Create a mathematics curriculum spanning grades 1 through 8 which is grade specific and sequential in complexity. Separate by grade. Group by modern pedagogy standards.
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Okay, here is a grade-specific mathematics curriculum for grades 1 through 8, designed to be sequential in complexity and grouped by common mathematical domains aligned with modern pedagogical standards.
Underlying Pedagogical Philosophy:
This curriculum emphasizes:
Conceptual Understanding: Students should understand why mathematical ideas work, not just memorize procedures.
Procedural Fluency: Students should develop efficiency and accuracy in carrying out procedures, built upon understanding.
Problem Solving & Application: Students should be able to apply mathematical concepts to solve real-world and mathematical problems.
Mathematical Practices: Throughout all grades, focus on reasoning, argumentation, modeling, using appropriate tools strategically, precision, looking for structure, and expressing regularity in repeated reasoning.
Connections: Highlighting connections between different mathematical topics and between mathematics and other disciplines/real life.
Multiple Representations: Using visuals, manipulatives, symbols, and language to represent mathematical ideas.
Productive Struggle: Creating opportunities for students to grapple with challenging problems.
Grade 1 Mathematics Curriculum
Overall Focus: Developing foundational understanding of whole numbers, addition, subtraction; describing shapes and space; basic measurement concepts.
Key Mathematical Domains & Topics:
Operations and Algebraic Thinking
Representing and solving problems involving addition and subtraction (within 20).
Understanding properties of operations (commutative, associative - informally).
Understanding the meaning of the equal sign (=).
Developing fluency for addition and subtraction within 10.
Determining unknown whole numbers in addition/subtraction equations (e.g., 8 + ? = 11).
Number and Operations in Base Ten
Counting to 120, starting from any number less than 120.
Reading and writing numerals to 120.
Understanding place value: tens and ones (bundles of ten).
Comparing two two-digit numbers based on meanings of tens and ones digits (using >, =, <).
Adding within 100 (two-digit + one-digit; two-digit + multiple of 10) using concrete models, drawings, and strategies based on place value.
Mentally finding 10 more or 10 less than a given two-digit number.
Subtracting multiples of 10 from multiples of 10 (in the range 10-90).
Measurement and Data
Ordering three objects by length; comparing lengths indirectly.
Expressing the length of an object as a whole number of non-standard length units (e.g., paper clips).
Telling and writing time in hours and half-hours using analog and digital clocks.
Organizing, representing, and interpreting data with up to three categories (tally charts, simple bar graphs).
Geometry
Distinguishing between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, size).
Composing two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, quarter-circles) or three-dimensional shapes (cubes, rectangular prisms, cones, cylinders) to create composite shapes.
Partitioning circles and rectangles into two and four equal shares; describing shares using words like halves, fourths, quarters.
Pedagogical Emphasis: Extensive use of manipulatives (counters, base-ten blocks, pattern blocks), drawings, acting out situations, number lines, and math talk to build foundational understanding.
Grade 2 Mathematics Curriculum
Overall Focus: Extending understanding of base-ten notation; building fluency with addition and subtraction; using standard units of measure; describing and analyzing shapes.
Key Mathematical Domains & Topics:
Operations and Algebraic Thinking
Using addition and subtraction within 100 to solve one- and two-step word problems.
Fluently adding and subtracting within 20 using mental strategies.
Foundations of multiplication: working with equal groups of objects (arrays, skip counting).
Number and Operations in Base Ten
Understanding place value: hundreds, tens, and ones.
Counting within 1000; skip-counting by 5s, 10s, and 100s.
Reading and writing numbers to 1000 using base-ten numerals, number names, and expanded form.
Comparing two three-digit numbers based on place value (using >, =, <).
Fluently adding and subtracting within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
Adding up to four two-digit numbers.
Adding and subtracting within 1000 using concrete models, drawings, and strategies; relate strategies to written methods.
Mentally adding/subtracting 10 or 100 to/from a given number 100-900.
Measurement and Data
Measuring the length of an object using standard units (inches, feet, centimeters, meters).
Estimating lengths.
Comparing lengths of two objects; relating addition and subtraction to length.
Telling and writing time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.
Solving word problems involving dollar bills, quarters, dimes, nickels, and pennies (using $ and ¢ symbols appropriately).
Generating measurement data and showing it on a line plot.
Drawing picture graphs and bar graphs (single-unit scale).
Geometry
Recognizing and drawing shapes having specified attributes (e.g., number of angles, number of faces). Identifying triangles, quadrilaterals, pentagons, hexagons, and cubes.
Partitioning rectangles into rows and columns of same-size squares and counting to find the total.
Partitioning circles and rectangles into two, three, or four equal shares; describing shares using words like halves, thirds, fourths, and phrases like a third of. Recognizing that equal shares of identical wholes need not have the same shape.
Pedagogical Emphasis: Continued use of models and drawings, developing mental math strategies, explaining reasoning, connecting different representations (e.g., base-ten blocks to written algorithms).
Grade 3 Mathematics Curriculum
Overall Focus: Developing understanding of multiplication and division and strategies for multiplication and division within 100; developing understanding of fractions, especially unit fractions; understanding area; analyzing two-dimensional shapes.
Key Mathematical Domains & Topics:
Operations and Algebraic Thinking
Interpreting products of whole numbers (e.g., 5 × 7 as 5 groups of 7).
Interpreting whole-number quotients (e.g., 56 ÷ 8 as the number of objects in each share or the number of shares).
Using multiplication and division within 100 to solve word problems.
Determining unknown whole numbers in multiplication/division equations.
Applying properties of operations (commutative, associative, distributive) as strategies to multiply and divide.
Understanding division as an unknown-factor problem.
Fluently multiplying and dividing within 100 (know from memory all products of two one-digit numbers by end of grade 3).
Solving two-step word problems using the four operations. Representing problems using equations with a letter for the unknown quantity. Assessing reasonableness of answers.
Identifying arithmetic patterns (including patterns in addition/multiplication tables).
Number and Operations in Base Ten
Using place value understanding to round whole numbers to the nearest 10 or 100.
Fluently adding and subtracting within 1000.
Multiplying one-digit whole numbers by multiples of 10 (e.g., 9 × 80).
Number and Operations—Fractions
Understanding a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
Understanding fractions as numbers on the number line.
Explaining equivalence of fractions in simple cases (e.g., 1/2 = 2/4) and comparing fractions by reasoning about their size (using visual fraction models). Must have the same whole for comparison. Comparing fractions with the same numerator or same denominator.
Expressing whole numbers as fractions (e.g., 3 = 3/1). Recognizing fractions equivalent to whole numbers.
Measurement and Data
Telling and writing time to the nearest minute. Solving word problems involving addition/subtraction of time intervals in minutes.
Measuring and estimating liquid volumes and masses of objects using standard units (grams, kilograms, liters). Solving one-step word problems involving masses or volumes.
Drawing scaled picture graphs and scaled bar graphs. Solving "how many more/less" problems using scaled graphs.
Understanding concepts of area and relating area to multiplication and addition. Measuring areas by counting unit squares.
Relating area to the operations of multiplication and addition (tiling, area formula for rectangles).
Recognizing perimeter as an attribute of plane figures and distinguishing between linear and area measures. Solving problems involving perimeters of polygons.
Geometry
Understanding that shapes in different categories (e.g., rhombuses, rectangles) may share attributes (e.g., four sides) and that shared attributes can define a larger category (e.g., quadrilaterals).
Partitioning shapes into parts with equal areas. Expressing the area of each part as a unit fraction of the whole.
Pedagogical Emphasis: Making connections between multiplication/division and area, using arrays and area models, extensive work with number lines for fractions, reasoning about fraction size, developing fluency through strategies and practice.
Grade 4 Mathematics Curriculum
Overall Focus: Developing understanding and fluency with multi-digit multiplication; developing understanding of dividing to find quotients involving multi-digit dividends; developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; understanding properties of geometric figures.
Key Mathematical Domains & Topics:
Operations and Algebraic Thinking
Interpreting multiplication equations as comparisons (e.g., 35 = 5 × 7 as 35 is 5 times as many as 7).
Multiplying or dividing to solve word problems involving multiplicative comparison.
Solving multistep word problems posed with whole numbers using the four operations, including problems with remainders. Using letters for unknown quantities. Assessing reasonableness.
Finding factor pairs for whole numbers 1-100. Recognizing prime and composite numbers.
Generating and analyzing patterns that follow a given rule.
Number and Operations in Base Ten
Generalizing place value understanding: recognizing that a digit in one place represents ten times what it represents in the place to its right.
Reading, writing, and comparing multi-digit whole numbers using base-ten numerals, number names, expanded form, and comparison symbols (>, =, <).
Rounding multi-digit whole numbers to any place.
Fluently adding and subtracting multi-digit whole numbers using the standard algorithm.
Multiplying a whole number of up to four digits by a one-digit number, and multiplying two two-digit numbers, using strategies based on place value and properties of operations. Illustrating calculations with equations, arrays, and/or area models.
Finding whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using various strategies.
Number and Operations—Fractions
Explaining why fractions are equivalent using visual fraction models. Generating equivalent fractions.
Comparing two fractions with different numerators and denominators (e.g., by creating common denominators/numerators, or comparing to a benchmark like 1/2).
Understanding addition and subtraction of fractions as joining/separating parts referring to the same whole.
Adding and subtracting fractions and mixed numbers with like denominators.
Solving word problems involving addition/subtraction of fractions with like denominators.
Applying understanding of multiplication to multiply a fraction by a whole number (e.g., 3 × (2/5) as 6 × (1/5)).
Solving word problems involving multiplication of a fraction by a whole number.
Understanding decimal notation for fractions (tenths and hundredths). Using decimal notation for fractions with denominators 10 or 100.
Comparing two decimals to hundredths by reasoning about their size.
Measurement and Data
Knowing relative sizes of measurement units within one system (km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec). Expressing measurements in a larger unit in terms of a smaller unit. Recording measurement equivalents in a two-column table.
Using the four operations to solve word problems involving distances, time intervals, liquid volumes, masses, and money, including problems involving simple fractions or decimals.
Applying the area and perimeter formulas for rectangles in real-world problems.
Making line plots to display data sets of measurements in fractions of a unit (1/2, 1/4, 1/8). Solving problems involving addition/subtraction of fractions using information from line plots.
Recognizing angles as geometric shapes formed wherever two rays share a common endpoint. Understanding concepts of angle measurement.
Measuring angles in whole-number degrees using a protractor. Sketching angles of specified measure.
Recognizing angle measure as additive. Solving addition/subtraction problems to find unknown angles.
Geometry
Drawing points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identifying these in two-dimensional figures.
Classifying two-dimensional figures based on the presence/absence of parallel or perpendicular lines, or the presence/absence of angles of a specified size. Identifying right triangles.
Recognizing a line of symmetry for a two-dimensional figure.
Pedagogical Emphasis: Connecting area models to multi-digit multiplication algorithms, using visual models extensively for fraction equivalence and operations, relating fractions and decimals, hands-on angle measurement.
Grade 5 Mathematics Curriculum
Overall Focus: Developing fluency with addition and subtraction of fractions; developing understanding of multiplication and division of fractions in limited cases; extending division to 2-digit divisors; integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths; developing understanding of volume.
Key Mathematical Domains & Topics:
Operations and Algebraic Thinking
Writing and interpreting numerical expressions using parentheses, brackets, or braces. Evaluating expressions with these symbols.
Writing simple expressions that record calculations with numbers.
Generating two numerical patterns using two given rules. Identifying apparent relationships between corresponding terms. Forming ordered pairs and graphing them on a coordinate plane.
Number and Operations in Base Ten
Recognizing that a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
Explaining patterns in the number of zeros of the product when multiplying by powers of 10; explaining patterns in the placement of the decimal point when multiplying/dividing by powers of 10. Using whole-number exponents to denote powers of 10.
Reading, writing, and comparing decimals to thousandths. Using base-ten numerals, number names, expanded form, and comparison symbols (>, =, <).
Rounding decimals to any place.
Fluently multiplying multi-digit whole numbers using the standard algorithm.
Finding whole-number quotients with up to four-digit dividends and two-digit divisors, using various strategies.
Adding, subtracting, multiplying, and dividing decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between operations; relating strategies to written methods.
Number and Operations—Fractions
Adding and subtracting fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions.
Solving word problems involving addition/subtraction of fractions. Assessing reasonableness.
Interpreting a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solving word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers.
Applying and extending previous understandings of multiplication to multiply a fraction or whole number by a fraction. Interpreting the product (a/b) × q.
Interpreting multiplication as scaling (resizing). Comparing the size of a product to the size of one factor based on the size of the other factor.
Solving real-world problems involving multiplication of fractions and mixed numbers.
Applying and extending previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. Interpreting division of a unit fraction by a non-zero whole number and vice versa. Solving related real-world problems.
Measurement and Data
Converting among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m). Using these conversions in multi-step problems.
Making line plots to display data sets of measurements in fractions of a unit (1/2, 1/4, 1/8). Using operations on fractions for this grade to solve problems involving information presented in line plots.
Understanding concepts of volume. Relating volume to multiplication and addition.
Measuring volumes by counting unit cubes.
Relating volume to the operations of multiplication and addition and solving real-world problems involving volume (finding volume of rectangular prisms). Finding volumes of composite figures made of rectangular prisms.
Geometry
Using a pair of perpendicular number lines (axes) to define a coordinate system. Graphing points in the first quadrant of the coordinate plane. Interpreting coordinate values in context.
Understanding that attributes belonging to a category of two-dimensional figures also belong to all subcategories (e.g., all rectangles have four right angles, and squares are rectangles, so all squares have four right angles).
Classifying two-dimensional figures in a hierarchy based on properties.
Pedagogical Emphasis: Connecting fraction operations to whole number operations, using visual models (area models, number lines) for fraction multiplication/division, developing decimal operation understanding through place value and connection to fractions, exploring volume with physical cubes.
Grade 6 Mathematics Curriculum
Overall Focus: Connecting ratio and rate to whole number multiplication and division; using concepts of ratio and rate to solve problems; completing understanding of division of fractions; extending the notion of number to the system of rational numbers (including negative numbers); writing, interpreting, and using expressions and equations; developing understanding of statistical thinking.
Key Mathematical Domains & Topics:
Ratios and Proportional Relationships
Understanding the concept of a ratio and using ratio language to describe relationships.
Understanding the concept of a unit rate a/b associated with a ratio a:b (b≠0). Using rate language.
Using ratio and rate reasoning to solve real-world problems (e.g., using tables, tape diagrams, double number lines, equations). Finding percent of a quantity; solving problems involving finding the whole, given a part and the percent. Using ratio reasoning for unit conversions.
The Number System
Interpreting and computing quotients of fractions. Solving word problems involving division of fractions by fractions.
Fluently dividing multi-digit numbers using the standard algorithm.
Fluently adding, subtracting, multiplying, and dividing multi-digit decimals using standard algorithms.
Understanding positive and negative numbers; using them to describe quantities having opposite directions or values (e.g., temperature, elevation, credits/debits). Using positive and negative numbers on the number line and coordinate plane.
Understanding rational numbers as points on the number line. Ordering and finding absolute value of rational numbers.
Solving real-world problems by graphing points in all four quadrants of the coordinate plane. Finding distances between points with the same first or second coordinate.
Expressions and Equations
Writing and evaluating numerical expressions involving whole-number exponents.
Writing, reading, and evaluating expressions in which letters stand for numbers.
Applying properties of operations to generate equivalent expressions (e.g., distributive property to write 2(4 + x) as 8 + 2x). Identifying equivalent expressions.
Understanding solving an equation or inequality as finding values that make it true. Using substitution to check solutions.
Using variables to represent numbers and writing expressions to solve real-world problems.
Solving one-step linear equations of the form x + p = q and px = q for non-negative rational numbers.
Writing inequalities of the form x > c or x < c. Representing solutions on number line diagrams.
Using variables to represent two quantities that change in relationship to one another (dependent and independent variables). Writing an equation to express one quantity in terms of the other. Analyzing the relationship using graphs and tables.
Geometry
Finding the area of right triangles, other triangles, special quadrilaterals, and polygons by composing/decomposing into rectangles and triangles.
Finding the volume of a right rectangular prism with fractional edge lengths using unit cubes. Applying formulas V = lwh and V = Bh.
Drawing polygons in the coordinate plane given coordinates for vertices. Using coordinates to find the length of a side.
Representing three-dimensional figures using nets made up of rectangles and triangles. Using nets to find the surface area of these figures.
Statistics and Probability
Recognizing a statistical question (one that anticipates variability in the data).
Understanding that data sets have a distribution described by center, spread, and overall shape.
Recognizing that a measure of center summarizes data with a single number, while a measure of variation describes spread.
Displaying numerical data in plots on a number line (dot plots, histograms, box plots).
Summarizing numerical data sets in relation to their context (reporting number of observations; describing nature of attribute; giving measures of center - median/mean - and variability - interquartile range/mean absolute deviation; describing overall pattern and deviations).
Pedagogical Emphasis: Developing ratio reasoning through various representations, extending arithmetic to rational numbers, formalizing algebraic expressions and one-step equations, connecting geometry to measurement (area, volume, surface area), introducing foundational statistical concepts.
Grade 7 Mathematics Curriculum
Overall Focus: Developing understanding of and applying proportional relationships; developing understanding of operations with rational numbers; working with expressions and linear equations; solving problems involving scale drawings and informal geometric constructions; working with two- and three-dimensional shapes to solve problems involving area, surface area, and volume; drawing inferences about populations based on samples.
Key Mathematical Domains & Topics:
Ratios and Proportional Relationships
Computing unit rates associated with ratios of fractions.
Recognizing and representing proportional relationships between quantities (deciding if two quantities are proportional via tables, graphs, equations; identifying the constant of proportionality - unit rate).
Using proportional relationships to solve multistep ratio and percent problems (simple interest, tax, markups/markdowns, gratuities, commissions, fees, percent increase/decrease, percent error).
The Number System
Applying and extending previous understandings of addition and subtraction to add and subtract rational numbers (integers, fractions, decimals). Representing operations on a number line.
Applying and extending previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. Understanding rules for multiplying signed numbers. Interpreting products/quotients of rational numbers in real-world contexts. Understanding division by zero is undefined.
Solving real-world problems involving the four operations with rational numbers.
Expressions and Equations
Applying properties of operations to add, subtract, factor, and expand linear expressions with rational coefficients.
Understanding that rewriting an expression in different forms can shed light on the problem (e.g., a + 0.05a = 1.05a means "increase by 5%" is same as "multiply by 1.05").
Solving multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, decimals), using tools strategically. Applying properties of operations. Assessing reasonableness using estimation.
Using variables to represent quantities and constructing simple equations (px + q = r, p(x + q) = r) and inequalities (px + q > r, px + q < r) to solve problems. Graphing solution sets of inequalities.
Geometry
Solving problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
Drawing geometric shapes with given conditions (freehand, ruler/protractor, technology). Focus on triangles from three measures of angles or sides.
Describing the two-dimensional figures that result from slicing three-dimensional figures (plane sections of right rectangular prisms and pyramids).
Knowing the formulas for the area and circumference of a circle and using them to solve problems. Giving informal derivation of the relationship between circumference and area.
Using facts about supplementary, complementary, vertical, and adjacent angles in multi-step problems to write and solve simple equations for an unknown angle.
Solving real-world problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
Statistics and Probability
Understanding that statistics can be used to gain information about a population by examining a sample; generalizations are valid only if the sample is representative. Understanding random sampling.
Using data from a random sample to draw inferences about a population with an unknown characteristic. Generating multiple samples to gauge variation in estimates/predictions.
Informally assessing the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between centers (e.g., using mean or median) as a multiple of a measure of variability (e.g., MAD or IQR).
Using measures of center and variability for numerical data from random samples to draw informal comparative inferences about two populations.
Understanding that probability is a number between 0 and 1 expressing likelihood.
Approximating probability by collecting data (experimental probability) and predicting relative frequency.
Developing probability models (uniform and non-uniform). Determining probabilities of events.
Finding probabilities of compound events using organized lists, tables, tree diagrams, and simulation. Understanding independent and dependent events (introduced).
Pedagogical Emphasis: Deepening understanding of proportionality, solidifying operations with all rational numbers, moving towards multi-step algebraic thinking, connecting geometry with algebra (angle equations), developing informal statistical inference and probability reasoning.
Grade 8 Mathematics Curriculum
Overall Focus: Formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations; grasping the concept of a function and using functions to describe quantitative relationships; analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem.
Key Mathematical Domains & Topics:
The Number System
Knowing that numbers that are not rational are called irrational. Understanding informally that every number has a decimal expansion; rational numbers terminate or repeat, irrational numbers do not.
Using rational approximations of irrational numbers to compare their size, locate them on a number line, and estimate the value of expressions (e.g., π²).
Expressions and Equations
Knowing and applying the properties of integer exponents to generate equivalent numerical expressions (e.g., 3² × 3⁻⁵ = 3⁻³ = 1/3³ = 1/27).
Using square root and cube root symbols to represent solutions to equations like x² = p and x³ = p, where p is a positive rational number. Evaluating square roots of small perfect squares and cube roots of small perfect cubes. Knowing √2 is irrational.
Using numbers expressed in scientific notation to estimate very large or very small quantities and to express how many times as much one is than the other. Performing operations with numbers in scientific notation.
Graphing proportional relationships, interpreting the unit rate as the slope of the graph. Comparing two proportional relationships represented differently.
Using similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane. Deriving the equation y = mx for a line through the origin and y = mx + b for a line intercepting the y-axis at b.
Solving linear equations in one variable (including cases with one solution, infinitely many solutions, or no solutions). Solving linear equations with rational number coefficients, including equations requiring expanding expressions using the distributive property and collecting like terms.
Analyzing and solving pairs of simultaneous linear equations (systems of linear equations). Understanding solutions correspond to points of intersection. Solving systems algebraically (substitution, elimination) and by inspection. Solving real-world problems leading to two linear equations.
Functions
Understanding that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs (input, output).
Comparing properties (e.g., rate of change, initial value) of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
Interpreting the equation y = mx + b as defining a linear function whose graph is a straight line. Giving examples of functions that are not linear.
Geometry
Verifying experimentally the properties of rotations, reflections, and translations (lines to lines, line segments to line segments of same length, angles to angles of same measure, parallel lines to parallel lines).
Understanding that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations. Describing a sequence that exhibits congruence.
Describing the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
Understanding that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations. Describing a sequence that exhibits similarity.
Using informal arguments to establish facts about the angle sum and exterior angle of triangles, about angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.
Explaining a proof of the Pythagorean Theorem and its converse.
Applying the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
Applying the Pythagorean Theorem to find the distance between two points in a coordinate system.
Knowing the formulas for the volumes of cones, cylinders, and spheres and using them to solve real-world problems.
Statistics and Probability
Constructing and interpreting scatter plots for bivariate measurement data to investigate patterns of association (clustering, outliers, positive/negative association, linear/non-linear association).
Knowing that straight lines are widely used to model relationships between two quantitative variables. For scatter plots suggesting linear association, informally fitting a straight line and informally assessing the model fit (judging closeness of data points to the line).
Using the equation of a linear model to solve problems in the context of bivariate data, interpreting the slope and intercept.
Understanding that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Constructing and interpreting two-way tables. Using relative frequencies to describe possible association.
Pedagogical Emphasis: Transitioning to more formal algebraic reasoning, introducing the core concept of a function, developing spatial reasoning through transformations and the Pythagorean Theorem, exploring relationships in bivariate data.
This curriculum provides a framework. Effective implementation requires flexibility, differentiation to meet diverse student needs, ongoing formative assessment, and the use of rich tasks that promote deep mathematical thinking.
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Okay, here is a grade-specific mathematics curriculum for grades 1 through 8, designed to be sequential in complexity and grouped by common mathematical domains aligned with modern pedagogical standards.
Underlying Pedagogical Philosophy:
This curriculum emphasizes:
Conceptual Understanding: Students should understand why mathematical ideas work, not just memorize procedures.
Procedural Fluency: Students should develop efficiency and accuracy in carrying out procedures, built upon understanding.
Problem Solving & Application: Students should be able to apply mathematical concepts to solve real-world and mathematical problems.
Mathematical Practices: Throughout all grades, focus on reasoning, argumentation, modeling, using appropriate tools strategically, precision, looking for structure, and expressing regularity in repeated reasoning.
Connections: Highlighting connections between different mathematical topics and between mathematics and other disciplines/real life.
Multiple Representations: Using visuals, manipulatives, symbols, and language to represent mathematical ideas.
Productive Struggle: Creating opportunities for students to grapple with challenging problems.
Grade 1 Mathematics Curriculum
Overall Focus: Developing foundational understanding of whole numbers, addition, subtraction; describing shapes and space; basic measurement concepts.
Key Mathematical Domains & Topics:
Operations and Algebraic Thinking
Representing and solving problems involving addition and subtraction (within 20).
Understanding properties of operations (commutative, associative - informally).
Understanding the meaning of the equal sign (=).
Developing fluency for addition and subtraction within 10.
Determining unknown whole numbers in addition/subtraction equations (e.g., 8 + ? = 11).
Number and Operations in Base Ten
Counting to 120, starting from any number less than 120.
Reading and writing numerals to 120.
Understanding place value: tens and ones (bundles of ten).
Comparing two two-digit numbers based on meanings of tens and ones digits (using >, =, <).
Adding within 100 (two-digit + one-digit; two-digit + multiple of 10) using concrete models, drawings, and strategies based on place value.
Mentally finding 10 more or 10 less than a given two-digit number.
Subtracting multiples of 10 from multiples of 10 (in the range 10-90).
Measurement and Data
Ordering three objects by length; comparing lengths indirectly.
Expressing the length of an object as a whole number of non-standard length units (e.g., paper clips).
Telling and writing time in hours and half-hours using analog and digital clocks.
Organizing, representing, and interpreting data with up to three categories (tally charts, simple bar graphs).
Geometry
Distinguishing between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, size).
Composing two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, quarter-circles) or three-dimensional shapes (cubes, rectangular prisms, cones, cylinders) to create composite shapes.
Partitioning circles and rectangles into two and four equal shares; describing shares using words like halves, fourths, quarters.
Pedagogical Emphasis: Extensive use of manipulatives (counters, base-ten blocks, pattern blocks), drawings, acting out situations, number lines, and math talk to build foundational understanding.
Grade 2 Mathematics Curriculum
Overall Focus: Extending understanding of base-ten notation; building fluency with addition and subtraction; using standard units of measure; describing and analyzing shapes.
Key Mathematical Domains & Topics:
Operations and Algebraic Thinking
Using addition and subtraction within 100 to solve one- and two-step word problems.
Fluently adding and subtracting within 20 using mental strategies.
Foundations of multiplication: working with equal groups of objects (arrays, skip counting).
Number and Operations in Base Ten
Understanding place value: hundreds, tens, and ones.
Counting within 1000; skip-counting by 5s, 10s, and 100s.
Reading and writing numbers to 1000 using base-ten numerals, number names, and expanded form.
Comparing two three-digit numbers based on place value (using >, =, <).
Fluently adding and subtracting within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
Adding up to four two-digit numbers.
Adding and subtracting within 1000 using concrete models, drawings, and strategies; relate strategies to written methods.
Mentally adding/subtracting 10 or 100 to/from a given number 100-900.
Measurement and Data
Measuring the length of an object using standard units (inches, feet, centimeters, meters).
Estimating lengths.
Comparing lengths of two objects; relating addition and subtraction to length.
Telling and writing time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.
Solving word problems involving dollar bills, quarters, dimes, nickels, and pennies (using $ and ¢ symbols appropriately).
Generating measurement data and showing it on a line plot.
Drawing picture graphs and bar graphs (single-unit scale).
Geometry
Recognizing and drawing shapes having specified attributes (e.g., number of angles, number of faces). Identifying triangles, quadrilaterals, pentagons, hexagons, and cubes.
Partitioning rectangles into rows and columns of same-size squares and counting to find the total.
Partitioning circles and rectangles into two, three, or four equal shares; describing shares using words like halves, thirds, fourths, and phrases like a third of. Recognizing that equal shares of identical wholes need not have the same shape.
Pedagogical Emphasis: Continued use of models and drawings, developing mental math strategies, explaining reasoning, connecting different representations (e.g., base-ten blocks to written algorithms).
Grade 3 Mathematics Curriculum
Overall Focus: Developing understanding of multiplication and division and strategies for multiplication and division within 100; developing understanding of fractions, especially unit fractions; understanding area; analyzing two-dimensional shapes.
Key Mathematical Domains & Topics:
Operations and Algebraic Thinking
Interpreting products of whole numbers (e.g., 5 × 7 as 5 groups of 7).
Interpreting whole-number quotients (e.g., 56 ÷ 8 as the number of objects in each share or the number of shares).
Using multiplication and division within 100 to solve word problems.
Determining unknown whole numbers in multiplication/division equations.
Applying properties of operations (commutative, associative, distributive) as strategies to multiply and divide.
Understanding division as an unknown-factor problem.
Fluently multiplying and dividing within 100 (know from memory all products of two one-digit numbers by end of grade 3).
Solving two-step word problems using the four operations. Representing problems using equations with a letter for the unknown quantity. Assessing reasonableness of answers.
Identifying arithmetic patterns (including patterns in addition/multiplication tables).
Number and Operations in Base Ten
Using place value understanding to round whole numbers to the nearest 10 or 100.
Fluently adding and subtracting within 1000.
Multiplying one-digit whole numbers by multiples of 10 (e.g., 9 × 80).
Number and Operations—Fractions
Understanding a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
Understanding fractions as numbers on the number line.
Explaining equivalence of fractions in simple cases (e.g., 1/2 = 2/4) and comparing fractions by reasoning about their size (using visual fraction models). Must have the same whole for comparison. Comparing fractions with the same numerator or same denominator.
Expressing whole numbers as fractions (e.g., 3 = 3/1). Recognizing fractions equivalent to whole numbers.
Measurement and Data
Telling and writing time to the nearest minute. Solving word problems involving addition/subtraction of time intervals in minutes.
Measuring and estimating liquid volumes and masses of objects using standard units (grams, kilograms, liters). Solving one-step word problems involving masses or volumes.
Drawing scaled picture graphs and scaled bar graphs. Solving "how many more/less" problems using scaled graphs.
Understanding concepts of area and relating area to multiplication and addition. Measuring areas by counting unit squares.
Relating area to the operations of multiplication and addition (tiling, area formula for rectangles).
Recognizing perimeter as an attribute of plane figures and distinguishing between linear and area measures. Solving problems involving perimeters of polygons.
Geometry
Understanding that shapes in different categories (e.g., rhombuses, rectangles) may share attributes (e.g., four sides) and that shared attributes can define a larger category (e.g., quadrilaterals).
Partitioning shapes into parts with equal areas. Expressing the area of each part as a unit fraction of the whole.
Pedagogical Emphasis: Making connections between multiplication/division and area, using arrays and area models, extensive work with number lines for fractions, reasoning about fraction size, developing fluency through strategies and practice.
Grade 4 Mathematics Curriculum
Overall Focus: Developing understanding and fluency with multi-digit multiplication; developing understanding of dividing to find quotients involving multi-digit dividends; developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; understanding properties of geometric figures.
Key Mathematical Domains & Topics:
Operations and Algebraic Thinking
Interpreting multiplication equations as comparisons (e.g., 35 = 5 × 7 as 35 is 5 times as many as 7).
Multiplying or dividing to solve word problems involving multiplicative comparison.
Solving multistep word problems posed with whole numbers using the four operations, including problems with remainders. Using letters for unknown quantities. Assessing reasonableness.
Finding factor pairs for whole numbers 1-100. Recognizing prime and composite numbers.
Generating and analyzing patterns that follow a given rule.
Number and Operations in Base Ten
Generalizing place value understanding: recognizing that a digit in one place represents ten times what it represents in the place to its right.
Reading, writing, and comparing multi-digit whole numbers using base-ten numerals, number names, expanded form, and comparison symbols (>, =, <).
Rounding multi-digit whole numbers to any place.
Fluently adding and subtracting multi-digit whole numbers using the standard algorithm.
Multiplying a whole number of up to four digits by a one-digit number, and multiplying two two-digit numbers, using strategies based on place value and properties of operations. Illustrating calculations with equations, arrays, and/or area models.
Finding whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using various strategies.
Number and Operations—Fractions
Explaining why fractions are equivalent using visual fraction models. Generating equivalent fractions.
Comparing two fractions with different numerators and denominators (e.g., by creating common denominators/numerators, or comparing to a benchmark like 1/2).
Understanding addition and subtraction of fractions as joining/separating parts referring to the same whole.
Adding and subtracting fractions and mixed numbers with like denominators.
Solving word problems involving addition/subtraction of fractions with like denominators.
Applying understanding of multiplication to multiply a fraction by a whole number (e.g., 3 × (2/5) as 6 × (1/5)).
Solving word problems involving multiplication of a fraction by a whole number.
Understanding decimal notation for fractions (tenths and hundredths). Using decimal notation for fractions with denominators 10 or 100.
Comparing two decimals to hundredths by reasoning about their size.
Measurement and Data
Knowing relative sizes of measurement units within one system (km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec). Expressing measurements in a larger unit in terms of a smaller unit. Recording measurement equivalents in a two-column table.
Using the four operations to solve word problems involving distances, time intervals, liquid volumes, masses, and money, including problems involving simple fractions or decimals.
Applying the area and perimeter formulas for rectangles in real-world problems.
Making line plots to display data sets of measurements in fractions of a unit (1/2, 1/4, 1/8). Solving problems involving addition/subtraction of fractions using information from line plots.
Recognizing angles as geometric shapes formed wherever two rays share a common endpoint. Understanding concepts of angle measurement.
Measuring angles in whole-number degrees using a protractor. Sketching angles of specified measure.
Recognizing angle measure as additive. Solving addition/subtraction problems to find unknown angles.
Geometry
Drawing points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identifying these in two-dimensional figures.
Classifying two-dimensional figures based on the presence/absence of parallel or perpendicular lines, or the presence/absence of angles of a specified size. Identifying right triangles.
Recognizing a line of symmetry for a two-dimensional figure.
Pedagogical Emphasis: Connecting area models to multi-digit multiplication algorithms, using visual models extensively for fraction equivalence and operations, relating fractions and decimals, hands-on angle measurement.
Grade 5 Mathematics Curriculum
Overall Focus: Developing fluency with addition and subtraction of fractions; developing understanding of multiplication and division of fractions in limited cases; extending division to 2-digit divisors; integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths; developing understanding of volume.
Key Mathematical Domains & Topics:
Operations and Algebraic Thinking
Writing and interpreting numerical expressions using parentheses, brackets, or braces. Evaluating expressions with these symbols.
Writing simple expressions that record calculations with numbers.
Generating two numerical patterns using two given rules. Identifying apparent relationships between corresponding terms. Forming ordered pairs and graphing them on a coordinate plane.
Number and Operations in Base Ten
Recognizing that a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
Explaining patterns in the number of zeros of the product when multiplying by powers of 10; explaining patterns in the placement of the decimal point when multiplying/dividing by powers of 10. Using whole-number exponents to denote powers of 10.
Reading, writing, and comparing decimals to thousandths. Using base-ten numerals, number names, expanded form, and comparison symbols (>, =, <).
Rounding decimals to any place.
Fluently multiplying multi-digit whole numbers using the standard algorithm.
Finding whole-number quotients with up to four-digit dividends and two-digit divisors, using various strategies.
Adding, subtracting, multiplying, and dividing decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between operations; relating strategies to written methods.
Number and Operations—Fractions
Adding and subtracting fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions.
Solving word problems involving addition/subtraction of fractions. Assessing reasonableness.
Interpreting a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solving word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers.
Applying and extending previous understandings of multiplication to multiply a fraction or whole number by a fraction. Interpreting the product (a/b) × q.
Interpreting multiplication as scaling (resizing). Comparing the size of a product to the size of one factor based on the size of the other factor.
Solving real-world problems involving multiplication of fractions and mixed numbers.
Applying and extending previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. Interpreting division of a unit fraction by a non-zero whole number and vice versa. Solving related real-world problems.
Measurement and Data
Converting among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m). Using these conversions in multi-step problems.
Making line plots to display data sets of measurements in fractions of a unit (1/2, 1/4, 1/8). Using operations on fractions for this grade to solve problems involving information presented in line plots.
Understanding concepts of volume. Relating volume to multiplication and addition.
Measuring volumes by counting unit cubes.
Relating volume to the operations of multiplication and addition and solving real-world problems involving volume (finding volume of rectangular prisms). Finding volumes of composite figures made of rectangular prisms.
Geometry
Using a pair of perpendicular number lines (axes) to define a coordinate system. Graphing points in the first quadrant of the coordinate plane. Interpreting coordinate values in context.
Understanding that attributes belonging to a category of two-dimensional figures also belong to all subcategories (e.g., all rectangles have four right angles, and squares are rectangles, so all squares have four right angles).
Classifying two-dimensional figures in a hierarchy based on properties.
Pedagogical Emphasis: Connecting fraction operations to whole number operations, using visual models (area models, number lines) for fraction multiplication/division, developing decimal operation understanding through place value and connection to fractions, exploring volume with physical cubes.
Grade 6 Mathematics Curriculum
Overall Focus: Connecting ratio and rate to whole number multiplication and division; using concepts of ratio and rate to solve problems; completing understanding of division of fractions; extending the notion of number to the system of rational numbers (including negative numbers); writing, interpreting, and using expressions and equations; developing understanding of statistical thinking.
Key Mathematical Domains & Topics:
Ratios and Proportional Relationships
Understanding the concept of a ratio and using ratio language to describe relationships.
Understanding the concept of a unit rate a/b associated with a ratio a:b (b≠0). Using rate language.
Using ratio and rate reasoning to solve real-world problems (e.g., using tables, tape diagrams, double number lines, equations). Finding percent of a quantity; solving problems involving finding the whole, given a part and the percent. Using ratio reasoning for unit conversions.
The Number System
Interpreting and computing quotients of fractions. Solving word problems involving division of fractions by fractions.
Fluently dividing multi-digit numbers using the standard algorithm.
Fluently adding, subtracting, multiplying, and dividing multi-digit decimals using standard algorithms.
Understanding positive and negative numbers; using them to describe quantities having opposite directions or values (e.g., temperature, elevation, credits/debits). Using positive and negative numbers on the number line and coordinate plane.
Understanding rational numbers as points on the number line. Ordering and finding absolute value of rational numbers.
Solving real-world problems by graphing points in all four quadrants of the coordinate plane. Finding distances between points with the same first or second coordinate.
Expressions and Equations
Writing and evaluating numerical expressions involving whole-number exponents.
Writing, reading, and evaluating expressions in which letters stand for numbers.
Applying properties of operations to generate equivalent expressions (e.g., distributive property to write 2(4 + x) as 8 + 2x). Identifying equivalent expressions.
Understanding solving an equation or inequality as finding values that make it true. Using substitution to check solutions.
Using variables to represent numbers and writing expressions to solve real-world problems.
Solving one-step linear equations of the form x + p = q and px = q for non-negative rational numbers.
Writing inequalities of the form x > c or x < c. Representing solutions on number line diagrams.
Using variables to represent two quantities that change in relationship to one another (dependent and independent variables). Writing an equation to express one quantity in terms of the other. Analyzing the relationship using graphs and tables.
Geometry
Finding the area of right triangles, other triangles, special quadrilaterals, and polygons by composing/decomposing into rectangles and triangles.
Finding the volume of a right rectangular prism with fractional edge lengths using unit cubes. Applying formulas V = lwh and V = Bh.
Drawing polygons in the coordinate plane given coordinates for vertices. Using coordinates to find the length of a side.
Representing three-dimensional figures using nets made up of rectangles and triangles. Using nets to find the surface area of these figures.
Statistics and Probability
Recognizing a statistical question (one that anticipates variability in the data).
Understanding that data sets have a distribution described by center, spread, and overall shape.
Recognizing that a measure of center summarizes data with a single number, while a measure of variation describes spread.
Displaying numerical data in plots on a number line (dot plots, histograms, box plots).
Summarizing numerical data sets in relation to their context (reporting number of observations; describing nature of attribute; giving measures of center - median/mean - and variability - interquartile range/mean absolute deviation; describing overall pattern and deviations).
Pedagogical Emphasis: Developing ratio reasoning through various representations, extending arithmetic to rational numbers, formalizing algebraic expressions and one-step equations, connecting geometry to measurement (area, volume, surface area), introducing foundational statistical concepts.
Grade 7 Mathematics Curriculum
Overall Focus: Developing understanding of and applying proportional relationships; developing understanding of operations with rational numbers; working with expressions and linear equations; solving problems involving scale drawings and informal geometric constructions; working with two- and three-dimensional shapes to solve problems involving area, surface area, and volume; drawing inferences about populations based on samples.
Key Mathematical Domains & Topics:
Ratios and Proportional Relationships
Computing unit rates associated with ratios of fractions.
Recognizing and representing proportional relationships between quantities (deciding if two quantities are proportional via tables, graphs, equations; identifying the constant of proportionality - unit rate).
Using proportional relationships to solve multistep ratio and percent problems (simple interest, tax, markups/markdowns, gratuities, commissions, fees, percent increase/decrease, percent error).
The Number System
Applying and extending previous understandings of addition and subtraction to add and subtract rational numbers (integers, fractions, decimals). Representing operations on a number line.
Applying and extending previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. Understanding rules for multiplying signed numbers. Interpreting products/quotients of rational numbers in real-world contexts. Understanding division by zero is undefined.
Solving real-world problems involving the four operations with rational numbers.
Expressions and Equations
Applying properties of operations to add, subtract, factor, and expand linear expressions with rational coefficients.
Understanding that rewriting an expression in different forms can shed light on the problem (e.g., a + 0.05a = 1.05a means "increase by 5%" is same as "multiply by 1.05").
Solving multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, decimals), using tools strategically. Applying properties of operations. Assessing reasonableness using estimation.
Using variables to represent quantities and constructing simple equations (px + q = r, p(x + q) = r) and inequalities (px + q > r, px + q < r) to solve problems. Graphing solution sets of inequalities.
Geometry
Solving problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
Drawing geometric shapes with given conditions (freehand, ruler/protractor, technology). Focus on triangles from three measures of angles or sides.
Describing the two-dimensional figures that result from slicing three-dimensional figures (plane sections of right rectangular prisms and pyramids).
Knowing the formulas for the area and circumference of a circle and using them to solve problems. Giving informal derivation of the relationship between circumference and area.
Using facts about supplementary, complementary, vertical, and adjacent angles in multi-step problems to write and solve simple equations for an unknown angle.
Solving real-world problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
Statistics and Probability
Understanding that statistics can be used to gain information about a population by examining a sample; generalizations are valid only if the sample is representative. Understanding random sampling.
Using data from a random sample to draw inferences about a population with an unknown characteristic. Generating multiple samples to gauge variation in estimates/predictions.
Informally assessing the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between centers (e.g., using mean or median) as a multiple of a measure of variability (e.g., MAD or IQR).
Using measures of center and variability for numerical data from random samples to draw informal comparative inferences about two populations.
Understanding that probability is a number between 0 and 1 expressing likelihood.
Approximating probability by collecting data (experimental probability) and predicting relative frequency.
Developing probability models (uniform and non-uniform). Determining probabilities of events.
Finding probabilities of compound events using organized lists, tables, tree diagrams, and simulation. Understanding independent and dependent events (introduced).
Pedagogical Emphasis: Deepening understanding of proportionality, solidifying operations with all rational numbers, moving towards multi-step algebraic thinking, connecting geometry with algebra (angle equations), developing informal statistical inference and probability reasoning.
Grade 8 Mathematics Curriculum
Overall Focus: Formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations; grasping the concept of a function and using functions to describe quantitative relationships; analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem.
Key Mathematical Domains & Topics:
The Number System
Knowing that numbers that are not rational are called irrational. Understanding informally that every number has a decimal expansion; rational numbers terminate or repeat, irrational numbers do not.
Using rational approximations of irrational numbers to compare their size, locate them on a number line, and estimate the value of expressions (e.g., π²).
Expressions and Equations
Knowing and applying the properties of integer exponents to generate equivalent numerical expressions (e.g., 3² × 3⁻⁵ = 3⁻³ = 1/3³ = 1/27).
Using square root and cube root symbols to represent solutions to equations like x² = p and x³ = p, where p is a positive rational number. Evaluating square roots of small perfect squares and cube roots of small perfect cubes. Knowing √2 is irrational.
Using numbers expressed in scientific notation to estimate very large or very small quantities and to express how many times as much one is than the other. Performing operations with numbers in scientific notation.
Graphing proportional relationships, interpreting the unit rate as the slope of the graph. Comparing two proportional relationships represented differently.
Using similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane. Deriving the equation y = mx for a line through the origin and y = mx + b for a line intercepting the y-axis at b.
Solving linear equations in one variable (including cases with one solution, infinitely many solutions, or no solutions). Solving linear equations with rational number coefficients, including equations requiring expanding expressions using the distributive property and collecting like terms.
Analyzing and solving pairs of simultaneous linear equations (systems of linear equations). Understanding solutions correspond to points of intersection. Solving systems algebraically (substitution, elimination) and by inspection. Solving real-world problems leading to two linear equations.
Functions
Understanding that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs (input, output).
Comparing properties (e.g., rate of change, initial value) of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
Interpreting the equation y = mx + b as defining a linear function whose graph is a straight line. Giving examples of functions that are not linear.
Geometry
Verifying experimentally the properties of rotations, reflections, and translations (lines to lines, line segments to line segments of same length, angles to angles of same measure, parallel lines to parallel lines).
Understanding that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations. Describing a sequence that exhibits congruence.
Describing the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
Understanding that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations. Describing a sequence that exhibits similarity.
Using informal arguments to establish facts about the angle sum and exterior angle of triangles, about angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.
Explaining a proof of the Pythagorean Theorem and its converse.
Applying the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
Applying the Pythagorean Theorem to find the distance between two points in a coordinate system.
Knowing the formulas for the volumes of cones, cylinders, and spheres and using them to solve real-world problems.
Statistics and Probability
Constructing and interpreting scatter plots for bivariate measurement data to investigate patterns of association (clustering, outliers, positive/negative association, linear/non-linear association).
Knowing that straight lines are widely used to model relationships between two quantitative variables. For scatter plots suggesting linear association, informally fitting a straight line and informally assessing the model fit (judging closeness of data points to the line).
Using the equation of a linear model to solve problems in the context of bivariate data, interpreting the slope and intercept.
Understanding that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Constructing and interpreting two-way tables. Using relative frequencies to describe possible association.
Pedagogical Emphasis: Transitioning to more formal algebraic reasoning, introducing the core concept of a function, developing spatial reasoning through transformations and the Pythagorean Theorem, exploring relationships in bivariate data.
This curriculum provides a framework. Effective implementation requires flexibility, differentiation to meet diverse student needs, ongoing formative assessment, and the use of rich tasks that promote deep mathematical thinking.
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