Math Curriculum

Palo Alto Unified School District MATHEMATICS IN GRADE TWO The goals for the K-12 Mathematics Program are to enable students to: use mathematics to solve problems; develop fluency in mathematical skills and procedures; reason and communicate mathematically; gain confidence in their own mathematical ability; and value mathematics -PAUSD Mathematics Curriculum Summary 5/95 At each grade level we provide a balanced, comprehensive program of basic skills, conceptual understanding, and problem solving to help students meet the PAUSD grade level standards in mathematics.

Major areas of study and highlights of the key mathematical ideas your child will be exploring in the second grade are described below. Throughout the year students work on internalizing their basic number facts, developing fluency in mental math and practicing and reviewing skills as necessary.

The Number System: The mathematical emphasis of this unit is on developing a sense of numbers as whole quantities, looking for patterns and relationships in the number system and gaining facility with addition combinations. Students explore 5 and 10 multiples of 5 and 10, and they are introduced to combining and separating problems.

Addition and Subtraction: The mathematical emphasis of this unit is on developing strategies for solving addition and subtraction problems based on an understanding of numbers, number relationships, and the operations themselves. Students solve a variety of word problems that involve combining, separating, and comparing two-digit numbers. Emphasis is placed on understanding the problem, deciding which operation to use, and selecting problem solving strategies.

Geometry and Fractions: The mathematical emphasis of this unit is on using a variety of materials to explore the structure of shapes and how they can be composed and decomposed into other shapes. Students investigate the structure of rectangular arrays, find halves of 2-D and 3-D shapes, and explore symmetry.

Sorting and Classifying Data: The mathematical emphasis of this unit is on investigating similarities and differences in sets of related objects, people, and data. Students collect, organize, and represent data to communicate information clearly, accurately, and in ways that make sense to them.

Collecting and Representing Data: The mathematical emphasis of this unit is on using a variety of materials to collect, keep track of, and organize numerical data. Students explore various types of data representations, including line plots and bar graphs, and organize and represent data in ways that makes sense to them.

Measurement: The mathematical emphasis of this unit is on exploring linear measurement using direct and indirect comparison and nonstandard units. Students construct, compare and measure simple paths in related activities.

Time: The mathematical emphasis of this unit is on exploring concepts of time and rhythm. Students sequence events in time and create visual representations of those sequences using timelines with regular scales. Students also explore patterns expressed in interpreting those patterns.

In the Classroom

If you peek into your child's classroom, you'll see students:

	working in a variety of groupings: as a whole class, individually, in pairs, and in small groups
	considering their own reasoning and the reasoning of other students as they figure out correct solutions
	communicating about mathematics orally, in writing, and using pictures, tables, graphs, and models
	sometimes completing many routine problems quickly; sometimes working thoughtfully on just a few problems in more depth
	using more than one strategy to double check
	using a variety of tools, including paper and pencil, calculators, cubes, blocks, and measuring tools.

Structure of the Curriculum

Units: The curriculum at each grade level is organized into units. Each unit offers three to six weeks of mathematical work on topics in number, data analysis, and geometry. Because of the many interconnections among mathematical ideas, units may revolve around two or three related ideas--for example, fractions and area, multiplication and division, measuring and data.

Pictures and unit descriptions adapted from TERC, grade 2, Implementing the Investigations in Number, Data, and Space Curriculum Dale Seymour Publications, 1995

Investigations: In each unit, students explore the central topics in depth through a series of investigations, exploring related problems and using important mathematical ideas. The investigations work together in a carefully planned way. In them students work in depth on a small number of problems, actively using mathematical tools and consulting with peers as they find their own ways to solve the problems. Significant time is allowed for students to think about the problems and to model, draw, write, and talk about their work.

Sessions: Each investigation lasts over several class sessions. The structure of the sessions is somewhat different from a more traditional approach. Rather than have the teacher begin the period by explaining the mathematics, demonstrating techniques, and talking students through the first problem, the bulk of the whole-group time is at the end of the session, when students are comparing their methods and results, analyzing their work, and sharing conclusions.

Through this type of curriculum, students come to understand, enjoy, and appreciate mathematics. When students are involved in designing a toy from interlocking cubes and then writing their own directions for building it, or playing a game that involves relationships among numbers under 100, or bringing their knowledge of the environment to a study of how quickly plants grow, they recognize their work as serious problem solving with a purpose. They develop a sense of appreciation for the power and beauty of mathematics as they learn to value their own thinking and strategies.

BY THE END OF KINDERGARTEN STUDENTS WILL:	BY THE END OF FIRST GRADE STUDENTS WILL:	BY THE END OF SECOND GRADE STUDENTS WILL:
Use pictures to record and share thinking	Begin to explain and justify solutions and strategies used to arrive at solution Create own word problems related to a particular topic or theme Use a calculator to solve simple arithmetic problems	Explain and justify solutions and strategies used to arrive at solutions Write own word problems related to a particular topic or theme, a given situation or equation or a teacher prompt Use a calculator to solve complex problems

BY THE END OF KINDERGARTEN STUDENTS WILL:	BY THE END OF FIRST GRADE STUDENTS WILL:	BY THE END OF SECOND GRADE STUDENTS WILL:
Estimate numbers of objects (e.g. beans in a jar) Count orally from 1 - 20 Count with 1:1 correspondence Arrange numbers in order from 1-20 Form the numerals 0-9 (with some reversals) Understand the numeral and the quantity it represents (1-10) Use ordinal numbers Count backwards from 10 Know the terms more, less, and equal Compare sets that are greater than less than, and equal based on counting Identify fair shares with concrete materials Identify 1/2	Estimate numbers of objects with increasing accuracy Demonstrate awareness of strategies used to estimate Count by ones past 100, by twos to 20, by fives and tens to 100 Understand sequence of numbers (Ex: 30 comes after 21) Read and write numbers 0 - 100 (with some reversals) Identify the ones, tens, and hundreds place Use ordinal numbers to describe position Count backwards from 20 Compare sets that are greater than, equal to, and less than, using symbols >, <, and = . Identify odd/even amounts Demonstrate understanding of fair shares Identify wholes and halves and represent them in various ways - pictorially and with manipulatives Substitute flexibly among different possible representations of quantity; 7 - 1, 2 + 4, 3 + 3	Use strategies for estimation (such as grouping and/or rounding number) Use flexibility in counting (e.g. by 1s, 2s, 5s, 10s) Represent two- and three-digit numbers in a variety of ways (with numbers, pictorially, with manipulatives, etc.) Read, write, order, and compare three-digit numbers (using < >) Understand concepts of odd and even Identify fractions (halves, quarters, thirds) and represent them in different ways pictorially and using manipulatives Use flexibility in representation of quantity (e.g.4 + 4; 5 + 3; 10 - 2)

BY THE END OF KINDERGARTEN STUDENTS WILL:	BY THE END OF FIRST GRADE STUDENTS WILL:	BY THE END OF SECOND GRADE STUDENTS WILL:
Create addition and subtraction problems with manipulatives Record various mathematical experiences with manipulatives and pictures Find missing addend up to 5 with manipulatives	Add, subtract numbers 0 - 20 using manipulatives Use numbers to write and solve single digit addition and subtraction problems Use manipulatives and/or pictorial representations to solve addition and subtraction story problems Regroup through meaningful experiences in large group setting Explain thinking used in solving plus and minus problems Uses equations to record and share thinking Use mental math to solve addition and subtraction story problems involving numbers 0 - 5	Know addition and subtraction facts to sum of 18 Begin to develop the concept of multiplication as repeated addition Add and subtract two-digit numbers with or without regrouping, using manipulatives or an algorithm with understanding Read, Select appropriate computational technique to solve problems Begin to consider when to use a calculator, when to use paper and pencil, and when to compute mentally Use equations to record and share thinking Recognize that a particular equation can represent many different problem situations Compute problems mentally with small or "friendly" numbers ("friendly" numbers are numbers that a child finds easy to work with, e.g. probably 40 or 25, but not 17 or 61)