This is part 3 of a series that will eventually be a book by the same name as this blog. While this is labeled Part 3, it will likely occur much later in the book, after presentation of topics for lower grades (K-6).
The math textbooks in use today have a dearth of good explanations, as well as word problems. Examples usually take the place of any kind of extended explanation, and while explicit instruction relies on worked examples, a textbook should have some additional explanation to go along with them. Indeed, the explanations provided in the teachers’ manuals for these books are instructive and should be included in the students’ textbooks as well. In addition, today’s textbooks generally include too many topics in one lesson, and the problems at the end of the lesson sometimes go beyond what was discussed in the lesson itself.
Compounding these difficulties are that many of these books tout a “balanced approach”. This usually means that preceding the lessons so described above there is an activity or “inquiry” for students in which they are to discover the general principles behind whatever the topic might be. For example, preceding a lesson on multiplying negative numbers, there may be thought problems on how much weight you would lose if you lost four pounds per week for five weeks, leading to the multiplication of -4 by 5, resulting in a loss of twenty pounds, or -20. The next day, they then have the more straightforward lesson made up mostly of more examples with the rules of multiplication of negative numbers stated formally.
While it is sometimes worthwhile to set aside a day to front load key concepts prior to a lesson, many times these inquiries can and should be incorporated in the main lesson. In a later chapter I will discuss the best way to proceed through such “balanced approach” textbooks. This chapter focuses on the sequence of topics that I and other teachers with whom I’ve worked believe make the most mathematical and logical sense.
This chapter also discusses textbooks and materials with which to supplement the textbooks that schools have adopted. These additional resources can be obtained from the internet and provide a source of better explanations as well as problems.
What order/sequence makes the most sense?
In one school where I taught, a sixth grade teacher who was hired at the same time as I noted that the math textbooks start off with ratios and proportions. She felt that this might be confusing to students. I noted that the seventh grade textbook did the same thing, but that I had reordered the sequence of topics to make more sense. I felt that starting with integers, and how to operate with negative numbers and then going into rational numbers would be a better way to start. We both agreed that students need some grounding in fractions to better understand ratios.
I’ve noticed many math books for seventh grade math begin with ratios and proportions including the US edition of JUMP Math. I believe this order is followed because it is also the order of topics in the Common Core Math Standards for that grade. There is nothing in the standards that prescribes that order, and I feel certain that teachers will not be terminated for using a different sequence than what appears in the Common Core standards, or in most textbooks that are aligned with the Common Core.
Similarly, I feel that an introduction to algebraic expressions and equations is necessary prior to ratios and proportions. Since ratios and proportions are one of the largest topics in seventh grade math, students need the appropriate background and tools with which to understand the concepts and operate with them to solve problems.
Scope and Sequence for Math 7
What follows is the scope and sequence for a regular seventh grade math course (Math 7), and an accelerated one. For the latter, the additional topics are introduced in a regular eighth grade math course (i.e, Math 8, not Algebra 1).
Key topics for which this series provides descriptions of their presentation in the traditional manner are: Integers, Rational Numbers, Expressions and Equations, Inequalities, Ratios and Proportions, and Percents
A. Integers and absolute value
B. Negative numbers (Adding, subtracting, multiplying and dividing)
II. Rational numbers
A. Rational numbers (general definition; ordering)
B. Adding rational numbers
C. Subtracting rational numbers
D. Multiplying and dividing rational numbers
III. Expressions and Equations
A. Algebraic expressions
B. Adding and subtracting linear expressions
C. Solving equations using addition or subtraction
D. Solving equations using multiplication or division
E. Solving two-step equations
F. Multi-step equations (as necessary; recommended for accelerated classes)
G. Word problems
A. Writing and graphing inequalities
B. Solving inequalities using addition or subtraction
C. Solving inequalities using multiplication or division
D. Solving two-step inequalities
E. Word problems
V. Ratios and proportions
A. Ratios and rates
C. Writing proportions
D. Solving proportions
F. Direct variation
A. Percents and decimals
B. Comparing and ordering fractions, decimals, and percents
C. The percent proportion
D. The percent equation
E. Percents of change: increase and decrease
F. Discounts and markups
G. Simple interest
H. Compound interest (for accelerated classes)
VII. Constructions and scale drawings
A. Adjacent and vertical angles
B. Complementary and supplementary angles
E. Scale drawings
A. Review of area of triangles and quadrilaterals
B. Circles and circumference
C. Perimeters of composite figures
D. Areas of circles
E. Areas of composite figures
IX. Surface area and volume
A. Surface areas of prisms
B. Surface areas of pyramids
C. Surface areas of cylinders
D. Volumes of prisms
E. Volumes of pyramids
X. Probability and Statistics
A. Outcomes and events
C. Experimental and theoretical probability
E. Combinations (for accelerated classes)
F. Compound events
E. Independent and dependent events
F. Samples and populations
G. Mean, Median and Mode
H. Comparing populations
Extra topics for accelerated classes (these are covered in Math 8)
XI. Transformation (Translations, Reflections, Rotations)
XII. Graphing and writing linear equations
XIII. Real Numbers and the Pythagorean Theorem
XIV. Volumes of Cylinders, Cones and Spheres
Additional Resources with Which to Supplement Textbooks
Singapore’s Primary Math Series (U.S. edition)
Pre-Algebra: An Accelerated Course (Dolciani, Sorgenfrey, Graham); Houghton Mifflin Company. 1988.