Florida ged plus college Preparation Program Curriculum and Resource Guide

Objective 1 – Problem Solving

Strategy – Use Multiple Steps to Solve Problems

• 
  Read the problem carefully, paying attention to detail
  
• 
  Define the type of answer that is required and eliminate extraneous information
  
• 
  Identify key words that will assist them in choosing the correct operation or in the case of multi-step problems, the correct operations
  
• 
  Use a graphic organizer
  
• 
  Set up the problem correctly and remember the order of operations
  
• 
  Use mental math and estimation skills
  
• 
  Check the answer for reasonableness
  
• 
  Use a calculator accurately and always double-check the answers
  

The following information can be used to help students understand the problem solving process and gain confidence in their own ability to solve math problems.

The first mistake students make when trying to solve a problem is to not read carefully. They breeze through the problem and think they know what the problem is asking them to do. Unfortunately, they usually decide the wrong thing. Reading comprehension skills are very important in math. The following graphic organizer can help students focus their attention on what the problem is really asking them to do. Have students use the organizer on different types of problems. As they become more comfortable with the process, they will no longer need the organizer but rather will think through the process in a logical manner. The graphic organizer forces students to look at the details of the problems and also helps them work through the information in an organized way.

One of the most difficult tasks for students is to determine what the question is asking them to do. Students should start by asking themselves: “What is the exact question I need to answer?” The second step is to see what information is provided and what the student needs to know in order to answer the question. Step three is to eliminate all the extraneous information that is included in the problem.

Look at the following example.

Harry Potter has asked his friend Hermione for a potion to turn them and their friend Ron into birds. They need to make a trip to Diagon Alley as birds because the flying car is in for repairs.

Diagon Alley is 9 miles away and a dose of the potion lasts 50 minutes. They only have enough potion for one dose each. If they can go 24 miles an hour as birds, and they start at 4:30 p.m. , can they get to Diagon Alley and back to Hogwarts again before the potion runs out at 5:20 p.m.? If so, how much time will they be able to spend in the Alley?

The next thing is to figure out what information you already have and what you need to know to answer the question(s).

1. 
   As birds, Harry and Ron can travel 24 miles in one hour.
   
2. 
   They want to go 9 miles twice in 50 minutes.
   

1. 
   They can leave at 4:30 p.m.
   
2. 
   They must return at 5:20 p.m.
   

1. 
   Can Harry and Ron get to Diagon Alley and back in 50 minutes?
   
2. 
   How long can they stay in Diagon Alley
   

In this case, the question that has to be answered is how much time they will be able to spend in the alley. Since it is a time question, the answer will need to be expressed in hours or minutes. Following these simple and organized steps, students can clearly identify the question they need to answer in any problem and eliminate extraneous information that can often result in a wrong answer.

Problem adapted from Ask Dr. Math: FAQ, Word Problems. Retrieved from the World Wide Web on February 4, 2006 at: http://www.mathforum.org/dr.math/faq/faw.word.problems.html.

Strategy – Identify Key Words

Math Term	Definition (In your own words)	Your Own Example
Example: Variable	A letter that stands for a number in an equation. It is a variable because its value can change.	6n x 2 n is the variable

The following is a list of commonly used math terms that will be helpful to students as they work with word problems.

Common Words that Indicate Math Operations
Addition	Subtraction	Multiplication	Division	Equals
Increased by More than Combined Together Total of Sum Added to	Decreased by Minus Less Difference Between/of Less than Fewer than	Times Multiplied by Product of Increased/decreased by a factor of (this is both addition/ subtraction and multiplication)	Of (as in half of) Times Multiplied by Product of Increased or decreased by a factor of (this is both addition/ subtraction and multiplication)	Is, are Was Were Will be Gives Yields Sold for

Basic Geometry Vocabulary Words
Acute Angle Area Center Chord Circumference Complementary Congruent Cube	Cube Cylinder Degree Equilateral Geometric Height Hexagon Intersect Isosceles	Length Obtuse Parallel Pentagon Perimeter Perpendicular Plane Plot Polygon	Pyramid Quadrilateral Radius Rectangle Rhombus Segment Scalene Sphere	Supplementary Symmetrical Triangle Trapezoid Vertex Volume Width

Selected Mathematic Vocabulary Words
Absolute value Approximately Base Commutative property Comparison Complex Convert Determine Diagram	Distributive property Equality Error of measurement Exponent Expression Face value Finite Formula	Frequency Generalization Graph Gross Imply Inequality Inference Interest rate Irrational Number Linear	Metric system Midpoint Mixed number Multiplier Notation Ordered pair Percentage Percentiles Permutation Probability	Quadrant Ratio Rational number Scientific notation Simplify Slope Square root Tally Underestimate Venn diagram

Strategy – Geometry Concentration

One way to assist students with using and applying vocabulary is to create games such as concentration, jeopardy, or matching activities. The following is an example of a geometry matching activity. You may wish to include additional terms.

Term	Definition
Right angle	A 90 degree angle.
Equilateral triangle	A triangle with all sides equal.
Scalene triangle	A triangle having three unequal sides and angles.
Vertex	The intersection of two sides.
Right triangle	A triangle with one internal angle equal to 90 degrees.
Pentagon	A polygon with 5 sides and 5 angles.
Square	A rectangle having all four sides of equal length.
Intersecting lines	Lines that cross each other.
Perpendicular lines	Two lines that cross each other to form a 90 degree angle.
Acute angle	An angle less than 90 degrees.
Chord	The line segment between two points on a given curve.
Radius	A straight line extending from the center of a circle to the surface.
Line segment	One part of a line.
Line	A continuous extent of length.
Point	A position in space.
Rectangle	A quadrilateral with opposite sides parallel.
Circle	A closed plane.
Hexagon	A six-sided figure.
Obtuse angle	An angle greater than 90 degrees.
Congruent angles	Two angles that have the same measure.
Bisect	To separate an angle into two congruent angles.
Degrees	The unit that angles are measured.
Isosceles triangle	A triangle with at least two sides congruent.
Similar	Figures that have the same shape, but not necessarily the same size.
Parallelogram	A quadrilateral with both pairs of opposite sides parallel.
Trapezoid	A quadrilateral with exactly one pair of opposite sides parallel.

Strategy – Math Translation Guide

The chart below gives you some of the terms that appear in many word problems. Use them in order to translate or “set-up” word problems into equations.

English	Math	Example	Translation
What, a number	x, n, etc.	Three more than a number is 8.	N + 3 = 8
Equals, is, was, has, costs	=	Danny is 16 years old. A CD costs 15 dollars.	d = 16 c = 15
Is greater than Is less than At least, minimum At most, maximum	> <  	Jenny has more money than Ben. Ashley’s age is less than Nick’s. There are at least 30 questions on the test. Sam can invite a maximum of 15 people to his party.	j > b a < n t  30 s  15
More, more than, greater, than, added to, total, sum, increased by, together	+	Kecia has 2 more video games than John. Kecia and John have a total of 11 video games.	k = j + 2 k + j = 11
Less than, smaller than, decreased by, difference, fewer	-	Jason has 3 fewer CDs than Carson. The difference between Jenny’s and Ben’s savings is $75.	j = c – 3 j – b = 75
Of, times, product of, twice, double, triple, half of, quarter of	x	Emma has twice as many books as Justin. Justin has half as many books as Emma.	e = 2 x j or e = 2j j = c x ½ or j = e/2
Divided by, per, for, out of, ratio of __ to __		Sophia has $1 for every $2 Daniel has. The ratio of Daniel’s savings to Sophia’s savings is 2 to 1.	s = d  2 or s = d/2 d/s = 2/1

Example 1
Jennifer has 10 fewer DVDs than Brad.
Step 1: j (has) = b (fewer) – 10

Remember, the word “has” is an equal sign and the word “fewer” is a minus sign, so:

Step 2: j = b – 10
Example 2
Clay got 10 fewer votes than Kimberly. Reuben got three times as many votes as Clay. The three contestants received a total of 90 votes. Write an equation in one variable that can be used to solve for the number of votes Kimberly received.
Step 1: Pick which unknown will be represented by the variable. Since you’re solving for Kimberly, let k be the number of votes Kimberly received.

Step 2: Represent the other two unknowns in terms of k. Clay got 10 fewer votes so it’s k - 10 and Reuben got three times that so it’s 3(k - 10).

Step 3: Set up the equation using all of the expressions to equal 90.

k + (k - 10) + 3(k - 10) = 90
Example 3
A school is having a special event to honor successful alumni. The event will cost $500, plus an additional $85 for each alum who is honored. Write an equation that best represents the number of alumni that can be honored.
Step 1: The amount the school can spend is equal to or less than $1,000, so it’s  1,000.

Step 2: The event has a fixed cost of $500 and a variable of $85 per alum so it’s 500 + 85a.

Step 3: The equation then becomes 500 + 85a  1,000.
Example 4
A computer repair company charges $50 for a service call plus $25 for each hour of work. Write an equation that represents the relationship between the bill, b, for a service call, and the number of hours spent on the call, h.
Step 1: Some questions include a situation where there is more than one cost. One of them is fixed and one is variable. First identify the sum of the fixed and variable costs so b equals the total.

Step 2: Next, identify the fixed cost of 50 and the variable cost of 25h (25 x the number of hours).

Step 3: The equation then becomes 50 + 25h = b.

Strategy – Use a Graphic Organizer

Graphic organizers are commonly used in reading and writing. However, they are rarely found in mathematics. The following graphic organizer was developed by the Texas Center for Adult Literacy and Learning (TCALL) and is part of their adult education toolkit at: http://www.tcall.tamu.edu/toolkit/CONTENTS.HTM. The original organizer has been modified at the request of adult education practitioners, in order to incorporate additional elements that may assist students in the problem-solving process.

Modified Word Problem Graphic Organizer

Main Idea (in your own words)
Question	Draw a Picture/Graph/Table
Pertinent Facts	Irrelevant Information
Relationship Sentence (no numbers)
Equation (number sentence)
Estimation (without computing)
Computation
Answer sentence

Texas Center for Adult Literacy and Learning, The Adult Basic Education Teacher’s Toolkit,

Word Problem Graphic Organizer

Main Idea (in your own words)
Question	Draw a Picture/Graph/Table
Pertinent Facts	Irrelevant Information
Relationship Sentence (no numbers)
Equation (number sentence)
Estimation (without computing)
Computation
Answer Sentence

Set Up the Problem Correctly and Remember the Order of Operations

Many students have a rough idea of how to solve a problem but then fail to set it up correctly. One area that causes a lot of concern for students is the order of operations. Students often forget the correct order in which an operation should be calculated, thus ending up with the wrong number. Take a look at the problem below:

3 + 2 x 5 = ?
Many students will work from left to right. 3 + 2 = 5 and then 5 x 5 = 25.

The correct order is 2 x 5 = 10, 10 + 3 = 13

Use the mnemonic – Please Excuse My Dear Aunt Sally to help students remember the correct order of operation.

• 
  Please – complete all operations within parentheses first
  
• 
  Excuse – next take care of any exponents that may be present
  
• 
  My Dear – complete all multiplication and division, working from left to right, before moving on to the last operations
  
• 
  Aunt Sally – lastly, perform all addition and subtraction working from left to right and you are done.
  

The Casio fx 260 automatically applies the correct order of operations. Students should be aware that they may use a calculator on Part I of the GED Test, but not in Part II. On Part II, they must remember the correct order without prompting.

Use Mental Math and Estimation Skills

Students should be able to look at the answer to a problem and use their mental math or estimation skills to determine if the answer is reasonable. Estimation skills can also be used to eliminate certain answers from the multiple-choice selections. Spend time during each math period working on mental math and estimation skills. This will help students gain confidence in their math ability.

Check the Answer for Reasonableness

Students often fail to check and see if the answer they provide for a problem is reasonable. Although it only takes a few seconds to go back and determine if the answer is in the correct unit of measure (minutes, seconds, hours, pounds, ounces, etc.), students often fail to take the time to re-check their answers. Set aside some time before the end of the class period just for checking work. This will help students get in the habit of doing so. Provide problems with answers that use the wrong units and have students find the errors. Reinforce with students the importance of checking for reasonableness.

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