Promoting Algebraic Thinking Guess and Check Sample Problems

Transcription

1 This is meant to accompany the audiobook on a mathematics problem solving strategy, and Check. It is meant for a teacher who has some experience teaching problem solving and wants to review the strategy before presenting it to a class. The eight problems in the handout should be a start. Students need problem solving on an almost daily basis, and the guess and check strategy needs to be visited more than eight times. Partial List of Problem Solving Strategies and Check Draw a Picture Look for a Pattern Build/Use a Model Make an Organized/Systematic List Make a Chart Work Backwards Change Your Point of Reference Use Friendly/Smaller Numbers Solve a Simpler Related Problem Act It Out Eliminate Possibilities 1

2 Promoting Algebraic Thinking and Check Sample Problems 1. A fishing pole is broken into two pieces. One piece is 26 inches shorter than the other. The fishing pole is sixty inches long. How long is each piece? 2. The sum of two numbers is 111. The larger exceeds the smaller by 17. What are the two numbers? 3. Find two consecutive even numbers whose sum is The product of Alvin s and Buzzy s age equals 225. Buzzy is 40 years older than Alvin. What are their ages? 5. A pair of trousers and a shirt cost $ The trousers are twice as expensive as the shirt. How much is each item? 6. A cable 84 meters long is cut into two pieces so that one piece is 18 meters longer than the other. Find the length of each piece of cable. 7. The total cost of a chair, a desk and a lamp is $562. The desk costs four times as much as the lamp and the chair costs $23 less than the desk. Find the cost of the chair and the desk. 8. Janelle has only quarters and dimes in her bank. She has 70 coins which are worth $ How many of each coin does she have? Make sure you: Show all work Have a chart for each problem. Have a sentence for each answer. Write an equation. 2

3 and Check Instructions: Problem 1: A fishing pole is broken into two pieces. One piece is 26 inches shorter than the other. The fishing pole is sixty inches long. How long is each piece? Step 1: Start with a Step 2: Identify the length of shorter piece Step 3: Use the to obtain information about the problem length of 2nd piece = Total length = 60 shorter piece 1st piece Step 4: Evaluate the (CHECK) length of 2nd piece = Total length = 60 Check shorter piece 1st piece Too Small Step 5: Keep Revising the and Check it until you succeed length of 2nd piece = shorter piece 1st piece + 26 Total length = 60 Check Too Small Too Large Too Large Too Small BINGO Step 6: Write a sentence that answers the question. 3

4 One piece is 17 inches and the other is 43 inches. Step 7: Write (not solve) an equation (G 4 and above) One possibility: length of shorter piece 2nd piece = 1st piece + 26 Total length = 60 Check Too Small Too Large Too Large Too Small BINGO x y x + y = 60 Another possibility: length of shorter piece 2nd piece = 1st piece + 26 Total Cards Should = 75 Check Too Small Too Large Too Large Too Small BINGO x x + 26 x+ x+26 = 60 or 2x + 26 = 60 Some things to ponder: Every chart should have at least 3 entries: one too big, one too little and one just right. Of course, the charts can have more than three entries. The idea is to encourage number sense and looking for patterns. Even if a child gets it right quickly, he/she can still make more entries. Some and Check Charts are far more complex. They might need many more columns It s a good bridge to starting algebraic representations 4

5 Problem 4: The product of Alvin s and Buzzy s age equals 225. Buzzy is 40 years older than Alvin. What are their ages? Step 1: Start with a Step 2: Identify the Alvin s age Step 3: Use the to obtain information about the problem Buzzy s age = Product = Step 4: Evaluate the (CHECK) Buzzy s age = Product = 225 Check Too large Step 5: Keep Revising the and Check it until you succeed Buzzy s age = Product = 225 Check Too large Too small BINGO Step 6: Write a sentence that answers the question. "

6 Alvin is 5 years old and Buzzy is 45 years old. Step 7: Write (not solve) an equation (G 4 and above) One possibility: Buzzy s age = Product = 225 Check Too large Too small BINGO x y x y = 225 Another possibility: Buzzy s age = Product = 225 Check Too large Too small BINGO x x + 40 x (x+40) = 225 or x x = 225 Some things to ponder: 300 Algebraic solution: x x = 225 x x 225 = 0 (x + 45) (x 5) = 0 x x = - 45 and x = y #

7 Problem 7: The total cost of a chair, a desk and a lamp is $562. The desk costs four times as much as the lamp and the chair costs $23 less than the desk. Find the cost of the chair and the desk. Step 1: Start with a Step 2: Identify the Lamp Step 3: Use the to obtain information about the problem Lamp Desk = 4 Lamp Chair = Desk Step 4: Evaluate the (Check) Desk = Chair = Lamp 4 Lamp Desk - 23 Total = 562 Check Too small Step 5: Keep Revising the and #$%& it until you succeed Desk = Chair = Lamp 4 Lamp Desk - 23 Total = 562 Check Too small Too large BINGO Step 6: Write a sentence that answers the question. "

8 The chair costs $237 and the desk costs $260. Step 7: Write (not solve) an equation (G 4 and above) One possibility: Desk = Chair = Lamp 4 Lamp Desk - 23 Total = 562 Check Too small Too large BINGO x y z x+y+z=562 Another possibility: Desk = Chair = Lamp 4 Lamp Desk - 23 Total = 562 Check Too small Too large BINGO x 4x 4x-23 9x-23=562 Some things to ponder: What other problem solving strategies were used? Not all guess and check charts are four columns. Why didn t the lamp get mentioned in the sentence above? If you begin with the chair, would the equation be different? (

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