Middle School Curriculum Map - Garfield Public Schools October 2014/7t… · · 2015-02-05Middle...

Garfield Middle School Aligned to the 2013 Common Core Curriculum Content Standards

ENGAGING STUDENTS • FOSTERING ACHIEVEMENT • CULTIVATING 21ST

CENTURY GLOBAL SKILLS

Middle School Curriculum Map

7th

Grade Math Apps Marking Period 1

Topic Chapters Number of Blocks Dates

Algebraic Reasoning Chapter 1 4 9/9 – 9/18

PRE-TEST 1 9/25

Integers and Rational

Numbers Chapter 2 9 9/19– 10/17

Applying Rational

Numbers Chapter 3 9 10/18– 11/15

MP 1

ASSESSMENT

Chapters 1, 2 and

half of 3 1 11/6

Marking Period 2 Topic Chapters Number of Blocks Dates

Proportional

Relationships Chapter 4 8 11/18– 12/11

Graphs Chapter 5 4 12/12-12/20

Percents Chapter 6 5 1/2 -1/15

MP 2

ASSESSMENT

Chapters- half of

3,4,5,6 1 1/24


Collecting, Displaying

and Analyzing Data Chapter 7 4 1/26– 1/31

Geometric Figures Chapter8 6 2/3 – 2/28

Measurement and

Geometry Chapter 9 6 3/3-3/18

NJ ASK REVIEW 4 3/19 – 3/27

MP 3

ASSESSMENT Chapters 7-9 3/28



Probability Chapter 10 10 5/5 – 5/30

Multi-step Equations

and Inequalities Chapter 11 5 6/2- 6/13

MP 4

ASSESSMENT Chapters 10-11 1 6/16

Topic Chapters Number of Blocks Dates


Graphing Lines Chapter 8 7 5/5 – 5/23

Data, Predications and

Linear Functions Chapter 9 6 5/27- 6/11

MP 4

ASSESSMENT Chapters 8 and 9 1 6/12




Unit Overview

Content Area: Math Applications

Unit Title: Algebraic Reasoning, Integers and Rational Numbers

Target Course/Grade Level 7th

Duration: 27 Blocks

Description

In this section students will apply prior knowledge of algebraic properties and number sense as they learn to compare and order integers, and to add, subtract, multiply, and divide them. Further, students will extend the order of operations to include exponents. Lastly, they learn why fractions and decimals are rational numbers.

Concepts & Understandings

Concepts

Order of operations.

Properties of numbers.

Variables and algebraic expressions.

Translating words into math

Simplifying algebraic expressions.

Integers.

Adding integers.

Subtracting integers.

Multiplying and dividing integers.

Solving equations containing integers.

Equivalent fractions and decimals.

Comparing and ordering rational numbers.

Understandings

Student understandings from the concepts include:

Simplify numerical expressions involving order of operations and exponents.

Using variables and symbols to translate words into math.

Simplifying algebraic expressions using properties of addition and multiplication.

Comparing and ordering integers and rational numbers.

Converting between fractions and decimals mentally, on paper, and with a calculator.

Using models to add, subtract, multiply and divide integers.

Learning Targets

CPI Codes

MA.CC - EE.7.01 MA.CC - EE.7.02 MA.CC - EE.7.0

,MA.CC -EE.7.04

MA.CC - G.7.03 MA.CC - NS.7.01 MA.CC - NS.7.02 MA.CC-NS. 7.03 MA.CC - RP.7.02

Math Practices-see addendum

21st Century Themes and Skills

See addendum




Guiding Questions What is the order of operations?

What are the properties of rational numbers?

How do you simplify numerical expressions? How do you evaluate algebraic expressions? How do you translate words into numbers, variable and operations?

Explain how to combine like terms? What is the difference between whole numbers and integers?

Explain absolute value? What are the rules for adding integers? Explain how to subtract integers?

What are the rules for multiplying and dividing integers? Explain which operation are the inverse of each other?

Why is it necessary to perform the same operation on both sides of the equation?

How do you write fraction as a decimal?

Explain how to use place value to convert a decimal into a fraction?

What needs to be the same before you can add or subtract fractions? How do you compare fractions with different denominators?

Unit Results

Students will ... The student will use the order of operations to simplify numerical expressions.

The student will identify properties of rational numbers and use them to simplify numerical expressions.

The students will evaluate algebraic expressions.

The students will translate words into numbers, variables, and operations.

The students will simplify algebraic expressions.

The student will compare and order integers and determine absolute value.

The student will add and subtract integers.

The student will multiply and divide integers.

The student will solve one step equations with integers. The student will write fractions as decimals, and vice versa, and determine whether a decimal is terminating or

repeating.

The student will compare and order fractions and decimals.

Suggested Activities The following activities can be incorporated into the daily lessons:

Do Now

Exploration worksheet

Challenge worksheet

Problem Solving worksheet

Journal Writing

Exit Ticket

Quiz




Test

Projects

Unit Overview


Unit Title Applying Rational Numbers


Duration: 19 Blocks

Description Students will build on fraction concepts to estimate and compute with fractions and mixed numbers. Students will evaluate, write, and solve equations that involve fraction, mixed numbers and decimals.


Concepts

Adding and subtracting decimals.

Multiplying decimals.

Dividing decimals.

Solving equations containing decimals.

Adding subtracting fractions.

Multiplying fractions and mixed numbers.

Dividing fractions and mixed numbers.

Solving equations containing fractions.

Understandings


Use addition, subtraction, multiplication, and division to solve problems involving fractions and decimals.

Solve equations with rational numbers.

Learning Targets

CPI Codes

MA.CC - EE.7.04 MA.CC - NS.7.03 MA.CC - NS.7.02 MA.CC - NS.7.01



See addendum Guiding Questions

Describe the steps necessary to add and subtract decimals?

How can you check an answer when adding or subtracting decimals?

Compare multiplication of decimals with multiplication of integers.

How do you determine the decimal placement in a product when multiplying decimals?

What are the rules when dividing an integer by a decimal?

Explain how to solve each type of decimal equation; addition, subtraction, multiplication, and division.

How do you obtain a common denominator?

Describe the process for subtracting/adding fractions with different denominators.

Describe how to multiply and divide mixed number and a fraction.

How do you determine the steps of a product?

Compare the steps used in multiplying mixed numbers with those used in dividing mixed numbers?

What is the process of solving an equation that involves mixed numbers?




Unit Results Students will ...

Add and subtract decimals.

Multiply decimals.

Divide decimals and integers by decimals.

Solve one step equations that contain decimals.

Add and subtract fractions.

Multiply fractions and mixed numbers.

Divide fractions and mixed numbers.

Solve one step equations that contain fraction.

Be assessed on students’ mastery of concepts and skills.

Suggested Activities

The following activities can be incorporated into the daily lessons:

Do Now


Challenge worksheet


Exit Ticket

Journal Writing

Quiz

Test

Projects




Unit Overview


Unit Title: Proportions and Ratios


Duration: 15 Blocks

Description Students will learn to find equivalent ratios and to see if ratios can form a proportion. They use unit rates to solve proportions involving variable, and they use proportions to solve problems involving scale. Further, they solve proportions to find missing lengths in missing figures.


Concepts

Rates

Identifying and writing proportions

Solving Proportions

Similar figures and proportions

Using similar figures

Scale drawings and scale models

Understandings


Use division to find unit rates and ratios in proportional relationships

Estimate and find solutions to application problems involving proportional relationships.

Use critical attributes to define similarity.

Use ratios and proportions in scale drawings and scale models.

Learning Targets

CPI Codes

MA.CC - RP.7.01 MA.CC - RP.7.02 MA.CC - G.7.01




Explain how you can tell whether a rate represents a unit rate?

What does it mean for ratios to be proportional?

Explain how the term cross product can help you remember how to solve a proportion. What are the characteristics of corresponding angles and corresponding sides in similar figures? How do you know if two shapes are proportional? How do you find a missing side length for one or two similar figures? How do find the actual height of an object whose scale drawing is smaller than the object?

Unit Results

Students will ... Find and compare unit rates, such as average speed and unit price.

Find equivalent ratios and identify proportions.




Solve proportions by using cross products.

Use ratios to determine if two figures are similar.

Use similar figures to find unknown measures.

Understand ratios and proportions in scale drawings and use ratios and proportions with scale.

Be assessed on students’ mastery of concepts and skills. Suggested Activities


Do Now


Challenge worksheet


Exit Ticket

Journal Writing

Quiz

Test

Projects




Unit Overview


Unit Title: Graphs

Target Course/Grade Level: 7th Grade

Duration: 7 Blocks

Description :

In this section students will study graphing in the coordinate plane. They explore linear relationships and the concept of slope as they graph point and lines in all four quadrants. Furthermore, they also investigate no-linear relationships, translations, reflections, rotations, and symmetry.


Concepts

The Coordinate Plane

Interpreting graphs

Slope and rates of change

Direct Variation

Understandings

Student understandings from the concepts include

Plotting and identifying ordered pairs of integers on a coordinate plane.

Graphing to demonstrate relationships between data sets.

Using rates of change to solve problems.

Writing and graphing linear equations to solve problems.

Learning Targets

CPI Codes

MA.CC - RP.7.02

MA.CC - RP.7.01 Math Practices-see addendum



Explain the meaning of a horizontal segment on a graph that compares distance to time?

Describe a real world situation that could be represented by a graph that has connected lines or curves?

Compare constant and variable rates of change? Describe a line with a negative slope?

Explain how to use the table of data to check whether the relationship between two variables is a direct variation.

Describe how to recognize a direct variation from an equation, from a table, and from a graph? Unit Results

Students will ... Plot and identify ordered pairs on a coordinate plane.

Relate graphs to situations.




Determine the slope of a line and recognize constant and variable rates of change?

Identify, write, and graph an equation of direct variation.




Do Now


Challenge worksheet


Exit Ticket

Journal Writing

Quiz

Test

Projects




Unit Overview


Unit Title: Percents

Target Course/Grade Level: 7th

Duration: 11 Blocks

Description :

Students will work with percents, including percents less than 1 and greater than 100. They begin by comparing percent to fractions and decimals. Then, using real-world applications they solve problems involving percents, including finding a percent of a number and finding a percent of change.


Concepts

Fractions, decimals, and percents

Estimating with percents

Using properties with rational numbers

Percent of change

Applications of percents

Simple interest

Understanding


Modeling and estimating percents

Writing equivalent fractions, decimals and percents, including percents less than 1 and greater than 100

Solving percent problems involving discounts, sales tax, tips, profit, and percent of change, commission, and simple interest.

Comparing fractions, decimals, and percents.

Learning Targets

CPI Codes

MA.CC,EE.7.02 MA.CC,EE.7.03 MA.CC - NS.7.01

MA.CC - NS.7.02 MA.CC - RP.7.03




Explain two methods for writing a decimal as a percents and for writing a fraction as a percent? Describe two ways to estimate with percents and tell when each is more appropriate to use?

Explain how the multiplication property of equality is used to write an equivalent equation without fractions or decimals?

Explain what is meant by 100 percent decrease? Explain how to find a price of an item if you know the total cost after 5 percent sales tax? How is finding commission similar to finding sales tax?




Explain the meaning of each variable in the interest formula? Name the variables in the simple interest formula that represent dollar amounts.

Unit Results

Students will ... Write decimals and fraction as fractions and percents.

Estimate percents.

Use properties of rational numbers to write equivalent expressions and equations.

Solve problems involving percent of change.

Find commission, sales tax, and percent of earnings.

Compute simple interest.




Do Now


Challenge worksheet


Exit Ticket

Journal Writing

Quiz

Test

Projects

.




Unit Overview


Unit Title: Collecting, Displaying, and Analyzing Data.


Duration: 9 Blocks

Description :

In this section students will build on their knowledge of graphing and statistics as they investigate different ways of obtaining and displaying data. They work with frequency tables and data displays like spreadsheets, histograms, stem-and-leaf plots, and scatter plots. Furthermore, they learn about sampling, surveys, and how to use data to make predictions.


Concepts

Mean median, mode, and range.

Box-and-whisker plots.

Populations and samples.

Understandings


Using an appropriate representation for displaying relationships among data.

Choosing among mean, median, mode, or range to describe a set of data.

Making inferences and convincing arguments based on analysis of data.

Learning Targets

CPI Codes

MA.CC.SP.7.01 MA.CC. SP 7.02, MA.CC. SP 7.03, MA.CC. SP 7.04




Explain how the outlier affects the mean, median, and mode of a data set?

Describe what you can tell about a data set from a box-and-whisker plot? Explain why it might be difficult to obtain a truly random sample of a very large population?

Unit Results

Students will ...




Find the mean, median, mode, and range of a data set.

Display and analyze data in box-and –whisker plots.

Compare and analyze sampling methods.




Do Now


Challenge worksheet


Exit Ticket

Journal Writing

Quiz

Test

Projects




Unit Overview


Unit Title: Geometric Figures

Target Course/Grade Level: 7th grade

Duration: 13 Blocks

Description :

In this section students will learn about the properties of lines and angles then, they classify angles according to their degree measures. They work with triangles, quadrilaterals, and other polygons, including congruent figures.


Concepts

Building blocks of geometry

Classifying angles

Line and angle relationships Angles in polygons Congruent figures

Understandings


Classifying pairs of angles as complementary or supplementary.

Identifying parallel and perpendicular lines. Using congruence to solve problems.

Learning Targets

CPI Codes

G.7.05 G.7.02 Math Practices –see addendum



How many ways can a line segment be named?

What are three different ways to classify an angle?

How do you find a complementary angle when one angle is given?

Explain why perpendicular lines can also be called intersecting lines?

Explain how to find the measure of an angle in a triangle when the measure of the two other angles are known?

Why are congruent figures always similar figures?

What does it mean when two polygons are congruent?

Unit Results

Students will ...

Identify and describe geometric figures Identify angles and angle pairs Identify parallel, perpendicular, and skew lines, and angles formed by a transversal.

Find the measures of angles in a polygon.




Identify congruent figures and use congruence to solve problem. Be assessed on students’ mastery of concepts and skills.


Do Now


Challenge worksheet


Exit Ticket

Journal Writing

Quiz

Test

Projects




Unit Overview


Unit Title : Measurement and geometry

Target Course/Grade Level:

Duration: 13 Blocks

Description :

In this section students will build on their knowledge of the geometric concepts to estimate and calculate areas of different types of polygons. Furthermore, they also calculate the circumference and area of circles, as well as the surface area and volume of prisms and cylinders.


Concepts

Perimeter and circumference.

Area of circles.

Area of irregular figures.

Three-dimensional figures.

Volume of prism and cylinders.

Surface area of prisms and cylinders.

Understandings


Finding the circumference and area of circles.

Finding the area of irregular figures. Finding the volume and surface area of three-

dimensional figures.

Learning Targets

CPI Codes

MA.CC,G.7.03,

MA.CC,G.7.04 MA.CC,G.7.06




Describe two ways to find a perimeter of a volleyball court?

How are the circumference of a circle and a perimeter of a polygon alike?

Compare finding the area of a circle when given a radius with finding an area when given a diameter? Describe two different ways to find the area of irregular figures?

Compare and contrast cylinders and prisms.

What is a cubic unit?

Compare and contrast the formulas for volume of a prism and volume of a cylinder. Explain how to use a net to find the surface area of a rectangular prism and a cylinder?


Find the perimeter of a polygon and the circumference of a circle.

Find the area of circles.

Find the area of irregular figures.

Identify various three-dimensional figures.




Find the volume of prisms and cylinders.

Find the surface area of prisms and cylinders.



Do Now


Challenge worksheet


Exit Ticket

Journal Writing

Quiz

Test

Projects




Unit Overview


Unit Title: Probability


Duration: 19 Blocks

Description :

In this section students will work with both theoretical and experimental probability. They find the probability of both simple and compound events. Furthermore, they conclude the chapter by learning about permutations and combinations.


Concepts

Probability

Experimental probability

Sample spaces

Theoretical probability

Making predictions.

Probability of independent and dependent events.

Combinations Permutations Probability of compound events

Understandings

Finding experimental and theoretical probabilities, including those of dependent and independent events.

Using lists and tree diagrams to find combinations and all possible outcomes of an experiment.

Using the fundamental counting principle and factorials to find permutations.

Learning Targets

CPI Codes

MA.CC,SP.7.05

MA.CC,SP.7.06 MA.CC,SP.7.07 MA.CC,SP.7.08

Math Practices –see addendum



Describe an event that has a probability of zero percent and an event that has a probability of 100 percent. Explain how experimental probability can be used to make predictions? Compare using a tree diagram and using the fundamental counting principle to find a sample space? Give an example of an experiment in which all of the outcomes are not equally likely?

Explain a difference between a prediction based on experimental probability and one based on theoretical probability?

Compare probabilities of independent and dependent events. Describe how combinations could help you find the probability of an event?




What are the possible ways to find the number of permutations of a group of objects? Explain how organized list, tree diagram, and tables help find the sample space in an experiment?


Use informal measures of probability Find experimental probability Use counting methods to determine all possible outcomes Find the theoretical probability of an event Use probability to predict events

Find the probability of an independent and dependent events

Find the numbers of possible combinations Find the number of possible permutations Find probability of compound events Be assessed on students’ mastery of concepts and skills.


Do Now


Challenge worksheet


Exit Ticket

Journal Writing

Quiz

Test

Projects




Unit Overview

Content Area: Math Application

Unit Title: Multi-Step equations and Inequalities


Duration: 11 Blocks

Description :

In this section students evaluate and write algebraic expressions and write and solve both one-step and two-step equations. Lastly, they draw upon their understandings of expressions and equations to graph, write, and solve inequalities.

.


Concepts

Solving two-step equations.

Solving multi-step equations.

Solving equations with variables on both sides.

Inequalities

Solving inequalities by adding or subtracting.

Solving inequalities by multiplying or dividing.

Solving multi-step inequalities

Understandings


Solving two-step and multi-step equations and equations with variables on both sides.

Reading, writing, and graphing inequalities on a number line.

Solving one-step and two-step inequalities Solving equations for a variable

Learning Targets

M.A.CC. EE 7.01,

M.A.CC. EE 7.04 Math Practices –see addendum


See addendum. Guiding Questions

Explain how you decide which inverse operations to use first when solving two-step equation?

Select a multi-step equation and challenge students to list the required steps.

Select an equation with variables on both sides and challenge students to decide which variable term to add or subtract.

Explain how to graph each type of compound inequalities.

Compare solving addition and subtraction equations with solving addition and subtraction inequalities.

Compare solving multiplication and division equations with solving multiplication and division inequalities.

Explain how to solve a simple two step inequality.

Unit Results

Students will ... Solve two- step equations

Solve multistep equations

Solve equations that have variables on both sides

Read and write inequalities and graph them on a number line

Solve one-step inequalities by adding or subtracting

Solve one-step inequalities by multiplying and dividing




Solve simple two- step inequalities



Do Now


Challenge worksheet

Exit Ticket


Journal Writing

Quiz

Test

Projects




MATH PRACTICES

1. CCSS.Math.Practice.MP1 Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem

and looking for entry points to its solution. They analyze givens, constraints, relationships,

and goals. They make conjectures about the form and meaning of the solution and plan a

solution pathway rather than simply jumping into a solution attempt. They consider analogous

problems, and try special cases and simpler forms of the original problem in order to gain

insight into its solution. They monitor and evaluate their progress and change course if

necessary. Older students might, depending on the context of the problem, transform

algebraic expressions or change the viewing window on their graphing calculator to get the

information they need. Mathematically proficient students can explain correspondences

between equations, verbal descriptions, tables, and graphs or draw diagrams of important

features and relationships, graph data, and search for regularity or trends. Younger students

might rely on using concrete objects or pictures to help conceptualize and solve a problem.

Mathematically proficient students check their answers to problems using a different method,

and they continually ask themselves, “Does this make sense?” They can understand the

approaches of others to solving complex problems and identify correspondences between

different approaches.

2. CCSS.Math.Practice.MP2 Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem

situations. They bring two complementary abilities to bear on problems involving quantitative

relationships: the ability to de-contextualize—to abstract a given situation and represent it

symbolically and manipulate the representing symbols as if they have a life of their own,

without necessarily attending to their referents—and the ability to contextualize, to pause as

needed during the manipulation process in order to probe into the referents for the symbols

involved. Quantitative reasoning entails habits of creating a coherent representation of the

problem at hand; considering the units involved; attending to the meaning of quantities, not

just how to compute them; and knowing and flexibly using different properties of operations

and objects.

http://www.corestandards.org/Math/Practice/MP1





3. CCSS.Math.Practice.MP3 Construct viable arguments and critique the reasoning of

others.

Mathematically proficient students understand and use stated assumptions, definitions, and

previously established results in constructing arguments. They make conjectures and build a

logical progression of statements to explore the truth of their conjectures. They are able to

analyze situations by breaking them into cases, and can recognize and use counterexamples.

They justify their conclusions, communicate them to others, and respond to the arguments of

others. They reason inductively about data, making plausible arguments that take into

account the context from which the data arose. Mathematically proficient students are also

able to compare the effectiveness of two plausible arguments, distinguish correct logic or

reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is.

Elementary students can construct arguments using concrete referents such as objects,

drawings, diagrams, and actions. Such arguments can make sense and be correct, even

though they are not generalized or made formal until later grades. Later, students learn to

determine domains to which an argument applies. Students at all grades can listen or read

the arguments of others, decide whether they make sense, and ask useful questions to clarify

or improve the arguments.

4. CCSS.Math.Practice.MP4 Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems

arising in everyday life, society, and the workplace. In early grades, this might be as simple as

writing an addition equation to describe a situation. In middle grades, a student might apply

proportional reasoning to plan a school event or analyze a problem in the community. By high

school, a student might use geometry to solve a design problem or use a function to describe

how one quantity of interest depends on another. Mathematically proficient students who can

apply what they know are comfortable making assumptions and approximations to simplify a

complicated situation, realizing that these may need revision later. They are able to identify

important quantities in a practical situation and map their relationships using such tools as

diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those

relationships mathematically to draw conclusions. They routinely interpret their mathematical

results in the context of the situation and reflect on whether the results make sense, possibly

improving the model if it has not served its purpose.






5. CCSS.Math.Practice.MP5 Use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a mathematical

problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a

calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic

geometry software. Proficient students are sufficiently familiar with tools appropriate for their

grade or course to make sound decisions about when each of these tools might be helpful,

recognizing both the insight to be gained and their limitations. For example, mathematically

proficient high school students analyze graphs of functions and solutions generated using a

graphing calculator. They detect possible errors by strategically using estimation and other

mathematical knowledge. When making mathematical models, they know that technology can

enable them to visualize the results of varying assumptions, explore consequences, and

compare predictions with data. Mathematically proficient students at various grade levels are

able to identify relevant external mathematical resources, such as digital content located on a

website, and use them to pose or solve problems. They are able to use technological tools to

explore and deepen their understanding of concepts.

6. CCSS.Math.Practice.MP6 Attend to precision.

Mathematically proficient students try to communicate precisely to others. They try to use

clear definitions in discussion with others and in their own reasoning. They state the meaning

of the symbols they choose, including using the equal sign consistently and appropriately.

They are careful about specifying units of measure, and labeling axes to clarify the

correspondence with quantities in a problem. They calculate accurately and efficiently,

express numerical answers with a degree of precision appropriate for the problem context. In

the elementary grades, students give carefully formulated explanations to each other. By the

time they reach high school they have learned to examine claims and make explicit use of

definitions.






7. CCSS.Math.Practice.MP7 Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Young

students, for example, might notice that three and seven more is the same amount as seven

and three more, or they may sort a collection of shapes according to how many sides the

shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in

preparation for learning about the distributive property. In the expression x2 + 9x + 14, older

students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an

existing line in a geometric figure and can use the strategy of drawing an auxiliary line for

solving problems. They also can step back for an overview and shift perspective. They can

see complicated things, such as some algebraic expressions, as single objects or as being

composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive

number times a square and use that to realize that its value cannot be more than 5 for any

real numbers x and y.

8. CCSS.Math.Practice.MP8 Look for and express regularity in repeated reasoning.

Mathematically proficient students notice if calculations are repeated, and look both for

general methods and for shortcuts. Upper elementary students might notice when dividing 25

by 11 that they are repeating the same calculations over and over again, and conclude they

have a repeating decimal. By paying attention to the calculation of slope as they repeatedly

check whether points are on the line through (1, 2) with slope 3, middle school students might

abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when

expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the

general formula for the sum of a geometric series. As they work to solve a problem,

mathematically proficient students maintain oversight of the process, while attending to the

details. They continually evaluate the reasonableness of their intermediate results.






Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content

The Standards for Mathematical Practice describe ways in which developing student

practitioners of the discipline of mathematics increasingly ought to engage with the subject

matter as they grow in mathematical maturity and expertise throughout the elementary,

middle and high school years. Designers of curricula, assessments, and professional

development should all attend to the need to connect the mathematical practices to

mathematical content in mathematics instruction.

The Standards for Mathematical Content are a balanced combination of procedure and

understanding. Expectations that begin with the word “understand” are often especially good

opportunities to connect the practices to the content. Students who lack understanding of a

topic may rely on procedures too heavily. Without a flexible base from which to work, they

may be less likely to consider analogous problems, represent problems coherently, justify

conclusions, apply the mathematics to practical situations, use technology mindfully to work

with the mathematics, explain the mathematics accurately to other students, step back for an

overview, or deviate from a known procedure to find a shortcut. In short, a lack of

understanding effectively prevents a student from engaging in the mathematical practices.

In this respect, those content standards which set an expectation of understanding are

potential “points of intersection” between the Standards for Mathematical Content and the

Standards for Mathematical Practice. These points of intersection are intended to be weighted

toward central and generative concepts in the school mathematics curriculum that most merit

the time, resources, innovative energies, and focus necessary to qualitatively improve the

curriculum, instruction, assessment, professional development, and student achievement in

mathematics.

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