# Guide to Leaving Certificate Mathematics Ordinary Level

Size: px
Start display at page:

Transcription

1 Guide to Leaving Certificate Mathematics Ordinary Level Dr. Aoife Jones Paper 1 For the Leaving Cert 013, Paper 1 is divided into three sections. Section A is entitled Concepts and Skills and contains four questions, all of which must be completed. These are designed to test the student s understanding of basic mathematical concepts. In Paper 1, this corresponds to an understanding of the basics of algebra and number systems. Questions that appear here tend to be on topics that do not lend themselves to the larger-scale problems that occur later on in the exam. Topics that can appear in this section include, but are not limited to: (i) prime numbers and factors (ii) inequalities and numberlines (iii) indices and scientific notation (iv) profit and loss, compound interest, income tax (v) quadratic equations and simultaneous equations (vi) and also topics that may appear in Section B. [See Example 1.] Section B is entitled Contexts and Applications and contains two questions. Again, no choice is given and both questions must be answered. These questions allow for more extensive testing of a student s understanding of particular subjects. It is important to note that the questions asked here may span over a number of topics that were traditionally asked in separate questions. As a result, students must be vigilant when approaching 1

2 these questions; they must now keep to mind their entire armoury of mathematical approaches when attempting any one question. Topics that can appear in this section include, but are not limited to: (i) complex numbers (ii) patterns (iii) series and sequences On this latter topic, students are expected to be able to identify patterns that are described in words or in diagrams, and to express the pattern in algebraic form. This topic is new to the syllabus with the introduction of Project Maths and aims to enable students to better understand the role that maths plays in natural world [see Example.] Section C is entitled Functions and Calculus (old syllabus) and contains three questions, of which students must answer two. This topic remains unchanged from previous years for the Leaving Cert in 013 only, and so the questions closely follow those asked in the past exam papers. Students should look at Questions 6, 7 and 8 from Paper 1 of the old exam papers, as the content has not changed and the questions still follow the same outline.

3 Example 1 - Paper 1 Section A Concepts and Skills 011 (Project Maths Trial Schools) Paper 1 Q1 (a) Explain what a prime number is. Definitions are being given a new emphasis with the introduction Project Maths. It is no longer sufficient to know whether a number is prime, but we must understand why. Solution A prime number is any number whose only factors are 1 and itself. (b) Express 65 as a product of prime numbers. All numbers can be written as a product of prime numbers. It is just a matter of breaking the factors down until they are all prime. Solution 65 = = = = (c) The number 61 1 is a prime number. Using your calculator, or otherwise, express its value, correct to two significant figures, in the form a 10 n, where 1 a < 10 and n N. Most calculators will do this for you. Solution 61 1 = = , correct to two significant figures. (d) Use your answer to part (c) to state how many digits there are in the exact value of Tip: If having difficulties with hard questions, we can try to think of a simple example and solve that. For example, is 3, which has two digits = 30, which has 3 digits. It is always one more than the power of the ten. Solution 61 1 = has = 19 digits. 3

4 Example - Paper 1 Section B Contexts and Applications 011 (Project Maths Trial Schools) Sample Paper 1 Q5 Síle is investigating the number of square grey tiles needed to make patterns in a sequence. The first three patterns are shown below, and the sequence continues in the same way. In each pattern, the tiles form a square and its two diagonals. There are no tiles in the white areas in the patterns there are only the grey tiles. Figure 1: (taken from the Leaving Cert paper, available from the State Examinations Commission website) (a) In the table below, write the number of tiles needed for each of the first five patterns. Patterns No. of tiles 1 33 The number of tiles for the 3 rd pattern can simply be counted. We should recognise that the first pattern is of width 5, the next of width 7 and the next of width 9. Therefore the next two will be of width 11 and 13. For the 4 th and 5 th patterns, a good approach is to actually draw out the tiles: 4

5 Figure : 4 th pattern 5 th pattern Solution Patterns No. of tiles (b) Find, in terms of n, a formula that gives the number of tiles needed to make the nth pattern. We note that the number of tiles goes up by 1 each time. This means it is an arithmetic sequence, where the common difference is d = 1. We then use the formula for the nth term of an arithmetic sequence T n. For this we must know a, the first term of the sequence. Solution T n = a + (n 1)d = 1 + (n 1)1 = 1 + 1n 1 = 9 + 1n 5

6 (c) Using your formula, or otherwise, find the number of tiles in the tenth pattern. When a question offers the choice of solving a problem by using a previous solution or otherwise, then the use of the previous solution is always preferable. The exception is if we have been unable to find the previous solution, in which case this phrasing is explaining that the problem can still be solved using an alternative method. In this case, that could involve either adding the 1 tiles to the number of tiles required for each pattern until reaching the tenth pattern. Alternatively, we could try to draw the pattern of width 3 tiles. T n = 9 + 1n T 10 = 9 + 1(10) T 10 = T 10 = 19 (d) Síle has 399 tiles. What is the biggest pattern in the sequence that she can make? This question clearly requires a whole number as the solution. Also, it is not possible to simply round the answer using the standard rounding-off convention. If we find it is possible to make the 3.6 th pattern, for example, then clearly she can only make the third pattern and not the fourth. For this reason, the number must be rounded down. Solution We use T n = 9 + 1n and we are told that T n 399. Therefore: T n = 9 + 1n n n 390 1n 390 n 1 n 3.5 So Síle can make the 3 nd pattern (and not the 33 rd ). 6

7 (e) Find, in terms of n, a formula for the total number of tiles in the first n patterns. When the terms of a sequence are added, this is known as a series. In this case, we have an arithmetic sequence where d = 1 and a = 1 as before, but we use the formula for the sum of n terms of an arithmetic sequence: S n = n [a + (n 1)d] = n [(1) + (n 1)(1)] = n [4 + 1n 1] = n [30 + 1n] = 30n + 1n = 15n + 6n (f) Síle starts at the beginning of the sequence and makes as many of the patterns as she can. She does not break up the earlier patterns to make the new ones. For example, after making the first two patterns, she has used up 54 tiles, (1 + 33). How many patterns can she make in total with her 399 tiles? This question simply requires using the formula we found in the previous section but with S n = 399. S n = 15n + 6n 399 = 15n + 6n 0 = 6n + 15n = n + 5n 133, dividing both sides by 3 0 = (n + 19)(n 7) n + 19 = 0 or n 7 = 0 n = 19 or n = 7 n = 19 or n = 7 n = 7, (as we disregard negative answers here) 7

8 So Síle can make the 1 st, nd, 3 rd, 4 th, 5 th, 6 th and 7 th patterns, which is 7 patterns in total. 8

9 Paper For the Leaving Cert 013, Paper is divided into just two sections. The entire of Paper has had a Project Maths re-lift, and there are no questions that follow the previous syllabus. As in Paper 1, Section A is entitled Concepts and Skills. It contains six questions, all of which must be completed. There is one choice given, as students are allowed the option of answering Question 6A or 6B. These six questions are designed to test the student s understanding of the basic mathematical concepts for examination in Paper. For the first five questions, topics that can appear include, but are not limited to: (i) coordinate geometry of the line and circle (ii) enlargements (iii) area and volume (iv) permutations (v) and also topics that may appear in Section B. [See Example 3.] Question 6 currently offers a choice of two topics, and will continue to do so for the duration of the three year roll-out of the new syllabus (01, 013, 014). After this time, there will be no choice given. Question 6A is based on synthetic geometry, which consists of being able to use certain terms related to logic and deductive reasoning (theorem, proof, axiom, corollary, converse, implies), as well as being able to carry out a number of geometric constructions (angle of 60 without using a protractor or set square, tangent to a given circle at a given point on it, parallelogram given the length of the sides and the measure of the angles, circumcentre and circumcircle of a given triangle using only straight-edge and compass, incentre and incircle of a given triangle using only straight-edge and compass, centroid of a triangle). Question 6B consists of answering a problem-solving question based on the geometry theorems studied for the Junior Certificate. In Paper, Section B is entitled Contexts and Applications and contains two questions, both of which must be answered. Again, these questions allow for more extensive testing of a student s understanding of particular subjects, and each question is likely to span a number of topics. In particular, students should be aware of the links 9

10 between probability and statistics and the strong possibility of these being tested together in one question. Topics that can appear in this section include, but are not limited to: (i) probability (ii) statistics (iii) trigonometry [See 011 (Project Maths Trial Schools) Paper Q7 as an example.] 10

11 Example (Project Maths Trial Schools) Paper Q3 A plastic toy is in the shape of a hemisphere. When it falls on the ground, there are two possible outcomes: it can land with the flat side facing down or with the flat side facing up. Two groups of students are trying to find the probability that it will land with the flat side down. (a) Explain why, even though there are two outcomes, the answer is not necessarily equal to 1. For questions like this that are asked in words, it is suitable for us to give an answer in words. Solution The answer would only be equal to 1 if both outcomes are equally likely, which we cannot know for definite. The two sides of the toy are shaped differently which may affect the outcome. (b) The students estimate the probability by experiment. Group A drops the toy 100 times. From this, they estimate that it lands flat side down with probability Group B drops the toy 500 times. From this, they estimate that it lands flat side down with probability (i) Which groups estimate is likely to be better, and why? Students are expected to know that probabilities estimated from experiments improve in accuracy as the number of trials increases. Solution Group B s estimate will be better as they have carried out more trials, leading to better accuracy. (ii) How many times did the toy land flat side down for Group B? If students get confused about what to do with the numbers, it is good practice to imagine what we would do if the numbers were easier. For example, if the toy was dropped 6 times and it was found to have a probability of 0.5 of landing flat side down, we would know it landed flat side down 3 times. This is because 0.5 is equal to 1, and a half of 6 is 3. Or, in other words, = 3. 11

12 Solution = 406 times (iii) Using the data from the two groups, what is the best estimate of the probability that the toy lands flat side down? Etimates are improved by using the most number of trials. Therefore the best estimate that can be made here is when we include the results from both experiments. Solution Group B found the toy landed flat side down = 406 times. Group A found the toy landed flat side down = 76 times. In total, the toy landed flat side down = 48 times out of 600 times. Best estimate of probability of landing flat side down =

### Grade 5 Math Content 1

Grade 5 Math Content 1 Number and Operations: Whole Numbers Multiplication and Division In Grade 5, students consolidate their understanding of the computational strategies they use for multiplication.

### 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,

### Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

### A Correlation of Pearson Texas Geometry Digital, 2015

A Correlation of Pearson Texas Geometry Digital, 2015 To the Texas Essential Knowledge and Skills (TEKS) for Geometry, High School, and the Texas English Language Proficiency Standards (ELPS) Correlations

### Prentice Hall Algebra 2 2011 Correlated to: Colorado P-12 Academic Standards for High School Mathematics, Adopted 12/2009

Content Area: Mathematics Grade Level Expectations: High School Standard: Number Sense, Properties, and Operations Understand the structure and properties of our number system. At their most basic level

### Paper 2 Revision. (compiled in light of the contents of paper1) Higher Tier Edexcel

Paper 2 Revision (compiled in light of the contents of paper1) Higher Tier Edexcel 1 Topic Areas 1. Data Handling 2. Number 3. Shape, Space and Measure 4. Algebra 2 Data Handling Averages Two-way table

### DELAWARE MATHEMATICS CONTENT STANDARDS GRADES 9-10. PAGE(S) WHERE TAUGHT (If submission is not a book, cite appropriate location(s))

Prentice Hall University of Chicago School Mathematics Project: Advanced Algebra 2002 Delaware Mathematics Content Standards (Grades 9-10) STANDARD #1 Students will develop their ability to SOLVE PROBLEMS

### Answer: 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

### In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data.

MATHEMATICS: THE LEVEL DESCRIPTIONS In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. Attainment target

### LESSON 4 Missing Numbers in Multiplication Missing Numbers in Division LESSON 5 Order of Operations, Part 1 LESSON 6 Fractional Parts LESSON 7 Lines,

Saxon Math 7/6 Class Description: Saxon mathematics is based on the principle of developing math skills incrementally and reviewing past skills daily. It also incorporates regular and cumulative assessments.

### Core 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

### Prentice Hall: Middle School Math, Course 1 2002 Correlated to: New York Mathematics Learning Standards (Intermediate)

New York Mathematics Learning Standards (Intermediate) Mathematical Reasoning Key Idea: Students use MATHEMATICAL REASONING to analyze mathematical situations, make conjectures, gather evidence, and construct

MTN Learn Mathematics Grade 10 radio support notes Contents INTRODUCTION... GETTING THE MOST FROM MINDSET LEARN XTRA RADIO REVISION... 3 BROADAST SCHEDULE... 4 ALGEBRAIC EXPRESSIONS... 5 EXPONENTS... 9

### Key Topics What will ALL students learn? What will the most able students learn?

2013 2014 Scheme of Work Subject MATHS Year 9 Course/ Year Term 1 Key Topics What will ALL students learn? What will the most able students learn? Number Written methods of calculations Decimals Rounding

### Mathematics. 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

### *&6( 0DWKHPDWLFV,QWURGXFWLRQ

2;)25 23(1 *&6( 0DWKHPDWLFV,QWURGXFWLRQ Maths GCSE Welcome to your Mathematics GCSE course! This introduction contains all the information you need to be able to start your course, and you can also use

### Introduction. Maths IGCSE. Which Syllabus does this Course follow?

Maths IGCSE Welcome to your Mathematics IGCSE course! This introduction contains all the information you need to be able to start your course, and you can also use it as a reference point as you work your

### G 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.

### Section 1: How will you be tested? This section will give you information about the different types of examination papers that are available.

REVISION CHECKLIST for IGCSE Mathematics 0580 A guide for students How to use this guide This guide describes what topics and skills you need to know for your IGCSE Mathematics examination. It will help

### Numeracy and mathematics Experiences and outcomes

Numeracy and mathematics Experiences and outcomes My learning in mathematics enables me to: develop a secure understanding of the concepts, principles and processes of mathematics and apply these in different

### Properties 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

### ALGEBRA 2/TRIGONOMETRY

ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Tuesday, June 1, 011 1:15 to 4:15 p.m., only Student Name: School Name: Print your name

### ALGEBRA 2/TRIGONOMETRY

ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Tuesday, January 8, 014 1:15 to 4:15 p.m., only Student Name: School Name: The possession

### MATH. ALGEBRA I HONORS 9 th Grade 12003200 ALGEBRA I HONORS

* Students who scored a Level 3 or above on the Florida Assessment Test Math Florida Standards (FSA-MAFS) are strongly encouraged to make Advanced Placement and/or dual enrollment courses their first choices

### Curriculum Overview YR 9 MATHS. SUPPORT CORE HIGHER Topics Topics Topics Powers of 10 Powers of 10 Significant figures

Curriculum Overview YR 9 MATHS AUTUMN Thursday 28th August- Friday 19th December SUPPORT CORE HIGHER Topics Topics Topics Powers of 10 Powers of 10 Significant figures Rounding Rounding Upper and lower

### Bridging Documents for Mathematics

Bridging Documents for Mathematics 5 th /6 th Class, Primary Junior Cycle, Post-Primary Primary Post-Primary Card # Strand(s): Number, Measure Number (Strand 3) 2-5 Strand: Shape and Space Geometry and

### IB Maths SL Sequence and Series Practice Problems Mr. W Name

IB Maths SL Sequence and Series Practice Problems Mr. W Name Remember to show all necessary reasoning! Separate paper is probably best. 3b 3d is optional! 1. In an arithmetic sequence, u 1 = and u 3 =

### Math Placement Test Sample Problems PRE-ALGEBRA

Math Placement Test Sample Problems The Math Placement Test is an untimed, multiple-choice, computer-based test. The test is composed of four sections: pre-algebra, algebra, college algebra, and trigonometry.

### Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

### Week 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

### Grade 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

### WORK SCHEDULE: MATHEMATICS 2007

, K WORK SCHEDULE: MATHEMATICS 00 GRADE MODULE TERM... LO NUMBERS, OPERATIONS AND RELATIONSHIPS able to recognise, represent numbers and their relationships, and to count, estimate, calculate and check

### Senior Phase Grade 8 Today Planning Pack MATHEMATICS

M780636110236 Senior Phase Grade 8 Today Planning Pack MATHEMATICS Contents: Work Schedule: Page Grade 8 2 Lesson Plans: Grade 8 4 Rubrics: Rubric 1: Recognising, classifying and representing numbers...22

### ALGEBRA 2/TRIGONOMETRY

ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Thursday, January 9, 015 9:15 a.m to 1:15 p.m., only Student Name: School Name: The possession

### Algebra I Credit Recovery

Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,

### SOLVING EQUATIONS WITH RADICALS AND EXPONENTS 9.5. section ( 3 5 3 2 )( 3 25 3 10 3 4 ). The Odd-Root Property

498 (9 3) Chapter 9 Radicals and Rational Exponents Replace the question mark by an expression that makes the equation correct. Equations involving variables are to be identities. 75. 6 76. 3?? 1 77. 1

### Answer: Quantity A is greater. Quantity A: 0.717 0.717717... Quantity B: 0.71 0.717171...

Test : First QR Section Question 1 Test, First QR Section In a decimal number, a bar over one or more consecutive digits... QA: 0.717 QB: 0.71 Arithmetic: Decimals 1. Consider the two quantities: Answer:

### What to Expect on the Compass

What to Expect on the Compass What is the Compass? COMPASS is a set of untimed computer adaptive tests created by the American College Test (ACT) Program. Because COMPASS tests are "computer adaptive,"

### Trigonometric Functions and Equations

Contents Trigonometric Functions and Equations Lesson 1 Reasoning with Trigonometric Functions Investigations 1 Proving Trigonometric Identities... 271 2 Sum and Difference Identities... 276 3 Extending

### Pre-Algebra 2008. Academic Content Standards Grade Eight Ohio. Number, Number Sense and Operations Standard. Number and Number Systems

Academic Content Standards Grade Eight Ohio Pre-Algebra 2008 STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express large numbers and small

### The Australian Curriculum Mathematics

The Australian Curriculum Mathematics Mathematics ACARA The Australian Curriculum Number Algebra Number place value Fractions decimals Real numbers Foundation Year Year 1 Year 2 Year 3 Year 4 Year 5 Year

### Precalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES

Content Expectations for Precalculus Michigan Precalculus 2011 REVERSE CORRELATION CHAPTER/LESSON TITLES Chapter 0 Preparing for Precalculus 0-1 Sets There are no state-mandated Precalculus 0-2 Operations

### MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab

MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring non-course based remediation in developmental mathematics. This structure will

### GEOMETRY. Constructions OBJECTIVE #: G.CO.12

GEOMETRY Constructions OBJECTIVE #: G.CO.12 OBJECTIVE Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic

### MATHS LEVEL DESCRIPTORS

MATHS LEVEL DESCRIPTORS Number Level 3 Understand the place value of numbers up to thousands. Order numbers up to 9999. Round numbers to the nearest 10 or 100. Understand the number line below zero, and

### COURSE SYLLABUS -----------------------------------------------------------------------------------

Last Reviewed by: Leslie Wurst Date Approved: Date Revised: Fall 2012 COURSE SYLLABUS Syllabus for: MATH 1010 Math for General Studies Former Course and Title: Former Quarter Course(s): Mat 1260 Contemporary

### Mathematics Georgia Performance Standards

Mathematics Georgia Performance Standards K-12 Mathematics Introduction The Georgia Mathematics Curriculum focuses on actively engaging the students in the development of mathematical understanding by

### Glencoe. correlated to SOUTH CAROLINA MATH CURRICULUM STANDARDS GRADE 6 3-3, 5-8 8-4, 8-7 1-6, 4-9

Glencoe correlated to SOUTH CAROLINA MATH CURRICULUM STANDARDS GRADE 6 STANDARDS 6-8 Number and Operations (NO) Standard I. Understand numbers, ways of representing numbers, relationships among numbers,

### Mathematical 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;

### NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS

NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document

### Charlesworth School Year Group Maths Targets

Charlesworth School Year Group Maths Targets Year One Maths Target Sheet Key Statement KS1 Maths Targets (Expected) These skills must be secure to move beyond expected. I can compare, describe and solve

### ALGEBRA. 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,

### Prentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)

Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify

### Mathematics Navigator. Misconceptions and Errors

Mathematics Navigator Misconceptions and Errors Introduction In this Guide Misconceptions and errors are addressed as follows: Place Value... 1 Addition and Subtraction... 4 Multiplication and Division...

### Incenter Circumcenter

TRIANGLE: Centers: Incenter Incenter is the center of the inscribed circle (incircle) of the triangle, it is the point of intersection of the angle bisectors of the triangle. The radius of incircle is

### Changes to GCSE assessment across subjects

OCR GCSE Mathematics (J560) now accredited ocr.org.uk/gcsemaths Introducing the new Mathematics GCSE for first teaching from 2015 In February 2013, the Secretary of State for Education Michael Gove wrote

### Performance Based Learning and Assessment Task Triangles in Parallelograms I. ASSESSSMENT TASK OVERVIEW & PURPOSE: In this task, students will

Performance Based Learning and Assessment Task Triangles in Parallelograms I. ASSESSSMENT TASK OVERVIEW & PURPOSE: In this task, students will discover and prove the relationship between the triangles

### Teaching & Learning Plans. Arithmetic Sequences. Leaving Certificate Syllabus

Teaching & Learning Plans Arithmetic Sequences Leaving Certificate Syllabus The Teaching & Learning Plans are structured as follows: Aims outline what the lesson, or series of lessons, hopes to achieve.

### FOREWORD. Executive Secretary

FOREWORD The Botswana Examinations Council is pleased to authorise the publication of the revised assessment procedures for the Junior Certificate Examination programme. According to the Revised National

### Geometry. Higher Mathematics Courses 69. Geometry

The fundamental purpose of the course is to formalize and extend students geometric experiences from the middle grades. This course includes standards from the conceptual categories of and Statistics and

### KEANSBURG SCHOOL DISTRICT KEANSBURG HIGH SCHOOL Mathematics Department. HSPA 10 Curriculum. September 2007

KEANSBURG HIGH SCHOOL Mathematics Department HSPA 10 Curriculum September 2007 Written by: Karen Egan Mathematics Supervisor: Ann Gagliardi 7 days Sample and Display Data (Chapter 1 pp. 4-47) Surveys and

### Consumer Math 15 INDEPENDENT LEAR NING S INC E 1975. Consumer Math

Consumer Math 15 INDEPENDENT LEAR NING S INC E 1975 Consumer Math Consumer Math ENROLLED STUDENTS ONLY This course is designed for the student who is challenged by abstract forms of higher This math. course

### Tennessee Mathematics Standards 2009-2010 Implementation. Grade Six Mathematics. Standard 1 Mathematical Processes

Tennessee Mathematics Standards 2009-2010 Implementation Grade Six Mathematics Standard 1 Mathematical Processes GLE 0606.1.1 Use mathematical language, symbols, and definitions while developing mathematical

### Grade 4 Unit 3: Multiplication and Division; Number Sentences and Algebra

Grade 4 Unit 3: Multiplication and Division; Number Sentences and Algebra Activity Lesson 3-1 What s My Rule? page 159) Everyday Mathematics Goal for Mathematical Practice GMP 2.2 Explain the meanings

### Mathematics (Project Maths)

Pre-Leaving Certificate Examination Mathematics (Project Maths) Paper 2 Higher Level February 2010 2½ hours 300 marks Running total Examination number Centre stamp For examiner Question Mark 1 2 3 4 5

### Number Sense and Operations

Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents

### Analytical Geometry (4)

Analytical Geometry (4) Learning Outcomes and Assessment Standards Learning Outcome 3: Space, shape and measurement Assessment Standard As 3(c) and AS 3(a) The gradient and inclination of a straight line

### MINISTRY OF EDUCATION

Republic of Namibia MINISTRY OF EDUCATION NAMIBIA SENIOR SECONDARY CERTIFICATE (NSSC) MATHEMATICS SYLLABUS ORDINARY LEVEL SYLLABUS CODE: 4324 GRADES 11-12 FOR IMPLEMENTATION IN 2006 FOR FIRST EXAMINATION

### Mathematics. Programme of study for key stage 3 and attainment targets (This is an extract from The National Curriculum 2007)

Mathematics Programme of study for key stage 3 and attainment targets (This is an extract from The National Curriculum 2007) Crown copyright 2007 Qualifications and Curriculum Authority 2007 Curriculum

### McDougal Littell California:

McDougal Littell California: Pre-Algebra Algebra 1 correlated to the California Math Content s Grades 7 8 McDougal Littell California Pre-Algebra Components: Pupil Edition (PE), Teacher s Edition (TE),

### Mathematical Induction. Mary Barnes Sue Gordon

Mathematics Learning Centre Mathematical Induction Mary Barnes Sue Gordon c 1987 University of Sydney Contents 1 Mathematical Induction 1 1.1 Why do we need proof by induction?.... 1 1. What is proof by

Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express

### 3. Logical Reasoning in Mathematics

3. Logical Reasoning in Mathematics Many state standards emphasize the importance of reasoning. We agree disciplined mathematical reasoning is crucial to understanding and to properly using mathematics.

### Determine If An Equation Represents a Function

Question : What is a linear function? The term linear function consists of two parts: linear and function. To understand what these terms mean together, we must first understand what a function is. The

### Mathematics (MAT) MAT 061 Basic Euclidean Geometry 3 Hours. MAT 051 Pre-Algebra 4 Hours

MAT 051 Pre-Algebra Mathematics (MAT) MAT 051 is designed as a review of the basic operations of arithmetic and an introduction to algebra. The student must earn a grade of C or in order to enroll in MAT

### Standards and progression point examples

Mathematics Progressing towards Foundation Progression Point 0.5 At 0.5, a student progressing towards the standard at Foundation may, for example: connect number names and numerals with sets of up to

### 2.5 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 x-axis and

### 1. The volume of the object below is 186 cm 3. Calculate the Length of x. (a) 3.1 cm (b) 2.5 cm (c) 1.75 cm (d) 1.25 cm

Volume and Surface Area On the provincial exam students will need to use the formulas for volume and surface area of geometric solids to solve problems. These problems will not simply ask, Find the volume

### National 5 Mathematics Course Assessment Specification (C747 75)

National 5 Mathematics Course Assessment Specification (C747 75) Valid from August 013 First edition: April 01 Revised: June 013, version 1.1 This specification may be reproduced in whole or in part for

### Math 115 Spring 2011 Written Homework 5 Solutions

. Evaluate each series. a) 4 7 0... 55 Math 5 Spring 0 Written Homework 5 Solutions Solution: We note that the associated sequence, 4, 7, 0,..., 55 appears to be an arithmetic sequence. If the sequence

### GCSE 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

### Sequences. 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

### Standards for Mathematical Practice: Commentary and Elaborations for 6 8

Standards for Mathematical Practice: Commentary and Elaborations for 6 8 c Illustrative Mathematics 6 May 2014 Suggested citation: Illustrative Mathematics. (2014, May 6). Standards for Mathematical Practice:

E XPLORING QUADRILATERALS E 1 Geometry State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space. Statement of Purpose: The activities in this

### BPS Math Year at a Glance (Adapted from A Story Of Units Curriculum Maps in Mathematics K-5) 1

Grade 4 Key Areas of Focus for Grades 3-5: Multiplication and division of whole numbers and fractions-concepts, skills and problem solving Expected Fluency: Add and subtract within 1,000,000 Module M1:

### Current Standard: Mathematical Concepts and Applications Shape, Space, and Measurement- Primary

Shape, Space, and Measurement- Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two- and three-dimensional shapes by demonstrating an understanding of:

### Math at a Glance for April

Audience: School Leaders, Regional Teams Math at a Glance for April The Math at a Glance tool has been developed to support school leaders and region teams as they look for evidence of alignment to Common

### PREPARING YOUR LEARNERS FOR THE MATHEMATICS (MAT) TEST

THE NATIONAL Dr Carol Bohlmann BENCHMARK TESTS: PREPARING YOUR LEARNERS FOR THE MATHEMATICS (MAT) TEST Dr Carol Bohlmann NBTP Mathematics Research Lead Centre for Educational Testing for Access and Placement

### Unit 6 Trigonometric Identities, Equations, and Applications

Accelerated Mathematics III Frameworks Student Edition Unit 6 Trigonometric Identities, Equations, and Applications nd Edition Unit 6: Page of 3 Table of Contents Introduction:... 3 Discovering the Pythagorean

### Parallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.

CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes

### NEW MEXICO Grade 6 MATHEMATICS STANDARDS

PROCESS STANDARDS To help New Mexico students achieve the Content Standards enumerated below, teachers are encouraged to base instruction on the following Process Standards: Problem Solving Build new mathematical

### MATHEMATICS (MATH) 3. Provides experiences that enable graduates to find employment in sciencerelated

194 / Department of Natural Sciences and Mathematics MATHEMATICS (MATH) The Mathematics Program: 1. Provides challenging experiences in Mathematics, Physics, and Physical Science, which prepare graduates

### Year 9 set 1 Mathematics notes, to accompany the 9H book.

Part 1: Year 9 set 1 Mathematics notes, to accompany the 9H book. equations 1. (p.1), 1.6 (p. 44), 4.6 (p.196) sequences 3. (p.115) Pupils use the Elmwood Press Essential Maths book by David Raymer (9H

### Friday, January 29, 2016 9:15 a.m. to 12:15 p.m., only

ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Friday, January 9, 016 9:15 a.m. to 1:15 p.m., only Student Name: School Name: The possession

### Grade 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

### Scope and Sequence KA KB 1A 1B 2A 2B 3A 3B 4A 4B 5A 5B 6A 6B

Scope and Sequence Earlybird Kindergarten, Standards Edition Primary Mathematics, Standards Edition Copyright 2008 [SingaporeMath.com Inc.] The check mark indicates where the topic is first introduced