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Mathematics TEKS Refinement 2006 – K-5 Tarleton State University Patterns, Relationships, and Algebraic Thinking Activity: How Much Does a Kilogram Cost? TEKS: (5.5) Patterns, relationships, and algebraic thinking. The student makes generalizations based on observed patterns and relationships. The student is expected to: (A) describe the relationship between sets of data in graphic organizers such as lists, tables, charts, and diagrams; (5.6) Patterns, relationships, and algebraic thinking. The student describes relationships mathematically. The student is expected to select from and use diagrams and equations such as y = 5 + 3 to represent meaningful problem situations. (5.3) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve meaningful problems. The student is expected to: (B) use multiplication to solve problems involving whole numbers (no more than three digits times two digits without technology); (5.14) Underlying processes and mathematical tools. The student applies Grade 5 mathematics to solve problems connected to everyday experiences and activities in and outside of school. The student is expected to: (A) identify the mathematics in everyday situations; (B) solve problems that incorporate understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness; (C) select or develop an appropriate problem-solving plan or strategy, including drawing a picture, looking for a pattern, systematic guessing and checking, acting it out, making a table, working a simpler problem, or working backwards to solve a problem; and (D) use tools such as real objects, manipulatives, and technology to solve problems. (5.15) Underlying processes and mathematical tools. The student communicates about Grade 5 mathematics using informal language. The student is expected to: (A) explain and record observations using objects, words, pictures, numbers, and technology; and (B) relate informal language to mathematical language and symbols. Patterns, Relationships, and Algebraic Thinking How Much Does a Kilogram Cost? Grade 5 Page 1 Mathematics TEKS Refinement 2006 – K-5 Tarleton State University (5.16) Underlying processes and mathematical tools. The student uses logical reasoning. The student is expected to: (A) make generalizations from patterns or sets of examples and nonexamples; and (B) justify why an answer is reasonable and explain the solution process. Note: Portions of this lesson address TEKS at other grade levels as well; however, the intent of the lesson fits most appropriately at the grade level indicated. Overview: Students will engage in the problem-solving process to discover patterns and to make generalizations about how many nickels and how many pennies it takes to equal the mass of one kilogram. Students will work with partners to choose a problem-solving strategy, determine a solution, and then share their approach with the class. Materials: Math Curse by Jon Scieszka Mr. Newton’s Problem – Handout/Transparency 1 How Much Does a Kilogram Cost? – Handout/Transparency 2 Possible Solution Strategies – Handout/Transparency 3 Mr. Newton Strikes Again – Handout/Transparency 4 Blank paper Large chart paper or blank transparencies Markers Calculators (optional) Grouping: Introductory activity – whole group Problem-solving process – students will have a working partner Time: 2 class periods Lesson: 1. 2. Procedures Share the book, Math Curse, by Jon Scieszka. Notes You may want to share only the introduction and a few problem pages that exemplify the problemsolving process along with some problem solving strategies. Review the 4-step problem-solving process with students. (5.14 B) Understand the problem, make a plan, carry out the plan, and evaluate the solution for reasonableness. Patterns, Relationships, and Algebraic Thinking How Much Does a Kilogram Cost? Grade 5 Page 2 Mathematics TEKS Refinement 2006 – K-5 3. Procedures Illustrate some of the problem-solving strategies that might have been employed by the main character in the book. Ask students to share additional problemsolving strategies (5.14 C). Record and post these strategies on chart paper, if not already posted in the room. 4. Tarleton State University Notes Sample illustrations: p. 6: Gallons/quarts/pints – make diagram like Gallon Guy or a table p. 7: Bus problem – write an equation p. 8: Birthdays – use a chart or list pp. 15-16: Find a pattern, etc. Use the last page to introduce the science teacher, Mr. Newton, and read his statement, “You know, you can think of almost everything as a science experiment!” Tell students that Mr. Newton is setting up a science experiment, and he needs our help with a problem. They will need to rely on their knowledge of the 4-step problemsolving process and one or more problemsolving strategies to solve it. 5. Share Mr. Newton’s Problem (Handout/Transparency 1) with the class. Read for comprehension. A method that can be used to help students comprehend the problem situation is: a. Have students read the problem chorally. You will set the pace and then fade out so that you can key in on the students’ oral reading behaviors. Make note of any difficulties (slowing down, voices dropping out, vocabulary, mispronunciations, etc.). b. Remove the problem, and ask students to tell you what they remember about the problem. Restate each student’s contribution. It is likely that students will have only partial or incorrect recall of the details in the problem. Emphasize the need for rereading any problem. Patterns, Relationships, and Algebraic Thinking How Much Does a Kilogram Cost? Grade 5 Page 3 Mathematics TEKS Refinement 2006 – K-5 Procedures Tarleton State University Notes c. Show the problem again, and repeat step a. d. Remove the problem again, and elicit details from the students, restating each student’s contribution. e. Finally, present the problem again and clarify anything that might have caused difficulty during the reading of the problem. 6. Ask guiding questions to check for understanding and summarization of the problem. Have students summarize what Mr. Newton wants to know. (number of nickels needed to equal the mass of one kilogram; number of pennies needed to equal the mass of 1 kilogram) Ask students what information in the problem will help them to find out. (nickels have a mass of 5 grams; pennies have a mass of 2.5 grams) At this point, it is important that students understand that we are not interested in the value of the coins. We are focusing on their mass. 7. Use the Think/Pair/Share method to have students determine which strategies might be used to solve the problem. Patterns, Relationships, and Algebraic Thinking How Much Does a Kilogram Cost? The “Think/Pair/ Share” method allows all students an opportunity to communicate. The teacher asks the question and allows everyone 2 minutes of quiet “think” time. Then students get with their working partner and share their answers. Finally the teacher gives several pairs the opportunity to share their thoughts with the entire class. Grade 5 Page 4 Mathematics TEKS Refinement 2006 – K-5 8. 9. 10. Procedures Elicit the strategies that students think might be useful. Tarleton State University Notes The following are some of the strategies that might be used with this problem: • make a table • look for a pattern • draw a picture or diagram • guess and check • formulate an equation Have students work with their partners to find For those students who arrive at a a solution to the problem. satisfactory solution early, have them demonstrate or justify their solution through a different strategy. Monitor students’ progress, giving as little guidance as is necessary. The teacher should determine if and when to allow the use of calculators. It may be helpful to allow their use when students are solving for the number of pennies. Since this problem involves a multiplicative relationship, and since 5th grade TEKS do not require students to multiply or divide with decimals, it may be beneficial to provide some students with calculators once they have exhibited a reasonable solution for the number of nickels needed to equal the mass of 1 kilogram. Some students may use proportional reasoning to determine that twice as many pennies as nickels would be equal the mass of 1 kilogram since the mass of two pennies equals the mass of one nickel. 11. Have students transfer their findings to a transparency or chart paper. Each pair should be ready to share their solutions with the group. Patterns, Relationships, and Algebraic Thinking How Much Does a Kilogram Cost? Ask pairs of students to share their strategies and solutions with the whole group. Have students with similar strategies contribute any further ideas or thoughts. Call Grade 5 Page 5 Mathematics TEKS Refinement 2006 – K-5 Procedures 12. Tarleton State University Notes on student pairs to share until all strategies have been exhausted. If different incorrect solutions are presented, allow the students to process their own work and look for explanations for the discrepancies. Discuss the interrelatedness of the various strategies that have been presented. Allow students to make these connections as much as possible. See Possible Solution Strategies (Handout/Transparency 3). Some possible points to bring out during the discussion are as follows: What does the table help us to see? A pattern What does a pattern allow us to do? Formulate a rule or equation If you used guess and check, what results match data in the table or match the equation? Which ones do not? Those that fell within the range of acceptable outputs without going over How does your drawing or diagram represent the data in the table or the solution equation? Answers will vary depending on the picture or diagram. 13. When students have satisfactorily arrived at the answers to Mr. Newton’s first two questions, present questions 3 and 4 to the students on Handout/Transparency 2. 14. Pose the problem question: So, how much does a kilogram cost? A kilogram of nickels cost $10.00 and a kilogram of pennies cost $4.00. Patterns, Relationships, and Algebraic Thinking How Much Does a Kilogram Cost? Now that students know the number of nickels (200) and pennies (400) needed to equal the mass of one kilogram, they can employ any strategy to calculate the values. Grade 5 Page 6 Mathematics TEKS Refinement 2006 – K-5 Procedures Have students share their strategies for arriving at these solutions. 15. Present the challenge questions to the students on Handout/Transparency 2. Answer: Mr. Newton will need 5 rolls of nickels or 8 rolls of pennies to make a kilogram. 16. Tarleton State University Notes You might tell students that besides knowing how much a kilogram of each coin costs, Mr. Newton is also interested in how many rolls of each coin would be needed. Have students share their strategies for arriving at their solutions. Homework: Have students complete Mr. Newton Strikes Again -Handout/Transparency 4. Assessment: One ton is equal to 2000 pounds. Measure your weight to the nearest 10 pounds. Based on this weight, about how many of you would it take to equal a “ton of kids?” Use words and pictures, diagrams, or tables to show how you arrived at your answer. Extensions: 1. Have students gather items from home that approximate the mass of one kilogram. Ask students to bring their items and objects to class. Let them use a balance to determine how closely they were able to approximate the mass of one kilogram. 2. Have students find other items/objects of uniform mass (washers, screws, nails, milliliters of water, binder clips, etc.) and determine how many it will take to approximate the mass of one kilogram. Let students use balances to determine the mass of each item. Then, students can calculate how many are needed to approximate one kilogram. Resources: Literature Connection: Counting on Frank by Rod Clement. Frank’s owner, the narrator of this book, might be called a measuring maniac. Students will enjoy his wacky way of counting and his methods for measuring the world around him. Modifications: This activity might be configured as a problem set for the week. Day 1: Read Math Curse and introduce the problem. Read the problem for comprehension and summarize the important information. Day 2: Have students generate possible strategies and select a Patterns, Relationships, and Algebraic Thinking How Much Does a Kilogram Cost? Grade 5 Page 7 Mathematics TEKS Refinement 2006 – K-5 Tarleton State University suitable strategy for solving the initial problem. Day 3: Have students share their strategies and solutions with the group. Connect the various strategies and representations shared by the students. Day 4: Have students solve questions 3 and 4. Let students share strategies and solutions. Day 5: Present the Challenge questions. Have students share strategies and solutions. Patterns, Relationships, and Algebraic Thinking How Much Does a Kilogram Cost? Grade 5 Page 8 Mathematics TEKS Refinement 2006 – K-5 Tarleton State University Mr. Newton’s Problem Mr. Newton, the science teacher, needs some kilogram masses for use in an upcoming science activity with the 4th grade students. Rather than order standard kilogram masses from a science supply store, he has decided to use coins because of their small size and uniform mass. Nickels have a mass of about 5 grams each, and pennies have a mass of approximately 2.5 grams each. Mr. Newton needs to know the following: 1. How many nickels does it take to equal the mass of 1 kilogram? 2. How many pennies does it take to equal the mass of 1 kilogram? Handout/Transparency 1 Patterns, Relationships, and Algebraic Thinking How Much Does a Kilogram Cost? Grade 5 Page 9 Mathematics TEKS Refinement 2006 – K-5 Tarleton State University How Much Does a Kilogram Cost? Continued: 3. How much would a kilogram of nickels cost? 4. How much would a kilogram of pennies cost? Challenge: 5. If nickels come in rolls of $2.00, how many rolls of nickels would Mr. Newton need to equal the mass of 1 kilogram? 6. If pennies come in rolls of $0.50, how many rolls of pennies would Mr. Newton need to equal the mass of 1 kilogram? Handout/Transparency 2 Patterns, Relationships, and Algebraic Thinking How Much Does a Kilogram Cost? Grade 5 Page 10 Mathematics TEKS Refinement 2006 – K-5 Tarleton State University Possible Solution Strategies 1. Make a table. Nickels Process Grams (N) (g) Pennies (P) Process Grams (g) 1 1x5 5 1 1 x 2.5 2.5 2 2x5 10 2 2 x 2.5 5 5 5x5 25 5 5 x 2.5 12.5 10 10 x 5 50 10 10 x 2.5 25 100 100 x 5 500 100 100 x 2.5 250 200 200 x 5 1000 200 200 x 2.5 500 N Nx5 g 400 400 x 2.5 1000 P P x 2.5 g 2. Write an equation (possible responses). 1000 ÷ 5 = 200 5 x ____ = 1000 1000 ÷ 2.5 = 400 2.5 x ____ = 1000 3. Guess and Check (possible guesses). 10 nickels would = 50 grams 20 nickels would = 100 grams 100 nickels would = 500 grams 200 nickels would = 1000 grams 2 pennies would = 5 grams 4 pennies would = 10 grams 40 pennies would = 100 grams 400 pennies would = 1000 grams Handout/Transparency 3 Patterns, Relationships, and Algebraic Thinking How Much Does a Kilogram Cost? Grade 5 Page 11 Mathematics TEKS Refinement 2006 – K-5 Tarleton State University Mr. Newton Strikes Again Now you know that 200 nickels or 400 pennies are approximately equal to the mass of one kilogram. Mr. Newton has some more questions for you to answer. Use your knowledge of patterns to help you answer his questions. 1. If 1 kilogram equals 1,000 grams, how many grams are equal to 2 kilograms? _____________ How many nickels would be needed to equal the mass of 2 kilograms? ______________________ How many pennies would be needed to equal the mass of 2 kilograms? ______________________ 2. If 1,000 grams equals one kilogram, how many grams would equal 1 kilogram? _____________ 2 How many nickels would be needed to equal the mass of 1 2 kilogram? ______________________ How many pennies would be needed to equal the mass of 1 2 kilogram? ______________________ 3. Mr. Newton has a mass of pennies equal to 250 grams. What fractional part of a kilogram is 250 grams? _________________________________ How many pennies would be equal in mass to 250 grams? _____________________________ How many nickels would be equal in mass to 250 grams? _____________________________ Handout/Transparency 4 Patterns, Relationships, and Algebraic Thinking How Much Does a Kilogram Cost? Grade 5 Page 12