Arizona Standards

Mathematics Standards
(Adopted 8/26/96)

Standard 5: Measurement And Discrete Mathematics

Students make and use direct and indirect measurement, metric and U.S. customary, to describe and compare the real world and to prepare for the study of discrete functions, fractals and chaos which have evolved out of the age of technology.

Students know and are able to do the following:

READINESS (Kindergarten)

  • 5M-R1.Recognize that a single object has different attributes (e.g., length, color, size, texture) that can be measured in different ways
  • 5M-R2.Compare and order objects according to observable attributes
  • 5M-R3.Use a variety of puzzles and games involving counting problems

FOUNDATIONS (Grades 1-3)

  • 5M-F1.Demonstrate that a single object has different attributes that can be measured in different ways (e.g., length, mass/weight, time, temperature, area and volume)

PO 1. Determine the characteristics (attributes) of an object that are measurable (e.g., length and weight are measurable; color and texture are not measurable)

PO 2. Identify the type of measure (e.g., weight, height, volume) for each attribute

  • 5M-F2.Explain the concepts related to units of measure and demonstrate the process of measurement with non-standard (e.g., using paper clip lengths), U.S. customary and metric units

PO 1. Select the appropriate unit of measure for a given characteristic of an object

length - inches, feet and yards;

centimeters and meters

capacity/volume - cups, gallons and liters

mass/weight - ounces, pounds, grams

and kilograms

PO 2. Select the appropriate tool (e.g., ruler, thermometer, measuring cup, scale) to measure the given characteristic of an object

PO 3. Measure a given characteristic of an object using non-standard units of measure

PO 4. Measure a given characteristic of an object using standard units of measure

PO 5. Tell time to the nearest minute on digital and traditional (analog) clocks

PO 6. Determine the passage of time (i.e., units of days, months and years) using a calendar

PO 7. Compare units of measure to determine more or less relationships

length - inches and feet

feet and yards

centimeters and meters

capacity - cups and gallons

mass - ounces and pounds

grams and kilograms

time - minutes and hours

hours and days

days and weeks

months and years

money - pennies, nickels, dimes,

quarters and dollars

PO 8. Compare units of measure to determine equivalent relationships

length - inches to feet

time - minutes to hours

days to weeks

months to years

money - pennies, nickels, dimes,

quarters to dollars

PO 9. Read a thermometer in Celsius and Fahrenheit to the nearest degree

  • 5M-F3.Make estimates of measurement

PO 1. Estimate a measurement

PO 2. Compare the estimation to actual measure

PO 3. Evaluate the reasonableness of the estimation

  • 5M-F4.Use discrete mathematical models for graphs to represent everyday situations (e.g., determine how many ways to move from point A to point B on a grid)

PO 1. Make a diagram to represent the number of combinations between two sets (e.g., "How many outfits can one make with three different colors of shirts and two different pairs of pants?")

ESSENTIALS (Grades 4-8)

  • 5M-E1.Estimate, make and use measurements (U.S. customary and metric) to describe and make comparisons

PO 1. Measure length, volume and weight in both U.S. customary and metric units (Grades 4-5)

PO 2. Convert measurement units to equivalent units within a given system (customary and metric) (e.g., 12 inches = 1 foot, 10 decimeters = 1 meter) (Grades 4-5)

PO 3. Estimate measurements for both U.S. customary and metric units within either system (Grades 4-5, 6-8)

PO 4. Compare estimated measurements between U.S. customary and metric systems (e.g., a yard is about a meter) (Grades 6-8)

  • 5M-E2.Select and use appropriate units and tools to measure to the degree of accuracy required in a particular problem-solving situation

PO 1. State the appropriate tool to measure in a particular situation (e.g., "What tool would you use to measure the top of your desk?") (Grades 4-5)

PO 2. State the appropriate unit of measurement in a particular situation (e.g., "What unit of measurement would you use to measure the top of your desk?" (Grades 4-5)

PO 3. Measure to the appropriate degree of accuracy to solve problems (e.g., measuring to the nearest sixteenth of an inch or using ounces, measuring to the nearest millimeter or using liters) (Grades 4-5, 6-8)

  • 5M-E3.Estimate, use and describe measures of distance, perimeter, area, volume, capacity, weight, mass and angles

PO 1. Differentiate between perimeter and area of quadrilaterals using concrete and illustrative models (Grades 4-5)

PO 2. Record estimates and measurements for

    1. distance (Grades 4-5)
    2. distance in scale drawings (Grades 6-8)
    3. perimeter (Grades 4-5)
    4. circumference (Grades 6-8)
    5. area (Grades 4-5, 6-8)
    6. volume (Grades 6-8)
    7. weight (Grades 4-5)
    8. mass (Grades 6-8)
    9. degrees of angles (Grades 6-8)
    10. capacity (Grades 6-8)

PO 3. Compare weight to mass and capacity to volume (Grades 6-8)

  • 5M-E4.Develop and use formulas and procedures to solve problems involving measurement

PO 1. Develop a procedure or formula to calculate

    1. area and perimeter of simple polygons (Grades 4-5)
    2. area of polygons and circles (Grades 6-8)
    3. surface area of rectangular prisms (Grades 6-8)
    4. volume of rectangular prisms (Grades 6-8)

PO 2. Use given formulas to find

    1. area and perimeter of simple polygons (Grades 4-5)
    2. circumference of a circle (Grades 6-8)
    3. area of polygons and circles (Grades 6-8)
    4. surface area of rectangular prisms (Grades 6-8)
    5. volume of prisms (Grades 6-8)
  • 5M-E5.Describe how a change in the linear dimension of an object affects its perimeter, area and volume

PO 1. Describe the change in perimeter and area when one dimension of an object is altered (Grades 4-5)

PO 2. Describe the effect on perimeter, area and volume when one dimension of an object is altered (Grades 6-8)

  • 5M-E6.Use calculators and computers to perform basic recursive and iterative processes

PO 1. Solve a problem using the iterative process

    1. doubling (e.g., "If you get paid 1 cent the first day, 2 cents the second day, each day doubling the previous day’s pay, how much would you get paid on the 20th day?") (Grades 4-5)
    2. designing a simple geometric pattern (e.g., design a basic quilt block; use it to generate the whole quilt) (Grades 6-8)

PO 2. Generate the interative sequence for the next six terms when given the first four terms (e.g., 4, 7, 10, 13, . . .) (Grades 4-5)

PO 3. Complete the interative sequence (e.g., given these terms and assuming a constant difference 21, -, -, -, -, -, 63, -, -, -!) (Grades 6-8)

PO 4. Generate subsequent terms of a recursive sequence (e.g., 3, 3, 6, 9, 15, . . .) (Grades 6-8)

PROFICIENCY (Grades 9-12)

  • 5M-P1.Represent problem situations using discrete structures such as finite graphs, matrices, sequences and recurrence relations

PO 1. Use matrices and finite graphs to display data

PO 2. Find a specified nth term of a simple arithmetic or geometric sequence, where the common difference or common ratio is an integer and n > 100

PO 3. Use simple or basic recursion formulas to solve real-life problems (e.g., compound interest)

  • 5M-P2.Represent and analyze finite graphs using matrices

PO 1. Interpret data using matrices and finite graphs (e.g., networks, street diagrams, tournament schedules, production schedules)

PO 2. Determine when a finite graph gives an accurate picture of a data set

PO 3. Translate a finite graph into a matrix and vice versa

  • 5M-P3.Develop and analyze algorithms

PO 1. Write an algorithm that explains a particular mathematical process (e.g., tell a younger child how to find the average of two numbers)

PO 2. Determine the purpose of a given algorithm

PO 3. Determine whether given algorithms are equivalent

  • 5M-P4.Solve enumeration and finite probability problems

PO 1. Find the outcome set of a situation

PO 2. Describe how likely it is that a specific event will happen

PO 3. Determine theoretical geometrical probabilities, given necessary formulas (e.g., "Given a circular target on a square base, what is the probability of hitting the circle with a dart, providing the dart goes inside the square?")

PO 4. Determine the number of possible outcomes in a real-world situation using the counting principle and tree diagrams

PO 5. Use critical path methods in problem solving (e.g., students can represent connections between cities, or some other objects, with graphs and illustrate the number of ways of moving from one vertex to another under various rules for movement [touch each vertex only once, touch each edge only once, begin and end at the same vertex, . . .])

DISTINCTION (Honors)

  • 5M-D1.Represent and solve problems using linear programming and difference equations
  • 5M-D2.Investigate problem situations that arise in connection with computer validation and the application of algorithms
  • 5M-D3.Describe, analyze and extend patterns produced by processes of geometric change such as fractals