Math


MATH

A MIND-READING TRICK GRADES: 5-12 I used this kind of trick when we are between topics or when a short day makes a regular lesson meaningless. I found that my students enjoy the challenge and some develop an interest in math that wasn't there before. MATERIALS: pencil paper three dice (optional if the students are familiar with dice--it is helpful if the students know that opposite sides of a die add to 7) some may need a calculator METHOD: Students roll (or pretend to roll) the three dice. The students should write the three digits showing in any order. Next he should write the numbers on the bottom of the dice, using the same order. For example if he rolls a 5, a 3, and a 1, he may write them in any order. Supposing he writes them 513. Then he subtracts each digit from 7 to get the numbers on the bottom of the dice and adds those numbers to his original numbers. This gives him a 6 digit number, 513264. Now have him divide that number by 37, then again by 3. Choose one student and ask him to give you the result. Now it's your turn. Subtract 7 from his result, then divide by 9. This gives the three original digits in the order he wrote them. (513264 divided by 37 equals 13872, and when divided by 3 equals 4624. Subtracting 7 gives 4617, and dividing by 9 gives 513, his original numbers in the order he wrote them.) You can complete the trick by supplying the last three digits of his 6-digit number if you wish. *Submitted by,* KENNETH ROBINSON NORDHOFF HIGH SCHOOL OJAI, CA kenben@wcc.net MAD MINUTE RELAY GRADES: 2-6 A variant of the twenty year old mad minute. Students learn timed math facts in a team environment. Math problems can be any single operation problem (ex: + - X / (for X and / use #s up to 12 X 12 or 144/12)) MATERIALS: timer pen paper METHOD: The students are split into two teams. Each team receives one sheet of problems. When the teacher says BEGIN, (or any other signal that means it) a five minute clock is started. Student one on each team begins working on problem one. After that student is done, the sheet is passed down and the procedure repeated. Each team goes 3x around (This may vary depending on class size. I generally get as close to 30 problems on one sheet as possible (ex. if you have 12 students you break it down 12/6 then 6x5= 30 and that works out even...In other words for 12 students number the problems 1-6 and have the students go 5x around.)) No conferring between teammates until the end. 7. If there is time left on the clock, teammates check the answers. 8. If sheets are handed in before the time limit expires, the team cannot take it back to continue to work on it. Most correct answers win. *Submitted by,* JOHN ORGOVAN no school listed no city listed jorgovan@suffolk.lib.ny.us EGG MATH GRADES: 1-3 Here is a fun spring game for the students to play. Tell the students to help Peter Rabbit find his egg basket. See how many eggs the students playing the game can put into the basket. MATERIALS: oval shapes for eggs (various colors). I made about 20 for the game. black marker to write the problems on the oval shapes golf tees manilla folder or poster board dice pocket on the basket for the eggs when answered correctly METHOD: Make and decorate a game board. Either use poster board or a manilla folder. Make oval shapes to use for the cards. Write math problems on the oval shapes. Place the math problem face up with a self check for the students on the back. On the game board make an oval shape a little bigger than the cards and write place eggs here. Make a start and a finish. I would put a rabbit at the start and then at the finish put a Easter basket with a pocket for the eggs. When a player answers the problem correctly they put the egg in the pocket. Put 3 oval shapes with skip on them. If a student draws that card then they loose a turn. To begin the students are to roll the dice to see who goes first. The player will draw a card and answer the question. If the player is correct they roll the dice and move that number of spaces. The player that gets to the Easter basket first wins. VARIATIONS: The oval shape math problems can also be used to just play a card like game. The student with the most cards after all the cards are gone wins. Make a manilla envelope and put about 4 pockets, each being a different color for the eggs to be placed when the player is correct. If the player is incorrect the oval shape goes back to the bottom of the deck. The player that wins is the one with the most eggs. I call this game Spring Fling. Make sentences on the oval shapes using the spelling words. The player reads the card and has to say the correct spelling word that goes in that blank. A self check is on the back. *Submitted by,* JANET HILL HOT SPRINGS SCHOOL DISTRICT HOT SPRINGS, AR llihpw@prodigy.net MAD MINUTE MATH PRACTICE GRADES: 1-6 I use this daily practice to reinforce basic math facts, improve speed of recall, and teach graphing skills. MATERIALS: Mad Minute (or some other sheet) of 30 basic math facts - addition, subtraction, multiplication, or division. I have a set of 5 worksheets that I rotate -- it doesn't matter that the kids see them weekly -- they don't even notice. one file folder per student one blank graph per student, stapled into the folder timer (a clock with a second hand will do if you don't have a timer) colored pencils (for graphing) METHOD: To maintain and improve my students' knowledge of their basic multiplication and division facts, we do a daily "mad minute." Each student has a two sided worksheet - one side is division and one side is multiplication. I set the timer for one minute and when everyone is ready, I say "Go." When the timer goes off, the students flip the page over (it doesn't matter what side they do first) and we repeat the procedure for the other side. At the end of the 2nd minute, they switch papers. I do a quick review of the vocabulary ("What is the answer to a multiplication problem called?" "What do we call the two numbers that are multiplied?" etc.) and then, going across the rows, I call out ONLY the answers . (Until they learn the difference between rows and columns, I review that vocabulary, too.) I always check the multiplication side first, but it doesn't matter, as long as you're consistent. As we finish checking each side, the students write the number correct as a fraction (N/30) at the top of the page. Once both sides are checked, the papers are returned and each student graphs his/her progress. We use a double line graph format, because line graphs are best at showing change over time. The students make up their own key and do their own graphing. Until they were proficient at the graphing, we practiced multiplication facts only and used a single line graph format. My students love the drill and the challenge of trying to beat their best time or "go up" on their graphs. It's a great way to reinforce facts, vocabulary, and graphing in less than 10 minutes per day and will work for any of the basic facts. Because the students do their own data collection for me, it's one less thing for me to do when it's IEP time. *Submitted by,* J. ALTERMAN SWEET APPLE ELEMENTARY SCHOOL ATLANTA, G no e-mail listed MAKING A BUDGET GRADES: 4-8 This activity can be used at the holiday time or for the students' birthdays MATERIALS: writing materials video: The Homecoming--A Christmas Story (optional) METHOD: I introduce the lesson by asking them to make a list of ALL the things they want for their birthday. (They go hog wild!). Then we discuss the difference between a want and a need. Next: they divide their lists into two categories, wants and needs. We discuss their lists and then they revise it (some things are deleted) Now we make a list of all of the standard household bills their parents receive each month. I impress upon them that I do not want to know their family's business, keep $$ amounts private. We form a class list on the board or overhead. Next we place estimated $$ amounts by each bill: ex. electricity $$, grocery $$, car payment $$, etc. Then we arrive at a grand total of the average monthly bills that their parents must pay before gifts may be purchased. (this is a real eye opener!!) We discuss wisdom and how it is obtained over a period of years, not learned form a text book. I assign them a project of finding a wise person (someone over the age of 60) to interview. They are to ask: What was Christmas/your birthdays like for you as a child? Then they are to record their story and any other information about their family traditions. Their information my be turned in as a video taped interview, written as a new paper article, written in story form, etc.You will be amazed at how these kids are touched by their experiences with these "wise" people. Hand back their original birthday list and ask them if the want to add or delete any of the items. They (9 out of 10 of them) will want to delete items. Rent a copy of the Film THE HOMECOMING - A CHRISTMAS STORY (Walton's Mountain family) to show the class. They will love it! (Even if it's not Christmas time) I have done this project for the past 7 years and find it to be a true learning experience for all kids!!! *Submitted by,* CAROLYN PRUITT REIDSVILLE MIDDLE SCHOOL REIDSVILLE, NC ctpruitt@yahoo.com CHILD-MADE JIGSAW GRADES: K-2 All children can do this activity, even at the kindergarten level with help. I have never had a problem with children completing this activity and the children take pride in their finished jigsaws. MATERIALS: old birthday or Christmas cards 1 sheet of white A4 copy paper per child wallpaper or other paste and glue brushes 1 pair of scissors for each child METHOD: Collect enough old Christmas, birthday and other greeting cards for each member of the class with pictures that will appeal to both sexes. The children take great pride in their work and choose their pictures with care. Cut the card so that the front is separate from the back. You should have one piece of card with a picture which is single thickness. Keep the commercial greeting part of the card for other activities. On the back of the card, draw 5 triangles. One should be the largest and will be made by placing a ruler from one corner diagonally across the card. The remainder of the card will consist of 4 triangles of different sizes. Each child looks carefully at the picture prior to cutting it out. Then each spice is cut out and placed carefully into position on the A4 paper. DO NOT ALLOW THE CHILDREN TO GLUE AT THIS STAGE. When the children have pieced their jigsaws together in the right position, then they can glue the card pieces onto the paper. Display on the pin board. *Submitted by,* SUZANNE SPIERS APPLECROSS PRIMARY SCHOOL PERTH, WESTERN AUSTRALIA, AUSTRALIA psyche@iinet.net.au THE TURKEY FARM GRADES: 4-8 Every week I try to incorporate a cooperative lesson into our math class. I teach 6th grade math and have about 43 students per period, so I must be prepared! This is a simple yet fun activity where students will find the mean, median, and mode of a given set of numbers.that I just did with my classes. It turned out great! MATERIALS: worksheet calculator marker crayons 2 sheets of construction paper scissors glue small paper plate METHOD: I distribute a worksheet that has a story that I made up : "Your group owns a turkey farm. The newly elected President has chosen your farm to supply them with 5 turkeys for their special Thanksgiving dinner. Only the 5 heaviest turkeys will be chosen." George 25lbs Millie 22lbs lulu 24lbs kiki 30lbs chi chi 24lbs wonka 14lbs Kyle 23lbs boo 28lbs The students choose the 5 heaviest and find the mean, median, and mode. The students then construct a turkey using the information and supplies given. Paper plate is the body where the mean, median, and mode are written The 2 sheets of construction paper are used for feathers. the 5 weights are written on each one in order from least to greatest. The feathers are glued onto the paper plate, a copy of a turkey head is given and colored, and also attached to the plate Any extra details are added by the group members I place it them on a bulletin board that reads, "Who let the turkeys out?" My students had a great time doing this project! *Submitted by,* MAIRA MAGUIRE CENTENNIAL MIDDLE SCHOOL MIAMI, FL mywonka@aol.com SOME IDEAS FOR SIMPLIFYING YOUR MATH CLASS GRADES: 4-12 MATERIALS: none METHOD: When teaching requires the use of tools, such as rulers, compasses, protractors, etc., try to get the whole class to have the same instrument. I had some class funds to use recently & bought enough protractors & compasses for everyone in the class to use. They will be re-used from year to year. When I went to teach the lesson, I didn't have to run around the room trying to show everyone how "theirs" worked. Also - try to use clear protractors. The new purple & green plastic ones are cute, but hard for beginners to use. Another idea for protractors: Use a small drill (like a Dremel tool) to put a small hole at the crosshairs. Some protractors come with a hole already there. Tie a string through the hole. When the knot is lined up over the crosshairs, the student can then pull the string up along the angle side they are measuring & the string points to the correct number of degrees. Use the overhead! Write to the publisher to get permission to make overhead transparencies of difficult lessons. This way everyone can watch what you're doing on the overhead. (I had a couple of difficult lessons on scale drawings & map distances that I taught this way. They weren't difficult lessons - just difficult to teach when everyone couldn't see what I was doing). Make a ruler out of transparency film or photocopy onto a transparency. Slide overhead transparencies into clear plastic sheet protectors. They can then be stored in a 3-ring binder, and the sheet protectors can be easily written on & erased. Use examples from real life, whenever possible. For a lesson on sales tax, photocopy a receipt from a recent purchase. Have the students figure out if the tax was correct. Copy your electric bill & talk about the way kilowatts are measured & billed. For a lesson on scale drawings, visit a new home development & take a copy of the floor plans of a new house. I found a really neat book about the way carpenters have to use math - such as measuring the angle & pitch of a staircase, etc. Challenge the students to think of a profession that doesn't use math (farmers have to measure acreage, pounds of fertilizer, etc., lawyers have to be able to bill accurately, etc. Every job requires that employees be able to check to see if their paycheck is correct!) Take math grades once a week instead of daily. I correct math lessons orally daily, usually with students marking their own mistakes, but I only collect them weekly - usually on test days. Of course I have to watch for cheating, but I know my kids' ability pretty well & it becomes obvious to spot. I record the grades while the students are testing. Since my school uses workbooks & I do not allow the students to tear out the pages, this is the only way I have found to be able to glance over their work for neatness, completeness, similar errors, skipping problems, etc. without keeping their books overnight. I also clip the corners of pages I've checked to help me go the the right lesson next time. Use manipulatives, even in middle school & high school. I was a straight-A student, but didn't really understand most math concepts until a college professor let us "play" with his 5th grade manipulatives. Use fraction pieces, counters, graph paper, etc. Go ahead & make 5 groups of 4 with edible manipulatives like Cheerios. It's the first time I really understood the concept of multiplication! Use "fun" manipulatives like m&m's, Skittles, pennies, etc. - they don't have to be boring bean counters. submitted by, C. DAMIGO no school listed SAN JOSE, CA thedamigos@aol.com HALLOWEEN PICTURE GRAPH GRADES: 2-7 After trick or treating, and while parents are looking over their contents for safe candy, the students categorize the candies in piles and name it on the chart. MATERIALS: the students' Halloween candy (information can be recorded at home) graph paper markers METHOD: The students color in one square for each one or every 5 candies of that category. Typical categories are: chocolate candies lollipops gummy candies sour candies fruit flavored candies cookies apples nuts round treats, etc. The children come up with their own categories and on their prepared grid they color in the squares for their totals and return the chart to school on November 1 or next day back. Some children will distinguish between chocolate with nuts and chocolate plain candies, snack size and regular and things like this. It makes it fun for them to categorize and name the rows before they color in the squares for each. submitted by SHERI RADOVICH HOLLADAY ELEMENTARY SCHOOL SALT LAKE CITY, UT slradovich@juno.com ELEMENTARY HALLOWEEN MATH GRADES: 2-5 BAT-O-GRAPH MATERIALS: black construction paper cut in shape of bats, or supply a bat pattern for them to cut white construction paper for the eyes brown, blue, red, green and yellow markers METHOD: Using the bat pattern, let each student make one bat with large white eyes. (Use 1/2" white circles to glue on black bat.) Children then color the eyes on their bats brown, blue, red, green, or yellow. Display the bats on a bulletin board and ask the following questions: How many bats have blue eyes? What is the difference in the number of bats with green eyes and the number yellow eyes? Which is greater, blue-eyed bats + brown-eyed bats or green-eyed bats + red-eyed bats? How many bats does each picture represent? THINK ABOUT IT WORD PROBLEMS MATERIALS: none METHOD: Give the following word problems for Halloween: There were 20 bats in Cabot Cave. One-fourth flew away. How many were left? Baby Billy Bat sleeps 20 hours a day. How many hours is he awake? Betty Bat can fly 10 miles per hour. How long will it take her to fly 30 miles at that speed? Bobby Bat left his cave at 8:00 P.m. He arrived at the haunted house at 11:58 P.m. How long did he fly? Big Bertram Bat has a wing span of 21 inches. Budd Bat's wing span is 2 1/2 feet. What is the difference? Big Bertha Bat ate 9 out of 10 pieces of Batpizza Supreme. What fractional part did she eat? What part was left? Barbara Bat worked at the haunted house cafeteria for 2 hours and 15 minutes. She arrived at 10:00 P.m. What time did she leave? submitted by ROCHELLE CHENOWETH ELKINS MIDDLE SCHOOL no city listed rchenoweth@neumedia.net MATH FACT CARD GAMES GRADES: 1-7 Even in today's age of technology, memorizing math facts is still very important. Memorization exercises the brain. Here's a few fun games that will provide a fun way to memorize math facts. MATERIALS: a deck of playing cards METHOD: Multiplication War--I have used this as a group game with about 4 students in a group. Students number or letter off with the first two students playing war, then students 2 and 3 play, then students 3 and 4, then 4 and 1, etc. until all cards are gone. That way the extra students act as judges on who answered first or got the correct answer, etc. Before you start, make a note that Ace = 1, Jack = 0, Queen = 11, and King = 12. Students pair up and shuffle their cards. Cards are dealt out evenly and are stacked face down in front of each student. Then war!! Both students turn over their top card at the same time. They multiply the two cards and shout the answer. The winner puts the cards in his/her winning pile. If a tie should occur, or both get the answer wrong, keep turning cards, one down and one up, until someone wins the pile. When all of their original stack has been played, use the winning pile until someone earns all of the cards. (Optional for a shorter game: When all of their original stacks have been played, students count their winnings, earning one point for each card, the most points win.) Facts War--(ace equals one, number cards equal their values, and each face card equals ten) To play: The dealer deals all of the cards facedown--half to her opponent and half to herself. Each player turns over her first two cards, multiplies the two values, and announces the product. The player with the higher product wins all four cards. If the products are the same ( 8x 5 =40 4 x king = 40 ) each player repeats Step 2. The winner keeps all eight cards. If there's still a tie, then Step 2 is repeated. The player with the highest product keeps all of the cards that have been turned over. Play continues until all of the cards have been used. The winner is the player who wins the most cards. These games can also be adapted for addition facts, subtraction facts (subtracting the smaller number from the larger) and even division, again using the larger number to be divided by the smaller one. Some will have remainders.) *submitted by* PATTY PASON SPRING CREEK ELEMENTARY SCHOOL SPRING CREEK, NV sunset@sierra.net GETTING TO KNOW YOU GRAPHING JOURNAL GRADES: 3-8 This lesson is a good one to do during the first week of school as it helps breaks the ice for your class and introduces or reinforces various forms of graphs. MATERIALS: graph sheets pencils or crayons METHOD: The first week of school I teach 5 major forms of graphs. I do one form a day. The first day I teach single bar graphs. Then double bar graphs, single line graphs, double line graphs, and circle graphs. Each day I have the class make 3 graphs. The first one we all make together, the second one is guided, and the third graph is made independently. To gather the information post the question and record the data on the board or overhead. On the first day the 3 bar graphs we make are: hair color, types of pets, and the year each child entered our school for the first time. When we make the double bar graph we use information gathered by the row. One of these is the number of brothers and sisters per row. This is set up across the x-axis using these identifiers: row 1 brothers, row 1 sisters, row 2 brothers, etc. Other graphs of this type that we do are # of aunts and uncles and # of cats and dogs. Single line graphs are very simple to do and there is a variety of info to be gathered. When we do double line graphs, we use information gathered by gender. The ones we are doing this year are birth month, favorite color, and height. Use two different colors to form the lines. Our final graphs are circle or pie ones. I created mine on the computer so that I could have a circle divided with the number of sections that correlated with the number of kids in my class. We are doing favorite soda pop, favorite outdoor activity, and favorite dessert. At the end of the week we have accomplished several major objectives. The class is now ready to use this essential tool throughout the year instead of waiting for the math book to introduce this. Also, put all of the wonderful graphs into a portfolio or journal which will be wonderful for Back to School Night. You can also use this data for a number of writing and language activities (biographies, interviews, web pages, etc.) *submitted by* SHELLEY BOWEN MITCHELL K-6 SCHOOL WINTON, CA fambowen@cyberlynk.com SOME IDEAS FOR SIMPLIFYING YOUR MATH CLASS GRADES: 4-12 MATERIALS: none METHOD: When teaching requires the use of tools, such as rulers, compasses, protractors, etc., try to get the whole class to have the same instrument. I had some class funds to use recently & bought enough protractors & compasses for everyone in the class to use. They will be re-used from year to year. When I went to teach the lesson, I didn't have to run around the room trying to show everyone how "theirs" worked. Also – try to use clear protractors. The new purple & green plastic ones are cute, but hard for beginners to use. Another idea for protractors: Use a small drill (like a Dremel tool) to put a small hole at the crosshairs. Some protractors come with a hole already there. Tie a string through the hole. When the knot is lined up over the crosshairs, the student can then pull the string up along the angle side they are measuring & the string points to the correct number of degrees. Use the overhead! Write to the publisher to get permission to make overhead transparencies of difficult lessons. This way everyone can watch what you're doing on the overhead. (I had a couple of difficult lessons on scale drawings & map distances that I taught this way. They weren't difficult lessons – just difficult to teach when everyone couldn't see what I was doing). Make a ruler out of transparency film or photocopy onto a transparency. Slide overhead transparencies into clear plastic sheet protectors. They can then be stored in a 3-ring binder, and the sheet protectors can be easily written on & erased. Use examples from real life, whenever possible. For a lesson on sales tax, photocopy a receipt from a recent purchase. Have the students figure out if the tax was correct. Copy your electric bill & talk about the way kilowatts are measured & billed. For a lesson on scale drawings, visit a new home development & take a copy of the floor plans of a new house. I found a really neat book about the way carpenters have to use math – such as measuring the angle & pitch of a staircase, etc. Challenge the students to think of a profession that doesn't use math (farmers have to measure acreage, pounds of fertilizer, etc., lawyers have to be able to bill accurately, etc. Every job requires that employees be able to check to see if their paycheck is correct!) Take math grades once a week instead of daily. I correct math lessons orally daily, usually with students marking their own mistakes, but I only collect them weekly – usually on test days. Of course I have to watch for cheating, but I know my kids' ability pretty well & it becomes obvious to spot. I record the grades while the students are testing. Since my school uses workbooks & I do not allow the students to tear out the pages, this is the only way I have found to be able to glance over their work for neatness, completeness, similar errors, skipping problems, etc. without keeping their books overnight. I also clip the corners of pages I've checked to help me go the the right lesson next time. Use manipulatives, even in middle school & high school. I was a straight-A student, but didn't really understand most math concepts until a college professor let us "play" with his 5th grade manipulatives. Use fraction pieces, counters, graph paper, etc. Go ahead & make 5 groups of 4 with edible manipulatives like Cheerios. It's the first time I really understood the concept of multiplication! Use "fun" manipulatives like m&m's, Skittles, pennies, etc. – they don't have to be boring bean counters. submitted by C. DAMIGO no school listed SAN JOSE, CA thedamigos@aol.com DAILY STORY PROBLEM GRADES: 3-8 This approach to story problems made a tremendous difference in my classroom this year. Test scores shot up both on proficiency tests and standardized tests. Although the instructions are designed for an elementary self-contained classroom, they can easily be adapted for middle school and departmentalized programs. MATERIALS: tagboard small incentive charts stickers METHOD: While this will take some preparation time, the pay-off is worth it! I have a daily story problem that is written on tagboard to put up in my class every morning. (It is worth the effort to put these on tag because there is no effort in future years to keep this going.) The problem is read aloud no matter what the grade level and students have until after lunch to solve the problem. Children keep a file folder with their answer papers inside. I give a new sheet a week and make sure the children are aware of having substantial space to work. All answers must have labels i.e. feet, puppies, centimeters, etc. After lunch, 3 or 4 students go to the board to solve the problem. They talk aloud as to how they solved the problem. When children use different methods to reach the same answer, we spend time discussing how and why that works. Each child has an incentive chart up in the class. Each day 2 students are assigned the task of collecting those papers with correct answers. A sticker is put on the chart for each student who was correct. When a child gets 20 stickers, he/she gets a prize and a new chart goes up on the wall. HELPFUL HINTS: I do not discourage children from talking to each other about ways to attempt to solve the problem. They may not copy each other though. I do not make up nonsense problems. If we are studying a specific unit, I look for information about that to create my problems. So, we did 3 weeks of problems about ancient Egypt and 3 weeks of insect problems. I vary the targeted math skill. So in one week, we may do one long division, one simple fraction, two on working with money, and one on decimals. I make sure that once and awhile the daily problem is very simple so that everyone is having success. I also made up a lot of trivia problems using the Guiness Book of World Records. My kids enjoyed reading about things like the largest pizza ever made. Finally, the students' ability to locate and use mathematical language improved tremendously. Many of my kids are second language learners and need constant practice in looking for key vocabulary--in addition to the daily review and practice of math skills. submitted by SHELLEY BOWEN MITCHELL SCHOOL ATWATER shellyb@cyberlynk.com THE BASIC PRACTICE MODEL GRADES K-12 The Basic Practice Model is the traditional behavioral approach utilized by many school districts which is a standard, traditional, direct lesson plan where the teacher presents to the whole class and the students practice. Many administrators evaluate teachers with this model in mind, so it is a good idea to have some good lessons prepared that utilize it. Besides, in this "day of constructivism," this model has its place and use. Here are the steps: ORIENTATION: Teacher establishes content, continuity with previous activities and future activities, establishes the objective of the lesson. PRESENTATION: The teacher presents both visually and orally to the whole class; students listen and watch. STRUCTURED PRACTICE: Teacher essentially presents again with the students working along with the presentation. GUIDED PRACTICE: Students work on another example while teacher circulates and offers assistance. INDEPENDENT PRACTICE: Students do another example without assistance. FEEDBACK: Hey, you "gotta" reflect and debrief. *submitted by* ROB SCHUCK PACOIMA MIDDLE SCHOOL LOS ANGELES, CA rschuck@glendale.edu SETTING A FOUNDATION FOR PROBLEM SOLVING GRADES 3-12 The beginning of the school year is a crucial time to begin the problem solving process--a process that is a central component of all new Math texts adopted today. The following are a number of stages, approaches and steps for problem. They should be discussed with the students, and if possible, put onto charts for display throughout the year. Examples should be chosen in accordance with the age and level of your students. 6 STAGES OF THE PROBLEM SOLVING PROCESS Define the problem Brainstorm possible solutions Evaluate and prioritize the possible solutions Choose the best solution Determine how to implement the solution Assess how well solution solved the problem 7 APPROACHES TO PROBLEM SOLVING Guess and check Find a pattern Use a systematic list (charts & tables) Use a drawing or a model Eliminate possibilities Work backwards Use a similar, simpler problem 5 STEPS TO PROBLEM SOLVING Read and understand the problem Organize the information Determine the operations needed, establish equation Solve and check answer State and label your answer submitted by ROB SCHUCK PACOIMA MIDDLE SCHOOL LOS ANGELES, CA rschuck@glendale.edu THE FOLLOWING ARE SOME VERY POPULAR PROBLEM SOLVING LESSONS THAT WE RAN LAST YEAR. THESE ARE AN EXCELLENT WAY TO CONDITION YOUR STUDENTS INTO HIGHER LEVEL THINKING SKILLS FROM THE BEGINNING OF THE YEAR! TEACHING THE "GUESS AND CHECK" METHOD GRADES 3-12 Guess and check is an important critical thinking process that is becoming increasingly prevalent within new math texts. It is usually introduced in some form in third grade, and is used in some form all the way up through senior high. There are four major steps involved in the "Guess and Check" method: Make a plan Create a chart or table Eliminate possibilities Look for a pattern The following are a number of examples you can use. (Additional examples can be found in virtually any math text book). They are listed in developmental order, less sophisticated to those more sophisticated. Pick those most appropriate to your students. (The numbers can easily be changed to provide additional examples). With practice your students will develop a self confidence that will enable them to obtain solutions ranging from a variety of correct answers to one correct answer. This will serve as a preparation for high order thinking skills as those used in Algebra, Geometry, etc. EXAMPLE 1 Using pennies, nickles and dimes, how many different combinations can be used to obtain 25 cents? (HINT: there are 12 ways) Make a chart with pennies, nickles, dimes and "total" as column headings. TEACHER NOTE: This problem introduces all four of the steps and adherence to ONE CONDITION--the combination must total 25 cents. The students should be able to put these combinations in any order they choose. As they practice this type of problem, they will find that using a particular system or order, (i.e. concentrating on pennies from greatest to least) will emerge as a faster, more accurate method. Initially, in the earlier grades, students should use actual coins and record their findings. EXAMPLE 2 Using nickles, dimes and quarters, how many different combinations (where at least one of each coin is used), can make 50 cents? Before you start, make a prediction. Compare your prediction to your findings. TEACHER NOTE: There are only 2 combinations. This example introduces TWO CONDITIONS--at least one of each coin AND a total of 50 cents. EXAMPLE 3 Using 17 coins--including AT LEAST ONE NICKLE, DIME AND QUARTER--how many different combinations can be used to make $2.25? Before you start, make a prediction. Compare your prediction to your findings. TEACHER NOTE: There are only 3 combinations. This example introduces THREE CONDITIONS--at least one of each coin, 17 coins AND a total of $2.25. EXAMPLE 4 Using 17 coins--including AT LEAST ONE NICKLE, DIME AND QUARTER--how many different combinations can be used to make $2.25--WHERE THERE ARE 4 MORE DIMES THAN NICKELS? Before you start, make a prediction. Compare your prediction to your findings. TEACHER NOTE: There is only 1 combination. This example introduces FOUR CONDITIONS--at least one of each coin, 17 coins, a total of $2.25 AND a relationship of one variable (dimes) to another (nickles). submitted by ROB SCHUCK PACOIMA MIDDLE SCHOOL LOS ANGELES, CA rschuck@glendale.edu USING A SYSTEMATIC APPROACH TO THE GUESS AND CHECK METHOD GRADES 3-12 Last time we traced the developmental stages of guess and check ("Teaching the Guess and Check Method"), utilizing four components. These components involved: Making a plan Creating a chart or table Eliminating possibilities Looking for a pattern The purpose of these components is to demonstrate to the student that through an organized, systematic process, answers to seemingly "impossible" problems can be found. The key is the systematic approach, because all four components evolve around the system. Having already explored the wonderful world of coin problems, the following examples are concerned with consecutive numbers and age problems. Also available are very basic problem solving examples that highlight each of the seven approaches originally addressed two weeks ago ("Setting a Foundation for Problem Solving"). EXAMPLE 1 5 years ago, Jay was seven times older than Mary. In five years, Mary will be half as old as Jay (or Jay will be twice as old as Mary). How old is each now? Make a horizontal chart with the following headings. (discuss the construction of the heading with the students): J5YA---M5YA---J7XOLDER?---JNOW---MNOW---JIN5Y---MIN5Y---M1/2J? KEY: J5YA (John's age 5 years ago) M5YA (Mary's age 5 years ago) J7XOLDER? (Is John 7 times Mary's age?) JNOW (John's age now) MNOW (Mary's age now) JIN5Y (John's age in five years) MIN5Y (Mary's age in five years) M1/2J? (Is Mary half of John's age?) TWO POSSIBLE ANSWERS: (Numbers are in order of the columns above) 7---1---YES---12---6---17---11---NO 14---2---YES---19---7---24---12---YES Therefore, Jay is 19 and Mary is 7 EXAMPLE 2 Make a chart similar to the one above. Let's make up a consecutive number problem--your choice. Using whole numbers: Four consecutive whole numbers have a sum of 14 and a product of 120. What is the second number? (2, 3, 4, 5) Using odd, whole numbers: Three consecutive odd, whole numbers have a sum of 9 and a product of 15. What is the third number? (1, 3, 5) Using even whole numbers: Three consecutive even, whole numbers have a sum of 12 and a product of 48. What are the numbers? (2, 4, 6) Using integers: Three consecutive integers have a sum of 0 and a product of 0. What is the first consecutive integer? (-1, 0, 1) *submitted by* ROB SCHUCK PACOIMA MIDDLE SCHOOL LOS ANGELES, CA rschuck@glendale.edu GUESS AND CHECK--FINAL PROJECT-- MULTIPLE VARIABLES AND CONDITIONS GRADES: 6-12 This is the last of the problem solving contributions that will be submitted, unless there is a sudden outcry for more! more! more! I hope that what has been presented so far has been of use for some of you. So...for the grand finale of problem solving utilizing the guess and check (trial and error) method, I present to you the infamous chickens, pigs, and sheep problem. TEACHER NOTE: Remember that the "guess and check" method utilizes four components. These components involve: Making a plan Creating a chart or table Eliminating possibilities Looking for a pattern The purpose of these components is to demonstrate to the student that through an organized, systematic process, answers to seemingly "impossible" problems can be found. The key is the systematic approach because all four components evolve around the system. Also, by using a systematic approach, it becomes increasingly easier to eliminate possibilities. This is especially true of the problem presented here. THE PROBLEM: You are given $100 to buy 100 farm animals (at least one each of three animals--chickens, pigs, and sheep). If chickens cost 10 cents, pigs cost $2, and sheep cost $5, how many of each animal must you purchase so that the total is 100 animals for $100? THE CHART: There should be five (5) column headings to represent the problem components. You might want to add a few more to make the students check to see which direction they need to make their "guesses". CHICKENS (.10)-----PIGS ($2)-----SHEEP ($5)-----100 ANIMALS?-----$100 CHICKENS: 35 ($3.50) PIGS: 40 ($80) SHEEP: 25 ($125) 100 ANIMALS?: YES $100: NO ($208.50) CHICKENS: 50 ($5) PIGS: 35 ($70) SHEEP: 15 ($75) 100 ANIMALS?: YES $100: NO ($150) TEACHER NOTE: These two lines represent a wealth of information. In addition to each column beginning to show a potential pattern of direction for future guesses, a viewer should be able to see the plan I am using. Also, what possibilities have already been eliminated? What other possibilities can be eliminated as a result? If your students become frustrated with their own attempts, you might consider using these two lines (or your own) to help them get back on track. THE SOLUTION: Do you really want me to tell you? Okay, I'll meet you half way. The number of chickens is a multiple of 10 (Why must this be so?). It is not 50 chickens. The number of pigs feet is almost = the number of chickens. There are less sheep than the other two animals (approximately 1/7 of chickens and 1/2 of pigs). *submitted by* ROB SCHUCK PACOIMA MIDDLE SCHOOL LOS ANGELES, CA rschuck@glendale.edu
Back to TEACHERS HELPING TEACHERS MAIN PAGE