Learning Math: Number and Operations
Divisibility Tests and Factors Homework
The number abcabc (where each letter represents one particular digit) is divisible by 7, 11, and 13 for all one-digit values of a, b, and c. Why is that?
Think about dividing abcabc by abc. Is the number you obtained divisible by 7, 11, and 13?
Use what you know about divisibility tests to find remainders for the following:
It is said that the mathematician Karl Gauss figured out how to find the sum of the first 100 counting numbers when he was in the second grade.
Then he added the 101s to get
100 • 101 = 10,100.
But that number is twice the sum, so the actual sum is 10,1002, or 5,050.
Examine Gauss’s process:
a. Are the missing totals 101? Why?
b. How many 101s are there in all? How do you know? Why did Gauss divide by 2?
Use Gauss’s method to find the sum of the first n counting numbers.
“Triangular numbers” describe the number of dots needed to make triangles like the ones below. The first triangular number is 1, the second is 3, and so on.
Use the results of Problem H4 to write a rule for the number of dots in the nth triangular number. See Note 6 below.
Think about how many dots the first triangle has and how many you need to add to make each new triangle.
Each * stands for any missing digit (i.e., they are not all the same digit). Decode these long-division problems:
Begin by looking for any additional digits that you can fill in, in order to have fewer unknowns. For example, the two asterisks below 5 have to be 5 as well, because there is no remainder.
We know that a two-digit number multiplied by a three-digit number produces a five-digit number. So think about the highest values you could have for those two numbers.
Also think about what numbers might work in the left-hand digit of the quotient and the right-hand digit of the divisor to produce a two-digit result that ends in 0.
This is true because abcabc divided by abc is 1,001. (Break up abcabc, the number, into [abc • 1,000] + abc, or abc • 1,001.) Since 1,001 is a multiple of 7, 11, and 13 (7 • 11 • 13 = 1,001), the number abcabc must be divisible by 7, 11, and 13.
a. Add the digits: 7 + 2 + 5 + 2 = 16. This is not a multiple of 3, but it is one more than a multiple of 3. Therefore, the remainder is 1.
b. Use the last two digits of the number: 34. Thirty-four is two more than a multiple of 4, so the remainder is 2.
c. Add the digits: 3 + 4 + 5 + 7 = 19. This is not a multiple of 9, but it is one more than a multiple of 9. Therefore, the remainder is 1.
d. Add the even power digits: 3 + 9 = 12. Add the odd power digits: 4 + 5 = 9. In order for these to be equal (and produce a multiple of 11), we would need to subtract 3 from the units digit, so the number 4,356 is a multiple of 11. This means that 4,359 is three more than a multiple of 11, so the remainder is 3.
a. Yes, each is formed by adding 1 to the first sequence and subtracting 1 from the second. The sum must stay constant, and this constant is 101.
b. There are 100 of the 101s, since there are 100 numbers in each sequence. Gauss divided his sum by 2 because each number appears twice in the sequence — once in the top row and once in the bottom row.
The sum can be written two ways:
The value (n + 1) appears n times. The total is n • (n + 1) 2. Testing this value for n = 100 gives the sum 100 • (101) 2 = 5,050, which is correct.
The number of dots is measured as 1 + 2 + 3 + … + n, so the total number of dots is n • (n + 1)2.
a. 13,09545 = 291
b. 354,39339 = 9,087
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.