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 nonzero 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 realworld 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 multidigit 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
 Unifix Flash Cards–Students will use the unifix flash cards to build a model, showing the relationship between repeated addition & multiplication and subtraction & division.
 Subtraction–There’s Got to be an Easier Way!–Students will work on subtraction with regrouping.
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.
 MultiDigit Multiplication–At the conclusion of this lesson, students will be able to successfully multiply multidigit 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
 TwoStep Equations–Students will work through TwoStep Equations using algebra tiles, drawing pictures, and writing the stepbystep process.
 The New Texas TwoStep–This activity gives a fun introduction to twostep equations.
UENLinked 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.