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Learning Standards for Mathematics

Mathematical Reasoning Number and Numeration
1. Students use mathematical reasoning to analyze
mathematical situations, make conjectures, gather
evidence, and construct an argument.

Students:
• use models, facts, and relationships to draw conclusions
about mathematics and explain their thinking.
• use patterns and relationships to analyze mathematical
situations.
• justify their answers and solution processes.
• use logical reasoning to reach simple conclusions.


This is evident, for example, when students:

  build geometric figures out of straws.
  find patterns in sequences of numbers, such as the triangular
numbers 1, 3, 6, 10, . . . .
  explore number relationships with a calculator (e.g., 12 + 6 = 18,
11 + 7 = 18, etc.) and draw conclusions.
2. Students use number sense and numeration to
develop an understanding of the multiple uses of
numbers in the real world, the use of numbers to
communicate mathematically, and the use of numbers
in the development of mathematical ideas.


Students:
• use whole numbers and fractions to identify locations,
quantify groups of objects, and measure distances.
• use concrete materials to model numbers and number
relationships for whole numbers and common fractions,
including decimal fractions.
• relate counting to grouping and to place-value.
• recognize the order of whole numbers and commonly
used fractions and decimals.
• demonstrate the concept of percent through problems
related to actual situations.


This is evident, for example, when students:

count out 15 small cubes and exchange ten of the cubes for a rod
ten cubes long.
use the number line to show the position of 1/4.
figure the tax on $4.00 knowing that taxes are 7 cents per $1.00.
 

Sample Problems

 

 

 
Key ideas are identified by numbers (1).
Performance indicators are identified by bullets (•).
Sample tasks are identified by triangles ().

Students will understand mathematics and become mathematically confident by
communicating and reasoning mathematically, by applying mathematics in
real-world settings, and by solving problems through the integrated study of
number systems, geometry, algebra, data analysis, probability, and trigonometry
.

Operations Modeling/Multiple
Representation
3. Students use mathematical operations and
relationships among them to understand mathematics.

Students:
• add, subtract, multiply, and divide whole numbers.
• develop strategies for selecting the appropriate
computational and operational method in problem-solving
situations.
• know single digit addition, subtraction, multiplication,
and division facts.
• understand the commutative and associative properties.


This is evident, for example, when students:

  use the fact that multiplication is commutative
(e.g., 2 x 7 = 7 x 2), to assist them with their memorizing of the
basic facts.
  solve multiple-step problems that require at least two different
operations.
  progress from base ten blocks to concrete models and then to
paper and pencil algorithms.
4. Students use mathematical modeling/multiple
representation to provide a means of presenting,
interpreting, communicating, and connecting
mathematical information and relationships.

Students:
• use concrete materials to model spatial relationships.
• construct tables, charts, and graphs to display and
analyze real-world data.
• use multiple representations (simulations, manipulative
materials, pictures, and diagrams) as tools to explain the
operation of everyday procedures.
• use variables such as height, weight, and hand size to
predict changes over time.
• use physical materials, pictures, and diagrams to explain
mathematical ideas and processes and to demonstrate
geometric concepts.


This is evident, for example, when students:

  build a 3 x 3 x 3 cube out of blocks.
  use square tiles to model various rectangles with an area of 24
square units.
  read a bar graph of population trends and write an explanation
of the information it contains.
 

Sample Problems

Measurement Uncertainty
5. Students use measurement in both metric and
English measure to provide a major link between the
abstractions of mathematics and the real world in
order to describe and compare objects and data.

Students:

• understand that measurement is approximate, never
exact.
• select appropriate standard and nonstandard
measurement tools in measurement activities.
• understand the attributes of area, length, capacity,
weight, volume, time, temperature, and angle.
• estimate and find measures such as length, perimeter,
area, and volume using both nonstandard and standard
units.
• collect and display data.
• use statistical methods such as graphs, tables, and charts
to interpret data.


This is evident, for example, when students:

  measure with paper clips or finger width.
  estimate, then calculate, how much paint would be needed to
cover one wall.
  create a chart to display the results of a survey conducted
among the classes in the school, or graph the amounts of survey
responses by grade level.
6. Students use ideas of uncertainty to illustrate that
mathematics involves more than exactness when
dealing with everyday situations.

Students:
• make estimates to compare to actual results of both
formal and informal measurement.
• make estimates to compare to actual results of
computations.
• recognize situations where only an estimate is required.
• develop a wide variety of estimation skills and strategies.
• determine the reasonableness of results.
• predict experimental probabilities.
• make predictions using unbiased random samples.
• determine probabilities of simple events.


This is evident, for example, when students:

  estimate the length of the room before measuring.
  predict the average number of red candies in a bag before
opening a group of bags, counting the candies, and then
averaging the number that were red.
  determine the probability of picking an even numbered slip from
a hat containing slips of paper numbered 1, 2, 3, 4, 5, and 6.



 

Sample Problems

   

 

Key ideas are identified by numbers (1).
Performance indicators are identified by bullets (•).
Sample tasks are identified by triangles ().

Students will understand mathematics and become mathematically confident by
communicating and reasoning mathematically, by applying mathematics in
real-world settings, and by solving problems through the integrated study of
number systems, geometry, algebra, data analysis, probability, and trigonometry.

Patterns/Functions

7. Students use patterns and functions to develop
mathematical power, appreciate the true beauty of
mathematics, and construct generalizations that
describe patterns simply and efficiently.

Students:
• recognize, describe, extend, and create a wide variety of
patterns.
• represent and describe mathematical relationships.
• explore and express relationships using variables and
open sentences.
• solve for an unknown using manipulative materials.
• use a variety of manipulative materials and technologies
to explore patterns.
• interpret graphs.
• explore and develop relationships among two- and three-dimensional
geometric shapes.
• discover patterns in nature, art, music, and literature.


This is evident, for example, when students:

represent three more than a number is equal to nine as
n + 3 = 9.
draw leaves, simple wallpaper patterns, or write number
sequences to illustrate recurring patterns.
write generalizations or conclusions from display data in charts
or graphs.

Sample Problem