Using Algebra: UNDERSTANDING WORD PROBLEMS: The Basics

Every day there are problems to solve and that is the purpose of algebra.

Thinking about the way you think *Planning Monitoring Evaluating

METACOGNITION THEORY Thinking about your thinking strategy. What is the plan to solve the problem? PLANNING – consists of understanding what the problem wants (implied by language or stated as facts) MONITORING – putting the steps in order that make sense EVALUATING – knowing if the answer is reasonable

HOW TO SOLVE WORD PROBLEMS STEP 1: Read the problem 3 times. STEP 2: Mark out information that is not needed. STEP 3: Design a script (Questions) What is the problem asking me? What is the problem telling me that is useful? (Get information you need to solve problem and delete what you don’t need.) What is the problem implying? Look for unasked questions. Is there a sequence or procedure to follow? STEP 4: What is the variable representing? Choose a letter x or n to represent what the problem is asking for Mark out any words are phrases represented by x

HOW TO SOLVE WORD PROBLEMS The Basics STEP 5: What is implied? Are there any unasked questions? Look for words are phrase that have meaning not stated as facts translate into an expression in math STEP 6: Write a sentence in English and translate to an equation & solve. Begin with writing a sentence in English then translate to an equation Use the English-Math Dictionary Watch for specific relationships STEP 7: Solve the equation. Use the solving technique appropriate for the equation STEP 8: Does the answer make sense? Is it reasonable? Go back to the script; reread your question, ask “did you answer the question” Don’t forget to check your work.

PROCEDURES FOR SCRIPTING QUESTIONS: PROCEDURES: 1. Write down your questions. 2. Review your script for possible missing questions –implied but not stated. 3. Develop a plan: decide what questions are helping you and which ones you need to change 4. Monitor and adjust Possible Questions: What is the question I am to find? Are there other questions that must be answered first? Are facts given or implied? What are the variables? Is there a special way to write them? Is there a formula or specific model? What learning technique, strategy or procedure do I need? Can I write an equation that mirrors my procedure to solve? Does the answer make sense in the real world?

Putting the Scripting Procedure to Work Read - Think - Write Questions / Equation - Solve

STEP 1: Read the problem 3 times PROBLEM: Price of an iPod. Kyle paid $120 for an iPod during a 20%-off sale. What was the regular price? STEP 2: Write a script to follow.  What is the problem asking me? Find the retail price  What is the problem telling me that is useful? You paid $120 on sale  What is the problem implying? You paid 80% of the regular price STEP 3: Can you draw a simple picture, and/or make a table of the problem, or use a formula. Make it more real or clarify. Some problems cannot be drawn but may have a formula. %paid / 100 % = part paid / original price HOW TO SOLVE WORD PROBLEMS

STEP 4: What is the variable x representing? X= the regular price of the I pod STEP 5: What is implied by “a 20% off sale”? 20% off means you pay 80% of regular price mark out 20% off sale and write in pd 80% STEP 6: Write a sentence and then an equation with remaining information. 80% of x = $120 paid or 80/100 (x) = 120 STEP 7: Solve the equation. 80/100 (x)= x = 120*100 x = 12000/80 or 150 dollars STEP 8: Does the answer make sense? Is it reasonable? $150 *80/100 = or $150 *.80 or 120 Did you check the answer? yes

STEP 1: Read the problem 3 times. Longest marriage. As half of the world’s longest-married couple, the woman was 2 yr younger than her husband. Together, their ages totaled 204 yr. How old were the man and the woman? STEP 2: Write a script to follow.  What is the problem asking me? How old is the man & the woman  What is the problem telling me that is useful? Woman’s age is 2yr. younger than the man  What is the problem implying? Younger implies less. STEP 3: Can you draw a simple picture, and/or make a table of the problem, or use a formula. Make it more real or clarify. Some problems cannot be drawn

STEP 4: What is the variable x representing? X= Man’s age STEP 5: What is implied by “2 years younger”? Implied that woman’s age is 2 years less than his or x - 2 STEP 6: Write an equation with remaining information. Man’s age + Woman’s age = 204 X + X-2 = 204 STEP 7: Solve the equation. 2X -2 =204 2X = or 206 X = 206/2 or 103 STEP 8: Does the answer make sense? Is it reasonable? If the man is 103 years, is that possible? If the woman is 2 years younger then she must be 101? Did you check the answer?

Special problem types for percent problems Sentence to Equation Model Proportion Model

STEP 1: Read the problem 3 times. Percent of tip: John is paying for dinner for Hannah’s birthday party and notices that a tip or gratuity of $21.00 has been included with the total bill of $ What percent of the dinner bill is the tip? STEP 2: Write a script to follow.  What is the problem asking me? To find the % of the tip charged  What is the problem telling me that is useful? The tip was $21 and the total bill with the tip was $161.  What is the problem implying? That the $161 is the total bill + the tip or 100 % + X% STEP 3: Can you draw a simple picture, and/or make a table of the problem, or use a formula. Make it more real or clarify. Some problems cannot be drawn Bill : all dinners+ Gratuity = $161 x% of dinners= $21

STEP 4: What is the variable x representing? X= tip percent STEP 5: What is implied by “a tip or gratuity of $21.00 has been included with the total bill of $161.00”? Implied $161 includes total of dinners + tip or $21 therefore, $ = $140 or cost of dinners STEP 6: Write an equation with remaining information. x % of $140 or part that is dip + $140 or cost of dinners = $161or total STEP 7: Solve the equation. (x/100 ) * = 161 (x/100) * 140 = 21 x/100 =21/140 or 3/20 x = 100 * 3 / 20 or 300/20 or 15 percent STEP 8: Does the answer make sense? Is it reasonable? 15 to 20 % of a bill is considered appropriate for a tip Did you check the answer? 15% of 140 = =161

Using the Proportion mode to solve: STEP 4: What is the variable x representing? X= tip percent STEP 5: What is implied by “a tip or gratuity of $21.00 has been included with the total bill of $161.00”? Implied $161 includes total of dinners + tip or $21 therefore, $ = $140 or cost of dinners STEP 6: Write an equation with remaining information. x/100 = 21/140 Model is %/100 = part/ total STEP 7: Solve the equation. x * 140 = 21 * 100 x = 2100/140 x = 30/2 or 15 percent STEP 8: Does the answer make sense? Is it reasonable? 15 to 20 % of a bill is considered appropriate for a tip Did you check the answer? 15% of 140 = =161

Some number problems have special ways to write variables Consecutive numbers Consecutive even or odd

STEP 1: Read the problem 3 times PROBLEM: The sum of three consecutive even numbers is 78. What are the numbers? STEP 2: Write a script to follow.  What is the problem asking me? Find 3 consecutive even numbers  What is the problem telling me that is useful? They add to 78  What is the problem implying? The numbers are 2 apart like 4, 6, and 8 STEP 3: Can you draw a simple picture, and/or make a table of the problem, or use a formula. no STEP 4: What is the variable x representing? X = the smallest or first one STEP 5: What is implied by “three consecutive even numbers”? since they are two apart, the next would be x + 2 and then x + 4 STEP 6: Write an equation. X + X+2 + X+4 = 78 STEP 7: Solve the equation. 3X + 6 = 78 3X = 72, so X = 72/3 or 24 STEP 8: Does the answer make sense? Is it reasonable? Yes but they want to know all 3 numbers. So then x +2 = 26 and x + 4 = 28 and adding =78

Solving Word Problems in Geometry Do you need a formula? Pythagorean theorem, Area, volume, perimeter, circumference Is there angles to be classified? vertical angles, alternate interior, exterior angles Is there a shape to be recognized? rhombus, kite, circle, similar or congruent

PROBLEM SOLVING IN GEOMETRY Is there a formula needed? A ramp 12 feet long is leaning against a raised platform which is 5 feet above the ground. What is the distance from the ramp's contact point with the ground and the base of the platform? Write a script for this problem after reading it 3 times. Is there a formula, angles or figure to be used?

Line AB is parallel to line CD. What is the sum of the measure of angle k and the measure of angle y? Angles: 1. Alternate interior angles are K and Z 2. Supplementary angles are Y and Z Therefore : implied that y + Z = 180 degrees and K =Z so the conclusion is that K and Y are 180 degrees when added