Mark D. Martin

Instruction Outline 7^th Grade Math Using State Standards

Text, Prentice Hall Mathematics, Course 2 (2004)

Introduction- Stella Maris Academy follows California State Standards and the diocesan standards. The California Standards are more specific, although consistent with the diocesan standards. The focus in this document is hence on the California Standards. Mr. Martin in general follows the topics as they are presented in the text for consistency and to ensure students have sufficient background knowledge to understand new topics. Where necessary, Mr. Martin deviates from that order, and frequently supplements materials, to ensure students are learning what is required in the State Standards. Tests are almost always hand designed by Mr. Martin to accurately assess students understanding of the concepts in the State Standards. The outline here, of course, is just that – a brief outline. In the notes column I have tried to highlight some of the key concepts or areas of difficulty. The timeline is approximate. Teaching is a day-by-day process with constant evaluation of how the students are doing.

Instruction is clear and straightforward. I introduce practical examples and applications as much as practical. I also try to introduce a sense of humor. Students are shown how to do small chunks of concepts consistent with their background knowledge. Alternately, and preferably, they are given situations or problems to help them discover a concept new to them consistent with their background knowledge. They are then given sufficient practice so that concept becomes part of their long-term memory. Part of that practice is homework. I try to make homework short enough to not be unduly burdensome while giving sufficient practice. Homework is posted on my class Web site.

Seventh grade math is tough. Concepts build upon those in all prior grades. Concepts come at a much higher pace than in prior years. More abstract concepts are being introduced. It is also probably the year in math that is most useful in real life. Most people indeed use fractions, decimals, percents, proportions, volume, area, probability, etc. in our daily lives even though we have forgotten how to, and don’t need to, factor polynomials, write geometric proofs, or do calculus. (Those things are vitally important to advancing math, science, engineering, technology and society in general. Most of us do not use them in our daily lives, however.)

Because the concepts are tough, and because middle school students’ brains are changing as fast as their physical development, and are as varied as their physical development, there is a wide range of abilities in seventh grade math. In one sense it would be great if I could teach each student one on one at exactly their own pace. When you think about it, however, this is probably not desirable. In the classroom setting, students are given the opportunity to see how others think and in doing so learn new problem solving skills. In a classroom students can also explain concepts to other students. A concept is not really mastered if you cannot explain it to someone else. The way to become the best learner is often to learn to be a good teacher. This benefits both the “teaching” student and the other student. Of course, in dealing with other students, students also learn valuable social skills – obviously important not only in their future professional life, but also in their life as a Christian. All students are expected to learn the concepts set forth here. Some will learn them with a high degree of mastery and will also be able to explore some of the nuisances of the concepts. Those students will be able to proceed to the Algebra class next year. Other students will have knowledge of the concepts but will have further opportunity in 8^th grade math to master the concepts before going on to Algebra in high school.

In the following chart the California State Standards sections are referred to by the following abbreviations: NS= Number Sense, A = Algebra and Functions, MG = Measurement and Geometry, S = Statistics, Data Analysis and Probability, MR = Mathematical Reasoning. Brief descriptions of the standards are given, but refer to the actual standard for a complete description. The 7^th Grade Math Standards are at: http://www.cde.ca.gov/be/st/ss/mthgrade7.asp. The Mathematical Reasoning Standards are meet throughout all topics and therefore are not separately set forth below.

	Topics	California State Standards	Notes
Qtr 1	Decimals and Integers (Chapter 1) Rounding and Estimation Add, subtract, multiply and divide decimals Metric Measurement and conversion (King Henry died Monday drinking chocolate milk.) Absolute value Inverse operations Add, subtract, multiply and divide integers Properties of Addition and Multiplication Distributive property Order of operations Mean, median and mode	NS1.2 decimal and integer operations NS2.5 absolute value A1.3 associative and commutative properties of + and x, identity properties of + and x, distributive property, inverse operations MG 1.1measurement conversions	-Rounding and estimation are review from prior years. Place value is also reviewed. -Decimal operations - also largely review, but frequently forgotten by students. We review the rules with plenty of practice. -Integer operations have previously been introduced, but are stressed in 7^th grade. Students are expected to memorize the following rules: -Addition * Same sign- add absolute values, give result sign of that the numbers have * Different signs – subtract absolute values, give result sign of the number with the greater absolute value. -Subtraction – add the opposite -Multiplication (two numbers) *same sign, product is positive *different signs, product neg -Division – same rules as multiplication -Order of Operations – PEMDAS
	Equations and Inequalities (Chapter 2) Evaluating and writing algebraic expressions Solving equations by adding or subtracting Solving equations by multiplying or dividing Solving two step equations Write equations Graph and write inequalities Solving inequalities by adding or subtracting Solving inequalities by multiplying or dividing	A1.1 writing equations A1.3 properties A1.4 algebraic terminology A4.1 two step equations and inequalities A4.2 multi step problems	Basic rules: Get variable by itself Whatever you do to one side of the equation do to the other side Same rules apply to inequalities except when multiplying or dividing by a negative number, you must reverse the inequality symbol.
	Exponents, Factors and Fractions (Chapter 3) Exponents Scientific notation Divisibility rules Prime factorization Finding Greatest Common Factor (GCF) Finding Least Common Multiple (LCM) Simplifying fractions by dividing by the GCF Equivalent fractions Comparing and ordering fractions by finding the Least Common Denominator (LCD) (i.e. the LCM of the denominators) Mixed numbers and improper fractions Writing fractions as decimals and writing decimals as fractions Defining rational numbers Comparing and ordering rational numbers	NS1.1 Scientific Notation NS1.3 decimal fraction conversions NS1.4 and 1.5 rational and irrational numbers NS2.1 Exponents (in part. Operations with common base done in 8^th grade math or algebra) A2.1 and 2.2 exponents (in part – exponent operations reserved until 8^th grade math or algebra)	A big challenge is getting students to understand the difference between a factor and a multiple. For example factors of 6 are 1, 2, 3, 6. Multiples of 6 include 12, 18, 24 . . . Likewise, a challenge is the difference in calculating the GCF and LCM. In both you can start by factoring each number using factor trees. To calculate the GCF, you find all the common factors; then multiply the common factors together. To calculate the LCM, you circle each factor where it occurs the greatest number of times, and then multiple all those factors together. Students also need to understand where we use GCF and LCM. For example, GCF is used to simplify fractions, and LCM is used to find a common denominator (LCD) in order to add or subtract fractions or to order fractions. Students learn that the GCF is always less than or equal to the smallest number and the LCM is always greater than or equal to the larger number. Students learn how to convert fractions to decimals and vice versa. They are also required to learn common fraction decimal equivalents by memory for halves, thirds, fourths, fifth, sixths, eighths, ninths and tenths. This is a real life skill for quick conversions in business or for standardized tests in the future.
Qtr 2	Operations With Fractions (Chapter 4) Estimating with fractions and mixed numbers Adding and subtracting fractions Adding and subtracting mixed numbers Multiplying fractions and mixed numbers Dividing fractions and mixed numbers Solving equations with fractions Changing customary units	NS2.2 Add subtract fraction with uncommon denominators NS2.3 in part – primarily reserved until 8^th grade math or algebra. MS1.1 measurement conversions	Building on what was learned in chapter 4, we use the LCM to add and subtract fractions. We learn that whole numbers and fractions in mixed numbers can be added separately. Unlike in prior years, however, students now combine their knowledge of integer operations and can calculate a problem with a result that has a negative sign. Students are reminded that mixed numbers must be converted to improper fractions to multiply or divide. To multiply, just multiply across. Cross cancel if you can. To divide, you multiply by the reciprocal of the second number. All of this is also used to solve equations involving fractions. Students also eliminate fractions in equations by multiplying both sides of the equation by the LCM.
	Ratios, Rates and Proportions (Chapter 5) Ratios (comparing two quantities by division) Writing ratios Writing ratios as decimals Writing equal ratios Writing ratios in simplest form Unit rates Dimensional analysis Proportions (two equal ratios) Cross products and solving proportions Proportional analysis Similar figures (same shape, not necessarily the same size) Maps and scaled drawings	A4.2 multi step problems involving rate MG1.1 Measurement conversions MG1.2 scaled drawings MG1.3 unit rates, dimensional analysis	Having learned about fractions and rational numbers, students now look at fractions as ratios. Students learn that two equal ratios are a proportion. They can then solve proportions using cross products. (E.g. find x in ¾=5/x.) They can also use cross products to determine if two ratios indeed form a proportion. Proportions have many practical uses such as in maps and drawings. With the help of shadows or photos, we find heights of telephone poles and trees without climbing them!
	Percents (Chapter 6) Understanding what a percent is Decimals, fractions and percents Percents greater than 100% or less than 1% Finding part, base or whole using percent proportion method and using “equation” method Finding sales tax, tips and commissions Finding percent of change Writing equations involving percent	NS1.6 percent increase or decrease (i.e. percent of change) NS1.7 discounts, markups, commissions, profit	Again building on prior chapters and prior grades, students convert between decimals, fractions and percents. In doing so they begin to understand why percents are such a useful comparison device in everyday life. Students learn to deal with percents greater than 100% (i.e. greater than 1) and percents less than 1% (i.e. less than .01). With careful attention and practice students will no longer be threatened by dreaded problems on standardized exams or elsewhere like: (1) Find 25% of 50, (2) 12.5 is what percent of 50? or (3) 25% of what number is 12.5. We do this by using both the percent proportion: and the equation or what I call the “translation method.” In the equation or translation method we translate from “English” to “Algebra” knowing some useful vocabulary. Find, what, etc. represents the variable. “Of” means multiplication. “Is” means equals. “Percent” or “%” means the rate expressed as a decimal. For example, in #1 above, x=.25(50). (#2) 12.5=x(50) (#3) .25x=12.5. Now, just solve for x. Percent of change is another useful topic for middle show students. For example, the iPhone, purse, video game, etc. was $300. Now it is $240. What is the percent of increase or decrease. Take the difference divided by the original amount.
Qtr 3	Geometry (Chapter 7) Points, lines, rays, segments Measuring and classifying angles (acute, right, obtuse, straight) – complementary and supplementary pairs of angles Constructions – perpendicular bisector, angle bisector, congruent angles using straight edge and compass Triangles (by sides – equilateral, scalene, isosceles; by angle – acute, right, obtuse) – sum of angles = 180° Quadrilaterals and other polygons Classification Sum of angles Congruent figures Circles Parts – chord, central angle, radius, diameter, arcs, semicircle Circle graphs	MG3.1 identifications, constructions, etc. MG3.4 Congruent figures MG3.6 elements of 3D objects, plane intersections, skew lines, etc.	Geometry is divided into two chapters. In this chapter students must master a lot of vocabulary. They get a break from the vocabulary section by being able to do constructions using only a straightedge and compass and also get a glimpse of what ancient civilizations had learned about geometry. They will build upon their knowledge in the next chapter.
	Geometry and Measurement (Chapter 8) Estimation length and area Area of parallelograms and triangles Area of trapezoids and irregular figures Circumference and areas of circles Square roots and irrational numbers Pythagorean Theorem Classifying three-dimensional figures Surface area of prisms and cylinders Volume of rectangular prisms and cylinders	NS2.4 roots MG2.1to 2.4 area, volume, surface area, etc. MG3.3 Pythagorean Theorem MG3.5 2-D nets of 3-D objects	Here the memorization turns to formulas instead of definitions. We try to make sense of the formulas, however, and some years even try to devise the formula for the area of a circle by dividing a circle into small pie shapes and making it into a parallelogram. Often concepts about the circumference of a circle fall around Pi-Day (3-14 get it!) and give rise to celebration. I try to use models so that the students can visualize surface area and volume. For example, it is much easier to find the surface area of a cylinder if you view it has a can with a rectangular label and two lids.
	Sequences, Patterns, Simple and Compound Interest, Equations (Part of Chapter 9) Number Sequences (sec. 9-2) Patterns (sec. 9-3) Simple Interest (sec. 9-7) Compound Interest (sec. 9-7) Writing equations from words (sec. 9-8) Solving equations with multiple variables for a particular variable (sec. 9-9)	NS 1.7 simple and compound interest A1.1writing equations	Chapter 9 is an introduction to several graphing concepts with several other concepts thrown in. I separate out the other concepts and put the graphing material with chapter 10 on graphing and functions.
Qtr 4	Graphing and Functions (Part of Chapter 9 and Chapter 10) The coordinate plane Definitions of x-axis, y-axis, origin, ordered pair, x-coordinate, y-coordinate, quadrants Writing coordinates and graphing points Horizontal and vertical lines Choosing scales and intervals (Sec. 9-1) Graphing linear equations by using a table Slope of a line Intercepts Slope Intercept Form of line Non-linear Relationships Functions Interpreting graphs Translations, symmetry, reflections, rotations (sometimes reserved until 8^th grade)	A1.5 represent quantitative relationships graphically and interpret graphs A3.1 graph functions to second and third powers A3.2 reserved until 8^th grade A3.3 graph linear functions and determine slope A3.4 plotting A4.2 solve multi step problems involving rate, average speed, distance time or a direct variation MG3.2 plot figures, translations, reflections	This is an important unit that students will use in the rest of their math studies throughout 8^th grade, high school and college. Students have previously had some introduction to graphs in prior years. Graphing becomes much more formalized now, however. Students go beyond just graphing points and learn how to graph linear equations using tables and the slope intercept form of the line, y=mx + b where m is the slope and b is the y intercept. They explore this and other forms of linear equations in more depth in either 8^th grade math or Algebra next year. Students also look at non-linear relationships by graphing from a table. They will delve into that topic in much greater detail either next year in Algebra or in Algebra in high school. I try to have students be able to practically understand how the graphing relates to the real world. For examples, some students will begin to understand that slope is equal to a rate of change giving them an initial glimpse into differential calculus. We supplement assignments with work in computer lab using Excel and Apple Grapher.
	Displaying and Analyzing Data (Chapter 11) Reporting frequency Spreadsheets and data display Stem and leaf plots Box-and-whisker plot Random samples and surveys Using data to persuade Exploring scatter plots	S1.1 data display S1.2 scatter plots S1.3 box and whisker plots – quartiles	This builds on introductory statistical analysis that students have been exposed to earlier in the text like mean, median and mean. Often, we work on spread sheets and data display in computer class. This, combined with the fact that it comes towards the end of the year, means this is often one of our shorter chapters.
	Probability (Chapter 12) Definition of outcome, even and probability Definition complement Definition odds Distinction between theoretical probability and experimental probability Probability of independent and compound events Permutations Combinations		Not part of 7^th grade standards. We cover only if we have sufficient time at the end of the year.