Introduction

discrete quantities. Individual, countable objects,such as cubes. See also continuous quantities.distributive property. The property that allowsnumbers in a multiplication or division expressionto be decomposed into two or more numbers. Thedistributive property applies to:• multiplication over addition, for example,6 x 47=(6 x 40)+(6 x 7)=240+42=282;• multiplication over subtraction, for example,4 x 98=(4 x 100) – (4 x 2)=400 – 8=392;• division over addition, for example, 72÷6=(60÷6)+(12÷6)=10+2=12);• division over subtraction, for example, 4700÷4=(4800÷4) – (100÷4)=1200 – 25=1175).dot plates. Paper plates with peel-off dotsapplied in various arrangements to representnumbers from 1 to 10. Dot plates are useful inpattern-recognition activities.doubles. Basic addition facts in which bothaddends are the same number (e.g., 4+4, 8+8).Students can apply a knowledge of doubles tolearn other addition facts (e.g., if 6+6=12, then6+7=13) and multiplication facts (e.g., if7+7=14, then 2 x 7=14).drill. Practice that involves repetition of a skillor procedure. Because drill often improves speedbut not understanding, it is important that conceptualunderstanding be developed before drillactivities are undertaken. See also automaticity.equal group problem. A problem that involvessets of equal quantities. If both the number andthe size of the groups are known, but the totalis unknown, the problem can be solved usingmultiplication. If the total in an equal groupproblem is known, but either the number ofgroups or the size of the groups is unknown,the problem can be solved using division.equality. The notion of having the same value,often expressed by the equal sign (i.e., =, as in8=3+5). In 3+5=8 and 8=3+5, the expressionson either side of the equal sign (3+5 and 8) havethe same value. It is important that students learnto interpret the equal sign as “is the same amountas” rather than “gives the answer”.estimation. The process of arriving at anapproximate answer for a computation, or at areasonable guess with respect to a measurement.Teachers often provide very young students witha range of numbers within which their estimateshould fall.estimation strategies. Mental mathematicsstrategies used to obtain an approximate answer.Students estimate when an exact answer isnot required and when they are checking thereasonableness of their mathematics work.Some estimation strategies are as follows:– clustering. A strategy used for estimatingthe sum of numbers that cluster around oneparticular value. For example, the numbers 42,47, 56, 55 cluster around 50. So estimate50+50+50+50=200.– ”nice” or compatible numbers. A strategythat involves using numbers that are easyto work with. For example, to estimate thesum of 28, 67, 48, and 56, one could add30+70+50+50. These nice numbers areclose to the original numbers and can beeasily added.– front-end estimation. (Also called “front-endloading”.) The addition of significant digits(those with the highest place value), withan adjustment of the remaining values.For example:Step 1 – Add the left-most digit in each number.193+428+253Think 100+400+200=700.92 A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6 – Volume One