Pan-Balance Problems with Two Balances

Transcription

1 Pan-Balance Problems with Two Balances Objective To develop a pan-balance approach for solving sets of two equations with two unknowns. epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Use addition and subtraction to solve [Operations and Computation Goal ] Use multiplication and division to solve [Operations and Computation Goal 3] Use a pan-balance model to solve linear equations with two unknowns. [Patterns, Functions, and Algebra Goal ] Key Activities Students solve pan-balance problems involving two balances. Ongoing Assessment: Recognizing Student Achievement Use the Math Message. [Patterns, Functions, and Algebra Goal ] Materials Math Journal, pp Study Link 0 slate for demonstration purposes: 00 standard " paper clips, 3 identical ballpoint pens, 3 identical compasses (pencils removed), identical plastic 6" half-circle protractors, plastic " ruler, or pan balances Making and Interpreting Line Plots Math Journal, p. 339 Students construct and interpret a line plot. Math Boxes 0 Math Journal, p. 30 Students practice and maintain skills through Math Box problems. Study Link 0 Math Masters, p. 97 Students practice and maintain skills through Study Link activities. READINESS Solving Logic Puzzles Class Data Pad (optional) Students solve logic puzzles related to ENRICHMENT Measuring Math Masters, p. 9 Students explore a sandglass model to solve a problem about time intervals. Advance Preparation For Part, use two pan balances for demonstration purposes. Alternately, the lesson includes instructions for a single pan balance. Teacher s Reference Manual, Grades 6 pp. 9 9 Lesson 0 79

2 Getting Started Mathematical Practices SMP, SMP, SMP3, SMP, SMP6, SMP Content Standards 5.NF.5a, 5.MD. Mental Math and Reflexes Write each riddle on the board or prepare a transparency. Students use their slates to write the number and then write the algebraic equation for the riddle: What number am I? If you double me and add 6, you get 0. 7; n + 6 = 0 If you add 5 to me, you get 0. -5; n + 5 = 0 If you double me, add, subtract, and then divide by, you get 9. 9; (n + - ) / = 9 If you take half of me and triple the result, you get 30. 0; n_ 3 = 30, or n _ 3 = 30 If you double me and double the result, you get 0. 0; n = 0 If you multiply me by 00 and add 5, you get 55. 5; (n 00) + 5 = 55 If you multiply me by,000 and add 3, you get a result that is,000 times as great as.003. ; (n,000) + 3 =.003,000 Math Message Answer Problems and on page 336 of your journal. Study Link 0 Follow-Up Have partners compare answers and resolve differences. Then have volunteers share their solution strategies. Ongoing Assessment: Recognizing Student Achievement Math Message Use the Math Message to assess students ability to solve equations using a pan-balance model. Students are making adequate progress if they correctly solve both problems. [Patterns, Functions, and Algebra Goal ] Teaching the Lesson 0 Math Message More Pan-Balance Problems Solve these In each figure, the two pans are in perfect balance.. as much as balls.. One ball weighs as much as 5 marbles. 3.. Solve these problems using both pan balances. In each problem, the pans are in perfect balance. The weights of objects, such as blocks, balls, marbles, and coins, are consistent within each problem. One coin weighs as much as 3 marbles. as much as 7 marbles. Math Message Follow-Up (Math Journal, p. 336) WHOLE-CLASS DISCUSSION Draw the Math Message pan-balance problems on the board or a transparency. Have volunteers share their answers and solution strategies. Under the pan balances, write: as much as how many marbles? Ask: Can you solve this problem using only one of the pan balances? No Discuss how the information from both pan balances needs to be combined to solve the problem. as much as balls. One ball weighs as much as 5 marbles. Therefore, one block weighs as much as 0 marbles. Because one ball weighs as much as 5 marbles, balls weigh as much as 0 marbles. Therefore, blocks weigh as much as 0 marbles, and block weighs as much as 0 marbles. One ball weighs as much as 30 marbles. as much as 5 marbles. 5. One ball weighs as much as 9 marbles. as much as 3 marbles. Math Journal, p. 336 Demonstrating How to Solve More Complex Pan-Balance Problems WHOLE-CLASS Explain that the class will look at two other examples where the information from two pan balances needs to be combined to solve the problems. 79 Unit 0 Using Data; Algebra Concepts and Skills

3 Example : Show the protractor, ruler, and paper clips. Tell students their goal is to determine the weight of the ruler and the protractor in terms of paper clips. (See margin.) Set up the first pan balance. Place a protractor in one pan. Then add paper clips to the other pan, one at a time, counting as you go, until the pans balance. If you have only one balance available, sketch the results of the first pan-balance situation on the board before setting up the second balance. Set up the second pan balance. Place a protractor and ruler in one pan. Then add paper clips to the other pan, one at a time, counting as you go, until the pans balance. Remind students that a protractor weighs 3 paper clips in the first pan balance. All that remains is to find the ruler s weight in paper clips. Ask: How can you change the pans to isolate the ruler in one pan, balanced by paper clips in the other pan? Tell students their solutions must follow one rule: Whatever you do, the pans must always remain balanced. Here are two possible approaches: Remove the protractor and replace it with 3 clips. Because the protractor weighs 3 clips, the pans remain balanced. Now remove 3 clips from each side. The pans remain balanced. So the ruler weighs as much as 39 paper clips. Remove the protractor from one pan. At the same time, remove 3 clips from the other pan. Because the protractor weighs 3 clips, the pans remain balanced. The ruler is balanced by 39 clips. Example : Show the ballpoint pens, compasses, and paper clips. Tell students their goal is to find the weight of a pen and the weight of a compass in terms of paper clips. Set up the first pan balance. Place pens in one pan. Then add clips to the other pan, one at a time, counting as you go, until the pans balance. (See margin.) Set up the second pan balance. Place compasses in one pan. Place compass and pen in the other, and add paper clips, one at a time, counting as you go, until the pans balance. (See margin.) Have students use the balances to test their solutions. Remove _ of the objects from each pan of the first pan balance. The pans remain balanced. So pen weighs 0 clips. Example balance # For the example here, add 3 clips. 3 balance # This might require 70 clips. 3 exchange balance # balance # balance # Exchange the protractor for 3 clips. remove 3 Then remove 3 clips from each pan. remove 3 Or remove the protractor from one pan and 3 clips from the other. Example balance # This might require 0 clips. balance # Remove pen from the second pan balance and replace it with 0 clips. The pans remain balanced. balance # exchange 0 balance # This might require clips. Lesson 0 793

4 0 More Pan-Balance Problems continued Remove compass from each pan. The pans remain balanced. So compass weighs 5 clips. 6. One coin weighs as much as clips. as much as clips. balance # 5 doughnut 7. One can weighs One doughnut weighs as much as blocks. as much as blocks. Solving Pan-Balance Problems (Math Journal, pp ) PARTNER PROBLEM SOLVING marbles 9 marbles One weighs One weighs 5 as much as marbles. as much as marbles. B B ounces B B A A A B B B Each can weighs B ounces. Each cube weighs A ounces. B 9 ounces A 3 ounces Math Journal, p. 337 Adjusting the Activity Have students write equations with variables to represent each of the pan-balance problems or match the problems to a prepared list of equations. They can record the correct equation above the illustration for each problem. Algebraic Thinking Have partners complete the journal pages. Some of the problems show pictures of objects in the balance pans; some show squares and triangles; and some show expressions with letter variables. Explain that some problems consist of two related parts and that students need to solve one of the parts before they have enough information to solve the other part. For example, in Problem 3, students need to complete the statement associated with the second pan balance before they can complete the statement associated with the first pan balance. Problem 3 Complete the second statement first. One coin weighs as much as 7 marbles. Since coin weighs as much as 7 marbles and block and coin weigh as much as 0 marbles, block must weigh as much as 3 marbles. Remove the coin and 7 marbles from the pans of the first pan balance. AUDITORY KINESTHETIC TACTILE VISUAL Adjusting the Activity ELL 0 More Pan-Balance Problems continued Draw a new pan balance underneath the one on the board to illustrate each step. Refer to the Math Message Follow-Up, and replace each ball with 5 marbles marbles One weighs One weighs 6 as much as marbles. as much as marbles.. 7 marbles x y marbles x y y y weighs x weighs as much as 5 marbles. as much as marbles. Then draw another pan balance underneath to show each side of the balance divided in half. Cross off one half on each side.. cup empty cup full If the cup is full, the coffee in the cup weighs as much as If the cup is full, the coffee plus the cup weighs as much as 6 marbles. marbles. 3. Two pens weigh as much as one compass. One pen and one compass together weigh 5 grams. Complete the pan-balance problems below. Find the weights of one pen and one compass. pen and pens compass compass 5g One pen weighs 5 grams. One compass weighs 30 grams. Math Journal, p. 33 A U D I T O R Y K I N E S T H E T I C T A C T I L E V I S U A L Encourage students to begin by deciding which of the two statements should be completed first. For example, in Problem 3, the second statement needs to be completed first. However, in Problem 5, the first statement should be completed first. Circulate and assist. A hint for Problem on journal page 33 is to determine the weight of the coffee separate from the cup. 79 Unit 0 Using Data; Algebra Concepts and Skills

5 Ongoing Learning & Practice Making and Interpreting Line Plots (Math Journal, p. 339) INDEPENDENT Students make and interpret a line plot consisting of a data set of measurements obtained from fraction cards. Students plot fractional units on the number line from 0 through. Remind students that they can use benchmarks to place the fractions on the number line. Review fraction operations as needed to find the data landmarks. Math Boxes 0 (Math Journal, p. 30) INDEPENDENT Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 0-. The skill in Problem 5 previews Unit content. Writing/Reasoning Have students write a response to the following: Explain your solution strategy for Problem a. Sample answer: The area of the rectangle is 3 or 6 ft. The area of a triangle is _ the base times the height. Since the base for both figures is the same, and _ of the height of the triangle is greater than the height of the rectangle, the triangle s area is greater because 3.5 = 7.5 ft. 0 Displaying Fractions on a Line Plot A full set of fraction cards was shuffled. Then 6 cards were dealt out showing the following fractions: _ 0 0 0_ 6 5 3_ 6 5_ 0 _ 3_ 9_ 6_ 6 0 6_ 6 9 9_ 6 6. Write each of the 6 fractions on the blanks below. If a fraction is not in simplest form, write it in simplest form. 5 0 _ 3 3. Make a line plot to display the data. Write the fractions below the number line in their approximate locations. 3. Find the following data landmarks. Then label them on the line plot. a. minimum b. maximum c. mode(s) d. median. a Find the sum of the 6 fractions. b Write the mean of the 6 fractions as a fraction. c Label the mean on the line plot. 5. Find the range of the data. _ 6 _ 6 7 X X X X X X X XX mean X X X X XXX min. median Math Journal, p max. mode Study Link 0 (Math Masters, p. 97) INDEPENDENT Home Connection Students solve pan-balance problems with one and two balances. 0 Math Boxes. Make a magnitude estimate for the product. Choose the best answer.,6 *.5 tenths. Circle the numbers below that are divisible by , Differentiation Options ones hundreds thousands ten-thousands Name the number for each point marked on the number line. READINESS Solving Logic Puzzles SMALL-GROUP 5 5 Min D A B F C E , or - _ -0., or - _ 0.5, or _.5, or _ -.9, or - 9_ 0.5, or _ 0 A = B = C = D = E = F = 57 To provide experience with deductive reasoning required to solve linear equations, have students solve logic puzzles. Write the following statements on the board or the Class Data Pad.. All K are M.. All M are B.. a. Which has the greater area, a 3 ft ft rectangle or a triangle with base 3 ft and height 5 ft? triangle b. Which has the greater area, a triangle with base 0 cm and height cm or a parallelogram with base 5 cm and height 6 cm? parallelogram Math Journal, p Find the volume of the prism. Fill in the circle next to the best answer. 5 ft Volume = length * width * height 5 ft A. 5 ft 3 B. 50 ft 3 C. 3 ft 3 D. 5 ft 3 3 ft 97 Lesson 0 795

6 Name STUDY LINK 0 Pan-Balance Problems In each figure below, the two pans are in perfect balance. Solve these One triangle weighs One pen weighs as much as 3 balls. as much as 3 paper clips. M weighs N weighs as much as 5 marbles. as much as 0 marbles. One weighs One weighs as much as s. as much as marbles. One cup of juice weighs One apple weighs as much as blocks. as much as 6 blocks. Practice Fill in the missing numbers to make true sentences. 6 Study Link Master M N 5 marbles N 0 marbles 5 marbles 6. (7 5) / 7. (( / 7) ) / 50. (( º 3) ) 6 9. (3 3) º ((3 / ) º 5) Math Masters, p Ask students to make a third statement that could be added about K and B. All K are B. Point out that to make the third statement, students had to put together the information from the first statements. Write the following equations next to the statements on the board or Class Data Pad.. K = M. M = B 3. K = B Ask students to substitute numbers for the letters so the sentences are still true, and write these next to the equations. Sample answers:. = 9. 9 = = Add the following pair of statements to the board or Class Data Pad.. 3 cubes and paper clip weigh as much as marbles.. paper clip weighs as much as marbles. Ask students to make a third statement that could be added about cubes and marbles. 3 cubes weigh as much as marbles, or cube weighs as much as marbles. Ask students to make another pair of statements that follow this pattern. Use students responses to clarify their reasoning approaches. Name 0 Teaching Master Measuring Franz buys two sandglasses from an antique dealer. However, when he gets home he realizes the sand in the sandglasses does not measure hour. The first sandglass measures a nine-minute interval, and the other sandglass measures a thirteen-minute interval. Franz wants to make a special cleaning solution to clean his new sandglasses. The solution needs to boil for 30 minutes. Can Franz use his sandglasses to measure 30 minutes from the time the solution starts to boil? Explain your solution by describing what Franz should do. When the solution starts to boil, set the 3-minute sandglass. When it runs out, 3 minutes will have passed. Set both the 3-minute and the 9-minute sandglasses. When the 9-minute sandglass runs out, 3 9 minutes will have passed. Start the 9-minute sandglass again. When the 3 minute sandglass runs out, 3 9 minutes will have passed. The 9-minute sandglass will have minutes of sand in its bottom. Start the 9-minute sandglass again. Now the minutes of sand are in the top. When the 9-minute sandglass runs out, 3 9 or 30 minutes will have passed. ENRICHMENT PARTNER Measuring (Math Masters, p. 9) 5 5 Min To further explore solving a problem with two unknowns, have students use a sandglass model to solve the problem on the Math Masters page. Students determine whether a sandglass with a 9-minute interval and one with a 3-minute interval can be used to time a 30-minute period. When partners have finished, discuss their solution strategies. One approach is to begin by setting the 3-minute sandglass. When it runs out, set the 3-minute and the 9-minute sandglasses at the same time. When the 9-minute sandglass runs out (3 + 9 minutes), start it again. After more minutes, the 3-minute sandglass will run out ( ), and there will be minutes in the bottom of the 9-minute sandglass. Turn the 9-minute sandglass again so that the minutes of sand on the bottom is now on the top. When this runs out, 30 minutes will have passed ( = 30 minutes). Math Masters, p Unit 0 Using Data; Algebra Concepts and Skills