Utah State Standards for Mathematics: Grade 8

UT.1. Pre-Algebra: Students will expand number sense to understand, perform operations, and solve problems with rational numbers.

1.1. Compute fluently with understanding and make reasonable estimates with rational numbers.

1.1.a. Compute fluently using all four operations with integers, and explain why the corresponding algorithms work.

1.1.b. Compute fluently using all four operations with rational numbers, including negative fractions and decimals, and explain why the corresponding algorithms work.

1.1.c. Check the reasonableness of results using estimation.

1.2. Analyze relationships among rational numbers, including negative rational numbers, and operations involving these numbers.

1.2.a. Order rational numbers in various forms, including scientific notation (positive and negative exponents), and place numbers on a number line.

1.2.b. Predict the effect of operating with fractions, decimals, percents, and integers as an increase or a decrease of the original value.

1.2.c. Recognize and use the identity properties of addition and multiplication, the multiplicative property of zero, the commutative and associative properties of addition and multiplication, and the distributive property of multiplication over addition.

1.2.d. Recognize and use the inverse operations of adding and subtracting a fixed number, multiplying and dividing by a fixed number, and computing squares of whole numbers and taking square roots of perfect squares.

1.3. Solve problems involving rational numbers using addition, subtraction, multiplication, and division.

1.3.a. Recognize the absolute value of a rational number as its distance from zero.

1.3.b. Simplify numerical expressions, including those with whole number exponents and absolute values, using the order of operations.

1.3.c. Solve problems involving rational numbers, percents, and proportions.

UT.2. Pre-Algebra: Students will use proportion and similarity to solve problems.

2.1. Model and illustrate meanings of ratios, percents, and decimals.

2.1.a. Compare ratios to determine if they are equivalent.

2.1.b. Compare ratios using the unit rate.

2.1.c. Represent percents as ratios based on 100 and decimals as ratios based on powers of ten.

2.1.d. Graph proportional relationships and identify the unit rate as the slope of the related line.

2.2. Solve a wide variety of problems using ratios and proportional reasoning.

2.2.a. Set up and solve problems involving proportional reasoning using variables.

2.2.b. Solve percent problems, including problems involving discounts, interest, taxes, tips, and percent increase or decrease.

2.2.c. Solve ratio and rate problems using informal methods.

2.3. Recognize similar polygons and use properties of similar triangles to solve problems and define the slope of a line.

2.3.a. Define similar polygons as polygons with corresponding angles congruent and corresponding sides that are proportional.

2.3.b. Identify pairs of similar triangles using two pairs of congruent angles, or two pairs of proportional sides with congruent included angles.

2.3.c. Find missing lengths of similar triangles, including inaccessible lengths, using proportions.

2.3.d. Define the slope of a line as the ratio of the vertical change to the horizontal change between two points, and show that the slope is constant using similarity of right triangles.

UT.3. Pre-Algebra: Students will develop fluency with the language and operations of algebra to analyze and represent relationships.

3.1. Generalize and express patterns using algebraic expressions.

3.1.a. Compare representations of a relation using tables, graphs, algebraic symbols, and mathematical rules.

3.1.b. Describe simple patterns using a mathematical rule or algebraic expression.

3.2. Evaluate, simplify, and solve algebraic expressions, equations, and inequalities.

3.2.a. Evaluate algebraic expressions, including those with whole number exponents, when given values for the variable(s).

3.2.b. Simplify algebraic expressions using the order of operations, algebraic properties, and exponent rules.

3.2.c. Solve single-variable linear equations and inequalities, including those that must be simplified on one side or those with variables on both sides of an equation.

3.3. Represent relationships using graphs, tables, and other models.

3.3.a. Identify approximate rational coordinates when given the graph of a point on a rectangular coordinate system.

3.3.b. Graph ordered pairs of rational numbers on a rectangular coordinate system.

3.3.c. Graph linear equations using ordered pairs or tables.

3.3.d. Recognize that all first order equations produce linear graphs.

3.3.e. Model real-world problems using graphs, tables, equations, manipulatives, and pictures, and identify extraneous information.

UT.4. Pre-Algebra: Students will use algebraic, spatial, and logical reasoning to solve geometry and measurement problems.

4.1. Apply the properties of proportionality of different units of measure.

4.1.a. Convert units of measure within the same system.

4.1.b. Create and interpret scale drawings and approximate distance on maps using scale factors.

4.1.c. Solve problems using scale factors.

4.2. Derive formulas for surface areas and volume of three-dimensional figures.

4.2.a. Derive formulas for and calculate surface area and volume of right prisms and cylinders using appropriate units.

4.2.b. Explain that if a scale factor describes how corresponding lengths in two similar objects are related, then the square of the scale factor describes how corresponding areas are related and the cube of the scale factor describes how corresponding volumes are related.

4.2.c. Find lengths, areas, and volumes of similar figures, using the scale factor.

4.2.d. Select appropriate two- and three-dimensional figures to model real-world objects, and solve a variety of problems involving surface areas and volumes of cylinders and prisms.

UT.5. Pre-Algebra: Students will understand concepts from probability and statistics and apply statistical methods to solve problems.

5.1. Calculate probabilities of events and compare theoretical and experimental probability.

5.1.a. Solve counting problems using the Fundamental Counting Principle.

5.1.b. Calculate the probability of an event or sequence of events with and without replacement using models.

5.1.c. Recognize that the sum of the probability of an event and the probability of its complement is equal to one.

5.1.d. Make approximate predictions using theoretical probability and proportions.

5.1.e. Collect and interpret data to show that as the number of trials increases, experimental probability approaches the theoretical probability.

5.2. Formulate questions and answer the questions by organizing and analyzing data.

5.2.a. Formulate questions that can be answered through data collection and analysis.

5.2.b. Determine the 25th and 75th percentiles (first and third quartiles) to obtain information about the spread of data.

5.2.c. Graphically summarize data of a single variable using histograms and box-and whisker plots.

5.2.d. Compute the mean and median of a numerical characteristic and relate these values to the histogram of the data.

5.2.e. Use graphical representations and numerical summaries to answer questions and interpret data.

1.1.d. Place rational and irrational numbers on a number line between two integers.

2.2.d. Interpret the slope of a linear function as a rate of change in real-world situations.

3.1.c. Multiply monomials by a polynomial.

3.1.d. Multiply binomials.

3.1.e. Simplify the quotient of monomials using positive exponents.

3.2.d. Solve proportions that include algebraic first-degree expressions.

3.4. Factor polynomials with common monomial factors and factor simple quadratic expressions.

3.4.a. Find the greatest common monomial factor of a polynomial.

3.4.b. Factor trinomials with integer coefficients of the form x^2 + bx + c.

3.4.c. Factor the difference of two squares and perfect square trinomials.

3.5. Solve quadratic equations using factoring or by taking square roots.

3.5.a. Solve quadratic equations that can be simplified to the form x^2 = a where a is greater than or equal to 0 by taking square roots.

3.5.b. Solve quadratic equations using factoring.

3.5.c. Write a quadratic equation when given the solutions.

1.2.e. Prove lines parallel or perpendicular using slope or angle relationships.

1.3.d. Apply triangle inequality theorems.

1.3.e. Identify medians, altitudes, and angle bisectors of a triangle, and the perpendicular bisectors of the sides of a triangle, and justify the concurrency theorems.

1.4. Analyze characteristics and properties of polygons and circles.

1.4.a. Use examples and counterexamples to classify subsets of quadrilaterals.

1.4.b. Prove properties of quadrilaterals using triangle congruence relationships, postulates, and theorems.

1.4.c. Derive, justify, and use formulas for the number of diagonals, lines of symmetry, angle measures, perimeter, and area of regular polygons.

1.4.d. Define radius, diameter, chord, secant, arc, sector, central angle, inscribed angle, and tangent of a circle, and solve problems using their properties.

1.4.e. Show the relationship between intercepted arcs and inscribed or central angles, and find their measures.

1.5. Perform basic geometric constructions, describing and justifying the procedures used.

1.5.a. Investigate geometric relationships using constructions.

1.5.b. Copy and bisect angles and segments.

1.5.c. Construct perpendicular and parallel lines.

1.5.d. Justify procedures used to construct geometric figures.

1.5.e. Discover and investigate conjectures about geometric properties using constructions.

1.6. Analyze characteristics and properties of three-dimensional figures.

1.6.a. Identify and classify prisms, pyramids, cylinders and cones based on the shape of their base(s).

1.6.b. Identify three-dimensional objects from different perspectives using nets, cross sections, and two-dimensional views.

1.6.c. Describe the symmetries of three-dimensional figures.

1.6.d. Describe relationships between the faces, edges, and vertices of polyhedra.

4.1.d. Calculate or estimate the area of an irregular region.

4.1.e. Find the length of an arc and the area of a sector when given the angle measure and radius.