Activities
Our school district operates on two 18-week semesters per year. Therefore, I developed 18 sets of sample problem sets that are included in Appendix C. I plan to present one to three questions per day, depending on the topic and type of question. The sample problem sets have between one and four days worth of problems. While I tried to arrange them in a logical order, they can be rearranged to suit the school schedule – shortened weeks, testing, etc. To complete a "normal" five-day week, I will simply model the remaining days' problems after the early ones, using different numbers.
Problem Set I focuses on place value. I will give instruction on all of the forms for expanding single place components of numbers. By the end of the week, I expect students to be able to write any number, including decimal fractions, as the sum of the digits-times-a-power-of-ten.
The objective of Problem Set II is to recognize that all three sums for each day are the same. By expanding each number to show place value, and adding the same place values, the digits are always the same, but in a different order. It also provides the opportunity to review or reinforce the Commutative Property of Addition.
Problem Set III is both a vocabulary check and a means for discussing the relative size of numbers, all based on place value. If students demonstrate proficiency with the vocabulary, I will supplement problems that practice the basic Rules of Exponents (Appendix B).
The objective of Problem Sets IV, V, and VII is to practice and/or learn mental arithmetic strategies for addition and subtraction. It is especially important for students to be able to verbalize their strategies and share them with classmates. Throughout the problem sets, I expect students to expand numbers to show place value, look for "friendly numbers" such as 10 or 100, use compensation (keep a constant difference), or "add up" in chunks to multiples of 10. I do not expect students to have names for the strategies, only to be able to explain and justify them.
Before starting Problem Set VI, I will give instruction on the Compensation Method (described in the Background section) that is useful for subtraction of multi-digit numbers that requires regrouping. I think my higher-level students will find it intriguing and will understand why it works. It could be optional for lower-level students as it could turn out to be another algorithm without meaning to them.
Problem Sets VIII and XIV both connect arithmetic to algebra. First students are asked to add, subtract and multiply polynomials as they have done in previous algebra classes. They can only add or subtract like terms (having the same power of x), and they add exponents of x when multiplying terms. After finding the sum, difference or product in simplest form, students will substitute x = 10, and, hopefully, recognize that the polynomials represent the expanded form of numbers showing place value for base ten numbers. If they don't make the connection themselves, I will help them see it!
Problem Sets IX, XI and XVI are designed to practice the Rules of Arithmetic. Students will use the Commutative and Associative Properties of Addition and Multiplication, and the Distributive Property, to simplify an assortment of arithmetic problems. Again, students will be asked to explain and justify their strategies.
Problem Sets X and XV are strictly number sense practice. Students will identify the relative order of numbers. In Problem Set X, they will demonstrate understanding of relative size based on place value of decimal fractions. If students demonstrate understanding early in the week, I will again supplement with exponent practice exercises. In Problem Set XV, they will demonstrate understanding of the effect of repeated multiplication (exponents) on different types of numbers – positive, negative, and rational. Some problems also evaluate understanding of negative exponents.
There are two objectives for Problem Set XII. The first objective is to estimate the product of multi-digit numbers using only the first digit (highest place value) of each. The second objective is to translate the estimated product into its Order of Magnitude from the highest power of ten. I will emphasize the estimation technique in this problem set as a way to check the reasonableness of answers quickly.
The purpose of Problem Set XIII is to demonstrate the area model/box method for multiplying multi-digit numbers in the same way algebra students use the area model to multiply polynomials. When numbers are expanded to show place values, they form the dimensions of the box, and each component in the first number is multiplied by each component of the second number ("Each with each rule"). This model also provides the opportunity to review/reinforce the Commutative and Associative Properties of Multiplication, the Distributive Property to combine like terms (having the same place value), and Rules of Exponents when multiplying powers of ten.
Problem Sets XVII and XVIII address the relative size of numbers, estimation and order of magnitude through problem solving. The problems primarily deal with very large numbers and ask students to relate them to smaller units they know. My primary source for these problems will be Lawrence Weinstein and John A. Adam's book entitled Guesstimation: Solving the World's Problems on the Back of a Cocktail Napkin. 8 The problems in the book are arranged by topics such as Animals and People, Energy, Transportation, etc. Not only do these problems ask students to estimate answers to within a power of 10, they require students to recognize relevant information. I will provide them the opportunity to search for the information they need using computers in the classroom or at home. These problems may need to be spread out over more than one day – brainstorming one day to determine what they need to know to solve the problem, collecting information and doing the rough calculations later. I expect these problems to be dynamic and change according to the interests of my students and items in the news. I may also use them randomly at other times throughout the semester.
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