MM//JJ MMaatthheemmaattiiccss 22,, AAddvvaanncceedd MATHEMATICS Curriculum Map

2014 - 2015

Mathematics Florida State Standards

M/J Mathematics 2, Advanced: Mathematics Florida State Standards

Mathematics Florida State Standards Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them. (MAFS.K12.MP.1) Solving a mathematical problem involves making sense of what is known and applying a thoughtful and logical process which sometimes requires perseverance, flexibility, and a bit of ingenuity.

2. Reason abstractly and quantitatively. (MAFS.K12.MP.2) The concrete and the abstract can complement each other in the development of mathematical understanding: representing a concrete situation with symbols can make the solution process more efficient, while reverting to a concrete context can help make sense of abstract symbols.

3. Construct viable arguments and critique the reasoning of others. (MAFS.K12.MP.3) A well-crafted argument/critique requires a thoughtful and logical progression of mathematically sound statements and supporting evidence.

4. Model with mathematics. (MAFS.K12.MP.4) Many everyday problems can be solved by modeling the situation with mathematics.

5. Use appropriate tools strategically. (MAFS.K12.MP.5) Strategic choice and use of tools can increase reliability and precision of results, enhance arguments, and deepen mathematical understanding.

6. Attend to precision. (MAFS.K12.MP.6) Attending to precise detail increases reliability of mathematical results and minimizes miscommunication of mathematical explanations.

7. Look for and make use of structure. (MAFS.K12.MP.7) Recognizing a structure or pattern can be the key to solving a problem or making sense of a mathematical idea.

8. Look for and express regularity in repeated reasoning. (MAFS.K12.MP.8) Recognizing repetition or regularity in the course of solving a problem (or series of similar problems) can lead to results more quickly and efficiently.

In Grade 7,instructional time should focus on four critical area: (1) developing understanding of and applying proportional relationships; (2) developing understanding of operations with rational numbers and working with expressions and linear equations; (3) solving problems involving scale drawings and informal geometric constructions, and working with two- and three-dimensional shapes to solve problems involving area, surface area, and volume; and (4) drawing inferences about populations based on samples. (1) Students extend their understanding of ratios and develop understanding of proportionality to solve single- and multi-step problems. Students use their understanding of ratios and proportionality to solve a wide variety of percent problems, including those involving discounts, interest, taxes, tips, and percent increase or decrease. Students solve problems about scale drawings by relating corresponding lengths between the objects or by using the fact that relationships of lengths within an object are preserved in similar objects. Students graph proportional relationships and understand the unit rate informally as a measure of the steepness of the related line, called the slope. They distinguish proportional relationships from other relationships.

(2) Students develop a unified understanding of number, recognizing fractions, decimals (that have a finite or a repeating decimal representation), and percents as different representations of rational numbers. Students extend addition, subtraction, multiplication, and division to all rational numbers, maintaining the properties of operations and the relationships between addition and subtraction, and multiplication and division. By applying these properties, and by viewing negative numbers in terms of everyday contexts (e.g., amounts owed or temperatures below zero), students explain and interpret the rules for adding, subtracting, multiplying, and dividing with negative numbers. They use the arithmetic of rational numbers as they formulate expressions and equations in one variable and use these equations to solve problems. (3) Students continue their work with area from Grade 6, solving problems involving area and circumference of a circle and surface area of three-dimensional objects. In preparation for work on congruence and similarity in Grade 8 they reason about relationships among two-dimensional figures using scale drawings and informal geometric constructions, and they gain familiarity with the relationship between angles formed by intersecting lines. Students work with three-dimensional figures, relating them to two-dimensional figures by examining cross-sections. They solve real-world and mathematical problems involving area, surface area, and volume of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes and right prisms. (4) Students build on their previous work with single data distributions to compare two data distributions and address questions about difference between populations. They begin informal work with random sampling to generate data sets and learn about the importance of representative samples for drawing inferences.

Fluency Recommendations 7.EE.3: Students solve multistep problems posed with positive and negative rational numbers in any form (whole numbers, fractions and decimals), using tools strategically. This work is the culmination of many progressions of learning in arithmetic, problem solving and mathematical practices. 7.EE.4: In solving word problems leading to one-variable equations of the form px + q = r and p(x + q) = r, students solve the equations

fluently. This will require fluency with rational number arithmetic (7.NS.1–3), as well as fluency to some extent with applying properties operations to rewrite linear expressions with rational coefficients (7.EE.1). 7.NS.1–2: Adding, subtracting, multiplying and dividing rational numbers is the culmination of numerical work with the four basic operations. The number system will continue to develop in grade 8, expanding to become the real numbers by the introduction of irrational numbers,

and will develop further in high school, expanding to become the complex numbers with the introduction of imaginary numbers. Because there are no specific standards for rational number arithmetic in later grades and because so much other work in grade 7 depends on rational number arithmetic (see below), fluency with rational number arithmetic should be the goal in grade 7. The following English Language Arts LAFS should be taught throughout the course: LAFS.68.RST.1.3: Follow precisely a multistep procedure when carrying out experiments, taking measurements, or performing technical tasks. LAFS.68.RST.2.4: 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. LAFS.68.RST.3.7: 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). LAFS.68.WHST.1.1: Write arguments focused on discipline-specific content. Introduce claim(s) about a topic or issue, acknowledge and distinguish the claim(s) from alternate or opposing claims, and organize the reasons and evidence logically. Support claim(s) with logical reasoning and relevant, accurate data and evidence that demonstrate an understanding of the topic or text, using credible sources. Use words, phrases, and clauses to create cohesion and clarify the relationships among claim(s), counterclaims, reasons, and evidence. Establish and maintain a formal style. Provide a concluding statement or section that follows from and supports the argument presented. LAFS.68.WHST.2.4: Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. LAFS.7.SL.1.1: Engage effectively in a range of collaborative discussions (one-on-one, in groups, and teacher-led) with diverse partners on Grade 7 topics, texts, and issues, building on others’ ideas and expressing their own clearly. Come to discussions prepared, having read or researched material under study; explicitly draw on that preparation by referring to evidence on the topic, text, or issue to probe and reflect on ideas under discussion. Follow rules for collegial discussions, track progress toward specific goals and deadlines, and define individual roles as needed. Pose questions that elicit elaboration and respond to others’

questions and comments with relevant observations and ideas that bring the discussion back on topic as needed. Acknowledge new information expressed by others and, when warranted, modify their own views.

LAFS.7.SL.1.2: Analyze the main ideas and supporting details presented in diverse media and formats (e.g., visually, quantitatively, orally) and explain how the ideas clarify a topic, text, or issue under study LAFS.7.SL.1.3: Delineate a speaker’s argument and specific claims, evaluating the soundness of the reasoning and the relevance and sufficiency of the evidence. LAFS.7.SL.2.4: Present claims and findings, emphasizing salient points in a focused, coherent manner with pertinent descriptions, facts, details, and examples; use appropriate eye contact, adequate volume, and clear pronunciation.

M/J Mathematics 2, Advanced: Common Core State Standards At A Glance

First Nine Weeks Second Nine Weeks Third Nine Weeks Fourth Nine Weeks

Course: M/J Mathematics 2, Advanced Unit One: Rational Numbers

Essential Question(s): In what ways can rational numbers be useful?

DSA Unit 1- Rational Numbers MAFS.8.NS.1.1 MAFS.8.NS.1.2 Unit 2- Exponents MAFS.8.EE.1.1 MAFS.8.EE.1.2 MAFS.8.EE.1.3 MAFS.8.EE.1.4

DIA- Unit 1& 2

Unit 3- Equations & Inequalities MAFS.7.EE.2.3 MAFS.7.EE.2.4

DIA – Unit 3

NOTE: Percent would need

to reviewed

Unit 4 – Functions MAFS.8.F.1.1 MAFS.8.F.1.2 MAFS.8.F.1.3 MAFS.8.F.2.4 MAFS.8.F.2.5 MAFS.8.EE.2.5 MAFS.8.EE.2.6 MAFS.8.EE.3.7 MAFS.8.EE.3.8

DIA – Unit 4 Unit 5-Two & Three Dimensional Figures MAFS.7.G.2.4 MAFS.7.G.2.6 MAFS.8.G.3.9 MAFS.7.G.1.3 MAFS.7.G.1.1

SSA

Unit 6-Angles/Pythagorean Theorem MAFS.7.G.1.2 MAFS.7.G.2.5 MAFS.8.G.1.5 MAFS.8.G.2.6 MAFS.8.G.2.7 MAFS.8.G.2.8

DIA – Unit 6

Unit 7 - Transformations MAFS.8.G.1.1 MAFS.8.G.1.2 MAFS.8.G.1.3 MAFS.8.G.1.4

DIA – Unit 7

Unit 8 – Probability MAFS.7.SP.3.5 MAFS.7.SP.3.6 MAFS.7.SP.3.7 MAFS.7.SP.3.8

DIA –Unit 8

Unit 9 - Statistics MAFS.7.SP.1.1 MAFS.7.SP.1.2 MAFS.7.SP.2.3 MAFS.7.SP.2.4 MAFS.8.SP.1.1 MAFS.8.SP.1.2 MAFS.8.SP.1.3 MAFS.8.SP.1.4

STATE EOC

Standard The students will:

Learning Goals I can:

Remarks

Resources

MAFS.8.NS.1.1: Know that numbers that are not rational are call 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. SMP # 2

• classify a number as a rational or irrational based on its decimal expansion.

• convert a repeating decimal into a rational number

Prior Knowledge MARS Task: Division http://map.mathshell.org/materials/tasks.php?taskid=368&subpage=apprentice http://mathstar.lacoe.edu/lessonlinks/menu_math/poly_power.html http://www.funbrain.com/cgi-bin/cr.cgi http://illuminations.nctm.org/ActivityDetail.aspx?id=64 http://www.uen.org/curriculumsearch/searchResults.action http://www.cpalms.org/Resources/PublicPreviewResource9955.aspx http://www.cpalms.org/Resources/PublicPreviewResource35460.aspx http://www.cpalms.org/Resources/PublicPreviewResource8647.aspx http://www.riversidepublishing.com/commoncore/pdf/A2KMathCCSItemSampler.pdf

MAFS.8.NS.1.2: Use 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). SMP # 6

• use reasoning to determine between which two consecutive whole numbers a square root will fall

• plot the estimated value of an irrational number on a number line

• estimate the value of an irrational number by rounding to a specific place value

• apply estimated values to compare two or more irrational numbers

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 Students locate rational and irrational numbers on the number line. Students compare and order rational and irrational numbers. Additionally, students understand that the value of a square root can be approximated between integers and that non-perfect square roots are irrational. Students also recognize that square roots may be negative and written as - 28.

Course: M/J Mathematics 2, Advanced Unit Two: Exponents

Essential Question(s):

How can algebraic expressions be used to model, analyze, and solve mathematical situations? Standard

The students will: Learning Goal

I can:

Remarks Resources

MAFS.8.EE.1.1: Know and apply the properties of integer exponents to generate equivalent numerical expressions SMP # 7

• determine the properties of integer exponents by exploring patterns and applying my understanding of properties of whole number exponents

• apply the properties of integer exponents to simplify expressions

For example, 32 × 3–5 = 3–3 = 1/33 = 1/27. Students may make the relationship that in scientific notation, when a number contains one nonzero digit and a positive exponent, that the number of zeros equals the exponent. This pattern may incorrectly be applied to scientific notation values with negative values or with more than one nonzero digit. Students may mix up the product of powers property and the power of a power property.

http://insidemathematics.org/problems-of-the-month/pom-pollygone.pdf http://katm.org/wp/wp-content/uploads/flipbooks/8thFlipFinaledited.pdf http://www.youtube.com/watch?v=xidvf2YwCvA http://www.cpalms.org/Standards/PublicPreviewBenchmark5492.aspx http://education.ti.com/calculators/downloads/US/Activities/Detail?id=5302&ref=%2fcalculators%2fdownloads%2fUS%2fActivities%2fSearch%2fSubject%3fs%3d5022%26sa%3d1008%26t%3d1134%26d%3d9%26size%3d15%26page%3d2

MAFS.8.EE.1.2: 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 SMP # 2

• recognize taking a square root as the inverse of squaring a number

• recognize taking a cube root as the inverse of cubing a number

• evaluate the square root of a perfect square

• evaluate the cube root of a perfect cube

• justify that the square root of a non-perfect square will be 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 3 are inverse operations. This understanding is used to solve equations containing square or cube numbers. Equations may include rational numbers such as x-2= 1 /4, x2 = 4 /9 or x3 = 1/8 (NOTE: Both the numerator and denominators are perfect squares or perfect cubes.) Students recognize perfect squares and cubes, understanding that non-perfect squares and non-perfect cubes are irrational.

MAFS.8.EE.1.3: 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. SMP # 7

• write an estimation of a large quantity by expressing it as the product of a single-digit number and a positive power of ten

• write an estimation of a very small quantity by expressing it as the product of a single-digit number and a negative power of ten

• compare quantities written as the product of a single-digit number and a power of ten by stating their multiplicative relationships

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. Students compare and interpret scientific notation quantities in the context of the situation. If the exponent increases by one, the value increases 10 times. Students understand the magnitude of the number being expressed in scientific notation and choose an appropriate corresponding unit. For example, 3 x 108 is equivalent to 30 million, which represents a large quantity. Therefore, this value will affect the unit chosen

Course: M/J Mathematics 2, Advanced Unit Two: Exponents (cont)

Essential Question(s):

How can algebraic expressions be used to model, analyze, and solve mathematical situations? Standard

The students will: Learning Goal

I can:

Remarks Resources

MAFS.8.EE.1.4: 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 SMP # 6

• add and subtract two numbers written in scientific notation

• multiply and divide two numbers written in scientific notation

• select the appropriate units for measuring derived measurements when comparing quantities written in scientific notation

• identify and interpret the various ways scientific notation is displayed on calculators and through computer software

http://www.cpalms.org/Resources/PublicPreviewResource42388.aspx http://www.cpalms.org/Resources/PublicPreviewResource10082.aspx http://www.khanacademy.org/math/arithmetic/exponents-radicals http://education.ti.com/calculators/downloads/US/Activities/Detail?id=1643&ref=%2fcalculators%2fdownloads%2fUS%2fActivities%2fSearch%2fSubject%3fs%3d5022%26sa%3d5022%26t%3d5036%26d%3d9

Course: M/J Mathematics 2, Advanced Unit Three: Expressions and Equations

Essential Question(s): How can algebraic expressions and equations be used to model, analyze, and solve mathematical situations?

Standard The students will:

Learning Goals I can:

Remarks

Resources

MAFS.7.EE.2.3 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.

• Simplify an expression following and using the order of operations.

• determine and explain the solution of an equation.

• perform operations; addition, subtraction, multiplication and division of integers, fractions, and decimals.

• solve multi-step linear equations, including equations with rational coefficients using different strategies

.

Examples of Opportunities for In-Depth FocusThis is a major capstone standard for arithmetic and its applications.

For example: If a woman making $25 an hour gets 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 9¾ inches long in the center of a door that is 27½ 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.

http://www.graniteschools.org/depart/teachinglearning/curriculuminstruction/math/secondarymathematics/Math%207%20Lessons/23-NewMath7LessonEJan1SolvingEquations.pdf http://www.graniteschools.org/depart/teachinglearning/curriculuminstruction/math/secondarymathematics/Math%207%20Lessons/24-NewMath7LessonEJan2SolvingEquationsPartII.pdf http://www.graniteschools.org/depart/teachinglearning/curriculuminstruction/math/secondarymathematics/Math%207%20Lessons/25-NewMath7LessonEJan3SolvingInequalities.pdf http://www.graniteschools.org/depart/teachinglearning/curriculuminstruction/math/secondarymathematics/Math%207%20Lessons/26-NewMath7LessonEJan4ModelRealWorldWithEquations.pdf

MAFS.7.EE.2.4 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. 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. b. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. SMP #2

• use a variable to represent a unknown quantity. • develop an understanding representing linear

inequalities in one variable. • construct and solve simple equations and

inequalities. • solve simple equations and inequalities fluently. • apply the concepts to set up equations and

inequalities in problem solving situations. • graph an inequality on the number line. • work with the properties of used to solve linear

equations. • develop an understanding of linear inequalities

in one variable. • develop an understanding of linear inequalities

in one variable.

Examples of Opportunities for In-Depth FocusWork toward meeting this standard builds on the work that led to meeting 6.EE.2.7 and prepares students for the work that will lead to meeting 8.EE.3.7.

For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?

Course: M/J Mathematics 2, Advanced Unit Four: Functions

Essential Question(s): How can algebraic expressions and equations be used to model, analyze, and solve mathematical situations?

How are functions useful? Standard

The students will: Learning Goal

I can:

Remarks Resources

MAFS.8.F.1.1: Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is a set of ordered pairs consisting of an input and the corresponding output. SMP # 3

• explain that a function represents a relationship between an input and an output where the output depends on the input; therefore, there can only be one output for each input

• show the relationship between the inputs and outputs of a function by graphing them as ordered pairs on a coordinate grid

Students sometimes consider an equations such as x = 10 a linear function. This is a linear equation, but it is not a linear function because there are infinite output values for the input value of 10. There is a misunderstanding about the notation f and f(x). The notation f is the name of the function (or rule) and f(x) is the output from the rule when x is the input. Students sometimes believe that f is the only letter that can be used to represent a function rule

http://www.youtube.com/watch?v=VUTXsPFx-qQ&safety_mode=true&persist_safety_mode=1&safe=active http://education.ti.com/calculators/timathnspired/US/Activities/Detail?sa=1008&t=9449&id=16899 http://education.ti.com/calculators/timathnspired/US/Activities/Detail?id=11347&sa=5022&t=5030 http://www.cpalms.org/Resources/PublicPreviewResource8617.aspx

MAFS.8.F.1.2: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). SMP# 7

• determine the properties of a function written in algebraic form (e.g., rate of change, meaning of y-intercept, linear, non-linear)

• determine the properties of a function when given the inputs and outputs in a table

• determine the properties of a function represented as a graph

• determine the properties of a function when given the situation verbally

• compare the properties of two functions that are represented differently (e.g., as an equation, in a table, graphically, or a verbal representation)

Examples of Opportunities for In-Depth FocusWork toward meeting this standard repositions previous work with tables and graphs in the new context of input/output rules.

Students may not relate that all three forms are representing the same information. Students may not understand that the x and y on a table are the ordered pairs on the graph. Students may have difficulty with the vocabulary words and may need to be reminded of them.

Course: M/J Mathematics 2, Advanced Unit Four: Functions (cont)

Essential Question(s): How can algebraic expressions and equations be used to model, analyze, and solve mathematical situations?

How are functions useful? Standard

The students will: Learning Goal

I can:

Remarks Resources

MAFS.8.F.1.3: Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. SMP #3

• explain why the equation y = mx + b represents a linear function and interpret the slope and y-intercept in relation to the function.

• give examples of relationships that are non-linear functions.

• analyze the rate of change between input and output values to determine if function is linear or non-linear.

• create a table of values that can be defined as a non-linear function.

For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Some students will mistakenly think of a straight line as horizontal or vertical only. Some students will mix up x- and y-axes on the coordinate plane, or mix up the ordered pairs.

http://education.ti.com/calculators/downloads/US/Activities/Detail?ID=8592&MICROSITE=ACTIVITYEXCHANGE http://education.ti.com/calculators/downloads/US/Activities/Detail?ID=5488&MICROSITE=ACTIVITYEXCHANGE http://education.ti.com/calculators/downloads/US/Activities/Detail?ID=9630&MICROSITE=ACTIVITYEXCHANGE

MAFS.8.F.2.4: Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

SMP #1

• write a linear function that models a given situation given verbally, as a table of x and y values, or as a graph.

• define the initial value of the function in relation to the situation.

• define the rate of change in relation to the situation.

• define the y-intercept in relation to the situation.

• explain any constraints on the domain in relation to the situation.

Some students may not pay attention to the scale on a graph, assuming that the scale units are always “one”. When making axes for a graph, some students may not using equal intervals to create the scale. Some students graph incorrectly because they don‘t understand that x usually represents the independent variable and y represents the dependent variable.

Course: M/J Mathematics 2, Advanced Unit Four: Functions (cont)

Essential Question(s): How can algebraic expressions and equations be used to model, analyze, and solve mathematical situations?

How are functions useful?

Standard The students will:

Learning Goal I can:

Remarks

Resources

MAFS.8.F.2.5: Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

SMP #4

• match the graph of function to a given situation.

• write a story that describes the functional relationship between two variables depicted on a graph.

• create a graph of function that describes the relationship between two variables.

MAFS.8.EE.2.5: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. SMP #8

• graph a proportional relationship in the coordinate plane.

• interpret the unit rate of a proportional relationship as the slope of a graph.

• justify that the graph of a proportional relationship will always intersect the origin (0, 0) of the graph.

• use a graph, a table, or an equation to determine the unit rate of a proportional relationship and use the unit rate to make comparisons between various proportional relationships.

Examples of Opportunities for In-Depth Focus When students work toward meeting this standard, they build on grades 6–7 work with proportions and position themselves for grade 8 work with functions and the equation of a line. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. Students may still have difficulty finding slope. They can still mix up the relationship between change of y and change of x. For example: dividing the change in x by the change in y subtracting the x coordinate from the y coordinate subtracting the x and y coordinates in different order.

Course: M/J Mathematics 2, Advanced Unit Four: Functions (cont)

Essential Question(s): How can algebraic expressions and equations be used to model, analyze, and solve mathematical situations?

How are functions useful? Standard

The students will: Learning Goal

I can:

Remarks Resources

MAFS.8.EE.2.6: 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. SMP #1

• create right triangles by drawing a horizontal line segment and a vertical line segment from any two points on a non-vertical line in the coordinate plane.

• justify that these right triangles are similar by comparing the ratios of the lengths of the corresponding legs.

• justify that since the triangles are similar, the ratios of all corresponding hypotenuses, representing the slope of the line, will be equivalent.

• justify that an equation in the form of y = mx will represent the graph of a proportional relationship with a slope of m and a y-intercept of 0.

• justify that an equation in the form of y = mx represents the graph of a linear relationship with a slope of m and a y-intercept of b.

Students must be sure to use corresponding sides of triangles when the figures are turned around.

MAFS.8.EE.3.7: Solve linear equations in one variable. a. give examples of linear equation 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 x=a, a=a, or a=b results. b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using he distributive property and collecting like terms.

• use the properties of real numbers to determine the solution of a linear equation

• 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 successfully 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)

• simplify a linear equation by using the distributive property and/or combining like terms

• solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

Examples of Opportunities for In-Depth Focus This is a culminating standard for solving one-variable linear equations. Students may think that checking solutions is redundant and useless. Disregarding signs when manipulating expressions. Students will sometimes combine unlike terms such as 2x + 5 combines to 7x. Students will sometimes forget to multiply the outside number by both numbers in the parentheses with distributive property. While students may understand to distribute the coefficient, they sometimes will forget to distribute the negative sign.

Course: M/J Mathematics 2, Advanced Unit Four: Functions (cont)

Essential Question(s):

How can algebraic expressions and equations be used to model, analyze, and solve mathematical situations? How are functions useful?

Standard The students will:

Learning Goal I can:

Remarks

Resources

MAFS.8.EE.3.8 : Analyze and solve pairs of simultaneous linear equations. a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. c. Solve real-world and mathematical problems leading to two linear equations in two variables. c. solve real-world and mathematical problems leading to two linear equations in two variable.

• explain how a line represents the infinite number of solutions to a linear equation with two variables.

Examples of Opportunities for In-Depth FocusWhen students work toward meeting this standard, they build on what they know about two-variable linear equations, and they enlarge the varieties of real-world and mathematical problems they can solve.

For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. 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. For example, given coordinates for two pairs of points, determine whether the line though the first pair of points intersects the line through the second pair.

Course: M/J Mathematics 2, Advanced Unit Five: Two and Three Dimensional Figures

Essential Question(s):

How does geometry better describe objects? Standard

The students will: Learning Goals

I can:

Remarks Resources

MAFS.7.G.2.4 Know the formulas for the area and circumference of a circle and solve problems; give an informal derivation of the relationship between the circumference and area of a circle. SMP #6

• state the formula for finding the circumference of a circle.

• state the formula for finding the area of a circle.

• calculate the circumference of a circle. • understand how the formula for the area

of a circle can be derived from the area of a parallelogram.

• understand the relationship between area and circumference of a circle.

• calculate the area of a circle. • determine the diameter or radius of a

circle when the circumference is given.

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).

Geometric constructions: http://www.opusmath.com/common-core-standards/7.g.1-solve-problems-involving-scale-drawings-of-geometric-figures-including http://www.opusmath.com/common-core-standards/7.g.2-draw-freehand-with-ruler-and-protractor-and-with-technology-geometric http://www.opusmath.com/common-core-standards/7.g.4-know-the-formulas-for-the-area-and-circumference-of-a-circle-and-use-them http://www.opusmath.com/common-core-standards/7.g.6-solve-real-world-and-mathematical-problems-involving-area-volume-and https://www.teachingchannel.org/videos/preparing-students-for-exams

MAFS.7.G.2.6 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. SMP #2

• determine the area of two-dimensional figures.

• determine the surface area and volume of three-dimensional figures.

• solve real-world problems involving area, surface area and volume.

• draw 3-dimentional figures.

Examples of Opportunities for In-Depth FocusWork toward meeting this standard draws together grades 3–6 work with geometric measurement.

Course: M/J Mathematics 2, Advanced Unit Five: Two and Three Dimensional Figures

Essential Question(s):

How does geometry better describe objects? Standard

The students will: Learning Goals

I can:

Remarks Resources

MAFS.8.G.3.9: Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. SMP#4

• describe the similarity between finding the volume of a cylinder and the volume of a right prism.

• recall the formula to find the volume of a cylinder, a cone, and a sphere.

• informally prove the relationship between the volume of a cylinder and the volume of a cone with the same base; and the volume of a sphere and the volume of a circumscribed cylinder.

• apply the formulas to find the volume of cylinders, cones, and spheres.

• solve real-world problems involving the volume of cylinders, cones, and spheres.

A common misconception among middle grade students is that “volume” is a “number” that results from substituting other numbers into a formula. For these students there is no recognition that “volume” is a measure – related to the amount of space occupied. It is important to provide opportunities for hands on experiences where students fill three dimensional objects.

MAFS.7.G.1.3 Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids SMP #4

• identify the two-dimensional figure that represents a particular slice of a three-dimensional figure.

Course: M/J Mathematics 2, Advanced

Unit Five: Two and Three Dimensional Figures

Essential Question(s): How does geometry better describe objects?

Standard The students will:

Learning Goals I can:

Remarks

Resources

MAFS.7.G.1.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas form a scale drawing and reproducing a scale drawing at a different scale.

• use a scale drawing to determine the actual dimensions and area of a geometric figure.

• use a different scale to reproduce a similar scale drawing.

MARS Task: Which is Bigger? http://insidemathematics.org/common-core-math-tasks/7th-grade/7-2004%20Which%20is%20Bigger.pdf Problem of the month: What’s your angle? http://insidemathematics.org/problems-of-the-month/pom-whatsyourangle.pdf

Course: M/J Mathematics 2, Advanced Unit Six: Angles/ Pythagorean Theorem

Essential Question(s): How can you convert from one measurement system to another?

How can you use similar triangles to solve problems? Standard

The students will: Learning Goals

I can:

Remarks Resources

MAFS.7.G.1.2 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

• draw a geometric shape with specific conditions.

• construct a triangle when given three measurement: 3 side lengths, 3 angle measurements, or a combination of side and angle measurements.

• determine when three specific measurements will result in one unique triangle, more than one possible triangle or no possible triangles.

MARS Task: Which is Bigger? http://insidemathematics.org/common-core-math-tasks/7th-grade/7-2004%20Which%20is%20Bigger.pdf Problem of the month: What’s your angle? http://insidemathematics.org/problems-of-the-month/pom-whatsyourangle.pdf

MAFS.7.G.2.5 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.

• state the relationship between supplementary, complementary, and vertical angles.

• use angle relationships to write algebraic equations for unknown angles.

• use algebraic reasoning and angle relationships to solve multi-step problems.

MAFS.8.G.1.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. SMP#4

• prove that the sum of any triangle’s interior angles will have the same measure as a straight angle.

• prove that the sum of any polygon’s exterior angles will be 360-degrees.

• make conjectures regarding the relationships and measurements of the angles created when two parallel lines are cut by a transversal.

• apply proven relationships to establish minimal properties to justify similarity.

Students will need reminders and review of the different types of angles formed by two lines cut by a transversal. Students will sometimes mistake angles that are supplementary with angles that are congruent, when two parallel lines are cut by a transversal. Students may have difficulty decomposing polygons into triangles. Students may struggle to develop the formula to find sums of interior angles of polygons.

Course: M/J Mathematics 2, Advanced Unit Six: Angles/ Pythagorean Theorem

Essential Question(s): How can you convert from one measurement system to another?

How can you use similar triangles to solve problems? Standard

The students will: Learning Goal

I can:

Remarks Resources

MAFS.8.G.2.6 Explain a proof of the Pythagorean Theorem and it converse. SMP#4, 7

• demonstrate the relationship of the three side lengths of any right angle by the use of visual models.

• apply algebraic reasoning to relate the visual model to Pythagorean Theorem.

• determine if a given triangle is a right triangle using Pythagorean Theorem.

Students often miss the idea that this works for only RIGHT triangles. It is important for students to explore the relationships of the areas of the side length of non-right triangles to clarify that a2 + b2 = c2 is only true for right triangles.

Students will sometimes substitute the hypotenuse in for a or b instead of c in the equation.

http://illuminations.nctm.org/LessonDetail.aspx?ID=L683 http://www.geom.uiuc.edu/~dwiggins/conj04.html http://similartriangles3.pbworks.com/w/page/23053498/Applying%20Similar%20Triangles%20to%20the%20Real%20World http://www.mathinterventions.org/files/uploads/Proportional_Reasoning_Similar_Polygons-1.pdf http://www.geom.uiuc.edu/~dwiggins/conj04.html http://www.mathwarehouse.com/geometry/triangles/index.php http://illuminations.nctm.org/LessonDetail.aspx?id=L745 http://www.sophia.org/angles-formed-by-transversal-lines-tutorial http://illuminations.nctm.org/LessonDetail.aspx?id=L765

MAFS.8.G.2.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. SMP#4

• apply the Pythagorean Theorem to find an unknown side length of a right triangle.

• draw a diagram and use the Pythagorean Theorem to solve real-world problems involving right triangles.

• draw a diagram to find right triangles in a three-dimensional figure and use the Pythagorean Theorem to calculate various dimensions.

Examples of Opportunities for In-Depth FocusThe Pythagorean theorem is useful in practical problems, relates to grade-level work in irrational numbers and plays an important role mathematically in coordinate geometry in high school.

Students will make the mistake of just adding the side lengths (a + b = c) Students will do a2 + b2 = c and forget to take the square root of c2 to find the length of the hypotenuse.

MAFS.8.G.2.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. SMP#2

• connect any two points on a coordinate grid to a third point so that the three points form a right triangle.

• apply the Pythagorean Theorem with the characteristics of a right triangle to find the distance between the original two points.

When finding a length of a segment drawn on dot paper, students count the dots instead of length in units.

Course: M/J Mathematics 2, Advanced

Unit Seven: Transformations

Essential Question(s): How are figures, angles, similarity and congruency related to Pythagorean Theorem?

Standard The students will:

Learning Goal I can:

Remarks

Resources

MAFS.8.G.1.1: Verify experimental properties of rotations, reflections and translations: a. Lines are taken to lines, and line

segments to line segments of the same length.

b. Angles are taken to angles of the same measure.

c. Parallel lines are taken to parallel lines.

SMP #3

• verify - by measuring and comparing lengths, angle measures, and parallelism of a figure and its image – that after a figure has been translated, reflected, or rotated, corresponding lines and line segments remain the same length, corresponding angles have the same measure, and corresponding parallel lines remain parallel.

Students sometimes believe that a reflection over y = x or y= -x is like a rotation of it. The correct reflection will actually look wrong compared to the original drawing. One way to help with this is to have the students physically fold their paper to make sure it does reflect over the correct line

http://www.shodor.org/interactivate/lessons/TranslationsReflectionsRotations/ http://www.math-drills.com/graphpaper.shtml http://www.math-drills.com/geometry.shtml http://www.opusmath.com/common-core-standards/8.g.3-describe-the-effect-of-dilations-translations-rotations-and-reflections http://education.ti.com/calculators/downloads/US/Activities/Detail?id=10278&ref=%2fcalculators%2fdownloads%2fUS%2fActivities%2fSearch%2fSubject%3fd%3d1007%26size%3d15%26page%3d94

MAFS.8.G.1.2: Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. SMP#4

• explain how transformations can be used to prove that two figures are congruent.

• perform a series of transformations (reflections, rotations, and/or translations) to prove or disprove that two given figures are congruent.

When labeling transformed images, students sometimes mistakenly apply the wrong label. Either they use the wrong letter, or forget the prime notation (e.g., A A instead of A A )

Course: M/J Mathematics 2, Advanced Unit Seven: Transformations (cont)

Essential Question(s): How does transformations relate to similarity and congruency?

Standard The students will:

Learning Goal I can:

Remarks

Resources

MAFS.8.G.1.3: Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. SMP#2

• describe the changes occurring to x- and y- coordinates of a figure after a translation, reflection, rotation, or dilation.

Students may identify the incorrect line of symmetry (x-axis vs. y-axis) when reflecting.

In translations, students can sometimes count to their points incorrectly or switch the order of the coordinates when writing the point down.

http://insidemathematics.org/common-core-math-tasks/8th-grade/8-2006%20Aaron's%20Designs.pdf http://education.ti.com/calculators/downloads/US/Activities/Detail?id=17252&ref=%2fcalculators%2fdownloads%2fUS%2fActivities%2fSearch%2fSubject%3fd%3d1007%26size%3d15%26page%3d100

MAFS.8.G.1.4: Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. SMP#4

• explain how transformations can be used to prove that two figures are similar.

• describe a sequence of transformations to prove or disprove that two given figures are similar.

Students may have difficulty working with rotations.

Students may only know terms used in earlier grades such as flip, slide, and turn.

Course: M/J Mathematics 2, Advanced Unit Eight: Probability and Statistical Graphs

Essential Question(s): How is probability used to make informed decisions about uncertain events?

Standard The students will:

Learning Goals I can:

Remarks

Resources

MAFS.7.SP.3.5 Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around ½ indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

• define probability as a ratio that compare favorable outcomes to all possible outcomes.

• recognize and explain that probabilities are expressed as a number between 0 to 1.

• interpret a probability near 0 as unlikely to occur and a probability near 1 as likely to occur.

• interpret a probability near ½ as being as equally to occur as to not occur.

http://www.graniteschools.org/depart/teachinglearning/curriculuminstruction/math/secondarymathematics/Math%207%20Lessons/42NewMath7LessonHApril4Probability.pdf

MAFS.7.SP.3.6 Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.

• collect data on a chance process to approximate its probability.

• use probability to predict the number of times a particular event will occur given a specific number of trials.

• use variability to explain why the experimental probability will not always exactly equal the theoretical probability.

For example, when rolling a number cure 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.

Course: M/J Mathematics 2, Advanced Unit Eight: Probability and Statistical Graphs (cont)

Essential Question(s): How is probability used to make informed decisions about uncertain events?

Standard The students will:

Learning Goals I can:

Remarks

Resources

MAFS.7.SP.3.7 Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. a. develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. b. develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.

• develop a simulation to model a situation in which all events are equally likely to occur.

• utilize the simulation to determine the probability of specific events.

• determine the probability of events that may not be equally likely to occur, by utilizing a simulation model.

For example, if a student is selected at random from a class, find the probability that a girl will be selected. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies? i) Simple events only.

http://www.graniteschools.org/depart/teachinglearning/curriculuminstruction/math/secondarymathematics/Math%207%20Lessons/40-NewMath7LessonHApr2CircleGraphs.pdf

MAFS.7.SP.3.8 Find probabilities of compound events using organized lists, tables, tree diagrams and simulation. a. understand that , just as with simple events, the probability of a compound event is the fraction of outcomes in the staple space for which the compound event occurs. b. represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (eg. “rolling double sixes”), identify the outcomes in the sample space which compose the event. c. design and use a simulation to generate frequencies of compound events.

• create a sample space of all possible outcomes for a compound event by using an organized list, a table or a tree diagram.

• use the sample space to compare the number of favorable outcomes to the total number of outcomes and determine the probability of the compound event.

• design and utilize a simulation to predict the probability of a compound event.

For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?

Course: M/J Mathematics 2, Advanced Unit Nine: Statistics

Essential Question(s): How can you use statistical displays to analyze, interpret and predict data?

How can you use measures of central tendency to distribute an amount evenly among a group of people? How can you use a formula for one measurement to write a formula for a different measurement?

Standard The students will:

Learning Goals I can:

Remarks

Resources

MAFS.7.SP.1.1 Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

• explain that inferences about a population can be made by examining a sample.

• explain why the validity of a sample depends on whether the sample is a representative of the population.

• explain that random sampling trends to produce representative samples.

MAFS.7.SP.1.2 Use data from a random sample to draw inference about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.

• draw inferences about a population based on data generated by a random sample.

• generate multiple samples from the same population and analyze the estimates or predictions based on the variation of each sample.

For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.

MAFS.7.SP.2.3 Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.

• find the difference in the mean or median of two different data sets.

• demonstrate how two data sets that are very different can have similar variabilities.

• draw inferences about the data sets by making a comparison of these difference relative to the mean absolute deviation or interquartile range of either set of data.

For example, the mean height of players on the basketball team is 1- cm greater than the mean height of players on the soccer team, about twice the variability ( mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.

MAFS.7.SP.2.4 Use measure of center and measure of variability for numerical data from random samples to draw informal comparative inference about two populations.

• compare two populations by using the means and/or medians of data collected from random samples.

• compare two populations by using the mean absolute deviations and/or interquartile ranges of data from random samples.

For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.

Course: M/J Mathematics 2, Advanced Unit Nine: Statistics (cont)

Essential Question(s): How can you use statistical displays to analyze, interpret and predict data?

How can you use measures of central tendency to distribute an amount evenly among a group of people? How can you use a formula for one measurement to write a formula for a different measurement?

Standard The students will:

Learning Goals I can:

Remarks

Resources

MAFS.8.SP.1.1: Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. SMP#4

• plot ordered pairs on a coordinate grid representing the relationship between two data sets.

• describe patterns in the plotted points such as clustering, outliers, positive or negative association, and linear or non-linear association and describe the pattern in the context of the measurement data.

• interpret the patterns of association in the context of the data sample.

When trying to decide correlation of a data set students will sometimes have confusion about whether there is positive, negative, or no correlation.

MAFS.8.SP.1.2: Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. SMP#3

• recognize whether or not data plotted on a scatter plot have a linear association.

• draw a straight trend line to approximate the linear relationship between the plotted points of two data sets.

• make inferences regarding the reliability of the trend line by noting the closeness of the data points to the line.

Students think the line of best fit must go through (0,0). This is sometimes true, but students need to be able to understand what the y-intercept means in the situation and determine if it makes sense given the situation. In general, students think there is only one correct answer in mathematics. Students may mistakenly think their lines of best fit for the same set of data will be exactly the same. Because students are informally drawing lines of best fit, the lines will vary slightly. Students will often times only chose data points that are plotted on the graph in order to find the line of best fit. Sometimes this will work because their line will go through two of the original data points. However, often times the line may NOT go through any of the data points and students still pick only the plotted points.

Course: M/J Mathematics 2, Advanced Unit Nine: Statistics (cont)

Essential Question(s): How can you use statistical displays to analyze, interpret and predict data?

How can you use measures of central tendency to distribute an amount evenly among a group of people? How can you use a formula for one measurement to write a formula for a different measurement?

Standard The students will:

Learning Goals I can:

Remarks

Resources

MAFS.8.SP.1.3: Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. SMP#2

• determine the equation of the trend line that approximates the linear relationship between the plotted points of two data sets.

• interpret the y-intercept of the equation in the context of the collected data.

• interpret the slope of the equation in the context of the collected data.

• use the equation of the trend line to summarize the given data and make predictions regarding additional data points.

For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. Students think the line of best fit must go through (0,0). This is sometimes true, but students need to be able to understand what the y-intercept means in the situation and determine if it makes sense given the situation.

MAFS.8.SP.1.4: Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two. SMP#3

• create a two-way table to record the frequencies of bivariate categorical values.

• determine the relative frequencies for rows and/or columns of a two-way table.

• use the relative frequencies and context of the problem to describe possible associations between the two sets of data.

Students may believe that bivariate data is only displayed in scatter plots. It is important to provide the opportunity to display bivariate, categorical data in a table.