_____________________

Rockwood Everyday Mathematics

Parent Resource Handbook

Kindergarten – 5th Grade ___________________________

Elementary Mathematics Rationale_______

The Rockwood Mathematics Curriculum reflects the importance of mathematical literacy for all students. The curriculum, based upon National Council of Teachers of Mathematics Standards, as well as Missouri Show-Me Standards, is student-centered and will allow students to explore, discover, conjecture, and apply mathematics. To facilitate student learning, teachers utilize a variety of techniques such as direct instruction, cooperative learning, and appropriate use of computers and calculators. Through numerous and interrelated mathematical experiences, students will work to attain the following goals: ♦ become mathematical problem-solvers ♦ communicate mathematically ♦ reason mathematically ♦ connect mathematics to their daily lives ♦ develop confidence in their own abilities to do mathematics ♦ appreciate and understand the role of mathematics in real-world situations The District’s Mathematics Curriculum has a multi-faceted focus, including problem solving, critical thinking, computation, and the integration of technology. These components and goals are an important part of each student’s educational experience. They provide the coherent viewpoint that mathematics is more than a body of knowledge; it is a way of thinking. The adoption of the Everyday Mathematics program supports the implementation of the Rockwood K-5 Math Curriculum through its rigorous design which supports our national, state, and district standards.

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Everyday Math Curriculum Features______

There are a number of features that distinguish the Everyday Mathematics curriculum. These include:

• Real-life Problem Solving- Everyday Mathematics emphasizes the application of mathematics to real world situations. Numbers, skills and mathematical concepts are not presented in isolation, but are linked to situations and contexts that are relevant to everyday lives.

• Balanced Instruction- Everyday Mathematics lessons include time for whole-group instruction as well as small group, partner, or individual activities. These activities balance teacher-directed instruction with opportunities for open-ended, hands-on explorations, long-term projects and on-going practice.

• Multiple Methods for Basic Skills Practice- Everyday Mathematics provides numerous methods for basic skills practice and review. These include written and oral fact drills, mental math routines, practice with fact triangles (flash cards of fact families), daily sets of review problems called math boxes, homework, and a wide variety of math games.

• Emphasis on Communication- Throughout the Everyday Mathematics curriculum, students are encouraged to explain and discuss their mathematical thinking, in their own words. Opportunities to verbalize their thoughts and strategies give children the chance to clarify their thinking and gain insights from others.

• Enhanced Home/School Partnerships- For grades 1-3, daily Home Links provide opportunities for family members to participate in the students' mathematical learning. Study Links are provided for most lessons in grades 4-6, and all grades include periodic letters to help keep parents informed about their children's experience with Everyday Mathematics.

• Appropriate Use of Technology- Everyday Mathematics teaches students how to use technology appropriately. The curriculum includes many activities in which learning is extended and enhanced through the use of calculators. At the same time, all activities in which calculators would function simply as crutches for basic computation are clearly marked with a "no calculator" sign.

*Adapted from UCSMP Everyday Mathematics Center Website

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Routines and Features_____________________ Mental Math and Reflexes

The term Mental Math and Reflexes refers to exercises, usually oral, designed to strengthen children’s number sense and to review and advance essential basic skills. Mental Math and Reflexes sessions usually last no longer than five minutes. For this type of activity, numerous short interactions are more effective than fewer prolonged sessions. There are several kinds of Mental Math activities. Some involve a choral counting routine; many are basic-skills practice with some problem-solving exercises.

Math Message

A Math Message is provided at the beginning of each lesson beginning with Unit 4 in first grade. The Math Message usually leads into the lesson for the day, including occasional reviews of topics previously covered. Children complete the Math Message before the start of each lesson.

The teacher might display the Math Message in a number of ways. The Math Message might be written on the board, on an overhead, on the Smart Board, or on a classroom television. Some teachers have children record their answers to the Math Message. Answers could be recorded in the Math Journals, on individual slates or whiteboards, on sheets of paper to be sent home, or in a spiral notebook.

Teachers often use Math Messages in Everyday Mathematics as guides to developing their own activities,

which they can design specifically for the needs of the children in their classroom.

Home Link

Home Links are the Everyday Mathematics version of homework assignments. Each lesson has a Home Link. The previous night’s Home Link is reviewed during each day’s lesson, so it is important for your child to bring the Home Link back to school completed each day. Home Links consist of active projects and ongoing review problems and serve three main purposes: (1) they promote follow-up, (2) they provide enrichment, and (3) they offer an opportunity for you to become involved in your child’s mathematics education. Many Home Links require children to interact with you or someone else at home.

Explorations and Projects

In Everyday Mathematics, the term Explorations means “time set aside for independent, small-group activities.” Besides providing the benefits of cooperative learning, small-group work lets everyone have a chance to use manipulatives such as pan balances, base-10 blocks, attribute blocks, and thermometers. Explorations can vary in length of time from one class to several days.

Name-Collection Boxes

Children use name-collection boxes to help manage equivalent names for numbers. These devices offer a simple way for children to experience the notion that numbers can be

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Routines and Features_____________________ expressed many different ways. Names for the same number can include sums, differences, products, quotients, the results of combining several operations, words in English or other languages, tally marks, arrays, Roman numerals, and so on.

In Kindergarten through third grade, a name-collection box is a diagram of a box with a label attached to it. The name on the label identifies the number whose names are inside the box.

16 20-4 10 less than 26 XVI 32 ÷ 2

(2 x 5) + 6 4 + 4+ 4+ 4 sixteen

1

7

Half of 32 8 twos 116 – 100

10 + 2 – 4 + 6 – 8 + 10 dieciséis

Beginning in fourth grade a simpler and more compact name-collection box is introduced.

14 1,400%

2 * 7

20 - 6

1 + 13

700/50

33 – 13

0.028 * 500

XIV

(3 *7) - 7

Frames-and-Arrows

Frames-and-Arrows diagrams provide children with a way to organize work with sequences. They are made up of shapes, called frames, connected by arrows to show the path for moving from one frame to another. Each frame contains a number in the sequence. Each arrow represents the rule that tells what number goes in the next frame. Frames-and-Arrows diagrams are also called chains. Here is a simple example of a Frames-and-Arrows diagram for the rule “Add 1.”

Rule: Add 1

Rule Add 1

2 3 4

In Frames-and-Arrows problems, some information is missing. Here are several examples:

1. The rule is given. Some of the frames are empty. Fill in the blank frames.

Rule: Add 2

Rule Add 2

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Solution: Write 9 and 13 in the blank frames.

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Routines and Features_____________________ 2. The frames are filled in. The rule is missing. Find the rule.

Rule

Solution: The rule is subtract 2, minus 2, or -2.

3. Some of the frames are empty. The rule is missing. Find the rule, and fill in the empty frames.

Rule

Solution: The rule is add 1. Write 8 and 10 in the empty frames.

A chain can have more than one arrow rule. If it does, the arrows for the rules must look different. For example, we can use a black arrow for one rule and a gray arrow for the other rule.

Rule Rule +3 + 2

In the following example, the rules and frames are given, but arrows are missing. Draw the arrows in the proper positions.

Rule Rule + 3 - 2

Solution: Draw the + 3 arrow from 5 to 8 and from 6 to 9, and draw the -2 arrow from 8 to 6.

8 6 4 2 What’s My Rule?

“What’s My Rule?” is an activity in which children analyze a set of number pairs to determine the rule that relates the numbers in each pair. Simple “What’s My Rule?” games begin in Kindergarten Everyday Mathematics. The first are attribute rules or activities that sort children into a specified group. For example, children with laces on their shoes belong, while children whose shoes don’t have laces don’t belong. 7 9

In first through third grades, this idea is extended to include numbers and rules for determining which numbers belong to specific sets of numbers. For example, the teacher might draw a circle on the board and begin writing numbers inside and outside the circle. The children suggest numbers and try to guess where they go. Once they can predict reliably which number belongs inside the circle, they propose rules for the sorting.

This idea evolves further to incorporate sets of number pairs in which the numbers in each pair are related to each other according to the same rule. A function machine can represent the connections between input, output, and the rule. Pairings are displayed in a table of values.

In a “What’s My Rule?” problem, two of the three parts ⎯ input, output, 5 8 6

6 9 11 4

9

5

Routines and Features_____________________ and rule ⎯ are known. The goal is to find the unknown part.

There are three types of “What’s My Rule?” problems.

1. The rule and the input numbers are known. Find the output numbers.

Rule: +10 In Out 39 54

163

Rule +10

2. The rule and the output numbers are known. Find the input numbers.

Rule: - 6 In Out 6 10 20

Rule - 6

3. The input and output numbers are known. Find the rule.

Rule: ? In Out 55 60 85 90

103 108

Rule ?

You can combine more than one type of problem in a single table. For instance, you could give the table in Problem 2 but give the input value of 26 and replace the 20 with a blank. If you give enough input and output clues, children can fill in the blanks as well as figure out the rule, as in the problem below.

Rule: ? In Out

12 6

10

26

Rule ?

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Routines and Features_____________________ Fact Power

“Knowing” the basic number facts is as important to learning mathematics as “knowing” words by sight is to reading. Students are often told that habits, good and bad, come from doing something over and over until they can do it without thinking. Developing basic number-fact reflexes can be likened to developing good habits.

In Everyday Mathematics, fact habits are referred to as fact power. Children in Grades 1-3 keep Fact Power tables of the facts they know. By the end of second grade most children should master the addition and subtraction facts. In third grade, the emphasis shifts to learning the multiplication and division facts. While some students may not be able to demonstrate mastery of all these facts, they should be well on their way to achieving this goal by the end of the year.

Practicing the facts is often tedious and traditionally involves many pages filled with drill-and-practice problems. In addition to number games and oral drills, teachers of Everyday Mathematics have had success with fact families.

+.- 0 1 2 3 4 5 6 7 8 9

0 0 1 2 3 4 5 6 7 8 9

1 1 2 3 4 5 6 7 8 9 10

2 2 3 4 5 6 7 8 9 10 11

3 3 4 5 6 7 8 9 10 11 12

4 4 5 6 7 8 9 10 11 12 13

5 5 6 7 8 9 10 11 12 13 14

6 6 7 8 9 10 11 12 13 14 15

7 7 8 9 10 11 12 13 14 15 16

8 8 9 10 11 12 13 14 15 16 17

9 9 10 11 12 13 14 15 16 17 18

Fact Families

Everyday Mathematics research has found that young children not only can understand the inverse relationships between arithmetic operations (addition “undoes” subtraction and vice versa; multiplication “undoes” division and vice versa), they often “discover” them on their own. In first- and second-grade, the inverses for sums and differences of whole numbers up to 10 are called the basic fact families. A fact family is a collection of four related facts linking two inverse operations. For example, the following four equations symbolize the fact family relating 3, 4, and 7 with addition and subtraction:

3 + 4 = 7 4 + 3 = 7

7 – 3 = 4 7 – 4 = 3

Basic fact families are modeled with fact triangles.

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Routines and Features_____________________ Fact Triangles

Fact Triangles are tools used to help build mental arithmetic reflexes. You might think of them as the Everyday Mathematics version of flash cards. Fact Triangles are more effective for helping children memorize facts, however, because of their emphasis on fact families.

In first grade, children play with Fact Triangles for addition/subtraction fact families through 9 + 9 and 18 – 9. These families are reviewed in second grade, and multiplication/ division Fact Triangles are introduced. In all grades, a useful long-term project is to have students write the appropriate four number models on the back of each Fact Triangle.

Fact Triangles are best used with partners. One player covers a corner with a finger, and the other player gives an addition or subtraction (or multiplication or division) fact that has the hidden number as an answer. This simple game makes it easy for children to play at home, so Fact Triangles are often recommended in Home Links.

Slates

Slates, or individual white boards, are used often, especially in the Mental Math and Reflexes ⎯ a brief five-minute activity that begins each lesson. Slate use provides a quick way to check children’s work as they respond to questions and then display their answers on their slates. Most children genuinely enjoy using slates. Slates allow everyone to quietly answer a question at the same time, and they help the teacher see at a glance which children may need extra help.

Calculators

Both teacher experience and educational research show that most children can develop good judgment about when to use and when not to use calculators. Students need to learn how to decide when it is appropriate to solve an arithmetic problem by estimating or calculating mentally, by using paper and pencil, or by using a calculator. The evidence indicates that children who use calculators are able to choose appropriately.

Calculators are useful teaching tools. They make it possible for young children to display numbers before they are skilled at writing. Calculators can be used to count forward or backward by any whole number or decimal ⎯ a particularly important activity in the primary grades because counting is so central to number and operations at this level. Calculators also allow children to solve interesting everyday problems requiring computations that 3 50

17

8 +, -

15

×, ÷

9

3 5 8

Routines and Features_____________________ might otherwise be too difficult for them to perform, including problems that arise outside of mathematics class. There is no evidence to suggest that this will cause children to become dependent on calculators or make them unable to solve problems mentally or with paper and pencils.

Before the availability of inexpensive calculators, the elementary school mathematics curriculum was designed primarily so that children would become skilled at carrying out algorithms. Thus there was little time left for children to learn to think mathematically and solve problems. Calculators enable children to think about the problems themselves, rather than thinking only about carrying out algorithms without mistakes.

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Games for Drill and Practice_______________

Frequent practice is necessary to attain strong mental arithmetic skills and reflexes. Although drill focused narrowly on rote practice with operations has its place, Everyday Mathematics also encourages practice through games.

Drill and games should not be viewed as competitors for class time, nor should games be thought of as time-killers or rewards. In fact, games satisfy many, if not most, standard drill objectives, and with many built-in options. Drill tends to become tedious and, therefore, gradually loses its effectiveness. Games relieve the tedium because children enjoy them. Indeed, children often wish to continue to play games during their free time, lunch, and even recess.

Drill exercises aim primarily at building fact and operations skills. Practice through games shares these objectives, but at the same time, games often reinforce other skills including calculator skills, money exchange and shopping skills, logic, geometric intuition, and intuition about probability and chance since many games involve numbers that are generated randomly.

Using games to practice number skills also greatly reduces the need for worksheets. Because the numbers in most games are generated randomly, the games can be played over and over without repeating the same problems. Many of the Everyday Mathematics games come with variations that allow

players to progress from easy to more challenging versions. Therefore, games practice, offers an almost unlimited source of problem material.

Games are fun and can be played by families to provide additional practice in an interesting way. Students can play some games across a variety of grade levels. Examples of some games that can be played at home are included on the following pages. The suggested grade levels are in parentheses. Random-Number Generators Many games involve generating numbers randomly. Several methods are possible. The Everything Math Deck: This deck of cards is really two decks in one: a whole number deck and a fraction deck using the backside of the cards. There are four cards each for the numbers 0-10, and one card each for the numbers 11-20. You can limit the range of numbers to be generated by removing some of the cards from the deck. These cards are used in many classrooms. Standard playing cards: Use the 2 through 10 cards as they are and use the aces to represent the number 1. Write the number 0 on the queens’ face cards, the numbers 11 through 18 on the remaining face cards (kings, jacks), and 19 and 20 on the jokers.

10

Games for Drill and Practice_______________

Dice: Use a regular die to generate numbers up to 6. A polyhedral die (die that has 8, 10, 12 sides, etc.) can be used to extend the range of numbers generated. Egg cartons: Label each cup with a number. For example, you might label the cups 0-11. Place one or more pennies or other small objects inside the carton, close the lid, shake the carton, and open the carton to see where the objects landed.

Directions for Selected Everyday Math Games

Disappearing Train (K) Concept: Number operations (+ and -) Players: 2 or more Materials: Blank die (or cube) with the sides marked -1, -2, -3, +1, +2, +3 At least 24 cubes, pennies, or other small objects to make trains. Directions: Explain the (-) and (+) signs on the cube: the minus sign before a number means “take away” (or subtract) that many objects and the plus sign before the number means “put together” or add that many objects. Players make trains of cubes (or blocks, bottle caps, buttons, etc.) equal in number. They take turns rolling the die (or cube) and removing, or adding, as many cars from their train as the number on the cube indicates. The game ends when the first train disappears. Players must roll the exact number needed to make the train disappear. If one car is left, the player needs to roll a -1 to finish.

Option: A non-competitive version of this game might be to work together on one train, (alternating turns rolling the die) to make it disappear. One and Only (K) Concept: Numeration; Counting Players: 3-5 Materials: Number Cards (1-10) Allow one set of ten cards for each player. Directions: Mix up a deck of number cards and pull out one card. The unpaired card is the “One and Only.” (Adults may remember “Old Maid” as a similar game.) Deal out all the cards. Players look at their hands and put down any pairs. The first player then draws a card from the person on the right. All subsequent pairs are laid down. The next player (to the right of the first player) then gets to pick a card from the person on his or her right, and so on. The game ends when one player puts down all his or her cards. The player who has the “odd” card at the end of the game is the “One and Only.” Penny- Nickel Exchange (K-1) Concept: Number Sense; Money Players: 2 or more (in pairs) Materials: 1 die (or number cube with 1-6 on it) for each pair of players; 40 pennies and 8 nickels for each pair of players Directions: Partners make a bank of 40 pennies and 8 nickels, using real money. Players take turns rolling the die and collecting the number of

11

Games for Drill and Practice_______________

pennies from the bank that matches the number rolled on the die. As players acquire 5 or more pennies, they say “Exchange” and turn in their 5 pennies for a nickel. The game ends when the bank is out of nickels. The partner with more nickels at the end wins the game. Option 1: Children play with a larger bank and two dice. This allows them to exchange coins more rapidly. At the end of any turn, each player should have fewer than five pennies. Option 2: Penny-Nickel-Dime Exchange: Use 1 die; 40 pennies, 8 nickels and 4 dimes for each partnership. Players take turns rolling the die and collecting the number of pennies from the bank that matches the number rolled on the die. Students first “exchange” 5 pennies for a nickel, and later exchange 2 nickels (or 5 pennies and 1 nickel) for a dime. The game ends when no more exchanges can be made. Students may add up their coin amounts to determine a winner or the objective may be to play the game until each partner has 60¢.

Top-It Games (K-6)

Number Top-It (K) Concept: Numeration, Number Operations; Basic Facts Players: 2 Materials: A set of number cards with four cards each of the numbers 0-10, a penny (optional) Directions: A player shuffles the cards and deals out the whole deck between them. The players place their stacks face down before them.

Each turns over his or her top card and reads the number aloud. Whoever turns over the larger number keeps both cards. If the cards match, they are put aside and the next cards are turned over until someone wins the round and takes all the cards for that round. When all the cards from both stacks have been used up, play ends. Players may toss a penny to determine whether the player with the most or least cards wins. Addition Top-It (1-3) Concept: Number Operations; Basic Facts Players: 2 or 3 Materials: Deck of cards: 4 each of 0-10 and 1 each of 11-20 Directions: A player shuffles the cards and places the deck number-side down on the playing surface. Each player turns over two cards and calls out their sum. The player with the largest sum wins the round and takes all the cards. In case of a tie for the largest sum, each tied player turns over two more cards and calls out their sum. The player with the largest sum takes all the cards of both plays. Play ends when not enough cards are left for each player to have another turn. The player with the most cards wins. Or players may toss a penny to determine whether the player with the most or the fewest cards wins. Option 1: Use a set of double-nine dominoes instead of a set of number cards. Place the dominoes facedown on the playing surface. Each player

12

Games for Drill and Practice_______________

turns over a domino and calls out the sum of the dots on the two halves. The winner of a round takes all the dominoes in play. Option 2: To practice addition with three addends, use three cards or three dice. Subtraction Top-It (1-3) Concept: Number Operations; Basic Facts Players: 2 or 3 Materials: Deck of cards: 4 each of 0-10 and 1 each of 11-20 Directions: This game is played the same way as Addition Top-It. Use the cards to generate subtraction problems. The player with the largest (or smallest) difference wins the round. Multiplication Top-It (3-6) Concept: Number Operations; Basic Facts Players: 2-4 Materials: Deck of cards: 4 each of 0-10 and 1 each of 11-20 Directions: This game is played just like Addition Top-It, except multiplication problems are generated from the 2 cards. The player with the largest product wins the round. Beat the Calculator (1-5) Concept: Number Operations; Basic Facts Players: 3

Materials: A set of number cards with four cards each of the numbers 0-10 Directions: One player is the “caller,” a second player is the “calculator,” and the third player is the “brain.” Shuffle the deck of cards and place it face down on the playing surface. The caller turns over the top two cards from the deck. These are the numbers to be added (or multiplied). The calculator finds the sum with a calculator, while the brain solves its without a calculator. The caller decides who got the answer first. Players trade roles every 10 turns or so. Option: To extend the facts, the caller attaches a 0 to either one of the numbers or both. For example, if the caller turns over a 4 and a 6, he or she may make up one of the following problems: 4*60, 40*6, or 40*60 Broken Calculator (1-5) Concept: Numeration Players: 2 Materials: Calculators Directions: Partners pretend that one of the number keys is broken. One partner says a number, and the other tries to display it on the calculator without using the “broken” key. For example, if the 8 key is “broken,” the player can display the number 18 by pressing 9 [+] 7[+] 2, 9 [*] 2, 72 [-] 50 [-] 2 [-] 2, etc. Scoring: A player’s score is the number of keys entered to obtain the goal. Scores for five rounds are

13

Games for Drill and Practice_______________

totaled, and the player with the lowest total wins. Two-Fisted Pennies Game (1-2) Concept: Number Operations; Basic Facts Players: 2 or more Materials: 10 pennies per player (or more) Directions: Children count out 10 pennies, and then split them between their two hands. (Help children identify their left hand and right hand.) Ask children to share their amounts. For example: “My left hand has 1 and my right hand has 9; left hand 3 and right hand 7; left hand 4 and right hand 6; left hand 5 and right hand 5.” The various splits for any given number can be recorded. Partners can continue to play using a different total number of pennies. For example, 9 pennies, 12 pennies, 20 pennies. Option 1: Partners take turns grabbing a part of a pile of 10 (or 20, etc.) pennies. The other partner takes the remainder of the pile. Both players count their pennies, secretly. The partner making the grab uses the count to say how many pennies must be in the partner's hand. (I have 2; you must have 8.) The eventual result is many addition names for 10, etc. Option 2: Use dimes instead of pennies. Have children tell the value of the money in each hand.

Pick-a-Coin (2-3) Concept: Money; Basic Facts Players: 2 or 3 Materials: A regular die or number cube; a recording sheet per player (see example below); Calculator Directions: Players take turns. At each turn, a player rolls a die five times. After each roll, he/she records the number that comes up on the die in any one of the empty cells for that turn on his/her own record sheet. Then they use their calculators to find the total amount and record it in the table. For example, player 1 rolls a 1, 2, 3, 4, and 5 and records this: Player 1 P N D Q $1 Total 1st Turn 1 3 4 2 5 $6.06 2nd Turn $_.__ 3rd Turn $_.__ 4th Turn $_.__ Total $_.__ After four turns, players use their calculators to find the grand total. The player with the highest total wins the game. Making Change Game (2-4) Concept: Money; Basic Facts Players: 2 or 3 Materials: 2 dice; a $1 bill, 6 quarters, 2 dimes, and 2 nickels for each player Directions: There is no money in the bank at the beginning of the game. Players take turns depositing money into the bank. To determine the amount that they are to deposit, they roll the dice and multiply the

14

Games for Drill and Practice_______________

total number of dots on the dice by 5 cents. At the beginning of the game, they will be able to count out the exact amount. Later, they make change from the money in the bank if they don't have the exact amount. The first player without enough money to put in the bank wins. Option 1: Use two different-colored dice to represent nickels and dimes. Each player starts with three $1 bills in addition to the coins. Option 2: Use three different-colored dice to represent nickels and dimes and quarters. Each player starts with six $1 bills in addition to the coins. Collection Game (2-4) Concept: Numeration; Money; Place Value Players: 2 or 3 Materials: Play money: 12 $1 bills; 12 $10 bills; and 1 $100 bill for each player; 2 dice; Place-Value Mat (see following sample) Directions: Players put all their money in the bank. At each turn, they roll the dice, take from the bank the amount they roll, and place the money on the game mat. Whenever possible, they trade ten $1 bills for a $10 bill or ten $10 bills for a $100 bill. The first player to trade for a $100 bill wins. Most games last 12-18 rounds.

Place-Value Mat One Hundred

Dollars $100

Ten Dollars

$10

One Dollar

$1

Take-Apart Game (2-4) Concept: Numeration; Money Players: 2 or 3 Materials: Play money: 12 $1 bills; 12 $10 bills; and 1 $100 bill for each player; 2 dice Directions: Each player begins with a $100 bill and the rest of the money goes in the bank. At each turn, players roll the dice and put the amount they roll into the bank. They exchange a bill of a higher denomination for bills of the next lower denomination, as needed. The first player with less than $12 wins. Option 1: Set larger or smaller goals. Option 2: Generate larger numbers (which shorten the number of rounds to reach a given goal) using 2 polyhedral dice or 3 regular dice. Option 3: Use dollar bills, dimes, and pennies instead of $100, $10, and $1 bills. Name that Number (2-6) Concept: Number Operations; Basic Facts Players: 2 or 3 Materials: Deck of cards: 4 each 0-10, 1 each of 11-20

15

Games for Drill and Practice_______________

Directions: Shuffle the deck of cards and deal 5 cards to each player. Turn over the top card. This is the target number for the round. Players try to name the target number by adding, subtracting, multiplying, or dividing the numbers on as many of their cards as possible. A card may only be used once. They write their solutions on a sheet of paper or slate. Then the players set aside the cards they used to name the target number and replace them with new cards from the top of the deck. They put the target number on the bottom of the deck and turn over the top card. This is the new target number. Play continues until there are not enough cards left in the deck to replace the players’ cards. The player who sets aside more cards wins the game. Sample turn: Player’s numbers: 7, 5, 8, 2, 10 Target number: 16 Some possible solutions: 7 * 2 = 14; 14 +10 =24; 24 - 8 = 16 (Four cards used) 8/2 = 4; 4 + 10 = 14; 14 + 7 = 21; 21 - 5 = 16 (All five cards used) Subtraction Pole Vault (4-5) Concept: Number Operations; Basic Facts Players: 1 or more Materials: Deck of cards: 4 each of 0-9; Scratch paper or slate to record results; Calculator to check answers Directions: Shuffle the cards and place the deck face down on the playing surface. Each player starts at 250.

They take turns doing the following: 1. Turn over the top 2 cards and make a 2-digit number. (There are 2 possible numbers.) Subtract this number from 250 on scratch paper. Check the answer on a calculator. 2. Turn over the next 2 cards and make another 2-digit number. Subtract it from the result in step 1. Check the answer on a calculator. 3. Do this 3 more times: (Take 2 cards, make a 2-digit number, subtract it from the last result and check the answer on a calculator.) The object is to get as close to 0 as possible, without going below 0. The closer to 0, the higher the pole-vault jump. If a result is below 0, the player knocks off the bar; the jump does not count. Sample jump: Turn 1: Draw 4 and 5. Subtract 45 (or 54). 250 - 45 = 205 Turn 2: Draw 0 and 6. Subtract 60 (or 6). 205 - 60 =145 Turn 3: Draw 4 and 1. Subtract 41 (or 14). 145 - 41 =104 Turn 4: Draw 3 and 2. Subtract 23 (or 32). 104 - 23 = 81 Turn 5: Draw 6 and 9. Subtract 69 (or 96) 81 - 69 = 12 Option 1: Players start by subtracting from 1000 and the target number could be -10 rather than 0.

16

Games for Drill and Practice_______________

Baseball Multiplication (3-6) Concept: Number Operations; Basic Facts Players: 2 Materials: 2 regular dice, 4 pennies Multiplication table or a calculator Directions: Take turns being the “pitcher” and the “batter”. 1. Draw a diamond and label “home plate,” “first base,” “second base,” and “third base.” 2. Make a score sheet that looks like the following: Innings 1 2 3 4 5 6 Total Team 1- outs runs Team 2 - outs runs 3. At the start of the inning, the batter puts a penny on home plate. 4. The pitcher rolls the 2 dice. The batter multiplies the 2 numbers that come up and tells the answer. The pitcher checks the answer in a multiplication table or on a calculator. 5. The batter looks up the product in the Hitting Table. If it is a hit, the batter moves all pennies on base as follows: Single: 1 base Double: 2 bases Triple: 3 bases Home Run: 4 bases or across home plate 6. A run is scored each time a penny crosses home plate. If a play is not a hit, it is an out. 7. A player remains the batter for 3 outs. Then players switch roles. The inning is over when both players have made 3 outs. 8. After making the third out, a batter records the number of runs scored in that inning on the scoreboard.

9. The player who has more runs at the end of 4 innings wins the game. If the game is tied at the end of 4 innings, play continues into extra innings until one player wins. 10. If, at the end of the first half of the last inning, the second player is ahead, there is no need to play the second half of the inning. The player who is ahead wins. Hitting Tables 1 to 6 Facts 1 to 9 Out 10 to 19 Single (1 base) 20 to 29 Double (2 bases) 30 to 35 Triple (3 bases) 36 Home Run (4 bases) 1 to 10 Facts 1 to 21 Out 22 to 45 Single 46 to 70 Double 71 to 89 Triple 90 to 100 Home Run 1 to 12 Facts 1 to 24 Out 25 to 49 Single 50 to 64 Double 65 to 79 Triple 80 to 144 Home Run Option 1: 1 to 10 Facts Game Use a number card deck with 4 each of the numbers 1 to 10 instead of dice. At each turn, draw 2 cards from the deck and find the product of the numbers. Use the 1 to 10 Facts Hitting Table to find out how to move the pennies. Option 2: 1 to 12 Facts Game At each turn, roll 4 regular dice. Separate them into 2 pairs. Add the numbers in each pair and multiply the sums. For example, suppose you roll a 2, 3, 5, and 6.

17

Games for Drill and Practice_______________

You could separate them as follows: 2 + 3 = 5 2 + 5 = 7 2 + 6 = 8 5 + 6 = 11 3 + 6 = 9 3 + 5 = 8 5 * 11 = 55 7 * 9 = 63 8 * 8 = 64 How you pair the numbers can make a difference in whether you make a base or an out. High-Number Toss (4 - 6) Concept: Numeration; Comparing Decimals Players: 2 Materials: Deck of cards: 4 each of the numbers 0 through 9 Directions: Begin by making a scorecard. See the example below. Shuffle the cards and place the deck face down on the playing surface. Each player has a scorecard on which to record his or her results. In each round: • Player A draws the top card from the deck and writes that number on any one of the three blanks on the scorecard. It need not be the first blank —it can be any one of them. • Player B draws the next card from the deck and writes the number on one of his or her blanks. • Players take turns doing this two more times. The player with the larger number wins the round. Scoring: The winner’s score for a round is the difference between the two players’ scores. The other player scores 0 for the round. Example: Player A: 0. 6 5 4 Player B: 0. 7 5 3 Player B has the larger number and wins the round.

Since 0.753 - 0.654 = 0.099, Player B scores 0.099 points and Player A scores 0 points for the round. Players take turns starting a round. At the end of 4 rounds, they find their total scores. The player with the larger total score wins the game. Game 1 Round 1 Score 0.___ ___ ___ ____________ Round 2 Score 0.___ ___ ___ ____________ Round 3 Score 0.___ ___ ___ ____________ Round 4 Score 0.___ ___ ___ ____________ Total: ____________ Doggone Decimal (6) Concept: Estimation; Numeration Players: 2 Materials: 1 deck of cards with 4 each of the numbers 0 through 9; 2 counters or coins per player (to use as decimal points); 4 index cards labeled 0.1, 1, 10, or 100; Calculator Directions: One player shuffles the number cards and deals 4 cards face down to each player. The other player shuffles the index cards, places them face down, and turns over the top card. The number that appears (0.1, 1, 10, or 100) is the Target Number. 1. Using 4 number cards and 2 decimal-point counters, each player

18

Games for Drill and Practice_______________

forms two numbers, each with two digits and a decimal point: • Each player tries to form numbers whose product is as close as possible to the Target Number. • The decimal point can go anywhere in a number. 2. Players compute the product of their numbers using a calculator to verify the correct answer. 3. The player whose product is closer to the Target Number wins all 8 number cards. 4. Four new number cards are dealt to each player, and a new Target Number is turned over. 5. The game ends when all four Target Numbers have been turned over. 6. The player with the most number cards wins the game. In the case of a tie, one tie-breaking round is played. Example: • The index card turned over is 10, so the Target Number is 10. • Belle is dealt 1, 4, 8, and 8. She forms the numbers 8.8 and 1.4. • Kate is dealt 2, 3, 6, and 9. She forms the numbers 6.9 and 3.2. • Belle’s product is 10.2 and Kate’s is 10.1. • Kate’s product is closer to 10. She wins the round and the cards. Fraction Action, Fraction Friction (5 - 6) Concept:Number Sense, Estimation Players: 2 or 3 Materials: Calculator; Make one set of 16 Fraction Action, Fraction Friction cards. The suggested set includes a card for each of the following fractions (for several

fractions there are 2 cards): 1/2, 1/3, 2/3, 1/4, 3/4, 1/6, 1/6, 5/6, 1/12, 1/12, 5/12, 5/12, 7/12, 7/12, 11/12, 11/12. Directions: Shuffle the Fraction Action, Fraction Friction cards. Deal one card to each player. The player with the fraction closest to 1/2 begins the game. Players take turns. At each turn: 1. The player takes a card from the top of the pile and places it face up on the playing surface. 2. At each turn, the player must announce one of the following:

1. “Action!” This means that the player wants an additional card. The player believes that the sum of the cards is not close enough to 2 to win the hand and that with an additional card; there is a good chance that the sum of the cards will not go over 2.“Friction!” This means that the player does not want an additional card. The player believes that the sum of the cards is close enough to 2 to win the hand, and that with an additional card, there is a good chance that the sum of the cards will go over 2. Play continues until all players have announced, “Friction!” or have a set of cards whose sum is greater than 2. The player whose sum is closest to 2 without going over is the winner of the hand. Players may check each other’s sums on their calculators. Reshuffle the cards and begin again. The winner of the game is the first player to win 5 hands.

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Games for Drill and Practice_______________

EM Games Online (K-6)- www.emgames.com This website can be accessed through a password provided by your child’s teacher. It has a variety of Everyday Mathematics games which are differentiated by grade level to support the learning of your child.

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Algorithms and Computation_______________ Even if you have never heard the word algorithm, it is a part of the working vocabulary of your children when they use the Everyday Mathematics program. An algorithm is a well-defined procedure or set of rules guaranteed to achieve a certain objective. Some people think of an algorithm only as one of the step-by-step operational procedures students learn in school mathematics. Actually, an algorithm is any reliable procedure or routine that, when followed properly, leads to a specific, expected, and guaranteed outcome. Furthermore, most people use many different algorithms every day such as operating a microwave oven, playing a computer game, or balancing a checkbook. Each of the algorithms involved in such everyday activities is, of course, in part a mathematical application. Teaching students to become comfortable with algorithmic and procedural thinking is essential to their growth and development as everyday problem solvers. After children have had plenty of opportunities to experiment with their own computational strategies, they are introduced to several algorithms for each operation. Some of these algorithms may be identical to or closely resemble methods children devised on their own. Others are simplifications or modifications of traditional algorithms or wholly new algorithms that may have significant advantages to children in today’s technological world. Still others are traditional algorithms, including the standard algorithms customarily taught in U.S. classrooms. As parents, try to be accepting and encouraging when your children attempt these computational procedures. As they experiment and share their solution strategies, please allow their ideas to flourish. Addition Algorithms

1. Partial-Sums Algorithm 1. Partial-Sums Algorithm: 268 + 483 600 1. Add the hundreds (200 + 400). 140 2. Add the tens (60 + 80). + 11 3. Add the ones (8 + 3).

751 4. Add the partial sums (600 + 140 + 11).

Add two or more numbers by calculating partial-sums, working one place-value column at a time, and adding all the sums to find the total. The partial sums are easier numbers to work with, and students feel empowered when they discover that, with practice, they can use this algorithm to add mentally.

2. Column Addition Algorithm: A. 2 6 8 + 4 8 3 6 14 11 7 4 11 7 5 1 1. Add the digits in each column.

1. Column Addition Algorithm A. Add one place-value column at a time. Write each place-value answer directly beneath the problem. Then go back and adjust each place-value answer, if necessary, one column at a time.

2. If necessary, adjust the 100’s and the 10’s. 3. If necessary, adjust 10’s and 1’s. B. 2 6 8 + 4 8 3 61 41 1 7 5 1

B. For some students the above process becomes so automatic that they start at the left and write the answer column by column, adjusting as they go without writing any in between steps. If asked to explain, they say something like this: “Well, 200 plus 400 is 600, but (looking at the next column) I need to adjust that, so write 7. Then, 60 and 80 is 140, but that needs adjusting, so write 5. Now, 8 and 3 is 11, no more to do, write 1.”

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Algorithms and Computation_______________ 3. Opposite Change Algorithm 3. Opposite Change Algorithm:

Rename the first addend, and then the second. 268 → (+2) → 270 →(+30) → 300 + 483 → (-2) → +481 →(-30) → +451 Add. 751 Explanation: Adjust by 2, and then by 30.

The opposite change rule says that if a number is added to one of the addends and the same number subtracted from the other addend, the sum will be unaffected. The purpose is to rename the addends so that one of the addends ends in zeros. Rename the second addend, and then the first.

268 → (-7) → 261 → (-10) → 251 + 483 → (+7) → +490 → (+10) →+500 Add. 751 Explanation: Adjust by 7, and then by 10.

Hundreds Tens Ones 9 3 2

- 3 5 6 Hundreds Tens Ones

2 12 9 3 2

- 3 5 6 Hundreds Tens Ones

12 8 2 12 9 3 2

- 3 5 6 5 7 6

1. Trade-First Algorithm: Examine the columns. You want to make trades so that the top number in each column is as large or larger than the bottom number. To make the top number in the ones column larger than the bottom number, borrow 1 ten. The top number in the ones column becomes 12, and the top number in the tens column becomes 2. To make the top number in the tens column larger than the bottom number, borrow one hundred. The top number in the tens column becomes 12, and the top number in the hundreds column becomes 8. Now subtract column by column in any order.

4. Short (Standard) Algorithm 4. Short (Standard) Algorithm:

1 1 2 6 8 1. Add the ones. (8 ones + 3 ones= 11 ones) + 4 8 3 Regroup. (11 ones= 1 ten and 1 one) 7 5 1 2. Add the tens. (1 ten + 6 tens + 8 tens= 15 tens) Regroup. (15 tens= 1 hundred and 5 tens)

3. Add the hundreds. (1 hundred + 2 hundreds + 4 hundreds= 7 hundreds)

The short algorithm for addition is the one that is familiar to most adults and children. In this algorithm, a person adds from right to left, one place-value column at a time, regrouping as necessary.

Subtraction Algorithms 1. Trade-First Algorithm This algorithm is similar to the traditional U.S. algorithm except that all the trading is done before the subtraction. This allows the students to focus on one step at a time.

22

Algorithms and Computation_______________ 2. Counting Up Algorithm 2. Counting Up

Algorithm: 356 + 4 360 + 40 400 + 500 900 + 32 932 Add. 576

Start with the number you are subtracting, the subtrahend, and “count up” to the number you are subtracting from, the minuend, in stages. To find 932 – 356, start with 356 and count up to 932. Find the sum of the numbers you added.

4. Same Change Algorithm: Add the same number. 932 → (+4) → 936 → (+40) → 976 - 356 → (+4) →-360 → (+40) → -400 Subtract. 576 Explanation: Adjust by 4, and then by 40. Subtract the same number. 932 → (-6) → 926 → (-50) → 876 - 356 → (-6) → -350 → (-50) → -300 Subtract. 576 Explanation: Adjust by 6, and then by 50.

3. Left-to-Right Subtraction Algorithm: 932 1. Subtract hundreds - 300 632 2. Subtract tens - 50 582 3. Subtract ones - 6

3. Left-to-Right Subtraction Algorithm Think of 356 as the sum of 300 + 50 + 6. Then subtract the parts of the sum one at a time, starting at the hundreds. 576

4. Same Change Algorithm If the same number is added to or subtracted from both parts of a subtraction problem, the result remains the same. The purpose is to rename both numbers so that the number being subtracted ends in zero.

5. Right-to-Left (Standard) Algorithm: Think: Can I subtract 6 ones from 2 ones? (no) Regroup the 3 tens and 2 ones as 2 tens and 12 ones. Then subtract 6 ones from 12 ones. Think: Can I subtract 5 tens from 2 tens? (no) Regroup the 9 hundreds and 2 tens as 8 hundreds and 12 tens. Then subtract 5 tens from 12 tens . Think: Can I subtract 3 hundreds from 8 hundreds? (yes) Then subtract the hundreds. 576 is the difference.

8 12 12 9 3 2 - 3 5 6 5 7 6

5. Right-to-Left (Standard) Algorithm This algorithm is most familiar to many adults and children. A person using this algorithm subtracts from right to left, one place-value column at a time, regrouping as necessary.

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Algorithms and Computation_______________ Multiplication Algorithms

1. Partial Products Algorithm: 67 x 53 50 x 60 3000 50 x 7 350 3 x 60 180 3 x 7 + 21 3551

1. Partial Products Algorithm The partial products algorithm for multiplication is based on the distributive, or grouping, property of multiplication. In this multiplication algorithm, each factor is thought as a sum of ones, tens, hundreds, and so on. For example, in 67 * 53, think 67 as 60 + 7, and 53 as 50 + 3. Then each part of one factor is multiplied by each part of the other factor, and all of the

resulting partial products are added together, as shown in the margin.

1. Partial Products Algorithm: 23 * 14 array 10 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 * 14 = (20 + 3) * (10 + 4) = (20 * 10) + (20 * 4) + (3 * 10) + (3 * 4) = 200 + 80 + 30 + 12 = 322

10 x 3

4 x 20

4 x 3

10 x 20

The algorithm can be demonstrated visually with arrays, which are among the first representations of products in Everyday Mathematics. The margin shows a 23-by-14 array which represents all of the partial products in 23 *14:

23 *14 = (20 + 3) * (10 + 4) = (20 * 10) + (20 * 4) + (3 * 10) + (3 * 4) = 200 + 80 + 30 + 12 = 322

One value of the partial product algorithm is that it previews a procedure for multiplication that is taught in high school algebra. Everything is multiplied by everything, and the partial products added. For example: (x + 2) * (x + 3) = (x * x) + (2 * x) + (x * 3) + (2 * 3)

= x2 + 5x + 6

1. Partial Products Algorithm: Algebra-Based Examples

With the rule (a + b)2 =a2+2ab + b2

252 can be calculated as (20 + 5)2 =400 + (2*100) + 25 = 625. With the rule (a + b) (a – b) = a2 – b2, 23 x 17 can be calculated as (20 + 3)(20 – 3) = 400-9= 391

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Algorithms and Computation_______________ 2. Lattice Method The use of the lattice method has been traced back to India before A.D. 1100. The person using the algorithm writes the partial products within a created lattice and then adds the numerals along each of the diagonals within the lattice. This is a student favorite because of the direct relationship to multiplication facts and its easy expandability to very large numbers.

Lattice Method Explanation:

353 * 4756 = 1,678,868

1

6

7

8

3 5 3

1 2

2 0

1 2 4

2 1

3 5

2 1 7

1 5

2 5

1 5 5

1 8

3 0

1 8 6

8 6 8

6 7

5

3

3

5

5 1

53 * 67 = 3551

2. Lattice Method

0 53

2 1

8 1

3

* Draw a lattice with as many boxes horizontally and vertically as digits in the factors of the problem. Write one factor along the top of the lattice and the other along the right side, placing one digit above or beside each box in the lattice. * Draw diagonals from the upper-right corner of each box, extending beyond the lattice. * Multiply each digit in one factor by each digit in the other. Write the product in the cell where the corresponding row and column meet. Write the tens digit of the product above the diagonal and the ones digit below the diagonal. For example, since 6 x 5 = 30, write 3 in the box above the diagonal and 0 in the box below the diagonal. If there is not a number in the tens place, a zero can fill the place above the diagonal. * Start at the bottom right corner and add the digits along each diagonal. Place the sum(s) at the bottom of each diagonal (outside the box) carrying the tens digit to the next diagonal, if needed. The first diagonal contains only 1, so the sum is 1. The sum on the second diagonal is 5 + 2 + 8 = 15. Write only the 5, and carry the one to the next diagonal. The sum along the third diagonal is then 1 + 3 + 0 + 1, or 5. The sum on the fourth diagonal is 3. * Read the product down the left side and across the bottom, left to right. The product is 3,551. The second example shows how the lattice method can be easily expanded to accommodate for larger factors.

3. Modified Repeated Addition 3. Modified Repeated

Addition: 67 Many children are taught to think of whole-number multiplication as repeated addition. However this strategy is inefficient for anything but small numbers. Using a modified repeated addition algorithm, in which multiples of 10, 100, and so on, are grouped together, can simplify the process. For example, it would be tedious to add 67 fifty-three times in order to solve 53 * 67. But if we think of ten 67’s as 670, then we can first add the 670’s (there are five of them) and then the three 67’s.

x 53 670 670 670 50 [67’s] 670 or 670 5 [670’s] 67 3 [67’s] 67

+ 67 3551 4. Short Algorithm (Standard) and Modified Versions

A. The standard U.S. algorithm. 4. Short Algorithm (Standard) and Modifications A. 67 B. 67 * 53 * 53 201 201 335 3350 3551 3551

B. Similar to the standard U.S. algorithm, except the blank is replaced with a zero, which makes it clear that in the second partial product, we are multiplying by 50 (five 10’s) and not just by 5.

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Algorithms and Computation_______________ Division Algorithms 1. Partial-Quotients Algorithm: In the partial-quotients method, it takes several steps to find the quotient. The partial-quotients algorithm begins with a series of “at least/less than” estimates of how many n’s are in m. After checking each estimate, if enough n’s have not been taken from the m’s, more are taken out. When all have been taken out, the previous estimates are added together. Even those students whose basic-facts knowledge and estimation skills are limited can find correct answers using this commonsense approach.

1. Partial-Quotients Algorithm:

Partial-Quotients Explanation: * Estimate the number of 12’s in 158. The student might begin with multiples of ten because they are simple to work with. There are at least ten 12’s in 158 (10 x 12 = 120), but there are fewer than twenty (20 x 12 = 240). 10 will be recorded as a first guess, and subtract ten 12’s from 158 leaving 38.

A. 12 10 158 - 120 38 3 - 36 2 13 158 12 = 13 R2 ÷ B. 12 10 158 - 120 38 2 - 24 14 1 12 2 13 158 12 = 13 R2 ÷

2. Column Division Algorithm: 1 3 R2 12 1 5 8

-0 15 38

1 -12 -36

3 2

* Now estimate the number of 12’s in 38. There are more than three (3 x 12 = 36) but less than four (4 x 12 = 48). Record 3 as the next guess, and subtract 36 from 38 leaving 2. * Since 2 is less that 12, estimation can stop. The final result is (10 + 3 = 13) plus the number that is left over (the remaining 2). In following this algorithm, students may not make the same series of estimates. In example b, a student used 2 as a second estimate, taking out just two 12’s and leaving 14. Another group of 12 is taken out in the next estimate and 2 more remain. The student would reach the final answer in three steps rather than two. One way is not better than another. One advantage of this algorithm is that students can use numbers that are easy for them to work with. Students who are good estimators and confident of their extended multiplication facts will need to make only a few estimates to arrive at a quotient, while others will be more comfortable taking smaller steps.

2. Column Division Algorithm This algorithm encourages students to apply place value concepts and divide the dividend into separate digits. For example, the number 158 would be thought of as 1 hundred, 5 tens, and 8 ones rather than 158. While working the problem, the student considers one place value column at a time. This graphic column presentation greatly reduces error.

Column Division Explanation: * To find 158/12, imagine sharing $158 among twelve people. Think about having 1 hundred-dollar bill, 5 ten-dollar bills, and 8 one-dollar bills. * First try to divide up the hundred-dollar bill. Each person could not get one. Therefore, there would be one left over. * Trade the remaining hundred dollar bill for 10 ten-dollar bills. There are now a total of 15 ten-dollar bills, of which each person can have 1. There are 3 ten-dollar bills remaining. * Trade the 3 remaining ten-dollar bills for 30 one-dollar bills. There are now a total of 38 one-dollar bills. Each person will get 3 one-dollar bills. There are 2 one-dollar bills remaining. Record the answer as 13 R2.

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Algorithms and Computation_______________ 3. Long Algorithm (Standard) for Division

This algorithm is familiar to most adults and children. 3. Long (Standard) Algorithm:

Long Division Explanation: * Think: How many 12’s are in 1? (0)

1 3 R 2 * Think: How many 12’s are in 15 (1) Multiply 12 x 1. (12)

12 1 5 8

1 2 3 8 - 3 6 2

Subtract 12 from 15. (3)

Bring down the 8 from the dividend. (to make 38) * Think: How many 12’s are in 38? (3) Multiply 12 x 3. (36) Subtract 36 from 38. (2) * 158/ 12 = 13 R 2

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Glossary____________________________________

absolute value- The absolute value of a positive number is the number itself. For example, the absolute value of 3 is 3. The absolute value of a negative number is the opposite of the number. For example, the absolute value of -6 is 6. The notation for the absolute value of a number n is n .

3

12 4

arcs

acre- A unit of area equal to 43,560 square feet. acute- See angle. addend- See addition. addition- A mathematical operation based on “putting things together.” Numbers being added are called addends; the result of addition is called the sum. In 12 + 33 = 45, 12 and 33 are addends, and 45 is the sum. Subtraction “undoes” addition: 12 +33 = 45; 45 -12 = 33, and 45 - 33 = 12. additive inverses- Two numbers whose sum is 0. The additive inverse of a number is also called its opposite. Example: 3 + (-3) = 0. The additive inverse of 3 is -3, and the additive inverse of -3 is 3. algebraic expression- An expression that contains a variable. For example, if Maria is 2 inches taller than Joe, and if the variable M represents Maria’s height, then the expression M – 2 represents Joe’s height. algorithm- A set of step-by-step instructions for doing something—carrying out a computation, solving a problem, and so on. analog clock- A clock that shows the time by the positions of the hour and minute hands. A digital clock shows the time in hours and minutes with a colon separating the two. angle- Two rays with a common endpoint. The common endpoint is called the vertex of the angle. An acute angle has a measure greater than 0° and less than 90°. An obtuse angle has a measure greater than 90° and less than 180°. A right angle has a measure of 90°. A straight angle has a measure of 180°. A reflex angle has a

measure greater than 180° and less than 360°. angles, adjacent- Two angles with a common side that do not otherwise overlap. In the diagram, angles 1 and 2 are adjacent angles. So are angles 2 and 3, angles 3 and 4, and angles 4 and 1. angles, vertical- Two intersecting lines form four angles. In the diagram, angles 2 and 4 are vertical angles. They have no sides in common. Their measures are equal. Similarly, angles 1 and 3 are vertical angles. apex- In a pyramid or cone, the vertex opposite the base. arc- Part of a circle from one point on the circle to another. For example, a semicircle is an arc; its endpoints are the endpoints of a diameter of the circle. area- The measure of the surface inside a closed boundary. The formula for the area of a rectangle is A =l x w where A represents the area, l the length, and w the width. The formula may also be expressed as A = b x h, where b represents the length of the base and h the height of the rectangle. arithmetic fact- Any of the basic addition and multiplication relationships and the corresponding subtraction and division relationships. There are

100 addition facts, from 0 + 0 = 0 to 9 + 9 = 18 100 subtraction facts, from 0 - 0 = 0 to 18 - 9 = 9 100 multiplication facts, from 0 x 0 = 0 to 9 x 9 = 81 90 division facts, from 0 1 0÷ = to 81 9 9÷ =An extended fact is obtained by multiplying some or all numbers in an arithmetic fact by a power of 10; for example, 20 + 30 = 50, 400 x 6 = 2400, 500 - 300 = 200, 240 60 = 4 ÷

reflex angle

obtuse angle right angle

acute angle

straight angle reflex angle

28

Glossary____________________________________

array- A rectangular arrangement of objects in rows and columns. associative property- The sum or product of an addition or multiplication problem stays the same when the grouping of the addends or factors is changed. Example: (5 + 4) + 6 = 5 + (4 + 6); (3 x 4) x 7 = 3 x (4 x 7) attribute- A common feature (size, shape, color, number of parts, and so on) of a set of figures. average- See mean. axis- Either of the two number lines used to form a coordinate grid. bar graph- A graph in which horizontal or vertical bars represent data. base- See exponential notation. base of a parallelogram- One of the sides

of a parallelogram; also, the length of this side. The shortest distance between the base and the side

opposite the base is the height of the parallelogram. base of a polygon- The side on which the polygon “sits”; the side that is perpendicular to the height of the polygon base of a polyhedron- The “bottom” face of a polyhedron; the face whose shape is the basis for classifying a prism or pyramid. base of a rectangle- One of the sides of a rectangle; also, the length of this side. The length of the side perpendicular to the base is the height of the rectangle.

base of a 3-dimensional figure- One face or a pair of faces on the figure. The height is the length of a line segment drawn perpendicular to a base of the figure that extends from that base to an opposite face or vertex. base of a triangle- One of the sides of a triangle; also, the length of this side. The shortest distance between the base and the vertex opposite the base is the height of the triangle. benchmark- A reference that is based on situations that are commonly known such as a dollar bill (six inches), the distance of a doorknob from the floor (about a meter or yard), a half-gallon of milk, or two-liter soda. bisect- To divide a segment, angle, or figure into two parts of equal measure. broken-line graph- See line graph. capacity- A measure of how much liquid a container can hold. See also volume. categorical data- Data that represents individuals or objects by one or more characteristics or traits they share, such as male or female, blue eyes or brown eyes. Categorical data is often treated as counts, proportions, or percentages of people or things in them. center point (of rotation)- The point that a geometric figure is rotated or turned around. The point can be on the figure, but does not have to be. chance- The probability of an outcome in an uncertain event. For example, in tossing a coin there is an equal chance of getting heads or tails. circle- The set of all points in a plane that are a given distance (the radius) from a given point (the center of the circle).

29

Glossary____________________________________

circle graph- A graph in which a circle and its interior are divided into parts to represent the parts of a set of data. The circle represents the whole set of data. Also called a pie graph. circumference- The distance around a circle or sphere. classify numbers- To group a set of numbers together by an attribute, such as odd, less than 20, more than 20, etc. recognizing that different types of numbers have particular characteristics. close to doubles- Number combinations such as 3 + 4, 6 + 7, etc. that are 1 whole number apart. common- Shared by two or more numbers. A common denominator of two fractions is any nonzero number that is a multiple of the denominators of both fractions. A common factor of two numbers is any number that is a factor of both numbers. commonly used fractions- Halves, thirds, fourths, fifths, sixths, eighths, and tenths. commutative property of addition or multiplication- Changing the order of the addends or factors does not change the sum or product. Example: 7 + 3 = 3 + 7, 9 x 4 = 4 x 9. compare- To examine two or more objects or amounts in order to note similarities and differences. complementary angles- Two angles whose measures total 90º. composing or decomposing numbers- Flexibly using or knowing numbers through creating and breaking numbers apart to form equivalent representations. Example; 41 as 30 + 10 + 1 or 36 can be thought of as 32 + 4, 20 + 16, 40 – 4, 12 x 3, 72 2, etc. ÷ composing or decomposing shapes- To put together or break apart given two- or three-dimensional shapes. Example: combine two squares to form a rectangle or break apart a square into two triangles.

composite number- A whole number that has more than two whole-number factors. For example, 10 is a composite number because it has more than two factors: 1, 2, 5, and 10. A composite number is divisible by at least three whole numbers. See also prime number. concentric circles- Circles that have the same center but radii of different lengths. cone- A 3-dimensional shape having a circular base, curved surface, and one vertex. congruent- Two figures that are identical—the same size and shape—are called congruent figures. If you put one on top of the other, they would match exactly. Congruent figures are also said to be congruent to each other. consecutive- Following one another in an uninterrupted order. For example, A, B, C, and D are four consecutive letters of the alphabet; 6, 7, 8, 9, and 10 are five consecutive whole numbers. consecutive angles- Two angles that are “next to each other”; they share a common side. constant- A number used over and over with an operation performed on many numbers. constant rate of change- See rate of change. conversion fact- A fact such as 1 yard = 3 feet or 1 gallon = 4 quarts. coordinate- A number used to locate a point on a number line, or either of two numbers used to locate a point on a coordinate grid.

30

Glossary____________________________________

coordinate grid- A device for locating points in a plane by means of ordered number pairs or coordinates. A rectangular coordinate grid is formed by two number lines that intersect at right angles at their 0 points. coordinate system- See coordinate grid. corresponding angles- Any pair of angles in the same relative position in two figures, or in similar locations in relation to a transversal intersecting two lines. In the diagram, <a and <e, <b and <f, <d and <h, and <c and <g are corresponding angles. If any two corresponding angles are congruent, then the lines are parallel. corresponding sides- Any pair of sides in the same relative position in two figures. counting numbers- The numbers used to count things. The set of counting numbers is {1, 2, 3, 4, . . .}. All counting numbers are integers and rational numbers, but not all integers or rational numbers are counting numbers. cube- See regular polyhedron. cubic centimeter (cm3)- A metric unit of volume; the volume of a cube 1 centimeter on a side. 1 cubic centimeter is equal to 1 milliliter. cubic unit- A unit used in a volume and capacity measurement. cubit- An ancient unit of length, measured from the point of the elbow to the end of the middle finger, or about 18 inches. The Latin word cubitum means “elbow.” customary system of measurement- The measuring system used most often in the United States. Units for linear measure

(length, distance) include inch, foot, yard, and mile; units for weight include ounce and pound; units for capacity (amount of liquid or other pourable substance a container can hold) include cup, pint, quart, and gallon. cylinder- A 3-dimensional shape having a curved surface and parallel circular or elliptical bases that are the same size. A can is a common object shaped like a cylinder. data- Information gathered by observation, questioning, or measurement. decimal- A number written in standard notation, usually one containing a decimal point, as in 2.54. A decimal that ends, such as 2.54, is called a terminating decimal. Some decimals continue a pattern without end, for example, 0.333. . . , or 0. 3 , which is equal to 1/3 . Such decimals are called repeating decimals. A terminating decimal can be thought of as a repeating decimal in which 0 repeats. degree (º)- A unit of measure for angles; based on dividing a circle into 360 equal parts. Also, a unit of measure for temperature. denominator- The number of equal parts into which the whole (or ONE or unit) is divided. In the fraction a

b, b is the

denominator. See also numerator. density- A rate that compares the mass of an object with its volume. For example, suppose a ball has a mass of 20 grams and a volume of 10 cubic centimeters. To find its density, divide its mass by its volume. (2 grams per cubic centimeter). diameter- A line segment that passes through the center of a circle (or sphere) and has endpoints on the circle (or sphere); also, the length of such a line segment. The diameter of a circle is twice its radius. See also circle. difference- See subtraction.

31

Glossary____________________________________

dividend /divisor= quotient dividend = quotient divisor

1

digit- In the base-10 numeration system, one of the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Digits can be used to write any number. For example, the numeral 145 is made up of the digits 1, 4, and 5. dimension- A measure in one direction, for example, length or width. distributive property- A property that relates two operations on numbers; usually multiplication and addition, or multiplication and subtraction.

Distributive property of multiplication over addition: a x (n + y) = (a x n) + (a x y)

2

Distributive property of multiplication over subtraction: a x (n - y) = (a x n) - (a x y)

This property gets its name because it “distributes” the factor outside the parentheses over the two terms within the parentheses. dividend- See division. divisibility test- A test to determine whether a whole number is divisible by another whole number, without actually doing the division. For example, to tell whether a number is divisible by 3, check whether the sum of its digits is divisible by 3. For example, 51 is divisible by 3 since 5 + 1 = 6, and 6 is divisible by 3. divisible by- One whole number is divisible by another whole number if the result of the division is a whole number (with a remainder of zero). For example, 28 is divisible by 7, because 28 divided by 7 is 4 with a remainder of zero. If a number n is divisible by a number x, then x is a factor of n. division- A mathematical operation based on “sharing” or “separating into equal parts.” The dividend is the total before sharing. The

divisor is the number of equal parts or the number in each equal part. The quotient is the result of division.

For example, in 28 7= 4, 28 is the dividend, 7 is the divisor, and 4 is the quotient. If 28 objects are separated into 7

equal parts, there are 4 objects in each part. If 28 objects are separated into parts with 7 in each part, there are 4 equal parts. The number left over when a set of objects is shared equally or separated into equal groups is called the remainder. For 28

÷

÷ 7, the quotient is 4 and the remainder is 0. For 29÷ 7, the quotient is 4 and the remainder is 1. Multiplication “undoes” division: 28÷ 7 = 4, and 4 x 7 = 28. divisor- See division. dodecahedron- See regular polyhedron. double bar graph- A graph in which horizontal or vertical bars represent data, and related data are displayed in adjacent bars. Also called a side-by-side bar graph. edge- The line segment where two faces of a polyhedron meet. elapsed time- The length of time between two fixed points in time. ellipse- A closed, oval, plane figure. An ellipse is the path of a point that moves so that the sum of its distances from two fixed points is constant. Each of the fixed points is called a focus of the ellipse. endpoint- The point at either end of a line segment; also, the point at the end of a ray. Endpoints are used to name line segments; for example, segment TL or segment LT names a line segment between and including points T and L. See also ray. equation- A mathematical sentence that states the equality of two quantities. equidistant marks- Marks equally distant from one to the next. equilateral polygon- A polygon in which all sides are the same length. equivalent- Equal in value, but in a different form. For example, , 0.5, and 50% are equivalent.

32

Glossary____________________________________

equivalent equations- Equations that have the same solution. For example, 2 + x = 4 and 6 + x = 8 are equivalent equations; their solution is 2. equivalent fractions- Fractions that have different numerators and denominators, but name the same number. For example, 1/2 and 4/8 are equivalent fractions. equivalent ratios- Ratios that can be named by equivalent fractions. For example, the ratios 12 to 20, 6 to 10, and 3 to 5 are

equivalent ratios, because 1220

= 610

= 35

.

estimate- A calculation of a close, rather than exact, answer; a “ballpark” answer; a number close to another number. even number- A whole number such as 2, 4, 6, and so on that can be evenly divided by 2 (divided by 2 with 0 remainder). See also odd number. expanded form/expanded notation- A way of writing a number as the sum of the values of each digit. For example, 356 is 300 + 50 + 6 in expanded form/expanded notation. exponential notation- A shorthand way of representing repeated multiplication of the same factor. For example, 23 is exponential notation for 2 x 2 x 2. The small, raised 3, called the exponent, indicates how many times the number 2, called the base, is used as a factor. expression- A mathematical phrase made up of numbers, variables, operation symbols, and/or grouping symbols. An expression does not contain relation symbols, such as =, >, or ≤. extended fact- See arithmetic fact. face- A flat surface on a 3-dimensional shape. fact- See arithmetic fact. fact family- A group of addition or multiplication facts together with the related subtraction or division facts. For example, 5 + 6 = 11, 6 + 5 = 11, 11 - 5 = 6, and

11- 6 = 5 form a fact family. 5 x 7 = 35, 7 x 5 = 35, 35÷7 = 5, and 35 5 = 7 form another fact family.

÷

factor (noun)- A number that is multiplied by another number. Factors may be whole numbers or rational numbers expressed as fractions or decimals. For example, 4, 3, and 2 are factors in the expression 4 x 3 x 2; 0.5 and 25 are factors in 0.5 x 25; 1/2 and 9 are factors in ½ x 9; and -2 and -5 are factors in -2 x (-5). See also multiplication. factor (verb)- To represent a number as a product of factors. factorial- A product of a whole number and all the smaller whole numbers except 0, for example, 3 x 2 x 1. The exclamation point, !, is used to write factorials. For example: 3! = 3 x 2 x 1= 6, 3! is read as “three factorial.” features (of the data set)- Features include the range, the outliers, the median, mean, and mode. flip/reflection- A transformation in which the image of a figure is a mirror image of the figure over a line of reflection. Each point A on the figure and its corresponding point A’ on the image are the same distance from the line of reflection on a line perpendicular to it. fluency- Efficient and accurate methods of computing and being able to demonstrate flexibility in computational methods chosen which result in students being able to explain their methods and produce accurate answers. formula- A general rule for finding the value of something. A formula is often written in abbreviated form with letters, called variables. For example, a formula for distance traveled can be written as d = r x t, where the variable d stands for distance, r for speed, and t for time.

33

Glossary____________________________________

ab

fraction- A number in the form or a/b, where a and b are whole numbers and b is not 0. Fractions are used to name part of a whole object or part of a whole collection of objects, or to compare two quantities. A fraction can represent division; for example, 23

can be thought of as 2 divided by 3.

generalizations- Reasoning about the structure of a pattern or rule. geometric solid- A 3-dimensional shape bounded by surfaces. Common geometric solids include the rectangular prism, square pyramid, cylinder, cone, and sphere. Despite its name, a geometric solid is “hollow”; it does not include the points in its interior. greatest common factor- The largest factor that two or more numbers have in common. For example, the common factors of 24 and 36 are 1, 2, 3, 4, 6, and 12. The greatest common factor of 24 and 36 is 12. growing patterns- Patterns that show an arithmetic change between pairs of elements in the pattern. For example, growing patterns may show numbers in decreasing order or buildings in decreasing size. Example: 3,5,8,12,… height of a parallelogram- See base of a parallelogram. height of a rectangle- See base of a rectangle. height of a 3-dimensional figure- See base of a 3-dimensional figure. height of a triangle- See base of a triangle. hypotenuse- In a right triangle, the side opposite the right angle. icosahedron- See regular polyhedron. identity property of addition- If a zero is added to a number, the sum is the same as that given number. Example: 7 + 0 = 7

identity property of multiplication- If a number is multiplied by one, the product is the same as the number. improper fraction- See top-heavy fraction. inequality- A number sentence stating that two quantities are not equal, or might not be equal. Relation symbols for inequalities include ≠,<, >, ≤, and ≥. inscribed polygon- A polygon, all of whose vertices are points on a circle or other figure. integer- A number without a decimal. Integer values can be less than, equal to, or greater than zero, for example, -2, 0, 6, 100, -23. intersect- To meet (at a point, line, and so on). irrational numbers- Numbers that cannot be written as fractions where both the numerator and denominator are integers and the denominator is not zero. For example, 2 andπ are irrational numbers. An irrational number can be represented by a nonterminating, nonrepeating decimal. For example, the decimal forπ , 3.141592653…, continues without a repeating pattern. The number 1.10100100010000…is irrational; there is a pattern in the decimal, but it does not repeat. justify- To defend or uphold as warranted or well-grounded. key sequence- A set of instructions for performing a particular calculation or function with a calculator. kite- A quadrilateral with two pairs of adjacent, congruent sides. The four sides cannot all have the same length, so a rhombus is not a kite. landmark numbers- Numbers that provide a foundation for extending number sense concepts. For example, the second grade level generally includes sums of tens and getting to the next ten or counting by fives.

34

Glossary____________________________________

least common denominator- The least common multiple of the denominators of every fraction in a given collection of fractions. See also least common multiple. least common multiple- The smallest number that is a multiple of two or more numbers. For example, some common multiples of 6 and 8 are 24, 48, and 72. 24 is the least common multiple of 6 and 8. leg of a right triangle- A side of a right triangle that is not the hypotenuse. line- A straight path that extends infinitely in opposite directions. line graph (broken-line graph)- A graph in which points are connected by a line or line segments to represent data. line of symmetry- A line through a symmetric figure. Each point in one of the halves of the figure is the same distance from this line as the corresponding point in the other half. line plot- A sketch of data in which check marks, X’s, or other marks above a number line show the frequency of each value. line segment- A straight path joining two points, called endpoints of the line segment. line symmetry- A figure has line symmetry if a line can be drawn through the figure that divides the figure into two parts so that both parts look exactly alike, but are facing in opposite directions. linear- Of or relating to a line. lowest terms- See simplest form. map legend- A diagram that explains the symbols, markings, and colors on a map. Also called a map key. map scale- A rate that compares the distance between two locations on a map with the actual distance between them. A labeled line segment, similar to a ruler, often represents the rate.

mathematics- A study of relationships among numbers, shapes, and patterns. Mathematics is used to count and measure things, to discover similarities and differences, to solve problems, and to learn about and organize the world. mean- A typical or middle value for a set of numbers. It is found by adding the numbers in the set and dividing the sum by the number of numbers. It is often referred to as the average. median- The middle value in a set of data when the data are listed in order from smallest to largest (or largest to smallest). If there is an even number of data points, the median is the mean of the two middle values. metric system of measurement- A measurement system based on the base-10 numeration system and used in most countries in the world. Units for linear measure (length, distance) include millimeter, centimeter, meter, kilometer; units for mass (weight) include gram and kilogram; units for capacity (amount of liquid or other pourable substance a container can hold) include milliliter and liter. minuend- See subtraction. mixed number- A number that is equal to the sum of a whole number and a fraction. For example, 2 ¼ is equal to 2 + 1/4. mode- The value or values that occur most often in a set of data. model- To represent a mathematical situation with maniputalives (objects), pictures, numbers or symbols. multiple of a number n- The product of n and a counting number. For example, the multiples of 7 are 7, 14, 21, 28,… multiplication- A mathematical operation. Numbers being multiplied are called factors. The result of multiplication is called the product. In 5 x 12 = 60, 5 and 12 are factors. 60 is the product. Division “undoes” multiplication; 60÷5 = 12 and 60 12 = 5. ÷

35

Glossary____________________________________

multiplicative inverses- Two numbers whose product is 1. For example, the multiplicative inverse of 5 is 1/5, and the multiplicative inverse of 3/5 is 5/3, or 1 2/3. Multiplicative inverses are also called reciprocals of each other. name-collection box- A boxlike diagram containing a number, used for collecting equivalent names for that number. negative number- A number less than 0; a number to the left of 0 on a horizontal number line. negative rational numbers- Numbers less than 0 that can be written as a fraction or a terminating or repeating decimal. For example, -4, -0.333…, and -4/5 are negative rational numbers. net of a prism- A flat 2-dimentional shape that can be folded into a 3-dimentional solid. nonstandard measurement- Measurement using units that have not been defined by a recognized authority. For example, paperclips, arm spans, base-10 blocks, and handfuls are all nonstandard units of measurement. nonstandard tool- A measurement tool that has not been defined by a recognized authority. Examples include paperclips, arm spans, base-10 blocks, and handfuls. number line- A line on which points correspond to numbers. number model- A number sentence that shows how the parts of a number story are related; for example: 5 + 8 = 13; 27 - 11 = 16; 3 x 30 = 90; 56÷8 = 7.

number sentence- A sentence that is made up of numerals and a relation symbol (=, <, >). Most number sentences also contain at least one operation symbol. Number sentences may also have grouping symbols, such as parentheses. numerator- In a whole divided into a number of equal parts, the number of equal parts being considered. In the fraction a/b , a is the numerator. numerical data- Data that represents objects or individuals by numbers assigned to certain measurable properties, such as their length or age. obtuse- See angle. odd number- A whole number that is not divisible by 2, such as 1, 3, 5, and so on. When an odd number is divided by 2, the remainder is 1. A whole number is either an odd number or an even number. opposite of a number- A number that is the same distance from zero on the number line as the given number, but on the opposite side of zero. See also additive inverses. order of operations- Rules that tell the order in which operations should be done. ordered number pair- Two numbers in a specific order used to locate a point on a coordinate grid. They are usually written inside parentheses: (2,3). ordinal number- A number used to express position or order in a series, such as first, third, tenth, and so on. origin- The point where the x-axis and y-axis intersect on a coordinate grid.

36

Glossary____________________________________

outlier- A number in a set of data that is much larger or smaller than most of the other numbers in the set. pan balance- A device used to compare the weights of objects or to weigh objects. parabola- The curve formed by the surface of a right circular cone when it is sliced by a plane that is parallel to a side of the cone. A parabola can also be described as the curve formed by all the points that are the same distance from a line and a point not on that line. parallel lines (segments, rays)- Lines (segments, rays) that are the same distance apart and never meet. parallelogram- A quadrilateral that has two pairs of parallel sides. Pairs of opposite sides of a parallelogram are congruent. part-to-part ratio- A ratio that compares a part of the whole to another part of the whole. For example, the statement “There are 8 boys for every 12 girls” expresses a part-to-part ratio. part-to-whole ratio- A ratio that compares a part of the whole to the whole. For example, the statement “8 out of 20 students are boys” expresses a part-to-whole ratio. The statement “12 out of 20 students are girls” also expresses a part-to-whole ratio. partitive- Distributive division that involves figuring out how many are in the group when the number of groups is known. Example: How would you divide 24 cookies equally among 6 children? (Think of dividing or partitioning the cookies into 6 equivalent subsets). percent (%)- Per hundred, or out of a hundred. For example, “48% of the students in the school are boys” means that out of every 100 students in the school, 48 are boys.

perimeter- The distance around a two-dimensional shape. A formula for the perimeter of a rectangle is P = 2 x (l + w), where l represents the length and w the width of the rectangle. perpendicular- Two rays, lines, or line segments that form right angles are said to be perpendicular to each other. per-unit rate- A rate that tells the quantity of items with a given unit for each item of a different unit. Two dollars per gallon, 12 miles per hour, and 4 words per minute are examples of per-unit rates. pi- The ratio of the circumference of a circle to its diameter. Pi is the same for every circle, approximately 3.14. Also written as the Greek letter π . picture (pictorial) graph- A graph that uses pictures or symbols to show data. place value- Determines the value of a digit in a number, written in standard notation, as determined by its position. Each place has a value ten times that of the place to its right and one-tenth the value of the place to its left. plane- A flat surface that extends forever. point- An exact location in space. Points are usually labeled with capital letters. point symmetry- The property of a figure that can be rotated 180° about a point in such a way that the resulting figure (the image) exactly matches the original figure (the preimage). polygon- A closed figure consisting of line segments (sides) connected endpoint to endpoint. polyhedron- A 3-dimensional shape, whose surfaces (faces) all are flat. Each face consists of a polygon and the interior of the polygon.

37

Glossary____________________________________

positive power of 10- See power of 10. positive rational numbers- Numbers greater than 0 that can be written as a fraction or a terminating or repeating decimal. For example, 7, 4/3, 8.125, and 5.111 . . . are positive rational numbers. power- Usually, a product of factors that are all the same. 5 x 5 x 5 (or 125) is called 5 to the third power, or the third power of 5, because 5 is a factor three times. 5 x 5 x 5 can also be written as 53. In general, a power of a number n is a number that can be represented in exponential notation as na, where a is any number. power of 10- A whole number that can be written as a product using only 10 as a factor; also called a positive power of 10. For example, 100 is equal to 10 x 10 or 102. 100 is called ten squared, the second power of 10, or 10 to the second power. A negative power of 10 is a number that can be written as a product using only 0.1, or 10-1, as a factor. 0.01 is equal to 0.1 x 0.1, or 10-2. Other powers of 10 include 101, or 10, and 100, or 1. prime factorization- A number, expressed as a product of prime factors. For example, the prime factorization of 24 is 2 x 2 x 2 x 3. prime number- A whole number greater than 1 that has exactly two whole-number factors, 1 and itself. For example, 7 is a prime number because its only factors are 1 and 7. A prime number is only divisible by 1 and itself. The first five prime numbers are 2, 3, 5, 7, and 11. See also composite number. prism- A polyhedron with two parallel faces (called bases) that are the same size and shape. Prisms are classified according to the shape of two parallel bases. The faces of a prism are always bounded by parallelograms, and are often rectangular. probability- A number from 0 to 1 that indicates the likelihood that something (an event) will happen. The closer a probability is to 1, the more likely it is that an event will happen.

product- See multiplication. proportion - A number model equating two fractions. For example, the problem, “Alan’s speed is 12 miles per hour. At the same speed, how far can he travel in three hours?” is modeled by the proportion: pyramid- A polyhedron in which one face (the base) is a polygon and the other faces are formed by triangles with a common vertex (the apex). A pyramid is classified according to the shape of its base. Pythagorean Theorem- The following famous theorem: If the legs of a right triangle have lengths a and b, and the hypotenuse has length c, then a2 x b2 = c2. quadrangle or quadrilateral- A polygon with four sides. qualitative change- A change (in the quality of something) that can be described by words such as taller, shorter, darker, lighter, warmer, etc. quantitative- Relating to number or quantity; can be counted or measured. quotient- See division. radius- A line segment from the center of a circle (or sphere) to any point on the circle (or sphere); also, the length of such a line segment. random number- A number that has the same chance of appearing as any other number. range- The difference between the maximum and minimum in a set of data. rate- A comparison of two quantities with unlike units. For example, a speed such as 55 miles per hour compares distance with time.

38

Glossary____________________________________

rates of change- Constant, change that occurs at a constant rate, or varying, change that occurs at an increasing or decreasing rate. ratio- A comparison of two quantities with like units. Ratios can be expressed with fractions, decimals, percents, or words; or they can be written with a colon between the two numbers being compared. For example, if a team wins 3 games out of 5 games played, the ratio of wins to total games is 3/5, 0.6, 60%, 3 to 5, or 3:5 (read “three to five”). rational number- Any number that can be represented in the form a

b, where a and b

are integers and b is not 0. Also, any number that can be represented by a terminating decimal or repeating decimal. 2/3 , -2/3 , 0.5, -0.5 and 0.333…are rational numbers. ray- A straight path that extends infinitely from a point, called its endpoint. real number- Any rational or irrational number. reciprocal- See multiplicative inverses. rectangle- A parallelogram with four right angles. rectangular prism- See prism. reference frame- A system of numbers, letters, or words to show quantities with reference to a zero point. Examples of reference frames are number lines, time lines, calendars, thermometers, maps, and coordinate systems. referent- A familiar object or place that a student can use as a basis for estimating the measurement of something; students might think of the length of their desks, the size of an orange, etc. reflection- See flip/reflection.

regular polygon- A convex polygon in which all the sides are the same length and all the angles have the same measure. regular polyhedron- A polyhedron with faces that are all congruent regular polygons. There are five regular polyhedrons:

tetrahedron: 4 faces, each formed by an equilateral triangle cube: 6 faces, each formed by a square octahedron: 8 faces, each formed by an equilateral triangle dodecahedron: 12 faces, each formed by a regular pentagon icosahedron: 20 faces, each formed by an equilateral triangle.

relation symbol- A symbol used to express the association between two quantities. The symbols used in number sentences are: = for equal to; ≠ for is not equal to; < for is less than; > for is greater than; for is less than or equal to; ≥ for is greater than or equal to.

≤

remainder- See division. repeating patterns- Patterns that are cyclical in nature, with each cycle repeating elements in the same order. Example: ABCABCABC rhombus- A parallelogram whose sides are all the same length. The angles are usually not right angles, but they may be right angles. right- See angle. right triangle- A triangle that has one right angle. rotation/turn- A turn around a center point or axis.

39

Glossary____________________________________

rotation symmetry- Property of a figure that can be rotated around a point in such a way that the resulting figure (the image) exactly matches the original figure (the preimage). The rotation must be more than 0 degrees, but less than 360 degrees. If a figure has rotation symmetry, its order of rotation symmetry is the number of different ways it can be rotated to match itself exactly. “No rotation” is counted as one of the ways. rounding- Replacing a number with a nearby number that is easier to work with or better reflects the precision of the data. 12,964 rounded to the nearest thousand is 13,000. scale drawing- An accurate picture of an object in which all parts are drawn to the same scale. If an actual object measured 33 by 22 yards, a scale drawing of it might measure 33 by 22 millimeters. scientific notation- A system for representing numbers in which a number is written as the product of a power of 10 and a number that is at least 1 but less than 10. Scientific notation allows writing big and small numbers with only a few symbols. For example, 4,000,000 in scientific notation is 4 x 106. 0.00001 in scientific notation is 1 x 10-5. sector- A region bounded by an arc and two radiuses of a circle. The word wedge is sometimes used instead of sector. segment- See line segment. semicircle- See circle. set- A collection of distinct elements or items. shape of data- An overview of numerical data- the highest and lowest points (range) of the data, where most of the data are clumped together, where there are no data, where there are no data located far from the rest of the data (outliers), as well as what the mode and median are.

simple equation- An equation that involves only one operation and one variable. Example: 6 + x = 11 simplest form- A fraction in which the numerator and denominator have no common factor except 1 and the numerator is less than the denominator. Also, a mixed number in which the fraction is in simplest form. Also called lowest terms of a fraction, or simple fractions. simple fraction- See simplest form. simplify an expression- To rewrite the expression by removing parentheses and by combining like terms. For example, 7y + 4 + 5 + 3y can be simplified as 10y + 9; 3 (2y + 5) – y can be simplified as 5y + 15. slide/translation- A transformation in which every point in the image of a figure is at the same distance in the same direction from its corresponding point in the figure. speed- A rate that compares distance traveled with the time taken to travel that distance. sphere- The set of all points in space that are a given distance (the radius of the sphere) from a given point (the center of the sphere). A ball is shaped like a sphere, as is Earth. square array- A rectangular array with the same number of rows as columns. For example, 16 objects will form a square array with 4 objects in each row and 4 objects in each column. square number- A number that is the product of a whole number multiplied by itself; a whole number to the second power. 25 is a square number, because 25 = 5 x 5. A square number can be represented by a square array.

40

Glossary____________________________________

square of a number- The product of a number multiplied by itself, symbolized by a raised 2. For example, 3.52 = 3.5 x 3.5 = 12.25. square root of a number- The square root of a number n is a number which, when multiplied by itself, results in the number n. For example, 4 is a square root of 16, because 4 x 4 = 16. The other square root of 16 is -4 because -4 x (-4) = 16. square unit- A unit used in area measurement. standard measurement- Measurement using units that have been defined by a recognized authority, such as a government or standards organization. For example, inches, meters, miles, seconds, pounds, grams, and acres are all standard units of measurement. standard notation- The most familiar way of representing whole numbers, integers, and decimals by writing digits in specified places. standard tool of measurement- A measurement tool that uses standard units of measurement, such as a ruler, a tape measure, a thermometer, and a clock. stem-and-leaf plot- A display of data in which digits with larger place values are named as stems, and digits with smaller place values are named as leaves.

step graph- A graph that looks like steps. Particularly useful when the horizontal axis represents time. straight angle- See angle. subdividing- See composing and decomposing shapes.

subtraction- A mathematical operation based on “taking away” or comparing (“How much more?”). The number being subtracted is called the subtrahend; the number it is subtracted from is called the minuend; the result of subtraction is called the difference. In 45 -12 = 33, 45 is the minuend, 12 is the subtrahend, and 33 is the difference. Addition “undoes” subtraction. 45 – 12 = 33, and 45 = 12 + 33. subtrahend- See subtraction. sum- See addition. supplementary angles- Two angles whose measures total 180º. symmetry- The balanced distribution of points over a line or around a point in a symmetric figure . tally mark- A mark used in a tally count, where a mark is made for each item as it is counted, usually in groups of five marks, with the fifth mark a diagonal line across the previous four vertical ones. term- In an algebraic expression or equation, a number or a product of a number and one or more variables. For example, the terms of the expression 5y + 3k – 8 are 5y, 3k, and 8. A variable term is a term that contains at least one variable. For example, in the equation 4b - 8 = b + 5, 4b and b are variable terms. A constant term is a term that does not contain a variable. For example, in the equation 4b – 8 = b + 5, 8 and 5 are constant terms.

Stems

tessellation- An arrangement of closed shapes that covers a surface completely without overlaps or gaps. tetrahedron- See regular polyhedron. theorem- A mathematical statement that can be proved to be true (or, sometimes, a statement that is proposed and needs to be proved). For example, the Pythagorean Theorem states that if the legs of a right triangle have lengths a and b, and the hypotenuse has length c, then a2 x b2 = c2.

10s Leaves 1s

2 4 4 5 6 7 7 8 3 1 1 2 2 6 6 6 4 1 1 3 5 8 5 0 2

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Glossary____________________________________

tiling- Covering a surface with uniform shapes so there are no gaps or overlaps, except possibly gaps around the edges. top-heavy fraction- A fraction that names a number greater than or equal to 1; a fraction whose numerator is equal to or greater than its denominator. Examples of top-heavy fractions are 7/5, 5/5, 9/7, and 16/4. Also called improper fraction. transformation– An operation on a geometric figure that produces a new figure (the image). Includes flips/reflections, slides/translations, and turns/rotations. translation- See slide/translation. trapezoid- A quadrilateral that has exactly one pair of parallel sides. No two sides need be the same length. triangle- A polygon with three sides.

equilateral triangle: has three sides of the same length. isosceles triangle: has two sides of the same length. scalene triangle: has no sides of the same length.

turn- See rotation/turn. tree diagram- A network of points connected by line segments and containing no closed loops. Factor trees and probability trees are tree diagrams used, respectively, to factor numbers and to represent probability situations in which there is a series of events unit- When counting, the word "unit" means "one." For example, if a car dealer expects a shipment of 20 units, that means 20 cars. variable- A letter or other symbol that represents a number. A variable need not

represent one specific number; it can stand for many different values. varying rate of change- See rate of change. Venn diagram- A picture that uses circles or rings to show relationships between sets. vertex- The point at which the rays of an angle, two sides of a polygon, or the edges of a polyhedron meet.

volume- The measure of the amount of space occupied by a 3-dimensional shape. whole- The entire object, collection of objects, or quantity being considered; the unit, 100%. whole number- Any of the numbers 0, 1, 2, 3, 4, and so on.

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Frequently Asked Questions______________

Everyday Mathematics in Rockwood Q: How does Everyday Mathematics relate to the Rockwood School District Curriculum? A: The curriculum writing committee aligned the Rockwood Mathematics Curriculum to the Missouri Grade Level Expectations and the NCTM Standards. To most effectively support the curriculum, the Everyday Mathematics Program was chosen because of its direct alignment to the NCTM Standards as well. Parent Involvement Q: How can I get involved? How can I reinforce my child’s mathematics learning at home? A: Communicate with your child’s teacher on a regular basis. If possible, volunteer to help with Explorations or Projects. Attend school functions, such as Family Math Night, to learn more about the Everyday Mathematics program. At home, talk with your child about real-life situations that involve mathematics, such as buying groceries or balancing the checkbook. Ask your child to “teach” you the mathematics lessons he is learning, including favorite games and creative solution strategies. Basic Facts Q: Will my child learn and practice basic facts? A: Your child will learn and practice all of the basic facts in many different ways without having to complete an overwhelming number of drill pages. She will play mathematics games in which numbers are generated randomly by dice, dominoes, spinners, or cards. She will work with Fact Triangles, which present fact families and stress the addition/subtraction and multiplication/division relationships. In fourth grade, she will take timed “50-facts” multiplication tests that will require her to learn the facts she does not already know. She will have continuing access to addition/subtraction and multiplication/division fact tables that will serve as references for the facts she does not yet know and as records of the facts she does. She will also take part in short oral drills to review facts with her classmates. Computation Q: Does my child have opportunities to learn, develop, and practice computation skills? A: Your child gains the fact knowledge he needs for computation from basic facts practice. He solves problems in a meaningful way through number stories about real-life situations that require him to understand the need for computation, which operations to use, and how to use those operations. He often has the opportunity to develop and explain his own strategies for solving problems

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Frequently Asked Questions______________

through algorithm invention. He practices mental arithmetic during Minute Math and 5-Minute math. He also performs activities that encourage him to round or estimate numbers mentally. Focus Algorithms Q: What are focus algorithms? A: Children spend a lot of time in the early stages of learning about computation experimenting with and sharing their own problem-solving methods instead of simply learning a set of prescribed standard algorithms. Everyday Mathematics also includes a focus algorithm for each operation—addition, subtraction, multiplication, and division. These algorithms are powerful and relatively efficient, and most are easier to understand and learn than traditional algorithms. All children are expected to master the focus algorithm for each operation. Once they show they have mastered it, they are free to use any method to solve problems. Given a choice, however, most children prefer their own procedures. Mastery Q: Why does my child have to move on to the next lesson if he hasn’t mastered skills in the current lesson? A: Mastery varies with each child and depends on his learning style and problem-solving style. Because people rarely master a new concept or skill after only one exposure, the program has a repeated-exposure approach that informally introduces topics for two years before formal study. This approach offers both consistent follow-up and a variety of experiences. If your child does not master a topic the first time it is introduced, he will have the opportunity to increase his understanding the next time it is presented. Your child will regularly review and practice new concepts by playing content-specific games and by completing written exercises and assessments. Addressing Individual Needs Q: My child has special needs. Will she be able to succeed in the program? How can the program address her individual needs? A: Everyday Mathematics is designed to be flexible and to offer many opportunities for teachers to meet the varying needs of each child. There are many open-ended activities that will allow your child to succeed at her current skill level. While playing games, inventing algorithms, writing number stories, and solving problems in Minute Math and Math Boxes exercises, your child will develop her strengths and improve in the weak areas. Furthermore, your child’s teacher may group students to best suit their needs. The teacher may also modify or adjust program material according to student needs.

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Frequently Asked Questions______________

Games Q: Why does my child play games in class? A: Everyday Mathematics games reinforce concepts in a valuable and enjoyable way. They are designed to help your child practice his basic facts and computation skills and develop increasingly sophisticated solution strategies. These games also lay the foundation for learning increasingly difficult concepts. Certain games give your child experience using a calculator, while other games emphasize the relationship between the money system and place value. Your child may play Everyday Mathematics games at home from time to time. Spend some time learning the games, and you will understand how much they contribute to your child’s mathematics progress. Assessment Q: How do you measure my child’s progress? What can you show me that demonstrates what she has learned? A: Ongoing and periodic assessments are used by Everyday Mathematics teachers to check student’s understanding. Teachers frequently make detailed written observations of students’ progress as they watch students working on Math Boxes or slate activities. They also evaluate students’ responses to Minute Math, interactions during group work or games, and written responses to Math Messages. Teachers use unit review and assessment pages to evaluate individual student progress. In addition to sending home traditional report cards, the teacher may show you a rubric—a framework for tracking your child’s progress. The rubric may be divided into categories describing different skill levels, such as Beginning, Developing, and Secure. Using these categories, the teacher indicates your child’s skill in and understanding of a particular mathematical topic. The teacher can use this record of progress to decide which areas need review and whether certain students need additional help or challenge. Calculators Q: Why is my child using a calculator? Will he become dependent on the calculator for solving problems? A: Your child uses a calculator to learn concepts, recognize patterns, develop estimation skills, and explore problem solving. He learns when a calculator can help solve problems beyond his current paper-and-pencil capabilities. He learns that in some situations, he can rely on his own problem-solving power to get an answer more quickly. Your child also uses basic fact and operations knowledge and estimation skills to determine whether the calculator’s solution is reasonable. He becomes comfortable with the calculator as one technological tool.

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Frequently Asked Questions______________

Standardized Tests Q: How do you help your class prepare for standardized tests? A: Everyday Mathematics teachers help students prepare for standardized tests in many ways. They spend more time on the Everyday Mathematics games that reinforce basic facts and skills. In addition, teachers of Everyday Mathematics promote higher-level thinking skills and application of skills in problem solving contexts. Teachers also review test-taking strategies, such as looking for reasonableness in an answer. These instructional practices prepare students for the critical thinking required to complete complex tasks presented on standardized tests.

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What Parents Can Do To Support Math___

Communicate With Your Child’s Teacher

Help with the Automatic Recall of Basic Facts

Be Open to Alternative Ways of Computing o Your child will be learning a number of new, research based ways

to add, subtract, multiply, and divide; as well as the traditional method you learned in school. Alternative Algorithms often reveal more underlying math concepts than traditional algorithms do.

Support Homework… Don’t Do It

o Ask Questions rather than give answers How did you do this in class? Can you explain what your teacher asked you to do? Can you draw a picture or make a diagram? What do you already know that can help you work through

the problem? What problems like this one have you had before? Tell me what you were thinking… how did you get that

answer?

Make Mathematics Fun o Play board games, solve puzzles, and ponder brain teasers with

your child. Children enjoy these activities while enhancing their mathematical thinking skills.

Be Positive

o Help your child have a “can do” attitude about mathematics by praising your child’s efforts and accomplishments. Acknowledge the fact that mathematics can be challenging at times and that persistence and hard work are the keys to success. Struggling at times in math is normal and necessary to understanding mathematics.

On-line Resources

o www.rockwood.k12.mo.us o www.everydaymath.uchicago.edu/ o www.ed.gov/pubs/parents/Math/index.html (US Dept. of Education) o www.emgames.com

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Do-Anytime Activities for Grades K-3_____

Mathematics means more when it is rooted in real-life situations. The following activities allow children to practice mathematics skills while riding in a car, doing chores, helping with shopping, and performing other everyday routines. These “do-anytime” activities are organized by topic and grade level. Visual Patterns, Number Patterns, and Counting K. Count the steps needed to walk from the sidewalk to the front door (or any two

places). Try to walk the same distance with fewer steps or with more steps. K. Practice counting past the “100 number barrier.” Start from different numbers,

such as 81, 92, 68, and so on. 1. Count orally by 2s, 5s, and 10s. 1. Count and pair objects found around the house, and determine whether

there’s an odd or even number of items. 2. Make a game out of doubling, tripling, and quadrupling small numbers. 2. Ask your child to count by certain intervals. For example, “Start at zero, and

count by 4s.” Addition, Subtraction, Multiplication, and Division K. Show your child three objects, and count them aloud together. Then put the

objects in your pocket, a box, or a bag. Put two more objects in with the three objects, and ask your child, “How many are in there now?” Repeat with other numbers and with subtraction (taking objects out of the pocket, box, or bag).

K. Make up “one more” and “one less” stories. Have your child use counters, such as pennies or raisins. For example, “The dinosaur laid 5 eggs.” (Your child puts down 5 counters). “Then the dinosaur laid one more egg.” (Your child puts down another counter). “How many eggs are there?”

1. Using the number grid, select a number and have your child point to the

number that is 1 more or 1 less than the selected number. Do problems like this: “Count back (or up) 5spaces. On which number do you land?”

1. Using the Fact Triangles, cover the sum for addition practice. Cover one of the other numbers for subtraction practice. Make this brief and fun.

2. Have your child explain how to use a facts table. 3. Practice addition and subtraction fact extensions. For example, 6 + 7 = 13 60 + 70 = 130 600 + 700 = 1,300 3. Provide your child with problems with missing factors for multiplication

practice. For example, “6 times what number equals 18?”

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Do-Anytime Activities for Grades K-3_____

Number Stories K. Encourage your child to figure out answers to real-life situations: “We have

one can of tuna, and we need five. How many more do we need to buy?” 1. Have your child tell you a number story that goes with a given number

sentence, such as 4 + 2 = 6. 1. Ask for answers to number stories that involve two or more items. For

example, “I want to buy a doughnut for 45 cents and a juice box for 89 cents. How much money do I need?” ($1.34)

2. Make up number stories involving estimation. For example, pretend that your

child has $2.00 and wants to buy a pencil marked 64¢, a tablet marked 98¢, and an eraser marked 29¢. Help your child estimate the total cost of the three items (without tax) and determine if there is enough money to buy them.

2. Take turns making up addition and subtraction number stories to solve. Share solution strategies.

3. Ask questions that involve equal sharing. For example, “Seven children share

49 baseball cards. How many cards does each child get?” 3. Ask questions that involve equal groups. For example, “Pencils are packaged

in boxes of 8. There are 3 boxes. How many pencils are there?” Place Value K. Have your child press the number 3 on a calculator. Have him or her press

another 3 and read the number. Repeat for 333 and 3,333. 1. Say a 2- or 3-digit number. Then have your child identify the actual value of

the digit in each place. For example, in the number 952, the value of the 9 is 900; the value of the 5 is 50; and the value of the 2 is 2 ones, or two.

2. Say a 3- or 4-digit number. Then have your child identify the actual value of

the digit in each place. For example, in the number 3,587, the value of the 3 is 3,000; the value of the 5 is 500; the value of the 8 is 80; and the value of the 7 is 7 ones, or 7.

3. Write decimals for your child to read, such as 0.32 (thirty-two hundredths) and

0.9 (nine-tenths). Thousands Hundreds Tens Ones . Tenths Hundredths Thousandths1,000 100 10 1 . .1 .01 .001

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Do-Anytime Activities for Grades K-3_____

Money and Time K. Start a family penny jar, and collect your family’s pennies. Count them from

time to time. K. Teach your child how to set the kitchen timer when you are cooking. 1. Count various sets of nickels and pennies together. 1. Have your child tell you the time as “minutes after the hour.” 2. Gather a handful of coins with a value less than $4.99. Have your child

calculate the total value. 2. Ask the time throughout the day. Encourage alternate ways of naming time,

such as twenty to nine for 8:40 and half past two for 2:30. 3. Have your child write the following amounts using a dollar sign and decimal

point: 4 dollar bills, 3 dimes, and 2 pennies; 4 dimes and 8 pennies; 3 dollar bills and 8 dimes; 8 pennies.

3. Draw an analog clock face with the hour and minute hands showing 8 o’clock. Ask your child to write the time shown. Repeat with other times, such as 3:30, 11:45, 7:10, and so on.

Measurement K. Arrange various objects (books, boxes, and cans) by various size and

measure (length, weight, and volume) attributes. Talk with your child about how they are arranged using comparison words like taller, shorter, narrower, wider, heaviest, lightest, more, less, about, and the same.

K. Record family heights by marking them on a doorframe. Record in centimeters as well as inches. Measure again in the same location several months later.

1. Use a standard measuring tool (a ruler, a tape measure, or a yardstick) to

measure objects located in the house. Keep an ongoing list of items measured and their approximate lengths and widths using inches.

2. Discuss household tools that can be used to measure things or help solve

mathematical problems. 2. Gather a tape measure, a yardstick, a ruler, a cup, a gallon container, and a

scale. Discuss the various things you and your child can measure with each. Compare to see which is the best tool for different types of measurement. For example, “What would you use to measure the length of a room: a tape measure, a yardstick, or a ruler?”

3. Review equivalent names for measurements. For example, “How many cups

in a pint?”

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Do-Anytime Activities for Grades K-3_____

Fractions K. As you cut a pizza into equal pieces, count the pieces, and describe the

pieces with their fraction names. For example, if you cut a pizza into four pieces, than each piece is ¼ of the whole pizza.

K. Compare the sizes of the pieces as you divide a pizza into smaller and smaller sections. “Is ½ of the pizza smaller or larger than ¼ of the pizza?”

1. Count out eight pennies (or any type of counter, such as beans or macaroni).

Ask your child to show you ½ of the pennies and then ¼ of the pennies. Do this with a variety of numbers.

1. Give your child several pieces of paper to fold into halves, fourths, or eights. He or she can label each part with the appropriate fraction symbol (1/2, 1/4, 1/8).

2. Read a recipe, and discuss the fractions in it. For example, ask, “How many

1/4 cups of sugar would we need to get 1 cup of sugar?” 2. Compare two fractions, and tell which is larger. For example, ask, “Which

would give you more of a pizza: 1/8 of it or 1/4 of it? 3. Help your child find fractions in the everyday world—in advertisements, on

measuring tools, in recipes, and so on. 3. Draw name-collection boxes for various numbers, and together with your child,

write five to ten equivalent names in each box. Include name-collection boxes for fractions and decimals. For example, a 1/2 name-collection box

might include 24

, 1020

, 0.5, 0.50, 5001,000

and so on.

Geometry K. Play “I Spy” with your child. Begin with easy clues, and work up to more

difficult ones. For example, “I spy something that has four legs and is a rectangle.”

K. Look around the house for different geometric shapes, such as triangles, squares, circles, and rectangles.

1. Look for geometric shapes around the house, at the supermarket, as part of

architectural features, and on street signs. Begin to call these shapes by their geometric names.

2. Look for 2- and 3-dimensional shapes in your home and neighborhood.

Explore and name the shapes, and brainstorm about their characteristics. 2. Use household items (such as toothpicks and marshmallows; straws; and

twist-ties, sticks, and paper) to construct shapes.

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Do-Anytime Activities for Grades K-3_____

3. Begin a Shapes Museum, a collection of common objects that represent a variety of 2- and 3-dimensional shapes. Label the shapes.

3. Search for geometric figures with your child. Identify them by name if possible, and talk about their characteristics. For example, a stop sign is an octagon, which has 8 sides and 8 angles. A brick is a rectangular prism, in which all faces are rectangles.

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Do Anytime Activities for Grades 4-6_________

Mathematics means more when it is rooted in real-life situations. The following activities allow children to practice mathematics skills while riding in a car, doing chores, helping with shopping, and performing other everyday routines. These “do-anytime” activities are organized by topic and grade level. Addition, Subtraction, Multiplication, and Division 4. Continue working on multiplication and division facts by using Fact Triangles and fact

families or by playing games in the Student Reference book or Games section of the Rockwood Everyday Mathematics Parent Handbook.

4. Give your child multi-digit numbers to add and subtract, such as 427 + 234, 72 – 35, and 815 – 377.

5. Practice extending multiplication facts. Write each set of problems so that your child

might recognize a pattern.

Set A 6 * 10 6 * 100 6 * 1,000 Set B 5 * 10 5 * 100 5 * 1,000 Set C 10 [7s] 100 [7s] 1,000 [7s]

5. When your child adds or subtracts multi-digit numbers, talk about the strategy that works best. Try not to impose the strategy that works best for you! Here are some problems to try: 467 + 343 = _______ ______ = 761 + 79 894 – 444 = _______ 842 – 59 = _______

6. Consider allowing your child to double or triple recipes for you whenever you are

planning to do that. Watch your child to make sure he or she does the math for every ingredient. Or your child can halve a recipe if your cooking plans call for smaller amounts.

6. Have your child calculate the tip of a restaurant bill through mental math and estimation. For example, if the bill is $25 and you intend to tip 15%, have your child go through the following mental algorithm: 10% of $25.00 is $2.50. Half of $2.50 (5%) is $1.25. $2.50 (10%) + $1.25 (5%) would be a tip of $3.75 (15%). The total amount to leave on the table would be $28.75.

Fractions, Decimals, and Percents 4. Have your child look for everyday uses of fractions and percents. Areas to explore

would be games, grocery or fabric stores, cookbooks, measuring cups and spoons, the evening news, and statistics in newspapers.

4. Encourage your child to incorporate such terms as whole, halves, thirds, and fourths into his or her everyday life.

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Do Anytime Activities for Grades 4-6_________

5. Write whole numbers and decimals for your child to read, such as 650 (six hundred fifty) and 42.5 (forty-two and five-tenths). Ask your child to identify digits in the various places—thousands place, hundreds place, tens place, ones place, tenths place, hundredths place, and thousandths place.

5. Help your child identify advertisements in signs, newspapers, and magazines that use percents. Help your child find the sale price of an item that is discounted by a certain percent. For example, a $40 shirt that is reduced by 25% is $30.

6. Encourage your child to incorporate the vocabulary of fractions and decimals into his

or her everyday speech. Make sure he or she understands that one-tenth is equivalent to 10%; quarter, to 25%; three-quarters, to 75%; and so on.

6. Encourage your child to read nutrition labels. Have him or her calculate the percent of fat in the item.

Fat calories = percent of fat (?) Total calories 100% Your child should use cross-multiplication to solve the problem. Measurement 4. Work with your child on drawing a scale map of your city, town, or neighborhood, or

have your child do a scale drawing of the floor plan of your house or apartment. 5. Encourage your child to develop his or her own set of personal measures for both

metric and U.S. customary units. 5. Encourage your child to create his or her own mnemonics, or sayings, to help in

remembering conversion measurements. Start with “A pint’s a pound the world ‘round,” and have your child create his or her own from there.

6. If you have carpentry hobbies, consider allowing your child to measure, cut, or add

and subtract measures for you. Expect him or her to be able to measure to the nearest eighth of an inch and to be able to add and subtract such measures.

6. If you are planning to paint or carpet a room, consider allowing your child to measure and calculate the area. Have him or her write the formula for Area (Area = length * width) and show you the calculations. If the room is an irregular shape, divide it into separate rectangular regions, and have your child find the area of each one. If a wall has a cathedral ceiling, imagine a line across the top of the wall to form a triangle. Your child can use the formula ½* base * height = A to calculate the area of the triangle.

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Do Anytime Activities for Grades 4-6_________

Geometry Explorations 4. Help your child recognize and identify real-world examples of right angles (the corner

of a book) and parallel lines (railroad tracks). 4. Encourage your child to identify and classify acute (less than 90°), obtuse (between

90° and 180°), right (90°), straight (180°), and reflex (between 180°and 360°) angles in everyday things (the architecture of a building, a bridge, a ramp, or a house).

4. Have your child compile a shapes portfolio or create a collage of labeled shapes. Images can be taken from newspapers, magazines, and photographs.

5. When you are at home or at a store, ask your child to identify different types of

polygons, such as triangles, squares, pentagons, and hexagons. 5. Ask your child to identify 2-dimensional and 3-dimensional shapes around the house. 6. Ask your child to find apparent right angles or other types of angles: acute (less than

90°) and obtuse (between 90° and 180°). Guide your child to look particularly at bridge supports for a variety of angles.

6. While you are driving in the car together, direct your child to look for congruent figures (figures with the same size and shape): Windows in office buildings, circles on stop lights, many street signs, and so on, are all congruent figures.

Patterns and Algebra Concepts 4. Have your child look for frieze patterns on buildings, rugs, floors, and clothing. A

frieze is a pattern which repeats in one direction. Friezes are often seen as ornaments in architecture. Have your child make sketches of friezes that he or she sees.

4. If your child has an interest in music, encourage him or her to study the mathematical qualities of the patterns of notes and rhythms. Composers of even the simplest tunes use reflections and translations of notes and chords (groups of notes).

5. Review tessellations with your child. Tessellations are designs which can fill a page,

without over-lapping, to form a pattern. Encourage your child to name the regular tessellations and to draw and name the eight semi-regular tessellations. Challenge your child to create non-polygonal Escher-type translation tessellations. You may want to go to the library first and show your child examples of Escher’s work.

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Do Anytime Activities for Grades 4-6_________

Data, Chance, and Probability 4. Help your child look up the population and land area of the state and city in which

you live and compare these facts with those of other states and cities. 4. Encourage your child to recognize the language of probability used in everyday

situations, such as weather reports and scientific findings. Have your child make a list of things that could never happen, things that might happen, and things that are sure to happen.

5. Visit the Web site for the U.S. Bureau of the Census at http://www.census.gov/.

Have your child write three interesting pieces of information that he or she learned from the Web site.

5. Have your child keep a running tally of when the school bus arrives. Or have your child time him- or herself to see how long it takes to walk to school in the morning compared to walking home in the afternoon. After a week, have your child plot the times, look for variations, and try to describe the times by using an equation.

6. While playing a game that uses a die, keep a tally sheet of how many times a certain

number lands. For example, find how many times during the game the number 5 comes up. Have your child write the probability for the chosen number. (1/6 is the probability that any given number on a six-sided die will land). The tally sheet should show how many times the die was rolled during the game and how many times the chosen number came up.

6. Watch with your child for events that occur without dependence on any other event. In human relationships, truly independent events may be difficult to isolate, but this observation alone helps to define the random events in games. Guide your child to see the difference between dependent events and random events. For example, “Will Uncle Mike come for dinner?” depends on whether or not he got his car fixed. However, “Will I get heads or tails when I flip this coin?” depends on no other event.

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