Tuesday, March 29, 2011

Place Value Game

Games provide extended place value practice for students and allow them to use their conceptual understanding to develop appropriate strategies to win the game.   The best games encourage students to try out many options in search of the best solution.   This search for the best solution prompts additional practice in a highly-motivational setting. 

Place Value Game:  Students use number cards to create the largest number possible.

  • Deck of digit cards for each set of partners (2-4 each of #0-9, depending on the level of students and the size of the numbers they will create)   Note: Spinners with #0-9 may be used instead of cards, if desired. 
  • Place value mat for each player.
  • Recording sheet, if desired.
  • Partner A turns over the first card and decides where to place that card on his/her place value mat.   Once the card is placed, it may not be moved.
  • Partner B turns over a card and decides where to place that card on his/her place value mat.   Again, the card may not be moved once it is placed.
  • Play continues with each partner turning over a card and deciding where to place it on the place value mat in hopes of building the largest number.
  • When all slots are filled on the place value mats, partners compare numbers to see who created the larger number.   That partner wins a point for the round.
  • Partners record both numbers on their recording sheet and circle the larger number
  • Students clear their mats, shuffle the cards and play additional rounds, as time allows. 
Variations:   Students try to form numbers to meet specified criteria (which will vary from these suggestions, based on the number of digits used): 
  • Students try to form the smallest number.
  • Students try to form a number that is closest to 500 (or 2000 or...)
  • Students try to form a number that is less than 1000.
  • Students form numbers and earn a different number of points, depending on the range within which the number falls (e.g. 1 point for numbers from 0-500, 2 points for numbers from 501-1000, etc.) 

Monday, March 28, 2011

Mathwire Who Has? Deck Collection

Several decks are available on Mathwire for downloading as PDFs.   All decks are 30 cards so that there are enough cards for most classes.   In fact, some students will probably need to have two cards to use the complete deck.   The decks are designed to print onto 2x4 inch labels (10 to a page).   These can then be affixed to index cards to create each deck.   If labels are not available, simply cut and paste the printout to create card decks. 
More or Less Deck:   this deck was developed to help students develop mental math proficiency.  Students will learn to mentally add or subtract a one-digit number or a multiple of  10.  Regular practice helps students develop proficiency in mental addition and subtraction.

Coins Deck:  Two decks are available for this series. 

  • Coin Deck uses clip art of actual coins on each card.  Students answer the question Who has 40 cents?  by stating I have 4 dimes, reading the coins on their cards to correctly answer the question.  Note that this 20-card version uses both heads and tails of coins.
  • Coin Deck with Antennas is the same deck as above, but antennas have been added to coins for an easy coin-counting modification for struggling learners.  Read more about the coin antenna strategy in Mathwire's Money Activities & Strategies collection.

Doubles Deck

This doubles deck was developed to provide practice in doubles facts.  This 20-card deck is straightforward practice of these important facts.  Regular practice helps students develop mastery of doubles.

 Base Ten Deck

The Base Ten deck provides practice in recognizing numbers pictured in base ten block representation. 

Mathwire Who Has? Collection

These are a few samples from the Mathwire collection.  See Who Has? Activities to read more about using these decks in the classroom and to download additional decks for fractions, geometry, etc. 

Thursday, March 24, 2011

Who Has? Decks: Multiplication Facts

Once students have developed conceptual understanding of the basic operations they need to develop fluency with the facts.   One quick way to include daily practice and motivate students to master these basic facts is through the use of the Who Has? card decks.   These decks can be created for virtually any topic and frequent use as both a whole class practice or as a center activity for partners or small groups will provide facts practice in a highly-motivating format.

Classroom Management Strategies

There are several strategies that have proven successful when implementing this activity:
  • Distributing Cards:   Distribute one card to each student, then distribute the extras to strong students in the beginning and to random students as the class becomes more familiar with the deck.
  • Class Play:   As you distribute the cards, encourage students to begin thinking about what the question for their card might be so that they are prepared to answer.   When all cards are distributed, select the "0" card or any student to begin.   Play continues until the game comes back to the original card.   That student answers and then says "stop" to signal the end of the game.
  • Timed Play:   Consider using a stopwatch to time the class game.   Record the time on the board so that students try each game to beat their current best time.   This practice encourages students to stay attentive and prompts students to practice basic facts so that the class time improves. 
  • Calling Out Answers:Discourage this practice by adding 5 seconds onto the class time whenever you hear an answer from someone who does not hold the card.   Use the same penalty for students who express vocal displeasure with delays by other students.
  • Partner or Small-Group Play:  
    • One student deals out the cards to all players.
    • Players arrange the cards face-up in front of them.   Students will find that arranging the cards in order from least to greatest will help them locate cards quickly.
    • Play begins with the "0" card or any card held by the player to the dealer's left.
    • Play continues as in the class game. Whoever has the card that answers the question reads that answer and then reads the question on that card.
    • Students turn over the cards after reading them.
    • The first person to turn over all his/her cards, wins the game. [Note: this is completely random but don't tell the students!]
    • Shuffle the cards and repeat the game.
The current record for the multiplication deck was held by a fourth grade class in New Jersey who completed the deck in 59 seconds. 
Mrs. Leanne Cherosky's 5th grade class at Buffalo Elementary just e-mailed that they broke the record with a time of :56.593.  Congratulations, 5th graders!!!  

If your class beats this record, be sure to leave a comment on this post with grade level, school and the time.

Wednesday, March 23, 2011

Writing in Mathematics

Getting Started

Often students who have difficulty writing in math class have less difficulty telling the teacher what they think.   Capitalize on this oral strength by incorporating the think-pair-share strategy more often into math lessons as a prelude to writing.  Using either of these options will increase the active participation of all students in the class.  The pause after a teacher's question provides think time for students, accommodating different learning styles and requiring all students to engage and reflect.

Think-Pair-Share: Some students are reluctant to write at first and benefit from practice sharing thoughts with a partner and hearing that partner put thoughts into words.   Reluctant students get to "practice" in a small setting with a partner before speaking to the whole class.   These students can also choose to share their thoughts, their partner's thoughts, or a combination of the two. 

The basic steps of Think-Pair-Share are:
  • Question:   Ask an open-ended question and tell students that they will think-pair-share the answer.
  • Think:   Give students 1-2 minutes to think quietly about their response to the question.   Walk around the room to reinforce this quiet, on-task response.
  • Pair:   Ask students to share thoughts with their partners and ask questions if they don't understand what their partner is saying.   Circulate around the room, listening to student conversations.
  • Share:   Ask for student volunteers to share as you begin this process. Later, you should call on non-volunteers to increase student accountability in this cooperative learning strategy.   Reinforce the expectation of active listening by requiring students to acknowledge the thoughts of classmates by saying:
    • I agree with [name's] answer...,
    • I don't agree with [name's] answer...,
    • I started the problem like [name] but then I...
  • NOTE: It is not necessary and, in fact, it is usually not time-effective to have each group share.   As you circulate around the room during the Pair share, identify students who have used different strategies or great models for thinking about an important concept.   Call on these students or their partners to share with the class. 

Think-Write-Pair-Share:   Once students are comfortable with the Think-Pair-Share strategy, introduce the Think-Write-Pair-Share strategy.   This strategy incorporates writing into the thinking process.   As students think about the question, they also write their response to the question using a variety of techniques: webbing, words, pictures, numbers, examples.   Teachers might start with a prompt poster that students can use for reference when they don't know where to start.   Effective prompts use successful pre-writing strategies such as:
  • Make a web.
  • Draw a picture and label.
  • Write a definition in your own words.
  • Create examples of the skill/concept and explain.
  • Write about a real-life use of this math concept or skill.
  • Connect the concept/skill to concepts/skills you already learned and use.
  • Reflect on your understanding of this concept/skill on a scale of 1-5 and explain.
  • Create a K-W-L chart of what you already know, what you want to know and what you have learned about the concept/skill.
Students then share their written responses with partners during which time students might elect to edit their own written response, choosing to replace certain words with better mathematical vocabulary, or add ideas and statements from their partner's writing.   Finally the teacher selects some students to share written responses with the class.   This process encourages students to get something down on paper and allows them some editing functions through the partner pairing.   Additionally, students benefit from regular listening to classmates sharing their own writing.

Tuesday, March 22, 2011

More Organized Counting

Houses in a Row

This problem challenges students to figure out how many different color combinations are possible for a row of houses, if no two houses can be painted exactly alike. Use large color squares and triangles to model this problem for students as an introduction to combinations.

Download Houses in a Row student worksheet and solution.

NOTE:  Have students explain their thinking and how they organized their information.  It is also important to ask students how they knew that they had found all of the possible combinations.

Monday, March 21, 2011

Problem Solving: Organized Counting

Stoplights challenges students to figure out how many different stoplights they can create with 3 different color linking cubes if each stoplight must have one cube of each color. Use linking cubes so that students can assemble all possible stoplights. 

NOTE:  You may choose to have students use any three different colors so that each student pair has enough linking cubes to assemble all possible combinations using one of each color to create the stoplight. 

Discussion:  In these types of problems, it is important that students know when they have found all of the possible combinations.  Be sure to ask students to explain how they know they have all of the combinations.  How did they know when they could stop looking?
Try to let students work on the problem without organizing their thinking.  Let pairs of students work together.  Ask students to explain their method and their thinking.  Peer explanations are so powerful in problem solving.  

You may be surprised at how students organize the challenge.  I remember a first grader telling me that he knew when he had them all because each color got to be on top twice.  He simply switched the bottom two cubes, then let another color be on top, etc.  To him, it was like being in the front of the line.  It was such a succinct explanation from a young learner.  Amazing!

Download Stoplights which includes the student handout and a brief explanation.  Be sure to make extra copies available to students,  so they aren't led into believing they must find 6, which is the correct answer. 

Suggestion:  To keep this task truly open-ended, hand out blank paper and have students draw their answers.  This method does not prompt students and also mimics state testing.


Friday, March 18, 2011

Scrambled Eggs Game

Successfully skip counting by 5s is an important mathematical foundation for the 5 times table, for counting nickels, for rounding numbers, etc. Counting by 5s and counting by 10s are often the first rote skip counting a student learns. While some students will naturally learn the count through classroom activities, other students will need deliberate intervention to master this sequence. 

 Scrambled Eggs Game

Distribute large cards with multiples of 5 to students. Ask students to quietly assemble themselves in correct order, holding the card in front of them. Time the class in this activity, if this motivates your class to master the sequence. 

NOTE: Cheap, white paper plates work well for this activity. Write large numbers with a black sharpie pen.  Paper plates stack easily for storage and also hold up well through repeated use. 

Partner Game: Each partner shuffles a set of count by five cards and turns them face down.  On the word "GO!" each partner turns over the first card and places it roughly before him/her.  Each student turns over one card at a time, ordering the cards before him/her until all cards are in correct order.  First student to correctly order the sequence of cards wins the game.  
  • Download Count by 5 Cards for individual student use. Copy on card stock, cut apart, and place in baggies for partner game use.

Thursday, March 17, 2011

Shamrock Paths

Shamrock Paths challenges students to use Pascal's Triangle to figure out how many different ways there are to spell Shamrock. Included in the PDF file are an explanation of how this problem relates to Pascal's Triangle and the numerical solution to the challenge. 

Download the Shamrock Paths student handout, explanation, and the solution.

Wednesday, March 16, 2011

Domino Flash Game

Play Domino Flash to help your students master the domino (and dice) patterns. Each student needs a domino mat and counters. Teachers may use overhead dominoes or Domino Flash Cards (copied on card stock, and cut apart) for this game. 

Domino Patterns:  The teacher shows a domino for a count of 5-10 seconds, depending on the ability level of the students, then covers it. Students look at the domino as it is shown, then build the domino from memory, using a domino mat and counters.  The teacher circulates around the room as students work, to observe student performance. After some time, the teacher asks students to describe the domino they saw and how they remembered the patterns to build. Finally, the teacher shows the domino again so that students are able to self-correct. 

Fact Family Practice:  The teacher shows a domino for a count of 5-10 seconds, depending on the ability level of the students, then covers it. Students look at the domino as it is shown, then build the domino from memory, using a domino facts template inserted in a sheet protector.  Students use dry erase markers to write the fact family for the given domino.

Center Activity: make the Domino Flash Game materials available for students to play as pairs or triads at center time. Students love to rotate playing teacher for this game!

Tuesday, March 15, 2011

Domino Parking Lot

Students use a set of regular dominoes and a domino parking mat. Each student selects a domino, counts the total number of dots (pips) and places the domino in that parking spot. Dominoes with the same number of dots may be stacked on top of each other in the parking spot, if necessary. 

Center Activity: Teachers may use craft foam and a sharpie marker to create the parking lot mat or download the Domino Parking Lot game mat.  Copy the game mat on card stock, and laminate the mat for student use or insert the game mat in a sheet protector. Provide sets of dominoes for small groups of students to sort. 

 Challenge: As a variation of the game, select target sums and give students a point for each domino they find to park in those spaces. This motivates students to search for those particular  combinations and heightens interest in finding those dominoes to win the most points for the group. Have one member of the group use a Domino Parking Lot Recording Sheet write in the day's winning numbers, then draw in the dots of the dominoes the group finds for those numbers.

Monday, March 14, 2011

Domino Fact Families

Dominoes may be used to introduce students to fact families. Students need a Domino Facts Template inserted in a sheet protector, dry erase marker, and some dominoes for this activity. 

The student selects a domino and draws it on the template (or uses counters to build the domino). He/she then counts the number of dots on each side of the domino, writing the numbers in the squares above the domino sides. The student figures out the total number of dots and writes this number in the rectangle below the domino. These three numbers are the number family the students will use to write the 4 number sentences for that fact family. 

Alternately, the teacher may display a Domino Flash Card and have the whole class use the same domino for the introductory activity. In this case, the teacher should use an overhead of the Domino Facts Template.   NOTE: inserting the overhead in a sheet protector allows the teacher to use dry erase markers and preserves the life of the overhead. For storage, many teachers elect to keep these often-used overheads in a binder. 

Differentiation: Vary the complexity of the dominoes students use to accommodate the varied needs of learners in the class. 

Center Activity: Make the materials available in the math center so that students practice fact families on a regular basis.

Do Now! Transition Activity: Many teachers opt to have students keep the Domino Facts Template in their desks so that they can use this activity as a daily part of math class, beginning or ending math class with fact family practice.

Friday, March 11, 2011

More Tangram Literature Connections

Grandfather Tang's Story by Ann Tempert

This book also uses tangrams to illustrate the story a grandfather tells his granddaughter.   Students will enjoy recreating the tangram creatures found throughout the book. 
  •  Suggestion:  Assemble the tangram pieces to replicate figures in the book.   Trace around the outside of the figures then fill in the shape with black marker.   These are best traced onto oaktag or card stock and laminated or placed in plastic sheet protectors before use.   Place the tangram characters and tangram pieces in your math center for students to try during center time.
  • Enrichment:  Ask students to create a new character for the book and make a tangram mat for the character that can be added to the tangram center for classmates to try. 

 Tangram Magician  by Lisa Campbell Ernst

This book is similar to Grandfather Tang's Story  in that the magician changes shapes throughout the story.  Students may use tangrams to reproduce the shapes.

The Tangram ABC Book  by T. Foster

The author presents several tangrams for each letter.  For example, the letter A is presented as an armadillo and an airplane tangram.  Once again, students may recreate these shapes using their own tangram sets.
  • Suggestion:   younger students may have eye-hand coordination difficulty trying to recreate a shape from a smaller drawing in the book.  Teachers may wish to assemble and trace the tangram shapes on card stock, showing the placement of each piece.  Younger learners are challenged to correctly identify, place and orient the tangram pieces to recreate each shape.

Then and Now on Old MacDonald's Farm by T. Foster

The author uses tangram shapes to retell the traditional story.  Teachers may add this book to a farm or animal unit and challenge students to recreate each animal using their own tangrams.

Thursday, March 10, 2011

Three Pigs, One Wolf and Seven Magic Shapes by Grace Maccarone is a mathematical take on the classic fairy tale.   Students are introduced to tangrams in this leveled reader.   The pigs use the seven magic shapes to form solutions to the problems they encounter.   Students will enjoy using tangrams to recreate the figures in the book as well as creating their own tangram figures for classmates to solve. 

  • Download the Seven Magic Shapes Templates that show individual pieces so that young students can position tangram pieces correctly to form the characters in the book.   These templates were designed to be used with commercial tangram pieces. 
  • Download the Seven Magic Shapes Shadow Templates  that show black shapes so that students must figure out how to position tangram pieces correctly to form the characters in the book.   These templates were designed to be used with commercial tangram pieces. 
  • Download the Tangram Shapes Template that can be used to create tangram pieces to use with templates above if commercial tangram pieces are not available. 
These activities help students develop spatial sense as they manipulate tangram pieces to form the figures in the different templates.  Students will also enjoy this retelling of a classic story.

Differentiation:  Teachers may use either set of templates to accommodate the varied needs of learners in the class.  Some students may need to begin with the first set that clearly shows each piece; other students should be challenged to form figures on the shadow templates.

Enrichment:  Challenge students to form their own original templates or supply additional tangram figures to challenge students.  NOTE:  Some young students have great difficulty solving tangram puzzles when they are supplied with a small figure commonly found in tangram books.  Teachers may form the figures on oaktag, trace the pieces then blacken the whole figure to create a template.  Make several copies of the templates for classroom use. 

Wednesday, March 9, 2011

Guess the Number

Guess the Number
Logic Number Puzzles

 Give students more practice in using the hundred board to solve number clue problems. Once again, students solve the problem clue by clue. With this set, however, students must use multiplication, money, and measurement facts to correctly solve the problem and guess the number.This is great practice for state testing as the clues draw on many different mathematical skills.

Sample Problem:Guess the Number-1
  • The number is greater than the number of pennies in a quarter.
  • The number is less than the number of pennies in five dimes.
  • The number is an odd number.
  • If you count by 5s, you say the number.
  • The sum of the digits is 8.
  • What is the number?
Suggested Uses:
  • Do Now! or warm-up activity
  • Math Center Activity
  • Problem solving activity on game days
  • Writing original Guess The Number problems for a class collection 
Download the Guess the Number teacher resource packet which includes 10 logic number puzzles, a blank template and answer key.

Tuesday, March 8, 2011

Problem Solving: Working Backwards

Bake Sale challenges students to work backwards to figure out how many cookies the girls actually baked for the school bake sale.  They left the cookies on cooling racks, but each got up at different times throughout the night, ate some cookies, then packaged some others to take home.  Lo and behold there are only 10 cookies left in the morning.  Each girl admits eating a few during the night but there are a lot of cookies missing.  Students must work backwards to figure out how many cookies they originally baked, then decide how many cookies each girl got and whether or not the cookies were fairly split.

Download the Bake Sale problem with possible solution.

Monday, March 7, 2011

Problem Solving: Remainders in Division

Top of the Morning to You!

This problem challenges students to identify the 100th and 1000th letter that will be printed on the electronic signboard. To solve this, students will grapple with a division problem, figuring out how many letters are in the message and how many times the message will repeat to identify the 100th letter.  This is a good introductory problem because there are no remainders, making this a straightforward division problem. Nonetheless, students must explain their thinking and the process they used to solve the problem, so it's effective practice for state testing.

Friday, March 4, 2011

Game Day: Number Line-Up

This activity is designed to actively involve students in using place value concepts to build numbers.   Student teams are given digit cards and asked to form numbers that fit specific conditions: 
  • Build the largest number.
  • Build the smallest number.
  • Build a number between 50,000 and 60,000.
  • Build a different number between 50,000 and 60,000. 
  • Build the largest number between 40,000 and 50,000.
  • Build the smallest multiple of 5.
  • Build an even number that is less than 25,000,
  • Continue to give conditions to tailor instruction to targeted skills.
Students at their desks should play with small decks of cards so that they are also actively engaged in thinking about each prompt.   Teachers should ask students to confirm the correctness of responses and add other possibilities.  Lead student discussion and reflection on how they thought about the problem, how they devised a solution and why they know their answer is correct.

Instructional Strategies:
  • Copy enough of the individual digit cards so that all students have a set in a plastic baggie in their desks.  Use these cards frequently to prompt active participation, asking all students to respond to appropriate prompts such as what number is in the hundreds place, the thousandths place, etc.
  • Copy the digit cards on a transparency and cut apart to create digit cards for the overhead.  Students may use these digit cards to explain their thinking.
  • These activities are perfect for a quick do-now review at the beginning of a class period.
  • Challenge students to develop original challenges.  They should provide the digit cards that may be used and appropriate challenge prompts.  Use student challenges in class and place them in the math center.  Be sure to label each challenge with the student's name and year they wrote it.  Siblings and friends will enjoy these challenges in future years.
These activities are very easily differentiated by varying both the number of digit cards used and the difficulty level of prompts to appropriately challenge the different ability levels within the classroom. 

Thursday, March 3, 2011

Problem Solving: Working Backwards

Monkey Business

NCTM published a similar problem several years ago.  The Monkey Business problem was created to help students practice the strategy of Working Backwards.  Review an earlier Mathwire Blog post on an easy Working Backwards strategy that uses circles and arrows to create a path forward and then reverses the operations to create a backwards path.

Monkey Business may be solved using this arrow path approach or using forward and backward tables as shown in the possible solution included with the Monkey Business handout.  No matter which method students use, the problem challenges them to think about mathematical operations very conceptually and use inverse operations to reverse the path. 

The NCTM Illuminations website hosts a lesson plan for two similar problems, Mangoes and Sailors and Coconuts, the one on which Monkey Business is based.  Be sure to check out the site to dowload the problems and read through an alternate problem solving strategy.  Students will benefit from adding these different problem solving strategies to their repertoire for future use.

Wednesday, March 2, 2011

Happy Birthday, Dr. Seuss!

Fish Out of Water

This is a basic counting game for younger students.  It's a great math activity to celebrate the birthday of Dr. Seuss, as the fish theme correlates well with this famous Dr. Seuss book.  Each student begins with 20 fish out of water.   Each player rolls a die and counts out that many fish to return to the fish bowl.   The first player to return all 20 of his/her fish to the bowl wins the game.   You may use foam fish cutouts and a plastic bowl for a more realistic version of the game or download the directions, game mat and center icons for Fish Out of Water.   This game was developed by Christine Sweeney, a Monmouth University student, for the ED 556 Probability Fair.  
  • Download the directions, game mat and center icons for the Fish out of Water game. 

Game Pieces:   Students may use any classroom counters as fish, but dollar stores often have collections of fish that may be used as game pieces for the game. Students may also  use goldfish crackers as a special treat for this birthday.

Data Analysis:  Even very young students can learn about probability by predicting how many dice rolls it will take them to get all of the fish back in the water.  A simple tally chart may be used to record both predictions and actual counts.  Lead a discussion with students to probe their understanding of the probability of a single die toss and to introduce basic concepts of probability -- equally likely, fair, probability, outcome.

Tuesday, March 1, 2011

Dr. Seuss Math Activities

Many schools celebrate the birthday of Theodor Geisel, better known as Dr. Seuss. Consider adding mathematical activities to these celebrations. One fish, two fish, red fish, blue fish is an especially good introduction to combinations. Use the Mathwire open-ended problems written to accompany this book. The problems may be used with older grades who enjoy hearing their childhood book read aloud.

  • Seussical Patterns: Primary students will enjoy completing the Seussical Patterns which feature the classic red and white striped Seuss hat.
  • Seussical NumbersAfter reading the book, challenge students to recall how numbers were used in the book. Create a class list of these uses, then reread the book and add to the list, as needed, to capture all of the Seussical numbers.  


    Seussical Combinations:

    The book states that not one of them is like another. These original Mathwire problems challenge students to use combinations to figure out how many unique fish could be created.  


    • Seussical Fish:  Challenge younger students to find all of the different combinations for Seussical Fish  using attributes from the story.  Making an orderly list is an effective strategy for solving this problem.  Other students might opt to draw the solution.

    • Fishy Combinations:  Older students will enjoy the challenge of the Fishy Combinations challenge problem which involves many more options.  The model solution uses the multiplication principle of counting to easily solve the problem.  Students may also use a tree diagram or list all of the possible red fish, then extend this list to the blue and black fish combinations.