Math Scope & Sequence

Overview of Scope and Sequence and Benchmarks

Number Sense

Kindergarten

count, recognize, write, order, and compare numbers
develop and discuss strategies for estimating quantities
understand number relationships such as more than, less than
solve problems using patterns, concrete materials, and logical thinking
explore addition and subtraction in various contexts

Grades 1 and 2

count, recognize, sequence, write, compare whole numbers to 500
model place value for two-digit numbers
solve addition and subtraction problems
develop proficiency with basic facts
develop algorithms to solve problems
use simple fractions to solve problems
explore beginning multiplication and division
write equations to solve story problems

Grades 3 and 4

recognize that a variety of approaches can be used to solve the same problem
use multiplication and division to solve problems
explore relationships between multiplication and division
describe numbers in relation to other numbers
explore operations with fractions
count and add money
estimate to determine the reasonableness of answers
apply number theory concepts such as odd, even, prime and composite numbers
determine fraction equivalencies
develop operation sense for fractions and whole numbers
model and solve two-digit multiplication and division problems
develop a repertoire of strategies to solve word problems
develop proficiency with basic addition, subtraction, multiplication, division facts

Algebraic Thinking: Patterns, Relations, and Functions

Kindergarten

find missing pattern pieces and sequence numerals
link written symbols to words, objects, numbers
skip count
use events and shapes to express relationships
recognize and complete pattern sequences
make predictions by using patterns
create and extend patterns
sequence numbers

Grades 1 and 2

model balancing in number sentences
model number relationships with concrete materials
explore relationships of operations
find the missing addend
write number sentences to match story problems
use models, rules, and graphs to show mathematical relationships
recognize, describe, create, and extend patterns
sort objects based on one or more characteristics to determine if a set is complete
find missing numbers and symbols in equations

Grades 3 and 4

use calculators to discover numerical patterns
apply values to variables
create a pattern that represents a numerical relationship
recognize patterns in multiplication
understand how change in one quantity results in change in another
explore the concept of the associative property
identify multiple attributes as a way of exploring algebra
write and apply a rule to explain a relationship or function
solve equations by using trial and error and numerical analysis
devise formulas for solving problems
create an attribute set, showing all possible unique combinations
use patterns to identify any element in a sequence
apply the concepts of variable, expression, and equation

Data Analysis: Statistics and Probability

Kindergarten

collect, sort, organize, and describe data
construct graphs using real objects and pictures
develop logical thinking skills
use pictures to collect and analyze data to solve problems
make predictions

Grades 1 and 2

use pictures and symbols to characterize and group objects
collect, display, and interpret data in Venn diagrams
collect, display, sort, and interpret data in graphs
record and discuss the results of activities related to chance
compare and contrast quantities in graphs
use a table or chart to solve a problem
explore probability
predict likely or unlikely outcomes based on collected data

Grades 3 and 4

conduct a survey
make and interpret graphs, charts, tables, and Venn diagrams
formulate and solve problems by collecting and analyzing data
carry out simple probability experiments
explore probability using spinners
use data to make predictions
understand the relationship between data and outcome

Geometry and Measurement

Kindergarten

compare and order objects by length, width, height, weight
estimate and use nonstandard units to measure objects
sort and classify objects by size and shape
recognize, identify, name, and draw shapes
use clocks to tell time

Grades 1 and 2

make two-dimensional models
recognize shapes in different orientations
explore symmetry and transformations
estimate and use nonstandard units to compare and order objects
estimate and compute measurements
investigate combining, subdividing, and changing shapes
identify coins
evaluate coin values, equivalencies
tell time by hour, half-hour, and minute

Grades 3 and 4

investigate rotations, slides, flips, and turns
measure objects to find the perimeter, area, and volume
explore the relationships between area and perimeter
demonstrate an understanding of the relationship between square and linear units
choose a reasonable measurement unit for a given situation
use geometric terminology to classify shapes
examine the relationship among the number of vertices, edges, and faces
use a protractor and ruler to construct simple plane figures, such as angles
classify and draw angles and triangles
tessellate shapes across a plane
use geometric arrays to explore multiples
draw objects using various scales
solve problems that involve time

Within all units of study, children are encouraged and expected to explain their thinking both orally and in writing to teachers and peers.

Number Systems

Grades 5 and 6

Understand properties of number systems: place value, exponents, negative numbers, square roots, powers
Understand integer concepts: add and subtract integers, order of operations
Understand the relationship between fractions, percents and decimals
Understand how to add, subtract, multiply and divide fractions and decimals

Grade 7

Understand integer concepts: add and subtract integers, order of operations
Understand the relationship between fractions, decimals and percentages
Understand how to calculate and estimate percentages
Understand and apply ratios and proportions and percents
Investigate exponents, negative exponents, and scientific notation

Probability and Statistics

Grades 5 and 6

Design and conduct surveys
Make frequency graphs (line plots), broken-line graphs, and single and double bar graphs.
Organize surveys and collect data
Create and interpret tables
Determine and analyze mode, median, mean and range
Describe data with words and numbers
Analyze graphs
Understand probability as a ratio

Grade 7

Organize data in tables
Create circle and line graphs
Use algebraic techniques to display and analyze real-world data
Represent data with spreadsheets
Interpret strip graphs and frequency graphs
Determine averages
Create and interpret line graphs
Conduct probability experiments
Determine experimental and theoretical probabilities involving two independent events
Use fractions, decimals and percents to represent probabilities

Grade 8

Collect real-world data by taking measurements
Analyze tables of real-world data
Compare mathematical models to real-world data
Use mathematical models to make predictions
Create tables, bar graphs, circle graphs, stem-and-leaf plots and scatter plots
Collect, organize, and graph data related to distance and times
Organize data in tables
Determine mode, median, mean and range
Compare and analyze mode, median, mean and range
Determine average speed
Determine and compare theoretical and experimental probabilities

Geometry

Grade 5 and 6

Visualize and move between 2-D and 3-D representations
Translate between orthogonal and isometric drawings
Visualize arrangements of cubes
Create isometric drawings of block structures
Identify regular shapes and shape names
Identify equal and parallel sides
Identify equal and right triangles
Explore properties of polyhedra
Create and interpret scale drawings
Use estimation and scale factor to rescale objects

Grade 7

Visualize orthogonal views
Draw accurate nets for triangular prisms and pyramids
Investigate which sets of 3 lengths can be used to create a triangle
Create and interpret scale drawings and scale models
Compare different scales
Visualize and draw lines of symmetry
Name and classify angles and triangles
Identify how many ways can three angles be combined to fit inside a circle
Find measure of angle, given two other angles in a triangle
Explore relationships among sides of triangle and angles
Investigate possible combinations of triangle types
Describe whether any triangle tessellates
Develop a classification method for sorting polygons
Classify quadrilaterals
Explore relationship between number of sides of a polygon and the sum of its angles
Explore why some polygons tessellate, and others do not
Create tessellations and other geometric designs
Explore relationships between number of symmetry lines and type of polygon

Grade 8

Use nets to make 3-D figures
Find nets for a cube
Draw nets and cross sections to see the connection between 2 and 3-D figures
Make 3-D models
Find cross-sections of various solids, including pyramids and cones
Build scale models of skyscrapers
Visualize slope lines, horizontal lines and vertical lines as parts of a right triangle
Relate 3-D models to 2-D representations
Visualize and understand the spatial relationship among horizontal, vertical, and slanted planes
Relate right triangle situations in the physical world to abstract drawings of right triangles
Understand why acute angles in right triangle always add up to 90 degrees
Develop and apply Pythagorean Theorem, including Pythagorean Triples
Connect ideas of slope angle, slope ratio, percent grade, and tangent ratio
Create and interpret scale drawings
Make scale drawings involving angle of elevation
Explore relationships between distance from screen and area of image of slide projector

Measurement

Grades 5 and 6

Measure perimeter
Devise tests for comparing angles and lengths of sides in a shape
Estimate area
Measure angles with a protractor
Create and interpret scale drawings
Measure using inches, feet, and fractions of inches
Compare uses of estimation and measurement
Describe and contrast the U.S. Customary and metric systems of measurement
Measure and estimate lengths of objects using U.S. Customary, metric, and nonstandard units
Select appropriate units of measurement
Explore length, area, and volume relationships
Understand the effect of scaling on linear, area, and volume measurements

Grade 7

Measure angles to create circle graphs
Estimate and measure dimensions of spaces and objects using the metric system
Determine the appropriate degree of precision required for measuring in a particular situation
Find area of rectangles use various methods (including formula) for finding area of: triangle, trapezoid, rectangles
Construct nets for prisms and pyramids with particular polygons as bases
Use a compass and ruler to construct triangles
Construct a net with any polygon as a base by using a compass and ruler to construct triangular panels
Estimate and measure angles
Explore relationship between radius, diameter, and circumference of circle
Estimate area of circle
Develop and apply formula for area of circle
Develop and apply formula for area of regular polygons
Use protractor, ruler, and compass to draw and measure angles, triangles, and polygons

Grade 8

Estimate and measure lengths with the metric system
Measure and estimate distance, time, and speed
Choose appropriate unit of measure to express area and volume
Convert between units of measure
Calculate or estimate the volume of a large object using standard and nonstandard units
Understand and convert units of surface area and volume
Find and estimate surface area and volume of rectangular and non rectangular prisms, and other geometric solids
Use nets to find surface area of pyramids
Understand volume formula for: prisms, pyramids, cones
Find surface area, cross sections, and volume of cylinders
Find area, volume, and surface area of compound figures
Use algebra lab gear to write expressions for the area of various rectangular figures
Discuss relationships among accuracy of measurements, quality of measuring tools, and limitations of human perception
Use method of sighting to find slopes of distance objects
Use a protractor to measure angles
Use a “Slope-o-meter” to find angles of elevation
Measure lengths using standard and nonstandard units
Relate standard and nonstandard units of measurement
Measure with a ruler

Algebra

Grades 5 and 6

Variables and expressions
Equivalent expressions
Using tables
Creating graphs on the coordinate plane
Identifying, describing, and generalizing patterns

Grade 7

Variables and expressions
Equivalent expressions
Writing equations with one variable
Solving simple equations
Algebraic inequalities
Graphing linear equations
Translating among words, tables, equations, and graphs
Tables and graphs
Algorithm for calculating profit
Concept of function

Grade 8

Solving problems with equations
Distance-time graphs
Speed-time graphs
Graphing change over time
Graphing average speed
Slope
Solving problems with graphs
Translating among words, tables, equations, and graphs
Relationships among distance, speed, and time
Describing growth sequences with equations and expressions
Graphing addition and multiplication growth sequences
Linear relationships
Exponential relationships
Modeling
Simplifying expressions
Equivalent expressions
Distributive property
Simplifying polynomials
Computing with polynomials
Solving linear equations
Pythagorean Theorem
Slope and right triangles
Tangent ratio
Connecting slope angle, slope ratio, percent grade, tangent ratio
Creating equations from graphs
Intercepts
Symmetry in graphs
Direct variation
Inverse variation
Quadratic functions
Exponential functions

Benchmarks

By the completion of 4th grade students should be able to:

understand that math is used in everyday situations, and is a useful tool to solve real problems
have developed a repertoire of strategies to choose from in order to solve problems
explain and share their mathematical thinking with others, both orally and in writing
understand the concepts of addition, subtraction, multiplication, and division
have basic addition, subtraction, multiplication, and division facts committed to memory
add, subtract, multiply, and divide multi-digit whole numbers
write word problems for the operations of addition, subtraction, multiplication, and division
make reasonable estimations to predict solutions to problems
understand place value
round numbers to tens, hundreds, and thousands place
understand basic fractions
use data to create graphs, pie charts, and diagrams
understand the differences between measurement systems
use measurement tools(rulers, balances, protractors, clocks) to find accurate measures
recognize, describe, extend, and create patterns
understand money

By the completion of 8th grade students should be able to:

With numbers and operations:

interchange fractions, decimals, and percents
place in appropriate order all types of numbers along a number line
use factors and multiples to solve problems
manipulate negative as well as positive numbers
understand order of operations
use inverse relationships of addition/subtraction, multiplication/division, and squares/square roots to solve problems
judge the reasonableness of numerical computations
use estimation to check reasonableness of numerical computations

With measurement:

use both English and Metric systems of measurement
select and use appropriate units for measuring angles, length, area, volume, weight, time and speed
convert among units within the same system

With algebra:

recognize patterns in series of numbers and use patterns to make predictions
solve simple equations and inequalities for variables
use variables to represent patterns and solve for different situations
model different problems using graphs and tables

With geometry:

recognize and name various two-and three-dimensional shapes
draw geometric objects with specific properties
determine circumference and area of circles
determine area of triangles, squares, rectangles, and more complex shapes
determine surface area and volume of cylinders and prisms

With statistics:

calculate and interpret the means, modes, medians, and ranges of various data sets
create and interpret histograms, stem-and-leaf plots, box plots, and scatterplots
compute probabilities of various simple events
distinguish between and calculate probabilities for various simple combinations and permutations