Activities 1 3 College and CareerReady Standard Addressed: Build fractions from unit fractions by applying and extending previous understandings of operations
More informationMathematics Online Instructional Materials Correlation to the 2009 Algebra I Standards of Learning and Curriculum Framework
Provider York County School Division Course Syllabus URL http://yorkcountyschools.org/virtuallearning/coursecatalog.aspx Course Title Algebra I AB Last Updated 2010  A.1 The student will represent verbal
More informationPerformance Assessment Task Which Shape? Grade 3. Common Core State Standards Math  Content Standards
Performance Assessment Task Which Shape? Grade 3 This task challenges a student to use knowledge of geometrical attributes (such as angle size, number of angles, number of sides, and parallel sides) to
More informationGMAT SYLLABI. Types of Assignments  1 
GMAT SYLLABI The syllabi on the following pages list the math and verbal assignments for each class. Your homework assignments depend on your current math and verbal scores. Be sure to read How to Use
More informationProblem of the Month Pick a Pocket
The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards: Make sense of problems
More informationCONTENTS. Please note:
CONTENTS Introduction...iv. Number Systems... 2. Algebraic Expressions.... Factorising...24 4. Solving Linear Equations...8. Solving Quadratic Equations...0 6. Simultaneous Equations.... Long Division
More informationALGEBRA. sequence, term, nth term, consecutive, rule, relationship, generate, predict, continue increase, decrease finite, infinite
ALGEBRA Pupils should be taught to: Generate and describe sequences As outcomes, Year 7 pupils should, for example: Use, read and write, spelling correctly: sequence, term, nth term, consecutive, rule,
More informationCOWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level 2
COWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level This study guide is for students trying to test into College Algebra. There are three levels of math study guides. 1. If x and y 1, what
More informationWritePlacer Sample Topic. WritePlacer. Arithmetic
. Knowledge of another language fosters greater awareness of cultural diversity among the peoples of the world. Individuals who have foreign language skills can appreciate more readily other peoples values
More informationOA413 Rounding on a Number Line Pages 80 81
OA413 Rounding on a Number Line Pages 80 81 STANDARDS 3.NBT.A.1, 4.NBT.A.3 Goals Students will round to the closest ten, except when the number is exactly halfway between a multiple of ten. PRIOR KNOWLEDGE
More informationDecimal Fractions. Grades 6 and 7. Teacher Document. We acknowledge the valuable comments of Hanlie Murray and Sarie Smit
Decimal Fractions Grades 6 and 7 Teacher Document Malati staff involved in developing these materials: Therine van Niekerk Amanda le Roux Karen Newstead Bingo Lukhele We acknowledge the valuable comments
More informationPearson Algebra 1 Common Core 2015
A Correlation of Pearson Algebra 1 Common Core 2015 To the Common Core State Standards for Mathematics Traditional Pathways, Algebra 1 High School Copyright 2015 Pearson Education, Inc. or its affiliate(s).
More informationAlgebra I. In this technological age, mathematics is more important than ever. When students
In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,
More informationSimple Examples. This is the information that we are given. To find the answer we are to solve an equation in one variable, x.
Worksheet. Solving Equations in One Variable Section 1 Simple Examples You are on your way to Brisbane from Sydney, and you know that the trip is 1100 km. You pass a sign that says that Brisbane is now
More information8 Square matrices continued: Determinants
8 Square matrices continued: Determinants 8. Introduction Determinants give us important information about square matrices, and, as we ll soon see, are essential for the computation of eigenvalues. You
More informationCreating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities
Algebra 1, Quarter 2, Unit 2.1 Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned
More informationBrain Game. 3.4 Solving and Graphing Inequalities HOW TO PLAY PRACTICE. Name Date Class Period. MATERIALS game cards
Name Date Class Period Brain Game 3.4 Solving and Graphing Inequalities MATERIALS game cards HOW TO PLAY Work with another student. Shuffle the cards you receive from your teacher. Then put them face down
More informationDecomposing Numbers (Operations and Algebraic Thinking)
Decomposing Numbers (Operations and Algebraic Thinking) Kindergarten Formative Assessment Lesson Designed and revised by Kentucky Department of Education Mathematics Specialists Fieldtested by Kentucky
More informationHandsOn Math Algebra
HandsOn Math Algebra by Pam Meader and Judy Storer illustrated by Julie Mazur Contents To the Teacher... v Topic: Ratio and Proportion 1. Candy Promotion... 1 2. Estimating Wildlife Populations... 6 3.
More informationMeasurement with Ratios
Grade 6 Mathematics, Quarter 2, Unit 2.1 Measurement with Ratios Overview Number of instructional days: 15 (1 day = 45 minutes) Content to be learned Use ratio reasoning to solve realworld and mathematical
More informationAnswer: The relationship cannot be determined.
Question 1 Test 2, Second QR Section (version 3) In City X, the range of the daily low temperatures during... QA: The range of the daily low temperatures in City X... QB: 30 Fahrenheit Arithmetic: Ranges
More informationF.IF.7e Analyze functions using different representations. Graph exponential and logarithmic functions, showing intercept and end behavior.
Grade Level/Course: Algebra and PreCalculus Lesson/Unit Plan Name: Introduction to Logarithms: A Function Approach Rationale/Lesson Abstract: This lesson is designed to introduce logarithms to students
More information26 Integers: Multiplication, Division, and Order
26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue
More informationMinnesota Academic Standards
A Correlation of to the Minnesota Academic Standards Grades K6 G/M204 Introduction This document demonstrates the high degree of success students will achieve when using Scott Foresman Addison Wesley
More information5 th Grade Common Core State Standards. Flip Book
5 th Grade Common Core State Standards Flip Book This document is intended to show the connections to the Standards of Mathematical Practices for the content standards and to get detailed information at
More informationFactoring Quadratic Trinomials
Factoring Quadratic Trinomials Student Probe Factor x x 3 10. Answer: x 5 x Lesson Description This lesson uses the area model of multiplication to factor quadratic trinomials. Part 1 of the lesson consists
More information23. RATIONAL EXPONENTS
23. RATIONAL EXPONENTS renaming radicals rational numbers writing radicals with rational exponents When serious work needs to be done with radicals, they are usually changed to a name that uses exponents,
More informationPrecalculus Orientation and FAQ
Precalculus Orientation and FAQ MATH 1011 (Precalculus) is a four hour 3 credit course that prepares a student for Calculus. Topics covered include linear, quadratic, polynomial, rational, exponential,
More informationPREPARATION FOR MATH TESTING at CityLab Academy
PREPARATION FOR MATH TESTING at CityLab Academy compiled by Gloria Vachino, M.S. Refresh your math skills with a MATH REVIEW and find out if you are ready for the math entrance test by taking a PRETEST
More informationMath Placement Test Study Guide. 2. The test consists entirely of multiple choice questions, each with five choices.
Math Placement Test Study Guide General Characteristics of the Test 1. All items are to be completed by all students. The items are roughly ordered from elementary to advanced. The expectation is that
More informationAlgebra II New Summit School High School Diploma Program
Syllabus Course Description: Algebra II is a two semester course. Students completing this course will earn 1.0 unit upon completion. Required Materials: 1. Student Text Glencoe Algebra 2: Integration,
More informationMathematics GradeLevel Instructional Materials Evaluation Tool
Mathematics GradeLevel Instructional Materials Evaluation Tool Quality Review GRADE 8 Textbooks and their digital counterparts are vital classroom tools but also a major expense, and it is worth taking
More informationNOT AN OFFICIAL SCORE REPORT. Summary of Results
From SAT TEST MARCH 8, 214 Summary of Results Page 1 of 1 Congratulations on taking the SAT Reasoning Test! You re showing colleges that you are serious about getting an education. The SAT is one indicator
More informationFractions as Numbers INTENSIVE INTERVENTION. National Center on. at American Institutes for Research
National Center on INTENSIVE INTERVENTION at American Institutes for Research Fractions as Numbers 000 Thomas Jefferson Street, NW Washington, DC 0007 Email: NCII@air.org While permission to reprint this
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More informationBeads Under the Cloud
Beads Under the Cloud Intermediate/Middle School Grades Problem Solving Mathematics Formative Assessment Lesson Designed and revised by Kentucky Department of Education Mathematics Specialists Fieldtested
More information5.3 SOLVING TRIGONOMETRIC EQUATIONS. Copyright Cengage Learning. All rights reserved.
5.3 SOLVING TRIGONOMETRIC EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Use standard algebraic techniques to solve trigonometric equations. Solve trigonometric equations
More informationGrade 7/8 Math Circles Sequences and Series
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Sequences and Series November 30, 2012 What are sequences? A sequence is an ordered
More information