Pan-Balance Equations, Part 2

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1 L E S S O N 84 Cumulative Overview Students cntinue t use the panbalance apprach t slve simple equatins. Mst f the weight prblems in this lessn are actually sets f tw equatins with tw unknwns and can be slved by substitutin. At first, the balance pans are shwn with pictures f bjects. Fllw-up examples put expressins with variables in the balance pans. Review; Pan-Balance Equatins, Part 2 TEACHER S REFERENCE MANUAL nne ADVANCE PREPARATION The balance demnstratins can be dne using a single pan balance, but they will be mre easily understd if yu use tw balances. Befre beginning the demnstratins, gather the fllwing items listed under Teacher Materials: paper clips, ballpint pens, cmpasses, half-circle prtractrs, a ruler, and r 2 pan balances. Prepare Teaching Aid 4 (the Cumulative Review Check-ff Sheet). Make ne cpy f Master 60 fr each student. TEACHER MATERIALS Teaching Aid 4 bx f 00 standard 4" paper clips 3 identical ballpint pens 3 identical cmpasses (pencils remved) 2 identical plastic 6" half-circle prtractrs plastic 2" ruler r 2 pan balances STUDENT MATERIALS Jurnal 2, pp Study Links 83 and 84 Master 60 ruler slate, chalk, eraser MATH MESSAGE One as 3 unces. weighs as much unces 23 unces Lessn 84 4

2 Master 60 MASTER Cumulative Review 60. Circle all the numbers belw that are greater than What is anther name fr 3 2? 3. Mrs. Jhnsn has 8 grandchildren a. Last week, Mrs. Jhnsn made $240 n a yard sale. She wants t give this mney t her grandchildren s that each gets the same amunt. Hw much will each child get? b. On their birthdays, Mrs. Jhnsn gives each f her grandchildren $2 fr their cllege savings accunts. Hw much des she spend n these birthday presents in a year? 4. Use yur ruler. a. Draw a line segment inches lng. b. Draw a line segment 2 2 inches lng. c. Which f the line segments that yu drew is clser in length t 8 centimeters? Nte Sample answers: 2,. $30 $200 (b) 2 2 in The ntatin used with variables ften causes students much difficulty. Fr example, in algebra, 3c means three times the value f c, r 3 º c. In the fllwing lessn, care is taken t help students realize that expressins like 3c and 3 º c are equivalent. Getting Started -Minute Math Read the fllwing number riddles. Yu may prefer t write them n the chalkbard r prepare an verhead transparency. What number am I? If yu duble me and add 6, yu get If yu take half f me and triple the result, yu get If yu add t me, yu get 0. Ç If yu duble me and duble the result, yu get 0. 0 If yu duble me, add 4, subtract 4, and divide by 2, yu get 9. 9 Independent written review (Master 60; Teaching Aid 4) Independent Activity Use Master 60 with, r smetime after, this lessn. Recrd students prgress n the Cumulative Review Check-ff Sheet (Teaching Aid 4). Prgress is being assessed in the fllwing areas: Fractins and decimals (Prblems and 2) Operatins: Multiplicatin and divisin number stries (Prblem 3) Measurement (Prblem 4) Study Link 83 Fllw-up Review answers as needed. Math Message Fllw-up It may help sme students if yu redraw the balance sketch as fllws, representing individual -unce weights with small s. Tell the class that is just anther way t write 42 Unit 8 Algebra Cncepts and Skills

3 Example balance # Add clips t balance the pans. Fr the example here, add 3 clips. Add clips t balance the pans. This might require 70 clips. 3 3 exchange remve 3 Instructin and Activities Demnstrate the pan-balance apprach. Whle-Class Activity Tw pan-balance examples are suggested belw. Mst students shuld catch n very quickly. Cntinue with similar examples if students appear t have difficulty. Example : Shw the prtractr, ruler, and paper clips. Tell students that their gal is t determine the weight f bth the ruler and the prtractr in terms f paper clips. Set up the first pan-balance. Place a prtractr in ne pan. Then add paper clips t the ther pan, ne at a time, cunting as yu g, until the pans balance. If yu have nly ne balance available, make a sketch f the results f this first pan-balance situatin n the chalkbard befre mving n t set up the secnd balance. Set up the secnd pan-balance. Place a prtractr and ruler in ne pan. Then add paper clips t the ther pan, ne at a time, cunting as yu g, until the pans balance. In this example, remind students that in the first pan balance, a prtractr weighs 3 clips. All that remains is t find the ruler s weight in clips. Ask students hw they might change the pans t islate the ruler in ne pan, balanced by paper clips in the ther pan. Tell them that there is nly ne rule t fllw: Whatever yu d, the pans must always remain balanced. Sme students may suggest the fllwing substitutin: Remve the prtractr and replace it with 3 clips. Since the prtractr weighs 3 clips, the pans remain balanced. Nw remve 3 clips frm each side. The pans remain balanced. S the ruler weighs 39 (70 3) clips. Other students may ntice a mre direct apprach: Remve the prtractr frm ne pan. At the same time, remve 3 clips frm the ther pan. The prtractr weighs 3 clips, s that s taking the same frm each side. The ruler remains, balanced by 39 clips. 70 remve 3 Lessn 84 43

4 Example 2 balance # Add clips t balance the pans. This might require 20 clips. Add clips t balance the pans. This might require 42 clips. Example 2: Shw the ballpint pens, cmpasses, and paper clips. Tell students that their gal is t determine the weight f bth a pen and a cmpass in terms f paper clips. Set up the first pan balance. Place 2 pens in ne pan. Then add clips t the ther pan, ne at a time, cunting as yu g, until the pans balance. Set up the secnd pan balance. Place 2 cmpasses in ne pan and cmpass and pen in the secnd pan. Then add clips t the secnd pan, ne at a time, cunting as yu g, until the pans balance. Have students test their slutins by actually using the balances. If necessary, guide students thrugh the fllwing manipulatins n the balance pans: balance # Remve ne-half f the bjects frm each pan. The pans remain balanced. S pen weighs 0 clips. Mre Pan-Balance Equatins Jurnal 2, p. 34 exchange 42 0 Slve these prblems, using bth pan balances in each prblem. In each prblem, the pans are in perfect balance. The weights f bjects, such as blcks, balls, marbles, and cins, may be different frm prblem t prblem but are cnsistent within each prblem. On, remve pen and replace it with 0 clips. The pans remain balanced.. One blck weighs One ball weighs One blck weighs One cin weighs 3 7 Remve cmpass frm each pan. The pans remain balanced. S cmpass weighs 2 clips. 3. One blck weighs One ball weighs One blck weighs One ball weighs as much as 9 marbles. as much as 3 marbles. 44 Unit 8 Algebra Cncepts and Skills

5 Jurnal 2, p. 346 Mre Pan-Balance Equatins (cntinued) Slve these pan-balance prblems. In each figure, the tw pans are in perfect balance.. One cin weighs One blck weighs as much as 2 clips. as much as 4 clips. 2 dughnut 6. One can weighs One dughnut weighs as much as 2 blcks. as much as 2 blcks. marbles 9 marbles 7. One weighs One weighs as much as marbles. as much as 4 marbles. B B 8 unces B A A A B B B 8. Each can weighs B unces. Each cube weighs A unces. B = 9 unces A = 3 unces B Jurnal 2, p. 347 Mre Pan-Balance Equatins (cntinued) Slve these pan-balance prblems. In each figure, the tw pans are in perfect balance marbles 9. One weighs One weighs as much as 4 marbles. as much as 6 marbles. x y 7 marbles 4 x y 8 marbles y 0. y weighs x weighs as much as marbles. as much as 2 marbles. cup empty cup 2 full. If the cup is full, the cffee in the cup weighs as much as marbles. If the cup is full, the cffee plus the cup weighs as much as 26 marbles. 2. Tw pens weigh as much as ne cmpass. One pen and ne cmpass tgether weigh 4 grams. 8 Slve pan-balance equatins. (Jurnal 2, pp ) Partner Activity Have students wrk in pairs t cmplete jurnal pages Sme f the jurnal page prblems use pictures f bjects in the balance pans. Sme use placehlders, such as and ; and sme use expressins with variables. D Prblems and 2 n jurnal page 34 with the class. Explain that each prblem cnsists f tw related parts and that students need t slve ne f the parts befre they have enugh infrmatin t slve the ther part. Fr example, in bth Prblems and 2, students need t cmplete the statement assciated with the secnd pan balance befre they can cmplete the statement assciated with the first balance. Slutin t Prblem : Cmplete the secnd statement first: One ball weighs as much as marbles. T cmplete the first statement, yu can reasn this way: Since ball weighs as much as marbles, 4 balls weigh as much as 20 marbles. Therefre, 2 blcks weigh as much as 20 marbles, and blck weighs as much as 0 marbles. Slutin t Prblem 2: Cmplete the secnd statement first: One cin weighs as much as 7 marbles. T cmplete the first statement, yu can reasn this way: Since cin weighs as much as 7 marbles and blck and cin weigh as much as 20 marbles, blck must weigh as much as 3 marbles (remve the cin frm the left side f the first pan and 7 marbles frm the right side f the first pan). Cautin students that they will have t figure ut which statement t cmplete first either the first r the secnd statement. (Fr example, in Prblem 4, students need t cmplete the first statement first in rder t cmplete the secnd statement.) If students are experiencing difficulty answering Prblem n jurnal page 347, encurage them t determine the weight f the cffee separate frm the cup. Cmplete the pan-balance sketches belw. Find the weight f ne pen and f ne cmpass. One pen weighs grams. One cmpass weighs 30 grams. Lessn 84 4

6 Math Bxes 84. Insert < r. a b. c d e. 4 < 3. Estimate the answer fr each prblem. Then slve each prblem. Estimate Slutin a. 603 º 7 b. 29 º 802 c. 48 Æ 202 d. 397 Æ ,000 24,000 0, ,000 42,83 23, ,309 e. 22 º 0,000 2,00 Jurnal 2, p a. Circle any triangles that lk like equilateral triangles. b. Write a definitin f an equilateral triangle. Sample answer: A triangle with three sides f the same length. Practice and Extensin Math Bxes 84 (Jurnal 2, p. 348) Math Bxes in this lessn are paired with Math Bxes in Lessn 86. The skill in Prblem is a prerequisite fr Unit 9. Study Link 84 Students cntinue t use tw pan balances t slve prblems. They als slve prblems invlving number sentences cntaining parentheses. Study Link 84 Study Link 84: Mre Pan-Balance Prblems. In each figure belw, the tw pans are in perfect balance. Slve these pan-balance prblems: 4. a. Make a stem-and-leaf plt f the fllwing numbers: 20,, 37, 44, 2, 20, 9, 87, 20, 0, 3, 90, 86, 43, 9 b. Find the landmarks. Mde: 20 Median: 20 Range: 8 Stems (0s) Leaves (s) M N marbles 2 N 20 marbles a. M weighs N weighs 8 marbles b. One weighs One weighs 2 as much as. as much as marbles c. One can f juice weighs One apple weighs 4 as much as blcks. as much as blcks True r false? a. ( + 6) º 3 = 63 T b. 30 = ((9 + 7) Ç ) º 2 c. 38 = 2 + ((8 º 6) Ç 0) d. 34 º (2 + 26) = 94 F T F 3. Fill in the missing numbers t make true sentences. a. 26 = (7 + 4)/2 b. ((28/7) + 2)/8 = 2 c. ((4 º 3) + 4) Ç 6 = 0 d. 0 = (3 Ç 3) º ((34/2) º ) 46 Unit 8 Algebra Cncepts and Skills

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