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# The California Mathematics Content Standards

## Grade Four Mathematics Content Standards

By the end of grade four, students understand large numbers and addition, subtraction,
multiplication, and division of whole numbers. They describe and compare simple fractions
and decimals. They understand the properties of, and the relationships between,
plane geometric figures. They collect, represent, and analyze data to answer questions.

Number Sense

1.0 Students understand the place value of whole numbers and decimals to two
decimal places and how whole numbers and decimals relate to simple fractions.
Students use the concepts of negative numbers: Read and write whole numbers in the millions. Order and compare whole numbers and decimals to two decimal places. Round whole numbers through the millions to the nearest ten, hundred,
thousand, ten thousand, or hundred thousand. Decide when a rounded solution is called for and explain why such a
solution may be appropriate.

Solve each of the following problems and observe the different roles played
by the number 37 in each situation:

1. Four children shared 37 dollars equally. How much did each get?

2. Four children shared 37 pennies as equally as possible. How many
pennies did each get?

3. Cars need to be rented for 37 children going on a field trip. Each car can
take 12 children in addition to the driver. How many cars must be
rented?

1.5 Explain different interpretations of fractions, for example, parts of a whole,
parts of a set, and division of whole numbers by whole numbers; explain
equivalents of fractions (see Standard 4.0).

True or false?

1. 1/4 > 2.54

2. 5/2 < 2.6

3. 12/18 = 2/3 (Note the equivalence of fractions.)

4. 4/5 < 13/15

1.6 Write tenths and hundredths in decimal and fraction notations and know the
fraction and decimal equivalents for halves and fourths (e.g., 1/2 = 0.5
or 0.50; 7/4 = 1 3/4 = 1.75).

1.7 Write the fraction represented by a drawing of parts of a figure; represent a
given fraction by using drawings; and relate a fraction to a simple decimal on
a number line.

TIMSS gr. 4, M-5)  Use concepts of negative numbers (e.g., on a number line, in counting, in
temperature, in “owing”).

True or false?

1. -9 > -10

2. -31 < -29 Identify on a number line the relative position of positive fractions, positive
mixed numbers, and positive decimals to two decimal places.

2.0 Students extend their use and understanding of whole numbers to the
addition and subtraction of simple decimals:

2.1 Estimate and compute the sum or difference of whole numbers and positive
decimals to two places.

Solve 55.73 - 48.25 = ?

2.2 Round two-place decimals to one decimal or the nearest whole number and
judge the reasonableness of the rounded answer.

Solve 17.91 + 2.18 = ? Students solve problems involving addition, subtraction, multiplication,
and division of whole numbers and understand the relationships among
the operations: Demonstrate an understanding of, and the ability to use, standard algorithms
for the addition and subtraction of multidigit numbers.

Solve 619,581 - 23,183 = ?

Solve 6,747 + 321,105 = ? Demonstrate an understanding of, and the ability to use, standard algorithms
for multiplying a multidigit number by a two-digit number and for
dividing a multidigit number by a one-digit number; use relationships
between them to simplify computations and to check results.

Solve:

1. 783 × 23 = ?

2. 8,988/6 = ?

3. 11,115/9 = ? Solve problems involving multiplication of multidigit numbers by two-digit
numbers. Solve problems involving division of multidigit numbers by one-digit
numbers.

4.0 Students know how to factor small whole numbers:

4.1 Understand that many whole numbers break down in different ways
(e.g., 12 = 4 × 3 = 2 × 6 = 2 × 2 × 3).

In how many distinct ways can you write 60 as a product of two numbers? Know that numbers such as 2, 3, 5, 7, and 11 do not have any factors except
1 and themselves and that such numbers are called prime numbers.

List all the distinct prime factors of 210.

Note: The sample
problems illustrate
the standards and
are written to help
clarify them. Some
problems are written
in a form that can be
used directly with
students; others will
need to be modified,
particularly in the
before they are
used with students.

The symbol identifies the key
standards for

Algebra and Functions

1.0 Students use and interpret variables, mathematical symbols, and properties to
write and simplify expressions and sentences:

1.1 Use letters, boxes, or other symbols to stand for any number in simple
expressions or equations (e.g., demonstrate an understanding and the use
of the concept of a variable).

Tanya has read the first 78 pages of a 130-page book. Give the number of
the sentence that can be used to find the number of pages Tanya must read
to finish the book. (Adapted from TIMSS gr. 4, I-7)

1. 130 + 78 = ___

2. ___ - 78 = 130

3. 130 - 78 = ___

4. 130 - ___ = 178 Interpret and evaluate mathematical expressions that now use parentheses.

Evaluate the two expressions: (28 - 10) - 8 = ___ and 28 - (10 - 8) = ___. Use parentheses to indicate which operation to perform first when writing
expressions containing more than two terms and different operations.

Solve Solve 1.4 Use and interpret formulas (e.g., area = length × width or A = lw) to answer
questions about quantities and their relationships.

There are many rules to get from Column A to Column B in the following
table. Can you state one rule? (Adapted from TIMSS, gr. 4, J-5)

 Column A Column B  Understand that an equation such as y = 3x + 5 is a prescription for determining
a second number when a first number is given. Students know how to manipulate equations: Know and understand that equals added to equals are equal. Know and understand that equals multiplied by equals are equal.

Measurement and Geometry

1.0 Students understand perimeter and area:

1.1 Measure the area of rectangular shapes by using appropriate units, such as
square centimeter (cm2), square meter (m 2), square kilometer (km2), square
inch (in 2), square yard (yd2), or square mile (mi 2).

1.2 Recognize that rectangles that have the same area can have different
perimeters.

Draw a rectangle whose area is 120 and whose perimeter exceeds 50. Draw
another rectangle with the same area whose perimeter exceeds 240.

1.3 Understand that rectangles that have the same perimeter can have different
areas.

Is the area of a 45 × 55 rectangle (in cm2) smaller or bigger than that of a
square with the same perimeter?

Draw a rectangle whose perimeter is 40 and whose area is less than 20.

1.4 Understand and use formulas to solve problems involving perimeters and
areas of rectangles and squares. Use those formulas to find the areas of more
complex figures by dividing the figures into basic shapes.

The length of a rectangle is 6 cm, and its perimeter is 16 cm. What is the
area of the rectangle in square centimeters? (Adapted from TIMSS gr. 8,
K–5) Students use two-dimensional coordinate grids to represent points and graph
lines and simple figures: Draw the points corresponding to linear relationships on graph paper
(e.g., draw 10 points on the graph of the equation y = 3x and connect them
by using a straight line). Understand that the length of a horizontal line segment equals the difference
of the x-coordinates.

What is the length of the line segment joining the points (6, -4) and
(21, -4)? Understand that the length of a vertical line segment equals the difference
of the y-coordinates.

What is the length of the line segment joining the points (121, 3) to
(121, 17)?

3.0 Students demonstrate an understanding of plane and solid geometric objects
and use this knowledge to show relationships and solve problems:

3.1 Identify lines that are parallel and perpendicular.

3.2 Identify the radius and diameter of a circle.

3.3 Identify congruent figures.

3.4 Identify figures that have bilateral and rotational symmetry.

Craig folded a piece of paper in half and cut out a shape along the folded
edge. Draw a picture to show what the cutout shape will look like when it is
opened up and flattened out (Adapted from TIMSS gr. 4, T-5).

Let AB, CD be perpendicular diameters of a circle, as shown. If we reflect
across the line segment CD, what happens to A and what happens to B
under this reflection? 3.5 Know the definitions of a right angle, an acute angle, and an obtuse angle.
Understand that 90°, 180°, 270°, and 360° are associated, respectively, with
1/4, 1/2, 3/4, and full turns.

3.6 Visualize, describe, and make models of geometric solids (e.g., prisms,
pyramids) in terms of the number and shape of faces, edges, and vertices;
interpret two-dimensional representations of three-dimensional objects; and
draw patterns (of faces) for a solid that, when cut and folded, will make a
model of the solid.

3.7 Know the definitions of different triangles (e.g., equilateral, isosceles,
scalene) and identify their attributes.

3.8 Know the definition of different quadrilaterals (e.g., rhombus, square,
rectangle, parallelogram, trapezoid).

Explain which of the following statements are true and why.

1. All squares are rectangles.

2. All rectangles are squares.

3. All parallelograms are rectangles.

4. All rhombi are parallelograms.

5. Some parallelograms are squares.

Statistics, Data Analysis, and Probability

1.0 Students organize, represent, and interpret numerical and categorical data and
clearly communicate their findings:

The following table shows the ages of the girls and boys in a club. Use the
information in the table to complete the graph for ages 9 and 10. (Adapted
from TIMSS gr. 4, S-1)

 Ages Number of Girls Number of Boys  Ages of students

1.1 Formulate survey questions; systematically collect and represent data on a
number line; and coordinate graphs, tables, and charts.

1.2 Identify the mode(s) for sets of categorical data and the mode(s), median,
and any apparent outliers for numerical data sets.

1.3 Interpret one- and two-variable data graphs to answer questions about a
situation.

2.0 Students make predictions for simple probability situations:

Nine identical chips numbered 1 through 9 are put in a jar. When a chip is
drawn from the jar, what is the probability that it has an even number?
(Adapted from TIMSS gr. 8, N-18)

2.1 Represent all possible outcomes for a simple probability situation in an
organized way (e.g., tables, grids, tree diagrams).

2.2 Express outcomes of experimental probability situations verbally and
numerically (e.g., 3 out of 4; 3/4).

Mathematical Reasoning

1.0 Students make decisions about how to approach problems:

1.1 Analyze problems by identifying relationships, distinguishing relevant
from irrelevant information, sequencing and prioritizing information,
and observing patterns.

1.2 Determine when and how to break a problem into simpler parts.

2.0 Students use strategies, skills, and concepts in finding solutions:

2.1 Use estimation to verify the reasonableness of calculated results.

2.2 Apply strategies and results from simpler problems to more complex
problems.

2.3 Use a variety of methods, such as words, numbers, symbols, charts, graphs,
tables, diagrams, and models, to explain mathematical reasoning.

2.4 Express the solution clearly and logically by using the appropriate mathematical
notation and terms and clear language; support solutions with
evidence in both verbal and symbolic work.

2.5 Indicate the relative advantages of exact and approximate solutions to
problems and give answers to a specified degree of accuracy.

2.6 Make precise calculations and check the validity of the results from the
context of the problem.

3.0 Students move beyond a particular problem by generalizing to other
situations:

3.1 Evaluate the reasonableness of the solution in the context of the original
situation.

3.2 Note the method of deriving the solution and demonstrate a conceptual
understanding of the derivation by solving similar problems.

3.3 Develop generalizations of the results obtained and apply them in other
circumstances.