# Mathematics

## Guiding Principles

The K-12 Medford Public Schools Mathematics Curriculum reflects and supports the vision of the *Common Core State Standards for Mathematics* and the *Massachusetts State Curriculum Framework for Mathematics (2017)*, the National Council of Teachers of Mathematics, and the Massachusetts Comprehensive Assessment System. The mathematics programs adopted by the district are based on the district’s K-12 philosophy that:

- Math class is about learning and not rote memorization
- Mistakes are valuable and give a reason to reflect
- Questions and their timing are important to have deeper learning
- Depth is achieved through interesting and engaging mathematical problems that encourage connections, conversations and modeling.

While the mathematics content standards change from one grade to the other, the mathematics practice standards stay the same for grades K -12 and are as follows:

- make sense of problems and persevere in solving them
- reason abstractly and quantitatively
- construct viable arguments and critique the reasoning of others
- model with mathematics
- use appropriate tools strategically
- attend to precision
- look for and make use of structure
- look for and express regularity in repeated reasoning.

#### DESE Curriculum Frameworks

## Curriculum Overview

## Elementary

### K–5 Mathematics Education at Medford Public Schools

The goal of mathematics instruction throughout the elementary grades is to have students develop both deep and complex understanding of mathematical concepts and skills and procedures they can use to solve problems accurately. Mathematics instruction emphasizes sense-making, problem solving, reasoning, and understanding relationships. Throughout each grade, our math program develops students mathematical habits-of-mind and ways of thinking like a mathematician.

- Kindergarten Unit Summaries
- Grade 1 Unit Summaries
- Grade 2 Unit Summaries
- Grade 3 Unit Summaries
- Grade 4 Unit Summaries
- Grade 5 Unit Summaries

## Kindergarten Unit Summaries

#### Unit 1: Counting People, Sorting Buttons

##### Classroom Routines and Materials

This unit introduces the processes, structures, and materials that form the basis of math instruction throughout the year and establishes the mathematical community.

#### Unit 2: Counting Quantities, Comparing Lengths

##### Counting and Measurement 1

This unit focuses on connecting number names, numerals, and quantities; counting and developing visual images of quantities up to 10; comparing and ordering two or more amounts, and describing and measuring the length of objects by direct comparison. Students develop strategies for accurately counting quantities up to 10.

#### Unit 3: Make a Shape, Fill a Hexagon

##### 2-D Geometry

This unit focuses on identifying, describing, and comparing attributes of 2-D shapes, naming shapes, considering the features of specific shapes (e.g., a triangle has three sides and three vertices), and composing and decomposing shapes from and into smaller shapes. Throughout the unit, students hear and use positional language to describe the location of the shapes they are working with.

#### Unit 4: Collect, Count, and Measure

##### Counting and Measurement 2

This unit focuses on counting and representing sets of up to 15 objects, applying counting skills by using multiple units to measure and compare lengths, decomposing numbers in many different ways, and beginning to make sense of the operations of addition and subtraction. Students act out story problems and play games that involve counting, comparing, and finding the total when a small amount is added or taken away.

#### Unit 5: Build a Block, Build a Wall

##### 3-D Geometry

This unit focuses on identifying, describing, and comparing attributes of 3-D shapes (e.g., a cube has 6 congruent faces), naming 3-D shapes, constructing 3-D shapes, and composing and decomposing 3-D shapes from and into smaller shapes. Students look for 3-D shapes in their environment and describe and compare the shapes they find.

#### Unit 6: How Many Now?

##### Addition, Subtraction, and the Number System 1

This unit focuses on counting sets of up to 20 objects; decomposing the numbers to 10 in a variety of ways (e.g., 7 can be seen as 5 and 2 or as 3 and 2 and 2); using notation to describe addition and subtraction situations; finding and exploring combinations of a number, and solving addition and subtraction story problems. Students work on accurate counting strategies and revisit addition and subtraction situations, including story problems, which they model, solve and represent on paper.

#### Unit 7: How Many Noses? How Many Eyes?

##### Modeling with Data

This unit focuses on describing attributes of objects and data and using this information to sort, classify, count, order, compare, and represent data, and to use this data to model real-world problems with mathematics. It also extends work with counting by 1s and introduces students to the counting sequences of 2s and 10s.

#### Unit 8: Ten Frames and Teen Numbers

##### Addition, Subtraction, and the Number System 2

This unit focuses on extending the counting sequence to 100, including counting from numbers other than 1 and by 10s, adding and subtracting in a variety of contexts, and making sense of the teen numbers as a group of ten ones and some number of leftover ones. Students solve story problems, relate a problem to the equation it represents, and make up and solve stories for given expressions. Students develop fluency with addition and subtraction within 5, find the complement of 10 when given one addend, and work with the teen numbers.

## Grade 1 Unit Summaries

#### Unit 1: Building Numbers and Solving Story Problems

##### Addition, Subtraction, and the Number System 1

This unit focuses on counting and comparing quantities, connecting counting to the operations of addition and subtraction, and developing an understanding of and strategies for solving addition and subtraction problems via story problems and games.

#### Unit 2: Comparing and Combining Shapes

##### 2-D Geometry

This unit focuses on observing and describing defining attributes of 2-D shapes, using those attributes to build, compare, and sort 2-D shapes, and composing and decomposing 2-D shapes. Students describe, name, and compare attributes of 2-D shapes, particularly triangles and quadrilaterals.

#### Unit 3: How Many of Each? How Many in All?

##### Addition, Subtraction, and the Number System 2

This unit focuses on counting on/back as a strategy for adding/subtracting; composing and decomposing numbers into two or more parts and playing games that involve an unknown change; adding more than two numbers; expanding students’ understanding of addition and subtraction notation, in particular, the meaning of the equal sign; and counting and comparing larger quantities.

#### Unit 4: Fish Lengths and Fraction Rugs

##### Measurement and Fractions

This unit focuses on developing accurate techniques for linear measurement, solving comparison problems about length, and ideas about time and fractions as equal parts of a whole. Students measure the lengths of various objects and compare two lengths. They observe and describe shapes (circles, squares, rectangles) that are (and are not) divided into halves and fourths, and partition shapes to show halves and fourths.

#### Unit 5: Number Games and Crayon Problems

##### Addition, Subtraction, and the Number System 3

This unit focuses on developing fluency with addition and subtraction within 10, including the 2-addend combinations of 10, understanding the meaning of the equal sign as a symbol of equivalence, and deepening students’ understanding of addition and subtraction through problems and games that involve finding an unknown addend or an unknown change.

#### Unit 6: Would You Rather Be an Eagle or a Whale?

##### Modeling with Data

This unit focuses on collecting, recording, representing, describing, and comparing data in two and three categories, and on conducting data investigations. As students engage in their own surveys, they use data to model real-world problems with mathematics.

#### Unit 7: How Many Tens? How Many Ones?

##### Addition, Subtraction, and the Number System 4

This unit focuses on counting by numbers other than 1, with an emphasis on groups of 10; adding and subtracting 10 from a 2-digit number, subtracting a multiple of 10 from a multiple of 10, and representing 2-digit numbers with tens and ones and using those models to add within 100. Students represent numbers as tens and ones to compare and add numbers within 100.

#### Unit 8: Blocks and Buildings

##### 3-D Geometry

This unit focuses on observing, describing, comparing, and building 3-D shapes and on developing a vocabulary for naming and describing defining attributes of 2-D and 3-D shapes. Students identify and compare attributes of 3-D shapes, compose and decompose 3-D shapes, and explore the relationship between 2-D and 3-D shapes.

## Grade 2 Unit Summaries

#### Unit 1: Coins, Number Strings, and Story Problems

##### Addition, Subtraction, and the Number System 1

This unit focuses on adding and subtracting single-digit numbers, focusing particularly on adding numbers in any order; shifting from counting by 1s to counting by groups, particularly groups of tens and ones, which lays the foundation for students' work with place value and the base-10 number system; and developing and refining strategies for solving a variety of addition and subtraction problems.

#### Unit 2: Attributes of Shapes and Parts of a Whole

##### Geometry and Fractions

This unit focuses on observing and describing defining attributes of 2-D and 3-D shapes (e.g., number and shape of faces, number and length of sides, and the number of angles and vertices), and using those attributes as they sort, construct, draw, and compare shapes. This unit also develops ideas about equal parts of a whole, focusing specifically on partitioning and describing halves, fourths, and thirds of one whole and recognizing that the same equal part of a whole (e.g. one half of a square) can be different shapes.

#### Unit 3: How Many Stickers? How Many Cents?

##### Addition, Subtraction, and the Number System 2

This unit focuses on the place value of 2-digit numbers and operating on those numbers within 100. They solve a variety of types of story problems and play games that involve combining amounts to get to 100 or $1. Work on fluency with addition and subtraction within 100 continues, with a focus on using known facts and knowledge of the operation. Students also identify, read, and write numbers up to 500, and mentally add and subtract 10 to numbers in that range.

#### Unit 4: Pockets, Teeth, and Guess My Rule

##### Modeling with Data

This unit focuses on sorting and classifying categorical data; ordering numerical data; and collecting and representing categorical and numerical data using a variety of representations: student-generated representations, picture graphs, bar graphs, Venn diagrams, cube towers, and line plots. Students describe the data and discuss what the data tell them about the group surveyed. In doing so students develop the ability to model with data, aspects of their world.

#### Unit 5: How Many Tens? How Many Hundreds?

##### Addition, Subtraction, and the Number System 3

This unit focuses on the place value of 3-digit numbers and operating on numbers within 100. Students solve a variety of types of story problems and play games that involve combining amounts to get to 100 or $1. Work on fluency with addition and subtraction within 100 continues, with a focus on using known facts and knowledge of the operations. Fluency with addition within 100 is a benchmark in this unit. Students also identify, read, and write numbers up to 1,000 and add and subtract 10 and 100 to numbers in that range.

#### Unit 6: How Far Can You Jump?

##### Linear Measurement

This unit focuses on developing strategies for accurately measuring length with nonstandard and standard units and tools and for considering the relationship between different units and tools (e.g., the larger the unit, the smaller the count will be). Students represent measurement data on a line plot and also solve story problems that involve adding, subtracting, and comparing lengths.

#### Unit 7: Partners, Teams, and Other Groups

##### Foundations of Multiplication

The focus of this unit is on working with equal groups as the foundation of multiplication by investigating even and odd numbers and by representing equal groups with arrays and tables.

#### Unit 8: Enough for the Class? Enough for the Grade?

##### Addition, Subtraction, and the Number System 4

This unit focuses on developing and achieving fluency with subtraction within 100, and on achieving fluency with addition and subtraction facts within 20. Students investigate how the strategies they know and use for adding and subtracting 2-digit numbers translate to adding and subtracting 3-digit numbers, represented with place-value notation.

## Grade 3 Unit Summaries

#### Unit 1: Understanding Equal Groups

##### Multiplication and Division 1

This unit focuses on understanding the meaning of multiplication, modeling multiplication using arrays, and understanding the inverse relationship between multiplication and division. Students use arrays, and known facts, to develop fluency with the multiplication facts, and are assessed on those that involve multiplying numbers to 10 by 1, 2, 5, and 10.

#### Unit 2: Graphs and Line Plots

##### Modeling with Data

This unit focuses on using bar graphs, pictographs, and line plots to represent, describe, and compare categorical and numerical data. They solve one- and two-step “how many more” and “how many less” problems using information presented in the graphs.

#### Unit 3: Travel Stories and Collections

##### Addition, Subtraction, and the Number System 1

This unit focuses on understanding and extending knowledge of place value and the number system to 1,000, and adding and subtracting accurately and efficiently. Students represent numbers as hundreds, tens, and ones, and find equivalent ways to use 100s, 10s, and 1s to represent a given number. They also develop strategies for adding and subtracting 2-digit and 3-digit numbers Students encounter a variety of different addition and subtraction problem types.

#### Unit 4: Perimeter, Area, and Polygons

##### 2-D Geometry and Measurement

This unit focuses on understanding and finding perimeter and area using standard units of measurement, and on classifying 2-D figures. Students use standard measurement tools to measure the length of objects and the perimeter of figures. They use square units to measure area. Students build on work connecting arrays and multiplication and find the area by multiplying length and width. Students also identify attributes of triangles and quadrilaterals and explore different categories of quadrilaterals.

#### Unit 5: Cube Patterns, Arrays, and Multiples of 10

##### Multiplication and Division 2

This unit focuses on the meaning and structure of multiplication and division and the relationship between them, solving multiplication and division problems, multiplying by multiples of 10, and learning the remaining multiplication facts to 10 × 10. Students solve story problems involving multiplying single-digit numbers by multiples of 10.

#### Unit 6: Fair Shares and Fractions on Number Lines

##### Fractions

This unit focuses on understanding the meaning of fractions as numbers and as equal parts of a whole; reasoning about equivalent fractions; comparing fractions, and using notation to model fractions and fraction relationships. Students build their knowledge of fractions and fraction equivalents as they represent and compare fractional quantities, including fractions greater than one, with pattern blocks and on the number line.

#### Unit 7: How Many Miles?

##### Addition, Subtraction, and the Number System 2

This unit focuses on understanding the operations of addition and subtraction and adding and subtracting fluently. Students add multiples of 10 and 100 and subtract them from, 3-digit numbers.

#### Unit 8: Larger Numbers and Multi-Step Problems

##### Multiplication and Division 3

This unit focuses on solving multiplication and division problems, learning division facts, identifying arithmetic patterns, and solving multi-step problems. Students develop strategies to solve multiplication and division problems, including problems with remainders.

## Grade 4 Unit Summaries

#### Unit 1: Arrays, Factors, and Multiplicative Comparison

##### Multiplication and Division 1

This unit focuses on using arrays and multiplicative comparison problems to understand multiplication, and gaining familiarity with factors and multiples. Students use arrays to model multiplication situations and find factors and identify prime numbers to 100. They use what they know about factors of 100 to find factors of multiples of 100 and examine the idea that factors of a number are also factors of a multiple of that number.#### Unit 2: Generating and Representing Measurement Data

##### Modeling with Data

This unit focuses on using line plots to represent, describe, and compare measurement data; modeling real-world problems with mathematics; and constructing arguments based on data. Students collect measurement data and use line plots to represent and analyze the data.#### Unit 3: Multiple Towers and Cluster Problems

##### Multiplication and Division 2

This unit focuses on solving multiplication problems with 2-digit numbers, understanding the meaning and structure of, and the relationship between, multiplication and division, and using that understanding to solve multiplication and division problems. Students use arrays to explore the distributive property of multiplication. They solve and represent division story problems, including problems with remainders.#### Unit 4: Measuring and Classifying Shapes

##### 2-D Geometry and Measurement

This unit focuses on investigating attributes of two-dimensional and three-dimensional shapes and how these attributes determine their classification. Students measure perimeter, area, and angles and explore mirror symmetry.#### Unit 5: Large Numbers and Landmarks

##### Addition, Subtraction, and the Number System

This unit focuses on understanding the meaning of addition and subtraction, understanding the base-10 number system with numbers up to 1,000,000, and adding and subtracting multi-digit numbers fluently, including with the U.S. standard algorithms.#### Unit 6: Fraction Cards and Decimal Grids

##### Fractions and Decimals

This unit focuses on understanding the meaning of fractions and decimals and comparing fractions and decimals including finding equivalents. Students use contexts and representations such as rectangles (an area model) and number lines (a linear model) to add, subtract, and multiply fractions.#### Unit 7: How Many Packages and Groups?

##### Multiplication and Division 3

This unit focuses on the operations of multiplication and division, including problems involving converting measurements. Students refine their strategies for solving multiplication problems with two 2-digit numbers and with a 4-digit number and a 1-digit number, and they use the relationship between multiplication and division to develop and practice strategies for solving division problems with up to a 4-digit dividend and a 1-digit divisor.#### Unit 8: Penny Jars and Towers

##### Analyzing Patterns and Rules

This unit focuses on solving multi-step problems, generating and analyzing patterns, and using symbolic notation to model situations.## Grade 5 Unit Summaries

#### Unit 1: Puzzles, Clusters, and Towers

##### Multiplication and Division 1

This unit focuses on the operations of multiplication and division. Students refine their strategies for solving multiplication problems with 2-digit numbers and use the relationship between multiplication and division to develop and practice strategies for solving division problems. They use the order of operations to solve computation problems.#### Unit 2: Prisms and Solids

##### 3-D Geometry and Measurement

This unit focuses on the structure and volume of three-dimensional (3-D) shapes, specifically on rectangular prisms and solids composed of rectangular prisms. Students build models and patterns for boxes that hold quantities of cubes and calculate the volume of these boxes, using a cube as a unit of measure. Students use standard units of measure for volume and apply formulas for volume.#### Unit 3: Rectangles, Clocks, and Tracks

##### Rational Numbers 1: Addition and Subtraction

This unit focuses on deepening and extending studentsâ€™ understanding of fractions and equivalent fractions and representing fractions using an area model (rectangles), a rotation model (a clock), and a linear model (number lines). They use these understandings to add and subtract fractions and mixed numbers.#### Unit 4: How Many People and Teams?

##### Multiplication and Division 2

This unit focuses on the operations of multiplication and division. Students refine their strategies for solving multiplication problems fluently, including using the U.S. standard algorithm. Students continue using the relationship between multiplication and division to efficiently solve division problems with 4-digit dividends and 2-digit divisors.#### Unit 5: Temperature, Height, and Growth

##### Analyzing Patterns and Rules

This unit focuses on using coordinate graphs, ordered pairs, tables, and symbolic notation to model real-world and mathematical situations. Students analyze arithmetic patterns in tables and the shapes of graphs to describe and compare these situations.#### Unit 6: Between 0 and 1

##### Rational Numbers 2: Addition and Subtraction

This unit focuses on deepening and extending studentsâ€™ understanding of decimals and the base-10 number system. Students represent decimals on grids and number lines. They use their understanding of decimals to compare, add, and subtract decimals.#### Unit 7: Races, Arrays, and Grids

##### Rational Numbers 3: Multiplication and Division

This unit focuses on multiplying and dividing rational numbers, which includes extending studentsâ€™ understanding of the meaning of those operations and of place value. Students use contexts and representations (fraction bars, arrays, and grids) to solve problems involving multiplication and division of fractions and decimals. Students also apply their understanding of multiplication and division to solve measurement conversion problems.#### Unit 8: Properties of Polygons

##### 2-D Geometry and Measurement

This unit focuses on classifying triangles and quadrilaterals based on their properties and on using patterns to describe how the perimeters and areas of rectangles change when the dimensions of the rectangle change. Students examine how categories of polygons are related and how a figure can belong to more than one category. As they build sequences of related rectangles, they analyze numerical relationships and practice adding and multiplying mixed numbers and decimals.## Middle School

### 6-8 Mathematics Education at Medford Public Schools

The goal of mathematics instruction throughout grades 6-8 is to have students build new mathematical ideas and experiences drawing from the connections and relationships with mathematics they learned in elementary school. All students engage in a three-year course of study that integrates content strands and mathematical practices.

## High School

### Mathematics Department at Medford High School

The Mathematics Department at MHS/MVTHS offers a core mathematics program for grade 9 -12 students that is rigorous in content, high in expectations, and accessible to all students. All students must take and pass four years of mathematics to include a course beyond Algebra II or Integrated Mathematics III. The recommended course sequence is Integrated Mathematics I/Integrated Mathematics II/Integrated Mathematics III, followed by a 5-credit mathematics elective. Students who complete 8th-grade Honors shall take Integrated Mathematics 1 Honors in grade nine. Students coming from 8th-grade Standard also have the opportunity to take Integrated Math 1 Honors. Computer programming courses offered in the mathematics department can count toward the 4-year requirement in the senior year with permission from the Curriculum Director and Director of Guidance. The department also offers mini-courses. The department offers the following main courses:

Algebra II (CP/Honors)

Integrated Mathematics I (CP/Honors)

Integrated Mathematics II (CP/Honors)

Integrated Mathematics III (CP)

Topics in Algebra and Trigonometry (CP)

Mathematical Decision Making (Honors)

Personal Finance (CP)

Consumer Math (CP)

Statistics (CP)

Pre-Calculus (CP/Honors)

Calculus (Honors)

Explorations in Computer Programming

AP Statistics

AP Calculus

AP Computer Science A Java