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22 Most Useful Creativity and Problem Solving Acronyms
- Study Again
1. SWOT Strengths, Weaknesses, Opportunities, Threats
Use it everyone – problem –solving, personal analysis, marketing planning, project planning, business planning…
2. IDEAL Identify, Define, Explore, Action, Lookback.
Process for solving problems: Identify the problem, Define it, Explore possible solutions and effects, Action the chosen solution, and Look back at the solution you brought about.
3. PEST Political, Economic, Social, Technological
Some use 'Environmental' used instead of 'Economic' depending on the context.
PEST is sometimes extended to 'PESTELI' in which the headings: Ecological (or Environmental), Legislative (or Legal), and Industry Analysis are added.
4. SLEPT Social, Legal, Economic, Political, Technological. 'SLEPT analysis' is a business review method similar to PEST or SWOT for assessing factors enabling or obstructing the business's performance, and typically its development potential.
5. TOTB (thus TOTBoxer and TOTBoxing) Think Outside The Box/Thinking Outside The Box. A TOTBoxer is a person who thinks outside the box - i.e., very creatively. TOTBoxing is thinking outside the box. Cleverer than a straightforward TOTB acronym, the expression elegantly describes a creative thinker, or the creative act.
6. SOSTAC Situation analysis, Objectives, Strategy, Tactics, Action, Control. SOSTAC is a business marketing planning system developed by writer and speaker PR Smith in the 1990s.
7. SCAMPER Creativity technique: Substitute Combine Adapt Modify, Magnify, Minify Put to other use Eliminate (Reverse, Rearrange).
8. PMI A decision-making strategy created by Edward de Bono. For any problem or solution, list these: Plus Points Minus Points Interesting Points
9. FFOE A creativity technique: Fluency (many ideas) Flexibility (variety of ideas) Originality (unique ideas) Elaboration (fully developed ideas).
10. DO IT A simple process for creativity:
Define problem Open mind and apply creative techniques Identify best solution Transform
11. PCD Possibilities, Consequences, Decision
12. MECE At McKinsey - every analysis is decomposed such that the issues are:
1. Mutually Exclusive: Each idea is distinct and separate; overlap represents muddled thinking and 2. Collectively Exhaustive: You've covered all the possibilities; you've thought of everything.
13. 5 C SITUATION ANALYSIS
Collaborators Customers Competitors Climate (or context)
14. PEST ANALYSIS Understanding 'big picture' forces of change
A PEST analysis is an analysis of the external macro-environment that affects all firms.
P.E.S.T. is an acronym for the Political, Economic, Social, and Technological factors of the external macro-environment.
15. GRASP Getting Results And Solving Problems
16. PACRA Purpose, Alternatives, Criteria, Resources, Action
17. ABCDEF Analyze - Brainstorm - Choose - Do -Evaluate - Finish
18. STAIR Steps S - State the problem T - Tools for the job A - Algorithm development I - Implementation of the algorithm R - Refinement
19. IDEAL identify the problem, D = define and represent the problem, E = explore possible strategies, A = act on the strategies, L = look back and evaluate the effects of your actions
20. CAUSED Can they do it, do they have a positive Attitude, is it Useful to them, are they Skilled in it, do they have similar Experience, is it Different.
21. S.O.D.A.S. S = Situation O = Options D = Disadvantages A = Advantages S = Solution
22. CAP Cover All Possibilities.
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Mnemonics for Solving Word Problems
I found two mnemonics that will help your child remember the steps to solving a problem. I suggest you choose the one that goes best with your math curriculum or your child’s learning style and make it into a poster to hang in your homeschool room or put in your child’s math folder. That way he can reference it easily until he memorizes it.
Here’s a mnemonic from Education World that uses the acronym SOLVE .
S tudy the problem. O rganize the facts. L ine up the plan. V erify the plan with computation. E xamine the answer.
And here’s another from Education World that uses the acronym STAR .
S earch the word problem. T ranslate the words into an equation. A nswer the problem. R eview the solution.
Know anyone else who could use these mnemonics for solving word problems? Share this post with them.
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Effective Word-Problem Instruction: Using Schemas to Facilitate Mathematical Reasoning
Sarah r. powell.
Department of Special Education, The University of Texas at Austin
Lynn S. Fuchs
Department of Special Education, Vanderbilt University
In a fourth-grade general education classroom, Mrs. Blanton posted her math lesson’s objective: Students will solve division word problems. During her instruction, Mrs. Blanton says, “In a word problem, the word share tells you to divide.” Mrs. Frank, a special education teacher, provides small-group instruction to Mrs. Blanton’s students with learning disabilities. During Mrs. Frank’s intervention time, she showed students the word problem of the day: On Wednesday, the coffee shop had 108 customers. The bookstore had 65 customers. How many more customers did the coffee shop have on Wednesday? Mrs. Frank reminds her students to use the Math Key Words Poster hanging in her resource room. The poster indicates that more means addition .
Many general and special education teachers across the U.S. teach word problems by defining problems as a single operation (e.g., “Today, we’re working on subtraction word problems”) and linking key words (e.g., more, altogether, share, twice ) to specific operations (e.g., share means to divide). Unfortunately, teaching students to approach word problems in these ways discourages mathematical reasoning and frequently produces incorrect answers. In Table 1 , we list eight common key words, identify the operation typically associated with each, and provide word problems that illustrate how reliance on key words can result in incorrect answers. Neither of these approaches—defining problems in terms of a single operation or linking key words to specific operations—has evidence to support its use.
Sample Key Words, Associated Operations, and Key Word Fails
In contrast, other approaches do promote mathematical reasoning and substantially boost word-problem performance among students with learning disabilities ( Fuchs et al., 2010 ; Griffin & Jitendra, 2009 ; Xin et al., 2011 ). Two practices that have emerged from high-quality research studies as particularly effective for word-problem instruction are: (a) attack strategies, which provide students with a general plan for processing and solving word problems ( Montague, 2008 ; Xin & Zhang, 2009 ), and (b) schema instruction, in which students learn to categorize word problems within problem types (i.e., schemas based on the word-problem’s mathematical structure); apply an efficient solution strategy for each word-problem schema; and understand the meaning of word-problem language ( Fuchs et al., 2014 ; Jitendra & Star, 2012 ). In this article, we focus on word problems commonly seen within textbooks and high-stakes assessments used within the U.S.
The Attack Strategy
An attack strategy is an easy-to-remember series of steps students use to guide their approach to solving word problems. A helpful attack strategy spans across schemas and grade levels. Researchers have determined that students’ use of an attack strategy is effective for improving word-problem performance ( Case et al., 1992 ; Fuchs et al., 2014 ; Jitendra, Griffin, Deatline-Buchman, Sczesniak, 2007 ; Jitendra & Star, 2012 ; Montague, 2008 ; Xin & Zhang, 2009 ). Some attack strategies address the first phase of word-problem solving—interpreting the word-problem’s meaning. During this phase, students read the problem, identify the question, and determine central idea of the problem (i.e., the schema or problem type). An attack strategy is important because many students skip this phase; instead, students will haphazardly select numbers from the word problem and rely on key words to identify an operation. Some attack strategies address the second phase of word-problem solving—finding the missing quantity. The second phase involves setting up a number sentence or using a graphic organizer, performing calculation(s), labeling the number answer, and checking whether the answer makes sense. In some cases, attack strategies address both phases.
In Figure 1 several different attack strategies are presented. The first four strategies make use of acronyms, which help students remember the attack strategy’s steps. An acronym is a mnemonic: a pattern of letters, ideas, or associations to help students remember something. Researchers have learned that mnemonics can help students with learning disabilities remember important information (e.g., Uberti, Scruggs, & Mastropieri, 2003 ) such as the steps of a general word-problem attack strategy. Although mnemonics can be helpful, attack strategies that do not make use of acronyms can also be effective. Students’ repeated use of the attack strategy facilitates retention. Although variations in attack strategies exist, the first part of word-problem solving across attack strategies is a thorough reading of the problem.
Sample attack strategies.
Whichever attack strategy the teacher selects, it is critical that the teacher explicitly models the attack strategy while explaining how it works. The teacher must also scaffold student learning of the attack strategy by decreasing levels of support until the attack strategy becomes a natural part of a student’s word-problem reasoning. Moreover, the teacher must also provide many opportunities for practice of the attack strategy with instructive corrective feedback. The exact amount of modeling, practice, and feedback depends on a student’s prior knowledge and skills and the quality of the teacher modeling and corrective feedback. Later in this article, we provide an example of Mrs. Frank modeling the RUN attack strategy as part of schema instruction.
Additive and Multiplicative Schemas
Schema instruction is a demonstrably effective instructional practice for promoting stronger word-problem performance for students with learning disabilities across grade levels (e.g., Fuchs, Craddock, et al., 2008 ; Fuchs, Seethaler, et al., 2008 ; Fuchs et al., 2010 ; Jitendra, Hoff, & Beck, 1999 ; Jitendra & Star, 2011 ; Powell et al., 2015 ). Whereas defining word problems by key words or operation has no research to support use for students with learning disabilities, schema instruction has a rich research base. Two categories of schemas that have broad usage for teachers are the additive and multiplicative schemas. These schemas can be used to solve word problems from kindergarten through eighth grade.
The three major additive schemas are combine, compare, and change problems. Each schema involves addition or subtraction concepts and procedures. Together, the three additive schemas (combine, compare, change) can be used to understand and solve any additive word problem. In Figure 2 , a definition, an equation with graphic organizer, an example problem, and variations for each schema is provided.
Combine problems put together two or more separate parts to make a sum or total (Part + Part = Total). Combine problems may also be called total or part-part-whole problems. In the upper elementary grades and middle school, combine problems often involve three or four parts (see variations in Figure 2 ). Combine problems require students to solve for the total or to find one of the parts. The top of Figure 3 provides two worked examples of a combine word problem: one requires the student to solve for the total (i.e., sum unknown problem), and the other requires students to solve for one of the parts (i.e., part unknown problem). Several validated schema instructional programs (e.g., Fuchs et al., 2009 ; Powell et al., 2015 ) employ the attack strategy RUN: R ead the problem; U nderline the label (i.e., what the problem is mostly about); and N ame the problem type. What follows is an example of Mrs. Frank teaching the second problem in Figure 3 .
Worked examples of additive word problems.
The combine equation (P1 + P2 = T) aids as an efficient solution strategy because students do not often understand how to organize the numbers presented in word problems. Note that, after setting up the equation 11 + ? = 29, students may add or subtract to solve the problem, depending on whether the missing information is one of the parts or the total. This shows that teachers should not describe this problem as an addition problem or subtraction problem during instruction; the deeper understanding of the problem is that it is a combine problem.
In compare problems, two sets are compared for a difference (Bigger − Smaller = Difference). Compare problems may also be called difference problems. Students may be asked to solve for the difference, the greater set, or the lesser set. To teach compare problems, such as the problem in Figure 3 , teachers should start with the RUN attack strategy. Then, teachers can use a graphic organizer to organize the word-problem information related to the compare schema. Teachers could also use a compare equation (B − s = D or G − L = D) to guide students in organizing word-problem information.
In change problems, an amount increases or decreases (i.e., changes) over time because something happens to change the starting amount (Start +/− Change = End). Change problems with an increase may be called join problems whereas change problems with a decrease may be called separate problems. Change problems can ask students to solve for an unknown start, change, or end amount. Change problems in the upper elementary and middle-school grades often involve multiple changes (see variations in Figure 2 ). In the worked examples of change problems of Figure 3 , a teacher starts with an attack strategy and then uses a change equation or change graphic organizer. Teachers should introduce these solution strategies (i.e., equation or graphic organizer) separately but allow students to choose the solution strategy they favor for daily use.
The three common multiplicative schemas, involving multiplication or division concepts, are equal groups, comparison, and proportions or ratios (see Figure 4 for definitions, graphic organizers, example problems, and variations). Students can use these three multiplicative schemas to represent and solve word problems in the upper elementary and middle-school grades.
In equal groups problems, a group or unit is multiplied by a specific number or rate for a product. Equal groups problems may also be called vary problems. The unknown may be the groups, the number or rate for each group, or the product. In the worked example in Figure 5 , the groups are unknown. A teacher should use an attack strategy to identify that the question is asking to determine the number of cartons of eggs. Then, the teacher models how to solve the problem. The equal groups graphic organizer allows for organization of the word-problem information. When solving the equation of ? × 12 = 60, students may multiply (i.e., what times 12 equals 60?) or divide (i.e., 60 divided by 12 equals what?). For this reason, teachers cannot describe these types of word problems as a multiplication or division problem. Instead, presenting word problems such as these through the equal groups schema promotes mathematical reasoning by encouraging students to solve the word problem algebraically (i.e., by balancing the two sides of the equation).
Worked examples of multiplicative word problems.
With comparison problems, a set is multiplied a number of times for a product. Even though the unknown may be the original set, the multiplier, or the product, students are most often asked to find the product in comparison problems. The worked example in Figure 5 is an example of a typical comparison problem. Teachers could use a graphic organizer to organize the information from the word problem. In addition, presenting this problem using a number line, with the set of 7 multiplied 3 times, could be helpful for students to understand the comparison of the word problem. The set of 7 is multiplied 3 times for a product of 21: 21 is compared to 7 as a multiple of 7.
With the proportions or ratio schema, students explore the relationships among quantities. This exploration helps students understand proportions, percentages, unit rate, or ratios. The unknown may be any part of the relationship. The worked examples in Figure 5 reflect the solving of a typical proportion or ratio word problems. When teaching both word problems, teachers should start with an attack strategy and then move to using a solution strategy (e.g., graphic organizer) that helps students understand how to organize the information presented in the word problem.
Grade Levels and Timelines for Introducing Schemas
As mentioned previously, an attack strategy should be introduced and practiced alongside schema instruction. Attack strategies are relatively simple and can be learned quickly; in contrast, understanding word-problem schemas and using a solution strategy (e.g., equation or graphic organizer) associated with each schema requires complex reasoning and a detailed set of skills. Developing mathematical reasoning related to the schemas takes sustained instruction that often spans the entire school year.
Additive schemas appear in mathematics materials as early as kindergarten, but typical schema introduction for additive schemas may start in first grade and continue across the elementary grades, depending upon the prior knowledge of students. Within a school year, we recommend introducing the additive schemas separately and providing mixed schema practice as new schemas are modeled and practiced. Among the three additive schemas, we recommend teaching combine problems first. This is because, when solving for missing parts in combine problems, the conceptual basis is the same no matter which of the parts is missing. Thus, the combine problem type is a relatively easy schema for establishing an understanding of the conceptual and procedural aspects of schema instruction.
With the additive schemas, we recommend teaching the compare problem type next. Compare problems are the most difficult of the three additive schemas. Teaching the compare problem schema after the combine schema allows students to benefit from the foundation of schema instruction achieved with combine problems. Teaching compare problems next means that students only need to distinguish between combine and compare problems (rather than among all three problem types). Teaching change problems last makes sense because the change problem’s central idea (increasing or decreasing) is the most story-like and intuitive of the three schemas.
In Table 2 , we provide a sample timeline, in weeks, for teaching the three additive schemas ( Fuchs et al., 2014 ; Powell et al., 2015 ). This timeline assumes the teacher is providing modeling and practice of word problems 2 or 3 times a week. Before schema instruction begins, the timeline includes an introductory unit in which the teacher teaches math skills foundational for schema instruction: single- or multi-digit addition and subtraction with and without regrouping; solving equations with missing information in any position (e.g., 4 + ? = 6, 2 + 4 = ?, ? − 4 = 2, 6 − ? = 2, 6 − 4 = ?); interpreting graphs and figures to find important information; and strategies for checking whether answers are reasonable. After the introductory unit, schema instruction begins with a dual focus on the attack strategy and word-problem schemas.
Sample Additive Schema Intervention
Unlike additive schemas, which are usually introduced and addressed within the same school year, equal groups and comparison schemas are typically featured during the elementary grades while the proportions or ratios schema is addressed more commonly in middle school. The equal groups schema is often introduced first because it represents the earliest explanations of multiplication and division (e.g., 3 × 2 is “three groups with two in each group”). Equal groups problems may initially be introduced in second or third grade. In third or fourth grade, the comparison schema should be explicitly taught. After the comparison schema is introduced, mixed practice should provide students with opportunities to distinguish between the equal groups and comparison schemas. In the middle school years, students should learn the proportions or ratio schema, with continued practice across the other additive and multiplicative schemas. The multiplicative schemas and additive schemas can be used to solve word problems with whole numbers or rational numbers. For example, the variations column in Figure 4 presents several multiplicative word problems with rational numbers.
Three Major Components of Effective Schema Instruction
Effective schema instruction incorporates the principles of explicit instruction, which have been shown to be necessary for students with learning disabilities ( Gersten et al., 2009 ). This includes providing explanations in simple, direct language; modeling efficient solution strategies instead of expecting students to discover strategies on their own; ensuring students have the necessary background knowledge and skills to succeed with those strategies; gradually fading support; providing multiple practice opportunities; and incorporating systematic cumulative review. As with attack strategies, the number of practice opportunities differs within schema instruction depending on the student’s incoming knowledge and skills, as well as the quality of teacher modeling, explanations, and corrective feedback.
Teaching What Each Schema Means
To explain the three components of effective schema instruction, we use the compare problem type, which is often the most difficult of the schemas for students to understand. Difficulty with the compare problem arises at least in part because its structure relies on subtraction that is conceptualized as a difference between two numbers. This is relatively or entirely unfamiliar to many students because subtraction is taught in schools primarily, or even exclusively, as taking away.
For Mrs. Frank, our special education teacher, her lesson’s goal is that students solve the problem introduced at the beginning of this article (see Figure 6 ). Before jumping to solving this compare problem, Mrs. Frank first presents intact compare stories with no missing quantities using concrete objects and actual student names. For example, Mrs. Frank introduces the compare schema by asking two students, Tina and Seth, to stand back to back, as she says: “Tina and Seth are students in my class. Tina is 43 inches tall. Seth is 48 inches tall. Seth is 5 inches taller than Tina.” Mrs. Frank then puts the compare graphic organizer (see the one aligned with height in Figure 2 ) on the board and leads a discussion in which she models and explains how to identify the boxes into which the bigger, smaller, and difference numbers go. Students discuss filling in the graphic organizer with a variety of compare stories, while Mrs. Frank gradually transfers responsibility to the students, all the time providing corrective feedback.
Sample compare problem.
When students are secure in their understanding of the central idea of the compare schema, Mrs. Frank proceeds by introducing the compare equation. Mrs. Frank uses the compare equation of B − s = D in which B stands for the bigger quantity, s for the smaller quantity, and D for the difference between the quantities. Mrs. Frank explains how the compare equation maps to the graphic organizer, and students use intact stories to practice filling in the graphic organizer and the equation. Mrs. Frank presents the equation and graphic organizer not only to confirm students’ understanding of the word-problem schema but also to help them organize the numbers in word problems.
Teaching a Solution Strategy for Each Schema
After students understand the meaning of a schema (e.g., compare problems compare two amounts for a difference), students learn to select a solution strategy and use the solution strategy to organize the information from the word problem. Teaching a solution strategy involves modeling from the teacher and practice opportunities in which students receive feedback from the teacher.
Now that Mrs. Frank’s students understand what the compare schema means and have mastered the RUN attack strategy within combine schema instruction, Mrs. Frank explicitly models how to solve compare problems. Initially, she uses a word problem with a difference missing. Students complete the same set of activities with the graphic organizer and equation, writing missing information into the graphic organizer and using a blank or question mark to represent the missing information in the equation. Gradually, Mrs. Frank omits the concrete manipulatives, integrates novel names into problems, and substitutes a hand gesture for easy reference to the graphic organizer (one of her hands parallel to the floor at about nose height; the other parallel to the floor at about chest height).
To practice an efficient solution strategy, Mrs. Frank begins to use the compare equation more often than the graphic organizer. She instructs students to write the compare equation as soon as students identify the word-problem schema. In Figure 6 , the compare equation is B − s = D. First, Mrs. Frank helps the students identify that the coffee shop is the bigger amount (marked with a B above “coffee shop”) and the bookstore is the smaller amount (marked with an s above “bookstore”). She then models how to rewrite the equation with quantities from the word problem as replacements for B, s, and D, using a question mark or a blank to stand in for the missing quantity (108 − 65 = ?). Then, she works with the students to do the computation in different ways (e.g., 108 − 65 or 65 + ? = 108). Mrs. Frank concludes by writing the answer (? = 43 more customers) and checking the reasonableness of the answer (108 − 65 = 43). As this instruction occurs, Mrs. Frank provides many practice opportunities for students and provides focused affirmative and corrective feedback.
After students learn to recognize word problems as belonging to schemas and are consistently using an efficient solution strategy (i.e., equation or graphic organizer) to organize the necessary word-problem information, the next phase of instruction involves explicitly teaching word-problem specific vocabulary and language.
Teach Important Vocabulary and Language Constructions
Word-problem solving relies heavily on reading and understanding language. Typically developing students often understand important math vocabulary prior to school entry and gradually learn to treat this language (e.g., all or more ) in a special, task-specific way involving more complicated constructions about sets ( in all and more than) . Many teachers assume that students have the necessary language comprehension to understand word problems and the problem’s schema. But for students with learning disabilities, this is a shaky assumption.
A strong focus on vocabulary and language is therefore important, especially for students with learning disabilities. Examples of vocabulary and constructions that require explicit instruction, focused on the meaning of the language, are (a) joining words (e.g., altogether, in all ) and superordinate categories (e.g., animals mean both dogs and cats ) in combine problems; (b) compare words (e.g., more, fewer, than , - er words) and adjective - er versus verb - er words (e.g., bigger vs. teacher ) in compare problems; and (c) cause-effect conjunctions (e.g., then, because, so ), implicit change verbs (e.g., cost, ate, found ), and time passage phrases (e.g., 3 hours later, the next day ) in change problems. We also recommend a focus on confusing cross-problem constructions (e.g., more than vs. then … more ) and “tricky” labels (e.g., questions with superordinate category words).
For multiplicative schemas, students should learn how words often featured in additive problems (e.g., more ) may be used within multiplicative problems (e.g., “How many times more flowers did Danica pick?”). It is also important for students to understand how to compare quantities with different units (e.g., minutes and hours ), and how in proportions, the units must be a focus of the organization of the problem (i.e., minutes compared to minutes ). For multiplicative problems, students must also learn math-specific vocabulary, such as ratio, rate , and percentage , the interpretation of such terms within word problems, and the variety of ways fractions and multiplicative relationships can be expressed.
We emphasize that word-problem specific language instruction should not teach students to rely on key words for recognizing schemas. As illustrated in Table 1 , key words do not help students become word-problem thinkers, and reliance on key words fails to produce correct answers much of the time. We recommend teaching students specifically how and why “grabbing numbers and key words” to form number sentences frequently produces wrong answers.
Teaching students to avoid using key words is accomplished in three ways. First, the teacher explicitly teaches how math words mean different things in the context of a story, so reading the full word problem is necessary to distinguish among meanings. Reading the entire word problem is one thing that students do not always do, and it is one of the reasons an attack strategy is necessary. For example, sharing a quantity in equal parts may refer to multiplicative problems (e.g., “Max had 80 dog biscuits and shared them equally among 10 dogs”), whereas sharing part of a unit or collection may refer to additive problems (e.g., “Max had 80 dog biscuits and shared 40 of them with his dogs”). Second, the teacher demonstrates solving problems using key words, while eliciting student discussion about how and why this approach produces mistakes (e.g., more does not reliably mean add; share does not reliably mean divide). Third, the teacher structures activities in which the class analyzes worked problems from “last year’s class” to identify how key words can lead students astray.
Multi-Step Word Problems
For solving one-step word problems, students learn to use a single schema. Solving word problems, however, is not always a one-step activity. To challenge students and engage students in mathematical reasoning, multi-step word problems are posed in many textbooks, on high-stakes tests, and in many authentic situations. Fortunately, when students understand word-problem schemas, solving multi-step word problems is much easier. This is because multi-step problems can incorporate more than one schema. For example, “Nathan bought 12 glazed donuts and 16 chocolate donuts for his class. The class ate 23 donuts. How many donuts does Nathan have left?” In this multi-step problem, students first use the combine schema to calculate that Nathan bought 28 donuts. Then, they apply the change schema to determine the change in number of donuts (i.e., 28 − 23 = ?).
Multi-step problems can also combine additive and multiplicative schemas. For example, “Nathan bought 12 glazed donuts and 16 chocolate donuts for his class. Each donut costs $1.10. How much did Nathan spend?” Students may first use the combine schema (i.e., 12 + 16 = 28 donuts) and then the equal groups schema (28 donuts × $1.10 each = ? cost). Note that there are other approaches to solving this problem. Some students may calculate the cost of the glazed donuts using an equal groups schema and then calculate the cost of the chocolate donuts using an equal groups schema. Finally, students may use the combine schema to determine the total cost of the glazed and chocolate donuts.
Key words, as mentioned near the start of this article, also fail as a strategy for solving multi-step word problems. For example, “For a bake sale, Katie baked 52 cupcakes but shared 4 of the cupcakes with her brother before taking the cupcakes to the sale. Buzz baked 42 cupcakes. How many cupcakes could Katie and Buzz sell altogether at the bake sale?” In this problem, some students may – without reading the problem – interpret share as meaning division or altogether as meaning addition. Neither word, processed in isolation and tied to an operation, produces a correct answer to this multi-step problem.
Summing Up: What to Do (and Not Do)
Schema instruction can be a powerful tool for helping students understand and solve word problems. Schema instruction facilitates mathematical reasoning by helping students understand the underlying structures within word problems that will be used across grade levels and with whole and rational numbers. To close, we summarize several key dos and do nots for teaching schemas.
Do not teach students to solve word problems by isolating key words and linking those words to operations. Don’t say things like “ share tells us to divide.” Teaching students what share means helps students understand the conceptual schema of the word problem but telling students to divide whenever they see share is error fraught. In a similar vein, do not define word problems by an operation. Do not say, “Today we’re working on division word problems.” There is no such thing as a “subtraction” word problem because some students may use addition to solve such a problem; others may use subtraction. Defining a word problem by operation undermines conceptual understanding.
On the other hand, to promote mathematical reasoning related to word problems, do explicitly teach word-problem solving. Students with learning disabilities benefit from explicit instruction on effective strategies for solving word problems. Do allocate sustained instructional time across the school year for teaching word problems. Do teach an attack strategy to help students understand how to work systematically through a word problem. Do teach the additive and multiplicative schemas, emphasizing what the schema means. Do use equations, graphic organizers, and hand gestures to help students understand the schema’s mathematical structure and organize word-problem information. Do include multi-step word problems that mix schemas. Do provide cumulative review across schemas, which mixes problems with and without irrelevant information, with and without problems that contain important information in graphs and figures, and with missing information in all slots of the schema’s equation. Finally, do provide explicit instruction on word-problem vocabulary and language constructions that provide students access to the meaning of word problems.
This research was supported in part by Grant R324A150078 from the Institute of Education Sciences in the U.S. Department of Education to the University of Texas at Austin and by Grants R01 HD053714 and P20 HD075443 from the Eunice Kennedy Shriver National Institute of Child Health & Human Development to Vanderbilt University. The content is solely the responsibility of the authors and does not necessarily represent the official views of the Institute of Education Sciences, the U.S. Department of Education, the Eunice Kennedy Shriver National Institute of Child Health & Human Development, or the National Institutes of Health.
Sarah R. Powell, Department of Special Education, The University of Texas at Austin.
Lynn S. Fuchs, Department of Special Education, Vanderbilt University.
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Journal of Science Education for Students with Disabilities
Home > JSESD > Vol. 16 (2013) > Iss. 1
Solving word problems: as easy as pies.
Mary Jane Heater , Fairfax Co. Public Schools Follow Lori A. Howard , Marshall University Follow Ed Linz Follow
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Many students are challenged when tasked to complete a word problem. While they may know the procedural steps to solve an equation, translating a word problem into an appropriate equation and producing a solution may often cause students to become confused or unwilling to try. This article provides a potential solution for teachers by discussing the use of a simple mnemonic tool to help organize the process. Mnemonics are a useful and effective strategy to help students with learning disabilities remember information and process steps. In the strategy presented, the mnemonic PIES is used to describe a 4-step process for solving word problems in which the acronym is described as P=Picture (draw a simple sketch) based on the situation described by the word problem), I=Information (circle key words in the problem and write next to picture), E=Equation (find an equation that fits the information), and S=Solve (solve the equation to produce an answer). PIES has been successfully used with all students in an inclusive high school Physics classroom, as well as self-contained high school science classes. Suggestions and an example for teachers are included.
Heater, Mary Jane; Howard, Lori A.; and Linz, Ed (2013) "Solving Word Problems: As Easy as PIES!," Journal of Science Education for Students with Disabilities : Vol. 16 : Iss. 1, Article 3. DOI: 10.14448/jsesd.05.0002 Available at: https://scholarworks.rit.edu/jsesd/vol16/iss1/3
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How to Teach Word Problems
Last Updated: September 3, 2020
This article was co-authored by Daron Cam and by wikiHow staff writer, Danielle Blinka, MA, MPA . Daron Cam is an Academic Tutor and the Founder of Bay Area Tutors, Inc., a San Francisco Bay Area-based tutoring service that provides tutoring in mathematics, science, and overall academic confidence building. Daron has over eight years of teaching math in classrooms and over nine years of one-on-one tutoring experience. He teaches all levels of math including calculus, pre-algebra, algebra I, geometry, and SAT/ACT math prep. Daron holds a BA from the University of California, Berkeley and a math teaching credential from St. Mary's College. This article has been viewed 8,921 times.
Word problems are a great way to prepare your students for using math in real life. However, many students find them intimidating.  X Research source First, introduce word problems so that your students will understand why they're used. Next, explain how to use the CUBES approach to solving word problems. You can then model the process for your students and have them practice it on their own.
Introducing Word Problems
- Ask students to work with a partner or group to identify ways that they might use math in real life. Next, have them turn that situation into a math-based story.
- For example, they might say they use math when dividing into teams at recess. They could turn that into a story like this: “If the class has 20 students and we need 2 teams to play, how many students will be on each team?”
- Suggest that they incorporate a hobby or favorite interest. For example, they might write a problem that has to do with a sport they play or a problem that centers around a favorite animal.
- Have them put their name in the problem.
- Here’s an example: “If Alex wants to score 10 soccer goals over the second half of the season and there are 5 more games left to play, how many goals should Alex score per game to stay on track?”
- You may need to do this several times for students to understand.
- Your sample problem might look like this: “Sarah is buying pizza for her slumber party. If each pizza has a certain number of slices and she knows how many guests are coming, how can she estimate the number of pizzas she needs. What other information might she need to arrive at the answer?” Your students should point out that Sarah would need to know how many pieces of pizza her guests will eat on average. She could then multiple that number of slices by the number of guests. To get the number of pizzas, she’d then divide the total number of slices by the number of slices in a pizza.
- For example, when determining how many pizzas Sarah will need to buy for her slumber party, the students could draw circles to represent the pizzas and stick figures to represent Sarah and her guests.
- However, stress that this is not a requirement for arriving at your answer, as some students may find this extra step frustrating.
- This can help visual learners better understand word problems.
- For example, students who can only add numbers should be given word problems that only require addition.
Using the CUBES Process
- C-Circle the numbers.
- U-Underline the question.
- B-Box the keywords.
- E-Eliminate unnecessary information.
- S-Show your work.
- Although some numbers may be extraneous information, it’s important that students circle all of them during this step. They can eliminate unnecessary information later.
- For example, your problem might look like this: “Katie asked 7 friends to sleepover on Friday, but only 4 can make it. She knows that she and each friend will eat 6 chicken nuggets, and each bag of frozen nuggets contains 15 nuggets. How many bags will she need to make sure she has enough nuggets?” Students should circle the 7, 4, 6, and 15.
- For example, they might need to know how much of something they’d need or how much of something they’d have left.
- When students are first learning word problems, the question should be at the end of the problem. It will often have a question mark at the end.
- In the problem above, the question is this: "How many bags will she need to make sure she has enough nuggets?”
- For example, addition could be indicated by words like “in total” or “all together.” Similarly, subtraction should be signified by “difference” or “less.”
- In the problem above, students would draw boxes around "only," which will indicate subtracting. They'll also draw boxes around "how many" and "enough," which suggest that they'll need to first multiply and then divide.
- For example, the sample problem has extra numbers: “Katie asked 7 friends to sleepover on Friday, but only 4 can make it. She knows that she and each friend will eat 6 chicken nuggets, and each bag of frozen nuggets contains 15 nuggets. How many bags will she need to make sure she has enough nuggets?” The students could cross out the 7, since it isn’t necessary for arriving at the correct answer.
- 4 friends are coming, plus Katie: 4 + 1 = 5 girls
- Each of them will eat 6 nuggets: 5 x 6 = 30 nuggets
- Each bag has 15 nuggets: 30 / 15 = 2 bags needed
- Answer: Katie needs to buy 2 bags of nuggets.
Modeling the Process
- You will need to work through several problems until your students start to grasp the concepts. Make sure the problems you choose match your students' language and math skills.
- You can work through the problems on the board or using a projector.
- For example, you could solve this problem: "Diego is baking cookies for his team's bake sale. Last year, he earned $100 to help his team. This year, he wants his sales to reach $120. If each cookie will cost $0.75, how many cookies will Diego need to bake?"
- Say, "I'm going to start by circling the numbers." Circle 100, 120, and 0.75.
- Next, say, "“Now I’m going to underline this, since I know it’s the question.” You'd underline this statement: "If each cookie will cost $0.75, how many cookies will Diego need to bake?"
- Say, “Now I’m going to cross this out, since the problem isn’t asking me about this." You'd cross out the sentence about Diego earning $100 last year.
- Write the steps in order to make it easier for students to copy down.
- Diego needs to make $120 by selling cookies for $0.75: 120 / .75 = 160
- Answer: He will need to bake 160 cookies.
- Not only does this show students what to watch out for, it also shows them that mistakes can happen to anyone.
- For example, you might have accidentally used the information about Diego earning $100 last year, which isn't relevant to the question.
- If a student gets the answer wrong, praise the effort, then ask another student to help. Say, “Great enthusiasm, James! We’re almost to that step, but we need to do something else first. Amy, could you help us out?”
Giving Students Practice
- Remind them to use CUBES to solve their problems.
- You can make sure students are on track by listening to what they are saying. Are they correctly explaining the process? Do they appear to be confused? Use your judgement to determine if you need to offer guidance.
- Praise groups that are on track. Say, "Excellent work!" or "I'm proud of the progress you're making!"
- It’s a good idea to have them do this in groups, as it increases the quality of the questions. Also, it’ll allow you to check each group’s questions before they switch papers, which would be harder if each student created questions.
- Not only is this fun for students, but it encourages higher-level learning.
- You can make up your own worksheets, or you can download them for free online.  X Research source Make sure that the word problems you give them match their language and math skills.
- When making your own worksheets, it’s best to work through the problems before you give them to students to make sure all of the information is present. Also, stick to topics your students will understand, such as sports, pets, and food.
Sample Word Problems and Illustrated CUBES Process
- Word problems are difficult for many learners, so have patience. Thanks Helpful 0 Not Helpful 0
- Encourage students to keep trying, even if they don’t get it at first. Thanks Helpful 0 Not Helpful 0
- You can find several different acronyms for solving word problems, but CUBES incorporates all of the steps. Thanks Helpful 0 Not Helpful 0
- Make sure that the practice problems you choose don’t include mathematical concepts that are beyond your students. When in doubt, work through the problem yourself before giving it to your young learners. Thanks Helpful 0 Not Helpful 0
You Might Also Like
- ↑ https://www.homeschoolmath.net/teaching/problem_solving.php
- ↑ https://www.teachingchannel.org/blog/2016/04/07/math-word-problems/
- ↑ Daron Cam. Academic Tutor. Expert Interview. 29 May 2020.
- ↑ https://www.weareteachers.com/struggling-with-math-word-problems/
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Algebra Topics - Introduction to Word Problems
Algebra topics -, introduction to word problems, algebra topics introduction to word problems.
Algebra Topics: Introduction to Word Problems
Lesson 9: introduction to word problems.
What are word problems?
A word problem is a math problem written out as a short story or scenario. Basically, it describes a realistic problem and asks you to imagine how you would solve it using math. If you've ever taken a math class, you've probably solved a word problem. For instance, does this sound familiar?
Johnny has 12 apples. If he gives four to Susie, how many will he have left?
You could solve this problem by looking at the numbers and figuring out what the problem is asking you to do. In this case, you're supposed to find out how many apples Johnny has left at the end of the problem. By reading the problem, you know Johnny starts out with 12 apples. By the end, he has 4 less because he gave them away. You could write this as:
12 - 4 = 8 , so you know Johnny has 8 apples left.
Word problems in algebra
If you were able to solve this problem, you should also be able to solve algebra word problems. Yes, they involve more complicated math, but they use the same basic problem-solving skills as simpler word problems.
You can tackle any word problem by following these five steps:
- Read through the problem carefully, and figure out what it's about.
- Represent unknown numbers with variables.
- Translate the rest of the problem into a mathematical expression.
- Solve the problem.
- Check your work.
We'll work through an algebra word problem using these steps. Here's a typical problem:
The rate to rent a small moving van is $30 per day, plus $0.50 per mile. Jada rented a van to drive to her new home. It took two days, and the van cost $360. How many miles did she drive?
It might seem complicated at first glance, but we already have all of the information we need to solve it. Let's go through it step by step.
Step 1: Read through the problem carefully.
With any problem, start by reading through the problem. As you're reading, consider:
- What question is the problem asking?
- What information do you already have?
Let's take a look at our problem again. What question is the problem asking? In other words, what are you trying to find out?
The rate to rent a small moving van is $30 per day, plus $0.50 per mile. Jada rented a van to drive to her new home. It took 2 days, and the van cost $360. How many miles did she drive?
There's only one question here. We're trying to find out how many miles Jada drove . Now we need to locate any information that will help us answer this question.
There are a few important things we know that will help us figure out the total mileage Jada drove:
- The van cost $30 per day.
- In addition to paying a daily charge, Jada paid $0.50 per mile.
- Jada had the van for 2 days.
- The total cost was $360 .
Step 2: Represent unknown numbers with variables.
In algebra, you represent unknown numbers with letters called variables . (To learn more about variables, see our lesson on reading algebraic expressions .) You can use a variable in the place of any amount you don't know. Looking at our problem, do you see a quantity we should represent with a variable? It's often the number we're trying to find out.
Since we're trying to find the total number of miles Jada drove, we'll represent that amount with a variable—at least until we know it. We'll use the variable m for miles . Of course, we could use any variable, but m should be easy to remember.
Step 3: Translate the rest of the problem.
Let's take another look at the problem, with the facts we'll use to solve it highlighted.
The rate to rent a small moving van is $30 per day , plus $0.50 per mile . Jada rented a van to drive to her new home. It took 2 days , and the van cost $360 . How many miles did she drive?
We know the total cost of the van, and we know that it includes a fee for the number of days, plus another fee for the number of miles. It's $30 per day, and $0.50 per mile. A simpler way to say this would be:
$30 per day plus $0.50 per mile is $360.
If you look at this sentence and the original problem, you can see that they basically say the same thing: It cost Jada $30 per day and $0.50 per mile, and her total cost was $360 . The shorter version will be easier to translate into a mathematical expression.
Let's start by translating $30 per day . To calculate the cost of something that costs a certain amount per day, you'd multiply the per-day cost by the number of days—in other words, 30 per day could be written as 30 ⋅ days, or 30 times the number of days . (Not sure why you'd translate it this way? Check out our lesson on writing algebraic expressions .)
$30 per day and $.50 per mile is $360
$30 ⋅ day + $.50 ⋅ mile = $360
As you can see, there were a few other words we could translate into operators, so and $.50 became + $.50 , $.50 per mile became $.50 ⋅ mile , and is became = .
Next, we'll add in the numbers and variables we already know. We already know the number of days Jada drove, 2 , so we can replace that. We've also already said we'll use m to represent the number of miles, so we can replace that too. We should also take the dollar signs off of the money amounts to make them consistent with the other numbers.
30 ⋅ 2 + .5 ⋅ m = 360
Now we have our expression. All that's left to do is solve it.
Step 4: Solve the problem.
This problem will take a few steps to solve. (If you're not sure how to do the math in this section, you might want to review our lesson on simplifying expressions .) First, let's simplify the expression as much as possible. We can multiply 30 and 2, so let's go ahead and do that. We can also write .5 ⋅ m as 0.5 m .
60 + .5m = 360
Next, we need to do what we can to get the m alone on the left side of the equals sign. Once we do that, we'll know what m is equal to—in other words, it will let us know the number of miles in our word problem.
We can start by getting rid of the 60 on the left side by subtracting it from both sides .
The only thing left to get rid of is .5 . Since it's being multiplied with m , we'll do the reverse and divide both sides of the equation with it.
.5 m / .5 is m and 300 / 0.50 is 600 , so m = 600 . In other words, the answer to our problem is 600 —we now know Jada drove 600 miles.
Step 5: Check the problem.
To make sure we solved the problem correctly, we should check our work. To do this, we can use the answer we just got— 600 —and calculate backward to find another of the quantities in our problem. In other words, if our answer for Jada's distance is correct, we should be able to use it to work backward and find another value, like the total cost. Let's take another look at the problem.
According to the problem, the van costs $30 per day and $0.50 per mile. If Jada really did drive 600 miles in 2 days, she could calculate the cost like this:
$30 per day and $0.50 per mile
30 ⋅ day + .5 ⋅ mile
30 ⋅ 2 + .5 ⋅ 600
According to our math, the van would cost $360, which is exactly what the problem says. This means our solution was correct. We're done!
While some word problems will be more complicated than others, you can use these basic steps to approach any word problem. On the next page, you can try it for yourself.
Let's practice with a couple more problems. You can solve these problems the same way we solved the first one—just follow the problem-solving steps we covered earlier. For your reference, these steps are:
If you get stuck, you might want to review the problem on page 1. You can also take a look at our lesson on writing algebraic expressions for some tips on translating written words into math.
Try completing this problem on your own. When you're done, move on to the next page to check your answer and see an explanation of the steps.
A single ticket to the fair costs $8. A family pass costs $25 more than half of that. How much does a family pass cost?
Here's another problem to do on your own. As with the last problem, you can find the answer and explanation to this one on the next page.
Flor and Mo both donated money to the same charity. Flor gave three times as much as Mo. Between the two of them, they donated $280. How much money did Mo give?
Problem 1 Answer
Here's Problem 1:
A single ticket to the fair costs $8. A family pass costs $25 more than half that. How much does a family pass cost?
Let's solve this problem step by step. We'll solve it the same way we solved the problem on page 1.
Step 1: Read through the problem carefully
The first in solving any word problem is to find out what question the problem is asking you to solve and identify the information that will help you solve it . Let's look at the problem again. The question is right there in plain sight:
So is the information we'll need to answer the question:
- A single ticket costs $8 .
- The family pass costs $25 more than half the price of the single ticket.
Step 2: Represent the unknown numbers with variables
The unknown number in this problem is the cost of the family pass . We'll represent it with the variable f .
Step 3: Translate the rest of the problem
Let's look at the problem again. This time, the important facts are highlighted.
A single ticket to the fair costs $8 . A family pass costs $25 more than half that . How much does a family pass cost?
In other words, we could say that the cost of a family pass equals half of $8, plus $25 . To turn this into a problem we can solve, we'll have to translate it into math. Here's how:
- First, replace the cost of a family pass with our variable f .
f equals half of $8 plus $25
- Next, take out the dollar signs and replace words like plus and equals with operators.
f = half of 8 + 25
- Finally, translate the rest of the problem. Half of can be written as 1/2 times , or 1/2 ⋅ :
f = 1/2 ⋅ 8 + 25
Step 4: Solve the problem
Now all we have to do is solve our problem. Like with any problem, we can solve this one by following the order of operations.
- f is already alone on the left side of the equation, so all we have to do is calculate the right side.
- First, multiply 1/2 by 8 . 1/2 ⋅ 8 is 4 .
- Next, add 4 and 25. 4 + 25 equals 29 .
That's it! f is equal to 29. In other words, the cost of a family pass is $29 .
Step 5: Check your work
Finally, let's check our work by working backward from our answer. In this case, we should be able to correctly calculate the cost of a single ticket by using the cost we calculated for the family pass. Let's look at the original problem again.
We calculated that a family pass costs $29. Our problem says the pass costs $25 more than half the cost of a single ticket. In other words, half the cost of a single ticket will be $25 less than $29.
- We could translate this into this equation, with s standing for the cost of a single ticket.
1/2s = 29 - 25
- Let's work on the right side first. 29 - 25 is 4 .
- To find the value of s , we have to get it alone on the left side of the equation. This means getting rid of 1/2 . To do this, we'll multiply each side by the inverse of 1/2: 2 .
According to our math, s = 8 . In other words, if the family pass costs $29, the single ticket will cost $8. Looking at our original problem, that's correct!
So now we're sure about the answer to our problem: The cost of a family pass is $29 .
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Most Used Actions
- How do you solve word problems?
- To solve word problems start by reading the problem carefully and understanding what it's asking. Try underlining or highlighting key information, such as numbers and key words that indicate what operation is needed to perform. Translate the problem into mathematical expressions or equations, and use the information and equations generated to solve for the answer.
- How do you identify word problems in math?
- Word problems in math can be identified by the use of language that describes a situation or scenario. Word problems often use words and phrases which indicate that performing calculations is needed to find a solution. Additionally, word problems will often include specific information such as numbers, measurements, and units that needed to be used to solve the problem.
- Is there a calculator that can solve word problems?
- Symbolab is the best calculator for solving a wide range of word problems, including age problems, distance problems, cost problems, investments problems, number problems, and percent problems.
- What is an age problem?
- An age problem is a type of word problem in math that involves calculating the age of one or more people at a specific point in time. These problems often use phrases such as 'x years ago,' 'in y years,' or 'y years later,' which indicate that the problem is related to time and age.
- High School Math Solutions – Systems of Equations Calculator, Elimination A system of equations is a collection of two or more equations with the same set of variables. In this blog post,... Read More
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Last updated on September 28, 2021 by Not So Wimpy Teacher
A Simple Problem Solving Strategy That Works Every Time
I love teaching math. And not just any math, but word problems especially.
There, I said it. I love teaching word problems.
I know not all teachers share my fondness for word problems. And many students dread them. But I’m going to share a simple problem-solving strategy that will have you and your students loving word problems too.
Okay, maybe you won’t all love them. But at least you won’t fear them anymore.
Abandon the Acronyms
You may be familiar with one of these popular problem-solving acronyms. STEP, PAWS, ROCKS, CUBES, RIDES, etc… You may even be using one of them.
These “strategies” go by a number of different names, but they all include circling, highlighting, underlining, and/or boxing key words and numbers. Each letter of the title stands for a different step that helps break down a complex process and keeps students moving toward a solution.
Unfortunately, these steps don’t really teach kids how to problem solve. These “helpful” tasks can actually hinder students’ efforts to solve problems and frustrate them in the process.
Key words can be confusing for kids. Often the same ones are used for multiple operations. For instance, addition and multiplication share many of the same keywords. And the word “total” can be used with any of the four operations.
Also, if students don’t recognize the key word, then this step isn’t helpful at all.
A strategy that asks kids to circle all the numbers becomes confusing when there are extra numbers included in a problem. And rather than focus on the math, students become preoccupied with checking off steps in the strategy.
Often kids are so programmed to circle, box and highlight they launch right into these steps before they even finish reading the problem. That often leads to confusion and lots of wrong answers.
Fast learners get frustrated with these processes. They view all the highlighting, circling and underlining as unnecessary. It feels like busywork and seems wholly unrelated to the math task at hand.
Instead of an acronym-based process what kids and teachers need is a simple strategy that works every time. And I’m going to share that process with you today.
A Problem Solving Strategy that Works
Problem solving step 1: read the entire problem.
This step might be obvious to us, but kids are impulsive and impatient. They want to jump right into doing the math.
Example: My teacher gave me three boxes of books. Each box has 5 books in it. How many books do I have in all?
Step one is to put down the pencil and just read.
You will need to model how to do this. I demonstrate this first step every time I solve a problem in front of the class. I put down my marker and read the entire problem out loud.
You may also want to have your students put their hands in the air or on their heads. Older kids can put a finger to their temples or their chins to show they are thinking.
Ask students to visualize what the problem looks like. Can they see a picture in their minds?
Problem Solving Step 2: Reread and Draw a Math Model
The next step is to reread the problem one sentence at a time. As students read, they draw a model that represents the words they are reading.
For the sample problem, students would draw three squares to represent the boxes. Then they would add five dots or lines to each square to represent the books.
This helps kids organize the information and see things in perspective. By drawing a picture, they can visualize how many of something they are dealing with. Pictures also help them make connections between the different parts of the problem.
Over time, kids will also make associations between different drawings and operations. They will learn that when they draw an array, they are multiplying. Or when they draw a set number of groups they are dividing.
Sometimes a model is off when students draw it step-by-step. But that’s okay. It teaches them trial and error, perseverance, and resilience. And, if kids can recognize that their model doesn’t match the problem, then they really understand the math.
Helpful hints for math models:
- Encourage students to make math models, not pictures—drawings don’t have to be pretty and they don’t have to be perfect. For instance if the problem says the classroom started the year with 100 pencils you don’t want kids drawing 100 pencils, or even making 100 tally marks. A box with the number 100 written on it is sufficient.
- Model ways to make simple models using dots or boxes to represent complex shapes.
- Over the course of the year introduce your class to many different types of models, including arrays, equal groups, bar/tape models, number lines, and area models.
Problem Solving Step 3: Write an Equation and Solve
This is everyone’s favorite step. The third step in the process is to use the model to write a number sentence. A well-drawn model should provide all the information the student needs to write the number sentence.
Example: Students can go back and count the three boxes they drew and the five dots inside each one and write the number sentence: 3 X 5=15
Once the number sentence is written, students solve for the unknown. And after they find the solution, they can use the model to check their answer.
In the sample problem students would do this by counting the total number of dots in each box they drew in the model.
Problem Solving Step 4: Write Your Answer in a Sentence
The last step is to write the answer in a complete sentence. Many teachers require their students to label the answer. But I go I step further. I want my students to write the answer in the sentence.
You will need to model how to go back to the word problem and use part of the question in their answer.
Example: I have fifteen books in all.
This step increases the odds that students will check to see if their answer makes sense and actually answer the question that was asked. It’s also great practice for writing and grammar skills.
How to Help Students Who Struggle with Word Problems
- Don’t panic, take your time
- Practice word problems every day and use these 4 steps every time you solve a problem
- Post the problem-solving strategy in your classroom and/or give students their own printed copies of each step
- Use word problems to introduce new concepts
- Identify common types of word problems
- Scaffold students with sentence starters
- Have students share ideas/strategies with one another
- Practice, practice, practice
Looking for more ways to get your students writing about math. Check out my Math Journals . They come with tons of great word problems that ask students to explain their mathematical thinking. They are available for grades 2, 3, and 4, in both printable and digital formats. There is also a printable version for grade 5 .
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Second Grade Math Journal (with Word Problems)
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February 15, 2021 at 11:11 am
I LOVE this. I think it is very important for kids to understand the problem before they start to solve. Getting away from following steps deepens their understanding of the math. One thing I do in my class is have my students write an answer statement first. They leave a blank for the answer, of course, but I feel this focuses their attention as little more on what they are trying to solve in the problem.
February 22, 2021 at 10:42 pm
Julie, Asking students to write an answer statement first goes right along with answering in a complete sentence. Sounds like it’s a great technique you’ve developed for your class!
February 17, 2021 at 9:12 am
I am one of those teacher that teacher using the acronym CUBES. It worked for some of my students but other it was a hard concept to grasp. So seeing there is another option that can really help student understand word problems I’m excited to try it out in my guided math, intervention as well as after school tutoring. What’s your take on using number less word problem?
February 22, 2021 at 10:38 pm
During episode 79 of the Not So Wimpy Teacher podcast I talk with Brittany Hege about wordless math problems. Wordless math problems are a great way for students to understand what action is happening in the problem.
February 27, 2021 at 4:51 pm
I ask myself…”Will the answer be larger or smaller that the number I started with?” That tells me what to do.
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Sat / act prep online guides and tips, the pemdas rule: understanding order of operations.
Everyone who's taken a math class in the US has heard the acronym "PEMDAS" before. But what does it mean exactly? Here, we will explain in detail the PEMDAS meaning and how it's used before giving you some sample PEMDAS problems so you can practice what you've learned.
PEMDAS Meaning: What Does It Stand For?
PEMDAS is an acronym meant to help you remember the order of operations used to solve math problems. It's typically pronounced "pem-dass," "pem-dozz," or "pem-doss."
Here's what each letter in PEMDAS stands for:
- P arentheses
- M ultiplication and D ivision
- A ddition and S ubtraction
The order of letters shows you the order you must solve different parts of a math problem , with expressions in parentheses coming first and addition and subtraction coming last.
Many students use this mnemonic device to help them remember each letter: P lease E xcuse M y D ear A unt S ally .
In the United Kingdom and other countries, students typically learn PEMDAS as BODMAS . The BODMAS meaning is the same as the PEMDAS meaning — it just uses a couple different words. In this acronym, the B stands for "brackets" (what we in the US call parentheses) and the O stands for "orders" (or exponents). Now, how exactly do you use the PEMDAS rule? Let's take a look.
How Do You Use PEMDAS?
PEMDAS is an acronym used to remind people of the order of operations.
This means that you don't just solve math problems from left to right; rather, you solve them in a predetermined order that's given to you via the acronym PEMDAS . In other words, you'll start by simplifying any expressions in parentheses before simplifying any exponents and moving on to multiplication, etc.
But there's more to it than this. Here's exactly what PEMDAS means for solving math problems:
- Parentheses: Anything in parentheses must be simplified first
- Exponents: Anything with an exponent (or square root) must be simplified after everything in parentheses has been simplified
- Multiplication and Division: Once parentheses and exponents have been dealt with, solve any multiplication and division from left to right
- Addition and Subtraction: Once parentheses, exponents, multiplication, and division have been dealt with, solve any addition and subtraction from left to right
If any of these elements are missing (e.g., you have a math problem without exponents), you can simply skip that step and move on to the next one.
Now, let's look at a sample problem to help you understand the PEMDAS rule better:
4 (5 − 3)² − 10 ÷ 5 + 8
You might be tempted to solve this math problem left to right, but that would result in the wrong answer! So, instead, let's use PEMDAS to help us approach it the correct way.
We know that parentheses must be dealt with first. This problem has one set of parentheses: (5 − 3). Simplifying this gives us 2 , so now our equation looks like this:
4 (2)² − 10 ÷ 5 + 8
The next part of PEMDAS is exponents (and square roots). There is one exponent in this problem that squares the number 2 (i.e., what we found by simplifying the expression in the parentheses).
This gives us 2 × 2 = 4. So now our equation looks like this:
4 (4) − 10 ÷ 5 + 8 OR 4 × 4 − 10 ÷ 5 + 8
Next up is multiplication and division from left to right . Our problem contains both multiplication and division, which we'll solve from left to right (so first 4 × 4 and then 10 ÷ 5). This simplifies our equation as follows:
Finally, all we need to do now is solve the remaining addition and subtraction from left to right :
The final answer is 22. Don't believe me? Insert the whole equation into your calculator (written exactly as it is above) and you'll get the same result!
Sample Math Problems Using PEMDAS + Answers
See whether you can solve the following four problems correctly using the PEMDAS rule. We'll go over the answers after.
Sample PEMDAS Problems
11 − 8 + 5 × 6
8 ÷ 2 (2 + 2)
7 × 4 − 10 (5 − 3) ÷ 2²
√25 (4 + 2)² − 18 ÷ 3 (3 − 1) + 2³
Here, we go over each problem above and how you can use PEMDAS to get the correct answer.
#1 Answer Explanation
This math problem is a fairly straightforward example of PEMDAS that uses addition, subtraction, and multiplication only , so no having to worry about parentheses or exponents here.
We know that multiplication comes before addition and subtraction , so you'll need to start by multiplying 5 by 6 to get 30:
11 − 8 + 30
Now, we can simply work left to right on the addition and subtraction:
11 − 8 + 30 3 + 30 = 33
This brings us to the correct answer, which is 33 .
#2 Answer Explanation
If this math problem looks familiar to you, that's probably because it went viral in August 2019 due to its ambiguous setup . Many people argued over whether the correct answer was 1 or 16, but as we all know, with math there's (almost always!) only one truly correct answer.
So which is it: 1 or 16?
Let's see how PEMDAS can give us the right answer. This problem has parentheses, division, and multiplication. So we'll start by simplifying the expression in the parentheses, per PEMDAS:
While most people online agreed up until this point, many disagreed on what to do next: do you multiply 2 by 4, or divide 8 by 2?
PEMDAS can answer this question: when it comes to multiplication and division, you always work left to right. This means that you would indeed divide 8 by 2 before multiplying by 4.
It might help to look at the problem this way instead, since people tend to get tripped up on the parentheses (remember that anything next to a parenthesis is being multiplied by whatever is in the parentheses):
Now, we just solve the equation from left to right:
8 ÷ 2 × 4 4 × 4 = 16
The correct answer is 16. Anyone who argues it's 1 is definitely wrong — and clearly isn't using PEMDAS correctly!
#3 Answer Explanation
Things start to get a bit trickier now.
This math problem has parentheses, an exponent, multiplication, division, and subtraction. But don't get overwhelmed — let's work through the equation, one step at a time.
First, per the PEMDAS rule, we must simplify what's in the parentheses :
7 × 4 − 10 (2) ÷ 2²
Easy peasy, right? Next, let's simplify the exponent :
7 × 4 − 10 (2) ÷ 4
All that's left now is multiplication, division, and subtraction. Remember that with multiplication and division, we simply work from left to right:
7 × 4 − 10 (2) ÷ 4 28 − 10 (2) ÷ 4 28 − 20 ÷ 4 28 − 5
Once you've multiplied and divided, you just need to do the subtraction to solve it:
28 − 5 = 23
This gives us the correct answer of 23 .
#4 Answer Explanation
This problem might look scary, but I promise it's not! As you long as you approach it one step at a time using the PEMDAS rule , you'll be able to solve it in no time.
Right away we can see that this problem contains all components of PEMDAS : parentheses (two sets), exponents (two and a square root), multiplication, division, addition, and subtraction. But it's really no different from any other math problem we've done.
First, we must simplify what's in the two sets of parentheses:
√25 (6)² − 18 ÷ 3 (2) + 2³
Next, we must simplify all the exponents — this includes square roots, too :
5 (36) − 18 ÷ 3 (2) + 8
Now, we must do the multiplication and division from left to right:
5 (36) − 18 ÷ 3 (2) + 8 180 − 18 ÷ 3 (2) + 8 180 − 6 (2) + 8 180 − 12 + 8
Finally, we solve the remaining addition and subtraction from left to right:
180 − 12 + 8 168 + 8 = 176
This leads us to the correct answer of 176 .
Another math acronym you should know is SOHCAHTOA. Our expert guide tells you what the acronym SOHCAHTOAH means and how you can use it to solve problems involving triangles .
Studying for the SAT or ACT Math section? Then you'll definitely want to check out our ultimate SAT Math guide / ACT Math guide , which gives you tons of tips and strategies for this tricky section.
Interested in really big numbers? Learn what a googol and googolplex are , as well as why it's impossible to write one of these numbers out.
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Hannah received her MA in Japanese Studies from the University of Michigan and holds a bachelor's degree from the University of Southern California. From 2013 to 2015, she taught English in Japan via the JET Program. She is passionate about education, writing, and travel.
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synonyms for problem-solving
How to use problem-solving in a sentence
“These are problem-solving products but that incorporate technology in a really subtle, unobtrusive way,” she says.
And it is a “ problem-solving populism” that marries the twin impulses of populism and progressivism.
“We want a Republican Party that returns to problem-solving mode,” he said.
problem-solving entails accepting realities, splitting differences, and moving forward.
It teaches female factory workers technical and life skills, such as literacy, communication and problem-solving .
Problem solving with class discussion is absolutely essential, and should occupy at least one third of the entire time.
In teaching by the problem-solving method Professor Lancelot 22 makes use of three types of problems.
Sequential Problem Solving is written for those with a whole brain thinking style.
Thus problem solving involves both the physical world and the interpersonal world.
Sequential Problem Solving begins with the mechanics of learning and the role of memorization in learning.
Choose the synonym for platform
Words Related To problem-solving
More analytical, most analytic.
Roget's 21st Century Thesaurus, Third Edition Copyright © 2013 by the Philip Lief Group.
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10 useful acronyms that help us be creative and solve problems
Posted on October 25, 2016 By thesuccessmanual Topic: Remarkable , Creativity , Acronyms
These useful writing acronyms are from our jumbo guide to 100+ usefulful acronyms . SWOT Strengths, Weaknesses, Opportunities, Threats Use it everywhere – problem –solving, personal analysis, marketing planning, project planning, business planning… IDEAL Identify, Define, Explore, Action, Lookback. Process for solving problems: Identify the problem, Define it, Explore possible solutions and effects, Action the chosen solution, and Look back at the solution you brought about. PEST Political, Economic, Social, Technological Some use 'Environmental' used instead of 'Economic' depending on the context. PEST is sometimes extended to ' PESTELI ' in which the headings: Ecological (or Environmental), Legislative (or Legal), and Industry Analysis are added. SLEPT Social, Legal, Economic, Political, Technological. 'SLEPT analysis' is a business review method similar to PEST or SWOT for assessing factors enabling or obstructing the business's performance, and typically its development potential. TOTB (thus TOTBoxer and TOTBoxing) Think Outside The Box/Thinking Outside The Box. A TOTBoxer is a person who thinks outside the box - i.e., very creatively. TOTBoxing is thinking outside the box. Cleverer than a straightforward TOTB acronym, the expression elegantly describes a creative thinker, or the creative act. SOSTAC Situation analysis, Objectives, Strategy, Tactics, Action, Control. SOSTAC is a business marketing planning system developed by writer and speaker PR Smith in the 1990s. SCAMPER Creativity technique: Substitute Combine Adapt Modify, Magnify, Minify Put to other use Eliminate (Reverse, Rearrange). PMI A decision-making strategy created by Edward de Bono. For any problem or solution, list these: Plus Points Minus Points Interesting Points FFOE A creativity technique: Fluency (many ideas) Flexibility (variety of ideas) Originality (unique ideas) Elaboration (fully developed ideas). DO IT A simple process for creativity: Define problem Open mind and apply creative techniques Identify best solution Transform Note: This compilation is from The Success Manual , 600+ pages of compiled wisdom on 125 important traits , skills and activities.
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Creative Problem Solving Acronym
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The Problem with Acronyms for Writing in Response to Reading
Written by: Carrie Rosebrock
With the absolute best of intentions, we have accidentally turned to a surface-building strategy to encourage our students to write something—anything!—in response to their reading. According to their research surrounding the most impactful literacy strategies, Doug Fisher and Nancy Frey reveal that mnemonic devices (aka acronyms) help students to consolidate surface understandings of material.
This makes sense and is incredibly helpful when teaching students what to do or which steps to follow. Why are most of us able to remember the order of the planets or units of measurement? Acronyms! I’m a fan of Roy G. Biv, the same as the next gal, but the problem with acronyms in response to reading is simple: They do not work—for several reasons.
Reason 1: Acronyms aid in building surface learning. The issue, of course, is that students need to analyze their texts in order to complete these tasks—and the acronym is not a learning tool that creates analysis. What does an acronym do instead? Label and confine.
Reason 2: Acronyms are too structured. Real writing does not follow one set structure. Nowhere in the reality of real writing do we operate with the five sentences in a paragraph model. And, we certainly do not write our paragraphs as topic sentence, three details, concluding sentence. When answering a question and then providing evidence to back up what we think (which is essentially the task required in written response questions), we do not randomly insert evidence, follow it up with the phrase, “This proves…” and then basically say the first sentence over again. No way. That’s not how we talk, and it’s not how we think.
Reason 3: They’re not hitting the target. Most responses to reading require students to elaborate, provide multiple details from the text, and communicate genuine thought. This task requires complexity in thought and response, and we’re struggling to find helpful ways to teach the thinking.
What do we find when we Google “constructed response?” A wide variety of anchor charts and clever terms. There are plenty of acronyms out there to attempt to teach this skill. The issue? These acronyms are asking students for one piece of evidence (not several) and often waste precious thinking on summarizing or restating prompts. Students are spending all of their time saying a whole lot of nothing in their writing—and it’s because we’re asking them to. There’s no reason to summarize my paragraph. I wrote a paragraph. It is already short. It doesn’t need a summary. I promise.
Reason 4: Acronyms create choppy, disjointed paragraphs. Ever read a constructed response where the student just slopped a random piece of evidence in out of nowhere? I certainly have; I see this all the time. Here’s the issue: kids see this acronym, and they know they need evidence—so they think it’s enough to simply have it. They aren’t spending any real time thinking or connecting to what they’ve included. They simply check the box of requirements and move on to the explanation (which is equally as jarring and choppy).
Transitions are being forgotten, and honestly, I don’t even know that students realize they should be transitioning from sentence to sentence. We’ve made what should be a coherent, well-developed task into broken, piece-meal thoughts. Why is the writing so choppy and disconnected? Because the thinking is choppy and disconnected.
Reason 5: Every acronym is a new language. If in 3rd grade students learn ACE, but in 4th grade they learn SASS, and in 5th grade it’s APE and in 6th grade it’s RACES…well, to a student, they’ve just learned four different ways their teachers wanted them to write. Students do not see the connection and transfer of these acronyms. They aren’t sitting there going, “Oh right! ACE and APE are the same—proof and cite are synonyms and we’re still doing what we know how to do.” Nope. Not at all.
They’re more likely thinking, “Okay, what does he want me to do this year? I don’t want to disappoint him and get my sentences out of order.” Why do they raise their hands seeking approval for their writing? We’ve trained them to believe there is one right way. We are teaching them to worry about order in place of critical thinking. This will fail us every time.
The acronym is a surface strategy, and surface strategies alone do not create deep literacy skills.
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Carrie Rosebrock is a District Alignment Coach who supports schools across the country with curriculum, assessment, leadership, and team processes. She is the co-author of Arrows: A Systems-Based Approach to School Leadership. She also currently serves as an adjunct professor for Butler University’s Experiential Program for Preparing School Principals. Previously Carrie has served as a learning specialist with the Central Indiana Educational Service Center, as well as the Secondary English Department Head for Brownsburg Community School Corporation in Brownsburg, Indiana. Carrie is a dynamic speaker, presenter, and coach who is passionate about empowering teams by clarifying their processes. Aside from supporting district teams and leaders, Carrie enjoys spending time outdoors with her husband, Brad and two children, Grace and Will.
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