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Essential Math Vocabulary This partial list of essential math concepts was compiled for educational purposes. It does not necessarily reflect the policy or position of the California After School Resource Center (CASRC) or the California Department of Education (CDE). Intended as a quick reference or tool for supporting mathematics, the list provides key terms, corresponding symbols, customary abbreviations, examples, related terms (“must-know” vocabulary), and instructional tips. (It is not a glossary). Educators are advised to seek more comprehensive resources (e.g., curricula and professional materials), such as those available from the CASRC library, to support students in after school programs. Key Term add Symbols, Abbreviations, or Examples + (plus) Related Terms & Instructional Tips addend, addition, sum, total, combine Tip: Young students starting school benefit from using manipulatives, such as counters, to practice addition. algebraic expression 9x (An expression for nine times a number) amount angle numbers, symbols, variables, unknown, operations Tip: Algebra is essential to real-life problem-solving in science, engineering and design, architecture, and routine tasks. Understanding the properties of addition and multiplication will help students to solve algebraic expressions. amt. quantity ∠ ray, vertex, intersect, point, trigonometry (the study of angles, triangles, and their functions), sine, cosine An Acute Angle Types of angles: right angle (equal to 90º) obtuse angle (greater than 90º) acute angle (less than 90º) complementary angles (add up to 90º) supplementary angles (add up to 180º) bisector (a ray dividing an angle into two equal parts) Tip: Use protractors to help elementary students measure a variety of angles, which are found in just about every corner of objects and places. Associative Property Associative Property of Addition (5+3)+9 = 5+(3+9) Associative Property of Multiplication (4x7)2 = 4(7x2) In addition, the sum (total) is not affected by the grouping of the addends. In multiplication, the product (answer) is not affected by the grouping of the factors. Tip: Help students understand this property by explaining that numbers “associate” with different “friends” inside the parenthesis, but the outcome is the same. 1 average avg. mean (the sum of a set of numbers divided by the number of elements in the set) Tip: Use real-life examples, such as sports statistics or class data, to help students compute averages. calendar Sun. (Sunday) Mon. (Monday) Tues. (Tuesday) Wed. (Wednesday) Thurs. (Thursday) Fri. (Friday) Sat. (Saturday) day, week/ly, month/ly, annual, yearly Seasons: spring, summer, fall/autumn, and winter Tip: Help young students develop their sense of time by having a daily calendar activity and talking about when different activities occur throughout the day. Discuss holidays and other special days as appropriate. Jan. (January) Feb. (February) Aug. (August) Sept. (Sept.) Oct. (October) Nov. (November) Dec. (December) cardinal numbers morning, noon/afternoon, midnight, evening, night 0, 1, 2, 3 … counting numbers Digits include 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Tip: Do your students know how to hyphenate numbers from 21 to 99 (e.g., seventy-three)? Use a math journal, word hunts and puzzles, and other fun ways to help students learn how to spell numbers. cent ¢ clock A.M. P.M. change (money left over after buying something) coins: penny (singular), pennies (plural) nickel, dime, quarter, half dollar, silver dollar Tip: Ask parents and school members to donate items for a class sale. Engage students in running the sale and counting the profit, if any. Use the money toward a worthy cause (e.g., a field trip or a charity donation). o’clock, watch, wristwatch half, quarter to, past, until Tip: Do your students know how to skip count by fives? If they do not, they may have a difficult time telling time on a clock. Lots of examples and practice telling time should help. Commutative Property Commutative Property of Addition 25+79 = 79+25 Commutative Property of Multiplication 11x3 = 3x11 In addition, the sum (total) is not affected by the order of the addends. In multiplication, the product (answer) is not affected by the order of the factors. Tip: Help students understand this property by explaining that numbers “commute” or travel back and forth, but the outcome is the same. 2 compare coordinates order, sequence, similar, different greater than > less than < equal = Tip: As a scaffold, pretend that the greater than (>) sign is an alligator that “eats” the bigger number to help young students compare two numbers. The Coordinates of Point A = 2, 3 Y A (2, 3) 3 2 1 grid, y-axis, x-axis, abscissa, point, ordered pair, vertical, horizontal Tip: Create math artwork by having students locate and connect several coordinates on a grid to reveal a hidden design. Graph paper can be useful for this. X 0 1 2 3 cost Price, dollar sign Tip: Manage the occasional behavior management problem by implementing a “token economy,” whereby students are “charged” a certain amount of make-believe “cash” for misbehaving. (Adding up their “infractions” will help them understand the concept of cost and increase their self-control). decimal point . ones, tenths, hundredths, thousandths, etc. Tip: Help students keep track of decimal points by using graph paper and placing each digit in a square. degree divide º ÷ and / (divided by) double dozen equal 2+2, 3+3, 4+4, 5+5, 6+6, 7+7, 8+8, 9+9, 10+10, etc. doz. = Tip: In math, the term “degree” can apply to measuring temperature, as well as to angle measurements. divisor, dividend, quotient, remainder dividend ÷ divisor = quotient Tip: Division requires students to know their addition, subtraction, and multiplication facts, and to apply all of them in a systematic manner. Using graph or lined paper turned sideways to keep the digits lined up as they divide may be helpful. duplicate, two of the same, pair Tip: Double facts help students solve operations faster and understand other fact relationships, such as: 4+4+1=9 (If students know that 4+4=8, they will add the 1 and arrive at 9 faster). twelve, half-dozen (6) Same, opposite of unequal (≠) 3 equation An Equation number sentence or two mathematical expressions that are equal Tip: Present equations as a game where the object is to find the “mystery number” (variable). Use the concept of a scale where both sides must balance equally. even number Examples: 34, 80, and 126 estimate 59 + 24 is about 85 divisible by 2 or ending in 0, 2, 4, 6, or 8 (opposite of odd numbers) Tip: Play a movement game by asking students to jump up and down if a number you call is odd, and to clap if it is even. Call out a few numbers, alternating between odds and evens. round, guess, approximation, mental math (computing in your head) (numbers rounded to 60 Tip: Help students develop their sense of quantity by and 25) estimating the number of items in a jar, or the actual cost of things. exponent 52 exponent base This problem is read as, “Five to the second power or five squared.” fact family Addition/Subtraction 9 + 3 = 12 3 + 9 = 12 12 – 3 = 9 12 – 9 = 3 Multiplication/Division 7 x 2 = 14 2 x 7 = 14 14 ÷ 7 = 2 14 ÷ 2 = 7 figure Fig. Flat Figures: exponent, power, base, square, cube, expanded form, standard form, repeated multiplication Tip: Help students experiment with exponents by writing out the factors in the problem. For instance, 5 9 =9x9x9x9x9 5= 9 59,049 related addition and subtraction or division and multiplication facts Tip: Use flash cards, repetition, educational toys, and games to help students memorize addition and subtraction facts. These are critical to more advanced operations, such as multiple-digit multiplication and division. shape, flat/plain, three-dimensional/solid, round, square, rectangular, polygon trapezoid parallelogram octagon Tip: Play a game of “human geometry” by asking students to make geometric figures using their fingers, arms, legs, or getting into small groups to position their bodies to form shapes. Solid Figures: cylinder rectangular prism 4 formula Formulas exist for: side, base, length, width, pi, square and cubic units • Perimeter (distance around a figure • Area (the square unit measure of the interior region of a figure • Circumference (the perimeter of a circle) • Volume (how many cubic units it takes to fill up a solid. Tip: Know these three basic formulas. Figure Polygon Formula Perimeter = P Example P = s1 + s2 + … (where s = side) 3cm P = 3+3+3+3+3 P = 15 cm Square Perimeter = P P = 4s 7in. P = 4(7) P = 28 in. Circle fraction Circumference = C C = 2Πd (where pi represented by Π is about 3.14 and d stands for diameter) 3m C = 2Πd C = 2 x 3.14 x 6 C = 37.68 m numerator, denominator, common denominator, shaded, part, whole, equivalent, proper, improper Tip: Help students in grades four through six compare fractions by cross-multiplying them, such as in the example below: Which fraction is greater, ¾ Cross-multiply: or ⅝? 3 x 8 = 24 and 4 x 5 = 20 3 4 frequency f graph 5 8 Therefore, ¾ >⅝ mean, median, mode, outlier, and histogram (a graph used to show numerical relationships) pictograph, bar graph, pie graph, etc. Note: A graph is a table or pictorial device used to show relationships between numbers. line Segment Lines can extend to infinity in both directions. Ray Tip: Explain that lines can extend to infinity in both directions, while rays start at a certain point along a line and continue, and segments refer to only a part of a line. Line Perpendicular lines Parallel lines 5 measurement Units of Measurement: centimeter (cm.) gram (g.) inch (in.) liter (L.) meter (m.) milliliter (mL.) ounce (oz.) multiply x and * (times) unit, area, distance, length, width, perimeter, volume Tip: Measurement may be abstract and challenging for students. Provide ample opportunities for them to measure real objects using various tools, including rulers, tape measures, containers, and thermometers. Invite them to hop, gallop, run, crawl, or skip from place to place to figure out how fast or slow they can move across small distances. group, product, multiple, factor, and array Tip: An array is a set of objects in equal rows or columns used to show groups. Allow students to make arrays using math unifix cubes to visually show groupings involved in multiplication. a 3-by-2 array of 6 stars or 3X2=6 negative - number line absolute value (the distance of a number from zero, always a positive number, represented by (II) Number Line with Positive and Negative Numbers odd number numbers less than zero Examples: 21, 79, and 345 Tip: Number lines help students understand locations, distance, intervals, and other relationships among numbers. Using sidewalk chalk, paint a big number line on the playground, and invite students to move back and forth along the line to represent addition and subtraction, negative and positive numbers, etc. numbers ending in 1, 3, 5, 7, or 9 (opposite of even numbers) Tip: Play a game of thumbs-up or thumbs-down by asking students to show thumbs-up if a number you call is odd, or thumbs-down if it is even. Call out a few numbers, alternating between odds and evens. one-to-one correspondence A Matching Game counting, representing, matching one object to another (critical to counting) Tip: Invite young students to play a variety of music instruments with distinct sounds. Have others clap or dance to the rhythms they hear. Then ask them to count and move simultaneously to help them understand oneto-one correspondence. opposite contrary, inverse (as in inverse operations, such as addition and subtraction, as well as multiplication and division 6 ordinal numbers first, second, third, fourth, fifth, sixth, seventh, eighth, ninth, tenth, etc. 1st 2nd 3rd pattern Tip: Ask students to line up and state their position along the line using ordinal numbers. Invite them to make signs to have them practice the spelling and/or abbreviations associated with ordinals. something that repeats itself or changes in a regular way (e.g., a design or configuration) Tip: Use pattern blocks to have students create patterns. Invite them to showcase their patterns and take turns figuring out their classmates’ patterns. percent % place value Tip: A percentage is simply a comparison of a number to 100. Use a hundreds chart to help students shade in different percentages (color 10 squares to show 10%, 25 squares to show 25%, and so on). Tip: Digits have a value according to their place/location within a number. Help students understand place value by starting with small numbers, and building on that for bigger numbers. Place mats and improvised charts are great for helping students put the correct digits on the correct place value location. Place Value Chart polygon Examples of Simple Polygons simple (one boundary), complex (intersects itself), convex (no angles pointing inwards), concave (internal angles greater than 180º), regular (all angles equal), and irregular (angles vary) Tip: Teach the meaning of prefixes associated with polygons to help students remember the number of sides in each figure: Prefix positive prepositions + 9, 10, 11 before between after triquad- Meaning/ # of Sides 3 4 pentahexaheptaoctanonadeca- 5 6 7 8 9 10 Polygon triangle quadrilateral (squares, rectangles, trapezoids) pentagon hexagon heptagon octagon nonagon decagon numbers greater than zero above, after, before, below, beneath, beside, between, next, next to, over, etc. Tip: These words help students understand math directions and problems. Play a game of “Simon Says” to help students understand the meanings of prepositions. 7 ratio regroup / Regrouping Tens 4 5 -3 1 Ones 17 7 9 8 ruler comparison of two numbers using division ones, tens, hundreds, thousands, ten thousands … Tip: Avoid using the terms “borrow” and “carry,” as these have been replaced with “regroup” because technically, that is what is happening (numbers are regrouped into a group of ten). Tip: A device used for measuring or drawing straight lines takes some practice for young students who are still developing their locomotor skills. Use small rulers first, and move bigger ones over time. season Tip: Discuss the look and feel of spring, summer, fall/autumn, and winter with young students. The Four Seasons biggest bigger size skip count Skip Counting by 5s and 12 Tip: Adjectives, comparatives, and superlatives, such as short(er/est), long(er/est), big(ger/est), small(er/est), are useful to students in making math comparisons. Tip: Skip counting helps students understand intervals, perform operations, and remember facts (e.g., addition and multiplication). Play a game of hopscotch to practice skip counting outdoors. Other ways to skip count: 2, 4, 6, 8, 10, 12 … 5, 10, 15, 20 … 10, 20, 30, 40 … 11, 22, 33, 44, 55 … solution strategy • • • • • • A solution of 3y = 18 is y = 6 To answer, solve, or figure out the value that makes an equation true is to find a solution. Math Strategy Examples understand, plan, solve, check, use logic Drawing conclusions Working backwards Finding clue words Find a pattern Make a table Draw a picture Tip: Encourage students to use a variety of problemsolving strategies, and to share their approaches with each other. 8 square root √ Subtract - (minus) Tip: Help students understand square roots as a special value that when multiplied by itself gives the same number. Having an understanding of exponents may help. difference, take away, less Tip: Have students line up around a sheet or a parachute. As a class, count several bean bags before adding them to the center of the parachute or sheet. Invite everyone to lift it, tossing the beanbags into the air. Catch as many as possible, prompting students to subtracting those that fall off. Repeat several times, checking or writing out the problems on the board as needed. symmetry symmetrical symmetrical (two halves match) asymmetrical (two halves do not match exactly) asymmetrical temperature Thermometer thermometer, degrees, weather Tip: People in many foreign countries measure how hot or cold it is using Celsius (C) scale, while the United States uses the Fahrenheit (F) scale. Help students understand temperatures by calling out a few weather forecasts, and inviting them to act out whether they will feel mild, cold, or hot. triangle Types of Triangles Equilateral: equal sides and angles measuring 60º Isosceles: two equal sides and angles Scalene: no equal sides or angles Right: contains one 90º angle variable Zero Property of Multiplication In 9n, the variable is n. nx0=0 equilateral, isosceles, scalene, right, obtuse, acute, base, height, arm, Pythagorean Theorem Tip: Help students understand how to compute the area of triangles using the formula Area = ½ bh, where b stands for the base, and h stands for the height of the triangle. h = 8 cm. A = ½ (b x h) A = ½ (5 cm x 8 cm) 2 A = 20 cm. b = 5 cm. unknown, a symbol that stands for a quantity (e.g., x or y) The product of any number and zero is zero. 9