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HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5472" \t "_top" MAFS.7.EE.2.3Solve multi-step real-life and mathematical problems posed with positive and negative HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" rational numbersin any form ( HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" whole numbers, HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" 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 HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" strategies. 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 9 3/4 inches long in the center of a door that is 27 1/2 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.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5473" \t "_top" MAFS.7.EE.2.4Use 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. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width? Solve word problems leading to inequalities of the form px + q r or px + q r, where p, q, and r are specific HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5474" \t "_top" MAFS.7.G.1.1Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5475" \t "_top" MAFS.7.G.1.2Draw (freehand, with ruler and protractor, and with HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" technology) geometric shapes with given conditions. HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" Focuson constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5476" \t "_top" MAFS.7.G.1.3Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5477" \t "_top" MAFS.7.G.2.4Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5478" \t "_top" MAFS.7.G.2.5Use 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.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5479" \t "_top" MAFS.7.G.2.6Solve 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.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5480" \t "_top" MAFS.7.SP.1.1Understand 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.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5481" \t "_top" MAFS.7.SP.1.2Use data from a random sample to draw inferences 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. 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.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5482" \t "_top" MAFS.7.SP.2.3Informally 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. For example, the mean height of players on the basketball team is 10 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.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5483" \t "_top" MAFS.7.SP.2.4Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. 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.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5484" \t "_top" MAFS.7.SP.3.5Understand 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 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5485" \t "_top" MAFS.7.SP.3.6 HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" Approximatethe 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. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5486" \t "_top" MAFS.7.SP.3.7Develop 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. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. 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?
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5487" \t "_top" MAFS.7.SP.3.8Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language ( HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" e.g., rolling double sixes), identify the outcomes in the sample space which compose the event. Design and use a simulation to generate frequencies for compound events. 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?
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5490" \t "_top" MAFS.8.EE.1.1Know and apply the properties of integer HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" exponentsto generate equivalent numerical expressions. For example, 3 ==1/3=1/27
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5491" \t "_top" MAFS.8.EE.1.2Use square root and cube root symbols to represent solutions to equations of the form x = p and x = 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.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5492" \t "_top" MAFS.8.EE.1.3Use numbers expressed in the form of a single digit times an integer HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" powerof 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 and the population of the world as 7 , and determine that the world population is more than 20 times larger.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5493" \t "_top" MAFS.8.EE.1.4Perform 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 ( HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" technology.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5494" \t "_top" MAFS.8.EE.2.5Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different 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 HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" speed.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5495" \t "_top" MAFS.8.EE.2.6Use 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 HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" axisat b.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5496" \t "_top" MAFS.8.EE.3.7Solve linear equations in one variable. 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 successively 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). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5497" \t "_top" MAFS.8.EE.3.8Analyze and solve pairs of simultaneous linear equations. 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. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. Solve real-world and mathematical problems 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.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5498" \t "_top" MAFS.8.F.1.1Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5499" \t "_top" MAFS.8.F.1.2Compare properties of two HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" functionseach represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5500" \t "_top" MAFS.8.F.1.3Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" functionsthat are not linear. For example, the function A = s 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.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5501" \t "_top" MAFS.8.F.2.4Construct 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.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5502" \t "_top" MAFS.8.F.2.5Describe qualitatively the functional relationship between two quantities by analyzing a graph ( HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" 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.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5503" \t "_top" MAFS.8.G.1.1Verify experimentally the properties of rotations, reflections, and translations: Lines are taken to lines, and line segments to line segments of the same length. Angles are taken to angles of the same measure. Parallel lines are taken to parallel lines.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5504" \t "_top" MAFS.8.G.1.2Understand 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.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5505" \t "_top" MAFS.8.G.1.3Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5506" \t "_top" MAFS.8.G.1.4Understand 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.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5507" \t "_top" MAFS.8.G.1.5Use 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. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5508" \t "_top" MAFS.8.G.2.6Explain a proof of the HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" Pythagorean Theoremand its converse.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5509" \t "_top" MAFS.8.G.2.7Apply the HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" Pythagorean Theoremto determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5510" \t "_top" MAFS.8.G.2.8Apply the HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" Pythagorean Theoremto find the distance between two points in a coordinate system.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5511" \t "_top" MAFS.8.G.3.9Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5488" \t "_top" MAFS.8.NS.1.1Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" rational numbersshow that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5489" \t "_top" MAFS.8.NS.1.2Use 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 ( HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" e.g., ). 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.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5512" \t "_top" MAFS.8.SP.1.1Construct 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.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5513" \t "_top" MAFS.8.SP.1.2Know 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.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5514" \t "_top" MAFS.8.SP.1.3Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. 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.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/5515" \t "_top" MAFS.8.SP.1.4Understand 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 variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?
The following overarching Mathematics Standards are included in the delivery of each of the above standards.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/6327" \t "_top" MAFS.K12.MP.1.1Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" algebraic expressionsor change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/6328" \t "_top" MAFS.K12.MP.2.1Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/6329" \t "_top" MAFS.K12.MP.3.1Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" progressionof statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument, explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/6331" \t "_top" MAFS.K12.MP.4.1Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/6332" \t "_top" MAFS.K12.MP.5.1Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" functionsand solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/6333" \t "_top" MAFS.K12.MP.6.1Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/6334" \t "_top" MAFS.K12.MP.7.1Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered 7 5 + 7 3, in preparation for learning about the distributive property. In the expression x + 9x + 14, older students can see the 14 as 2 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/6335" \t "_top" MAFS.K12.MP.8.1Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x + x + 1), and (x 1)(x + x + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.
The following Language Arts Standards are embedded in the delivery of all the Mathematics Standards.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/6164" \t "_top" LAFS.68.RST.1.3Follow precisely a multistep procedure when carrying out experiments, taking measurements, or performing technical tasks.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/6165" \t "_top" LAFS.68.RST.2.4Determine 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 68 texts and topics.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/6168" \t "_top" LAFS.68.RST.3.7Integrate quantitative or technical information expressed in words in a text with a version of that information expressed visually ( HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" e.g., in a flowchart, diagram, model, graph, or table).
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/6224" \t "_top" LAFS.68.WHST.1.1Write 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.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/6226" \t "_top" LAFS.68.WHST.2.4Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/6096" \t "_top" LAFS.7.SL.1.1Engage 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.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/6097" \t "_top" LAFS.7.SL.1.2Analyze the main ideas and supporting details presented in diverse media and formats ( HYPERLINK "http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10286" \o "" e.g., visually, quantitatively, orally) and explain how the ideas clarify a topic, text, or issue under study.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/6098" \t "_top" LAFS.7.SL.1.3Delineate a speakers argument and specific claims, evaluating the soundness of the reasoning and the relevance and sufficiency of the evidence.
HYPERLINK "http://www.cpalms.org/Public/PreviewStandard/Preview/6099" \t "_top" LAFS.7.SL.2.4Present 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.
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