“The power of choice” menu activities
Math Problem of the Day
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A daily (or weekly) exercise in problem-solving for Kivalliq Math Month and throughout the year.
Preface
During Math Month challenge your class to a “Math Problem of the Day” each morning. Give students a few minutes to work on it and let them return to it throughout the day when their other work is finished. Students who have solved the problem may complete an entry form with their name and solution. At the end of the day randomly select an entry (a correct one!) and award a small prize to the student. Note: if the selected entry does not have the correct answer, choose another until a correct solution is found. Always go through the solution with your entire class, even if no correct answers are submitted. If once a day is too much, consider doing a “Problem of the Week.”
Suitable problems may be found in the Math to the Max resources, the Problem Solver Binders, or on math-related web sites, They may also be created by teachers to directly match the content and difficulty level familiar to their classes.
The problems in this document are suitable for grades 5 to 8 and were taken from the web site: (). Solutions for these problems and more can be found on this web site as well.
Resources and Supports (that are in your school)
|[pic] |Math to the Max (Gr. 1-6) |[pic] |Problem Solver (Gr.1- 8) |
| |Edmonton Public Schools | |By Shirley Hoogeboom & Judy Goodnow |
| |Resource Development Services (2000) | |Creative Publications |
| |ISBN #1-894522-41-9 | |Wright Group/McGraw-Hill (1989) |
| | | |ISBN 0-88488-582-8 |
| | | | |
|[pic] |Mathemagics |[pic] |Key Skills Computer Software (K-Gr. 6) |
| |Kivalliq Math Ed. Panel | |Sunburst Technology Corp. |
| |FirstClass Server: | | |
| |Teacher Conferences( Kivalliq Math Forum | |(2005) |
|[pic] |Classroom Puzzles |[pic] |100 Classroom Riddles |
| |Kivalliq Math Ed. Panel | |Kivalliq Math Ed. Panel |
| |FirstClass Server: | |FirstClass Server: |
| |Teacher Conferences( Kivalliq Math Forum | |Teacher Conferences( Kivalliq Math Forum |
| | | | |
| | | | |
Web-Based Resources
Figure This! Math Challenge for Families—This web site is developed by the National Council of Teachers of Mathematics (NCTM) and contains eighty different mathematical challenges suitable for home and school. The activities are organized into five pdf documents that may be downloaded, viewed and/or printed. (These five pdf files are also available in the First Class folder for Math Month) The problems are suitable for upper elementary and junior high school, but are also challenging for high school students and teachers alike. An index is included that maps the challenges to its specific math content area.
Math Problem Solving--Featuring original mathematics problem solving worksheets for teachers and parents to copy for their kids. Use them for teaching, reinforcement, and review. They are most appropriate for grades four and five, but many are designed to be challenging and informative to older and more advanced students as well.
ABC Teach—This site contains many sets of word problems in downloadable pdf format for elementary and middle school.
Teachnology—This is an on-line teacher resource complete with free lesson plans, worksheets and assessment ideas. For a paid membership, you can have access to even more resources.
Jim Kreuger
Baker Lake, October/09
Problem Solving Through Literacy and Routine
The Value of Teaching with Problems
Elementary and Middle School Mathematics—5th Ed (p. 37) by John Van De Walle’s (2004) Pearson Education.
➢ Problem solving places the focus of the students’ attention on ideas and sense making.
➢ Problem solving develops the belief in students that they are capable of doing mathematics and that mathematics makes sense.
➢ Problem solving provides ongoing assessment data that can be used to make instructional decisions, help students succeed, and inform parents.
➢ Problem solving develops “mathematical power” “Remember that good mathematics is not how many answers you know, but how you behave when your don’t know.”
➢ Problem Solving allows multiple entry points for a wide range of students
➢ A problem-based approach engages students and is fun!
Problem Solving Strategies
See The Problem Solver by Shirley Hoogeboom and Judy Goodnow—McGraw-Hill (1987). This series consists of eight binders for Grades 1-8. Copies of this resource was purchased for each Kivalliq School and is kept in the Student Support Room.
Act Out or Use Manipulatives
Some children may find it helpful to act out a problem or to move objects around while they are trying to solve a problem. It allows them to develop visual images of both the data in the problem and the solution process. By taking an active role in finding the solution, children are more likely to remember the process they used and be able to use it again for solving similar problems. The dramatizations and objects need not be elaborate: small scraps of paper and colored chips or counters will usually work quite well. This strategy is especially helpful when the problem solver wants to visualize relationships.
Work Backwards
To solve certain problems, the solver must make a series of computations, starting with data presented at the end of the problem and ending with data presented at the beginning of the problem.
Draw or Use a Diagram
For some children, it may be helpful to use an available picture or make one when trying to solve a problem. The pictures or diagrams need not be beautiful or well drawn. It is most important that they help the problem solver understand and manipulate the data in the problem. Using pictures is almost a necessity for some problems, particularly those which involve mapping.
Use or Look for a Pattern
A pattern is a regular, systematic repetition. A pattern may be numerical, visual, or behavioral. By identifying the pattern, the problem solver can predict what will "come next" and what will happen again and again in the same way. Looking for patterns is a very important strategy for problem solving, and is used to solve many different kinds of problems. Sometimes students can solve a problem just by recognizing a pattern, but often they will have to extend a pattern to find a solution. Making a number table often reveals patterns, and for this reason is frequently used in conjunction with the "look for a pattern" strategy.
Use Logical Reasoning
Logical reasoning is really used for all problem solving. However, there are types of problems that include or imply various conditional statements such as: "if. . . then," or "if. . . then. . . else," or "if something is true, then. . ." or "if something is not true, then. . .." The data given in the problems can often be displayed in a chart or matrix. This kind of problem requires formal logical reasoning as the problem solver steps his or her way through the statements given in the problem.
Guess and Check
Guessing and checking is helpful when a problem presents large numbers or many pieces of data, or when the problem asks the solver to find one solution but not all possible solutions to a problem. When problem solvers use this strategy, they guess the answer, test to see if it is correct, and make another guess if the previous one was incorrect. In this way, they gradually come closer and closer to a solution by making increasingly more reasonable guesses. Problem solvers can also use this strategy to get started, and may then find another strategy which can be used. Guessing and checking is particularly helpful when a problem presents so many pieces of data that making an organized list becomes a major task
Make and Organized List
Making an organized list helps problem solvers organize their thinking about a problem. Recording work in an organized list makes it easy to review what has been done and to identify important steps that must yet be completed. It also provides a systematic way of recording computations made with given data or recording combinations of given items.
Use or Make a Table
A table is an orderly arrangement of data, such as numbers. Problem solvers find that making tables helps them keep track of data, spot missing data, and identify data that is asked for in the problem. Because patterns often become obvious when data is organized in a table, this strategy is often used in conjunction with other strategies. In the example below, the table is used to keep track of data and could also be used for identifying a number pattern.
Make it Simpler
Students will find it helpful to be able to make problems simpler, especially when they begin to solve complex problems. Making a problem simpler may mean reducing large numbers to small numbers, or reducing the number of items given in a problem. The simpler representation of the problem, then, may suggest what operation or process can be used to solve the more complex problem. The simpler representation may even reveal a pattern which can be used to solve the problem.
Brainstorm
This strategy is often used when all else fails. When the problem solver cannot think of a similar problem that he or she has solved before, and cannot think of another strategy to use, brainstorming is a good strategy to try. Brainstorming means looking at a problem in new and inventive ways. There are always problems that stretch people beyond their experience and expertise. When students encounter problems that they cannot solve, they must be encouraged to open up, stretch, allow for inspiration, be creative, be flexible, and keep on trying!
Teaching Word Problems: Steps to Success
Starting off on the right foot.
1. Choose easy questions, perhaps one or two grades lower.
2. Establish a four-step process. (see template)
3. Use overhead projector: class examples
4. Practice with the same type of problem used in your class example.
5. Use mixed ability groups. (make sure you have a strong reader in each group)
6. Assess group work as well as problem solving.
Word Problem Wednesday
Problem solving is a key skill and should be integrated throughout your entire math year plan. However, since word problems are as much a literacy challenge as they are mathematical, they can take time to get into. Rather than trying to do word problems on a daily basis, KMEP recommends that teachers schedule word problems on a weekly or biweekly basis, such as “Word Problem Wednesday”. By devoting an entire period to working with word problems, you allow students to learn the routine of the problem solving approach and practice their literacy skills & problem solving strategies.
Problems may be solved individually or in groups and teachers should alternate these strategies from time to time. If routines are established and roles are clear, Word Problem Wednesday can accommodate mixed ability groups.
Roles for Problem Solving Groups
When using a group problem solving approach, assign specific roles to each member. For example: • Problem Reader • Solution Recorder
• Solution Explainer • Manipulative Collector
All group members are responsible for clean-up.
Problem Solving Through Literacy and Routine
KMEP’s Word Problem Primer
An exercise in literacy, routine, attitude, and math
Word problems present the greatest challenge for most math teachers in the Kivalliq. For students to have success at solving word problems they must be:
➢ Good readers
➢ Knowledgeable in the vocabulary of Mathematics.
➢ Experienced in the structure and routine of problem solving.
➢ Proficient in the basic math skills.
➢ Confident in taking risks.
➢ Persistent in finding the solution.
➢ Optimistic about their chances for success.
By identifying these precursors for success, KMEP hopes to help teachers to structure their word problem instruction and activities in such a way that builds towards success incrementally. To this end, KMEP makes the following recommendations.
1. Weekly Sessions
Schedule enough time to address the literacy elements of word problems. KMEP recommends focusing one period each week on word problems (Word Problem Wednesdays). By using a whole period, you can focus on word attack skills and reinforce the problem solving routine that you want your students to follow.
2. Problem Solving Template
KMEP suggests a simple four step process that stresses reading, perseverance, showing work and writing complete sentences. Math to the Max and The Problem Solver each have practical templates. See page 46 for KMEP’s suggested template.
3. Math Vocabulary Word Wall
Math has its own language and students need to be taught its specialized and general vocabulary. For example: “tens place” is a specialized math term, while “plus”, “add”, “increase by”, and “more than” can all indicate the basic operation of addition.
4. Problem Solving Strategies
Students should be taught the different problem solving strategies as they need them and be given enough practice with the strategy to become confident with it. The Problem Solver organizes its problems by strategies and so it can be useful for reinforcing a specific strategy.
5. Translating English or Inuktitut into Math
Students should be encouraged to read the problem over at least twice and then underline the words that are math-related. Their challenge then becomes “how do these words relate to the question asked?”
6. Introduce Problems in a Development Sequence
Begin problem solving as a literacy activity and increase the mathematics as students become more confident in reading and the routine of problem solving.
KMEP has identified five types of word problems and has sequenced them as indicated below in its Word Problem Development Sequence.
|Type |Description |Examples |
|1. Simple Literacy |Simple Literacy word problems do not involve any |Jade goes to the store. He buys three apples and |
| |complex mathematics or difficult vocabulary. The |two bananas. How many apples does Jade buy? |
| |context is simple so that students can | |
| |successfully solve problems and learn the problem |Annie has three dogs. A black one named Rex, a |
| |solving routine. Students are challenged to read |white one named Snowy, a spotted one named Patches.|
| |the problem, select the correct information, and |What colour is Rex? |
| |write out a sentence answer. Teachers should | |
| |create their own problems to match the literacy |On Monday the temperature was -23oC, on |
| |level of their students. |Tuesday the temperature was -21oC, and on Wednesday|
| | |the temperature was -25oC. What was the |
| | |temperature on Wednesday? |
|2. Math Vocabulary |Math Vocabulary word problems are very similar to |Elizapee is 143 cm tall. Alice is 139 cm tall. |
| |Simple Literacy problems with the addition of |Candace is 147 cm tall. Who is the tallest? |
| |increasingly new and specific math vocabulary. | |
| |Students are challenged to read the problem, |TJ has 1,354 pennies in his piggy bank. Identify |
| |identify and understand the new vocabulary, select|the tens digit in this number. |
| |the correct information, and write out a sentence | |
| |answer. Teachers should create their own |Stops signs are shaped like an octagon. How many |
| |problems to specifically match any new vocabulary |sides does an octagon have? Draw a picture of an |
| |that has been recently introduced. |octagon. |
| | | |
| | |Using the numerals 4, 9 and 3, how many different |
| | |2-digit numbers can be made? (Don’t use the |
| | |numerals twice as in numbers like 33 or 44). |
|Type |Description |Examples |
|3. Single |Simple Operation word problems require students to |The Peewee hockey team sold three cases of |
|Operation |perform simple mathematical operations in addition |chocolate almonds. Each case holds twenty-five |
| |to reading the problem, understanding the vocabulary|boxes of chocolate almonds. How many boxes of |
| |and identifying the math. These problems challenge |chocolate almonds did the Peewees sell? |
| |students to identify which mathematical operation is| |
| |needed to solve the problem based on word cues and |Vinnie, Keith, and Jason buy a pizza at the Quick |
| |context. Students must perform the operation (show |Stop. The pizza has eight slices. How many |
| |their work) and communicate their answer. Teachers |complete slices does each boy get? How many are |
| |may find problems like this in Math to the Max or |left over? |
| |The Problem Solver, however careful selection of | |
| |appropriate problems must be made or teachers must |Sarah has $5.00. Peter has two dollars and fifteen|
| |create their own original problems to match their |cents less than Sarah. How much money does Peter |
| |students’ level. |have? |
| | | |
| | |How many days are in five weeks? How did you |
| | |figure this out? |
|4. Compound Operations |Compound Operations word problems require students |Sarah has $15.00. Peter twice as much money as |
| |to perform more than one mathematical operations in |Sarah. Trina has twenty dollars more than Sarah. |
| |order to solve the problem. Students must be |Who has the most money? |
| |careful readers and know their math vocabulary in | |
| |order to “find the math” in this type of problem. |I am thinking of two 2-digit numbers. These two |
| |Student success is dependent on both literacy and |numbers have the same digits. The sum of the |
| |mathematical skills. Teachers may find problems like|digits in each number is 9. The difference between|
| |this in Math to the Max or The Problem Solver, |the two numbers is 9. |
| |however careful selection of appropriate problems | |
| |must be made or teachers must create their own |James and Judy want to fundraise by selling juice |
| |original problems. |at a fleamarket. They have a big jug to mix the |
| | |juice in and they want to charge 50¢ for each |
| | |glass. How could they find out how much each jug |
| | |of juice is worth? |
|5. Complex Context |Complex Context word problems are tricky to read |You have about 10 sugar cubes. Your problem is to |
| |because of complex (usually long) instructions or |figure out how many sugar cubes would fit on the |
| |story that surrounds the problem. These problems |surface of your desk, stacked to a depth of 5 |
| |can be open ended and require multi-step logic to |layers of cubes. |
| |find the solution. Strong reading skills and good | |
| |math skills are necessary to successfully undertake |Clear everything else off your desk before you |
| |these types of problems. |begin. Develop a plan for attacking this problem. |
| | |Show your calculations below. |
KMEP Problem Solving Template
Name:______________________ Strategy Used:__________________________
Steps to Problem Solving:
1. Read the problem, then read it again and identify the math.
2. Solve the Problem (do the math) and show your work.
3. Write out your solution in a complete sentence.
4. Read the problem to check your answer! (Is it reasonable?)
Show your work here
Solution:
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Simple Literacy
Single Operation
Complex Context
Compound Operations
Math Vocabulary
Success with Word Problems
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