## Problem Solving in Mathematics

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The main reason for learning about math is to become a better problem solver in all aspects of life. Many problems are multistep and require some type of systematic approach. There are a couple of things you need to do when solving problems. Ask yourself exactly what type of information is being asked for: Is it one of addition, subtraction, multiplication , or division? Then determine all the information that is being given to you in the question.

Mathematician George Pólya’s book, “ How to Solve It: A New Aspect of Mathematical Method ,” written in 1957, is a great guide to have on hand. The ideas below, which provide you with general steps or strategies to solve math problems, are similar to those expressed in Pólya’s book and should help you untangle even the most complicated math problem.

## Use Established Procedures

Learning how to solve problems in mathematics is knowing what to look for. Math problems often require established procedures and knowing what procedure to apply. To create procedures, you have to be familiar with the problem situation and be able to collect the appropriate information, identify a strategy or strategies, and use the strategy appropriately.

Problem-solving requires practice. When deciding on methods or procedures to use to solve problems, the first thing you will do is look for clues, which is one of the most important skills in solving problems in mathematics. If you begin to solve problems by looking for clue words, you will find that these words often indicate an operation.

## Look for Clue Words

Think of yourself as a math detective. The first thing to do when you encounter a math problem is to look for clue words. This is one of the most important skills you can develop. If you begin to solve problems by looking for clue words, you will find that those words often indicate an operation.

Common clue words for addition problems:

Common clue words for subtraction problems:

- How much more

Common clue words for multiplication problems:

Common clue words for division problems:

Although clue words will vary a bit from problem to problem, you'll soon learn to recognize which words mean what in order to perform the correct operation.

## Read the Problem Carefully

This, of course, means looking for clue words as outlined in the previous section. Once you’ve identified your clue words, highlight or underline them. This will let you know what kind of problem you’re dealing with. Then do the following:

- Ask yourself if you've seen a problem similar to this one. If so, what is similar about it?
- What did you need to do in that instance?
- What facts are you given about this problem?
- What facts do you still need to find out about this problem?

## Develop a Plan and Review Your Work

Based on what you discovered by reading the problem carefully and identifying similar problems you’ve encountered before, you can then:

- Define your problem-solving strategy or strategies. This might mean identifying patterns, using known formulas, using sketches, and even guessing and checking.
- If your strategy doesn't work, it may lead you to an ah-ha moment and to a strategy that does work.

If it seems like you’ve solved the problem, ask yourself the following:

- Does your solution seem probable?
- Does it answer the initial question?
- Did you answer using the language in the question?
- Did you answer using the same units?

If you feel confident that the answer is “yes” to all questions, consider your problem solved.

## Tips and Hints

Some key questions to consider as you approach the problem may be:

- What are the keywords in the problem?
- Do I need a data visual, such as a diagram, list, table, chart, or graph?
- Is there a formula or equation that I'll need? If so, which one?
- Will I need to use a calculator? Is there a pattern I can use or follow?

Read the problem carefully, and decide on a method to solve the problem. Once you've finished working the problem, check your work and ensure that your answer makes sense and that you've used the same terms and or units in your answer.

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- Learn About Natural Numbers, Whole Numbers, and Integers
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## Teaching Mathematics Through Problem Solving

By Tom McDougal, Akihiko Takahashi

What do your students do when faced with a math problem they don't know how to solve? Most students give up pretty quickly. At best, they seek help from another student or the teacher. At worst, they shut down, seeing their failure as more evidence that they just aren't good at math. Neither of these behaviors will serve students in the long run. Inevitably, someday, every one of your students will encounter problems that they will not have explicitly studied in school and their ability to find a solution will have important consequences for them.

In the Common Core State Standards for Mathematics, the very first Standard for Mathematical Practice is that students should “understand problems and persevere in solving them.”1 Whether you are beholden to the Common Core or not, this is certainly something you would wish for your students. Indeed, the National Council of Teachers of Mathematics (NCTM) has been advocating for a central role for problem solving at least since the release of Agenda for Action in 1980, which said, “Problem solving [must] be the focus of school mathematics… .”2

The common instructional model of “I do, we do, you do,” increases student dependence on the teacher and decreases students’ inclination to persevere. How, then, can teachers develop perseverance in problem solving in their students?

First we should clarify what we mean by “problem solving.” According to NCTM, “Problem solving means engaging in a task for which the solution is not known in advance.”3 A task does not have to be a word problem to qualify as a problem — it could be an equation or calculation that students have not previously learned to solve. Also, the same task can be a problem or not, depending on when it is given. Early in the year, before students learn a particular skill, the task could be a problem; later, it becomes an exercise, because now they know how to solve it.

In Japan, math educators have been thinking about how to develop problem solving for several decades. They studied George Polya's How to Solve It ,4 NCTM's Agenda for Action , and other documents, and together, using a process called lesson study , they began exploring what it would mean to make problem solving “the focus of school mathematics.” And they succeeded. Today, most elementary mathematics lessons in Japan are organized around the solving of one or a very few problems, using an approach known as “teaching through problem solving.”

“Teaching through problem solving” needs to be clearly distinguished from “teaching problem solving.” The latter, which is not uncommon in the United States, focuses on teaching certain strategies — guess-and-check, working backwards, drawing a diagram, and others. In a lesson about problem solving, students might work on a problem and then share with the class how using one of these strategies helped them solve the problem. Other students applaud, the students sit down, and the lesson ends. These lessons are usually outside the main flow of the curriculum; indeed, they are purposely independent of any curriculum.

In “teaching through problem solving,” on the other hand, the goal is for students to learn precisely that mathematical idea that the curriculum calls for them to learn next.

A “teaching through problem solving” lesson would begin with the teacher setting up the context and introducing the problem. Students then work on the problem for about 10 minutes while the teacher monitors their progress and notes which students are using which approaches. Then the teacher begins a whole-class discussion. Similar to a “teaching problem solving” lesson, the teacher may call on students to share their ideas, but, instead of ending the lesson there, the teacher will ask students to think about and compare the different ideas — which ideas are incorrect and why, which ideas are correct, which ones are similar to each other, which ones are more efficient or more elegant. Through this discussion, the lesson enables students to learn new mathematical ideas or procedures. This approach is represented in Figure 1.

Let's illustrate this with an example from a hypothetical fifth-grade lesson based on the most popular elementary mathematics textbook in Japan. (This textbook has been translated into English as Mathematics International and is available at http://GlobalEdResources.com . 5) During most Japanese lessons, the textbook is closed, but the textbook shows how the authors think the lesson might play out.

When the lesson begins, the blackboard is completely empty. The teacher starts by displaying, either with a poster or using a projector, the picture from the textbook of four different rabbit cages, shown in Figure 2 (it is not uncommon for Japanese elementary students to care for rabbits in several rabbit hutches, so this is a familiar context).

## Figure 2 (Mathematics International, Grade 5, p. A93)

“What do you notice about the cages?” the teacher asks. Some students notice that some of the cages are different sizes. The teacher then asks, “Should each cage have the same number of rabbits?” No, say the students, smaller cages should have fewer rabbits, so the rabbits aren't too crowded.

The teacher then displays the pictures in Figure 3. “What do you think?” the teacher asks, as he puts them up one at a time for dramatic effect. “Are these equally crowded, or do you think some cages are more crowded than others?” There is some discussion about the rabbits in cage B, and students decide that just because they are bunched together right now, they probably won't stay that way. Students recognize that cages A and B are the same size, and since cage A has more rabbits (9 vs. 8), it is more crowded. The teacher writes that observation on the board: “When two cages are the same size, the one with more rabbits is more crowded.”

## Figure 3 (Mathematics International, Grade 5, p. A93)

“What about the others?” he asks. “How can we decide which are more crowded?” This last question becomes the key mathematical question of the lesson, and the teacher writes it on the board: “Let's think about how to compare crowdedness.” Students copy this problem in their notebooks while he writes.

The teacher gives students a piece of paper with the pictures from Figure 3 to glue in their notebooks and gives them 5 minutes to think about the problem. Several students take a ruler and begin measuring. “Why are you doing that?” the teacher quietly asks one of them. “I want to figure out the area,” the student says. “Oh! You think the area might be important. Write that idea in your notebook.” Other students count the rabbits and decide that B and C are equally crowded because they look like they are the same size, but they are unsure about D.

The teacher stops the students and asks for ideas. He first calls on a student who thinks that B and C are the same size. He records her idea on the board: “Arthi says B and C look like they are the same size and have the same number of rabbits, so they are equally crowded.” A student who found the areas says that they are not. The teacher records this idea on the board: “Karen thinks you need to know the area.” He turns to the first student. “Arthi, what do you think?” he asks. She and other students agree. The teacher posts a table with the areas of the four cages (Figure 4). “Let's copy this table into our notebooks, and think about the problem some more.”

## Figure 4 (Mathematics International, Grade 5, p. A94)

Students work independently for another 5 minutes while the teacher monitors their progress, encourages them to keep thinking, and reminds them to record their ideas in their notebook. He anticipates the following five ideas and notes which students are using them:

Idea 1: B and C have the same number of rabbits, but C has a smaller area, so C is more crowded. Unsure about A vs. C.

Idea 2: If you make 5 copies of A and 6 copies of C, they would have the same area (30 m2). A would then have 45 rabbits while C would have 48 rabbits, so C is more crowded.

Idea 3: If you make 8 copies of A and 9 copies of C, they would have the same number of rabbits (72). A would have an area of 48 m2 while C would have an area of 45 m2, so B is more crowded.

Idea 4: Divide: (area) ÷ (# of rabbits) = amount of area per rabbit

Idea 5: Divide: (# of rabbits) ÷ (area) = number of rabbits per unit area

The teacher invites students to explain their ideas to the class, selecting students based on the order above, while he records each idea on the blackboard. He asks students to compare Idea 1 to the thinking used to compare A and B. He writes on the board: “If either the area or the number of rabbits is the same, it's easy to compare.” The student with Idea 2 says, “I found a way to make the area the same,” and explains. This prompts the student with Idea 3 to say, “I used kind of the same approach to make the number of rabbits the same.”

When a student with Idea 4 comes up, she begins, “I decided to divide the area by the number of rabbits.” The teacher stops her. He writes: “(area) ÷ (# of rabbits).” Then he asks the class, “Why is she doing this? Who can explain her thinking?” Another student says, “That gives the amount of area for each rabbit.” He lets the student finish her idea:

A: 9÷6 = 1.5 C: 8÷5 = 1.6

The teacher asks the class to clarify what the 1.5 and 1.6 mean (m2 per rabbit) and what that says about the crowdedness of each cage.

He then invites a student to explain Idea 5: “I divided the other way…”

A: 6÷9 = 0.66… C: 5÷8 = 0.625

“Why is he doing this?” the teacher asks the class. “What does this 0.66… mean? What does 0.625 mean?” (“Rabbits per square meter,” the students answer.)

The teacher then asks the class to look for similarities across the five ideas, which are all visible on the blackboard. Some students note that Ideas 2 and 3 use multiplication while Ideas 4 and 5 use division, a superficial similarity. But some students notice the more significant connection that 2 and 5 are both about making the area the same, while 3 and 4 are both about making the number of rabbits the same.

“We haven't talked about cage D yet,” the teacher points out. “How shall we compare A, C, and D? Please try using one of these ideas.”

Students work in their notebooks for a few minutes. Students who try using multiplication (Idea 2 or 3) discover that the method is cumbersome. The teacher invites students who used Ideas 4 and 5 to share their calculations, adding them to the lists from before: Idea 4:

A: 9÷6 = 1.5 C: 8÷5 = 1.6 D: 15÷9 = 1.66… (m2/rabbit) Idea 5: A: 6÷9 = 0.66… C: 5÷8 = 0.625 D: 9÷15 = 0.6 (rabbits/m2)

“What do you think about these ideas?” asks the teacher, and students respond, “They are easy!” So the teacher writes a summary on the board, “Using division, it is easy to compare crowdedness.” He asks the students to write a reflection in their notebooks. One student who used multiplication writes, “I tried using multiplication, but dividing is easier. Next time I want to try that.” And the lesson ends.

In the students’ previous experience with comparing quantities, a single quantity was important, such as the number of apples or kilograms or square meters. Their prior experience with division was about finding a missing multiplier or multiplicand, which was itself a single quantity. This problem presented students for the first time with a situation in which two numbers needed to be considered. So by working on a problem about rabbits and cages, students learn that division can be used to compute a new type of quantity, a per unit quantity, that expresses the relationship between rabbits and area and can be used to compare crowdedness. In subsequent lessons, students will see how division can be used to compute other types of per unit quantities, such as the productivity of two farms in crops grown per acre of land or the cost per pencil.

What was the teacher's role in helping students learn this new mathematical idea? He never explained anything to the students, but the task had to be carefully constructed, and the teacher had to be very deliberate in how he directed the lesson, or the lesson wouldn't have worked.

The task was accessible to all students in the beginning by the fact that two cages had the same area (A and B) and two cages had the same number of rabbits (B and C), but since it wasn't clear whether B and C were the same size, students were pushed to think formally about area. And, while using multiplication was feasible for comparing cages A and C, the area of cage D was such that multiplication was cumbersome for comparing all three cages. Students who might have been happy with using multiplication and uncomfortable with the decimal values that result from division were pushed by cage D to appreciate the efficiency of using division.

The teacher's role in the lesson can be compared to the role of a film director, who carefully stages each scene and makes cuts between cameras to create the desired effect. Early in the lesson, the teacher highlighted the idea, raised by students, that equal areas or equal numbers of rabbits made comparisons easier. This was the foundation for the idea of dividing to find a “per unit quantity,” square meters per one rabbit or rabbits per one square meter. By starting with a discussion of incorrect or partially correct ideas and writing them on the board, the teacher valued those ideas. This encourages students to try: Even if they can’t solve the whole problem, they might come up with something to contribute. When a student first suggested the idea of dividing, the teacher asked other students to explain the thinking behind it. This enabled students who did not themselves think of dividing to make the idea their own. And by carefully organizing student ideas on the board (Figure 5), the teacher made it easier for students to compare those ideas with each other and to follow the flow of learning in the lesson.

## Figure 5 (includes items from Mathematics International, Grade 5, pp. A93-94)

Although the lesson vignette above is fictional, videos of lessons like it can be found at http://tinyurl.com/kuwb4bg . The grade 3 lesson “Multiplication Algorithm” and the grade 5 lesson “Do I Have a Window Seat or an Aisle Seat?” are particularly good, both for the quality of the lessons and for the quality of the videos themselves.

Japanese educators believe that regular lessons that teach through problem solving, interspersed with occasional practice days, help their students learn mathematics more thoroughly than didactic instruction coupled with a greater amount of practice. Certainly Japanese students have performed very well on the TIMSS and PISA international studies of mathematics achievement. But perhaps more important, teaching through problem solving habituates students to being confronted with unfamiliar problems, to struggling at length with those problems, and to learning from those problems. This is a way to cultivate perseverance in problem solving.

Reading this article and watching videos, however, will not equip most teachers to incorporate teaching through problem solving into their practice. The teacher who wishes to do so is faced with several challenges. The first challenge is that few curricula are designed to support such lessons; most are designed to support fairly direct instruction by the teacher. The second problem is that students are not used to learning this way and may resist. And the third problem is that teaching this way is hard. It requires ways of thinking about a lesson that are unfamiliar to almost all U.S. teachers. One needs to be absolutely clear about what the mathematical goal of the lesson is; that goal is never for students to simply solve a problem. One needs to anticipate the various solutions, correct and incorrect, that are likely to come from students, as well as the ways students will get stuck. One needs to plan how the discussion around the various student ideas will address misconceptions and build toward the mathematical goal of the lesson. One needs to think about how the ideas will be organized on the board so that students can easily compare them.

Japanese teachers certainly did not learn to teach this way by reading articles or watching videos. They learned it — and continue to learn it — by trying it, together, one lesson at a time through a process called lesson study .6,7 A full treatment of lesson study would be another article in itself, but U.S. teachers who are interested in learning to teach through problem solving can find more information about lesson study at http://LessonStudyGroup.net and at http://LSAlliance.org . Lesson Study Alliance organizes the annual Chicago Lesson Study Conference, which features live lessons by teachers who are working to incorporate teaching through problem solving into their practice.

1. National Governors Association Center for Best Practices, Council of Chief State School Officers, Common Core State Standards for Mathematics (Washington, DC: National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010); online at www.corestandards.org/math/ . 2. National Council of Teachers of Mathematics, An Agenda for Action: Recommendations for School Mathematics of the 1980s (Washington, DC: NCTM, 1980); online at www.nctm.org/standards/content.aspx?id=17278 . 3. National Council of Teachers of Mathematics, Principles and Standards for School Mathematics (Washington, DC: NCTM, 2000); online at http://www.nctm.org/standards/content.aspx?id=16909 . 4. George Polya, How to Solve It: A New Aspect of Mathematical Method (Princeton, NJ: Princeton University Press, 1945). 5. T. Fujii and S. Iitaka, Mathematics International , Grades 1-6 (Tokyo: Tokyo Shoseki Co., Ltd., 2012). 6. Akihiko Takahashi, “Implementing Lesson Study in North American Schools and School Districts” (no date); online at http://hrd.apec.org/images/a/ae/51.2.pdf . 7. Akihiko Takahashi and Makoto Yoshida, “Ideas for Establishing Lesson-Study Communities.” Teaching Children Mathematics , May 2004.

Tom McDougal is executive director of Lesson Study Alliance in Chicago, a nonprofit organization that promotes and supports Lesson Study. He taught middle and high school mathematics and was an elementary math specialist.

Akihiko Takahashi is associate professor of mathematics education at DePaul University in Chicago. He taught students in grades 1-6 for 19 years in Japan, where he helped lead the national shift to teaching mathematics through problem solving.

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## Why It's So Important to Learn a Problem-Solving Approach to Mathematics

was invited to the Math Olympiad Summer Program (MOP) in the 10th grade. I went to MOP certain that I must really be good at math. But in my five weeks at MOP, I encountered over sixty problems on various tests and I didn’t solve a single one. That’s right—I was 0-for-60+. I came away no longer confident that I was good at math. I assumed that most of the other kids did better at MOP because they knew more tricks than I did. My formula sheets were pretty thorough, but perhaps they were missing something. By the end of MOP, I had learned a somewhat unsettling truth. The others knew fewer tricks than I did, not more. They didn’t even have formula sheets!

At another contest later that summer, a younger student, Alex, from another school asked me for my formula sheets. In my local and state circles, students’ formula sheets were the source of knowledge, the source of power that fueled the top students and the top schools. They were studied, memorized, revered. But most of all, they were not shared. But when Alex asked for my formula sheets I remembered my experience at MOP and I realized that formula sheets are not really math . Memorizing formulas is no more mathematics than memorizing dates is history or memorizing spelling words is literature. I gave him the formula sheets. (Alex must later have learned also that the formula sheets were fool’s gold—he became a Rhodes scholar.)

The difference between MOP and many of these state and local contests I participated in was the difference between problem solving and what many people call mathematics. For these people, math is a series of tricks to use on a series of specific problems. Trick A is for Problem A, Trick B for Problem B, and so on. In this vein, school can become a routine of learn tricks for a week, use tricks on a test, forget most tricks quickly. The tricks get forgotten quickly primarily because there are so many of them, and also because the students don’t see how these ‘tricks’ are just extensions of a few basic principles.

I had painfully learned at MOP that true mathematics is not a process of memorizing formulas and applying them to problems tailor-made for those formulas. Instead, the successful mathematician possesses fewer tools, but knows how to apply them to a much broader range of problems. We use the term problem solving to distinguish this approach to mathematics from the memorize, use, forget approach.

After MOP I relearned math throughout high school. I was unaware that I was learning much more. When I got to Princeton I enrolled in organic chemistry. There were over 200 students in the course, and we quickly separated into two groups. One group understood that all we would be taught could largely be derived from a very small number of basic principles. We loved the class—it was a year-long exploration of where these fundamental concepts could take us. The other, much larger, group saw each new destination not as the result of a path from the building blocks, but as yet another place whose coordinates had to be memorized if ever they were to visit again. Almost to a student, the difference between those in the happy group and those in the struggling group was how they learned mathematics. The class seemingly involved no math at all, but those who took a memorization approach to math were doomed to do it again in chemistry. The skills the problem solvers developed in math transferred, and these students flourished.

We use math to teach problem solving because it is the most fundamental logical discipline. Not only is it the foundation upon which sciences are built, it is the clearest way to learn and understand how to develop a rigorous logical argument. There are no loopholes, there are no half-truths. The language of mathematics is as precise as it is ‘right’ and ‘wrong’ (or ‘proven’ and ‘unproven’). Success and failure are immediate and indisputable; there isn’t room for subjectivity. This is not to say that those who cannot do math cannot solve problems. There are many paths to strong problem-solving skills. Mathematics is the shortest .

Problem solving is crucial in mathematics education because it transcends mathematics. By developing problem-solving skills, we learn not only how to tackle math problems, but also how to logically work our way through any problems we may face. The memorizer can only solve problems he has encountered already, but the problem solver can solve problems she’s never seen before. The problem solver is flexible; she can diversify. Above all, she can create .

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## 5 Teaching Mathematics Through Problem Solving

Janet Stramel

In his book “How to Solve It,” George Pólya (1945) said, “One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles. The student should acquire as much experience of independent work as possible. But if he is left alone with his problem without any help, he may make no progress at all. If the teacher helps too much, nothing is left to the student. The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.” (page 1)

What is a problem in mathematics? A problem is “any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method” (Hiebert, et. al., 1997). Problem solving in mathematics is one of the most important topics to teach; learning to problem solve helps students develop a sense of solving real-life problems and apply mathematics to real world situations. It is also used for a deeper understanding of mathematical concepts. Learning “math facts” is not enough; students must also learn how to use these facts to develop their thinking skills.

According to NCTM (2010), the term “problem solving” refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development. When you first hear “problem solving,” what do you think about? Story problems or word problems? Story problems may be limited to and not “problematic” enough. For example, you may ask students to find the area of a rectangle, given the length and width. This type of problem is an exercise in computation and can be completed mindlessly without understanding the concept of area. Worthwhile problems includes problems that are truly problematic and have the potential to provide contexts for students’ mathematical development.

There are three ways to solve problems: teaching for problem solving, teaching about problem solving, and teaching through problem solving.

Teaching for problem solving begins with learning a skill. For example, students are learning how to multiply a two-digit number by a one-digit number, and the story problems you select are multiplication problems. Be sure when you are teaching for problem solving, you select or develop tasks that can promote the development of mathematical understanding.

Teaching about problem solving begins with suggested strategies to solve a problem. For example, “draw a picture,” “make a table,” etc. You may see posters in teachers’ classrooms of the “Problem Solving Method” such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no evidence that students’ problem-solving abilities are improved when teaching about problem solving. Students will see a word problem as a separate endeavor and focus on the steps to follow rather than the mathematics. In addition, students will tend to use trial and error instead of focusing on sense making.

Teaching through problem solving focuses students’ attention on ideas and sense making and develops mathematical practices. Teaching through problem solving also develops a student’s confidence and builds on their strengths. It allows for collaboration among students and engages students in their own learning.

Consider the following worthwhile-problem criteria developed by Lappan and Phillips (1998):

- The problem has important, useful mathematics embedded in it.
- The problem requires high-level thinking and problem solving.
- The problem contributes to the conceptual development of students.
- The problem creates an opportunity for the teacher to assess what his or her students are learning and where they are experiencing difficulty.
- The problem can be approached by students in multiple ways using different solution strategies.
- The problem has various solutions or allows different decisions or positions to be taken and defended.
- The problem encourages student engagement and discourse.
- The problem connects to other important mathematical ideas.
- The problem promotes the skillful use of mathematics.
- The problem provides an opportunity to practice important skills.

Of course, not every problem will include all of the above. Sometimes, you will choose a problem because your students need an opportunity to practice a certain skill.

Key features of a good mathematics problem includes:

- It must begin where the students are mathematically.
- The feature of the problem must be the mathematics that students are to learn.
- It must require justifications and explanations for both answers and methods of solving.

Problem solving is not a neat and orderly process. Think about needlework. On the front side, it is neat and perfect and pretty.

But look at the b ack.

It is messy and full of knots and loops. Problem solving in mathematics is also like this and we need to help our students be “messy” with problem solving; they need to go through those knots and loops and learn how to solve problems with the teacher’s guidance.

When you teach through problem solving , your students are focused on ideas and sense-making and they develop confidence in mathematics!

## Mathematics Tasks and Activities that Promote Teaching through Problem Solving

## Choosing the Right Task

Selecting activities and/or tasks is the most significant decision teachers make that will affect students’ learning. Consider the following questions:

- Teachers must do the activity first. What is problematic about the activity? What will you need to do BEFORE the activity and AFTER the activity? Additionally, think how your students would do the activity.
- What mathematical ideas will the activity develop? Are there connections to other related mathematics topics, or other content areas?
- Can the activity accomplish your learning objective/goals?

## Low Floor High Ceiling Tasks

By definition, a “ low floor/high ceiling task ” is a mathematical activity where everyone in the group can begin and then work on at their own level of engagement. Low Floor High Ceiling Tasks are activities that everyone can begin and work on based on their own level, and have many possibilities for students to do more challenging mathematics. One gauge of knowing whether an activity is a Low Floor High Ceiling Task is when the work on the problems becomes more important than the answer itself, and leads to rich mathematical discourse [Hover: ways of representing, thinking, talking, agreeing, and disagreeing; the way ideas are exchanged and what the ideas entail; and as being shaped by the tasks in which students engage as well as by the nature of the learning environment].

The strengths of using Low Floor High Ceiling Tasks:

- Allows students to show what they can do, not what they can’t.
- Provides differentiation to all students.
- Promotes a positive classroom environment.
- Advances a growth mindset in students
- Aligns with the Standards for Mathematical Practice

Examples of some Low Floor High Ceiling Tasks can be found at the following sites:

- YouCubed – under grades choose Low Floor High Ceiling
- NRICH Creating a Low Threshold High Ceiling Classroom
- Inside Mathematics Problems of the Month

## Math in 3-Acts

Math in 3-Acts was developed by Dan Meyer to spark an interest in and engage students in thought-provoking mathematical inquiry. Math in 3-Acts is a whole-group mathematics task consisting of three distinct parts:

Act One is about noticing and wondering. The teacher shares with students an image, video, or other situation that is engaging and perplexing. Students then generate questions about the situation.

In Act Two , the teacher offers some information for the students to use as they find the solutions to the problem.

Act Three is the “reveal.” Students share their thinking as well as their solutions.

“Math in 3 Acts” is a fun way to engage your students, there is a low entry point that gives students confidence, there are multiple paths to a solution, and it encourages students to work in groups to solve the problem. Some examples of Math in 3-Acts can be found at the following websites:

- Dan Meyer’s Three-Act Math Tasks
- Graham Fletcher3-Act Tasks ]
- Math in 3-Acts: Real World Math Problems to Make Math Contextual, Visual and Concrete

## Number Talks

Number talks are brief, 5-15 minute discussions that focus on student solutions for a mental math computation problem. Students share their different mental math processes aloud while the teacher records their thinking visually on a chart or board. In addition, students learn from each other’s strategies as they question, critique, or build on the strategies that are shared.. To use a “number talk,” you would include the following steps:

- The teacher presents a problem for students to solve mentally.
- Provide adequate “ wait time .”
- The teacher calls on a students and asks, “What were you thinking?” and “Explain your thinking.”
- For each student who volunteers to share their strategy, write their thinking on the board. Make sure to accurately record their thinking; do not correct their responses.
- Invite students to question each other about their strategies, compare and contrast the strategies, and ask for clarification about strategies that are confusing.

“Number Talks” can be used as an introduction, a warm up to a lesson, or an extension. Some examples of Number Talks can be found at the following websites:

- Inside Mathematics Number Talks
- Number Talks Build Numerical Reasoning

## Saying “This is Easy”

“This is easy.” Three little words that can have a big impact on students. What may be “easy” for one person, may be more “difficult” for someone else. And saying “this is easy” defeats the purpose of a growth mindset classroom, where students are comfortable making mistakes.

When the teacher says, “this is easy,” students may think,

- “Everyone else understands and I don’t. I can’t do this!”
- Students may just give up and surrender the mathematics to their classmates.
- Students may shut down.

Instead, you and your students could say the following:

- “I think I can do this.”
- “I have an idea I want to try.”
- “I’ve seen this kind of problem before.”

Tracy Zager wrote a short article, “This is easy”: The Little Phrase That Causes Big Problems” that can give you more information. Read Tracy Zager’s article here.

## Using “Worksheets”

Do you want your students to memorize concepts, or do you want them to understand and apply the mathematics for different situations?

What is a “worksheet” in mathematics? It is a paper and pencil assignment when no other materials are used. A worksheet does not allow your students to use hands-on materials/manipulatives [Hover: physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics]; and worksheets are many times “naked number” with no context. And a worksheet should not be used to enhance a hands-on activity.

Students need time to explore and manipulate materials in order to learn the mathematics concept. Worksheets are just a test of rote memory. Students need to develop those higher-order thinking skills, and worksheets will not allow them to do that.

One productive belief from the NCTM publication, Principles to Action (2014), states, “Students at all grade levels can benefit from the use of physical and virtual manipulative materials to provide visual models of a range of mathematical ideas.”

You may need an “activity sheet,” a “graphic organizer,” etc. as you plan your mathematics activities/lessons, but be sure to include hands-on manipulatives. Using manipulatives can

- Provide your students a bridge between the concrete and abstract
- Serve as models that support students’ thinking
- Provide another representation
- Support student engagement
- Give students ownership of their own learning.

Adapted from “ The Top 5 Reasons for Using Manipulatives in the Classroom ”.

any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method

should be intriguing and contain a level of challenge that invites speculation and hard work, and directs students to investigate important mathematical ideas and ways of thinking toward the learning

involves teaching a skill so that a student can later solve a story problem

when we teach students how to problem solve

teaching mathematics content through real contexts, problems, situations, and models

a mathematical activity where everyone in the group can begin and then work on at their own level of engagement

20 seconds to 2 minutes for students to make sense of questions

Mathematics Methods for Early Childhood Copyright © 2021 by Janet Stramel is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

## Share This Book

## Solving Equations

Solving equations involves finding the value of the unknown variables in the given equation. The condition that the two expressions are equal is satisfied by the value of the variable. Solving a linear equation in one variable results in a unique solution, solving a linear equation involving two variables gives two results. Solving a quadratic equation gives two roots. There are many methods and procedures followed in solving an equation. Let us discuss the techniques in solving an equation one by one, in detail.

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## What is the Meaning of Solving Equations?

Solving equations is computing the value of the unknown variable still balancing the equation on both sides. An equation is a condition on a variable such that two expressions in the variable have equal value. The value of the variable for which the equation is satisfied is said to be the solution of the equation. An equation remains the same if the LHS and the RHS are interchanged. The variable for which the value is to be found is isolated and the solution is obtained. Solving an equation depends on what type of equation that we are dealing with. The equations can be linear equations, quadratic equations, rational equations, or radical equations.

## Steps in Solving an Equation

The aim of solving an equation is to find the value of the variable that satisfies the condition of the equation true. To isolate the variable, the following operations are performed still balancing the equation on both sides. By doing so LHS remains equal to RHS, and eventually, the balance remains undisturbed throughout.

- Addition property of equality : Add the same number to both the sides. If a = b, then a + c = b + c
- Subtraction property of equality : Subtract the same number from both sides. If a = b, then a - c = b - c
- Multiplication property of equality: Multiply the same number on both sides. If a = b, then ac = bc
- Division property of equality : Divide by the same number on both sides. If a = b, then a/c = b/c (where c ≠ 0)

After performing this systematic balancing method of solving an equation by a series of identical arithmetical operations on both sides of the equation, we separate the variable on one of the sides and the ultimate step is the solution of the equation.

## Solving Equations of One Variable

A linear equation of one variable is of the form ax + b = 0, where a, b, c are real numbers. The following steps are followed while solving an equation that is linear.

- Remove the parenthesis and use the distributive property if required.
- Simplify both sides of the equation by combining like terms.
- If there are fractions , multiply both sides of the equation by the LCD (Least common denominator) of all the fractions.
- If there are decimals , multiply both sides of the equation by the lowest power of 10 to convert them into whole numbers.
- Bring the variable terms to one side of the equation and the constant terms to the other side using the addition and subtraction properties of equality.
- Make the coefficient of the variable as 1, using the multiplication or division properties of equality.
- isolate the variable and get the solution.

Consider this example: 3(x + 4) = 24 + x

We simplify the LHS using the distributive property.

3x + 12 = 24 + x

Group the like terms together using the transposing method. This becomes 3x - x = 24-12

Simplify further ⇒ 2x = 12

Use the division property of equality, 2x/2 = 12/2

isolate the variable x. x = 6 is the solution of the equation.

Use any one of the following techniques to simplify the linear equation and solve for the unknown variable. The trial and error method, balancing method and the transposing method are used to isolate the variable.

## Solving an Equation by Trial And Error Method

Consider 12x = 60. To find x, we intuitively try to find that 12 times what number is 60. We find that 5 is the required number. Solving equations by trial and error method is not always easy.

## Solving an Equation by Balancing Method

We need to isolate the variable x for solving an equation. Let us use the separation of variables method or the balancing method to solve it. Consider an equation 2x + 3 = 17.

We first eliminate 3 in the first step. To keep the balance while solving the equation, we subtract 3 from either side of the equation.

Thus 2x + 3 - 3 = 17 - 3

We have 2x = 14

Now to isolate x, we divide by 2 on both sides. (Division property of equality)

2x/2 = 14/2

Thus, we isolate the variable using the properties of equality while solving an equation in the balancing method.

## Solving an Equation by Transposing Method

While solving an equation, we change the sides of the numbers. This process is called transposing. While transposing a number, we change its sign or reverse the operation. Consider 5y + 2 = 22.

We need to find y, so isolate it. Hence we transpose the number 2 to the other side. The equation becomes,

Now taking 5 to the other side, we reverse the operation of multiplication to division. y = 20/5 = 4

## Solving an Equation That is Quadratic

There are equations that yield more than one solution. Quadratic polynomials are of degree two and the zeroes of a quadratic polynomial represent the quadratic equation.

Consider (x+3) (x+2)= 0. This is quadratic in nature. We just equate each of the expressions in the LHS to 0.

Either x+3 = 0 or x+2 =0.

We arrive at x = -3 and x = -2.

A quadratic equation is of the form ax 2 + bx + c = 0. Solving an equation that is quadratic, results in two roots : α and β.

Steps involved in solving a quadratic equation are:

## By Completing The Squares Method

By factorization method, by formula method.

Solving an equation of quadratic type by completing the squares method is quite easy as we apply our knowledge of algebraic identity: (a+b) 2

- Write the equation in the standard form ax 2 + bx + c = 0.
- Divide both the sides of the equation by a.
- Move the constant term to the other side
- Add the square of one-half of the coefficient of x on both sides.
- Complete the left-hand side as a square and simplify the right-hand side.
- Take the square root on both sides and solve for x.

For more information about solving equations (quadratic) by completing the squares, click here .

Solving an equation of quadratic type using the factorization method , follow the steps discussed here. Write the given equation in the standard form and by splitting the middle terms, factorize the equation. Rewrite the equation obtained as a product of two linear factors. Equate each linear factor to zero and solve for x. Consider 2x 2 + 19x + 30 =0. This is of the standard form ax 2 + bx + c = 0.

Split the middle term in such a way that the product of the terms should equal the product of the coefficient of x 2 and c and the sum of the terms should be b. Here the product of the terms should be 60 and the sum should be 19. Thus, split 19x as 4x and 15x (as the sum of 4 and 15 is 19 and their product is 60).

2x 2 + 4x + 15x + 30 = 0

Take the common factor out of the first two terms and the common factors out of the last two terms.

2x(x + 2) + 15(x + 2) = 0

Factoring (x+2) again, we get

(x + 2)(2x + 15) = 0

x = -2 and x = -15/2

Solving an equation that is quadratic involves such steps while splitting the middle terms on factorization.

Solving an equation of quadratic type using the formula

x = [-b ± √[(b 2 -4ac)]/2a helps us find the roots of the quadratic equation ax 2 + bx + c = 0. Plugging in the values of a, b, and c in the formula, we arrive at the solution.

Consider the example: 9x 2 -12 x + 4 = 0

a= 9, b = -12 and c = 4

x = [-b ± √[(b 2 -4ac)]/2a

= [12 ± √[((-12) 2 -4×9×4)] / (2 × 9)

= [12 ± √(144 - 144)] / 18

= (12 ± 0)/18

x = 12/18 = 2/3

## Solving an Equation That is Rational

An equation with at least one polynomial expression in its denominator is known as the rational equation. Solving an equation that is rational involves the following steps. Reduce the fractions to a common denominator and then solve the equation of the numerators .

Consider x/(x-1) = 5/3

On cross-multiplication, we get

3x = 5(x-1)

3x = 5x - 5

3x - 5x = - 5

## Solving an Equation That is Radical

An equation in which the variable is under a radical is termed the radical equation. Solving an equation that is a radical involves a few steps. Express the given radical equation in terms of the index of the radical and balance the equation. Solve for the variable.

Consider √(x+1) = 4

Now square both the sides to balance it. [ √(x+1)] 2 = 4 2

Thus x = 16-1 =15

Important Notes on Solving Equations:

- Solving an equation is finding the value of the variable in the equation.
- The solution of an equation satisfies the condition of the given equation.
- Solving an equation of linear type can be also done graphically .
- If the right side part of an equation is zero, then for solving equation, just graph the left side of the equation and the x-intercept (s) of the graph would be the solution(s).

☛ Related Articles:

- Solving Equations Calculator
- Simultaneous linear equations
- One variable linear equations and inequations
- Simple equations and their applications

## Examples of Solving an Equation

Example 1. Use the balancing method of solving equations: (x-2) / 5 - (x-4) / 2 = 2

The given equation is (x-2) / 5 - (x-4) / 2 = 2.

Solving an equation that is rational involves the following steps.

Simplify the LHS. Take the LCD of the denominators. LCD is 10.

[2(x-2) -5(x-4)]/10 = 2

Use distributive property and simplify the numerator.

We get [2x- 4 -5x+20]/10 = 2

Use the multiplication property of equality to get rid of the denominator.

10 × [2x- 4 -5x+20]/10 = 10 × 2

Simplifying we get - 3x + 16 = 20

isolate the term with the variable using the addition property of equality

-3x + 16 - 16 = 20 - 16

isolate the variable using the division property of equality

-3x/3 = 4/3

Answer: Thus solving (x-2)/ 5 - (x-4)/2 = 2, x = -4/3

Example 2. Use the transposing method of solving an equation 0.4(a+10)= 2 - 0.6a

Solving equations that are linear involving decimals involves the following steps.

Given 0.4(a+10)= 2 - 0.6a

0.4 a + 0.4 × 10 = 2- 0.6a

0.4 a + 4 = 2- 0.6a

0.4 a + 0.6a = 2-4

Answer: The solution is a = -2

Example 3. What is the value of p on solving an equation: 4 (p - 3) - (p - 5) = 4?

Given: 4 (p - 3) - (p - 5) = 4

Let us use the transposing method in solving equations.

4p - 12 - p + 5 = 4 (distributive property)

Answer: The value of p = 11/3

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## Practice Questions on Solving an Equation

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## FAQs on Solving Equations

What is solving an equation.

Solving an equation is finding the value of the unknown variables in the given equation. The process of solving an equation depends on the type of the equation .

## What are The Steps in Solving Equations?

Identify the type of equation: linear, quadratic, logarithmic, exponential, radical or rational.

- Remove the brackets, if any in the given equation. Apply the distributive property.
- Add the same number to both the sides
- Subtract the same number from both the sides
- Multiply the same number on both the sides
- Divide by the same number on both sides.

## What are The Golden Rule in Solving an Equation?

The type of the equation is identified. If it is a linear equation, separating the variables method or transposing method is used. If it is a quadratic equation, completing the squares, splitting the middle terms using factorization is used or by formula method.

## How Do You use 3 Steps in Solving an Equation?

The 3 steps in solving an equation are to

- remove the brackets, if any using the distributive property,
- simplify the equation by adding or subtracting the like terms ,
- isolating the variable and solving it.

## How Do You Solve Linear Equations?

While solving an equation that is linear, we isolate the variable whose value is to be found. We either use transposing method or the balancing method.

## How Do You Solve Quadratic Equations?

While solving an equation that is quadratic, we write the equation in the standard form ax 2 + bx + c = 0, and then solve using the formula method or factorization method or completing the squares method.

## How Do You Solve Radical Equations?

While solving an equation that is radical , we remove the radical sign, by raising both the sides of the equation to the index of the radical, isolate the variable and solve for x.

## How Do You Solve Rational Equations?

While solving an equation that is rational, we simplify the expression on each side of the equation, cross multiply , combine the like terms and then isolate the variable to solve for x.

## What Does It Mean to "Make Sense of Problems and Persevere in Solving Them?"

If there is one mathematical practice that rules them all, this one is it. Perhaps that’s why it’s listed first. To me this practice is a short but accurate description of what mathematics is, and what we should strive for in our mathematics classrooms. First, we’ll offer some key descriptions of what it looks like to engage in MP1. Then, we’ll take some time to unpack several words and phrases in this practice so we can get a clear idea of what is meant, rather than repeating the somewhat nebulous phrase of “problem solving”. Let’s dig in.

## Students who are proficient at MP1 learn to:

Interpret the meaning of the problem, including its constraints, relationships, and goals.

Look for entry points to a solution.

Plan a solution pathway.

Consider similar problems, simpler cases, and edge cases.

Connect information given in multiple representations or construct an alternate representation.

Identify what concepts or prior knowledge are relevant to the problem and how to leverage them to move toward a solution.

Use the results of an unfruitful attempt to plan a new attempt.

Check the reasonableness of their answers.

You may have noticed that some of these skills sound very similar to our lists from MP2 (Reason abstractly and quantitatively), MP7 (Look for and make use of structure) and MP8 (Look for and express regularity in repeated reasoning). This is because the 8 mathematical practices are actually nested. If MP1 is the main goal of mathematics, then MP2, MP7, and MP8 are the three major pathways to arrive there.

## Unpacking MP1

“make sense”.

Note here that this phrase is being used as a verb. It represents an active process of students wrestling with ideas and working to connect them to things they already know. We often use this as an adjective, as in, ‘this question makes sense’ to denote that we understand something, or that it is logical . When we take the adjective perspective, we often assume that what we present to students should already make sense to them, because we taught them the appropriate procedure or algorithm, or because we gave a thorough explanation of why something works. They should look at a problem and know what to do. However, true problem solving is about actively making sense.

## “Problem Solving”

Although not combined quite like this in the original phrasing, it is clear that this is at the heart of MP1. Problem solving has become such a buzzword that it can mean everything and nothing. Not every task done in a mathematics classroom can be considered problem solving. Before we become clear about what problem solving is, let’s establish what it is not.

It is not a synonym for “real world application.”

It is not a synonym for “answering questions” (like those you might assign out of a textbook or other resource).

It is not confined to the end of a unit to be used only when students already have a firm grasp of all the definitions, formulas, and procedures.

I have the following 3 criteria when it comes to identifying tasks that actually require problem solving.

Features a question students haven’t seen before or haven’t been asked in that way before. Students don’t immediately know what steps to take based on notes or worked examples.

Centered on conceptual understanding of the topic or background knowledge that allows students to make sense of the problem or manipulate the parts of the problem to try to find a solution.

Requires identifying what prior knowledge is relevant, i.e. what should I be thinking about to make sense of this problem?

When students immediately know what path to take to arrive at a solution, there is no problem solving involved. There is no problem at all, only an exercise. When the path to the destination is clear, there is no productive struggle .

When we breadcrumb students toward a solution with too much scaffolding, we rob them of opportunities for genuine problem solving. While scaffolding is helpful and necessary at times, as teachers we can inadvertently eliminate the problem solving aspect of a problem by telling students exactly what they should be thinking about to solve a problem. Consider the following problem.

Let’s suppose that students have never been given this combination of information before about a linear function, or perhaps not represented in this way (function notation, rather than a verbal description of the change in outputs and y-intercept). Students don’t have a worked example they can look at to find an answer.

Students who are problem solving will have to:

Understand the definition of a linear function. An equation for f(x) will be the equation of a line, which in its simplest form has a y-intercept and a slope.

Understand the properties of a linear function. There is a constant change in outputs over same size input intervals. If the function goes down 20 over 5 units, it will do so over the entirety of the graph. The choice of x=12 and x=17 might have been arbitrary, but it is sufficient information to calculate the constant rate of change.

Make sense of the function notation. They will have to identify that the first statement is telling them that the y-values at x=12 and x=17 are 20 units apart, and that the y-values are decreasing. They will have to identify that f(0)=2.5 is function notation for the y-intercept.

Notice how many times students are having to identify relevant information_ both from the problem and their own prior learning. The ideas about constant rate of change are conceptual ideas. Furthermore, identifying the expression f(17)-f(12) as one component of the rate of change (the change in y-values or outputs) and understanding how this expression can be used to determine the slope is highly conceptual.

Now consider this scaffolded version of the same question.

Return to the bulleted list above. What problem solving aspects do these scaffolds remove? How do these scaffolds tell students what they need to be thinking about?

I am not arguing that there is no place for using scaffolded questions to help students learn to navigate a complex task or prompt. But we should be hesitant to call this genuine problem solving because the aspects that are intrinsic to problem solving, like generating a solution path and accessing relevant prior knowledge, are missing.

## “Persevere”

Perseverance is not the same as endurance. Endurance is about bearing something, getting through it, staying the course, like when someone endures watching a long, terrible movie. Endurance is what a student needs when they’re asked to solve 50 one-step equations. Perseverance is not just about sheer will or determination, but about courage to keep going even in the face of obstacles or hardship. Perseverance is what a student needs when they’ve tried 3 different strategies that didn’t work and need to go back to the drawing board yet again. We can teach students to persevere by giving them tasks that are challenging yet accessible, where an initial solution path might be apparent but not necessarily correct. The task should have an easy entry point and evolving complexity. This means the task itself is easy to explain and there is room for exploration. Perseverance is the friendly neighbor of productive struggle , and both are skills that must be taught! For an in-depth explanation of how to do this, we recommend reading “Productive Math Struggle” by John SanGiovanni, Kevin J. Dykema, and Susie Katt.

## Problem Solving Can’t Be Taught with a Poster

I wholeheartedly agree with teachers’ desire to help their students learn how to become effective problem solvers. But allow me to let the cat out of the bag and tell you that a poster with the “five steps of problem solving” that is some combination of circling the question and underlining the important numbers is not going to cut it. Problem solving can be taught in as far as it can be developed . It is not something that we can check off a list of standards as something that students have mastered or have not yet mastered. Problem solving takes time and is learned by doing , not merely by observing.

How might you incorporate more true problem solving into your own classroom?

- Math Medic Core

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## Problem-Solving vs Word Problems

I remember preparing for an interview for my first teaching position in the 90’s. I was told that I would likely be asked to explain my approach to teaching problem-solving. I jumped on the Internet to research problem-solving and craft my response. What I found was that problem-solving in math basically meant teaching students to solve word problems. I ended up getting the job and, for a number of years, taught what I thought was problem-solving. What I’ve come to find out, however, is that while we certainly need to teach students strategies for solving word problems, problem-solving is so much more than solving word problems.

## Problem-Solving > Word Problems

Think for a minute about a problem you’ve solved recently. I’ll give you a personal example. My current car lease ends next month, and I have to decide what to do. Usually, I just turn in my old car and lease another one. This year, however, is different. We are in the midst of an unprecedented shortage of new cars, driving new car prices way up. Not the best time to buy or lease a new car. At the same time, used car prices are surging and many used cars are selling at close to their original MSRP. Once again I jumped on the Internet to research the situation. I found out that I might be able to purchase my car at lease-end and turn around and sell it at a higher price! But that would leave me without a car. So, I have decided to purchase my car at lease-end and hold onto it until new car prices start to come back down. I should still get a trade-in value on my current car higher than what it will cost me to purchase it at lease-end. Of course, all this sounds great in theory and seems to be the right decision based on the data, but I won’t really know if I made the best decision until sometime next year.

This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.

I think you’d agree that what I described was some heavy-duty problem-solving with pretty significant consequences. Yet not a word problem in sight. You see, true problem-solving is messy and goes way beyond solving word problems.

George Polya is often called the Father of Problem-Solving. In 1945, he outlined a 4-step process for solving problems in his ground-breaking book How to Solve It . You can see the four steps pictured below.

Now think about the process I went through while solving my car problem. Don’t you see the four steps in what I did?

The problem is that well-intentioned teachers have tried to turn the problem-solving process, which is inherently messy, into an algorithm—if you do these steps, then you can easily solve problems. This is why we see students boxing numbers, underlining questions, and looking for “key” words, all shortcuts that basically give students permission to not read and understand word problems.

So how do we teach students to become problem solvers? Well, it might sound simplistic, but we give them rich problems to solve and get out of the way. Again, with the best of intentions, teachers often provide too much support and students come to depend on it.

I recently facilitated a book study on the book Productive Math Struggle: A 6-Point Action Plan for Fostering Perseverance , by John J. SanGiovanni, Susie Katt, and Kevin J. Dykema. It is a fabulously useful and easy-to-read book, chock full of implementable ideas. In other words, it’s a book you will use and not just read. It’s no coincidence that the first three chapters all deal with creating a climate where productive struggle can thrive. Let’s face it, many math classrooms still run on the premise that math is about regurgitating a memorized procedure. Not much thinking involved. First, we as teachers need to embrace the idea of teaching through productive struggle. Then, we need to set students up for success as we introduce struggle into our lessons.

One way to increase productive struggle and thinking in our classroom is to flip the sequence of our instruction. Rather than the traditional direct teaching approach of I do, We do, and You do , we flip the process so students are given a problem to solve before direct instruction.

Here’s an example. Say students have been using a part/whole diagram to represent join/result unknown word problems . So they have been practicing identifying if each number in a word problem represents one of the parts or the whole and creating part/whole diagrams, such as this one. By looking at the diagram, you can probably construct the word problem they were solving, right?

Now you would like to introduce a new structure—join/change unknown. It’s a more complicated type of problem. Here’s an example of this type of problem.

Mariana had $20. Her grandmother gave her some money for her birthday. Now Mariana has $28. How much money did Mariana’s grandmother give her for her birthday?

I could proceed to teach this new structure with a scripted lesson: Boys and girls, you have been using a diagram to solve join/result unknown problems. Today, I’m going to show you how to use the diagram to solve a new structure—Join/change unknown. I would model a problem or two, we would work a couple together, and then they could practice more on their own. A typical I do, We do, You do lesson.

But instead, what if we read the new problem out loud together, and then I commented, Huh. This problem sounds a little different. Work with a partner to see if you can solve it using your part/whole diagram. In other words, the You do comes first! Would every pair of students be successful in solving the problem? Probably not. But after all students have the opportunity to struggle with it, think of the rich discussions we can have. It’s likely that at least some students will determine that it’s a new structure and then I can come along behind and put a name to it.

So I hope you will commit to thinking of problem-solving as something beyond just solving word problems and give students the opportunity to productively struggle in your classroom. I think you’ll see engagement soar!

If you want more information on addition/subtraction structures check out this post .

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Let’s be clear—students need to have automaticity with their basic facts. What’s changed is the way we teach facts, moving away from rote memorization toward a strategy-based approach. Students learn…

## Identifying Geometric Attributes

Students begin identifying geometric attributes at a fairly early age. Our 1st Grade teachers introduce vertex, edge, and face. Still, it seems that kids have trouble answering questions about geometric…

Cy Fair does model drawing!! It works

Howdy, neighbor! Yes, model drawing is so powerful. I’m looking forward to next year when this whole group comes up knowing it!

I’ve been using model drawing with my first graders and it’s been powerful to see how they understand what’s happening in the problem, especially those tricky missing addend problems. This is the most confident I’ve felt with first graders and problems in a very long time!

I totally agree, Nilda! It really helps them to visualize the math that is happening in the story!

Looks like a great lesson!

Tara The Math Maniac

Thanks, Tara!

Hi Donna I am a math coach in Massachusetts and want institute bar models in all grade levels, K-5, for the 2017-2018 school year. Can you recommend a resource for me to use for research prior to professional development? Thanks Meg

A book that I have used and think is a great resource is Char Forsten’s book on model drawing .

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## MathBootCamps

What does it mean to “solve for x” or “solve the equation”.

Probably half of all of the directions in any algebra book say something along the lines of “solve the following for x”. We go through the steps we see in the examples and then move on right? But what are you really being asked to do?

Solve for x means find the value of x that would make the equation you see true.

Think about this equation: x + 1 = 3. If you were asked to solve it, that would mean finding some value for x that gives you three when you add one to it. Something plus one is three – what is the something? Well it must be two right? Since 2 + 1 = 3, x = 2. That’s it. That’s what solving an equation is all about!

This works no matter how complicated the equation gets. The only difference is that with a simple equation like this, it was possible to just think about it and get the answer. The more complicated the equation gets, well, there will be some more work involved. Either way – the idea is the same: Find the x that makes it true.

By the way, this is one of the great things about algebra – you can ALWAYS check your answers!

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## To Solve the Equations

Methods for solving equations, solving equations with a single variable, using the trial and error technique to solve an equation, using the balancing method to solve an equation, using the transposing method to solve an equation, solve|definition & meaning.

To solve means to find the solution to a problem. Solving equations with variables means finding the set of values that we can put in place of the variable(s) so that the equation becomes true (i.e., LHS = RHS). For example, consider the equation x + 9 = 10. If x = 0, this equation does not hold since 9 $\neq$ 10. But x = 1 is a valid solution since 1 + 9 = 10.

Figure 1 below shows the solution where the left-hand side is equal to the right-hand side.

Figure 1 – Representation of LHS and RHS.

The process of solving equations entails determining the value of an unknown element in the given formula. The value for the variable satisfies the criterion that the two alternatives are equal.

A linear equation with one variable has a single solution; a linear equation with two variables has two answers. When you solve a quadratic equation , you get two roots. When solving an equation, numerous approaches and processes are used.

Solving equations entails calculating the value of an unknown variable while keeping the equation balanced on both sides. An expression is a variable condition in which two alternatives in the parameter have the same value.

The answer to the problem is the quantity of the parameter for which the solution is fulfilled. If the LHS & RHS of an equation are swapped, the equation stays the same. The variable whose value is to be determined is isolated, and the answer is achieved.

The sort of equation we are working with influences how we solve it. Quadratic, linear, radical, and rational equations are all examples of equations.

Figure 2 below shows the solution of an equation.

Figure 2 – Representation of solving an equation.

The goal of solving an expression is to identify the values of parameters that make the equation’s condition true. To isolate the parameter, conduct the following actions while keeping the equation balanced on both sides. By doing so, LHS remains equivalent to RHS, and the equilibrium eventually remains unchanged throughout.

- The additional feature of equality states that you should add the same number to both sides. If x = y, x + z Equals y + z.
- Equality’s subtraction property: Take the same number off both sides. If x = y, x – z Equals y – z.
- Equality’s multiplication property: Multiply both sides by the same number. If x = y, xz Equals yz.
- Equality’s division property: On both sides, divide by the exact number. If an Equals b, then $\dfrac{x}{z}$ = $\dfrac{y}{z}$ (c = 0).
- Equality’s division property: On both sides, divide by the exact number. If an Equals b, then $\dfrac{x}{z}$ = $\dfrac{y}{z}$ (c = 0). Following this methodical balancing strategy of answering an expression using comparable mathematical functions across both sides of the expression, we separate the component into one side. The last process is the answer to the problem.

Following this methodical balancing strategy of answering an expression by a succession of comparable arithmetic on both aspects of the equation, we split the factor into one side, and the final step is indeed the result of the problem.

Figure 3 shows the methods of solving equations.

Figure 3 – Methods to solve an equation.

A one-variable linear equation of the form ax + b = 0, in which a, b, and c are real values. When solving a linear equation, the following stages are taken.

- If necessary, remove the parentheses and apply the distributive property.
- Combine comparable terms to simplify both sides of the problem.
- If the equation contains fractions, multiply all sides even by the least common denominator of each of the fractions.
- To convert decimals to whole numbers, multiply the equation’s sides by the smallest power of ten.
- Using the subtracting and adding properties of equality, move the variable parts to one side of the expression and the consistent terms to the other.
- To use equality’s multiplication and division characteristics, set the variable’s coefficient to 1.
- Find the solution by isolating the variable.

Consider the expression 12x = 60. To determine x, we calculate 12 times whatever number equals 60. We discover that the needed number is 5. It is only sometimes straightforward to solve equations through trial and error.

To solve an equation, we must isolate the variable x. Let us solve it using the removal of variables approach or the balancing method. Consider the following equation:

2x + 4 = 17

In the first stage, we remove three. To maintain the equation balanced when solving it, we deduct four from each of the sides. Therefore:

2x + 4 – 4 = 17 – 4

We shift the sides of the integers while solving an equation. This is known as transposing . We alter the sign of a number or reverse the action when transposing it. Consider the expression 5y + 2 = 22.

We must isolate y to discover it. As a result, we flip the value 2 to the opposite side. The formula becomes:

5y = 22 – 2

Taking 5 to the opposite side, we invert the multiplication to the division process. y Equals 20 divided by 5 equals 4.

## Examples of Solving Equations

Below are some examples of solving the equations

Solve the expressions 0.4(a+10) = 2 – 0.6a by transposing it.

The steps for solving linear equations with decimals are as follows.

Given that:

0.4(a + 10) = 2 – 0.6a

Solving for a:

0.4a + 0.4 × 10 = 2 – 0.6a

(0.4a) + 4 = 2 – 0.6a

(0.4a) + 0.6a = 2 – 4

So the solution is -2.

When calculating the equation 4 [(p – 3)] – (p – 5) = 5, figure out the value of p?

4 (p – 3) – 5 (p – 5) = 5

Let us solve equations using the transposing approach. Applying the distributive property:

4p – 12 – p + 5 = 5

3p – 8 = 4

The answer is that p equals 4.

All images are made using GeoGebra.

## Solution Definition < Glossary Index > Speed Definition

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## What does it mean to solve or find solutions in mathematics?

Something that has been really confusing me lately is that this equation has four solutions $$3x(x+1)(x^2+x+2)=16x(x+1)(2x+1)$$

But what does that mean? Until now solutions to me has meant, what are the coordinates of $x$ when $y$ equals a given value, normally $y=0$.

But this equation has kind of thrown me off because when plotting this equation on a graph I get only two points at which the line crosses the $x$-axis.

I hope someone understands my point.

- terminology

- 2 $\begingroup$ The solutions are the values of $x$ for which $3x(x+1)(x^2+x+2)-16x(x+1)(2x+1)=0$. Plot the graph of the function $f(x)=3x(x+1)(x^2+x+2)-16x(x+1)(2x+1)$: it crosses the axis in 4 points (although you'll have to zoom in close to see them clearly!) $\endgroup$ – user64687 Commented Feb 4, 2015 at 12:44
- 7 $\begingroup$ How did you "plot" this equation? It only has 1 variable. The output is true or false. $\endgroup$ – DanielV Commented Feb 4, 2015 at 12:46
- 1 $\begingroup$ If you set it equal to zero and plot with WolframAlpha, you can see that it has 4 uniques solutions, but three of them are really close together, so if you use a large scale, you might not see all of them. $\endgroup$ – user141592 Commented Feb 4, 2015 at 13:21
- 1 $\begingroup$ @Johanna thanks mate, never knew about this site $\endgroup$ – Thomas Winkworth Commented Feb 4, 2015 at 14:45
- $\begingroup$ Does this answer your question? What does it mean to solve an equation? $\endgroup$ – Jam Commented Oct 2, 2022 at 0:09

## 5 Answers 5

In general, a solution to an equation means the values which make the equation true.

According to this definition, if $y = f(x)$, then finding the coordinates of $x$ that make $y = 0$ means finding the values that make the following equation true:

The solution to several simultaneous equations, called a system of equations, are the values that make all equations true at the same time.

Also note that some equations have no solutions, while others might have solutions for every $x$ in the domain of the equation.

The definition is similar for inequalities.

edit: you can also find solutions to things like $y \equiv 4 \bmod 7$, which means that $y$ has a remainder of four when divided by seven. Or things like $\sqrt{2x+9} \text{ is a real number}$, or $x \text { and } y + 1 \text{ have the same absolute value}$. You're finding the values of the variables that make the statement true.

Also note that it is incorrect to find solutions to things that cannot be called true or false, as then there is nothing to solve. For example, it doesn't make sense to find the solutions to $2$, $x + 1$, or $\text{divisible by } 5$.

You can think of it as two function:

$f(x) = 3x(x+1)(x^2+x+2)$

$g(x) = 16x(x+1)(2x+1)$

When do the graphical repentations of these functions intersect?

Exactly when $f(x) = g(x)$ or,

$3x(x+1)(x^2+x+2) = 16x(x+1)(2x+1)$.

Now you can solve for $x$, to find the four $x$-values for which the two curves intersect.

Solving $3x(x+1)(x^2+x+2)=16x(x+1)(2x+1)$

is equivalent in solving $3x(x+1)(x^2+x+2) - 16x(x+1)(2x+1) = 0$

This is further equivalent in solving $x(x + 1)[(Ax + B)(Cx + D)] = 0$; for some $A, B, C$ and $D$.

Solving the last equation, we therefore get 4 roots, namely $x = 0,$ or $x= -1,$ or $x = -B/A,$ or $x = …$ .

It must be clear that cancellations [of the factors $x$ and $(x + 1)$] must NOT be done to the original from the start; Otherwise we will have only 2 roots left [with the other two missed as described.]

Definition : Solve. We say an equation is solved if and only if we have listed the set of all objects which make the equation true. We say we have found a solution if we have found an element of the solution set.

Example: If $x= 0$, then \begin{align} 3x(x+1)(x^2+x+2)=16x(x+1)(2x+1) \end{align} becomes \begin{align} 0 = 0 \end{align} which is true, and hence $x =0$ solves the equation.

Example: If $x =1$, then \begin{align} 3x(x+1)(x^2+x+2)=16x(x+1)(2x+1) \end{align} becomes \begin{align} 3\cdot 2 \cdot 4 = 16\cdot 2\cdot 3 \end{align} which is false, and hence $x=1$ is not a solution.

The four roots of the problem given are $-1$, $-1/3$, $0$, and $10$. They should be visible by graphing the fourth order equation resulting from collecting terms and setting same equation to zero. That equation is $3x^4 - 26x^3 - 39x^2 - 10x$.

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## How to Find the Mean

The mean is the average of the numbers.

It is easy to calculate: add up all the numbers, then divide by how many numbers there are.

In other words it is the sum divided by the count .

## Example 1: What is the Mean of these numbers?

- Add the numbers: 6 + 11 + 7 = 24
- Divide by how many numbers (there are 3 numbers): 24 / 3 = 8

## The Mean is 8

Why does this work.

It is because 6, 11 and 7 added together is the same as 3 lots of 8:

It is like you are "flattening out" the numbers

## Example 2: Look at these numbers:

3, 7, 5, 13, 20, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29

The sum of these numbers is 330

There are fifteen numbers.

The mean is equal to 330 / 15 = 22

## The mean of the above numbers is 22

Negative numbers.

How do you handle negative numbers? Adding a negative number is the same as subtracting the number (without the negative). For example 3 + (−2) = 3−2 = 1.

Knowing this, let us try an example:

## Example 3: Find the mean of these numbers:

3, −7, 5, 13, −2

- The sum of these numbers is 3 − 7 + 5 + 13 − 2 = 12
- There are 5 numbers.
- The mean is equal to 12 ÷ 5 = 2.4

## The mean of the above numbers is 2.4

Here is how to do it one line:

Mean = 3 − 7 + 5 + 13 − 2 5 = 12 5 = 2.4

## Try it yourself!

Now have a look at The Mean Machine .

Advanced Topic: the mean we have just looked at is also called the Arithmetic Mean , because there are other means such as the Geometric Mean and Harmonic Mean .

## IMAGES

## VIDEO

## COMMENTS

Problem-solving requires practice. When deciding on methods or procedures to use to solve problems, the first thing you will do is look for clues, which is one of the most important skills in solving problems in mathematics. If you begin to solve problems by looking for clue words, you will find that these words often indicate an operation.

In "teaching through problem solving," on the other hand, the goal is for students to learn precisely that mathematical idea that the curriculum calls for them to learn next. A "teaching through problem solving" lesson would begin with the teacher setting up the context and introducing the problem. Students then work on the problem for ...

Problem-solving is the ability to use appropriate methods to tackle unexpected challenges in an organized manner. The ability to solve problems is considered a soft skill, meaning that it's more of a personality trait than a skill you've learned at school, on-the-job, or through technical training. While your natural ability to tackle ...

We use math to teach problem solving because it is the most fundamental logical discipline. Not only is it the foundation upon which sciences are built, it is the clearest way to learn and understand how to develop a rigorous logical argument. There are no loopholes, there are no half-truths.

The very first Mathematical Practice is: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of ...

Problem-solving is often described as figuring out what to do when you don't know what to do. My power went out unexpectedly this morning, and I have work to do. That's a problem that I had to solve. I had to think about what the problem was, what my options were, and formulate a plan to solve the problem. No keywords.

This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. ... Problem Solving Strategy 3 (Using a variable to find the sum of a sequence.) Gauss's strategy for sequences. last term = fixed number (n-1) + first term.

Solving Equations

Teaching about problem solving begins with suggested strategies to solve a problem. For example, "draw a picture," "make a table," etc. You may see posters in teachers' classrooms of the "Problem Solving Method" such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no ...

Mathematical Problem | Definition, Principles & Uses

Brief. Problem solving plays an important role in mathematics and should have a prominent role in the mathematics education of K-12 students. However, knowing how to incorporate problem solving meaningfully into the mathematics curriculum is not necessarily obvious to mathematics teachers. (The term "problem solving" refers to mathematical ...

Solve. more ... To find a value (or values) we can put in place of a variable that makes the equation true. Example: x + 2 = 7. The variable is x, when we put 5 in place of x we get 5 + 2 = 7, and 5 + 2 = 7 is true, so x = 5 and the equation is solved. (Note: if we said x=6 then the equation is not solved because 6+2=7 is not true)

Solving an Equation - Methods, Techniques, and Examples

Problem solving has become such a buzzword that it can mean everything and nothing. Not every task done in a mathematics classroom can be considered problem solving. Before we become clear about what problem solving is, let's establish what it is not.

Polya (1945/1973) posited four problem-solving steps in How to Solve It: understanding. the problem, devising a plan, carrying out the plan and looking back. As obvious as this may seem, we should not take for granted that mathematics educators'. views of problem solving are universally accepted.

Problem-Solving vs Word Problems

Mathway | Algebra Problem Solver

Equation solving

Solve for x means find the value of x that would make the equation you see true. Think about this equation: x + 1 = 3. If you were asked to solve it, that would mean finding some value for x that gives you three when you add one to it. Something plus one is three - what is the something?

Definition. To solve means to find the solution to a problem. Solving equations with variables means finding the set of values that we can put in place of the variable(s) so that the equation becomes true (i.e., LHS = RHS). For example, consider the equation x + 9 = 10. If x = 0, this equation does not hold since 9 $\neq$ 10. But x = 1 is a valid solution since 1 + 9 = 10.

IMO solving an equation is giving the solution set in extension (S = {− 1, 1}) rather than in intention (S = {x ∈ R | x2 = 1}). The elements of the set can be specified via mathematical expressions, preferably that allow effective computation. This makes the solution constructive.

Something that has been really confusing me lately is that this equation has four solutions $$3x(x+1)(x^2+x+2)=16x(x+1)(2x+1)$$ But what does that mean? Until now solutions to me has meant, what a...

The mean is the average of the numbers. It is easy to calculate: add up all the numbers, then divide by how many numbers there are. In other words it is the sum divided by the count. Example 1: What is the Mean of these numbers? 6, 11, 7. The Mean is 8.