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Integrated Math I
Instructional Focus Documents
Introduction:
The purpose of this document is to provide teachers a resource which contains:
• The Tennessee grade level mathematics standards
• Evidence of Learning Statements for each standard
• Instructional Focus Statements for each standard
Evidence of Learning Statements:
The evidence of learning statements are guidance to help teachers connect the Tennessee Mathematics Standards with evidence of learning
that can be collected through classroom assessments to provide an indication of how students are tracking towards grade-level conceptual
understanding of the Tennessee Mathematics Standards. These statements are divided into four levels. These four levels are designed to
help connect classroom assessments with the performance levels of our state assessment. The four levels of the state assessment are as
follows:
• Level 1: Performance at this level demonstrates that the student has a minimal understanding and has a nominal ability to apply the
grade-/course-level knowledge and skills defined by the Tennessee academic standards.
• Level 2: Performance at this level demonstrates that the student is approaching understanding and has a partial ability to apply the
grade-/course-level knowledge and skills defined by the Tennessee academic standards.
• Level 3: Performance at this level demonstrates that the student has a comprehensive understanding and thorough ability to apply
the grade-/course-level knowledge and skills defined by the Tennessee academic standards.
• Level 4: Performance at this level demonstrates that the student has an extensive understanding and expert ability to apply the
grade-/course-level knowledge and skills defined by the Tennessee academic standards.
The evidence of learning statements are categorized in this same way to provide examples of what a student who has a particular level of
conceptual understanding of the Tennessee mathematics standards will most likely be able to do in a classroom setting.
Instructional Focus Statements:
Instructional focus statements provide guidance to clarify the types of instruction that will help a student progress along a continuum of
learning. These statements are written to provide strong guidance around Tier I, on-grade level instruction. Thus, the instructional focus
statements are written for level 3 and 4.
Quantities (N.Q)
Standard M1.N.Q.A.1 (Supporting Content) Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
Scope and Clarifications: (Modeling Standard) There are no assessment limits for this standard. The entire standard is assessed in this course.
Evidence of Learning Statements
Students with a level 1 understanding of this standard will most likely be able to:
Students with a level 2 understanding of this standard will most likely be able to:
Students with a level 3 understanding of this standard will most likely be able to:
Students with a level 4 understanding of this standard will most likely be able to: Choose units appropriately when solving a simple problem.
Choose a graphical representation to represent a real-world problem.
Choose a data display that describes the values and units in a problem.
Choose and interpret units appropriately when solving a simple problem.
Identify when units need to be converted to the same unit within a contextual problem.
Connect values to the units to represent given information.
Choose appropriate units in order to evaluate a formula, given an input value.
Choose an interpretation of the graph that represent a real-world problem.
Choose a data display with
Use units as a way to understand problems and to guide the solution path for a multi-step problem.
Choose and interpret units appropriately when solving a multi- step problem, including problems that contain real-world formulas.
Recognize the relationship between the units for all variables in a formula.
Choose and interpret the scale and the origin in graphs and data displays.
Determine the most appropriate data display based on the units given in a problem.
Explain if the information is represented appropriately using mathematical justification, given a numerical and/or a graphical representation of a real-world problem.
Create a real-world problem involving formulas and data represented either in a table or graph in which the data must be analyzed for appropriate units and scale. Explain the interpretation of the units, scale, and origin with respect to the contextual situation using precise mathematical vocabulary.
Standard M1.N.Q.A.2 (Supporting Content) Identify, interpret, and justify appropriate quantities for the purpose of descriptive modeling.
Scope and Clarifications: (Modeling Standard) Descriptive modeling refers to understanding and interpreting graphs; identifying extraneous information; choosing appropriate units; etc. There are no assessment limits for this standard. The entire standard is assessed in this course.
Evidence of Learning Statements
Students with a level 1 understanding of this standard will most likely be able to:
Students with a level 2 understanding of this standard will most likely be able to:
Students with a level 3 understanding of this standard will most likely be able to:
Students with a level 4 understanding of this standard will most likely be able to: Identify the units in a problem.
Connect the units to the values in a real-world problem.
Identify individual quantities in context of the real-world problem and label them with appropriate units.
Determine if quantities are labeled with the correct units in the context of a real-world problem.
Recognize extraneous information in a real-world problem.
Identify and interpret necessary information in order to select or create a quantity that models a real- world problem.
Explain the meaning of individual quantities in the context of the real- world problem.
Attend to precision when defining quantities and their units embedded in context.
Explain and justify the relationship between solutions to contextual problems and the values used to compute the solutions.
Appropriately interpret, explain the meaning of, and draw conclusions about the quantities in a real-world problems.
Identify, interpret, and justify complex information embedded in a real-word problem containing a variety of descriptors or units in order to solve contextual problems for the purpose of descriptive modeling.
Represent quantities in descriptive modeling situations and explain their relationship using numeric, algebraic, and graphical representations.
Students with a level 1 understanding of this standard will most likely be able to:
Students with a level 2 understanding of this standard will most likely be able to:
Students with a level 3 understanding of this standard will most likely be able to:
Students with a level 4 understanding of this standard will most likely be able to: Make observations about quantities given a graph or model.
Explain why information is extraneous in a real-world problem.
Instructional Focus Statements
Level 3:
In grades K-8, students developed an understanding of measuring, labeling values, and understanding how the value of a number relates to the described quantity. In the high school Numbers and Quantity (NQ) domain, students develop an understanding of reasoning quantitatively and solving problems requiring the evaluation of the appropriateness of the form in which quantities are provided. Instruction for this standard should be integrated with a wide variety of standards throughout the course. Students should extend their understanding of using appropriate quantities in descriptive modeling situations where they can make comparisons between two distinct quantities and justify the quantities appropriately in order to describe or to solve a contextual problem. Descriptive modeling refers to understanding and interpreting graphs; identifying extraneous information; choosing appropriate units; etc. Instruction should focus on providing opportunities for students to select appropriate quantities embedded in real-world contextual problems and attend to precision by describing the quantities in descriptive modeling situations. The study of dimensional analysis is an excellent avenue to help students understand how critical values, units, and quantities are used in interpreting information and modeling a real-world problem. Furthermore, students must be given opportunities to write and create appropriate labels for quantities and explain the meaning of the quantities in a context. Being able to identify, interpret, and justify quantities is a skill that will serve students well to have mastered during this course as this standard lays the foundation for using units as a way to understand problems.
Level 4:
Instruction should focus on providing opportunities for students to work with problems that have a variety of descriptors and units embedded in the context. Students should be asked to extend their knowledge of quantities by representing them in multiple formats such as a graphical representation of the given information, algebraic representation of the quantities, and multiple representations to predict or draw conclusions about the solution of the real-world problem. Instruction should provide opportunities for students to analyze and critique the interpretation of quantities in a descriptive modeling problem. Additionally, students should be given ample opportunities to design their own contextual problem in which they would have to use quantities appropriately in order to describe the modeled contextual situation.
As this is a modeling standard, students should solve contextual problems and be able to choose a level of accuracy of measurement quantities that is reasonable and makes sense to the contextual situation. For example, when solving a multi-step problem or using graphing technology, students should determine when it is appropriate and not appropriate to use precise values (values that are not rounded or truncated), rounded values, or truncated values. Students should be able to justify their reasoning for using certain values and explain why their choice is important with respect to the context. Additionally, students should experience different solution paths that involve using different forms of values and explain how accuracy does or does not have an impact on the solution within the context of the problem.
Instruction should focus on providing students a plethora of opportunities to use a variety of measurements including measuring tools and graphing technology. Students should have ample time to explore traditional, physical tools as well as electronic, and digital tools. During this exploration, class discussion should focus on helping students ascertain the difference between precision and accuracy and when it is appropriate to apply each of them in certain problems. Furthermore, instruction should be infused with a broad spectrum of different types of units that describe tiny to very large quantities. This is a modeling standard and students should make connections to other disciplines such as science.
Level 4:
Students have a great opportunity to support their understanding of this standard through the lens of a wide variety of other disciplines. Instruction should focus on providing students with experiences involving problem situations that interest them. Include inquiry with this standard and allow students ample time to explore repeated measurement in order to determine an acceptable level of accuracy when reporting quantities. Also, instruction should provide the opportunity for students to analyze and critique the level of accuracy chosen by others to report quantities.
This modeling standard is a great way to make connections to other disciplines, specifically science. An extension of this standard can include applying the concept of significant figures, especially in science related contexts. Additionally, in science contexts, students can apply their knowledge of significant digits and scientific notation to explore tasks and report quantities appropriately.
SEEING STRUCTURE in EXPRESSIONS (A.SSE)
Standard M1.A.SSE.A.1 (Major Work of the Grade) Interpret expressions that represent a quantity in terms of its context. M1.A.SSE.A.1a Interpret parts of an expression, such as terms, factors, and coefficients. M1.A.SSE.A.1b Interpret complicated expressions by viewing one or more of their parts as a single entity.
Scope and Clarifications: (Modeling Standard) For example, interpret P (1 + r) n as the product of P and a factor not depending on P. Tasks are limited to linear and exponential expressions, including related numerical expressions.
Evidence of Learning Statements
Students with a level 1 understanding of this standard will most likely be able to:
Students with a level 2 understanding of this standard will most likely be able to:
Students with a level 3 understanding of this standard will most likely be able to:
Students with a level 4 understanding of this standard will most likely be able to: Identify parts of an expression (i.e., factor, coefficient, term).
Define the formal definition of the terms: factor, coefficient, and term.
Define the formal definition of the term expression.
Label the single entities in an expression.
Recognize arithmetic operations in an expression in order to see the structure of the expression.
Understand and use the definitions of terms, factors, coefficients, and like terms in order to describe the structure of the individual parts of the expression.
Identify parts of an expression as a single entity.
Recognize that individual parts of an expression affect the whole expression.
State arithmetic operations performed within an expression.
Interpret parts of an expression (i.e., term, factor, coefficient) embedded in a real-world situation and explain each part in terms of the context.
Interpret parts of an expression (i.e,. term, factor, and coefficient) and explain each part in terms of the function the expression defines.
Explain the structure of an expression and how each term is related to the other terms by interpreting the arithmetic meaning of each term in the expression and recognizing when combining like terms is appropriate.
Interpret expressions in a variety of forms by explaining the relationship between the terms and the structure of the expression.
Interpret parts of complex expressions with varying combinations of arithmetic operations and exponents by viewing one or more of their parts as a single entity.
Write and interpret expressions that represent a real-world context and use the expressions to solve contextual problems.
Write expressions in a wide variety of formats and then for each
Level 4:
Students need to be presented with complex expressions that include a combination of different arithmetic operations and interpret in terms of a real- world context. The pinnacle of level 4 understanding is being able understand, interpret , and explain the relationship between equivalent representations of an expression. Students should be able to explain not only the expression in terms of a contextual situation, but also how each term within the expression connects back to the contextual situation. . Additionally, instruction should focus on relating expressions to real world contexts. For example, students should be given problems that describe contextual situations from multiple perspectives. Students should interpret the contextual situation for each individual perspective and write an expression that represents the context for each. Students should be challenged to interpret the meaning of the expressions created and use them to predict outcomes and solve problems. Instruction should expose students to multiple representations of the expressions by making connections between the equivalent expressions, which will in turn help students recognize the most useful form of an expression depending on context. Students should be challenged to justify why other formats are equivalent and which format is most relevant given the context of the problem.
Standard M1.A.SSE.B.2 (Major Work of the Grade) Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. A1.A.SSE.B.2a Use the properties of exponents to rewrite exponential expressions.
Scope and Clarifications: (Modeling Standard) For example, the growth of bacteria can be modeled by either f(t) = 3(t+2)^ or g(t) = 9(3t)^ because the expression 3(t+2)^ can be rewritten as (3t) (3^2 ) =9(3t). Tasks have a real-world context. As described in the standard, there is an interplay between the mathematical structure of the expression and the structure of the situation such that choosing and producing an equivalent form of the expression reveals something about the situation.
Evidence of Learning Statements
Students with a level 1 understanding of this standard will most likely be able to:
Students with a level 2 understanding of this standard will most likely be able to:
Students with a level 3 understanding of this standard will most likely be able to:
Students with a level 4 understanding of this standard will most likely be able to: Recognize an exponential expression.
Recognize properties of exponents.
Without a context, choose an equivalent form of an exponential expression.
Choose an equivalent form of an exponential expression and choose the properties used to transform the expression, from a real-world context.
Generate an equivalent form of an exponential expression and identify the properties of exponents used to generate the expression, From a real-world context.
Generate equivalent forms of an exponential expression, justify each transformation with a property, and explain the benefits of the equivalent expression, from a real- world context.
Instructional Focus Statements
Level 3:
The introduction of rational exponents and practice with the properties of exponents in high school further widens the field of operations students will be manipulating. It is important to note that this is a modeling standard and that the exponential expressions should be embedded in real-world situations. This provides a context for seeing structure in the expression and allows students to see when and why it is beneficial to view them in different forms.
Additionally, it’s important to note that the focus is not on writing expressions in simplest form as there really is no simplest form. The form that expressions are written in should be driven by what is being done with the expression in the first place.
CREATING EQUATIONS* (A.CED)
Standard M1.A.CED.A.1 (Major Work of the Grade) Create equations and inequalities in one variable and use them to solve problems.
Scope and Clarifications: (Modeling Standard) i. Tasks are limited to linear or exponential equations with integer exponents. ii. Tasks have a real-world context. iii. In the linear case, tasks have more of the hallmarks of modeling as a mathematical practice (less defined tasks, more of the modeling cycle, etc.).
Evidence of Learning Statements
Students with a level 1 understanding of this standard will most likely be able to:
Students with a level 2 understanding of this standard will most likely be able to:
Students with a level 3 understanding of this standard will most likely be able to:
Students with a level 4 understanding of this standard will most likely be able to: Choose a linear equation in one variable that represents a simple, real-world situation.
Solve a one variable linear equation.
Solve a one variable linear inequality.
Identify if a real-world situation can be represented by a linear or exponential equation.
Determine if the solution to a real- world situation requires a one- variable or two variable equation or inequality.
Create and solve a one variable linear equation that represents a simple, real-world situation.
Create and solve a one variable linear inequality that represents a simple, real-world situation.
Choose an exponential equation to represent a simple, real-world situation.
Choose an exponential inequality to represents a simple, real-world situation.
Create and solve a one variable linear, or exponential equation that represents a real-world situation.
Create and solve a one-variable linear inequality that represents a real-world situation.
Create and solve a one-variable exponential inequality that represents a simple real-world situation.
Create a real-world situational problem to represent a given linear or exponential equation or inequality.
Create and solve a one-variable exponential inequality that represents a real-world situation.
Instructional Focus Statements
Level 3:
In Integrated Math I, the variety of function types that students encounter allows students to create more complex equations and work within more complex situations than what has been previously experienced.
As this is a modeling standard, students need to encounter equations and inequalities that evolve from real-world situations. Students should be formulating equations and inequalities, computing solutions, interpreting findings, and validating their thinking and the reasonableness of attained solutions in order to justify solutions to real-world problems. Real-world situations should elicit equations and inequalities from situations which are linear and exponential in nature. It is imperative that students have the opportunity to work with each of these function types equally.
Level 4:
When given an equation or inequality, students can generate a real-world situation that could be solved by a provided equation or inequality demonstrating a deep understanding of the interplay that exists between the situation and the equation or inequality used to solve the problem. Additionally, students should continue to encounter real-world problems that are increasingly more complex. Students should be using the modeling cycle to solve real-world problems.
As this is a modeling standard, students need to encounter equations that evolve from both mathematical and real-world situations. Students should be formulating equations, computing solutions, interpreting findings, and validating their thinking and the reasonableness of attained solutions in order to justify solutions to mathematical and real-world problems. Mathematical situations should elicit equations from situations which are exclusively linear in nature.
Level 4:
One of the most natural situations for students to create an equation or graph from is a real-world situation. Students need to be exposed to variety of real world situations that illicit varying linear functions. Students should encounter real-world problems that are increasingly more complex over time. They should be using the modeling cycle in order to develop and provide justification for their solutions.
Additionally, students should be posed with an equation and then asked to generate a real-world situation that could be solved by a provided equation. Students with this capability are demonstrating a deep understanding of the interplay that exists between the situation and the equation used to solve the problem.
Standard M1.A.CED.A.3 (Major Work of the Grade) Represent constraints by equations or inequalities and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.
Scope and Clarifications: (Modeling Standard) For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. There are no assessment limits for this standard. The entire standard is assessed in this course.
Evidence of Learning Statements
Students with a level 1 understanding of this standard will most likely be able to:
Students with a level 2 understanding of this standard will most likely be able to:
Students with a level 3 understanding of this standard will most likely be able to:
Students with a level 4 understanding of this standard will most likely be able to: Define variables that represent unknown values in a real-world problem.
Describe the difference of a viable solution and a non-viable solution.
Choose an equation or inequality that models the constraint on a variable given a contextual problem.
Determine when a solution would viable or non-viable solution, given an equation or inequality that represents a real-world problem.
Determine the viability of each solution, given an equation or inequality that represents a contextual situation and a set of possible solutions.
Write an equation or inequality that models the constraint on a variable given a contextual problem.
Write a system of equations or inequalities that models the constraint on a variable given a contextual problem.
Explain constraints on a variable in context of a real-world problem and interpret solutions to determine the viability by using a graph, table, and equation.
Justify the solution that models a real-world problem where there is a limitation on a variable.
Interpret solutions as viable or nonviable options in a modeling
Create and provide a solution to a real-world problem that has natural limitations on variables. Explain the solution and its viability using multiple representations (i.e. table, graph, equation) and precise mathematical language.
Explain examples of both viable and nonviable solutions in context of a real-world problem.
Use multiple representations to justify a solution’s viability and explain when one representation elicits a more efficient justification.
the unknown variables. Furthermore, students should be provided the opportunity to critique the solutions of others, and instruction should require students to justify and develop logical arguments for the viability of solutions.
Standard M1.A.CED.A.4 (Major Work of the Grade) Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
Scope and Clarifications: (Modeling Standard) i. Tasks are limited to linear equations. ii. Tasks have a real-world context.
Evidence of Learning Statements
Students with a level 1 understanding of this standard will most likely be able to:
Students with a level 2 understanding of this standard will most likely be able to:
Students with a level 3 understanding of this standard will most likely be able to:
Students with a level 4 understanding of this standard will most likely be able to: Choose equivalent forms of a given linear real-world formula.
Rearrange simple real-world linear formulas to highlight a quantity of interest.
Rearrange complex real-world linear formulas to highlight a quantity of interest.
Rearrange real-world linear formulas and explain the benefit of solving the formula for the various variables.
Instructional Focus Statements
Level 3:
In previous grades, students have focused on rearranging simple linear formulas to highlight a quantity of interest. In Integrated Math I, the linear formulas student work with should be fairly complex.
As this is a modeling standard, student should be encountering formulas that come from real-world situations. Additionally, students need to be developing a conceptual understanding of why they might need to write formulas in different ways and what the benefit would be to these various representations of the same real-world formula.
Level 4:
Students need to be exposed to a wide variety of real-world formulas increasing in complexity over time. Additionally, it is imperative that they are able to explain why formulas might need to be expressed in different ways and the benefit that each form provides.
