This study guide page lists the content standards for mathematics in the state of California. I have also added some definitions for key vocabulary terms, and given examples and additional explanations for many standards. The key standards for fifth grade are marked with an asterisk.
(This page is still being modified to go with our new curriculum: Everyday Mathematics, but much new material has been added. I have removed the page number references from our old textbook and have added definitions used in the new text. I will continue to revise it as we progress through the units of Everyday Mathematics.)
Students compute with very large and very small numbers, positive integers, decimals, and fractions and understand the relationship between decimals, fractions, and percents. They understand the relative magnitudes of numbers.
1.1) Estimate, round, and manipulate very large (e.g., millions) and very small (e.g., thousandths) numbers.
• Estimate: An estimate is a number close to an exact number. It is used when an exact number cannot be found or is not needed. To estimate is to find such a number. The term estimate is both a noun and a verb.
• Rounding: Replacing a number with one that tells about how many or how much.
• There are ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. If the number you're looking at is in the bottom half (0, 1, 2, 3, or 4), then you "round down." If the number you're looking at is in the top half (5, 6, 7, 8, or 9), then you round up.
For example: If you are told to round 478 to the nearest hundred, then you look at the 7 in the tens place. Since it is in the top half of our ten digits, then you round up to 500.
I put "round down" in quotation marks because I do not like that term. For example: If you are told to round 832 to the nearest hundred, then you would round it to 800. The 8 in the hundreds place doesn't go down to 7, it remains an 8. This is quite unlike our previous example, where the 4 in the hundreds place in 478 does actually go up to the next number, 5.
• Standard Form, Expanded Form, Word Form: This is what they mean by "manipulate" in the standards, using different forms to express numbers. One example of real-life application of this is when writing a check; you use both standard and word forms to confirm the amount of the check.
• (Roman numerals)
1.2) Interpret percents as a part of a hundred; find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number. [This is a key standard for fifth grade.]
• Percent: A ratio of a number to 100.
• Ratio: The comparison of two quantities.
• (Decimal and percent equivalents for common fractions)
• (Compute a given percent of a whole number)
1.3) Understand and compute positive powers of nonnegative integers; compute examples as repeated multiplication.
- Power of a number - The product of factors that are all the same. For example, 5 * 5 * 5 (or 125) is called "5 to the third power" or "the third power of 5" because 5 is a factor three times. 5 * 5 * 5 can also be written as 53.
- Square number - A number that is the product of a counting number with itself. For example, 25 is a square number because 25 = 5 * 5. The square numbers are 1, 4, 9, 16, 25, and so on.
- Exponential notation - A way to show repeated multiplication by the same factor. For example, 23 is exponential notation for 2 * 2 * 2. The small raised 3 is the exponent. It tells how many times the number 2, called the base, is used as a factor.
- Exponent - A small, raised number used in exponential notation to tell how many times the base is used as a factor. For example, in 53, the base is 5, the exponent is 3, and 53 = 5 * 5 * 5 = 125.
- Base (in exponential notation) - The number that is raised to a power. For example, in 53, the base is 5.
- "Unsquaring" a number - Finding the square root of a number.
- Square root (of a number) - The square root of a number n is a number that, when multiplied by itself, gives n. For example, 4 is the square root of 16 because 4 * 4 = 16.
• Integers: The set of whole numbers and their opposites. Don't forget that zero is a whole number, and therefore an integer, but it does not have an opposite.
1.4) Determine the prime factors of all numbers through 50 and write the numbers as the product of their prime factors using exponents to show multiples of a factor (e.g., 24 = 2 x 2 x 2 x 3 = 23 x 3). [This is a key standard for fifth grade.]
- Composite number - A counting number that has more than 2 different factors. For example, 4 is a composite number because it has three factors: 1, 2, and 4.
- Prime number - A counting number that has exactly two different factors: itself and 1. For example, 5 is a prime number because its only factors are 5 and 1. The number 1 is not a prime number because that number has only a single factor, the number one itself. The number 2 is the only even prime number.
- Factor - (1) Whenever two or more numbers are multiplied to give a product, each of the numbers that is multiplied is called a factor. For example, in 4 * 1.5 = 6, 6 is the product and 4 and 1.5 are called factors. (2) A counting number whose product with some other number equals n. For example, 2 and 3 are factors of 6 because 2 * 3 = 6. But 4 is not a factor of 6 because 4 * 1.5 = 6 and 1.5 is not a counting number.
- Product - The result of multiplying two numbers, called factors. For example, in 4 * 3 = 12, the product is 12.
- Multiplication: factor x factor = product
- Factor string - A counting number written as a product of two or more of its factors. The number 1 is never part of a factor string. For example, a factor string for 24 is 2 * 3 * 4. This factor string has three factors so its length is three. Another factor string for 24 is 2 * 3 * 2 * 2 (length four).
- Prime factorization - A counting number expressed as a product of prime factors. Every counting number greater than 1 can be written as a product of prime factors in only one way. For example, the prime factorization of 24 is 2 * 2 * 2 * 3. (The order of the factors does not matter; 2 * 3 * 2 * 2 is also the prime factorization of 24.) The prime factorization of a prime number is that number. For example, the prime factorization of 13 is 13.
1.5) Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers. [This is a key standard for fifth grade.]
Students perform calculations and solve problems involving addition, subtraction, and simple multiplication and division of fractions and decimals.
2.1) Add, subtract, multiply, and divide with decimals; add with negative integers; subtract positive integers from negative integers; and verify the reasonableness of the results. [This is a key standard for fifth grade.]
• (Add and subtract decimals)
• (Multiply decimals)
• (Divide decimals)
• (Add integers)
• (Subtract integers)
• Integers: The set of whole numbers and their opposites. [Zero is neither positive nor negative.]
• Opposites: Integers that are the same distance, but in opposite directions, from zero on a number line. [The sum of a number and its opposite is zero.]
• Absolute value: The numerical value of a given number, or the distance of an integer from zero on a number line.
- Addition: addend + addend = sum
- Subtraction: minuend – subtrahend = difference
2.2) Demonstrate proficiency with division, including division with positive decimals and long division with multidigit divisors. [This is a key standard for fifth grade.]
- Quotient - The result of dividing one number by another number. For example, in 35 / 5 = 7, the quotient is 7.
- Remainder - An amount left over when one number is divided by another number. For example, if 7 children share 38 cookies, each child gets 5 cookies and 3 are left over. We may write 38 / 7 = 5 R3, where R3 stands for the remainder.
- Even number - A counting number that can be divided by 2 with no remainder. The even numbers are 2, 4, 6, 8, and so on. 0, -2, -4, -6 and so on are also usually considered even.
- Odd number - A counting number that cannot be evenly divided by 2. When an odd number is divided by 2, there is a remainder of 1. The odd numbers are 1, 3, 5, and so on.
- Divisible by - If one counting number can be divided by a second counting number with a remainder of 0, then the first number is divisible by the second number. For example, 28 is divisible by 7 because 28 divided by 7 is 4, with a remainder of 0.
- Division: dividend / divisor = quotient (and remainder)
2.3) Solve simple problems, including ones arising in concrete situations, involving the addition and subtraction of fractions and mixed numbers (like and unlike denominators of 20 or less) and express answers in simplest form. [This is a key standard for fifth grade.]
• (Add and subtract fractions)
• (Add and subtract mixed numbers)
• Simplest form: A fraction is in simplest form when the greatest common factor of the numerator and denominator is 1.
• Like fractions: Fractions that have the same denominator.
• Unlike fractions: Fractions that have different denominators.
• Least common multiple (LCM): The least number, other than zero, that is a common multiple of two or more numbers.
• Least common denominator (LCD): The least common multiple of two or more denominators.
• Greatest common factor (GCF): The greatest factor that two or more numbers have in common. [Finding the GCF is helpful in reducing fractions to their simplest form.]
2.4) Understand the concept of multiplication and division of fractions.
• The product of a fraction and a whole number is larger than the fraction and smaller than the whole number. If fraction = small and whole number = large then small x large = medium. Example: 1/4 x 8 = 2
• The product of two fractions is smaller than either of the two factors. If fraction = small then small x small = smaller. Example: 1/4 x 1/3 = 1/12
• Dividing a whole number by a fraction is the same as multiplying that whole number by the reciprocal of the fraction. Example: 3 / 1/6 is the same as 3 x 6.
In other words, how many times does 1/6 go into 3? Eighteen times.
• Reciprocal: One of two numbers whose product is 1. Example: 5 or 5/1 is the reciprocal of 1/5. 5/1 x 1/5 = 1.
• Dividing a fraction by a fraction works exactly the same way. Invert the second fraction (write its reciprocal) and then multiply the two fractions. Example: 1/2 / 1/12 is the same as 1/2 x 12/1 = 6. In other words, how many times does 1/12 go into 1/2? Six times.
2.5) Compute and perform simple multiplication and division of fractions and apply these procedures to solving problems.
• (Multiplication of fractions)
• (Division of fractions)
Students use variables in simple expressions, compute the value of the expression for specific values of the variable, and plot and interpret the results.
1.1) Use information taken from a graph or equation to answer questions about a problem situation.
1.2) Use a letter to represent an unknown number. Write and evaluate algebraic expressions with one variable by substitution. [This is key standard for fifth grade.]
• (Write and evaluate expressions)
• Variable: A letter or symbol that stands for a number or quantity.
• Expression: A mathematical phrase or the part of a number sentence that combines numbers, operation signs, and sometimes variables, but doesn't have an equal sign. [Expressions can be evaluated, not solved.]
• Evaluate: To find the value; in an algebraic expression, to replace a variable with a value and then perform the operation(s).
- Number model - A number sentence or expression that models or fits a number story or situation. For example, the story Sally had $5, and then she earned $8, can be modeled as the number sentence 5 + 8 = 13, or as the expression 5 + 8.
• Equation: An algebraic or numerical sentence that shows that two quantities are equal. An equation gives two names for the same value. [Equations can be solved; a solution can be found.]
• Solution: A value that, when substituted for the variable, makes an equation true.
1.3) Know and use the distributive property in equations and expressions with variables.
- Commutative Property - A property of addition and multiplication (but not of subtraction or division) that says that changing the order of the numbers being added or multiplied does not change the answer. These properties are often called turn-around facts.
• Distributive property of multiplication: The property that states that multiplying a sum by a number is the same as multiplying each addend in the sum by the number and then adding the products.
1.4) Identify and graph ordered pairs in the four quadrants of the coordinate plane. [This is a key standard for fifth grade.]
• Ordered pair: A set of two numbers used to locate a point on a grid. The first number tells the left-right (horizontal) position and the second number tells the up-down (vertical) position.
1.5) Solve problems involving linear functions with integer values; write the equation; and graph the resulting ordered pairs on a grid. [This is a key standard for fifth grade.]
Students understand and compute the volumes and areas of simple objects.
1.1) Derive and use the formula for the area of a triangle and of a parallelogram by comparing it with the formula for the area of a rectangle (i.e., two of the same triangles make a parallelogram with twice the area; a parallelogram is compared with a rectangle of the same area by cutting a pasting a right triangle on the parallelogram. [This is a key standard for fifth grade.]
• Area of a rectangle = length x width
(A = lw)
• Area of a triangle = 1/2(base x height)
(A = 1/2bh) or (A = bh / 2)
• Area of a parallelogram = base x height (A = bh)
• The base of a plane figure (like a triangle or parallelogram) is usually thought of as the side on which the figure "sits."
• The height of a plane figure (like a triangle or parallelogram) is the length of a perpendicular line from the base to the top of the figure.
1.2) Construct a cube and rectangular box from two-dimensional patterns and use these patterns to compute the surface area for these objects. [This is a key standard for fifth grade.]
• Face: A polygon that is a flat surface of a solid figure.
• Net: A two-dimensional pattern that can be folded into a three-dimensional prism or pyramid.
• Surface area: The sum of the areas of all the faces of a solid figure. [The surface area of a rectangular prism is found by adding the areas of its six faces.]
1.3) Understand the concept of volume and use the appropriate units in common measuring systems (i.e., cubic centimeter [cm3], cubic meter [m3], cubic inch [in3], cubic yard [yd3]) to compute the volume of rectangular solids. [This is a key standard for fifth grade.]
• Volume: The measure of the space a solid figure occupies.
• Volume = length x width x height
(V=lwh)
1.4) Differentiate between, and use appropriate units of measure for, two- and three-dimensional objects (i.e., find the perimeter, area, volume).
• Perimeter: The distance around a closed plane figure.
• Perimeter is one-dimensional (length) and is measured in units. Examples: cm, m, in, ft, yd, etc.
• Area: The area of a plane figure is the space enclosed by that figure. It is measured as the number of square units needed to cover the surface of that figure.
• Area is two-dimensional (length and width) and is measured in square units. Examples: cm2, m2, in2, ft2, yd2, etc.
• Volume is three-dimensional (length, width, and height) and is measured in cubic units. Examples: cm3, m3, in3, ft3, yd3, etc.
Students identify, describe, and classify the properties of, and the relationships between, plane and solid geometric figures.
2.1) Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by using appropriate tools (i.e., straightedge, ruler, compass, protractor, drawing software). [This is a key standard for fifth grade.]
• Angle: A figure formed by two rays that meet at a common endpoint.
• Perpendicular lines: Two lines that intersect (i.e., meet) to form right angles.
• Parallel lines: Lines in the same plane that never intersect.
• Rectangle: A polygon with opposite sides that are equal and with four right angles. [All squares are rectangles, but not all rectangles are squares.]
• Triangle: A polygon with three sides and three angles.
2.2) Know that the sum of the angles of any triangle is 180° and the sum of the angles of any quadrilateral is 360° and use this information to solve problems. [This is a key standard for fifth grade.]
2.3) Visualize and draw two-dimensional views of three-dimensional objects made from rectangular solids.
Students display, analyze, compare, and interpret different data sets, including data sets of different sizes.
1.1) Know the concepts of mean, median, and mode; compute and compare simple examples to show how they may differ.
• Mean: To find the mean, first find the sum of the set of numbers, and then divide by the number of addends. [The mean is the most common type of average. There is only one mean for any data set.]
• Median: The middle number in a set of data that are arranged in order. [You can arrange the numbers from least to greatest or greatest to least. If the data set has an even number of data, then find the mean of the two middle numbers. Example: If the two middle numbers are 4 and 5, then the median is 4.5 (4 + 5 = 9. 9 / 2 = 4.5) There is only one median for any data set.]
• Mode: The number or numbers that occur most often in a set of data. [You may have one mode, multiple modes, or no modes for any data set.]
1.2) Organize and display single-variable data in appropriate graphs and representations (e.g., histogram, circle graphs) and explain which types of graphs are appropriate for various data sets.
1.3) Use fractions and percentages to compare data sets of different sizes.
1.4) Identify ordered pairs of data from a graph and interpret the meaning of the data in terms of the situation depicted by the graph. [This is a key standard for fifth grade.]
1.5) Know how to write ordered pairs correctly, for example, (x, y). [This is a key standard for fifth grade.]
