Practice: Math Games

What's My Rule? (1-3)

Duration: 10 to 20 minutes

• Sort and classify objects
• Recognize and describe a rule for the classification of objects

• Manipulatives (such as buttons)
• 2 paper plates for sorting

• This game requires no physical preparation.
• Set norms for group play.

• Explain that you are thinking of a rule—one thing that students have in common—that you want students to guess. You will put students who meet your rule in a small group to your right, and those who don't in a small group to your left.
• Choose a rule that is easy to recognize, such as students who are boys, or students who are girls. Remember not to say what your rule is; you want students to guess.
• As students take turns nominating others, ask guiding questions that will help students think about patterns and classification:
  • What made you choose that person?
  • What does your choice have in common with others in the "meets my rule" group?
  • How is your choice different from students in the "meets my rule" group?
• Once students understand the game, let them take turns coming up with a rule (students who are wearing red) and challenge them to come up with new and more challenging rules.

For a whole-group activity

• Identify two areas in the room where students will cluster once the game begins. One area should be labeled "does not satisfy this rule" and the other "does satisfy this rule."
• To begin, the activity leader thinks of a rule (for instance, wearing red) but does not reveal the rule to the group.
• Students will nominate someone from the group that they believe may satisfy the rule.
• Direct that student to join the "does not satisfy this rule" or "does satisfy this rule" cluster, based on the predetermined rule.
• As the clusters increase in size, encourage all students to discover the rule. Ask students to refrain from guessing, and to make sure they check their suspicions as to what the rule may be before asking the instructor whether their discovery indeed is the rule.
• Ask students questions such as,"Why do you think that arrangement satisfies my rule?" "What do you notice?"
• The activity leader should initially be the instructor, who will model how the game is to be played; however, students can then take turns as the activity leader.

For a small-group activity, students can sort buttons, cards, or numbers into two groups, trying to find the rule of the activity leader.

• Student engagement and participation
• Comments and answers that reflect students' ability to sort and classify objects
• Comments and answers that indicate that students can recognize and articulate rules for classification
• Comments and answers that indicate that students are listening to, monitoring, and applying problem solving strategies of peers

Teaching Tips for ELL

• Do a practice round in which the afterschool instructor tells students the category or rule, and students have to divide themselves. This will help ELLs understand why the group is being divided.
• Keep the initial groupings familiar and concrete so that ELLs can understand and visualize the criteria for forming groups independent of language ability. Use real objects that are available in the classroom such as buttons, writing utensils, counters, or clothing whenever possible or have pictures available.
• Be sure to demonstrate or literally point out the criteria or concept being applied after each sequence. This is a great opportunity for oral language development.
• Be sensitive to classroom diversity issues when choosing or allowing students to choose categories or rules by which students themselves might be categorized. This is a great activity to bring diverse groups of students together rather than have them separated by eye color or hair color or language. To avoid such situations, the afterschool instructor may want to have several categories written on slips of paper that students can pull out of a hat.
• When dividing the room in half and labeling each side, try substituting a more familiar phrase such as "meets the rule" or "follows the rule" instead of "satisfies the rule" and "does not satisfy the rule."

Language Goals

• What's My Rule is a great opportunity for native speakers to model and ELLs to practice positive and negative sentences. Consider the sentence structures below:
  

  Sam is wearing blue.	Tina is not (isn't) wearing blue.
  They are wearing blue.	They are not (aren't) wearing blue.
  This button is round.	This button is not (isn't) round.
  These buttons are round.	These buttons are not (aren't) round.

• Be aware of the verbs and language forms used when modeling for ELLs. The language should naturally flow out of the context or situations as shown in the examples. The idea is to limit the language structures being modeled so that ELLs can recognize and practice familiar structures. If too many sentence patterns are used interchangeably to describe one rule, ELLs may not understand the basic rule being described and may be reluctant to speak themselves.
• Contractions and pronouns are okay. ELLs need to learn English as it is spoken, but usage should conform to general standards.

24® (4-6)

Duration: 5 to 20 minutes

• Develop fluency in adding, subtracting, multiplying, and dividing whole numbers
• Recognize and generate equivalent representations for the same number
• Understand and use properties of operations, such as the commutative property, 2 + 3 = 3 + 2, and 5 x 6 = 6 x 5
• Develop fluency with basic number combinations for multiplication and division
• Apply and adapt a variety of strategies to solve problems

• 24® game cards (see Resources tab)
• Paper
• Pencils
• Calculators (optional)

• Familiarize yourself with the game's instructions and goals.

• You may want to begin by reviewing the factors of 24 to familiarize students with the game. This will help get them warmed up to approaching one number through a variety of operations and all the ways that you can get to 24.
• One of the common mistakes students make is repeating an operation they have already used. For example, 4 x 2 x 2 + 8 = 24 is the same as 8 + 2 x 2 x 4 = 24. But rather than pointing this out as a mistake, use questioning strategies to help students see for themselves if they have or have not come up with a new way of getting to 24, and encourage them to keep trying.

For a whole-group activity

• Present students with a game card which has four numbers on it, making sure all students can view the card.
• Ask students to create number sentences using the four numbers and any combination of addition, subtraction, multiplication, and/or division to make 24. Students can use each number on the card only once in their number sentence.
• Encourage students to use mental math but allow them access to tools, if necessary.
• Have one student share a number sentence, allowing other students to check and see if the calculations were correct.
• Ask students if there is more than one number sentence that solves the problem. Provide time for students to share all the number sentences that are created.
• In addition to requesting a solution, ask students how they developed the solution. This will provide you and their peers a glimpse of their problem-solving strategies.
• These steps can be repeated with other game cards time permitting.

For a small-group activity

• Give students a deck of cards (each card has four numbers on it).
• Using the four numbers on the card, encourage students to record all of the number sentences they find that make 24.
• Encourage students to share their findings with their group and with you when you join their group.
• While you are with each group, make sure that you ask students questions that guide them to additional strategies and number sentences where appropriate. For example, "Is there another way to make 24?" or "How did you figure this one out?"
• Students can continue this process for an allotted time or until they have completed their deck of cards.

• Student engagement and participation
• Comments and answers that reflect an understanding of the meanings of operations to get to 24 (add, subtract, multiply, and divide whole numbers)
• Strategies for discovering different ways to make 24
• Working together to problem solve

Teaching Tips for ELL

• To clarify the rules and objectives of the game, try presenting beginning ELLs with four individual number cards rather than one single card with all four numbers written on it. Have another colored or slightly different card imprinted with the number 24. Create more cards with parentheses, the equal sign, and symbols representing the operations of addition, subtraction, multiplication, and division. Ask students if they can arrange the order of the cards so that the final answer is 24. You may need several of each of the operations cards.
• In addition to listening to instructions, it is important for ELLs to receive new information and instructions by seeing and touching (if possible). When ELLs can see and demonstrate for themselves that the numbers and symbols can be rearranged to equal 24, they are also demonstrating their understanding of the problem to the instructor.

Language Goals

• Have English-speaking partners model reading the number sentences. Encourage ELLs to read the number sentences with their partners and, if they are confident, for the entire class.

Hide and Seek (6-8)

Duration: 10 to 20 minutes

• Use coordinate systems to specify locations
• Use coordinate geometry to represent and examine the properties of geometric shapes
• Use coordinate geometry to examine specific geometric shapes, such as regular polygons or those with pairs of parallel or perpendicular sides

• Copies of Geometry: Hide and Seek (PDF)
• Watercolor, dry erase, or overhead markers

• Gather materials.
• Determine how you would like to generate the groups. You might have students count off by threes, or assign them to a group of three.

Getting Started

Students may have a question about how to start, and it may be helpful to talk with them about how the playing grid works and the vocabulary that relates to it: x-axis (the horizontal axis), y-axis (the vertical axis), ordered pair, (x-coordinate, y-coordinate), vertex, edge, and the names of common geometric shapes, from triangle through octagon.

Students may begin by using their own informal terms, which is fine, but be sure to check their understanding of the underlying concepts, and encourage them to order their points in an x,y sequence. For instance, even if they are using terms like "four points up" or "three to the right," ask them to put the horizontal direction first.

During Play

As students play, encourage the group seeking the figure to talk about how they will keep track of each guess, and which of the four conditions each guess met. The hider should tally the number of guesses.

The students' language to refer to the points may be ambiguous. Phrases like, "the point all the way at the left" or, "just below the last point" may come up. If you hear something like this, ask the seekers how the hider will know how to answer and whether there is a more concise, less ambiguous way to refer to a point. This way, students may see the value of the (x,y) ordered-pair approach, which does not depend on outside references and is less ambiguous.

If someone draws a circle, or another shape that is not a polygon, ask them to draw a regular polygon instead, but as time allows discuss circles or curves after the play.

After Play

Ask students to review their strategies. Were some figures harder to find than others? Why? How did they keep track of what they had discovered?

• Group students into threes. One student hides and the other two seek geometric figures in a coordinate system.
• Provide the hider with one grid and the "seekers" with an identical grid. During play, the hider is back-to-back with the seekers.
• Have the hiders draw a geometric shape on the grid, making sure the vertices, the intersections of two or more lines of the geometric figure (for instance, a corner), have integer coordinates.
• After the hider draws the geometric shape, he or she begins the game by telling the seekers what type of geometric shape was drawn.
• The goal of the game is for the seekers to cooperatively locate the points corresponding to the vertices of the hidden shape. The hider provides clues after each guess that tells the seekers where the guess is relative to the shape. There are four clues that can be given during the game: (1) inside the shape; (2) outside the shape; (3) on an edge but not a vertex; or (4) a vertex. The seekers use the clues, their knowledge of a coordinate system, and their knowledge of the properties of geometric shapes to find the vertices in as few guesses as possible.
• Once the seekers have determined all the vertices of the shape, students rotate positions so that each student has the opportunity to be the hider.

• Student engagement and participation
• Comments and answers that reflect students' use of the properties of geometric shapes to help them find coordinates
• Comments and answers that reflect students' use of coordinate systems to specify locations
• Students work cooperatively and discuss strategies for selecting the next coordinate

Teaching Tips for ELL

• Have an easily readable poster or word wall with the necessary vocabulary and illustrations for students to refer when needed. Or let students use picture dictionaries or other resources that include both the word and a visual referent for each term.
• In order to demonstrate how the game is played, play a whole-class practice game using the overhead projector with the instructor providing a shape for students to guess. The instructor can also help students strategize about how to make a good guess.
• Write and color-code the four clues on the board so that students can refer back to and discuss them as needed. Show students how they can use each respective color on their Hide and Seek grid to color-code points that are inside or outside the shape, on an edge or on a vertex

Language Goals

• Don't worry about ELLs' sentence structure or correctness apart from the ordered number pairs, the clues, and the math vocabulary. This is a great activity for them to practice vocabulary in context and gain fluency speaking with their peers.
• Many mathematical terms in English have cognates in other languages. For example, the words "axis" is the same in English and in Spanish. Other words may vary only slightly from one European language to another. For example, "rectangle," "rhombus" and "vertex" are rectángulo, rombo, and vértice, respectively, in Spanish. Some ELLs, like English-speaking students, may not yet know many of these geometric terms and concepts in their native language. Translation will not help those students who do not yet understand the underlying concept.
• Have beginning ELLs practice matching simplified versions of each clue with a sample shape. Model by pointing and repeating Inside the shape or Outside the shape. Next, have them point when you repeat those phrases.
• Try substituting the words "line" and "corner" for "edge" and "vertex." The goal is for them to eventually use the more precise terms, but students can often understand and build upon the more familiar terms.

What's the Chance? (9-12)

Duration: 1 hour

• Communicate about mathematics (e.g., use mathematical language, compare their own thinking with other students' thinking, construct logical arguments to support their thinking)
• Use strategies to understand new math content
• Select and use the best method of representing and describing a set of data
• Use experimental methods to determine theoretical probability

• Teacher's Guide (PDF)
• Student Worksheet (PDF)
• Pencils
• Paper (graphing and other as needed)
• Calculators available
• Game marker (e.g., coin, counter)
• Number cube (die) with an even number of sides

• Print out accompanying PDFs and familiarize yourself with the scenarios students will be involved with. If necessary, make time to share the lesson with a day-time mathematics instructor to converse about the standards and mathematics involved.
• Decide how you will organize students into pairs (allowing choice or assigning partners).
• Make sure all materials are available for all students.
• Prepare a brief introduction of the game and students' mission during the game. Students have probably had experiences playing dice games as well as games of chance. You may decide to ask students to share these experiences and to think about whether they felt the games were set up fairly. The goals of this discussion are to clarify the task and pique students' interest in the problem.

• Give a brief introduction of the game and the expectations of the lesson.
• Ask students to familiarize themselves with the instructions and to get started as soon as their twosome is ready. Before playing, students are asked to predict which player they think will have the most wins. You can collect the predictions by putting tally marks on the board for Player 1 or Player 2. Be sure to ask students to talk about the reasoning behind their predictions.
• Circulate and stay involved while students play and answer the questions on the worksheet. While talking with students, try to generate student thinking without giving away answers or leading students down a particular line of thinking.
• For example, you can ask students, How did you arrive at this answer? or What happened when you modified the game board?
• Ask, Are you surprised by the results? Why or why not? What is surprising to you?
• Once students realize that Player 2 wins most of the time, ask, Why would it be that Player 2 wins more often than Player 1?
• To get students ready to answer the questions on the theoretical probability of the game, you can ask, Based on what you know about probability, how often do you think Player 2 should win? All the time? Half the time? How could you figure this out?
• As students play the game, encourage them to bounce ideas around about who is winning and why. Student dialogue will enhance the richness of the game and the learning experiences associated with playing it.
• Be prepared for some students to move forward to the extensions and/or save the extensions for another day. You may decide that some students can take the extensions home to work on (and play with a family member) if they wish.
• After students have played the game and answered the questions, lead a class discussion on fairness in games and what it means. This will allow students to affirm their learning by thinking through and comparing what they discovered while they played. (e.g. their thinking, their discoveries, their theories, their ideas).

For a small-group activity, students can sort buttons, cards, or numbers into two groups, trying to find the rule of the activity leader.

• All students are engaged and actively seeking answers
• Students communicate effectively about mathematics (e.g., use mathematical language, compare their own thinking with other students, gain clarification from each other)
• Students use strategies effectively
• Students represent and describe data effectively
• Students use experimental methods and determine theoretical probability
• Students exercise their imaginations and creativeness to modify/develop games.

Race to the Finish (K-2)

Duration: At least 10 minutes

• Count with understanding and recognize "how many" in sets of objects
• Develop a sense of whole numbers and represent and use them in flexible ways
• Understand data and how data can be represented in bar graphs
• Connect number words and numerals to the quantities they represent, using physical models and representations on number cubes
• Develop fluency with basic number combinations for addition

• Data Analysis and Probability: Race to the Finish (PDF)
• Water color, dry erase, or overhead markers
• Dice for each small group

• Print out the PDF to use as the game board. Laminate or cover it with clear contact paper.
• If you plan to introduce the game to the whole group, make an overhead copy of game board.

As students play, ask guiding questions to assess student understanding. Use the sample questions below or develop your own:

During Play

• Who is ahead? Which bicyclist is in first place right now? Who is in second place? How far behind is the second place bicyclist? How do you know?
• How far from the finish line is bicyclist 5?
• Which bicyclist moves ahead this roll? How do you know?
• How many times have you rolled a 5? How do you know?

After Play

• For whom did you cheer in this game? If you play the game again, are you going to pick this bicyclist to cheer on again?
• Which bicyclists are more/less likely to win? Why?
• Will bicyclist number one ever win? How do you know?

• Set the game up, making sure that all students can see the game board (an overhead projector might help).
• Have students select a number (1-12) that they will cheer for; each number represents a cyclist on the board game.
• Ask students to predict who will win and why.
• Provide two number cubes and have students take turns rolling dice.
• After each roll, students add the two numbers rolled and mark an X in the next empty space above the bicyclist whose number corresponds with the sum of the two numbers rolled.
• Each X represents the bicyclist moving closer toward the finish line.
• Students continue taking turns rolling the cubes and charting the outcome until one of the bicyclists reaches the finish line.
• As students play, encourage them to predict who they think will win, who is ahead, and why. Ask questions throughout and after the game. See Teaching Tips for sample questions.

• Student engagement and participation
• Comments and answers that reflect students' use of their charts to discuss the bicyclists' standings in the race
• Comments, answers, and choices that reflect an understanding of impossible and more/less likely outcomes
• Students count with understanding and recognize "how many" in sets of objects
• Comments and answers that reflect a developing understanding and fluency in addition

Teaching Tips for ELL

• In order to set the scene of a bike race, ask if any students have ever raced their bikes with a friend. If possible, show a picture of a bike race or racers to clarify the context of this activity.
• Give each student a copy of the Race to the Finish game board so they can practice listening and demonstrate their understanding of the course of events. Beginning ELLs may not contribute orally, but they can often demonstrate their understanding with a non-verbal, physical response—in this case by marking an X in the proper column for each bicyclist.
• Pair beginning ELLs with English-speaking partners initially so that they can check whether the ELL is tallying correctly and offer help if needed. It is important to encourage English-speaking partners not to automatically share answers with ELLs but to wait, watch, and offer help only when necessary.

Language Goals

• Provide frequent opportunities for students to practice their understanding and pronunciation of ordinal numbers: Who is first? Who is fifth? ELLs who understand numerals and ordinals can practice pronunciation of ordinal numbers, which can be difficult for all young students but especially for those speakers of languages in which the final "-th" sound does not exist.

The Size Is Right (K-8)

Duration: 10 to 20 minutes

For preK-2

• Sort and classify objects
• Recognize and describe a rule for the classification of objects
• Understand length, volume, weight, area, and time
• Compare and order objects according to these attributes
• Understand how to measure using nonstandard and standard units
• Select an appropriate unit and tool to measure

For grades 3-5

• Understand such attributes as length, area, weight, volume, and size of angle, and select the appropriate type of unit for measuring each attribute
• Understand the need for measuring with standard units and become familiar with standard units in the customary and metric systems
• Carry out simple unit conversions, such as from centimeters to meters, within a system of measurement
• Understand that measurements are approximations and how differences in units affect precision

For grades 6-8

• Understand both metric and customary systems of measurement
• Understand the relationships among units and convert from one unit to another within the same system
• Understand, select, and use units of appropriate size and type to measure angles, perimeter, area, surface area, and volume

• Items to estimate length, volume, weight, area, time, and other age-appropriate attributes
• Stop watch
• Tape measure, scales and other tools to measure items

• Gather a variety of items to measure, and tools to conduct the measurement.
• Develop age-appropriate questions regarding the materials.
• Determine the order in which students will participate (for example, by picking names from a hat).
• Set group norms for play.
• Develop limitations for guessing amounts. (For example, within what amount is a correct answer?)

Sample Round Questions

• How long is this pencil? Is the actual length longer or shorter than 15 cm?
• How long (in seconds) does it take ... (me) to .... (count to 20) at a normal rate? Is the time it will take me to count to 20 higher or lower than 30 seconds?
• How far (in inches) can I jump? Is the length that I can jump more or less than 18 inches?
• How many of these cups does it take to fill this jar? Is the number of cups that it will take to fill this jar larger or smaller than 4?
• How many fluid ounces are in this container? Is the number of fluid ounces in this container higher or lower than 8?
• How large is this angle? Is the measurement of this angle greater or lesser than 30 degrees?
• How heavy is this rock? Is the weight of this rock more or less than 10 grams?

• Select a student or two to be contestants and to "come on down" to play the game.
• Explain to students that they will be presented with an object and asked to answer a given question about the object (relating to its size).
• Present the players with the item, then ask your pre-selected question (see sample questions in the Teaching Tips). Choose items and corresponding sizes appropriate for the age of the player.
• After contestants make a higher/lower decision, have them check if they are correct by selecting a tool and completing the measurement to determine the actual size of the item.
• Encourage the audience to coach the player as they try to come up with the correct answer.

• Student engagement and participation
• Estimates that become increasingly accurate, illustrating that students are becoming more familiar with customary systems of measurement and better estimators of size
• Students can measure using standard and nonstandard units of measure
• Students select appropriate tools for measuring attributes
• Students share as well as listen to, apply, and learn from problem-solving strategies of peers

Teaching Tips for ELL

• ELLs, like many other students, probably aren't familiar with the format of The Price is Right game show, so you may have to model an example or two with students who might already be familiar with the program.
• Since the metric system is more commonly used throughout the world, students from outside the United States may not be familiar with U.S. customary units such as ounces (liquid and weight), feet, and pounds. This presents an opportunity for ELLs to provide their metric-system expertise while their English-speaking partners can help with concepts such as gallons and miles per hour.
• Show ELLs what the unfamiliar base unit looks like before asking them the question. For example, show students what a liquid ounce looks like before asking them the question "Does this cup hold more than 10 ounces or less than 10 ounces?"
• In this activity, ELLs will have to grapple with unfamiliar language in addition to a new measuring system. Comparative words such as "faster/slower," "higher/lower," and "greater/less than" may be difficult for ELLs to understand and use independently. Use hand and arm gestures to demonstrate the quantity being compared and allow students to do the same when giving their answers.

Language Goals

• Model and encourage ELLs to use new comparative vocabulary words and phrases. With more advanced students, model and encourage them to try using the words in a complete sentence, such as A pencil is shorter than one foot.
• Encourage all students to give their final answers using the number and unit, such as "two cups or five minutes."