Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16

Content Area Mathematics Grade Level 7th Grade

Standard Grade Level Expectations (GLE) GLE Code

1. Number Sense, Properties, and Operations

1. Proportional reasoning involves comparisons and multiplicative relationships among ratios

MA10-GR.7-S.1-GLE.1

2. Formulate, represent, and use algorithms with rational numbers flexibly, accurately, and efficiently

MA10-GR.7-S.1-GLE.2

2. Patterns, Functions, and Algebraic Structures

1. Properties of arithmetic can be used to generate equivalent expressions MA10-GR.7-S.2-GLE.1

2. Equations and expressions model quantitative relationships and phenomena MA10-GR.7-S.2-GLE.2

3. Data Analysis, Statistics, and Probability

1. Statistics can be used to gain information about populations by examining samples MA10-GR.7-S.3-GLE.1

2. Mathematical models are used to determine probability MA10-GR.7-S.3-GLE.2

4. Shape, Dimension, and Geometric Relationships

1. Modeling geometric figures and relationships leads to informal spatial reasoning and proof

MA10-GR.7-S.4-GLE.1

2. Linear measure, angle measure, area, and volume are fundamentally different and require different units of measure

MA10-GR.7-S.4-GLE.2

Colorado 21st Century Skills Critical Thinking and Reasoning: Thinking Deeply, Thinking Differently Information Literacy: Untangling the Web Collaboration: Working Together, Learning Together Self-Direction: Own Your Learning Invention: Creating Solutions

Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

Module Titles Length of Unit Dates

Ratios and Proportional Relationships 29 Days 8/22 – 9/30 Acc: 8/29 - 9/21 (17 days)

Rational Numbers 24 Days 10/1 – 11/4 Acc: 9/22 – 10/10 (13 days)

Expressions and Equations 29 Days 11/7 – 12/20 Acc: 10/11 – 11/8 (20 days)

Percent and Proportional Relationships 20 Days 1/4 – 2/1 Acc: 11/9 – 11/22 (12 days)

Statistics and Probability 15 Days 2/2 – 2/24 Acc: 11/28 – 12/13 (11 days)

Geometry 14 Days 2/27 – 3/17 Acc: 1/4 – 1/31 (19 days)

Module 4, 5 and 6 revisited 35 Days 3/27 – 5/16

Invention

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16

MATHEMATICAL PRACTICES

MATHEMATICAL PRACTICE 1: 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.

MATHEMATICAL PRACTICE 2: 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 decontextualize—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.

MATHEMATICAL PRACTICE 3: 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.

MATHEMATICAL PRACTICE 4: 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,

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16 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.

MATHEMATICAL PRACTICE 5: 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.

MATHEMATICAL PRACTICE 6: 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.

MATHEMATICAL PRACTICE 7: 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.

MATHEMATICAL PRACTICE 8: 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.

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16

Name of Assessment

Purpose of Assessment

Location of Assessment

Suggestions

Performance Assessments

OPTIONAL

Pre and Post of each topic with in each module. These are useful for data cycles.

Schoology:

follow these clicks: Groups, Secondary Teachers, Resources, Math, 7th Grade, Assessments, Pre/Post

If you want to condense and create one comprehensive pre and post assessment for each module, the suggestion would be to use only question 1 from each pre and post-test within the desired module.

Accelerated:

When using the performance assessments, give only question 2 for the pre-test to check for understanding. If question 2 has no access for them, they can do question 1 to identify misconceptions. For the post-test, only give question 2. If you only want to give one pre and one post-test for each module concerning all Topics, you may condense each Topic pre and post-test using all of the question 2s.

End of Module Common Assessments

REQUIRED

Post assessment used to assess student understanding of priority standards for modules.

School City:

District Tab, 2016-2017, Math Grade 7 Collections

Testing window will be open for all assessments at the beginning of the school year. Testing windows will close 10 days after the module completion date on the cover of the curriculum guide.

General classes and accelerated classes will take the same assessment.

Prerequisite Assessments

OPTIONAL

Used to assess student understanding of the foundational standards for the priority standards in each module.

School City:

District Tab, 2016-2017, Math Grade 7 Collections, Prerequisite Assessments

These assessments are useful for data cycles in general 7th Grade math classes. They help to identify holes in learning from prior grades.

The assessments are not necessary in accelerated classes.

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16 Suggested Big Idea Module 3: Expressions and Equations

Content Emphasis Cluster Use properties of operations to generate equivalent expressions

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Solve real-life and mathematical problems involving angle measure, area, surface area, and

Volume

Mathematical Practices MP.1. Make sense of problems and persevere in solving them.

MP.2. Reason abstractly and quantitatively.

MP.3. Construct viable arguments and critique the reasoning of others.

MP.4. Model with mathematics.

MP.5. Use appropriate tools strategically.

MP.6. Attend to precision.

MP.7. Look for and make use of structure.

MP.8. Look for and express regularity in repeated reasoning.

Common Assessment End of Module Assessment

Graduate Competency Prepared graduates use critical thinking to recognize problematic aspects of situations, create mathematical

models, and present and defend solutions

Prepared graduates understand quantity through estimation, precision, order of magnitude, and comparison.

The reasonableness of answers relies on the ability to judge appropriateness, compare, estimate, and

analyze error

CCSS Priority Standards Cross-Content

Connections Writing Focus Language/Vocabulary Misconceptions

CCSS.MATH.CONTENT.7.EE.B.3 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. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets

Literacy Connections

RST.6-8.4

Determine the

meaning of

symbols, key terms,

and other domain-

specific words and

phrases as they are

used in a specific

scientific or

technical context

relevant to grades

6-8 texts and

topics.

RST.6-8.5

Writing Connection

WHST.6-8.2

Write

informative/explanatory

texts, including the

narration of historical

events, scientific

procedures/

experiments, or

technical processes.

a. Introduce a

topic clearly,

previewing what is to

follow; organize

ideas, concepts, and

information into

Academic Vocabulary- apply, convert, represent,

identify, interpret,

expanded form , standard

form factored form,

coefficient, circle, diameter,

circumference, radius, pi,

circular region or disk

Technical Vocabulary- Equivalent, expressions,

inequalities, properties of

operations, addition,

subtraction, multiplication,

division, factoring,

expansion, arithmetic

solution strategy, algebraic

solution strategy, arithmetic

As students begin to build and

work with expressions

containing more than two

operations, students tend to set

aside the order of operations. For

example having a student

simplify an expression like 8 +

4(2x - 5) + 3x can bring to light

several misconceptions. Do the

students immediately add the 8

and 4 before distributing the 4?

Do they only multiply the 4 and

the 2x and not distribute the 4 to

both terms in the parenthesis? Do

they collect all like terms 8 + 4 –

5, and 2x + 3x? Each of these

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16 a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

CCSS.MATH.CONTENT.7.EE.B.4 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. CCSS.MATH.CONTENT.7.G.B.6 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.

Analyze the

structure an author

uses to organize a

text, including how

the major sections

contribute to the

whole and to an

understanding of

the topic.

RST.6-8.7

Integrate

quantitative or

technical

information

expressed in words

in a text with a

version of that

information

expressed visually

(e.g., in a

flowchart, diagram,

model, graph, or

table).

RST.6-8.8

Distinguish among

facts, reasoned

judgment based on

research findings,

and speculation in a

text.

broader categories as

appropriate to

achieving purpose;

include formatting

(e.g., headings),

graphics (e.g., charts,

tables), and

multimedia when

useful to aiding

comprehension.

b. Develop the

topic with relevant,

well-chosen facts,

definitions, concrete

details, quotations, or

other information and

examples.

c. Use appropriate

and varied transitions

to create cohesion and

clarify the

relationships among

ideas and concepts.

d. Use precise

language and domain-

specific vocabulary to

inform about or

explain the topic.

e. Establish and

maintain a formal

style and objective

tone.

f. Provide a concluding

statement or section

that follows from and

supports the

information or

operations, algebraic

equations, correctness,

algebraic manipulations,

operation, both sides,

negative number, reverse

L.6-8.6

Acquire and use

accurately grade-

appropriate general

academic and domain-

specific words and

phrases; gather

vocabulary knowledge

when considering a

word or phrase

important to

comprehension or

expression.

L.6-8.4

Determine or clarify

the meaning of

unknown and multiple-

meaning words and

phrases choosing

flexibly from a range

of strategies.

show gaps in students’

understanding of how to simplify

numerical expressions with

multiple operations.

Students may believe:

Pi is an exact number rather than

understanding that 3.14 is just an

approximation of pi. Many

students are confused when

dealing with circumference

(linear measurement) and area.

This

confusion is about an attribute

that is measured using linear

units (surrounding) vs. an

attribute that is measured using

area units (covering).

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16

explanation presented.

WHST.6-8.4

Produce clear and

coherent writing in

which the development,

organization, and style

are appropriate to task,

purpose, and audience.

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16

Strand(s): EE Topic A. Use Properties of Operations to Generate Equivalent Expressions (7.EE.A.1, 7.EE.A.2) November 7 – November 14 (6 days) ACCELERATED PACE and SUGGESTIONS: Eliminate lesson 3. (5 days)

Previous Grade Standard Standard(s) for Grade/Course: Next Grade Standard

6.EE.A.3 Apply the properties of operations to generate equivalent expressions. 6.EE.A.4 Identify when two expressions are equivalent

Priority Standard CCSS.MATH.CONTENT.7.EE.B.4 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. Focus Standards: CCSS.MATH.CONTENT.7.EE.A.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

8.EE.7A

Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). 8.EE.7B Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

Changes Changes

In 6th grade students are fluent with one step equations, in 7th they become fluent at solving 2 step equations. Order of operation is extended from evaluating numerical values to using them to construct and deconstruct equations. Students move from a very concrete model of distributing and factoring (area model/tape diagrams) to a more fluent procedural method of distributing 𝑎(𝑏 + 𝑐) = 𝑎𝑏 +𝑎𝑐 and begin to include rational coefficients.

In 8th grade students begin to solve more complex multi-step linear equations and must be able to determine the type of solution (zero, one or infinite). They also extend linear equations to systems of equations by substitution and elimination.

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16

Anchor Problem: Preparing the Learner - make sure you pretest the following concepts and address these before beginning this module and spend the time shoring these up before you begin this module. The bottom line is that students have to be exposed to how to solve 2 step equations BEFORE beginning this unit. Take the time you need to make every single student can do this. Other skills required to solve 2 step equations could also include: Evaluating using order of operation, Simplifying expressions – combining like terms, distributing integer values, Solving one step equations, Translating expressions (math to English and English to math - more than, less than, sum, difference, is, was, per, in addition to, etc.) Inverse operations (How would you “undo” addition? What is the reciprocal of 10? What is the additive inverse of 5?) This is definitely a unit many teachers discussed splitting and offering a 3A common assessment just on the algebraic equations and a 3B assessment for the surface area and volume portions. Let’s talk about it if it is still a problem. Big Ideas for Module 3 Topic A

Equivalent expressions (Proving them equivalent 3 different ways: modeling, substituting, creating identical expressions Recall I can deconstruct an equation using the order of operation.

Ex: 2𝑥 + 3 = 7

Deconstruct the equation 2(3𝑥 + 1) − 2 = 42

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16

1 X 2 Times 2 3 Plus 3 4 Equals 7

I can determine whether a value is a solution to an

equation by substituting and proving both expressions have equal value

I can distribute a product to create a sum or difference and

I can factor a sum or difference into a product of a rational number and a quantity

I can determine if two expressions are equivalent by evaluation, representation, and algebraic simplification.

Rewrite 3(𝑥 − 1) as an equivalent sum.

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16 Making Connections

I can write an expression that represents a contextual situation; or translate from words to mathematical notation and mathematical notation to words.

Write the sum and then rewrite the expression in standard form by removing parentheses and collecting like terms.

a. 6 and 𝑝 − 6

b. 10𝑤 + 3 and – 3

c. −𝑥 − 11 and the opposite of – 11

d. The opposite of 4𝑥 and 3 + 4𝑥

e. 2𝑔 and the opposite of (1 − 2𝑔)

Write the product and then rewrite the expression in standard form by removing parentheses and collecting like terms.

f. 7ℎ − 1 and the multiplicative inverse of 7

g. The multiplicative inverse of −5 and 10𝑣 – 5

h. 9 − 𝑏 and the multiplicative inverse of 9

i. The multiplicative inverse of 1

4 and 5𝑡 −

1

4

j. The multiplicative inverse of −1

10𝑥 and

1

10𝑥−

1

10

Teacher Ideas for Interaction Eureka

In Grade 6, students interpreted expressions and equations as they reasoned about one-variable equations (6.EE.A.2). This module consolidates and expands upon students’ understanding of equivalent expressions as they apply the properties of operations (associative, commutative, distributive) to write expressions in both standard form (by expanding products into sums) and in factored form (by expanding sums into products). They use linear equations to solve unknown angle problems and other problems presented within context to understand that solving algebraic equations is all about the numbers. It is assumed that a number already exists to satisfy the equation and context; we just need to discover it. A number sentence is an equation that is said to be true if both numerical expressions evaluate to the same number; it is said to be false otherwise. Students use the number line to understand the properties of inequality and recognize when to preserve the inequality and when to reverse the inequality

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16 when solving problems leading to inequalities. They interpret solutions within the context of problems. Students extend their sixth-grade study of geometric figures and the relationships between them as they apply their work with expressions and equations to solve problems involving area of a circle and composite area in the plane, as well as volume and surface area of right prisms. In this module, students discover the most famous ratio of all, 𝜋, and begin to appreciate why it has been chosen as the symbol to represent the grades 6–8 mathematics curriculum, A Story of Ratios.

To begin this module, students will generate equivalent expressions using the fact that addition and multiplication can be done in any order with any grouping and will extend this understanding to subtraction (adding the inverse) and division (multiplying by the multiplicative inverse) (7.EE.A.1). They extend the properties of operations with numbers (learned in earlier grades) and recognize how the same properties hold true for letters that represent numbers. Knowledge of rational number operations from Module 2 is demonstrated as students collect like terms containing both positive and negative integers.

An area model is used as a tool for students to rewrite products as sums and sums as products and can provide a visual representation leading students to recognize the repeated use of the distributive property in factoring and expanding linear expressions (7.EE.A.1). Students examine situations where more than one form of an expression may be used to represent the same context, and they see how looking at each form can bring a new perspective (and thus deeper understanding) to the problem. Students recognize and use the identity properties and the existence of inverses to efficiently write equivalent expressions in standard form (2𝑥 + (−2𝑥) + 3 = 0 + 3 =3)(7.EE.A.2). By the end of the topic, students have the opportunity to practice Module 2 work on operations with rational numbers (7.NS.A.1, 7.NS.A.2) as they collect like terms with rational number coefficients (7.EE.A.1).

Topic A focuses on generating equivalent expressions. L1,L2, L3 provide good practice and discussion for combining like terms and delves into the concept of inverse as it relates to combining

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16 like terms (adding the opposite, multiplying by the reciprocal). Students are taught to both model and evaluate to prove expressions are equivalent. L4-5 focuses on distributing and factoring using the area model to generate equivalent expressions. L6 extends combining like terms and distributing with rational coefficients. Throughout this entire topic, have students show the expressions are equivalent 3 different ways: modeling, evaluating, and simplifying algebraically. Blended Resources, Personal Learning Resources, Differentiated Learning Resources CCSS Math Resources Common Core stations 7th grade Quia (google Quia “solving 2 step equation word problems” for jeopardy, rags to riches, matching, concentration, or quizzes) what number makes the equation true Pre-Assessment topics AB Representations:

“Number tricks” – the inverse method Area Model Tape Diagrams Hands on equations 3D geometric models, 2D geometric models

Vocabulary

An Expression in Expanded Form (description) (An expression that is written as sums (and/or differences) of products whose factors are numbers, variables, or variables raised to whole number powers is said to be in expanded form. A single number, variable, or a single product of numbers and/or

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16

variables is also considered to be in expanded form.)

An Expression in Standard Form (description) (An expression that is in expanded form where all like-terms have been collected is said to be in standard form.)

An Expression in Factored Form (middle school description) (An expression that is a product of two or more expressions is said to be in factored form.)

Coefficient of the Term (The number found by multiplying just the numbers in a term together is called the coefficient of the term.)

Circle (Given a point 𝐶 in the plane and a number 𝑟 > 0, the circle with center 𝐶 and radius 𝑟 is the set of all points in the plane that are distance 𝑟 from the point 𝐶.)

Diameter of a Circle (The diameter of a circle is the length of any segment that passes through the center of a circle whose endpoints lie on the circle. If 𝑟 is the radius of a circle, then the diameter is 2𝑟.)

Circumference (The length around a circle.)1

Pi (The number pi, denoted 𝜋, is the value of the ratio given by the circumference to the diameter, that is, 𝜋 = (circumference)/(diameter).)

Circular Region or Disk (Given a point 𝐶 in the plane and a number 𝑟 > 0, the circular region (or disk) with center 𝐶 and radius 𝑟 is the set of all points in the plane whose distance from the point 𝐶 is less than or equal to 𝑟. The interior of a circle with center 𝐶 and radius 𝑟 is the set of all points in the plane whose distance from the point 𝐶 is less than 𝑟.)

Familiar Terms and Symbols2

Variable (middle school description) letter than represents and unknown value

Numerical Expression (middle school description) 2

1 “Distance around a circular arc” is taken as an undefined term in G-CO.1. .

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16

or more numbers including one or more operation

Value of a Numerical Expression – the value of the expression when all variables are replaced with numerical values

Expression (middle school description)- 2 or more numbers or variables including one or more operation

Linear Expression – an expression that contains a variable

Equivalent Expressions – 2 expressions that when all variables are replaced with numerical values the values of the expressions are equal

Equation – 2 expressions including an equal sign

Number Sentence – an equation including all numbers and at least one operation and an equal sign that creates equal expressions

True or False Number Sentence – when substituting values in for all variables and the expression on the left side of the equal sign has the same value as the value of the expression on the right side of the equal side. If they do not, it is a false number sentence.

Truth Values of a Number Sentence – these are values which make the equation or inequality true

Identity – the identity for addition is zero because 3+0=3, the identity didn’t change. For multiplication it is 1 because 3x1=3.

Term – a single value or variable that could be added or subtracted with other terms

Distribute- allows you to write a product as a sum or difference

Factor-a number that when multiplied by another number creates a product.

Properties of Operations (distributive, commutative, associative)

Inequality – a sentence that produces multiple solutions because answers are greater than or less

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16

than a given number

Figure-combination of points, lines, or planes

Segment – part of a line with two endpoints

Length of a Segment – the absolute value of the difference between the two points

Measure of an Angle – the portion of a 360 degree circle bounded by two radii

Adjacent Angles- two angles that share a same side (ray)

Vertical Angles – angle formed by two intersecting lines

Triangle – a 3 sided polygon

Square – a four sided polygon

Right Rectangular Prism – a 3 dimensional object with 6 faces where the top and bottom meet the side faces at 90 angles

Cube – a 3 dimensional object with 6 congruent faces

Surface of a Prism – the faces of the prism (surface area – the number of square units it takes to cover the surface)

Probing Questions : Why does order matter when simplifying expressions? Which comes first multiply or divide?

Why did we use the associative and commutative properties of addition?

Can we use any order, any grouping when subtracting expressions? Explain.

How does 3(𝑥 + 𝑦) = 3𝑥 + 3𝑦

What is the opposite of a quantity? What is an inverse?

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16 Priority Standards:

Strand(s): EE, GB Topic B: Solve Problems Using Expressions, Equations, and Inequalities (7.EE.B.3, 7.EE.B.4, 7.G.B.5) November 15 – November 30 (9 days) ACCELERATED PACE and SUGGESTIONS: Combine 8-9, 10-11, and 14-15. (6 days)

Previous Grade Standard Standard(s) for Grade/Course: Next Grade Standard

6.EE.B.6 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.B.7 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.

CCSS.MATH.CONTENT.7.EE.B.3 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. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For CCSS.MATH.CONTENT.7.EE.B.4.A Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width? Focus Standard: CCSS.MATH.CONTENT.7.G.B.5 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.

8.EE.7b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. 8.EE.8 Analyze and solve pairs of simultaneous linear equations.

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16

Changes Changes

In 6th grade students are fluent with one step equations, in 7th they become fluent at solving 2 step equations/inequalities. Equation solving then extends into multiple steps requiring distributing and combining like terms. Students also practice solving equations through geometry curricula involving triangle sum theorem, linear pairs, supplementary and complementary angles, and vertical angles. There is lots of geometry vocabulary that must be taught while students are practicing solving equations.

In 8th grade students develop fluency around solving multistep linear equations and then extend into solving systems of linear equations algebraically with substitution and elimination methods. Students extend the geometry content by solving for angles formed by parallel lines cut by a transversal.

Prepare the Learner: the lower the student, the more they need modeling. Please do what is best for students not yourself. Be willing to be flexible and offer several methods of solving although our primary method is the “inverse” method.

Big Ideas for Module 3 Topic B: (approximately 2 weeks)

Solving multi-step equations; solving problems by writing and solve multi-step problems

Recall/Skills I can FLUENTLY solve 2 step equations and

inequalities I can solve a multistep linear equation.

• Solve: 5

4𝑛+5=20

Solve 1

2𝑥+3>2 and graph your solution on a number line.

−13 = 5(1 − 14𝑚) − 2𝑚

Making Connections I can solve for a single variable in real world

geometry formulas. Example: Given the volume, length and width of a rectangular prism, I can find the height.

• A rectangle is twice as long as wide. One way to write an expression to find the perimeter would be 𝑤𝑤+𝑤𝑤+2𝑤𝑤+2𝑤𝑤. Write the expression in two other ways.

Solution: 6𝑤 OR 2(𝑤)+2(2𝑤).

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16

I can solve real world problems by writing a linear equation and solving it.

I can find the measures of missing angles by solving

an equation.

•

An equilateral triangle has a perimeter of 6𝑥+15. What is the length of each of the sides of the triangle? Solution: 3(2𝑥+5), therefore each side is 2𝑥+5 units long. ∠𝐴 𝑎𝑛𝑑 ∠𝐵 are supplementary angles. 𝑚∠𝐴 = 130°, 𝑚∠𝐵 = (𝑥 + 10)°. Find

the value of 𝑥.

Teacher Ideas for Interaction Eureka

In Topic B, students use linear equations and inequalities to solve problems. They continue to use bar diagrams from earlier grades where they see fit but will quickly discover that some problems would more reasonably be solved algebraically (as in the case of large numbers). Guiding students to arrive at this realization on their own develops the need for algebra. This algebraic approach builds upon work in Grade 6 with equations (6.EE.B.6, 6.EE.B.7) to now include multi-step equations and inequalities containing rational numbers (7.EE.B.3, 7.EE.B.4). Students solve problems involving consecutive numbers, total cost, age comparisons, distance/rate/time, area and perimeter, and missing angle measures. Solving equations with a variable is all about numbers, and students are challenged with the goal of finding the number that makes the equation true. When given in context, students recognize that a value exists, and it is simply their job to discover what that value is. Even the angles in each diagram have a precise value, which can be checked with a protractor to ensure students that the value they find does indeed create a true number sentence.

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16 Topic B is all about solving lots of equations and having students practice writing expressions and equations from context. This is where students must develop fluency solving 2 step equations/inequalities. Engage provides sprint exercises for this purpose in L12 and L15. L7-9 focus on solving problems in context by writing equations and solving them. L10-11 focus specifically on geometry problems where students are using equations to find measures of angles and sides. L12-15 extend the concept of solving equations (where there is a single solution (no solution and infinite solutions have not been discussed) to inequalities where there are infinite solutions. Take time developing the graphing portion where solutions are represented by dots on a number line. Model 2𝑥 = 6, |3|, 2𝑥 < 6. Good activity provided in L15 for inequalities. Lots and lots of practice. Blended Resources, Personal Learning Resources, Differentiated Learning Resources CCSS Math Resources Common Core stations 7th grade Quia (google “quia solving multi-step equations” for jeopardy, rags to riches, matching, concentration, or quizzes)

Inside Mathematics • Growing Staircases (click here) • Once Upon A Time (click here) MARS Shell Center • Fencing (click here) Youcubed.org Number transformer challenge: writing an equation from cubes/pictures (click here)

Post Assessment topic AB module 3

Pre-Assessment topic C

Probing Questions :

What are some of the methods used to write products as sums?

Why is the inequality reversed when you multiply or divide by a negative?

What does it mean for an inequality to be preserved? What does it mean for the inequality to be reversed?

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16

Why is it possible to have more than one solution?

What is the difference between consecutive integers and consecutive even or odd integers?

What inequality symbol represents “is more than”? Why?

What is a solution set of an inequality?

How do you know when you need to use an inequality instead of an equation to model a given situation?

Is it possible for an inequality to have exactly one solution? Exactly two solutions? Why or why not?

When does a greater than become a less than? How are equations and expressions similar? How are the different?

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16 Priority Standards:

Strand(s):EE, GB Topic C: Use Equations and Inequalities to Solve Geometry Problems (7.G.B.4, 7.G.B.6) December 1 – December 15 (11days) ACCELERATED PACE and SUGGESTIONS: Condense lessons 21-24 to one day. Combine 22-23. (7 days)

Previous Grade Standard Standard(s) for Grade/Course: Next Grade Standard

6.G.A.1

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.A.2 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.A.4 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.

CCSS.MATH.CONTENT.7.G.B.6 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. Focus Standards: CCSS.MATH.CONTENT.7.G.B.4 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.

The next time these topics are revisited is in High School Geometry

Changes Changes

In 6th grade students learn area is the number of squares that covers and volume is the number of cubes that fills. They find surface area of 3D objects by making nets. Volume is limited to right prisms. In 7th, students learn to define 𝜋 as circumference divided

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16

by diameter instead of a memorized 3.14. Students must memorize the formulas for area of circle, triangle, and parallelogram and the formula for circumference of a circle. The work around area extends to composite areas and area/perimeter on the coordinate plane, while the work around surface area continues with nets. The work around volume continues to focus on right prisms but extends to composite solids.

Recall/Skills- Level I

I can find the area of any polygon by dividing it into rectangular or triangular pieces and finding the sum of the areas.

I can find composite area of figures.

Find the area of the trapezoid shown below using the formulas for rectangles and triangles.

Find the area of the shaded region.

Choose one of the figures shown below and write a step by step procedure for determining the area. Find another person that chose the same figure as you did. How are your procedures the same and different? Do they yield the same result?

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16

I can find the volume of right rectangular prisms

A cereal box is a rectangular prism. What is the volume of the cereal box? What is the surface area of the cereal box? (Hint: Create a net of the cereal box and use the net to calculate the surface area.) Make a poster explaining your work to share with the class.

Making Connections

I can define pi as the quotient of the circumference and the diameter.

(Students measure the circumference and diameter of several circular objects in the room (clock, trash can, door knob, wheel, etc.). Students organize their information and discover the relationship between circumference and diameter by noticing the pattern in the ratio of the measures. Students write an expression that could be used to find the circumference of a circle with any diameter and check their expression on other circles.)

I can give an informal derivation of the relationship between the circumference and area of a circle. (Students will use a circle as a model to make several equal parts as you would in a pie model. The greater the number of cuts, the better. The pie pieces are laid out to form a shape similar to a parallelogram. Students will then write an expression for the area of the parallelogram related to the radius (note: the length of the base of the parallelogram is half the circumference, or πr, and the height is r, resulting in an area of πr2. Extension: If students are given the circumference of a circle, could they write a formula to determine the circle’s area or given the area of a circle, could they write the formula for the circumference?)

Pi lab

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16

I can solve real life problems using area or

volume.

The seventh grade class is building a mini golf game for the school carnival. The end of

the putting green will be a circle. If the circle is 10 feet in diameter, how many square feet of grass carpet will they need to buy to cover the circle? How might you communicate this information to the salesperson to make sure you receive a piece of carpet that is the correct size?

Preparing the Learner Make sure you pre-assess students to determine if they understand basic concepts of area, perimeter, surface area, and volume. Teacher Ideas for Interaction Eureka

In Topic C, Students continue work with geometry as they use equations and expressions to study area, perimeter, surface area, and volume. This final topic begins by modeling a circle with a bicycle tire and comparing its perimeter (one rotation of the tire) to the length across (measured with a string) to allow students to discover the most famous ratio of all, pi. Activities in comparing circumference to diameter are staged precisely for students to

recognize that this symbol has a distinct value and can be approximated by 22

7 or 3.14 to give students an intuitive sense of the relationship that exists.

In addition to representing this value with the 𝜋 symbol, the fraction and decimal approximations allow for students to continue to practice their work with rational number operations. All problems are crafted in such a way to allow students to practice skills in reducing within a problem, such as using 22

7 for finding circumference with a given diameter length of 14 cm, and recognize what value would be best to approximate a solution. This

understanding allows students to accurately assess work for reasonableness of answers. After discovering and understanding the value of this special ratio, students will continue to use pi as they solve problems of area and circumference (7.G.B.4).

In this topic, students derive the formula for area of a circle by dividing a circle of radius 𝑟 into pieces of pi and rearranging the pieces so that they are

lined up, alternating direction, and form a shape that resembles a rectangle. This “rectangle” has a length that is 1

2 the circumference and a width of 𝑟.

Students determine that the area of this rectangle (reconfigured from a circle of the same area) is the product of its length and its width: 1

2𝐶 ∙ 𝑟 =

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16 1

22𝜋𝑟 ∙ 𝑟 = 𝜋𝑟2 (7.G.B.4). The precise definitions for diameter, circumference, pi, and circular region or disk will be developed during this topic with

significant time being devoted to student understanding of each term.

Students build upon their work in Grade 6 with surface area and nets to understand that surface area is simply the sum of the area of the lateral faces and the base(s) (6.G.A.4). In Grade 7, they continue to solve real-life and mathematical problems involving area of two-dimensional shapes and surface area and volume of prisms, e.g., rectangular, triangular, focusing on problems that involve fractional values for length (7.G.B.6). Additional work (examples) with surface area will occur in Module 6 after a formal definition of rectangular pyramid is established.

Lessons 16-18 focus heavily on moving students from 𝜋 = 3.14 to =𝐶

𝑑 . Students should do the pi lab where they measure the circumference and

diameter of multiple real life circular objects and start to compare the results of 𝐶

𝑑 . Discussion should include inaccuracy of measurement but students

should get the idea. They also focus heavily on applications of circumference and area in context (including quarter circles and half circles) and on area of composites involving circles. The focus is still on solving equations so they are provided parts of the geometric formulas and must solve for a single variable. Make sure students are memorizing basic area and circumference formulas. Lesson 19 moves students to the coordinate plane to derive area formulas for parallelogram and triangles and students begin to understand height must be perpendicular to the base while lesson 20 provides many composite area problems to practice with. Lessons 21-22 focus on surface area of solids via making nets and finding the sum of the areas. Lessons 23-24 focus on volume of prisms by finding the area of the base and multiplying by the height while lessons 25-26 involve solving real problems involving both surface area and volume. Students will find both fractional parts of solids as well as the remaining space when a solid is inside a solid. Problems can range from very low to very high. Blended Resources, Personal Learning Resources, Differentiated Learning Resources CCSS Math Resources Common Core stations 7th grade Quia (google “quia area or circumference of circle” for jeopardy, rags to riches, matching, concentration, or quizzes) MARS Shell Center • Using Dimensions: Designing a Sports Bag Post Assessment topic C Module 3 Module 3 Common Assessment

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/10/16 Probing Questions :

In the pi lab, why didn't you get exactly 3.1415.... for your constant of proportionality? Is it possible to get a number that is greater than 3.5 for pi (if you measured accurately)? Explain. How are the diameter and the radius related? Students often say that area is the 'inside' of a circle and circumference is the 'outside.' Why isn't this accurate? Explain your thinking? Give real life examples of perimeter or circumference? Why isn't circumference units squared? How is the area of a circle like the area of a rectangle? If you have the diameter of a circle how can you find the area? How is using grid paper to find area easier or harder? In what situations is knowing surface area useful? What was your thought process as you took the net and attempted to draw a sketch of the shape without knowing what the actual object is? How is surface area like covering and volume like filling? How is area related to surface are and nets? When is knowing volume helpful? What strategy do you use to determine the face that is the Base of a 3D object? When is it better to think about Volume as Bh instead of lwh?