8th GradeCommon Core State Standards Alignment Livonia Public Schools
Revised 6/4/12
Mathematics | Grade 8
Grade 8 Overview
The Number System (NS)- Know that there are numbers that are not rational, and approximate them by rational numbers.
- Work with radicals and integer exponents.
- Understand the connections between proportional relationships, lines, and linear equations.
- Analyze and solve linear equations and pairs of simultaneous linear equations.
- Define, evaluate, and compare functions.
- Use functions to model relationships between quantities.
- Understand congruence and similarity using physical models, transparencies, or geometry software.
- Understand and apply the Pythagorean Theorem.
- Solve real-world and mathematical problems involving volume of cylinders, cones and spheres.
- Investigate patterns of association in bivariate data.
- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.
Grade 8 Critical Areas (from CCSS pg. 52)
The Critical Areas are designed to bring focus to the standards at each grade by describing the big ideas that educators can use to build their curriculum and to guide instruction. The Critical Areas for eighth grade can be found on page 52 in the Common Core State Standards for
Mathematics.
- Formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations
Students use linear equations and systems of linear equations to represent, analyze, and solve a variety of problems. Students recognize equations for proportions (y/x = m or y = mx) as special linear equations (y = mx+ b), understanding that the constant of proportionality (m) is the slope, and the graphs are lines through the origin. They understand that the slope (m) of a line is a constant rate of change, so that if the input or x-coordinate changes by an amount A, the output or y-coordinate changes by the amount m·A. Students also use a linear equation to describe the association between two quantities in bivariate data (such as arm span vs. height for students in a classroom). At this grade, fitting the model, and assessing its fit to the data are done informally. Interpreting the model in the context of the data requires students to express a relationship between the two quantities in question and to interpret components of the relationship (such as slope and y-intercept) in terms of the situation.
Students strategically choose and efficiently implement procedures to solve linear equations in one variable, understanding that when they use the properties of equality and the concept of logical equivalence, they maintain the solutions of the original equation. Students solve systems of two linear equations in two variables and relate the systems to pairs of lines in the plane; these intersect, are parallel, or are the same line. Students use linear equations, systems of linear equations, linear functions, and their understanding of slope of a line to analyze situations and solve problems.
- Grasping the concept of a function and using functions to describe quantitative relationships
Students grasp the concept of a function as a rule that assigns to each input exactly one output. They understand that functions describe situations where one quantity determines another. They can translate among representations and partial representations of functions (noting that tabular and graphical representations may be partial representations), and they describe how aspects of the function are reflected in the different representations.
- Analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem
Students use ideas about distance and angles, how they behave under translations, rotations, reflections, and dilations, and ideas about congruence and similarity to describe and analyze two-dimensional figures and to solve problems. Students show that the sum of the angles in a triangle is the angle formed by a straight line, and that various configurations of lines give rise to similar triangles because of the angles created when a transversal cuts parallel lines. Students understand the statement of the Pythagorean Theorem and its converse, and can explain why the Pythagorean Theorem holds, for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points on the coordinate plane, to find lengths, and to analyze polygons. Students complete their work on volume by solving problems involving cones, cylinders, and spheres.
Standards for Mathematical Practice
The Common Core State Standards for Mathematical Practice are expected to be integrated into every mathematics lesson for all students Grades K-12. Below are a few examples of how these Practices may be integrated into tasks that Middle School students complete.
Standards for Mathematical / Practice Explanations and Examples1. Make sense of problems
and persevere in solving
them. / In grade 8,students solve real world problems through the application of algebraic and geometric concepts. Studentsseek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinkingby asking themselves, “What is the most efficient way to solve the problem?”, “Does this make sense?”, and “Can Isolve the problem in a different way?”
2. Reason abstractly and
quantitatively. / In grade 8,students represent a wide variety of real world contexts through the use of real numbers and variables inmathematical expressions, equations, and inequalities. They examine patterns in data and assess the degree oflinearity of functions. Students contextualize to understand the meaning of the number or variable as related to theproblem and decontextualize to manipulate symbolic representations by applying properties of operations.
3. Construct viable arguments
and critique the reasoning of
others. / In grade 8, students construct arguments using verbal or written explanations accompanied by expressions,equations, inequalities, models, and graphs, tables, and other data displays (i.e. box plots, dot plots, histograms,etc.). They further refine their mathematical communication skills through mathematical discussions in which theycritically evaluate their own thinking and the thinking of other students. They pose questions like “How did you getthat?”, “Why is that true?” “Does that always work?” They explain their thinking to others and respond to others’thinking.
4. Model with mathematics. / In grade 8, students model problem situations symbolically, graphically, tabularly, and contextually. Students form expressions, equations, or inequalities from real world contexts and connect symbolic and graphical representations. Students solve systems of linear equations and compare properties of functions provided in different forms. Students use scatterplots to represent data and describe associations between variables. Students need many opportunities to connect and explain the connections between the different representations. They should be able to use all of these representations as appropriate to a problem context
5. Use appropriate tools
strategically. / Students consider available tools (including estimation and technology) when solving a mathematical problem and decide when certain tools might be helpful. For instance, students in grade 8 may translate a set of data given in tabular form to a graphical representation to compare it to another data set. Students might draw pictures, use applets, or write equations to show the relationships between the angles created by a transversal.
6. Attend to precision. / In grade 8, students continue to refine their mathematical communication skills by using clear and precise language in their discussions with others and in their own reasoning. Students use appropriate terminology when referring to the number system, functions, geometric figures, and data displays
7. Look for and make use of
structure. / Students routinely seek patterns or structures to model and solve problems. In grade 8, students apply properties to generate equivalent expressions and solve equations. Students examine patterns in tables and graphs to generate equations and describe relationships. Additionally, students experimentally verify the effects of transformations and describe them in terms of congruence and similarity.
8. Look for and express
regularity in repeated
reasoning. / In grade 8, students use repeated reasoning to understand algorithms and make generalizations about patterns.Students use iterative processes to determine more precise rational approximations for irrational numbers. Theyanalyze patterns of repeating decimals to identify the corresponding fraction. During multiple opportunities to solve and model problems, they notice that the slope of a line and rate of change are the same value. Students flexibly make connections between covariance, rates, and representations showing the relationships between quantities.
Domain: The Number System (8.NS)
Common Core Cluster: Know that there are numbers that are not rational, and approximate them by rational numbers.
Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: Real Numbers, Irrational numbers, Rational numbers, Integers, Whole numbers, Natural numbers, radical, radicand, square roots, perfect squares, cube roots, terminating decimals, repeating decimals, truncate
Common Core State Standards / Deconstruction Notes and Examples / Math Practices / Cluster
Vocabulary / ALIGNMENT
Unit/Investigation
8.NS.1. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. / 8.NS.1Students understand that Real numbers are either rational or irrational. They distinguish between rational and irrational numbers, recognizing that any number that can be expressed as a fraction is a rational number. The diagram below illustrates the relationship between the subgroups of the real number system.
Students recognize that the decimal equivalent of a fraction will either terminate or repeat. Fractions that terminate will have denominators containing only prime factors of 2 and/or 5. This understanding builds on work in 7th grade when students used long division to distinguish between repeating and terminating decimals.
Students convert repeating decimals into their fraction equivalent using patterns or algebraic reasoning. One method to find the fraction equivalent to a repeating decimal is shown below.
Example 1:
Change 0.4 to a fraction.
- Let x = 0.444444…..
- Multiply both sides so that the repeating digits will be in front of he decimal. In this example, one digit repeats so both sides are multiplied by 10, giving 10x= 4.4444444….
- Subtract the original equation from the new equation.
– x = 0.444444…..
9x= 4
- Solve the equation to determine the equivalent fraction.
9 9
x =
Additionally, students can investigate repeating patterns that occur when fractions have denominators of 9, 99, or 11.
Example 2:
is equivalent to 0., is equivalent to 0., etc. / 5, 6 / Real Numbers Irrational numbers Rational numbers Integers
Whole numbers Natural numbers radical
radicand
square roots
perfect squares
cube roots terminating decimals
repeating decimals truncate / Looking for Pythagoras
8.NS.2Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example,
by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get
better approximations. / 8.NS.2Students locate rational and irrational numbers on the number line. Students compare and order rational and irrational numbers. Students also recognize that square roots may be negative and written as -.
Example 1:
Compare and
Solution: Statements for the comparison could include:
and are between the whole numbers 1 and 2
is between 1.7 and 1.8
is less than
Additionally, students understand that the value of a square root can be approximated between integers and that nonperfect square roots are irrational.
Example 2:
Find an approximation of
- Determine the perfect squares is between, which would be 25 and 36.
- The square roots of 25 and 36 are 5 and 6 respectively, so we know that is between 5 and 6.
- Since 28 is closer to 25, an estimate of the square root would be closer to 5. One method to get an estimate is to divide 3 (the distance between 25 and 28) by 11 (the distance between the perfect squares of 25 and 36) to get 0.27.
- The estimate of would be 5.27 (the actual is 5.29).
Domain: Expressions and Equations (8.EE)
Common Core Cluster: Work with radicals and integer exponents.
Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: laws of exponents, power, perfect squares, perfect cubes, root, square root, cube root, scientific notation, standard form of a number. Students should also be able to read and use the symbol: ±
Common Core State Standards / Deconstruction Notes and Examples / Math Practices / Cluster
Vocabulary / ALIGNMENT
Unit/Investigation
8.EE.1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 = 3–3 = 1/33 = 1/27. / 8.EE.1In 6th grade, students wrote and evaluated simple numerical expressions with whole number exponents (ie. 53 = 5 • 5 • 5 = 125). Integer (positive and negative) exponents are further developed to generate equivalent numerical expressions when multiplying, dividing or raising a power to a power. Using numerical bases and the laws of exponents, students generate equivalent expressions.
Students understand:
- Bases must be the same before exponents can be added, subtracted or multiplied. (Example 1)
- Exponents are subtracted when like bases are being divided (Example 2)
- A number raised to the zero (0) power is equal to one. (Example 3)
- Negative exponents occur when there are more factors in the denominator. These exponents can be expressed as a positive if left in the denominator. (Example 4)
- Exponents are added when like bases are being multiplied (Example 5)
- Exponents are multiplied when an exponents is raised to an exponent (Example 6)
- Several properties may be used to simplify an expression (Example 7)
=
Example 2:
= =
Example 3:
Students understand this relationship from examples such as . This expression could be simplified as = 1.
Using the laws of exponents this expression could also be written as = . Combining these gives.
Example 4:
Example 5:
(32) (34) = () = 36 = 729
Example 6:
(43)2 = 43x2 = 46 = 4, 096
Example 7:
/ 1, 7 / laws of exponents
power
perfect squares perfect cubes
root
square root
cube root
scientific notation standard form of a
number / Mini unit
8.EE.2. 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 and cube roots of small perfect cubes. Know that √2 is irrational. / 8.EE.2Students recognize perfect squares and cubes, understanding that non-perfect squares and non-perfect cubes are irrational.
Students 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.
Example 1:
= 16 and = 4
NOTE: = 16 while = -16 since the negative is not being squared. This difference is often problematic forstudents, especially with calculator use.
Example 2:
and
NOTE: there is no negative cube root since multiplying 3 negatives
would give a negative.
This understanding is used to solve equations containing square or cube numbers. Rational numbers would haveperfect squares or perfect cubes for the numerator and denominator. In the standard, the value of p for square rootand cube root equations must be positive.
Example 3:
Solve:
Solution:
NOTE: There are two solutions because 5 • 5 and -5 • -5 will both equal 25.
Example 4:
Solve:
Solution:
Example 5:
Solve:
Solution:
Example 6:
Solve:
Solution:
Students understand that in geometry the square root of the area is the length of the side of a square and a cube rootof the volume is the length of the side of a cube. Students use this information to solve problems, such as findingthe perimeter.
Example 7:
What is the side length of a square with an area of 49 ft2?
Solution: = 7 ft. The length of one side is 7 ft. / 5, 6 / Looking for Pythagoras
8.EE.3. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger. / 8.EE.3Students use scientific notation to express very large or very small numbers. Students compare and interpret scientific notation quantities in the context of the situation, recognizing that if the exponent increases by
one, the value increases 10 times. Likewise, if the exponent decreases by one, the value decreases 10 times. Students solve problems using addition, subtraction or multiplication, expressing the answer in scientific notation.
Example 1:
Write 75,000,000,000 in scientific notation.
Solution: 7.5 x 1010
Example 2:
Write 0.0000429 in scientific notation.
Solution: 4.29 x 10-5
Example 3:
Express 2.45 x 105 in standard form.
Solution: 245,000
Example 4:
How much larger is 6 x 105 compared to 2 x 103
Solution: 300 times larger since 6 is 3 times larger than 2 and 105 is 100 times larger than 103.
Example 5:
Which is the larger value: 2 x 106 or 9 x 105?
Solution: 2 x 106 because the exponent is larger / 4, 5, 6 / Mini unit
8.EE.4. 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 (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. / 8.EE.4Students understand scientific notation as generated on various calculators or other technology. Students enter scientific notation using E or EE (scientific notation), * (multiplication), and ^ (exponent) symbols.