 ## Mathematics Overview

The Primary Years Programme (PYP) views mathematics not as a fixed body of knowledge to be transmitted, but as a way of thinking and a language for understanding meaning. Traditionally, mathematical knowledge has been disassembled in schools broken into unrelated skills-based activities. Our vastly expanded knowledge base about learning mathematics however, tells us that people assemble, or construct, mathematical knowledge. This requires us to look at mathematics not as a fixed body of knowledge to be transmitted, but as a language and a way of thinking. By engaging in various activities and discussions, students construct meaning, using multiple strategies, developing an understanding of which strategies are most effective and efficient.

Cognitive psychologists have described the stages through which children learn mathematics:

 Constructing meaning Transferring meaning into signs and symbols Understanding and applying Teachers plan activities, through which students construct meaning from direct experience, by using manipulatives and conversation. Teachers connect the notation system with the concrete objects and associated mathematical processes. The teacher provides the symbols for the students. Students begin to describe their understanding using signs and symbols. Teachers plan authentic activities in which students independently select and use appropriate symbolic notation to process and record their thinking.

 The following expectations are arranged in five interwoven strands of knowledge. In number and pattern and function, students inquire into our number system, and its operations, patterns and functions. The remaining strands, data handling, measurement and shape and space, are the areas of mathematics that other disciplines use to research, describe, represent and understand aspects of their domain.

These are the expectations for each year group in Mathematics

 Jump to the expectations for Early Childhood    Year 1    Year 2    Year 3    Year 4     Year 5     Year 6 Number Pattern and function Data handling Measurement Shape and Space The students in Early childhood will: Read, write and model numbers to 20 Count, compare and order numbers to 20 Estimate quantities to 10 Use ordinal numbers to describe the position of things in a sequence Model number relationships to 10: “Show me one more than three, take two cubes away from these cubes” Use language of mathematics: more, less, number names, total Use 1:1 correspondence Explore the conservation of number through the use of manipulatives (xxxx = x x x x) Select and explain an appropriate method for solving a problem Find and describe simple patterns Create simple patterns using real objects Sort and label real objects into sets by attributes Create a graph of real objects and compare quantities using number words Discuss and identify outcomes that will happen, won’t happen and might happen Identify, compare and describe attributes of real objects and situations: longer, shorter, heavier, empty, full, hotter, colder Identify, compare and sequence events in their daily routine: before, after, bedtime, storytime, today, tomorrow Sort, describe and compare 3-D shapes according to attributes such as size or form Explore and describe the paths, regions and boundaries of their immediate environment (inside, outside, above, below) and their position (next to, behind, in front of, up, down) Number Pattern and function Data handling Measurement Shape and Space The students in Year 1 will: Begin to read, write and order all numbers from 0 to 100; Understand and use the vocabulary of comparing and ordering these numbers Count forwards and backwards from any small number, and in fives and tens from zero to one hundred Estimate quantities to 50 Understand the operation of addition, and of subtraction (as 'take away' or 'difference'), and use the related vocabulary and symbols Begin to memorize simple number facts Use fraction names (half and quarter) to describe part and whole relationships Create, describe and extend patterns Recognize, describe and extend patterns in numbers: odd and even, skip counting by 5s and 10s Identify simple patterns and rules for addition and subtraction Solve simple problems by sorting, classifying and organizing information in various ways Discuss, interpret and compare data represented in teacher generated diagrams: tree, Carroll and Venn Begin to understand the purpose of graphing data Estimate, measure, label and compare using non-standard units of measurement: length, mass or capacity Begin to understand why we use standard units of measurement to measure Use a calendar to determine the date and to identify and begin to sequence days of the week and months of the year Tell time to the hour Use everyday language to describe features of familiar 3-D and 2-D shapes Sort and label 2D and 3D shapes using appropriate vocabulary: sides, corners, circle, sphere, cube Create 2D shapes Number Pattern and function Data handling Measurement Shape and Space The students in Year 2 will: Count, read, write and order whole numbers to at least 100; know what each digit represents (including 0 as a place holder). Read, write and model numbers, using the base 10 system. Read, write and model addition and subtraction to 20 (with and without regrouping). Understand that subtraction is the inverse of addition. ? Recall addition and subtraction facts to 10. Use knowledge that addition can be done in any order to do mental calculations more efficiently. Understand the operation of multiplication as repeated addition or as describing an array in real world contexts, using manipulatives, diagrams, and symbols. Use fraction names (half and quarter) to describe part and whole relationships. Create, describe and extend patterns. Recognize, describe and extend patterns in numbers: odd and even, skip counting, 2s, 5s and 10s, counting on, counting back. Explore patterns in fact families. Record, organize, classify, display and understand data in a variety of simple ways Discuss, interpret and compare data represented in teacher generated diagrams: tree, Carroll and Venn Understand the purpose of graphing data Create a pictograph and simple bar graph from a graph of real objects, and interpret data by comparing quantities: more, fewer, less than, greater than Discuss, identify, predict and place outcomes in order of likelihood: impossible, unlikely, likely, certain Estimate, measure and compare lengths, masses and capacities, using standard units Understand why we use standard units of measurement to measure Use a calendar to determine the date and to identify and sequence days of the week and months of the year Read a simple scale to the nearest labelled division, including using a ruler to draw and measure lines to the nearest centimetre Estimate, identify and compare lengths of time, second, minute, hour, day, week, month Read and write the time to the hour, half hour and quarter hour Use the mathematical names for common 2-D and 3-D shapes; sort shapes and describe some of their features Find and explain symmetry in their immediate environment Create and explain simple symmetrical patterns Use mathematical vocabulary to describe position, direction and movement Number Pattern and function Data handling Measurement Shape and Space The students in Year 3 will: Read, write and order numbers to 1000 and understand what each digit represents Estimate quantities to at least 100 Count on or back in tens or hundreds from any two- or three-digit number Count in 3s, 4s, 5s, and explore other numbers Recognize unit fractions such as 1/2, 1/3, 1/4, 1/5, 1/10, and begin to use them to find fractions of shapes and numbers Add and subtract mentally a 'near multiple of 10' to or from a two-digit number Reasonably estimate answers to 100, using rounding and approximation Know by heart all addition and subtraction facts for each number to 20 Know by heart facts for the 2, 3, 4, 5 and 10 multiplication tables Choose and use appropriate operations to solve word problems, explaining methods and reasoning Understand and be able to explain the relationship between the number operations., addition and subtraction, multiplication and division Identify patterns in number systems to 100 Discuss, compare and create sets from data that has subsets using tree, Carroll, Venn and other diagram Design a survey, process and interpret the data Solve a given problem by organizing and interpreting numerical data in simple lists, tables and graphs Collect and display data in a bar graph and interpret results Use the scale on the vertical axis of a bar graph to represent large quantities Read and write the time to the minute using intervals of fifteen minutes, ten minutes and five minutes on a 12-hour clock Estimate, measure, label and compare using formal methods and standard units of measurement: length, mass, time and temperature Select appropriate tools and units of measurement Begin to understand and use monetary notation Identify right angles Identify lines of symmetry in simple shapes and recognize shapes with no lines of symmetry Number Pattern and function Data handling Measurement Shape and Space The students in Year 4 will: Read, write and model numbers, using the base 10 system, to 1000 Count, compare and order numbers to 1000 Estimate quantities to 1000 Count in 3s, 4s, 5s, and explore other numbers Use number patterns to learn multiplication tables: 1s, 2s, 5s, 10s Automatically recall basic addition and subtraction facts Model addition and subtraction equations to 1000 (with and without regrouping) Use mathematical vocabulary and symbols of multiplication and division: times, divide, product, factors, x ÷ Use and describe multiple strategies to solve addition, subtraction, multiplication and division problems Read, write and model multiplication and division problems Compare fractions using manipulatives and using fractional notation Use mathematical vocabulary and symbols of fractions: numerator, denominator, equivalence Understand and model the concept of equivalence to 1: two halves = 1, three thirds = 1 Reasonably estimate answers: rounding and approximation Select and explain an appropriate method for solving a problem Analyse patterns in number systems to 100 Recognize, describe and extend more complex patterns in numbers Understand and use the relationship between addition and subtraction: 4 + 3 = 7, 7 – 3 = 4 Identify patterns and rules for multiplication and division: 4 x 3 = 12, 3 x 4 = 12 and 12 ÷ 3 = 4, 12 ÷ 4 = 3 Model, with manipulatives, the relationship between multiplication and division model, with manipulatives, the relationship between multiplication and addition (repeated addition) Model, with manipulatives, the relationship between division and subtraction Model multiplication as an array Understand and use number patterns to solve problems (missing numbers) Discuss, compare and create sets from data that has subsets using tree, Carroll, Venn and other diagrams Collect and display data in a bar graph and interpret results Use the scale on the vertical axis of a bar graph to represent large quantities Find, describe and explain the mode in a set of data and its use Understand the purpose of a database by manipulating the data to answer questions and solve problems Use probability to determine mathematically fair and unfair games and to explain possible outcomes  Estimate, measure, label and compare using formal methods and standard units of measurement: length, mass, time and temperature Select appropriate tools and units of measurement Describe measures that fall between numbers on a measure scale: 31/2kg, between 4cm and 5cm Estimate, measure, label and compare perimeter and area Model addition and subtraction using money Read and write the time to the minute, using intervals of 10 minutes, 5 minutes and 1 minute, on 12-hour analogue and digital clocks Sort, describe and model regular and irregular polygons: triangles, hexagons, trapeziums Identify, describe and model congruency in 2-D shapes Combine and transform 2-D shapes to make another shape Create symmetrical patterns, including tessellation Identify lines and axes of reflective and rotational symmetry Understand an angle as a measure of rotation by comparing and describing rotations: whole turn; half turn; quarter turn; north, south, east and west on a compass Locate features on a grid using coordinates Number Pattern and function Data handling Measurement Shape and Space The students in Year 5 will: Read, write and model numbers using base ten system up to one million Count, compare and order numbers up to one million Estimate quantities beyond a thousand Automatically recall addition, subtraction, multiplication and division facts Create and solve using multiple strategies, addition, subtraction multiplication and division problems Use number patterns to solve problems involving larger numbers Read, write, model and compare fractions Understand and model equivalence Reasonably estimate answers using rounding and approximation Select, explain and defend various methods for solving problems Recognise, describe and extend more complex patterns in numbers Model and explain number patterns and use patterns to solve problems Understand and use the relationship among the four operations - addition, subtraction, multiplication and division Use real-life problems to create a number pattern following a rule Design surveys and collect, organise and record data in displays Create, interpret, discuss and compare data displays Use a scale on a graph to represent large quantities Find, describe and explain the mode in a set of data Discuss, identify, predict and place outcomes in order of likelihood: impossible, unlikely, likely and certain Estimate, measure, label and compare length, mass, time, temperature Select and use appropriate standard units of measurement Use measuring tools accurately Develop procedures for finding perimeter, area and volume Measure and construct angles in degrees using a protractor Use and construct time-lines Read and write the time to the minute and second, using intervals of 10, 5 and 1 minute on twelve and 24-hour clocks Combine and transform 2-D shapes to make another shape Create symmetrical patterns, including tessellation Identify lines and axes of reflective symmetry Understand that an angle is a measure of rotation Locate features on a grid using co-ordinates Use geometric vocabulary of 2-D and 3-D shapes and angles Classify, sort and label different types of triangles and quadrilaterals Use a pair of compasses Number Pattern and function Data handling Measurement Shape and Space The students in Year 6 will: Read, write and model numbers to one million and beyond Automatically use number facts Read, write, model and compare fractions, decimals and percentages Interchange fractions, decimals and percentages Add and subtract decimals to the thousandths Find and use ratios Read, write and model additional and subtraction of integers Use exponential notation Describe strategies to create and solve more complex problems Understand and use the relationship between the four operations Model and explain number patterns and use real-life problems to create a number pattern following a rule Develop, explain and model simple algebraic formulas Model exponents as repeated multiplication Understand and use exponents and roots as inverse functions Display and interpret data in a variety of ways, compare data display Find, describe and explain the range, mode, median and mean in a set of data Use a numerical probability scale 0-1 or 1%-100% Determine theoretical probability of an event and explain why it might differ from the experimental probability Estimate, measure, label and compare perimeter, area and volume Develop procedures for finding perimeter, area and volume Use the correct tools for any measurement with accuracy Measure and construct angles in degrees using a protractor Use and construct 12-hour and 24-hour timetables Be able to determine time world-wide Use the mathematical vocabulary of 2-D and 3-D shapes and angles Classify, sort and label all types of triangles and quadrilaterals Turn a 2-D net into a 3-D shape and vice versa Use scale and ratio to enlarge and reduce shapes Read and plot coordinates in four quadrants 5concepts
 Demonstrations of deeper learning in responsible behaviour through positive action and service; a manifestation in practice of the other essential elements Powerful ideas which have relevance within and across the disciplines and which students must explore and re-explore in order to develop understanding Significant, relevant, subject matter which students are expected to explore and know aboutClick here to find out more about the 6 themesClick here to find out more about the 6 subject areas Those things which the students need to be able to do to succeed in a changing, challenging world Dispositions which are expressions of fundamental values, beliefs and feelings about learning, the environment and people

home

updated 16 fÈvrier, 2007