12 days 4.1.7A1 Extend understanding of the number system by constructing meanings for the following (unless otherwise noted, all indicators for grade 7 pertain to these sets of numbers as well): Rational numbers; Percents; Whole numbers with exponents. 4.1.7A2 Demonstrate a sense of the relative magnitudes of numbers. 4,1.7.A4 Compare and order numbers of all named types. 4.1.7.B2 Use exponentiation to find whole number powers of numbers. 4.1.7.B3Understand and apply the standard algebraic order of operations, including appropriate use of parentheses. Topic: Number Relationships and Fractions How do mathematical ideas interconnect and build on one another to produce a coherent whole? How can we compare and contrast numbers? How can counting, measuring, or labeling help to make sense of the world around us? Enduring One representation may sometimes be more helpful than another; and, used together, multiple representations give a fuller understanding of a problem. A quantity can be represented numerically in various ways. Problem solving depends upon choosing wise ways. Numeric fluency includes both the understanding of and the ability to appropriately use numbers. Chapter 2 Be sure to do Lab 2.2 Number Properties Supplement Section 2.2 Powers and Exponents - rewrite / or division sign as a fraction with numerator over denominator Can do page 69 Doubling Numbers a good informal introduction to exponential growth Do Lab 2.4 Investigating Factors Supplement Section 2.4 Number Patterns: suggestion for Prime Factorization - don t extend all branches on factor trees Could combine 2.5 Greatest Common Factor and 3.1 Least Common Multiple - use an algebra model with variables raised to powers; GCF is the smaller (smallest) exponent the terms share; LCM is the larger (largest) exponents terms share: Ex. GCF of a b 2 c 5 and a 3 c 4 is a c 4 LCM of a b 2 c 5 and a 3 c 4 is a 3 b 2 c 5 Be sure to do Lab 2.7 Investigating Models for Fractions
10 days 4.1.7.B1 Use and explain procedures for performing calculations with integers and all number types named above with: Pencil-andpaper; Mental math; Calculator. Topic: Fractions and their Operations What makes a computational strategy both effective and efficient? How do operations affect numbers? Enduring Computational fluency includes understanding the meaning and the appropriate use of numerical operations. The magnitude of numbers affects the outcome of operations on them. Materials: McDougal Littelll Passport 2 Chapter 3 Be sure to do Lab 3.1 Investigating Common Multiples Can skip page 123 Accuracy and Rounding May want to do Lab 3.3 Addition and Subtraction of Fractions May want to do Lab 3.5 Investigating Multiplication and Area Models May want to do Lab 3.7 Investigating Division of Fractions Supplement with real life application problems involving operations with fractions
12 days 4.1.7A3 Understand and use ratios, proportions, and percents (including percents greater than 100 and less than 1) in a variety of situations. 4.2.7.A2 Understand and apply the concept of similarity: Using proportions to find missing measures; Scale drawings; Models of 3D objects. 4.2.7.D1 Solve problems requiring calculations that involve different units of measurement within a measurement system (e.g., 4 3 plus 7 10 equals 12 1 ). 4.2.7.D2 Select and use appropriate units and tools to measure quantities to the degree of precision needed in a particular problemsolving situation. Topic: Ratios and Proportions How can measurements be used to solve problems? Enduring Everyday objects have a variety of attributes, each of which can be measured in many ways. What we measure affects how we measure it. Measurements can be used to describe, compare, and make sense of phenomena. Chapter 6 Be sure to do Lab 6.1 Investigating Ratios Be sure to do Lab 6.3 Proportions Be sure to do Lab 6.6 Similar Polygons Be sure to do page 281 Rates of Speed Always make connections of proportions to other topics Could supplement Section 6.7 - Scale Drawings Supplement solving problems requiring calculations that involve different units of measurement within a measurement system (e.g. 4 3 plus 7 19 equals 12 1 )
12 days 4.1.7.A5 Use whole numbers, fractions, decimals, and percents to represent equivalent forms of the same number. 4.1.7.C1 Use equivalent representations of numbers such as fractions, decimals, and percents to facilitate estimation. Topic: Percents and Decimals How can counting, measuring, or labeling help to make sense of the world around us? Enduring One representation may sometimes be more helpful than another; and, used together, multiple representations give a fuller understanding of a problem. A quantity can be represented numerically in various ways. Problem solving depends upon choosing wise ways. Chapter 7 Be sure to do Lab 7.2 Modeling Percents Be sure to do Lab 7.3 Investigating Large and Small Percents Be sure to do Lab 7.4 Investigating Percent Be sure to do page 333 What Percent is Good Enough? Easiest and most effective way to solve percent problems is to use the general form for every percent problem: the equation a % of b is (or =) c and set up a proportion from that: a / 100 = c / b then cross multiply and divide to solve for any missing value; rephrase it as % number over 100 equals is number (part) over of number (total) Supplement all sections with real life applications Can skip Section 7.7 Percent Increase and Percent Decrease not in NJ CCCS Suggestion: Combine Chapters 4 and 10 and perhaps parts of Chapter 12
15 days 4.1.7.B1 Use and explain procedures for performing calculations with integers and all number types named above with: Pencil-andpaper; Mental math; Calculator. 4.3.7A1 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. 4.3.7.D2 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. 4.3.7.D3 Create, evaluate, and simplify algebraic expressions involving variables: Order of operations, including appropriate use of parentheses; Substitution of a number for a variable. 4.3.7.D4Understand and apply the properties of operations, numbers, equations, and inequalities: Additive inverse; Multiplicative inverse. Topic: Using Integers What makes an algebraic algorithm both effective and efficient? Enduring Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole. Reasoning and/or proof can be used to verify or refute conjectures or theorems in algebra. Chapter 4 & 10 Be sure to do Lab 4.1 Investigating Patterns and Expressions/Be sure to do page 177 Be sure to do Lab 4.4 Investigating Integer Addition Can skip Lab 4.6 _ investigating Algebra Tiles Supplement with Hands On Equations if desired; Temperature project if desired Be sure to do Lab 10.1 Distances on a Number Line Supplement graphing functions and understanding / describing their general behavior for equations involving two variables Supplement analysis of functional relationships to explain how a change in one quantity can result in a change in another Supplement use of patterns to model situations; use manipulatives, tables, graphs, verbal rules, algebra; growth situations such as population growth and compound interest using recursive definitions Supplement creating, evaluating, simplifying algebraic expressions involving variables; order of operations, including appropriate use of parentheses; substitution of a number for a variable May want to do Lab 10.5 Multiplying Integers
10 days 4.3.7.B1 Graph functions, and understand and describe their general behavior: Equations involving two variables. 4.3.7.C1 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.C2 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). Topic: Algebra How are patterns of change related to the behavior of functions? How can mathematical models be used to describe physical relationships? Enduring Patterns and relationships can be represented graphically, numerically, symbolically, or verbally. Mathematical models can be used to describe and quantify physical relationships. Chapter 12 Be sure to do Lab 5.1 Gathering and Interpreting Data Be sure to do Lab 5.2 Investigating Range and Scale Should do Lab 5.3 Investigating Quartiles Supplement with TI 83 if desired for box and whisker plots and data analysis Be sure to do Lab 5.7 Investigating Probability
15 days 4.2.7.A1 Understand and apply properties of polygons: Quadrilaterals, including squares, rectangles, parallelograms, trapezoids, rhombi; Regular polygons. 4.2.7.B1 Understand and apply transformations: Finding the image, given the pre-image, and vice-versa; 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.C1 Use coordinates in four quadrants to represent geometric concepts. 4.2.7C2 Use a coordinate grid to model and quantify transformations (e.g., translate right 4 units). 4.2.7E1 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. Topic: Geometry in the Plane How can spatial relationships be described by careful use of geometric language? How do geometric relationships help to solve problems and/or make sense of phenomena? How can measurements be used to solve problems? Enduring Geometric properties can be used to construct geometric figures. Geometric relationships provide a means to make sense of a variety of phenomena. Measurements can be used to describe, compare, and make sense of phenomena. Chapter 8 Could do Lab 8.1 Investigating Angle Measures Section 8.1 Intersecting Lines and Angle Measures in 6th grade NJ CCCS Be sure to do Lab 8.2 Slides in the Coordinate Plane Supplement Section 8.2 what sequence of transformations is needed to map one figure onto another; reflections, rotations and translations that result in images congruent to the pre-image; Dilations (stretching / shrinking) that result in images similar to the pre-image Supplement use of coordinates in four quadrants to represent geometric concepts Sections 8.3 8.6 are in the 6th grade NJ CCCS Can do Lab 8.8 Supplement development and application of strategies for finding perimeter and area of geometric figures that are made by combining triangles, rectangles and circles or parts of circles (composite figures) Supplement properties of quadrilaterals
15 days 4.2.7.E2 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). Topic: Geometry in Space How can spatial relationships be described by careful use of geometric language? How do geometric relationships help to solve problems and/or make sense of phenomena? Enduring Geometric properties can be used to construct geometric figures. Geometric relationships provide a means to make sense of a variety of phenomena. Chapter 9 Section 9.1 - Exploring Polyhedrons in 6th grade NJ CCCS Section 9.2 Surface area of a prism in 8th grade NJ CCCS Section 9.3 Surface area of a cylinder in 8th grade NJ CCCS Section 9.4 Spatial Visualization in 12th grade NJ CCCS Section 9.5 Volume of a prism in 8th grade NJ CCCS Section 9.6 Volume of a cylinder in 6th grade NJ CCCS Section 9.7 Similar prisms in 12th grade NJ CCCS May want to do Lab 9.5 Investigating Volume May want to do Lab 9.7 Investigating Similar Prisms Supplement recognizing 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
12 days 4.4.7.B3 Estimate probabilities and make predictions based on experimental and theoretical probabilities. 4.4.7.B4 Play and analyze probability-based games, and discuss the concepts of fairness and expected value. 4.4.7.C1 Apply the multiplication principle of counting: Permutations: ordered situations with replacement (e.g., number of possible license plates) vs. ordered situations without replacement (e.g., number of possible slates of 3 class officers from a 23 student class). 4.4.7.C3 Apply techniques of systematic listing, counting, and reasoning in a variety of different contexts. 4.4.7.D1 Use vertex-edge graphs to represent and find solutions to practical problems: Finding the shortest network connecting specified sites; Finding the shortest route on a map from one site to another; Finding the shortest circuit on a map that makes a tour of specified sites. Topic: Probability & Discrete Math How can experimental and theoretical probabilities be used to make predictions or draw conclusions? What counting strategy works best here? Enduring Experimental results tend to approach theoretical probabilities after a large number of trials. Grouping by attributes (classification) can be used to answer mathematical questions. Algorithms can effectively and efficiently be used to quantify and interpret discrete information. Chapter 11 Be sure to do Lab11.2 Investigating Tree Diagrams Supplement estimating probabilities and making predictions based on experimental and theoretical probabilities Be sure to do Lab 11.3 Investigating Order Be sure to do Lab 11.5 Investigating Fair Games Section 11.5 Expected Value in both 7th and 8th grade NJ CCCS Supplement probability using real life applications
15 days 4.4.7.A1 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; Box-and-whisker plot, upper quartile, lower quartile; Scatter plot; Calculators and computer used to record and process information. 4.4.7A2 Make inferences and formulate and evaluate arguments based on displays and analysis of data. Topic: Data Analysis & Statistics How can the collection, organization, interpretation, and display of data be used to answer questions? Enduring The message conveyed by the data depends on how the data is collected, represented, and summarized The results of a statistical investigation can be used to support or refute an argument. Chapter 5 Be sure to do Lab 5.1 Gathering and Interpreting Data Be sure to do Lab 5.2 Investigating Range and Scale Should do Lab 5.3 Investigating Quartiles Supplement with TI 83 if desired for box and whisker plots and data analysis Be sure to do Lab 5.7 Investigating Probability