Strategies
My first strategy requires that the students learn basic math terminology. So I will begin my lesson by going over the language of math. I think that there are too many assumptions by math teachers. It would be unwise of me to assume that everyone has been introduced to linear equations and the language associated with them. That is why I think is important to clarify the obvious sometimes, I will go over all the various ways to describe the operations we will be using to solve our problems. I look at this lesson as one of those "stitch in time, saves nine" ideas. It will also send a very powerful message to the students and that is that there is no question that is to simple to ask. This quick review will make it easier for my students to understand the questions being asked in word problems. I will increase my students ability to recognize the mathematical operation by increasing their familiarity with the language.
Students will list all the ways they know of to say add. The list should include the mathematical terms along with a mathematical phrase to show understanding. The first mathematical term on the list will be "plus", and the mathematical phrase will be something like "three plus two". The mathematical term "increased by" will have a mathematical phrase something like "the number is increased by 7". The mathematical term "more than" can be accompanied with the mathematical phrase "Thomas has 8 more than John". The mathematical term "combined" may be accompanied by the mathematical phrase "Trish and Jes combined their money". The mathematical term "together" and a mathematical phrase such as "together we have 7 books" will demonstrate understanding, as would the mathematical term " total" combined with the mathematical phrase "the total is 14". The list should also include the mathematical term "sum" with a mathematical phrase "the sum is". The mathematical term "added to" along with the mathematical phrase "five added to nine" will give us a rather complete list of mathematical term/phases to helps students recognize addition in a word problem. Students will be encouraged to add to the list at any time in the future we encounter another example.
Student will make a second list that connects the mathematical terms for subtraction to a mathematical phrase that demonstrate understanding of both concepts. The mathematical term "minus" along with the mathematical phrase "three minus nine" demonstrates understanding. The mathematical term "decreased by" and the mathematical phrase "The class was decreased by 14 leaving only 31 remaining" also demonstrates understanding. The mathematical term "less" and the mathematical phrase "my pay is less than yours" demonstrates not only understanding but disappointment. Finally the mathematical terms "fewer than" and "take away" when paired with the mathematical phrases "there are fewer flowers than plants" and "if I take away 7 from the original group I will only have 91 remaining", should give the students a beginning list of terms for subtraction.
Students will make a third list once again connecting mathematical terms with mathematical phrases but this time we will be covering the mathematical operation of multiplication. The mathematical term "times" and the mathematical phrase "two times" demonstrates understanding. Pairing the mathematical term "multiplied by" with the mathematical phase "multiply three by" along with the mathematical terms "double, triple and quadruple and the mathematical phrases "twice as many, three times as many and four times as many" will also show understanding. The final and often the most overlooked or misunderstood mathematical term "of" will be strongly emphasized because of its importance in fractions. If my students can understand that 1/2 of 1/2 is 1/2 times 1/2, or 1/4, and understand the math that they need to arrive at the answer, then they will have a much easier time in the future when we are doing a/b. Once again, this list may be added as the lesson progresses, but for now I think we have enough information to continue on with the final operation and that is division. We will use division to "undo" multiplication because they are inverses of each other so this connection must be noted here.
Just like before, students will compile a list of a the mathematical terms that define division and connect them with a mathematical phrase to demonstrate understanding. The mathematical term "quotient" and the mathematical phrase "the quotient of ten and five" demonstrates understanding as does the mathematical terms "shared/split" and the mathematical phrase "we split or shared the pizza".The mathematical term "percent" can be represented by the mathematical phrase "piece of 100". This list will be a good start for students understanding of language and can be added to as the lesson progresses.
The next tool or strategy I will introduce is the power of a sketch. This is second only to understanding the vocabulary in its importance as a tool for solving math problems. When you were in first grade you were given visuals to help you understand the idea of addition and subtraction. It helped you then, why wouldn't it help now? The drawing or sketch can be as simple as the ones you drew in first grade too. These visuals will give us an image of the problem. I believe that we have learned all of the basic math skills we need (addition, subtraction, multiplication, division) by the time we are in third grade. So when in doubt go back to basics. I will use simple line sketches to demonstrate something like, Kristen's book and Paula's book together are 36 books, or ____ + ____ = 36. The same simplistic idea can be used for many problems and students can use this to scaffold information as they gather it. The idea is to start with a simple true representation of the problem. When I learn more about who has more books or if the books are evenly divided then I can alter my sketch.
My third strategy is to help students to identifying the variable, and also stress the importance of clarifying what the variable really represents. We will become the detectives in the story and eliminate the suspects. This narrowing down of information will ultimately make our job easier so that we can "solve the case". To get my students to do this we will begin by identifying what they do not know. I know this seems backwards but just trust me on this. The usual response will be something like, "I don't know anything about the second side," or "I don't know how many apples there are." Great, now we all know that in algebra we call the unknown X. Once they have identified the unknown they have their entry into the problem and we can begin scaffolding. If I know that the first side is two times longer than the second side, or I have twice as many oranges as apples - all I have to do is take the second side or the number of apples (which we are calling X ) and double it. This we will call 2X. Using this process, my students will have an easy way to approach the problem and be less likely to give in to the knee jerk reaction of just making no attempt.
I will give my students one final tool - success! I know as adults we find it hard to believe that students would not want to see if their answer is correct, but trust me, students are sometimes just so happy to have come up with any answer, that they gladly accept it, right or wrong. Let us have a little sympathy for them. Instead of getting upset with them, show them that checking answers is like getting paid at the end of a long work week. The satisfaction in getting the correct answer and knowing it is correct may not be as rewarding as a paycheck, but it is a mental pat on the back when a student checks his/her work and knows it is correct. It just feels good. I want to make sure that my students feel good about the hard work that they do. I want them to experience success.
Clearly, I plan on using very basic tools to assist my students in the process of unraveling word problems. Know the language, draw a sketch, organize the facts, tell me what you do not know, and reward yourself!
Let us put this plan to the test. It is time to take it to the classroom and check out the strategies. I will begin with three simple problems in which the students can use the tools outlined above. I will refer to the first sample problem as a Picture Problem because we will draw a sketch or picture to make solving it easy. I will have another problem that I will refer to as an Equal Problem because we will be setting it equal to each other to solve. Finally, I will refer to my last problem as a Two Types Problem. For this problem we will be marrying two different types of information together before we can solve it. Below are examples of each type of problem. I have attached many more examples in the appendix that I will use and refer to through out this lesson.
Picture Problems
Students should look for the "red flag" to help them identify Picture Problems. Picture problem are problems that have several pieces that are combined together to get one total. In the case below we are trying to find the lengths of two pieces of string that together total 126 centimeters. This type of problem is easy to solve with the assistance of a simple sketch. A more extensive list of Picture Problems is listed in the appendix.
Amy cut a string that was 126 centimeters long into two pieces so that one piece is twice as long as the other. How long is each piece?
Students will be asked to find any of the mathematical terms or phrases that we have discussed. Relying on their mathematical term/phrase list (stratagie1- language), "twice as long" is the intended response.
Students will be instructed to sketch a simple line to represent the two pieces of string (strategy 2 - sketch). The sketch should have this information _______/______=126, and represent a true statement. It is not complete, but it is true, because it shows the two pieces of string and they are equal to 126. We will adjust the sketch as we learn more information.
Stratagie3 or identifying the variable, is asking the student to become a detective. I will ask them what they don't know. If they reread the problem they will realize that they know nothing about the one piece of string, but they do know something about the other piece. They know that the one piece is "twice as long" as the other piece. We can scaffold this information and add it to our simple sketch, resulting in the equation 2x + x = 126. Combining like terms is the mathematical term for the next step but I can not assume that my students know it. I will discuss the meaning and have students add that to the list of mathematical term for addition. This is also an example of the distributive property because we can also look at 2X + X as X(1 + 2). We have now created a new equation or 3x = 126. I would now have my students focus in on the x, we have in front of us three x's, but we are only interested in one x. What is the three doing to the x? We can refer bask to our mathematical term/phrase list again and see that the three is multiplying the x. What undoes multiplication? Division since they are inverses of each other. If I divide both sides by three, I will have the x by itself, resulting in the new equation x = 42.
There is to much math here to be ignored though. I need to make sure that my students understand why they did what they just did and why it is correct. We just used math magic to turn 3x into x by dividing both sides of the equation by 3. We have already discussed the idea about changing the look of something by doing the same operation to everything. Now I want to emphasize another point. What is the secret that allows us to change 3X into X? Since the symbols or saying three divided by three, or how many three's are there in three, our answer is 1. Therefore 5/5, 6/6, x/x and 1/2/1/2 or ANY number divided by itself, are all just another representation of 1 because equals divided by equals are equal. This is just one more opportunity to explain the obvious and increase understanding and not just having students do the steps. This will be helpful when we begin using fractions in our problems because, yes a fraction divided by the same fraction is still one. Fractions are numbers too, so they follow the same rules.
Success (strategy 4) is the final step. I can not stress enough the importance of teaching your students the satisfaction of knowing you are correct. It is a great confidence building skill and that confidence produces students who are more willing to step out of the box and look at math problems and other problems from different view points. The original question was about the length of two pieces of string whose total was 126 centimeters. We found that x = 42, so 2x ( one piece of string) + x (second piece of string) = 126, or 2(42) + 42 = 126, when we distribute the 2, results in 84 + 42 = 126 , or 126 = 126. Since all our statements are correct, we are must be correct. Congratulations!
Equal problems
This next type of word problem once again relies on the student's understanding of language to help make setting up and solving the problem easy. Equal problems can be identified by the term "the same" appearing somewhere in the problem. Students will first be asked what does X = 2 mean. The symbols do not look at all the same but they are equal in value, so "the same value" is represented by the equal sign (=). This subtle reminder will make solving the Equal Problems a simple process of setting the equations equal to each other. Students will already have experience making and solving linear equation. The following is an example of an Equal Problem along with a path to follow for solving the problem. The appendix has a more examples of Equal Problems that I will refer to and use through out this lesson.
Sam and Hector are gaining weight for the football season. Sam weighs 205 pounds and is gaining two pounds per week. Hector weighs 195 but is gaining three pounds per week. In how many week will they both weight the same amount?
We can identify this problem as an Equal Problem because we are being asked when Hector's weight will be equal to ("the same as") Sam's weight. The language once again is important because students must understand that "the same" is the mathematical phrase for the mathematical symbol "=". Once this is understood we can concentrate on defining the meaning of the X. In this case the X represents the weeks. Finally we can think about the question we are being asked. When is Sam's weight the same as Hector's weight? When we translate the questions into symbols the result is: Sam's weight (2x +205 ) the same (=) as Hector's weight ( 3x + 195) We can now just write the question in symbols and the result is (2x + 205) = (3x + 105).
The next step of getting the variables together is what I have termed the "bully problem". I have lovingly named it this because that is what you do, you pick on, or start with the smaller of the two variables. This will eliminate a negative variable. If I start with 2x and simply eliminate it by adding -2x to both side, the result will be 205 = x + 195. I will show that I could have started with the 3x but if I did I would have had to deal with negative numbers so that is why whenever possible starting with the smaller of the two numbers is the easiest way. The same idea of elimination by adding the opposite number is used to eliminate 195. This will give us the final answer of x = 10. The x represent the number of weeks, so the final answer is in ten weeks Hector and Sam will be the same weight.
Two Type Problems
The Equal Problems above will show the students that understanding the question and being able to translate words to symbols makes solving the problem easy. This lays the foundation for moving on to the type of equation the I have called Two Type Problems. The reason I have named them this is because, as you will see, we will need to satisfy two different conditions. The trick to recognizing these problems is counting the actual numbers given in the problem, there will be a total of four numbers given. Three of the numbers will pertain to the same thing and one will represent a total. The question will be asking about the total or quantity that we are give the least amount of information. The following is an example of a Two Type Problem.
Lilly sold 75 tickets worth $ 462.50 for the school play. The cost was $ 7.50 for adults tickets and $ 3.50 for student tickets. How many of each kind of ticket did she sell?
We need to organize the many numbers that we are given and identify what their values represent. It is important to note the differences between the numbers, not only because they must be noted in the problem but because this distinction will help students identify Two Type Problems in the future.
The number 75 represent the total number of tickets sold, and the remaining numbers represent money. The $ 462.50 represents the total dollars collected. The 7.50 represent the cost of an adult ticket and the 3.50 represents the cost of the student ticket. We have identified the two different conditions(tickets/dollars), so we can make two separate equations now that represent our problem.
The total number of adult and the total number of student tickets sold was 75, or A + S = 75. The money earned from the adult tickets and the students tickets was $ 462.50, and we know the price of each ticket, so 7.5A + 3.5S = 462.50. Lets make sure that our symbols match our sentences. A + S = 75, that is true, and 7.5A + 3.5S = 462.50 is also true. Now that we have two accurate equations, we can make a system of equations and either use substation or multiplication/distribution to eliminate one of the variables. The equations will be 7.5A + 3.5S = 462.50
A + S = 75
Rewriting the equation to A = 75 - S, I would stop here and ask my students if the statement A = 75 - S is still a true statement, since it is still true, we can substitute this new equation into the first equation which would result in 7.5(75-S) + 3.5S = 462.5. Distribution of 7.5 and combining like terms will result in a new equation (562.50 - 4S = 462.50) by subtracting 562.50 to both sides of the equation we would get -4S = -100 and finally divide both side by -4 to solve for S. The number of students (S) would be 25.
I would also show students that they could have distributed either a -7.5 or a -3.5 to the first equation and eliminated a variable, using this opportunity to go over the idea that if you do something to one side of an equation you can do the same to the other side and not change the value. You may change the look but not the value stressing the idea that 1/2 is the same value as 5/10 even though they do not look the same. I hate to pass up an opportunity to reinforce basic ideas that students may not have thought about for a while.
Students will have mastered the necessary math skills needed to solve this problem because of the progression of problems presented. There were two ways to solve this problem, substitution or distribution both resulting in correct answers, the choice is left to the students.
The examples shown above are a natural progression from a simplistic approach to a more complex approach to solving word problems. The student will have mastered identification of problems which will give them a direction to begin and ultimately solve the problem. This solid foundation will make categorizing and solving more complex word problem an achievable goal for students.
I have added several detailed examples of each type of problem at the end of this lesson along with many unsolved examples of the three types of problems. It is beneficial to have students determine what type of problem it is before they begin working. It is necessary to give the students the "red flags" in order for them to be able to complete this task, this is just another way to help students be successful.
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