OREGON MATH STANDARDS (2021): [8.AEE]
Overview
The intent of clarifying statements is to provide additional guidance for educators to communicate the intent of the standard to support the future development of curricular resources and assessments aligned to the 2021 math standards.
Clarifying statements can be in the form of succinct sentences or paragraphs that attend to one of four types of clarifications: (1) Student Experiences; (2) Examples; (3) Boundaries; and (4) Connection to Math Practices.
2021 Oregon Math Guidance: 8.AEE.A.1
Cluster: 8.AEE.A - Expressions and Equations Work with radicals and integer exponents.
STANDARD: 8.AEE.A.1
Standards Statement (2021):
Apply the properties of integer exponents using powers of 10 to generate equivalent numerical expressions.
Connections:
Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |
6.AEE.A.1 | 8.AEE.A.3, 8.AEE.A.4 | N/A | 8.EE.A.1 8.AEE.A Crosswalk |
Standards Guidance:
Boundaries
- Exploration of integer exponents can be limited to base 10 for this standard. Additional bases could be explored as a possible extension.
Teaching Strategies
- Students should use numerical reasoning to identify patterns associated with properties of integer exponents.
- The following properties should be addressed: product rule, quotient rule, power rule, power of product rule, power of a quotient rule, zero exponent rule, and negative exponent rule.
Progressions
- Students have been denoting whole number powers of 10 with exponential notation since Grade 5, and they have seen the pattern in the number of zeros when powers of 10 are multiplied. They express this as 10a∙10b=10a+b for whole numbers a and b. Requiring this rule to hold when a and b are integers leads to the definition of the meaning of pwers with 0 and negative exponents. For example, we define 100=1 because we want 10a∙100=10a+0=10a, so 100 must equal 1. (Please reference page 11 in the Progression document)
Examples
- Generate equivalent numerical expressions. For example,
- Illustrative Mathematics:
- Student Achievement Partners:
2021 Oregon Math Guidance: 8.AEE.A.2
Cluster: 8.AEE.A - Expressions and Equations Work with radicals and integer exponents.
STANDARD: 8.AEE.A.2
Standards Statement (2021):
Represent solutions to equations using square root and cube root symbols.
Connections:
Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |
6.AEE.B.4 | HS.AEE.B.5, 8.GM.B.7 | 7.NS.A.3, 8.NS.A.2, 8.GM.B.6, 8.GM.C.9, HS.NQ.A.1 | 8.EE.A.2 8.AEE.A Crosswalk |
Standards Guidance:
Clarifications
- Equations should include those with irrational number solutions, such as the solution for x2 = 14 would include
Boundaries
- Use square root and cube root symbols to represent solutions to equations of the form 𝑥2=𝑝 and 𝑥3=𝑝, where 𝑝 is a positive rational number.
- Evaluate square roots of small perfect squares up to 225 and cube roots of small perfect cubes up to 1000.
- Know irrational numbers include square roots of non-perfect squares, such as , and cube roots of non-perfect cubes.
Teaching Strategies
- Students should be able to find patterns within the list of square numbers and then with cube numbers.
- Students should be able to recognize that squaring a number and taking the square root of a number are inverse operations; likewise, cubing a number and taking the cube root are inverse operations.
Progressions
- Notice that students do not learn the properties of rational exponents until high school. However, they prepare in Grade 8 by starting to work systematically with the square root and cube root symbols, writing, for example, and .
- Since is defined to mean the positive solution to the equation x2 = p (when it exists). It is not mathematically correct to say (as is a common misconception). In describing the solutions to x2 = 64, students should write (Please reference page 11 in the Progression document).
Examples
- and
- Student Achievement Partners:
2021 Oregon Math Guidance: 8.AEE.A.3
Cluster: 8.AEE.A - Expressions and Equations Work with radicals and integer exponents.
STANDARD: 8.AEE.A.3
Standards Statement (2021):
Estimate very large or very small quantities using scientific notation with a single digit times an integer power of ten.
Connections:
Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |
4.OA.A.2, 5.NBT.A.2, 8.AEE.A.1 | 8.AEE.A.4 | 5.NBT.A.4 | 8.EE.A.3 8.AEE.A Crosswalk |
Standards Guidance:
Clarifications
- Students should use place value reasoning which supports the understanding of digits shifting to the left or right when multiplied by a power of 10.
- Product and quotient rules for powers is relevant at 8th grade, and only for powers of 10
Teaching Strategies
- Students should use the magnitude of quantities to compare numbers written in scientific notation to determine how many times larger (or smaller) one number written in scientific notation is than another.
- Students should have opportunities to compare numbers written in scientific notation in contextual problems.
Examples
- Compare two quantities written in this format. For example, estimate the population of the United States as and the population of the world as , and determine that the world population is more than 20 times larger.
- Illustrative Mathematics:
2021 Oregon Math Guidance: 8.AEE.A.4
Cluster: 8.AEE.A - Expressions and Equations Work with radicals and integer exponents.
STANDARD: 8.AEE.A.4
Standards Statement (2021):
Perform operations with numbers expressed in scientific notation.
Connections:
Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |
7.AEE.B.3, 8.AEE.A.1, 8.AEE.A.3 | N/A | HS.NQ.B.3 | 8.EE.A.4 8.AEE.A Crosswalk |
Standards Guidance:
Clarifications
- Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used.
- Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities.
- Interpret scientific notation that has been generated by technology.
Teaching Strategies
- Students should use place value reasoning, which supports the understanding of digits shifting to the left or right when multiplied by a power of 10.
- Students combine knowledge of integer exponent rules and scientific notation to perform operations with numbers expressed in scientific notation.
- Students should solve problems involving real-life contexts.
Progressions
- Include authentic contexts where both standard and scientific notation are used. Use scientific notation to choose units of appropriate size for measurements of very large or very small quantities. (Please reference page 11 in the Progression document).
Examples
- Use millimeters per year for seafloor spreading.
- Interpret scientific notation that has been generated by technology such as may be displayed in a calculator as “1.2E6”.
- Illustrative Mathematics:
- Student Achievement Partners:
2021 Oregon Math Guidance: 8.AEE.B.5
Cluster: 8.AEE.B - Understand the connections between proportional relationships, lines, and linear equations.
STANDARD: 8.AEE.B.5
Standards Statement (2021):
Graph proportional relationships in authentic contexts. Interpret the unit rate as the slope of the graph, and compare two different proportional relationships represented in different ways.
Connections:
Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |
7.RP.A.1, 7.RP.A.2 | 8.AFN.A.2, HS.AEE.D.9 | HS.GM.A.3 | 8.EE.B.5 8.AEE.B Crosswalk |
Standards Guidance:
Terminology
- Various forms of linear functions include standard and slope-intercept forms.
- Key features include rate of change (slope), intercepts, strictly increasing or strictly decreasing, positive, negative, and end behavior.
Teaching Strategies
- Use verbal descriptions, tables and graphs created by hand and/or using technology.
Progressions
- As students start to build a unified notion of the concept of function they are able to compare proportional relationships presented in different ways.
- For example, the table on the right shows 300 miles in 5 hours, whereas the graph shows more than 300 miles in the same time. (Please reference page 12 in the Progression document).
Examples
- Interpret the unit rate as the slope of the graph. Compare one or more proportional relationships represented in different ways.
- For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
- Illustrative Mathematics:
2021 Oregon Math Guidance: 8.AEE.B.6
Cluster: 8.AEE.B - Understand the connections between proportional relationships, lines, and linear equations.
STANDARD: 8.AEE.B.6
Standards Statement (2021):
Write the equation for a line in slope intercept form y = mx + b, where m and b are rational numbers, and explain in context why the slope m is the same between any two distinct points.
Connections:
Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |
7.RP.A.2 | 8.AEE.C.8, 8.AFN.A.2, 8.AFN.A.3, HS.AEE.D.9 | HS.GM.A.3 | 8.EE.B.6 8.AEE.B Crosswalk |
Standards Guidance:
Clarification
- Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane
- Derive the equation 𝑦=𝑚𝑥 for a line through the origin and the equation 𝑦=𝑚𝑥+𝑏 for a line intercepting the vertical axis at 𝑏.
Boundaries
- Content expecation would include students using both standard and slope-intercept forms for a linear equation. Students should be able to rewrite linear equations written in different forms depending on the given context.
- This work could also include generating equations using the point-slope form to generate an equation for a line that passes through a point with a given slope.
Terminology
- Forms of linear equations:
- Standard Form:
- Slope-Intercept Form:
- Point-Slope Form: for a line with slope m, that passes through the point (x1,y1)
Progressions
- Content expecation would be for a student to calcluate the slope between two points (x1,y1) and (x2,y2) as the difference between change in y (e.g. “rise”) over the change in x (eg. “run”) as . A fact that a line has a well-defined slope where the ratio between the rise and run for any two points on the line is always the same—depends on similar triangles. (Please reference page 12 in the Progression document).
Examples
- Know that the slope m is the same between any two distinct points on a non-vertical line and be able to explain or demonstrate why.
- Derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
- Illustrative Mathematics:
- Student Achievement Partners:
2021 Oregon Math Guidance: 8.AEE.C.7
Cluster: 8.AEE.C - Analyze and solve linear equations and pairs of simultaneous linear equations.
STANDARD: 8.AEE.C.7
Standards Statement (2021):
Solve linear equations with one variable including equations with rational number coefficients, with the variable on both sides, or whose solutions require using the distributive property and/or combining like terms.
Connections:
Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |
7.AEE.A.1, 7.AEE.B.4 | HS.AEE.A.3, HS.AEE.A.2, HS.AEE.C.8, HS.AEE.D.11 | N/A | 8.EE.C.7 8.AEE.C Crosswalk |
Standards Guidance:
Clarifications
- To achieve fluency, students should be able to choose flexibly among methods and strategies to solve mathematical problems accurately and efficiently.
- Students should rearrange formulas to highlight a quantity of interest using the same reasoning as in solving equations. Interpret and explain the results.
Terminology
- Parts of an expression include terms, factors, coefficients, and operations.
Boundaries
- This standard also includes solving or giving examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions.
- Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
Teaching Strategies
- Students should use algebraic reasoning in their descriptions of the solutions to linear equations.
- Problems should be practical and contextual providing a purpose for analyzing equivalent forms of an expression.
Progressions
- Building upon skills from grade 7, students combine like terms on the same side of the equal sign and use the distributive property to simplify the equation when solving. Emphasis in this standard is also on using rational coefficients. Solutions of certain equations may elicit infinitely many or no solutions. Include linear equations and inequalities with rational number coefficients and whose solutions require expanding expressions using the distributive property and collecting like terms.
Examples
- Given ax + 3 = 7, solve for x.
- Illustrative Mathematics:
- Student Achievement Partners:
2021 Oregon Math Guidance: 8.AEE.C.8
Cluster: 8.AEE.C - Analyze and solve linear equations and pairs of simultaneous linear equations.
STANDARD: 8.AEE.C.8
Standards Statement (2021):
Find, analyze, and interpret solutions to pairs of simultaneous linear equations using graphs or tables.
Connections:
Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |
6.AEE.B.4, 7.AEE.B.4, 8.AEE.B.6 | HS.AEE.B.4, HS.AEE.B.6, HS.AEE.D.9, HS.AEE.C.7, HS.AEE.D.11 | HS.GM.A.3 | 8.EE.C.8 8.AEE.C Crosswalk |
Standards Guidance:
Clarification
- Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs.
- Estimate solutions by graphing the equations; solve simple cases by inspection, or by using tables.
Teaching Strategies
- Include mathematical problems in authentic contexts leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.
- Students should be able to analyze and find solutions to systems of equations presented numerically in tables, or graphically.
Progressions
- Students should be provided with opportunities to explore systems of equations represented using technology such as interactive graphs to analyze and interpret the solutions to the systems.
- Students should have the opportunity to explore visual graphs of equations that are parallel, perpendicular or neither parallel nor perpendicular to develop a deep, conceptual understanding. (Please reference page 13 in the Progression document).
Examples
- Given coordinates for two pairs of points, a student can determine whether the line through the first pair of points intersects the line through the second pair.
- A student can graph two linear equations that represent a culturally relevant problem using digital graphing tools (e.g., Desmos, graphing calculators, or other) and visually make sense of the graphed lines in context. A student can provide a verbal or written explanation of their reasoning.
- A student can recognize that there is no solution to the system of equations formed by 3x + 2y = 5 and 3x + 2y = 6 because the lines are parallel and 3x + 2y cannot simultaneously be 5 and 6.
- Illustrative Mathematics:
- Student Achievement Partners: