Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Aligning NJ Grade 7 Mathematics Curricula to the Common Core State Standards NEW OLD Common Core State Standards (CCSS) adopted June 16, 2010 2008 NJ Core Curriculum Content Standards (NJ cccs) How is it related to the old content? If not related, where did old content go? Ratios and Proportional Relationships 7.RP Analyze proportional relationships and use them to solve real-world and mathematical problems. 7.RP.1. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour. 7.RP.2. Recognize and represent proportional relationships between quantities. a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. This CCSS moves “rates” from grade 8 (4.1.8.A.3) to grade 7. It also moves “compound measurement units” from grade 8 (4.2.8.D.6) to grade 7. NEW (to grade 7) 4.1.7.A.3. Understand and use ratios, proportions, and percents (including percents greater than 100 and less than 1) in a variety of situations. 4.2.7.C.1. Use coordinates in four quadrants to represent geometric concepts. 4.3.7 B.1. Graph functions, and understand and describe their general behavior. Equations involving two variables 4.3.7.C.1. Analyze functional relationships to explain how a change in one quantity can result in a change in another, using pictures, graphs, charts, and equations. 4.3.7.A.1. Recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers. Descriptions using tables, verbal and symbolic rules, graphs, simple equations or expressions Finite and infinite sequences Generating sequences by using calculators to repeatedly apply a formula Related, but the new CCSS contain more specific expectations regarding proportional relationships. Related, but the new CCSS move “rates” from grade 8 to grade 7. c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. Although the new CCSS postpone functions until grade 8, the use of symbolic algebra (equations) to model (represent) relationships is explicit in bullet 1 of NJ cccs 4.3.7.C.2. 4.3.7.C.2. Use patterns, relations, symbolic algebra, and linear functions to model situations. Using manipulatives, tables, graphs, verbal rules, algebraic expressions/ equations/inequalities Growth situations, such as population growth and compound interest, using recursive (e.g., NOW-NEXT) formulas (cf. science standards and social studies standards) d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. 7.RP.3. Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. The new CCSS move “rates” from grade 8 to grade 7. NEW (to grade 7) Although possibly new in many 7th grade classes, the content may have been included in others as “a variety of situations” under NJ cccs 4.1.7.A.3. The new CCSS postpone introducing the concept of a function until grade 8. Without the formal terminology, sequences are introduced in grades 4 and 5 in the CCSS. The formal study of arithmetic and geometric sequences is postponed until HS. Growth functions and the formal study of arithmetic and geometric sequences are postponed until HS. [“Understand and use ratios, proportions, and percents (including percents greater than 100 and less than 1) in a variety of situations” from 4.1.7.A.3 above.] Related, but the new CCSS contain more specific expectations regarding proportional relationships. Shaded content is not a focus in this grade because it has already been covered in previous grades, because it will be covered in later grades, or because it is not included at any grade in the new CCSS. April 2011 1 The Number System 7.NS Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. 7.NS.1. Apply and extend previous Similar, but the understandings of addition and subtraction new CCSS contain to add and subtract rational numbers; more specific represent addition and subtraction on a expectations horizontal or vertical number line diagram. regarding operations a. Describe situations in which opposite involving negative quantities combine to make 0. For numbers. example, a hydrogen atom has 0 4.1.7.A.1. Extend understanding of the number system by constructing meanings for the following: Rational numbers Percents Whole numbers with exponents 4.1.7.B.1. Use and explain procedures for per-forming calculations with integers and all number types named above (Rational numbers, Percents, Whole numbers with exponents) with: Pencil-and-paper Mental math Calculator charge because its two constituents are oppositely charged. b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. 4.3.7 D.1. Use graphing techniques on a number line. Absolute value Arithmetic operations represented by vectors (arrows) (e.g., “-3 + 6” is “left 3, right 6”) c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. 4.3.7.D.4. Understand and apply the properties of operations, numbers, equations, and inequalities. Additive inverse Multiplicative inverse 4.1.7.B.1. Use and explain procedures for performing calculations with integers and all number types named above (Rational numbers, Percents, Whole numbers with exponents) with: Pencil-and-paper Mental math Calculator d. Apply properties of operations as strategies to add and subtract rational numbers. 7.NS.2. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world contexts. c. Apply properties of operations as strategies to multiply and divide rational numbers. Related, but the new CCSS contain more specific expectations regarding operations involving negative numbers. d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. Similar 4.3.7.D.4. Understand and apply the properties of operations, numbers, equations, and inequalities. Additive inverse Multiplicative inverse 4.1.7.A.6. Understand that all fractions can be represented as repeating or terminating decimals. Related but much more specific expectation Exponents get more attention in grades 6 and HS in the CCSS. Also linked to CCSS 7.NS.2 below for multiplication and division In Transition: Students coming to seventh grade from classes in which the 2008 standards were used may not yet have been introduced to absolute value. Until the curriculum change has been implemented at grade 6, teachers will need to continue introducing this concept at grade 7. Also linked to CCSS 7.NS.1 above for addition and subtraction Also linked to the new CCSS 7.EE.1 above Shaded content is not a focus in this grade because it has already been covered in previous grades, because it will be covered in later grades, or because it is not included at any grade in the new CCSS. April 2011 2 4.1.7.A.5. Use whole numbers, fractions, decimals, and percents to represent equivalent forms of the same number. From the introduction to Grade 7, critical area (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.” 7.NS.3. Solve real-world and mathematical problems involving the four operations with rational numbers. [“Computations with rational numbers extend the rules for manipulating fractions to complex fractions.” (Footnote to Common Core State Standards)] 4.1.7.A.2. Demonstrate a sense of the relative magnitudes of numbers (as applied to rational numbers and percents). 4.1.7.A.4. Compare and order numbers of all named types Rational numbers Percents Whole numbers with exponents While not explicitly articulated in the CCSS for grade 7, these expectations from the 2008 NJ cccs are part of the “understanding of number” as explained in the CCSS Grade 7 introduction. Compare & order are not explicitly articulated in the grade 7 CCSS. Exponents get more attention in grades 6 and HS in the CCSS. 4.5.7.A.2. Solve problems that arise in mathematics and in other contexts. Open-ended problems Non-routine problems Problems with multiple solutions Problems that can be solved in several ways [An application of CCSS 7.NS.1 and 7.NS.2 (NJ cccs 4.1.7.B.1) above] Similar, except that manipulating complex fractions is new at this grade level. Expressions and Equations 7.EE 4.3.7.D.3. Create, evaluate, and simplify algebraic expressions involving variables. Order of operations, including appropriate use of parentheses Substitution of a number for a variable In CCSS, this is 6th grade content (6.EE.2). 4.3.7.D.4. Understand and apply the properties of operations, numbers, equations, and inequalities. Additive inverse Multiplicative inverse 4.5.7.E.2. Select, apply, and translate among mathematical representations to solve problems. Also linked to the new CCSS 7.NS.1d and 7.NS.2c above 4.1.7 B.2. Use exponentiation to find whole number powers of numbers. CCSS move this from grade 7 to grade 6 (6.EE.1 & 2c). In Transition: Students coming to seventh grade from classes in which the 2008 standards were used may not yet have mastered this content. Until the curriculum change has been implemented at grade 6, teachers will need to continue including this material at grade 7. Use properties of operations to generate equivalent expressions. 7.EE.1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. Related but more specific expectation that goes beyond the 2008 NJ cccs for this grade level. 7.EE.2. Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.” Instructional guidance beyond the level of specificity provided in the NJ cccs In Transition: Students coming to seventh grade from classes in which the 2008 standards were used may not yet have mastered this content. Until the curriculum change has been implemented at grade 6, teachers will need to continue including this material at grade 7. Shaded content is not a focus in this grade because it has already been covered in previous grades, because it will be covered in later grades, or because it is not included at any grade in the new CCSS. April 2011 3 4.1.7 B.3. Understand and apply the standard algebraic order of operations, including appropriate use of parentheses. [Also including exponents, from NJ cccs 4.1.7.A.1 and 4.1.7.B.2] In CCSS, this is grade 6 content (6.EE.1 & 6.EE.2c). See also 5.OA.1 & 2. In Transition: Students coming to seventh grade from classes in which the 2008 standards were used may not yet have mastered this content. Until the curriculum change has been implemented at grade 6, teachers will need to continue including this material at grade 7. Solve real-life and mathematical problems using numerical and algebraic expressions and equations. 7.EE.3. Solve multi-step real-life and Similar but more mathematical problems posed with positive specific and negative rational numbers in any form expectation (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. 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. 7.EE.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. 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? b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific 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. 4.5.7.A.2. Solve problems that arise in mathematics and in other contexts. Open-ended problems Non-routine problems Problems with multiple solutions Problems that can be solved in several ways 4.1.7.C.1. Use equivalent representations of numbers such as fractions, decimals, and percents to facilitate estimation. 4.5.7.D.4. Rely on reasoning, rather than answer keys, teachers, or peers, to check the correctness of their problem solutions. Somewhat similar, although CCSS move fluent use of algebraic methods from grade 8 to grade 7. Note that the new CCSS are more limiting in terms of types of equations to be solved. Note also that this CCSS allows rational coefficients. 4.3.7.D.2. Solve simple linear equations informally and graphically. Multi-step, integer coefficients only (although answers may not be integers) Using paper-and-pencil, calculators, graphing calculators, spreadsheets, and other technology Somewhat related but much more specific expectation. CCSS move the solving of linear inequalities from grade 8 (4.3.8.D.3) to grade 7. 4.3.7.D.4. Understand and apply the properties of operations, numbers, equations, and inequalities. Additive inverse Multiplicative inverse Also linked to the new CCSS 7.EE.1, 7.NS.1d, and 7.NS.2c above NEW (to grade 7) Geometry 7.G Draw, construct, and describe geometrical figures and describe the relationships between them. Shaded content is not a focus in this grade because it has already been covered in previous grades, because it will be covered in later grades, or because it is not included at any grade in the new CCSS. April 2011 4 7.G.1. Solve 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. Similar although slightly more specific expectation 4.2.7 A.2. Understand and apply the concept of similarity. 7.G.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. 7.G.3. Describe the two-dimensional figures that result from slicing threedimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. Instructional guidance beyond the level of specificity provided in the NJ cccs 4.2.7.A.3. Use logic and reasoning to make and support conjectures about geometric objects. [Related to Using proportions to find missing measures Scale drawings Models of 3D objects 3-D objects are not explicitly included at this grade level. Mathematical Process No. 3 description, that students “make conjectures and build a logical progression of statements to explore the truth of their conjectures.”] The new CCSS move this from grade 8 (NJ cccs 4.2.8.A.1) to grade 7. NEW (to grade 7) 4.2.7.A.1. Understand and apply properties of polygons. Quadrilaterals, including squares, rectangles, parallelograms, trapezoids, rhombi Regular polygons In the CCSS, Identification and classification of twodimensional figures, including quadrilaterals, are in grade 5 (5.G.3 and 4). Applying those properties is not explicitly articulated in CCSS at any grade. 4.2.7 B.2. Understand and apply transformations. CCSS move this from grade 7 to grade 8. Finding the image, given the pre-image, and vice-versa Sequence of transformations needed to map one figure onto another Reflections, rotations, and translations result in images congruent to the pre-image Dilations (stretching/shrinking) result in images similar to the pre-image 4.2.7.C.2. Use a coordinate grid to model and quantify transformations (e.g., translate right 4 units). 4.2.7.D.1. Solve problems requiring calculations that involve different units of measurement within a measurement system (e.g., 4’3” plus 7’10” equals 12’1”). CCSS move this from grade 7 to grade 8 (8.G.1, 2, 3, & 4). Although not precisely articulated in the new CCSS, it is strongly related to 5.MD.1 and 6.RP.3d. Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. 4.2.7.D.3. Recognize that all measurements of continuous quantities are approximations. 4.2.7.D.2. Select and use appropriate units and tools to measure quantities to the degree of precision needed in a particular problem-solving situation. 7.G.4. Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation Although not explicitly articulated in the CCSS, this remains a critical understanding for students to solve real-life problems. It is implicitly related to Mathematical Practice Standard #6. The new CCSS for Mathematical Practice include selection of tools and precision: “Mathematically proficient students consider the available tools when solving a mathematical problem” (#5). “They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context” (#6). In Transition: CCSS move area and circumference of a circle from NEW (to grade 7) Students coming to seventh grade from classes in which the Shaded content is not a focus in this grade because it has already been covered in previous grades, because it will be covered in later grades, or because it is not included at any grade in the new CCSS. April 2011 5 of the relationship between the circumference and area of a circle. 2008 standards were used may already know this content. Once the curriculum change has been implemented at grade 6, teachers can no longer assume previous familiarity with circumference and area of a circle. grade 6 (NJ cccs 4.2.6.E.2) to grade 7. 4.2.7 E.1. Develop and apply strategies for finding perimeter and area. Geometric figures made by combining triangles, rectangles and circles or parts of circles Estimation of area using grids of various sizes 7.G.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. 7.G.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. CCSS move this from grade 8 (4.2.8.A.1) to grade 7. Although this 2008 CPI is somewhat related to CCSS 7.G.4 above and 7.G.6 below, the emphasis is very different. NEW (to grade 7) CCSS move this from grade 8 (4.2.8.E.3) to grade 7. NEW (to grade 7) 4.2.7 E.2. Recognize that the volume of a pyramid or cone is one-third of the volume of the prism or cylinder with the same base and height (e.g., use rice to compare volumes of figures with same base and height). CCSS move volume of a cone from grade 7 to grade 8 (8.G.9). Finding the volume of a pyramid is postponed until HS. Statistics and Probability 7.SP Use random sampling to draw inferences about a population. 7.SP.1. Understand that statistics can be Related but much used to gain information about a population more specific by examining a sample of the population; expectations 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. 7.SP.2. Use 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. 4.4.7 A.2. Make inferences and formulate and evaluate arguments based on displays and analysis of data. CCSS move this from grade 8 (4.4.8.A.4) to grade 7. NEW (to grade 7) Draw informal comparative inferences about two populations. 7.SP.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. 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 NEW Shaded content is not a focus in this grade because it has already been covered in previous grades, because it will be covered in later grades, or because it is not included at any grade in the new CCSS. April 2011 6 twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable. 7.SP.4. Use 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. 4.4.7 A.1. Select and use appropriate representations for sets of data, and measures of central tendency (mean, median, and mode). Type of display most appropriate for given data Related but much more specific and more demanding expectation. Box-and-whisker plot, upper quartile, lower quartile Scatter plot Calculators and computer used to record and process information Investigate chance processes and develop, use, and evaluate probability models. 7.SP.5. Understand that the probability of a CCSS move this chance event is a number between 0 and 1 from grade 6 that expresses the likelihood of the event (4.4.6.B.1) to occurring. Larger numbers indicate greater grade 7. 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. 7.SP.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. 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. 7.SP.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. 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. b. 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? Identification of appropriate data displays is not explicitly included in the CCSS at any grade, but is critical throughout. In CCSS, box plots and interquartile range are in grade 6 In CCSS, scatter plots are in grade 8 Supportive of CCSS Mathematical Practice #5 NEW (to grade 7) 4.4.7.B.1. Interpret probabilities as ratios, percents, and decimals. 4.4.7.B.2. Model situations involving probability with simulations (using spinners, dice, calculators and computers) and theoretical models. Frequency, relative frequency Similar, but slightly more specific expectation. 4.4.7.B.3. Estimate probabilities and make predictions based on experimental and theoretical probabilities. [“Estimate probabilities and make predictions based on experimental and theoretical probabilities” from 4.4.7.B.3 above.] Related to and an extension of NJ cccs 4.4.7.B.3 above 4.4.7.B.4. Play and analyze probabilitybased games, and discuss the concepts of fairness and expected value. In CCSS, concepts of fairness and expected value are postponed until HS. Shaded content is not a focus in this grade because it has already been covered in previous grades, because it will be covered in later grades, or because it is not included at any grade in the new CCSS. April 2011 7 7.SP.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 sample 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 (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event. c. 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? 4.4.7.C.3. Apply techniques of systematic listing, counting, and reasoning in a variety of different contexts. CCSS move this from grade 8 (4.4.8.B.2) to grade 7. “Explore compound events” was included in 2008 NJ cccs at grade 6 (4.4.6.B.3) Bullet b is related to 4.4.7.C.3. NEW (to grade 7) 4.4.7.C. Discrete MathematicsSystematic Listing and Counting 4.4.7.D. Discrete MathematicsVertex-Edge Graphs and Algorithms In CCSS, Systematic Listing and Counting (with the exception of 4.4.7.C.3 as indicated above) is postponed until HS Vertex-Edge Graphs are not included in the new CCSS at any grade level Shaded content is not a focus in this grade because it has already been covered in previous grades, because it will be covered in later grades, or because it is not included at any grade in the new CCSS. April 2011 8