Private: Learning Math: Number and Operations
Meanings and Models for Operations Homework
Explain the diagrams that illustrate 124 using the following models:
a. The partitive (or equal groups) model
b. The quotative (or measurement or repeated subtraction) model
Which two of the following models represent the same multiplication problem?
Model (2) illustrates two groups of six, and model (4) illustrates six groups of two.
Show how to compute 19513 using two different models: the area model and long division. Show how the area model relates to long division.
Explain how adding zeros with the colored-chip model helps you understand subtraction problems like +2 – (-4).
Why doesn’t the colored-chip model work for all division problems with positive and negative integers?
Chapin, Suzanne and Johnson, Art (2000). Chapters 3 and 4 in Math Matters: Understanding the Math You Teach, Grades K-6 (pp. 40-72). Sausalito: CA: Math Solutions Publications.
Reproduced with permission from the publisher. © 2001 by Math Solutions Publications. All rights reserved.
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Math Matters: Understanding the Math You Teach, Grades K-6
In the partitive model, we interpret 124 as dividing 12 into four equal groups. The result is the number in each group. Here, there are four columns, so a natural division is to group each column. The result is 3, the number in each column.
In the quotative model, we interpret 124 as dividing 12 into groups containing fours. The result is the number of groups. Here, there are four in each row, so a natural division is to group each row. The result is 3, the number of groups required to make 12.
The correct answer is (2) and (3), which each illustrate the multiplication problem 2 • 6 (i.e., two groups of six). The first model, (1), illustrates the problem 3 • 4 (or 4 • 3), and the last model, (4), illustrates the problem 6 • 2 (i.e., six groups of two). Note that all four models represent the same solution, 12, which does not imply that they represent the same problem.
Using the area model, we start with one flat, nine longs, and five units, and we wish to make a rectangle with 13 rows.
This cannot be done without first exchanging one of the longs for 10 units.
After doing this, we can arrange a 13-by-15 rectangle, so the result of the division is 15.
Using long division, 13 goes into 19 once, with a remainder of 6. Carry down the 5 for 65. Thirteen goes into 65 five times evenly. The quotient is 15.
The area model relates to long division in that it matches the long division algorithm. In long division, you first subtracted 130, or 10 • 13, and then subtracted 65, or 5 • 13, for a total result of 15. Thus, 15 • 13 = 195. Similarly, the area model gave you the rectangle 13 by 15, which consists of two rectangles: 10 • 13 and 5 • 13.
In problems like this one, there is no way to “subtract” chips that are not there. Here, adding four zero pairs does not change the value of +2, but it gives us the four red chips we need in order to “subtract.”
To use the colored-chip model for division, we need to use either partitive or quotative division to do the computations. When dividing a positive number by a negative number, we cannot use either of those models. We cannot “partition” a group into a negative number of parts, nor can we “count” the number of negatives within a positive number.
Session 1 What Is a Number System?
Understand the nature of the real number system, the elements and operations that make up the system, and some of the rules that govern the operations. Examine a finite number system that follows some (but not all) of the same rules, and then compare this system to the real number system. Use a number line to classify the numbers we use, and examine how the numbers and operations relate to one another.
Session 2 Number Sets, Infinity, and Zero
Continue examining the number line and the relationships among sets of numbers that make up the real number system. Explore which operations and properties hold true for each of the sets. Consider the magnitude of these infinite sets and discover that infinity comes in more than one size. Examine place value and the significance of zero in a place value system.
Session 3 Place Value
Look at place value systems based on numbers other than 10. Examine the base two numbers and learn uses for base two numbers in computers. Explore exponents and relate them to logarithms. Examine the use of scientific notation to represent numbers with very large or very small magnitude. Interpret whole numbers, common fractions, and decimals in base four.
Session 4 Meanings and Models for Operations
Examine the operations of addition, subtraction, multiplication, and division and their relationships to whole numbers. Work with area models for multiplication and division. Explore the use of two-color chips to model operations with positive and negative numbers.
Session 5 Divisibility Tests and Factors
Explore number theory topics. Analyze Alpha math problems and discuss how they help with the conceptual understanding of operations. Examine various divisibility tests to see how and why they work. Begin examining factors and multiples.
Session 6 Number Theory
Examine visual methods for finding least common multiples and greatest common factors, including Venn diagram models and area models. Explore prime numbers. Learn to locate prime numbers on a number grid and to determine whether very large numbers are prime.
Session 7 Fractions and Decimals
Extend your understanding of fractions and decimals. Examine terminating and non-terminating decimals. Explore ways to predict the number of decimal places in a terminating decimal and the period of a non-terminating decimal. Examine which fractions terminate and which repeat as decimals, and why all rational numbers must fall into one of these categories. Explore methods to convert decimals to fractions and vice versa. Use benchmarks and intuitive methods to order fractions.
Session 8 Rational Numbers and Proportional Reasoning
Begin examining rational numbers. Explore a model for computations with fractions. Analyze proportional reasoning and the difference between absolute and relative thinking. Explore ways to represent proportional relationships and the resulting operations with ratios. Examine how ratios can represent either part-part or part-whole comparisons, depending on how you define the unit, and explore how this affects their behavior in computations.
Session 9 Fractions, Percents, and Ratios
Continue exploring rational numbers, working with an area model for multiplication and division with fractions, and examining operations with decimals. Explore percents and the relationships among representations using fractions, decimals, and percents. Examine benchmarks for understanding percents, especially percents less than 10 and greater than 100. Consider ways to use an elastic model, an area model, and other models to discuss percents. Explore some ratios that occur in nature.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants) who have adapted their new knowledge to their classrooms.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants) who have adapted their new knowledge to their classrooms.