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
Step 2 – Adjust the estimate to reflect the sizeof the remaining digits.93+28+53 is approximately 175.Think 700+175=875.– rounding. A process of replacing a numberby an approximate value of that number.For example, 106 rounded to the nearest tenis 110.evaluation. A judgement made at a specific,planned time about the level of a student’sachievement, on the basis of assessment data.Evaluation involves assigning a level, grade,or mark.expectations. The knowledge and skills thatstudents are expected to learn and to demonstrateby the end of every grade or course, as outlined inthe Ontario curriculum documents for the varioussubject areas.extension. A learning activity that is related toa previous one. An extension can involve a taskthat reinforces, builds upon, or requires applicationof newly learned material.family math night. An event designed to bringparents and children together for the purpose ofengaging in mathematics activities and to helpparents learn more about the Ontario mathematicscurriculum.figure. See three-dimensional figure.five frame. A 1 by 5 array onto which countersor dots are placed, to help students relate a givennumber to 5 (e.g., 7 is 2 more than 5) and recognizethe importance of 5 as an anchor in ournumber system. See also ten frame.flat. In base ten blocks, the representation for 100.flow chart. A graphic organizer in which linesand/or arrows show the relationships among ideasthat can be represented by words, diagrams,pictures, or symbols.♥ ♥♥ ♥♥ ♥♥ ♥array3 x 4♥ ♥♥ ♥equal groupsMultiplicationfactor3 x 4=12repeated additionproductfluency. Proficiency in performing mathematicalprocedures quickly and accurately. Althoughcomputational fluency is a goal, students shouldbe able to explain how they are performingcomputations, and why answers make sense.See also automaticity.fractional sense. An understanding that wholenumbers can be divided into equal parts that arerepresented by a denominator (which tells howmany parts the number is divided into) and anumerator (which indicates the number of thoseequal parts being considered). Fractional senseincludes an understanding of relationships betweenfractions, and between fractions and whole numbers(e.g., knowing that 1 /3 is bigger than 1 /4 andthat 2 /3 is closer to 1 than 2 /4 is).front-end estimation. See under estimationstrategies.front-end loading. See front-end estimationunder estimation strategies.4+4+4fast way to addGlossary 93
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In general, students first need to
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Respect How Each Student LearnsTeac
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- Page 119 and 120: Payne, J.N. (Ed.). (1990). Mathemat