Mathematics

• How do math tools help me understand mathematics
• How can I tell how many or how much is in a group?
• How can I tell if there are enough objects needed for a situation?
• What does it mean to be a member of a mathematical community?

Students explore and use mathematical tools while teachers gather information through observations and questions about students’ counting knowledge and skills.  Students also have opportunities to work with math tools and topics related to geometry, measurement, and data through a variety of centers.  In the last section of the unit, students are expected to count up to 10 using various mathematical tools as support.

Unit 2: Numbers 1-10

Essential Questions

• How do numbers and quantities relate to each other?
• How do I write numbers 1-10?
• How do I know if a number is less than another number?

Students continue to develop counting concepts and skills, including comparing, while learning how to write numbers.  Students use fingers and five frames as well as familiar activity structures to build their counting skills and concepts.  Students build their math vocabulary as they start to use the terms “fewer” and “more” when comparing the numbers of objects or images.

Unit 3: Flat Shapes All Around Us

• What shapes are in my environment?
• How can shapes connect to make different changes?

In this unit, students will be introduced to the foundational concept of geometry, with a focus on flat (two-dimensional) shapes.  Students will explore differences in shapes and use informal language to describe, compare, and sort them.  Students reinforce counting and comparison skills by using pattern blocks to make larger shapes as well as positional words (above, below, next to, beside) to describe the shapes they compose.

Unit 4: Understanding Addition and Subtraction

• Why do I need to add?
• Why do I need to subtract?
• How can I represent and solve problem situations using objects, pictures, words and numbers?

Students develop their understanding of addition and subtraction as they represent and solve story problems within 10.  They relate counting to either putting objects together or taking objects away.  Students develop understanding of mathematical expressions and connect expressions to pictures and story problems and find the value of addition and subtraction expressions within 10.

Unit 5: Composing and Decomposing Numbers to 10

• How can I use place value to decompose numbers to find sums or differences?
• How do I take apart and recombine numbers in a variety of ways for finding sums and differences?

Students will explore different ways to compose and decompose numbers within 10 and how to represent the compositions and decompositions.  Students link 10 frames and their fingers as tools to think about pairs of numbers that make 10.  Students will also practice writing numbers and develop the understanding of balanced equations, “5 is 3 plus 2”.

Unit 6: Numbers 11-20

• How can you know a quantity without counting each object?
• How do you know how many objects you have?
• What is an efficient way to count an amount greater than ten?

In this unit, students count and represent collections of objects and images within 20.  They use the 10-frame as a tool to see teen numbers as 10 ones and some more ones, with emphasis on the structure of the numbers 11-19.  Students practice tracing and writing numbers 11-20 and practice equations with the addend first, using mathematical tools to reinforce understanding.

Unit 7: Solid Shapes All Around Us

• How can shapes help us count and compare numbers?
• What shapes are around us?
• How can we compose shapes using smaller shapes?

Students explore solid shapes (three-dimensional) while reinforcing their knowledge of counting, number writing and comparison, as well as flat shapes.  They compose figures with pattern blocks and continue to count up to 20 objects, write and compare numbers, and solve story problems.  Students use their own language to describe attributes of solid shapes as they identify, sort, compare, and build them, while also learning the names for cubes, cones, spheres, and cylinders.

Unit 8: Putting it All Together

• How do numbers and quantities relate to each other?
• What does it mean to be a member of a mathematical community?
• How do I take apart and recombine numbers in a variety of ways for finding sums and differences?
• How can I represent and solve problem situations using objects, pictures, words and numbers?

In this culminating unit, Kindergarten students revisit major work and fluency goals of the grade, applying their learning from the year.  They revisit concepts of counting and comparing, math in the community, practice composing and decomposing within 5 as well as within 10.  This unit lays the foundation for grade 1, where students add and subtract fluently within 10 and count and compare larger quantities.

Unit 1: Adding, Subtracting and Working with Data

Essential Questions

Unit 1 Overview

• How is adding and subtracting like counting?
• What is the best way to represent data I collect in a survey so that it is clear?
• How does data help me to explain or describe real life situations?
• What does it mean to be a member of a mathematical community?

In this unit, students in grade 1 deepen their understanding of addition and subtraction within 10 and extend what they know about organizing objects into categories and representing quantities.  Activities in this unit reinforce kindergarten understandings of addition and subtraction word problems and initiate the year-long work of developing fluency with sums and differences within 10.  Students also extend their understanding of engaging in data by using drawings, symbols, tally marks, and numbers to represent data, as well as ask and answer questions about the data.

Unit 2: Addition and Subtraction Story Problems

• How can I use addition or subtraction to show how to solve a story problem where I can add to/take from a total or where I need to figure out the change?
• How can I use equations to show different ways to make a total amount?
• How can I show “how many more” and “how many fewer” using addition and subtraction equations?
• How can I write and solve story problems using drawings, pictures, words, or equations?

Students expand on their understanding of story problem types that were established in Kindergarten and work to solve the majority of story problem types.  The focus in this unit is for students to interpret and understand the meaning of the story problem and build their fluency of addition and subtraction within 10.  A large focus of this unit is for students to represent story problems with multiple equations, deepening their understanding of addition and subtraction, and to explain the relationship between their equations and the story problem.

Parent video: Modeling with tape diagrams

Unit 3: Adding and Subtracting Within 20

• How can I use what I know about tens and ones to add and subtract two-digit numbers?
• How do I recognize what strategy to use for a specific problem?
• How can using number relationships help me solve addition and subtraction problems?

In this unit, students develop an understanding of 10 ones as a unit called “a ten” and use the structure to add and subtract within 20.  Students decompose and recompose addends to find the sum of two or three numbers, for example to find the value of 6 plus 9, they may decompose 6 into 1 and 5, compose the 1 and 9 into 10, and find 5 plus 10.  Students work on subtraction by using their knowledge of addition to find the difference of two numbers and learn two new story problems through the unit.

Unit 4: Numbers to 99

• How can we find the total when we join two quantities?
• How can we find what is left when we take one quantity from another?
• How can we represent a number using tens and ones?
• What do less than, greater than, and equal to mean?
• What is estimating and when can you use it?

Students develop an understanding of place value for numbers up to 99 as well as an understanding of the structure of numbers in our base ten system, allowing them to see that two digits of a two-digit number represent how many tens and ones there are.  As they develop their understanding of tens and ones, they will learn to transition from counting by one to counting by ten and then counting on for numbers greater than 10.  Students will use drawings and mathematical tools to represent numbers up to 99 and will compare two-digit numbers.

Unit 5: Adding Within 100

• How can I estimate the answers for operations involving two-digit numbers?
• How can I use what I know about tens and ones to add two-digit numbers?
• What strategies do I use to compute sums mentally?

Students use place value and properties of addition to add within 100.  They make sense of methods for adding, like composing a ten when adding ones and ones, and work with a variety of representations- connecting cubes, drawings, expressions, and equations.  The focus for students is to make sense of the numbers and ways of adding rather than applying an algorithm.

Unit 6: Length Measurements Within 120 Units

• How do I measure length using non-standard units of measure?
• How do I choose the appropriate tool and unit when measuring?
• How can objects be measured, compared, and ordered by length?

In this unit, students extend their knowledge of linear measurement while continuing to develop their understanding of operations, algebraic thinking, and place value.  Students compare the length of objects by lining them up at their endpoints and explore ways to compare lengths of two objects that cannot be lined up.  Students develop precision with different measuring tools, solve story problems and are introduced to new story problem types, and reason how to count and represent groups of objects over 99 up to 120.

Unit 7: Geometry and Time

• How can I tell time on an analog and digital clock?
• How can we break shapes into equal shares and what these shares are called?
• How do I distinguish shapes between defining and non-defining attributes?

In this unit, students focus on geometry and time, expanding their knowledge of two and three-dimensional shapes, partition shapes into halves and fourths, and tell time to the hour and half an hour.  Students extend the foundation they build about shapes in Kindergarten to develop more precise vocabulary to sort shapes into categories and use shapes to begin to learn the language of fractions.  Students also learn to use the circle as a clock and how hour and minute hands partition the clock to the hour and to “half past” or __:30.

Unit 8: Putting it All Together

How can using number relationships help me solve addition and subtraction problems?

How can I use an addition or subtraction equation to show how to solve a story problem where I can add to/take from a total or where I need to figure out the change?

How can I use numbers to 120?

In this last unit for Grade 1 math, students will work on solidifying their understanding of the major concepts and skills for the year to prepare them for Grade 2.  The sections in this unit include adding and subtracting within 20, and fluently within 10.  Students will also practice solving story problems they were introduced to during the year.  Additionally, they will count and represent numbers within 120.

Unit 1: Wrapping up Adding and Subtraction Within 1,000

Unit Overview

• Add and subtract within 1,000 using strategies and algorithms based on place value, properties of operations, or the relationship between addition and subtraction.
• Use place value understanding to round whole numbers to the nearest multiple of 10 or 100.
• Use and explain place value relationships to compose and decompose numbers.
• Fluently subtract within 1,000, using algorithms based on place value, properties of operations, and the relationship between addition and subtraction.

This unit progresses students toward a third grade goal of fluently adding and subtracting within 1,000. Students build on their knowledge of addition and subtraction strategies that they learned in second grade. Students will use place value understanding to round, estimate, and build their fluency in adding and subtracting whole numbers. They also use expanded form to add and subtract within 1,000 as they move toward the standard algorithm.

• Parent video: Regrouping to subtract (place value disks)
• Parent video: Rounding to the nearest 10 or 100

• Represent, solve, and create scaled picture graphs and bar graphs to represent a data set.
• Solve one- and two-step story problems using addition and subtraction.
• Represent and solve multiplication problems using equal groups, arrays, expressions, and equations with a symbol for the unknown.
• Use multiplication within 100 to solve word problems in situations involving equal groups and arrays.

Students expand on their grade 2 knowledge of representing data with graphs as they are introduced to multiplication with one picture in a picture graph equaling 2 or 5 units. As students expand on their understanding of equal size groups and multiplication, they relate the idea of a x b through both groups of objects and amount in each group as well as rows and columns of arrays. Students also make sense of the meaning of multiplication expressions before solving them.

Parent video: Multiplication Fluency

Unit 3: Area, Multiplication and Perimeter

Unit Goals

Unit Overview

• Understand the concept of area and relate it to multiplication and division.
• Describe “area” as the number of unit squares that cover a plane figure, without gaps and overlaps.
• Measure the area of a rectangle by counting unit squares.
• Represent multiplication using area diagrams, expressions, and equations with a symbol for the unknown.
• Explain why the area of a rectangle can be determined by multiplying the side lengths.
• Solve problems involving the area of a rectangle.
• Find the areas of figures composed of rectangles.
• Solve problems involving perimeter and area, in and out of context.
• Fluently multiply within 100

Unit 4: Relating Multiplication to Division

Unit Overview

• Learn about and apply understandings of the relationship of multiplication and division.
• Represent and solve “how many groups?” and “how many in each group?” problems.
• Represent multiplication and division using equal groups drawings, arrays, area diagrams, number line diagrams, expressions, and equations with a symbol for the unknown
• Represent and solve word problems using all four operations.
• Understand division as an unknown-factor problem and use the properties of operations to develop fluency with related multiplication and division problems.
• Multiply one-digit whole numbers by multiples of 10 in the range 10–90

• Add and subtract within 1000 using strategies and algorithms based on place value
• Assess the reasonableness of answers.
• Solve two-step word problems, using addition, subtraction, and multiplication.

This mini unit allows students to spend time applying skills they learned in units 1-4. They use their adding, subtracting, multiplying, and dividing skills to solve two step word problems. Additionally, they practice their reasoning and estimation skills to determine if answers are reasonable.

Parent video: Solving 2-step Word Problems

Unit 5: Fractions as Numbers

Unit Overview

• Understand that unit fractions are formed by partitioning shapes into equal parts.
• Understand that fractions are built from unit fractions, such that a fraction 𝑎𝑏 is the quantity formed by 𝑎 parts of size 1/b
• Represent fractions on diagrams and number lines
• Generate equivalent fractions using diagrams, number lines, and models.
• Express whole numbers as fractions and fractions as whole numbers

In this unit, students work to make sense of fractions, with a focus in modeling and using diagrams to represent and compare fractions and relate them to whole numbers. Students use different representations to identify 1 whole and reason about the size of fractional parts. Later in the unit, students compare fractions with the same denominator as well as those with the same numerator to recognize that as the numerator gets larger, more parts are being counted and as the denominator gets larger, the size of each part in a whole gets smaller.

Parent video: Fractions on a number line

Unit 6: Measuring Length, Time, Liquid Volume, and Weight

Unit Overview

• Measure and estimate weights and liquid volumes of objects and represent length data in halves and fourths of an inch on line plots.
• Measure lengths using rulers marked with halves and fourths of an inch.
• Estimate relative units of measure (weight, liquid volume, and time) and use the four operations to solve problems involving measurement
• Solve problems involving addition and subtraction of time intervals in minutes
• Tell time to the minute
• Solve problems involving in the four operations and measurement contexts.

Students measure length, weight, liquid volume and time in this unit. They begin with length measurement, building on their previous units' work of fractions, by exploring length in halves and fourths of an inch on a ruler, learning about mixed numbers and equivalent fractions as they work. Next, students learn about standard units for measuring weight (kilograms and grams) and liquid volume (liters), finishing the unit by measuring time to the minute. In the final section of the unit, they solve problems related to all of the measurements learned through the unit.

Parent Video: Units of Measure

Unit 7: Two-dimensional Shapes and Perimeter

Unit Goals

Unit Overview

• Reason about shapes and their attributes, with a focus on quadrilaterals.
• Apply geometric understanding to solve problems
• Solve problems involving perimeter and area of shapes.

• Use place value relationships to show understanding of the base ten system within 1 million
• Read, represent, and describe the relative magnitude of multi-digit whole numbers up to 1 million in different forms
• Identify digit equivalence based on place value location
• Round, order, and compare digits within 1 million (and represent digits in multiple ways)
• Add and subtract using the algorithm within 1 million

Unit 2: Factors, Multiples and Multiplicative Comparison

Unit Goals

Unit Overview

• Identify and apply understanding of factors and multiples and their relationship
• Interpret, represent, and solve multiplicative comparison problems
• Use and apply multiplication fluency within 100

• Multiply whole numbers using partial products and area models, including 1 digit by up to 4 digits and 2 digits by 2 digits.
• Divide whole numbers using partial quotients and concrete representations with and without remainders by up to 4 digits divided by 1 digit.
• Generate patterns that follow a given rule

In this unit, students multiply and divide multi-digit whole numbers using partial products and partial quotient strategies, and apply this understanding to solve multi-step problems using the four operations. Students multiply up to four digits by single-digit numbers, and to multiply a pair of two-digit numbers, transitioning from using diagrams to using algorithms to record partial products. In division, students see that it helps to decompose a dividend into smaller numbers and find partial quotients, relying on place value application and understanding. Students apply their knowledge to solve multi-step problems about measurement in various contexts.

• Parent video: Area model for multiplication
• Parent video: Partial product multiplication
• Parent video: Solving 2 step word problems

Unit 4: Fractions and Equivalence

Unit Goals

Unit Overview

• Generate and reason about equivalent fractions (denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100)
• Compare and order fractions (denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100)

In this unit, students expand on their fractional understanding. They use fraction strips, tape diagrams, and number lines to make sense of the size of fractions, generate equivalent fractions, and compare and order fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100. Students generalize that a fraction’s equivalency can be represented with expressions and the concepts that link the mathematical models to the mathematical expressions.

Unit 5: Fractions and Decimals

Unit Goals

Unit Overview

• Learn that the fraction form of a number divided by b is a product of a whole number n and a unit fraction 1 divided by b (). 𝑛𝑏= 𝑛 𝑥 1𝑏
• Learn that a number times a divided by b is the same as the number times a divided by b ( 𝑛 𝑥 𝑎 ÷ 𝑏 = 𝑛 𝑥𝑎𝑏)
• Learn to add and subtract fractions and mixed numbers with like denominators and learn to add and subtract tenths and hundredths.
• Relate tenths and hundredths (fraction form) to tenths and hundredths (decimal form)
  • Write tenths and hundredths in decimal notation.
  • Represent, compare, and order decimals to the hundredths by reasoning about their size

In this unit, students deepen their understanding of how fractions can be composed and decomposed, and learn about operations on fractions. Students multiply fractions by whole numbers, add and subtract fractions with the same denominators, and add tenths and hundredths, using familiar concepts and representations (ex: tape diagrams and number lines). Students take a closer look at the relationship between tenths and hundredths and learn to express them in decimal notation, reason about the size of tenths and hundredths written as decimals, locate decimals on a number line, and compare and order them. Students will then apply these skills in the context of measurement and data by analyzing line plots with fractional lengths, answering questions about data.

Parent video: Decomposing fractions

Unit 6: Using Measurement Concepts

Unit Goals

Unit Overview

• Use the relationship between multiplication and division to convert units of measure, within a given system, from larger to smaller units.
• Understand the relative sizes of kilometers, meters and centimeters, liters and milliliters, kilograms and grams, and pounds and ounces.
• Multiply and divide multi-digit whole numbers using partial products and partial quotients strategies, and apply this understanding to solve multi-step problems using the four operations.
• Use the four operations to solve problems that involve multi-digit whole numbers and assess the reasonableness of answers.

In this unit, students make sense of multiplication as a way to compare quantities. They use this understanding to solve problems about measurement.. Through this unit, they use their new knowledge to apply their learning to various units of length, mass, capacity, and time to convert units within the same system of measurement.

• Parent video: Units of measure
• Parent video: Area Model
• Parent video: Partial Products

Unit 7: Angles and Angle Measurement

Unit Goals

Unit Overview

• Draw and identify points, rays, segments, angles, and lines, including parallel and perpendicular lines.
• Use a protractor to measure angles and to draw angles of given measurements.
• Identify acute, obtuse, right, and straight angles in two-dimensional figures.
• Write equations to represent angle relationships, and reason about and find unknown measurements.

Unit 8: Properties of Two-Dimensional Shapes

Unit Goals

Unit Overview

• Classify triangles and quadrilaterals based on the properties of their side lengths and angles.
• Use understanding of geometrical attributes to solve problems, including problems involving perimeter and area.
• Identify and draw lines of symmetry in two-dimensional figures.

In this unit, students deepen their understanding of the attributes and measurement of two-dimensional shapes. Students classify triangles and quadrilaterals based on the properties of their side lengths and angles, and learn about lines of symmetry in two-dimensional figures. They use their understanding of these attributes to solve problems, including problems involving perimeter and area.

Unit 1: Compare, Order, Round, Add and Subtract Decimals

Unit Goals

Unit Overview

• Understand and represent with expanded form that a digit one place to the left is ten more, and a digit to the right is one tenth.
• Compare, round, and order decimals through the thousandths place based on the value of the digits in each place.
• Read, write, and represent decimals to the thousandths place, including the expanded form.
• Round, compare, add and subtract decimals to the thousandths.

Unit 2: Wrapping up Multiplication and Division with Multi-Digit Numbers

Unit Goals

Unit Overview

• Multiply multi-digit whole numbers, using the standard algorithm.
• Divide whole numbers up to four digits by two-digit divisors, using strategies based on place value and the properties of operations.
• Multiply and divide to solve real-world and mathematical problems involving area.

Unit 3: Multiplying and Dividing Decimals

Unit Goals

Unit Overview

• Multiply decimals with products resulting in the hundredths using place value reasoning and properties of operations.
• Divide decimals with quotients resulting in the hundredths using place value reasoning and properties of operations.

Unit 4: Fractions as Quotients and Fraction Multiplication

Unit Goals

Unit Overview

• Develop an understanding of fractions as the division of the numerator by the denominator (a divided by b is equivalent to a in the numerator and b in the denominator)
• Solve problems involving division of whole numbers leading to answers that are fractions.
• Represent and explain the relationship between division and fractions.
• Connect division to multiplication of a whole number by a non-unit fraction as well as a unit fraction.
• Represent and solve problems involving the multiplication of a whole number by a fraction or mixed number.
• Write, interpret, and evaluate numerical expressions that represent multiplication of a whole number by a fraction or mixed number.

Unit 5: Multiplying and Dividing Fractions

Unit Goals

Unit Overview

• Multiply mixed numbers and fractions
• Understand that 𝑎𝑏𝑥 𝑐𝑑=𝑎 𝑥 𝑐𝑏 𝑥 𝑑
• Divide a unit fraction by a whole number using division concepts
• Divide a whole number by a unit fraction using division concepts
• Use the understanding of fraction multiplication to represent to area concepts

Students extend multiplication and division of whole numbers to multiply fractions by fractions and divide a whole number and a unit fraction. Students find the product of two fractions, divide a whole number by a unit fraction, and divide a unit fraction by a whole number.

Unit 6: The Size of a Product and Conversions

Unit Goals

Unit Overview

• Learn, understand, and apply conversion between different measurement systems including metric, time, and standard.
• Interpret multiplication as scaling (resizing).
• Make generalizations about multiplying a whole number by a fraction greater than 1, a fraction less than 1, and a fraction equal to 1.
• Explain patterns when multiplying and dividing by powers of 10.
• Solve multi-step problems involving measurement conversions.

Unit 7: Add and Subtract Fractions with Unlike Denominators

Unit Goals

Unit Overview

• Add and subtract fractions with unlike denominators.
• Create line plots to display fractional measurement data, and use the information to solve problems.
• Solve problems involving addition and subtraction of fractions.

Unit 8: Finding Volume

Unit Goals

Unit Overview

• Learn volume is an additive attribute of a solid figure with cubic units of measure
• Describe area as length times width times height or base times height
• Find volume using lwh or bh
• Apply multiplicative and addition strategies to find volumes of figures, including volumes where sides are fractional or decimals.
• Find the volume of a figure composed of rectangular prisms.

This math unit allows students to apply and practice multiple skills of fractional and decimal multiplication and division through the application of volume concepts. Students connect to the concept of volume by building on their understanding of area and multiplication. Students represent their volume prisms with numerical expressions and discover the rules for finding volume. Toward the end of the unit, students apply their understanding of volume to find the volume of complex shapes as well as apply their knowledge to real-world problems.

• How can coordinates be used to navigate?
• How can patterns be displayed on a coordinate grid?
• How can the relationship between sets of numbers be used to solve problems?

Unit 1: Scale Drawings

Essential Questions

Unit 1 Overview

• How can I decide if two images are scaled copies?
• How can I use scale factor to understand lengths in scaled drawings?

In this unit, students study scaled copies of pictures and plane figures, then apply what they have learned to scale drawings (maps and floor plans).  Through the unit, students learn that all lengths in scaled copies are multiplied by scale factor while measure of angles stays the same.  Students work with scale drawings to discover principles and strategies to reason about scale, and learn to express scales in units (1 cm represents 10 km) as well as non-units (the scale is 1 to 100).  The culminating activity for this unit is for students to apply what they have learned and create a floor plan.

Unit 2: Introducing Proportional Relationships

Essential Questions

Unit 2 Overview

• How can I use tables, graphs, and descriptions to identify proportional relationships? 
• How can I find and use scale factor with rational number lengths and areas?

In this unit, students learn to understand and use proportional relationship terms and recognize when a relationship is or is not proportional.  They represent proportional relationships with tables, equations, and graphs and use terms and representations in reasoning about situations that involve constant speed, unit pricing, and measurement conversions.

Unit 3: Measuring Circles

Essential Questions

Unit 3 Overview

• How can one part of a circle determine the measure of another part? 
• How are circumference and area connected?

In this unit, students extend their knowledge of circles and geometric measurement, applying their knowledge of proportional relationships to the study of circles. They extend their grade 6 work with perimeters of polygons to circumferences of circles, and recognize that the circumference of a circle is proportional to its diameter, with constant of proportionality. They encounter informal derivations of the relationship between area, circumference, and radius.

Unit 4: Proportional Relationships & Percentages

Essential Questions

Unit 4 Overview

• How can we use equations with fractions, decimals, and percentages to represent increase and decrease? 
• How can we calculate important quantities, such as sales tax, tips, and discounts? 
• How can we use information about increases and decreases to find original amounts?

In this unit, students deepen their understanding of ratios, scale factors, unit rates (also called constants of proportionality), and proportional relationships, using them to solve multi-step problems that are set in a wide variety of contexts that involve fractions and percentages.

Unit 5: Rational Number Arithmetic

Essential Questions

Unit 5 Overview

• How are signed numbers used to represent quantities, opposites, and absolute value? 
• What concrete and pictorial models can be used to represent operations with integers? 
• What strategies can be used to predict that the sum of two integers is positive, negative or zero?  
• What strategies can be used to determine if the product or quotient of two integers is positive or negative?

In this unit, students interpret signed numbers (all rational numbers in either decimal or fractional form) in context together with their sums, differences, products, and quotients.

Unit 6: Expressions, Equations, and Inequalities

Essential Questions

Unit 6 Overview

• How can we use representations to model equations? 
• How do mathematicians use symbols to represent greater than/equal to/less than situations? 
• What methods can we use to find unknown solutions of equations and inequalities?

In this unit, students solve equations of the forms px + q = r and p(x + q) = r, and solve related inequalities, e.g., those of the form px + q > r and px + q ≥ r, where p, q and r are rational numbers.

Unit 7: Angles, Triangles, and Prisms

Essential Questions

Unit 7 Overview

• What are the characteristics of angles and sides that will create geometric shapes, especially triangles? 
• How can the special angle relationships – supplementary, complementary, vertical, and adjacent – be used to write and solve equations for multi-step problems? 
• What measurements and calculations can be used to tell how much space is in three-dimensional objects and how much area is needed to cover their surfaces?

In this unit, students investigate whether sets of angle and side length measurements determine unique triangles or multiple triangles, or fail to determine triangles. Students also study and apply angle relationships, learning to understand and use the terms “complementary,” “supplementary,” “vertical angles,” and “unique” (MP6). The work gives them practice working with rational numbers and equations for angle relationships. Students analyze and describe cross-sections of prisms, pyramids, and polyhedra. They understand and use the formula for the volume of a right rectangular prism, and solve problems involving area, surface area, and volume.

Unit 8: Probability and Sampling

Essential Questions

Unit 8 Overview

• What influences the probability that a given event will occur? 
• What tools can we use to model probabilities with real data? 
• How can predictions be made based on data?

In this unit, students understand and use the terms “event,” “sample space,” “outcome,” “chance experiment,” “probability,” “simulation,” “random,” “sample,” “random sample,” “representative sample,” “overrepresented,” “underrepresented,” “population,” and “proportion.” They design and use simulations to estimate probabilities of outcomes of chance experiments and understand the probability of an outcome as its long-run relative frequency.

Unit 1: Rigid Transformations and Congruence

Essential Questions

Unit 1 Overview

• How are figures affected by various transformations?
• How can we prove congruence using a variety of methods?
• What is the relationship between reflections, rotations, and transformations?

In this unit, students expand on their knowledge of geometric figures to include rotations and mirror orientations.  Students progress through learning of transformations on the plane, to transformations of an object while learning about rigid transformations (translations, reflections, and rotations).  They learn about the properties related to plane figures, and how to reason using these properties.

Unit 2: Dilation, Similarity, Introducing Slope

Essential Questions

Unit 2 Overview

• How can I use tables, graphs, and descriptions to identify proportional relationships? 
• How can I find and use scale factor with rational number lengths and areas?

In grade 8, students study pairs of scaled copies that have different rotation or mirror orientations, examining how one member of the pair can be transformed into the other, and describing these transformations. Initially, they view transformations as moving one figure in the plane onto another figure in the plane. As the unit progresses, they come to view transformations as moving the entire plane.

Unit 3: Linear Relationships

Essential Questions

Unit 3 Overview

• How can we recognize and model proportional relationships? 
• What is a linear relationship and how is it represented on a graph? 
• How do we calculate slope, and what does it mean if the slope is positive, negative, zero or undefined? 
• What does it mean to be a solution to a linear equation in two variables?

In this unit, students gain experience with linear relationships and their representations as graphs, tables, and equations through activities designed and sequenced to allow them to make sense of problems and persevere in solving them (MP1).  Students will have opportunities to use language to interpret situations involving proportional relationships, interpret graphs using different scales, interpret slopes and intercepts of linear graphs, justify reasoning about linear relationships, justify correspondences between different representations, and justify which equations correspond to graphs of horizontal and vertical lines.

Unit 4: Linear Equations and Linear Systems

• What does it mean to say an equation has no solution? One solution? Infinitely many solutions? 
• What methods can you use to solve a system of equations? 
• What is the meaning of a “solution” to a system of linear equations?

In this unit, students build on their grades 6 and 7 work with equivalent expressions and equations with one occurrence of one variable, learning algebraic methods to solve linear equations with multiple occurrences of one variable. Students learn to use algebraic methods to solve systems of linear equations in two variables, building on their grades 7 and 8 work with graphs and equations of linear relationships. Understanding of linear relationships is, in turn, built on the understanding of proportional relationships developed in grade 7 that connected ratios and rates with lines and triangles.

Unit 5: Functions and Volume

Essential Questions

Unit 5 Overview

• What is a function? 
• How do the terms “independent variable” and “dependent variable” connect to the inputs and outputs of a function? 
• What is the difference between a linear function and a piecewise function? 
• How can you derive the formula for the volume of a cone from the formula for the volume of a cylinder?

In this unit, students are introduced to the concept of a function as a relationship between “inputs” and “outputs” in which each allowable input determines exactly one output. In the first three sections of the unit, students work with relationships that are familiar from previous grades or units (perimeter formulas, proportional relationships, linear relationships), expressing them as functions. In the remaining three sections of the unit, students build on their knowledge of the formula for the volume of a right rectangular prism from grade 7, learning formulas for volumes of cylinders, cones, and spheres.

Unit 6: Associations in Data

Essential Questions

Unit 6 Overview

• How does a scatterplot help us make predictions about trends in data? 
• What does it mean when we say that data in a scatterplot has a positive association, a negative association, or neither? 
• How do we create a relative frequency table from a two-way table?

In this unit, students analyze bivariate data—using scatter plots and fitted lines to analyze numerical data, and using two-way tables, bar graphs, and segmented bar graphs to analyze categorical data.

Unit 7: Exponents and Scientific Notation

Essential Questions

Unit 7 Overview

• How can you use inductive reasoning to observe patterns and write general rules involving properties of exponents? 
• How can you evaluate a nonzero number with an exponent of zero? How can you evaluate a nonzero number with a negative integer exponent? 
• How can you perform operations with numbers written in scientific notation? 
• How can you estimate how many times larger (or smaller) one number written in scientific notation is than another?

In grade 6, students studied whole-number exponents. In this unit, they extend the definition of exponents to include all integers, and in the process codify the properties of exponents. They apply these concepts to the base-ten system, and learn about orders of magnitude and scientific notation in order to represent and compute with very large and very small quantities.

Unit 8: Pythagorean Theorem

Essential Questions

Unit 8 Overview

• What is the meaning of the “square root” of a number? 
• How do you use the Pythagorean Theorem in two and three dimensions, e.g., to determine lengths of diagonals of rectangles and right rectangular prisms? 
• How do you use the Pythagorean Theorem to estimate distances between points in the coordinate plane? 
• How do you make decimal approximations of irrational numbers?

Work in this unit is designed to build on and connect students’ understanding of geometry and numerical expressions. The unit begins by foreshadowing algebraic and geometric aspects of the Pythagorean Theorem and strategies for proving it. In the second section, students work with figures shown on grids, using the grids to estimate lengths and areas in terms of grid units.  In the third section, students work with edge lengths and volumes of cubes and other rectangular prisms.  In the fourth section, students work with decimal representations of rational numbers and decimal approximations of irrational numbers.

Unit 9: Putting it All Together

Essential Questions

Unit 9 Overview

• How can shapes create patterns called tessellations? 
• How can scatterplots and lines of best fit help to organize and understand data?

In these lessons, students solve complex problems. In the first several lessons, they consider tessellations of the plane, understanding and using the terms “tessellation” and “regular tessellation” in their work, and using properties of shapes to make inferences about regular tessellations. In the later lessons, they investigate relationships of temperature and latitude, climate, season, cloud cover, or time of day. In particular, they use scatter plots and lines of best fit to investigate the question of modeling temperature as a function of latitude

Unit 1: One Variable Statistics

Essential Questions

Unit 1 Overview

• How can the representations and analysis of data inform and influence decisions?
• How can you collect, organize, and display data?
• What is the most appropriate way to display a given set of data?

In this first unit of Algebra, students are building upon their knowledge of summarizing data and data displays that were formed in middle school mathematics.  The students represent and interpret data using different data displays such as box plots, histograms, and dot plots.  Data vocabulary is developed through the unit and students use technology to create data displays and calculate summary statistics.  Students progress through the unit to explore measures of variability and analyze values of data.

Unit 2: Linear Equations, Inequalities and Systems

Essential Questions

Unit 2 Overview

• What can we do with a system of equations/inequalities that we cannot do with a single equation/inequality?  
• What types of relationships can be modeled by linear graphs? 
• How can we utilize equations to solve problems?

In this unit, students further develop their capacity to create, manipulate, interpret, and connect these representations (algebraic, verbal, tabular, and graphical) and to use them for modeling. Through this unit, students see that graphs of equations can help us make sense of constraints and identify values that satisfy them, investigate different ways to express the same relationship or constraint—by analyzing and writing equivalent equations, and realize that some equations are more helpful than others, depending on what we want to know.  Students see that a solution to an inequality (in one or two variables) is a value or a pair of values that makes the inequality true, and a solution to a system of inequalities in two variables is any pair of values that make both inequalities in the system true. The solution set of a system of inequalities, they learn, can be best represented by graphing.

Unit 3: Two Variable Statistics

Essential Questions

Unit 3 Overview

• How can you determine if there is an association between two variables? 
• How do statisticians make predictions based on data?

In grade 8, students informally constructed scatter plots and lines of fit, noticed linear patterns, and observed associations in categorical data using two-way tables. In this unit, students build on this previous knowledge by assessing how well a linear model matches the data using residuals as well as the correlation coefficient for best-fit lines (found using technology). The unit also revisits two-way tables to find associations in categorical data using relative frequencies.

Unit 4: Functions

Essential Questions

Unit 4 Overview

• What are some types of mathematical relationships that can be modeled as functions? 
• How can you represent and describe functions? 
• How can we use functions to model data and make predictions?

In grade 8, students learned that a function is a rule that assigns exactly one output to each input. They represented functions in different ways—with verbal descriptions, algebraic expressions, graphs, and tables—and used functions to model relationships between quantities, linear relationships in particular.  

In this unit, students expand and deepen their understanding of functions. They develop new knowledge and skills for communicating about functions clearly and precisely, investigate different kinds of functions, and hone their ability to interpret functions. Students also use functions to model a wider variety of mathematical and real-world situations.

Unit 5: Introduction to Exponential Functions

Essential Questions

Unit 5 Overview

• How do I use different representations to analyze exponential functions? 
• How do I build an exponential function that models a relationship between two quantities? 
• How can we use real-world situations to construct and compare exponential models and solve problems?

In this unit, students are introduced to exponential relationships. Students learn that exponential relationships are characterized by a constant quotient over equal intervals, and compare it to linear relationships which are characterized by a constant difference over equal intervals. They encounter contexts that change exponentially. These contexts are presented verbally and with tables and graphs. They construct equations and use them to model situations and solve problems. Students investigate these exponential relationships without using function notation and language so that they can focus on gaining an appreciation for critical properties and characteristics of exponential relationships.

Unit 6: Introduction to Quadratic Functions

Essential Questions

Unit 6 Overview

• What are the characteristics of quadratic functions and what can they tell you about real-world relationships? 
• How can you use different forms of quadratic functions to represent conditions in real world situations?

Prior to this unit, students have studied what it means for a relationship to be a function, used function notation, and investigated linear and exponential functions. In this unit, they begin by looking at some patterns that grow quadratically. They contrast this growth with linear and exponential growth. They further observe that eventually these quadratic patterns grow more quickly than linear patterns but more slowly than exponential patterns.

Unit 7: Quadratic Equations

Essential Questions

Unit 7 Overview

• How do you decide which method is best when solving quadratic equations? 
• How can solving a quadratic equation connect to real-world situations?

In this unit, students interpret, write, and solve quadratic equations. They see that writing and solving quadratic equations enables them to find input values that produce certain output values.

Unit 1: Sequences and Functions

Essential Questions

Unit 1 Overview

• What are the different ways to represent sequences and functions?
• How do sequences and series model real-world problems and their solutions?
• How can function describe real-world situations,model predictions and solve problems?

In this uit, students are given an opportunity to revisit representations of functions, using the example of sequence as a particular type of function.  Students learnt that sequences are a type of function in which the input variable is the position, and the output variable is the term at that position.  Students learn how expressing regulation in repeated reasoning is present in linear and exponential functions.  At the end of the unit, students use mathematical modeling to represent mathematical situations.

Unit 2: Polynomials and Rational Functions

Essential Questions

Unit 2 Overview

• What is the relationship between the zeros and factors of polynomials? 
• What is the relationship between polynomial functions and rational functions?

In previous courses, students learned about linear and quadratic functions. They rewrote expressions for these functions in different forms to reveal structure and identified key features of their graphs, such as intercepts. In this unit, students will expand their earlier work as they investigate polynomials of higher degree and the features that all polynomial functions have in common. They will engage in practice to establish the Remainder Theorem, and transition to working with rational functions and solve rational equations, and hone skills to manipulate polynomials expressions while proving or disproving that two expressions are equivalent.

Unit 3: Complex Numbers and Rational Exponents

Essential Questions

Unit 3 Overview

• How can we use exponents to compare sizes of real-life objects? 
• How are expressions involving radicals and exponents related? 
• Why are complex numbers needed to supplement the real number system?

In this unit, students use what they know about exponents and radicals to extend exponent rules to include rational exponents, solve various equations involving squares and square roots, develop the concept of complex numbers by defining a new number i whose square is -1, and use complex numbers to find solutions to quadratic equations.

Unit 4: Exponential Functions and Equations

Essential Questions

Unit 4 Overview

• How can we use logarithms to generalize patterns in real-life measurements? 
• How do you model a quantity that changes regularly over time by a percentage?

This unit begins by activating students’ prior knowledge.  Students recall that an exponential function involves a change by equal factors over equal intervals and can be expressed as f(x)=a࣪bx, where a is the initial value of the function (the value when x is 0), and b is the growth factor.  They review the use of verbal descriptions, tables, and graphs to represent exponential functions.

Unit 5: Transformations and Functions

Essential Questions

Unit 5 Overview

• How can we use transformations of functions to model real-life situations? 
• How can transformations of functions help make predictions?

Prior to this unit, students have worked with a variety of function types, such as polynomial, radical, and exponential. The purpose of this unit is for students to consider functions as a whole and understand how they can be transformed to fit the needs of a situation, which is an aspect of modeling with mathematics (MP4). An important takeaway of the unit is that we can transform functions in a predictable manner using translations, reflections, scale factors, and by combining multiple functions. Throughout the unit students analyze graphs, tables, equations, and contexts as they work to connect representations and understand the structure of different transformations (MP7).

Unit 6: Trigonometric Functions

Essential Questions

Unit 6 Overview

• How do trigonometric and circular functions model real-world data? 
• How can we adjust trigonometric models using transformations to resemble real-life situations and make predictions?

In this unit, students are introduced to trigonometric functions. While they have previously studied a variety of function types with different key features, this is the first time students are asked to consider periodic functions, that is, functions whose output values repeat at regular intervals. This unit also builds directly on the work of the previous unit by having students apply their knowledge of transformations to trigonometric functions and use these functions to model periodic situations.