Benchmark 9A: The Mathematical World - Numbers

• There have been systems for writing numbers other than the Arabic system of place values based on tens. The very old Roman numerals are now used only for dates, clock faces, or ordering chapters in a book. Numbers based on 60 are still used for describing time and angles. (2 of 7)
• Standard 6-3, page 91, Grades 5-8
Extend their understanding of whole number operations to fractions, decimals, integers, and rational numbers
• A number line can be extended on the other side of zero to represent negative numbers. Negative numbers allow subtraction of a bigger number from a smaller number to make sense, and are often used when something can be measured on either side of some reference point (time, ground level, temperature, budget). (2 of 7)
• Standard 4-1 page 32, Grades K-4

Standard 5-1, page 87, Grades 5-8
Understand, represent, and use numbers in a variety of equivalent forms (integer, fraction, decimal, percent, exponential, and scientific notation) in real-world and mathematical problem situations

Standard 6-1, page 91, Grades 5-8
Understand and appreciate the need for numbers beyond the whole numbers

Standard 6-2, page 91, Grades 5-8
Develop and use order relations for whole numbers, fractions, decimals, integers, and rational numbers

Standard 6-3, page 91, Grades 5-8
Extend their understanding of whole number operations to fractions, decimals, integers, and rational numbers

• Numbers can be written in different forms, depending on how they are being used. How fractions or decimals based on measured quantities should be written depends on how precise the measurements are and how precise an answer is needed. (3 of 7)
• Standard 12-1 page 57, Grades K-4
Develop concepts of fractions, mixed numbers, and decimals

Standard 12-2 page 57, Grades K-4
Develop number sense for fractions and decimals

Standard 4-2 page 32, Grades K-4
Relate various representations of concepts or procedures to one another

Standard 5-4, page 87, Grades 5-8
Investigate relationships among fractions, decimals, and percents

Standard 6-3, page 91, Grades 5-8
Extend their understanding of whole number operations to fractions, decimals, integers, and rational numbers

• The operations + and – are inverses of each other one undoes the other; likewise x and / . (4 of 7)
• Standard 6-4, page 91, Grades 5-8
Understand how the basic arithmetic operations are related to one another
• The expression a/b can mean different things: a parts of size 1/b each, a divided by b, or a compared to b. (5 of 7)
• Standard 6-4, page 91, Grades 5-8
Understand how the basic arithmetic operations are related to one another
• Numbers can be represented by using sequences of only two symbols (such as 1 and 0, on and off); computers work this way. (6 of 7)
• Standard 4-2 page 32, Grades K-4
Relate various representations of concepts or procedures to one another

Standard 6-3, page 91, Grades 5-8
Extend their understanding of whole number operations to fractions, decimals, integers, and rational numbers

Standard 14-3 page 184, Grades 9-12
Appreciate that seemingly different mathematical systems may be essentially the same