Description I. The Structure and Calculation of Decimals 01 The Structure of Decimals 1. Decimals are formed using 10 digits and place values less than 1. Writing decimals in base ten | The structure of decimals | Place value of decimals | Expanded notation in base ten 2. Zeros can be added infinitely to the end of a decimal. Zeros at the end of a decimal 3. The higher the place value, the greater the value. Comparing the size of decimals 02 Calculating with Decimals 1. The decimal point is the standard for aligning decimal places. Addition and subtraction of decimals 2. A 1 in a higher place is equal to 10 in the place below it. Addition with carrying, subtraction with borrowing | Zeros at the end of a decimal 3. ●.■ means there are ●■ copies of 0.1. (Decimal) × (Whole Number) 4. The decimal point shifts according to the number of decimal places being multiplied. The position of the decimal point in multiplication | × (Decimal) 5. Multiplying doesn't always make a value larger. The relationship between the multiplier and the product's size 6. ●.■ means there are ●■ copies of 0.1. (Decimal) ÷ (Whole Number) 7. It's easier to divide by a whole number. The relationship between dividend, divisor, and quotient | ÷ (Decimal) 8. Dividing doesn't always make a value smaller. The relationship between the divisor and the quotient's size 9. Zeros can be added infinitely to the end of a decimal. Bringing down zeros after the decimal point to calculate | Division where the decimal digits of the quotient continue endlessly | Terminating and non-terminating decimals 10. The remainder depends on the quotient. Quotient and remainder | The decimal point of the remainder II. The Structure and Calculation of Fractions 01 The Structure of Fractions 1. A fraction represents how much of a part makes up a whole. Fractions | The value of a fraction of a whole 2. When the whole is 1, a fraction represents its own value. The value of a fraction 3. The value of a fraction equals the quotient of the numerator divided by the denominator. The relationship between fractions and division 4. The name of a fraction is determined by the relative size of the numerator and denominator. Proper and improper fractions | Mixed numbers | The relationship between improper fractions and mixed numbers 5. There are infinitely many fractions with equal value. Equivalent fractions | Reducing fractions | Finding common denominators 6. The size of fractions is compared by making the denominators or numerators equal. Comparing by equalizing denominators | Comparing by equalizing numerators 7. Writing a fraction in base ten turns it into a decimal. Expressing fractions as decimals (1) | Expressing fractions as decimals (2) | Expressing decimals as fractions | Comparing the size of fractions and decimals 8. Setting a standard allows comparison of the size of numbers. Ratio and rate | Percentage 02 Calculating with Fractions 1. In addition and subtraction, the two numbers are values; in multiplication, one number is a rate. The principles of addition and multiplication of fractions 2. Add and subtract the value of the part relative to the whole. Addition and subtraction of fractions with the same denominator 3. The value of a fraction is based on its denominator. Calculations with whole numbers and fractions | Addition and subtraction of fractions with carrying and borrowing 4. The standard must be the same in order to add or subtract values. Addition and subtraction of fractions with different denominators 5. When the value of a fraction is treated as the whole, find the amount corresponding to a fraction of it. Multiplication of fractions | Multiplication of whole numbers and fractions | Multiplication of mixed numbers | Reciprocals 6. Division is multiplication by the reciprocal. Division of fractions