Learning Math: Number and Operations
Rational Numbers and Proportional Reasoning Session 8: Homework
Show how to use Cuisenaire Rods to model 3/4 2/3.
a. The shaded part is 5 1/4. Specify the unit:
b. Using that same unit, what would three small rectangles represent?
c. List other values that the shaded part could represent, and name the unit for each value.
Under what conditions does 2/3 + 3/4 = 5/7?
Which shows a greater change: a 6-by-6 square becoming a 4-by-8 rectangle, or a 4-by-8 rectangle becoming a 6-by-6 square? Explain how you know.
If you have a $1,000 investment that decreases by 50% in value and then increases by 50%, how much would it then be worth? Did you use relative or absolute reasoning to solve this problem?
Using the same model as in Problem B3 (d) to divide 3/4 by 2/3, we need to ask, “How many browns (2/3) are there in a blue rod (3/4)?” The answer is 1 1/8, or 9/8:
a. The unit is four small rectangles.
b. Three small rectangles represent 3/4.
c. Some other values the shaded part could represent are 3 1/2 (the unit is six small rectangles), 21 (the unit is one small rectangle), and 10 1/2 (the unit is two small rectangles). Other values are also possible.
The equation will be satisfied when these fractions represent part-part ratios, and we are finding the proportion of the combined group by adding the numerators and denominators.
Changing a 4-by-8 rectangle into a 6-by-6 square shows a greater relative change.
Here is how we know: When a 6-by-6 square becomes a 4-by-8 rectangle, its area changes from 36 square units to 32 square units — a decrease of four square units. This decrease represents 4/36, or 1/9, of the square’s original area.
When a 4-by-8 rectangle becomes a 6-by-6 square, its area changes from 32 square units to 36 square units — an increase of four square units. This increase represents 4/32, or 1/8, of the rectangle’s original area.
Since 1/8 is greater than 1/9, the rectangle showed a greater change.
However, you could also say that in an absolute sense, the change for both was equal, as the area of both the rectangle and the square changed by four square units.
After it decreases 50% in value, the investment would be worth $500. Then, after the 50% increase, it would be worth $750. Solving this involves multiplicative, or relative type of reasoning.
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.