Posted by: degarcia | April 2, 2009

Big Idea: Basic Facts and Algorithms

Basic Facts and Algorithms


As Defined in Big Ideas and Understandings as the Foundation for Elementary and Middle School Mathematics by Randall I. Charles

Basic facts and algorithms for operations with rational numbers use notions of equivalence to transform calculations into simpler ones.

Examples of Mathematical Understanding

Mental Calculations

  • Number relationships and sequences can be used for mental calculations (one more, one less; ten more, ten less; 30 is two more than 28; counting back by thousands from 50,000 is 49,000, 48,000, 47,000 etc.)

  • Numbers can be broken apart and grouped in different ways to make calculations simpler.


Whole Number Basic Facts and Algorithms

  • Some basic addition and multiplication facts can be found by breaking apart the unknown fact into known facts. Then the answers to the known facts are combined to give the final value.

  • Subtraction facts can be found by thinking of the related addition fact.
  • Division facts can be found by thinking about the related multiplication fact.
  • When 0 is divided by any non-zero number, the quotient is zero, and 0 cannot be a divisor.
  • Addition can be used to check subtraction, and multiplication can be used to check division.

  • Powers of ten are important benchmarks in our numeration system, and thinking about numbers in relation to powers often can make addition and subtraction easier.

  • When you divide whole numbers sometimes there is a remainder; the remainder must be less than the divisor.

  • The real-world situation determines how a remainder needs to be interpreted when solving a problem.

Rational Number Algorithms

  • Fractions with unlike denominators are renamed as equivalent fractions with like denominators to add and subtract.
  • The product of two fractions can be found by multiplying numerators and multiplying denominators.
  • A fraction division calculation can be changed to an equivalent multiplication calculation (i.e., a/b ÷ c/d = a/b x d/c, where b, c, and d = 0).

  • Division with a decimal divisor is changed to an equivalent calculation with a whole number divisor by multiplying the divisor and dividend by an appropriate power of ten.

  • Money amounts represented as decimals can be added and subtracted using the same algorithms as with whole numbers.

Measurement

  • Algorithms for operations with measures are modifications of algorithms for rational numbers.

  • Length measurements in feet and inches can be added or subtracted where 1 foot is regrouped as 12 inches.

  • Times in minutes and seconds can be added and subtracted where 1 minute is regrouped as 60 seconds.

NCTM Standards

Numbers and Operations

  • Understand numbers, ways of representing numbers, relationships among numbers, and number systems.
  • Understand meanings of operations and how they relate to one another.
  • Compute fluently and make reasonable estimates.

Algebra

  • Understand patterns, relations, and functions.
  • Represent and analyze mathematical situations and structures using algebraic symbols.

Measurement

  • Understand measurable attributes of objects and the units, systems, and processes of measurement.
  • Apply appropriate techniques, tools, and formulas to determine measurements.

Problem Solving

  • Solve problems that arise in mathematics and in other contexts.
  • Apply and adapt a variety of appropriate strategies to solve problems.

Connections

  • Recognize and use connections among mathematical ideas.
  • Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.

Utah Education Network Standards

Note:  Each link includes multiple lesson plans and activities, though specific ones are linked below.

Kindergarten

  • Standard 1, Objective 1, Indicator b
  • Standard 1, Objective 2, Indicators a, b, c, d

First Grade

  • Standard 1, Objective 1, Indicators a, b, c
  • Standard 1, Objective 2, Indicator c
  • Standard 1, Objective 3, Indicators b, c, d
  • Standard 2, Objective 2, Indicator c

Second Grade

  • Standard 1, Objective 1, Indicators a, c
  • Standard 1, Objective 3, Indicators a, b, d, e
  • Standard 2, Objective 1, Indicator b
  • Standard 2, Objective 2, Indicator c

Third Grade

  • Standard 1, Objective 1, Indicators b, e
  • Standard 1, Objective 2, Indicator c
  • Standard 1, Objective 3, Indicators a, b, c, d
  • Standard 1, Objective 4, Indicators a, b, c
  • Standard 2, Objective 2, Indicators b, d
  • Standard 4, Objective 1, Indicator a
  • Standard 4, Objective 2, Indicator a

Fourth Grade

  • Standard 1, Objective 1, Indicator b
  • Standard 1, Objective 3, Indicators a, c, d, e
  • Standard 1, Objective 4, Indicators a, b, d
  • Standard 1, Objective 5, Indicators a, b, c, d
  • Standard 2, Objective 2, Indicators a, b, c, d
  • Standard 4, Objective 2, Indicators c, d, e

Fifth Grade

  • Standard 1, Objective 2, Indicators a, e
  • Standard 1, Objective 3, Indicators a, b, c, d
  • Standard 1, Objective 4, Indicators a, b, c
  • Standard 1, Objective 5, Indicator a
  • Standard 1, Objective 6, Indicators a, b, c, d, e
  • Standard 2, Objective 2, Indicators a, b
  • Standard 4, Objective 1, Indicators a, b, c
  • Standard 4, Objective 2, Indicators c, d, e
  • Standard 5, Objective 1, Indicator d

Sixth Grade

  • Standard 1, Objective 1, Indicators b, c, d
  • Standard 1, Objective 2, Indicators c, e
  • Standard 1, Objective 3, Indicators b, c
  • Standard 1, Objective 4, Indicator a
  • Standard 1, Objective 5, Indicator a
  • Standard 1, Objective 6, Indicator a, b, c
  • Standard 2, Objective 2, Indicator c

UEN Lesson Plans

Note:  There are many more lesson plans found by following the links listed with the standards above.

Kindergarten

  • Counting Photo Book–Students create their own counting books.
  • M&Ms Game–Students will connect the numeral to an appropriate quantity of M&M candies.

First Grade

  • Add It Up–This activity is designed to teach students to use manipulatives to solve addition problems while recognizing the symbols of addition and equal value (i.e., add, “and,” plus, +, sum, equals, =, same as).
  • Out of Sight Missing Addends–Students will learn how to use manipulatives to solve missing addends in math problems.

Second Grade

  • Circle the Wagons a Number is Missing–Students have an opportunity to work in pairs as they search for the missing addend that will complete a mathematical sentence. Students will demonstrate the ability to change the order of the addends and still produce the same sum.
  • It’s a Fact–This lesson contains many student activities to help students master addition and subtraction skills.
  • Mathematical Roadmaps–Students will learn about adding multi-digit numbers.
  • What’s Your Function?–Students are presented with conditions that will require the ability to make sense of a mathematical situation with missing information. They will recognize that some function has taken place and their assignment will be to figure out what it was.

Third Grade

Fourth Grade

  • Long Division–The goal of this lesson is to further develop the sharing concept of division by using objects. Students will have one and two digit quotients with and without remainders.
  • Multi-Digit Multiplication–At the conclusion of this lesson, students will be able to successfully multiply multi-digit numbers using rectangular arrays, and a variety of mental math strategies.

Fifth Grade

  • Partial Quotient–This activity is designed to help students understand what occurs in the division process, rather than just following prescribed steps.
  • Multiplication Strategy Review–Students will play a game to practice all of the strategies for multiplication they have learned.

Sixth Grade

  • Two-Step Equations–Students will work through Two-Step Equations using algebra tiles, drawing pictures, and writing the step-by-step process.
  • The New Texas Two-Step–This activity gives a fun introduction to two-step equations.

UEN-Linked Interactive Websites

Note:  There are more links on each grade level found organized by Standard.  These are the links for the first Standard at each grade level.


All Information from Big Ideas and Understandings as the Foundation for Elementary and Middle School Mathematics by Randall I. Charles and http://www.uen.org.

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