6 Grade Mathematics - P.S.I.S. 187 Hudson Cliffs School · 2019-10-26 · Understand a rational...

6th Grade Mathematics

Unit 1: Multiplying and Dividing Time: September-October

Common Core Standards

Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.

6.NS.1

Multiply and divide multi-digit numbers using the standard algorithm. 6.NS.2

Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

6.NS.3

Find greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor.

6.NS.4

Unit 2: Fractions Time: 1 October

Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.

5.NF.1

Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

5.NF.4

Interpret multiplication as scaling (resizing), by: Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.

5.NF.5

Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

5.NF.6

Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

5.NF.7

Unit 3: Ratios Time: October-November

Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.

6.RP.1

Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations

6.RP.2

Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

6.RP.3

Unit 4: Introduction to Rational Numbers Time: November-December

Understand that positive and negative numbers are used together to describe quantities having opposite directions or values; use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation

6.NS.5

Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

6.NS.6

Understand ordering and absolute value of rational numbers. 6.NS.7

Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

6.NS.8

Unit 5: Expressions Time: December-January

Write and evaluate numerical expressions involving whole-number exponents. 6.EE.1

Write, read, and evaluate expressions in which letters stand for numbers. 6.EE.2

Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

6.EE.3

Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).

6.EE.4

Unit 6: Equations Time: January-February

Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

6.EE.5

Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

6.EE.6

Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

6.EE.7

Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

6.EE.8

Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.

6.EE.9

Unit 7: Geometry Time: March

Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems

6.G.1

Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real world and mathematical problems.

6.G.2

Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

6.G.3

Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

6.G.4

Unit 8: Statistics Time: March-April

Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.

6.SP.1

Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

6.SP.2

Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

6.SP.3

Display numerical data in plots on a number line, including dot plots, histograms, and box plots. 6.SP.4

Summarize numerical data sets in relation to their context. 6.SP.5

Unit 9: Integers Time: May-June

Apply properties of operations as strategies to add and subtract rational numbers 7.NS.1d

Review of the coordinate plane 6.NS.6

Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative

7.NS.1

Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

7.NS.1c

Apply and extend previous understandings of multiplication and division to multiply and divide rational numbers

7.NS.2

Review of Order of Operations with integers 6.EE.2


Unit 1: Integers Time: 2 Weeks


Apply properties of operations as strategies to add and subtract rational numbers 7.NS.1

Review of the coordinate plane 6.NS.6

Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative

7.NS.1

Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

7.NS.1

Apply and extend previous understandings of multiplication and division to multiply and divide rational numbers

7.NS.2

Review of Order of Operations with integers 6.EE.2

Unit 2: Decimals Time: 1 Week

Addition and subtraction of positive and negative decimals 7.NS.1, 7.NS.3

Multiplication of positive and negative decimals 7.NS.2, 7.NS.3

Division of positive and negative decimals 7.NS.2, 7.NS.3

Mixed review of all 4 operation with positive and negative decimals 7.NS.1 , 7.NS.3,

7.NS.2

Unit 3: Fractions Time: 3 Weeks


Add and subtract fractions with like and unlike denominators (positive and negative)

o Convert between mixed numbers and improper fractions.

o Use a number line to model computations.

o Reduce fractions to simplest form.

7.NS.1, 7.NS.3

Multiplication of rational and/or mixed numbers 7.NS.2, 7.NS.3

Division of rational and/or mixed numbers 7.NS.2, 7.NS.3

Converting fractions to decimals and connecting that terminating and repeating decimals are rational numbers

7.NS.2, 7.NS.3

Unit 4: Introduction to Algebra Time: 2 Weeks

Review of one-step rational equations with addition, subtraction, multiplication, and division 6.EE.7

Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically

7.EE.3

Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations to solve problems by reasoning about the quantities

7.EE.4

Unit 5: Algebra Continued Time: 4 Weeks

Translate sentences into algebraic equations: construct simple (1- and 2-step) equations and inequalities of the form px + q = r and p(x + q) = r, to solve problems

7.EE.4

Writing “let” statements with 1- and 2-step equations 7.EE.4

Combining like terms: Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related

7.EE.2

Simplify expressions using the distributive property 7.EE.1

Factoring using GCF: 5x + 30= 5(x + 6). 7.EE.1

Unit 6: Inequalities Time: 2 Weeks

Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities

7.EE.4

7.EE.4

7.EE.4

Unit 7: Ratio and Proportion Time: 4 Weeks

Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.

7.RP.3

Use proportional relationships to solve multistep ratio problems

7.RP.3

Recognize and represent proportional relationships between quantities

Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.

Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship

7.RP.2

between the total cost and the number of items can be expressed as t = pn.

Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate

Solve 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.

7.G.1

Unit 8: Percent Time: 2 Weeks

Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error

7RP.3

Unit 9: Statistics Time: 2 Weeks

Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.

Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.

Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.

7.SP.1

7.SP.2

7.SP.3

7.SP.4

Unit 10: Probability Time: 3 Weeks

Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.

Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.

Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.

7.SP.1

7.SP.2

7.SP.3

7.SP.4

Unit 11: Circles Time: 1 Week

Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle

7.G.4

NYS MATH TEST

Unit 12: Introduction to Geometry Time: 2 Weeks


Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle

7.G.2

Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure

7.G.5

Unit 13: Area, Surface Area, & Volume Time: 2 Weeks

Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids

7.G.3

Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms

7.G.6


Unit CCLS Integrated Algebra TIME

Unit 1: Algebra 5 WEEKS

Solve 1-Step Equations 8.EE.7 AA22—Solve all types of linear equations in one variable.

Solve 2-Step Equations 8.EE.7 AA22—Solve all types of linear equations in one variable. AA25—Solve equations involving fractional expressions.

Combine Like Terms 8.EE.7b

Distributive Property 8.EE.7b AN1—Identify and apply the properties of real numbers (closure, communitive, distributive, identity, inverse)

Variables on both sides 8.EE.7b AA22—Solve all types of linear equations in one variable.

Multi-Step Equations 8.EE.7 AA22—Solve all types of linear equations in one variable. AA25—Solve equations involving fractional expressions.

No solutions, infinite solutions, one solutions, etc.

8.EE.7a AA21—Determine whether a given value is a solution to a given linear equation in one variable or linear inequality in one variable.

Translate algebraic expressions and equations

8.EE.7 AA1—Translate verbal to algebraic expression AA2—Write a verbal expression to match a mathematical expression AA3—distinguish between expression & equation AA4--Translate verbal to algebraic equation/inequality

Define variables 8.EE.7

Applications of equations in solving verbal problems

8.EE.7 AA6—Write algebraic equations/inequalities to represent a situation


Unit 2: Ratios and Proportions 3 WEEKS

Identifying ratios and rates/unit rates

8.EE.5 AM1—Calculate rates using appropriate units AM2—Solve problems involving conversions within measurement systems, given the relationship between the units

Identify a proportion using cross products, equivalent fractions, and tables

8.EE.5 AN5—Solve algebraic problems arising from situations that involve fractions, decimals, percents, and proportionality/direct variation.

AA26—Solve algebraic proportions in one variable which result in linear or quadratic equations

Solve a proportion 8.EE.5 AN5—Solve algebraic problems arising from situations that involve fractions, decimals, percents, and proportionality/direct variation.


Identify direct variation using tables



Applications of proportions and direct variation (include currency, scale drawings, better buy, and other applications)




Similar triangle application 8.G.5 AA36—Write the equation of a line parallel to the x- or y- axis.

AA38—Determine if two lines are parallel, given their equations in any form.

Unit 3: Introduction to Functions 5 WEEKS

Function vocabulary 8.F.1

Identify a function using tables and vertical line test

8.F.1 AG3--Determine when a relation is a function, by examining ordered pairs and inspecting graphs of relations

Transform equations isolating the “y” variable

8.F.3

Graph a function using a table of values

8.F.3

Derive a function rule from a given table of values

8.F.1

Identify types of slope (i.e. positive, negative, undefined, zero)

8.F.2

Calculate slope from a given table, two points, or line

8.F.2, 8.F.4, 8.EE5

AA33—Determine the slop of a line, given the coordinates of two points on the line.

Understand the “rise over run” using similar right triangles can be used to establish the slope and that it is constant for a non-vertical line

8.EE.6 AA32—Explain slope as a rate of change between dependent and independent variables.


Identify x- and y- intercepts from a table graph, and equation

8.F.2

Given the slope and y-intercept, derive the equation

8.F.4 AA34—write the equation of a line, given its slope and the coordinates of a point on the line.

AA36—Write the equation of a line, given the coordinates of two points on a line.

Given a linear graph, derive the equation

8.F.4

Graph the function using slope-intercept form

8.F.3

Discuss parallel linear functions 8.F.2 AA36—Write the equation of a line parallel to the x- or y- axis.

AA38—Determine if two lines are parallel, given their equations in any form.

Graph direct variation (proportional relationships); interpret the unit rate as the slope of the graph

8.EE.5

8.F.5

AA32—Explain slope as a rate of change between dependent and independent variables.

Unit 4: Linear Systems of Equations

2 WEEKS

Solve a linear system of equations graphically (integer solutions) and check Include one, no and infinite

8.EE.8a AG7—Graph and solve systems of linear equations and inequalitites with rational coefficients in two variables


solutions.

Solve a linear system of equations algebraically (use substitution, elimination, and choosing the most efficient method)

8.EE.8b AA10—Solve systems of linear equation in two variables algebraically

Solve a verbal system of linear equations graphically and/or algebraically

8.EE.8b AA7—Analyze and solve verbal problems whose solution requires solving systems of linear equations in two variables

Unit 5: Plane Geometry 2 WEEKS

Complementary and Supplementary

8.G.5

Vertical and adjacent 8.G.5

Parallel lines cut by a transversal (alternate interior, alternate exterior, corresponding, vertical, supplementary)

8.G.5

Perpendicular Lines 8.G.5

Sum of interior angles of a triangle (triangle can appear in parallel line diagram)

8.G.5

Sum of exterior angles of a triangle

8.G.5

Measure of the exterior angle of 8.G.5


a triangle equals the sum of the non-adjacent interior angles

Unit 6: Transformations 2 WEEKS

Identify the four transformations (reflections, rotations, translations, and dilations)

8.G.1

Preservation of non-preservation of properties with transformations (i.e., orientation , size, shape, angle measure, etc.)

8.G.1

Manipulate shapes to investigate the transformations (use technology to explore transformations)

8.G.1

Know proper transformation vocabulary and notation (pre-image, image, original, prime, double prime)

8.G.1

Identify and perform a sequence of transformations (using any combination of the four transformations)

8.G.2

Write a statement of congruence between the pre-image and the image

8.G.2


Know and be able to apply “rules” for reflections, rotations, translations, and dilations (reflection over any line, rotation of 90, 180, 270 degrees clockwise and counterclockwise)

8.G.3

Understand the differences between similarity and congruence

8.G.4

Determine the scale factor of a dilation (greater and less than 1)

8.G.4

Discuss Angle-Angle criteria with dilations to prove similarity of shapes

8.G.5

Unit 7: Exponents 2 WEEKS

Distinguish among base, power, and exponent

8.EE.1

Develop and use the Laws of Exponents (product/quotient, negative exponents, zero exponents, power of powers)

8.EE.1 AN6—Evaluate expressions involving factorial, absolute value, and exponential expressions

AA6—Analyze and solve verbal problems that involve exponential growth and decay

AA12—Multiply and divide monomial expressions with a common base, using properties of exponents (integral exponents only)

Apply a combination of Laws of 8.EE.1


Exponents to show equivalency

Write a number in scientific notation

8.EE3

Convert a number from scientific to standard form

8.EE.3

Estimate and compare numbers in scientific notation; express how many times as much one is than the other (use the ≈ symbol).

8.EE.3

Multiply and divide with numbers expressed in scientific notation

8.EE.4 AN4—Understand and use scientific notation to compute products and quotients of numbers.

Choose the appropriate units of measure based on size context.

8.EE.4

Unit 8: Geometry—Volume 2 WEEKS

Identify a cone, cylinder, and sphere

8.G.9

Memorize the formulas for a cone, cylinder, and sphere

8.G.9

Use the formulas to calculate volume

8.G.9 AG2—Use formulas to calculate volume and surface area of rectangular solids and cylinders

Solve application problems 8.G.9 AG2—Use formulas to calculate volume and surface area of rectangular solids and cylinders


Unit 9: Statistics 2 WEEKS

Statistics vocabulary (i.e. univariate, bivariate, correlation, outlier, independent, dependent, scatter plot, line of best fit, qualitative, quantitative)

8.SP.1 AS1—Categorize data as qualitative or quantitative

AS2—Determine whether the data to be analyzed is univariate or bivariate

AS3—Determine when collected data or display of data may be biased

AS8—Construct manually a reasonable line of best fit for a scatter plot and determine the equation of that line

AS10—Evaluate published reports and graphs that are based on data by considering: experimental designs, appropriateness of the data analysis, and the soundness of the conclusions

AS12—Identify the relationship between the independent and dependent variables from a scatter plot (positive, negative, or none)

AS13—Understand the difference between correlation and causation

AS14—Identify variables that might have a correlation but not a causal relationship

Construct a scatter plot from given data

8.SP.1 AS7—Create a scatter plot of bivariate data

AS8—Construct manually a reasonable line of best fit for a scatter plot and determine the equation of that line

Interpret patterns of association between two quantities

8.SP.1 AS12—Identify the relationship between the independent and dependent variables from a scatter plot (positive, negative, or none)

AS13—Understand the difference between correlation and


causation


Interpolate data from a scatter plot

8.SP.2 AS12—Identify the relationship between the independent and dependent variables from a scatter plot (positive, negative, or none)



Analyze data in a scatter plot and informally fit a line of best fit; approximate its slope and interpret its meaning in context

8.SP.2, 8.SP.3 AS8—Construct manually a reasonable line of best fit for a scatter plot and determine the equation of that line




Compare and contrast data in a given scatter plot

8.SP.2, 8.SP.3 AS12—Identify the relationship between the independent and dependent variables from a scatter plot (positive, negative, or none)




Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects

S.SP.4 AS1—Categorize data as qualitative or quantitative

AS2—Determine whether the data to be analyzed is univariate or bivariate

AS3—Determine when collected data or display of data may be biased




Use relative frequencies calculated for rows or columns to describe possible association between the two variables

8.SP.4 AS5—Construct a histogram, cumulative frequency histogram, and a box-and-whisker plot, given a set of data

AS9—Analyze and interpret a frequency distribution table or histogram, a cumulative frequency distribution table or histogram, or a box-and-whisker plot

Unit 10: Rational and Irrational Numbers

1 WEEK

Graphically organize the following number systems: Natural and Counting Numbers (i.e. whole numbers, integers, rational and irrational numbers and real numbers)

8.NS.1


Know definitions and classify rational and irrational numbers

8.NS.1

Convert fractions into decimals 8.NS.1

Convert repeating and termination decimals into fractions

8.NS.1

Approximate a square root between two integers and locate them on a number line

8.NS.2

Compare and order rational and irrational numbers

8.NS.2

Find square roots and cube roots (using technology); know how to square and cube a number)

8.EE.2

Use square root and cube root symbols to express solutions to equations

8.EE.2 AN2—Simplify radical terms (no variable in the radicand)

AN3—Perform the four arithmetic operations using like and unlike radical terms and express the result in simplest form

Explain a proof of the Pythagorean Theorem and its converse

8.G.6 AA45—Determine the measure of a third side of a triangle using the Pythagorean Theorem, given the lengths of any two sides

Given the side lengths of a triangle, prove that it is (or is not) a right triangle (converse)



Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real0world and mathematical problems in two and three dimensions


Apply the Pythagorean Theorem to find the distance between two points in a coordinate system


6 Grade Mathematics - P.S.I.S. 187 Hudson Cliffs School · 2019-10-26 · Understand a rational...

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Transcript of 6 Grade Mathematics - P.S.I.S. 187 Hudson Cliffs School · 2019-10-26 · Understand a rational...