What is a word problem with examples

Last Updated: December 14, 2021 | Author: Angela Durant

Contents

  • 1 How do you write a word problem?
  • 2 What is a word problem in math definition?
  • 3 What are the different types of word problems?
  • 4 How do I solve this word problem?
  • 5 What is a one step word problem?
  • 6 What is your definition of the word problem?
  • 7 What are the 4 steps in solving word problems?
  • 8 What are the 5 steps to solving word problems?
  • 9 How do you solve 5th grade word problems?
  • 10 What are the 7 steps to problem solving?
  • 11 How do you solve a simple problem?
  • 12 How do you explain word problems to 4th graders?
  • 13 How do you teach a two step word problem?
  • 14 How do you do a 3 step word problem?
  • 15 Why do students struggle with word problems?
  • 16 How can I help my child understand math word problems?
  • 17 What are multi-step word problems?
  • 18 How do you solve math word problems video?

How do you write a word problem?

Problem-Solving Strategy

  1. Read the word problem. Make sure you understand all the words and ideas. …
  2. Identify what you are looking for.
  3. Name what you are looking for. …
  4. Translate into an equation. …
  5. Solve the equation using good algebra techniques.
  6. Check the answer in the problem. …
  7. Answer the question with a complete sentence.

What is a word problem in math definition?

Definition of word problem

: a mathematical problem expressed entirely in words typically used as an educational tool.

What are the different types of word problems?

You can use three common types of word problems — part-part whole, separate and join and multiply and divide — for everything from counting pennies to calculating a tip.

How do I solve this word problem?

What is a one step word problem?

A one-step equation is an algebraic equation you can solve in only one step. You’ve solved the equation when you get the variable by itself, with no numbers in front of it, on one side of the equal sign.

What is your definition of the word problem?

What is a basic definition of problem? A problem is a situation, question, or thing that causes difficulty, stress, or doubt. A problem is also a question raised to inspire thought. In mathematics, a problem is a statement or equation that requires a solution. Problem has a few other senses as a noun and an adjective.

What are the 4 steps in solving word problems?

The 4 Steps to Solving Word Problems

  • Read through the problem and set up a word equation — that is, an equation that contains words as well as numbers.
  • Plug in numbers in place of words wherever possible to set up a regular math equation.
  • Use math to solve the equation.
  • Answer the question the problem asks.

What are the 5 steps to solving word problems?

5 Steps to Word Problem Solving

  • Identify the Problem. Begin by determining the scenario the problem wants you to solve. …
  • Gather Information. …
  • Create an Equation. …
  • Solve the Problem. …
  • Verify the Answer.

How do you solve 5th grade word problems?

Here are the seven strategies I use to help students solve word problems.

  1. Read the Entire Word Problem. …
  2. Think About the Word Problem. …
  3. Write on the Word Problem. …
  4. Draw a Simple Picture and Label It. …
  5. Estimate the Answer Before Solving. …
  6. Check Your Work When Done. …
  7. Practice Word Problems Often.

What are the 7 steps to problem solving?

Effective problem solving is one of the key attributes that separate great leaders from average ones.

  1. Step 1: Identify the Problem. …
  2. Step 2: Analyze the Problem. …
  3. Step 3: Describe the Problem. …
  4. Step 4: Look for Root Causes. …
  5. Step 5: Develop Alternate Solutions. …
  6. Step 6: Implement the Solution. …
  7. Step 7: Measure the Results.

How do you solve a simple problem?

The Problem-Solving Process

  1. Define the problem. Differentiate fact from opinion. …
  2. Generate alternative solutions. Postpone evaluating alternatives initially. …
  3. Evaluate and select an alternative. Evaluate alternatives relative to a target standard. …
  4. Implement and follow up on the solution.

How do you explain word problems to 4th graders?

Making Math Word Problems Accessible for Fourth Graders

  1. Teach a Logical Process. If your child is struggling with 4th grade math word problems, teach him a logical process to go through to determine what needs to be done. …
  2. Teach Common Clue Words. …
  3. Provide Practice. …
  4. Use Manipulates or Diagrams.

How do you teach a two step word problem?

How do you do a 3 step word problem?

Why do students struggle with word problems?

One of the biggest reasons why some students struggle with word problems is because they aren’t just regular math problems – they involve reading! And more than that, students have to be able to fully comprehend what is happening in the problem in order to figure out how to solve it.

How can I help my child understand math word problems?

Strategies to Help Children With Word Problems

  • Use highlighters to mark important information. Word problems are overwhelming, but you can help your child break the problem down by only highlighting the parts of the problem that matter to them. …
  • Remove the numbers from the equation. …
  • Use visuals.

What are multi-step word problems?

Multi-step word problems are maths problems that require multiple calculations to solve them. They will usually will involve more than one operation and often more than one strand from the curriculum. For example a multi-step word problem on area and perimeter may also involve ratio and multiplication.

How do you solve math word problems video?

Источник

This article is about algorithmic word problems in mathematics and computer science. For other uses, see Word problem.

In computational mathematics, a word problem is the problem of deciding whether two given expressions are equivalent with respect to a set of rewriting identities. A prototypical example is the word problem for groups, but there are many other instances as well. A deep result of computational theory is that answering this question is in many important cases undecidable.[1]

Background and motivationEdit

In computer algebra one often wishes to encode mathematical expressions using an expression tree. But there are often multiple equivalent expression trees. The question naturally arises of whether there is an algorithm which, given as input two expressions, decides whether they represent the same element. Such an algorithm is called a solution to the word problem. For example, imagine that   are symbols representing real numbers — then a relevant solution to the word problem would, given the input  , produce the output EQUAL, and similarly produce NOT_EQUAL from  .

The most direct solution to a word problem takes the form of a normal form theorem and algorithm which maps every element in an equivalence class of expressions to a single encoding known as the normal form — the word problem is then solved by comparing these normal forms via syntactic equality.[1] For example one might decide that   is the normal form of  ,  , and  , and devise a transformation system to rewrite those expressions to that form, in the process proving that all equivalent expressions will be rewritten to the same normal form.[2] But not all solutions to the word problem use a normal form theorem — there are algebraic properties which indirectly imply the existence of an algorithm.[1]

While the word problem asks whether two terms containing constants are equal, a proper extension of the word problem known as the unification problem asks whether two terms   containing variables have instances that are equal, or in other words whether the equation   has any solutions. As a common example,   is a word problem in the integer group ℤ,
while   is a unification problem in the same group; since the former terms happen to be equal in ℤ, the latter problem has the substitution   as a solution.

HistoryEdit

One of the most deeply studied cases of the word problem is in the theory of semigroups and groups. A timeline of papers relevant to the Novikov-Boone theorem is as follows:[3][4]

  • 1910: Axel Thue poses a general problem of term rewriting on tree-like structures. He states «A solution of this problem in the most general case may perhaps be connected with unsurmountable difficulties».[5][6]
  • 1911: Max Dehn poses the word problem for finitely presented groups.[7]
  • 1912: Dehn presents Dehn’s algorithm, and proves it solves the word problem for the fundamental groups of closed orientable two-dimensional manifolds of genus greater than or equal to 2.[8] Subsequent authors have greatly extended it to a wide range of group-theoretic decision problems.[9][10][11]
  • 1914: Axel Thue poses the word problem for finitely presented semigroups.[12]
  • 1930 – 1938: The Church-Turing thesis emerges, defining formal notions of computability and undecidability.[13]
  • 1947: Emil Post and Andrey Markov Jr. independently construct finitely presented semigroups with unsolvable word problem.[14][15] Post’s construction is built on Turing machines while Markov’s uses Post’s normal systems.[3]
  • 1950: Alan Turing shows the word problem for cancellation semigroups is unsolvable,[16] by furthering Post’s construction. The proof is difficult to follow but marks a turning point in the word problem for groups.[3]: 342 
  • 1955: Pyotr Novikov gives the first published proof that the word problem for groups is unsolvable, using Turing’s cancellation semigroup result.[17][3]: 354  The proof contains a «Principal Lemma» equivalent to Britton’s Lemma.[3]: 355 
  • 1954 – 1957: William Boone independently shows the word problem for groups is unsolvable, using Post’s semigroup construction.[18][19]
  • 1957 – 1958: John Britton gives another proof that the word problem for groups is unsolvable, based on Turing’s cancellation semigroups result and some of Britton’s earlier work.[20] An early version of Britton’s Lemma appears.[3]: 355 
  • 1958 – 1959: Boone publishes a simplified version of his construction.[21][22]
  • 1961: Graham Higman characterises the subgroups of finitely presented groups with Higman’s embedding theorem,[23] connecting recursion theory with group theory in an unexpected way and giving a very different proof of the unsolvability of the word problem.[3]
  • 1961 – 1963: Britton presents a greatly simplified version of Boone’s 1959 proof that the word problem for groups is unsolvable.[24] It uses a group-theoretic approach, in particular Britton’s Lemma. This proof has been used in a graduate course, although more modern and condensed proofs exist.[25]
  • 1977: Gennady Makanin proves that the existential theory of equations over free monoids is solvable.[26]

The word problem for semi-Thue systemsEdit

The accessibility problem for string rewriting systems (semi-Thue systems or semigroups) can be stated as follows: Given a semi-Thue system   and two words (strings)  , can   be transformed into   by applying rules from  ? Note that the rewriting here is one-way. The word problem is the accessibility problem for symmetric rewrite relations, i.e. Thue systems.[27]

The accessibility and word problems are undecidable, i.e. there is no general algorithm for solving this problem.[28] This even holds if we limit the systems to have finite presentations, i.e. a finite set of symbols and a finite set of relations on those symbols.[27] Even the word problem restricted to ground terms is not decidable for certain finitely presented semigroups.[29][30]

The word problem for groupsEdit

Given a presentation   for a group G, the word problem is the algorithmic problem of deciding, given as input two words in S, whether they represent the same element of G. The word problem is one of three algorithmic problems for groups proposed by Max Dehn in 1911. It was shown by Pyotr Novikov in 1955 that there exists a finitely presented group G such that the word problem for G is undecidable.[31]

The word problem in combinatorial calculus and lambda calculusEdit

One of the earliest proofs that a word problem is undecidable was for combinatory logic: when are two strings of combinators equivalent? Because combinators encode all possible Turing machines, and the equivalence of two Turing machines is undecidable, it follows that the equivalence of two strings of combinators is undecidable. Alonzo Church observed this in 1936.[32]

Likewise, one has essentially the same problem in (untyped) lambda calculus: given two distinct lambda expressions, there is no algorithm which can discern whether they are equivalent or not; equivalence is undecidable. For several typed variants of the lambda calculus, equivalence is decidable by comparison of normal forms.

The word problem for abstract rewriting systemsEdit

Solving the word problem: deciding if   usually requires heuristic search (red, green), while deciding   is straightforward (grey).

The word problem for an abstract rewriting system (ARS) is quite succinct: given objects x and y are they equivalent under  ?[29] The word problem for an ARS is undecidable in general. However, there is a computable solution for the word problem in the specific case where every object reduces to a unique normal form in a finite number of steps (i.e. the system is convergent): two objects are equivalent under   if and only if they reduce to the same normal form.[33]
The Knuth-Bendix completion algorithm can be used to transform a set of equations into a convergent term rewriting system.

The word problem in universal algebraEdit

In universal algebra one studies algebraic structures consisting of a generating set A, a collection of operations on A of finite arity, and a finite set of identities that these operations must satisfy. The word problem for an algebra is then to determine, given two expressions (words) involving the generators and operations, whether they represent the same element of the algebra modulo the identities. The word problems for groups and semigroups can be phrased as word problems for algebras.[1]

The word problem on free Heyting algebras is difficult.[34]
The only known results are that the free Heyting algebra on one generator is infinite, and that the free complete Heyting algebra on one generator exists (and has one more element than the free Heyting algebra).

The word problem for free latticesEdit

Example computation of x∧z ~ x∧z∧(x∨y)

x∧z∧(x∨y) ≤~ x∧z
by 5. since x∧z ≤~ x∧z
by 1. since x∧z = x∧z
 
 
x∧z ≤~ x∧z∧(x∨y)
by 7. since x∧z ≤~ x∧z and x∧z ≤~ x∨y
by 1. since x∧z = x∧z by 6. since x∧z ≤~ x
by 5. since x ≤~ x
by 1. since x = x

The word problem on free lattices and more generally free bounded lattices has a decidable solution. Bounded lattices are algebraic structures with the two binary operations ∨ and ∧ and the two constants (nullary operations) 0 and 1. The set of all well-formed expressions that can be formulated using these operations on elements from a given set of generators X will be called W(X). This set of words contains many expressions that turn out to denote equal values in every lattice. For example, if a is some element of X, then a ∨ 1 = 1 and a ∧ 1 = a. The word problem for free bounded lattices is the problem of determining which of these elements of W(X) denote the same element in the free bounded lattice FX, and hence in every bounded lattice.

The word problem may be resolved as follows. A relation ≤~ on W(X) may be defined inductively by setting w ≤~ v if and only if one of the following holds:

  1.   w = v (this can be restricted to the case where w and v are elements of X),
  2.   w = 0,
  3.   v = 1,
  4.   w = w1 ∨ w2 and both w1 ≤~ v and w2 ≤~ v hold,
  5.   w = w1 ∧ w2 and either w1 ≤~ v or w2 ≤~ v holds,
  6.   v = v1 ∨ v2 and either w ≤~ v1 or w ≤~ v2 holds,
  7.   v = v1 ∧ v2 and both w ≤~ v1 and w ≤~ v2 hold.

This defines a preorder ≤~ on W(X), so an equivalence relation can be defined by w ~ v when w ≤~ v and v ≤~ w. One may then show that the partially ordered quotient set W(X)/~ is the free bounded lattice FX.[35][36] The equivalence classes of W(X)/~ are the sets of all words w and v with w ≤~ v and v ≤~ w. Two well-formed words v and w in W(X) denote the same value in every bounded lattice if and only if w ≤~ v and v ≤~ w; the latter conditions can be effectively decided using the above inductive definition. The table shows an example computation to show that the words x∧z and x∧z∧(x∨y) denote the same value in every bounded lattice. The case of lattices that are not bounded is treated similarly, omitting rules 2 and 3 in the above construction of ≤~.

Example: A term rewriting system to decide the word problem in the free groupEdit

Bläsius and Bürckert
[37]
demonstrate the Knuth–Bendix algorithm on an axiom set for groups.
The algorithm yields a confluent and noetherian term rewrite system that transforms every term into a unique normal form.[38]
The rewrite rules are numbered incontiguous since some rules became redundant and were deleted during the algorithm run.
The equality of two terms follows from the axioms if and only if both terms are transformed into literally the same normal form term. For example, the terms

 , and
 

share the same normal form, viz.  ; therefore both terms are equal in every group.
As another example, the term   and   has the normal form   and  , respectively. Since the normal forms are literally different, the original terms cannot be equal in every group. In fact, they are usually different in non-abelian groups.

Group axioms used in Knuth–Bendix completion

A1    
A2    
A3        
Term rewrite system obtained from Knuth–Bendix completion

R1    
R2    
R3    
R4    
R8    
R11    
R12    
R13    
R14    
R17        

See alsoEdit

  • Conjugacy problem
  • Group isomorphism problem

ReferencesEdit

  1. ^ a b c d Evans, Trevor (1978). «Word problems». Bulletin of the American Mathematical Society. 84 (5): 790. doi:10.1090/S0002-9904-1978-14516-9.
  2. ^ Cohen, Joel S. (2002). Computer algebra and symbolic computation: elementary algorithms. Natick, Mass.: A K Peters. pp. 90–92. ISBN 1568811586.
  3. ^ a b c d e f g Miller, Charles F. (2014). Downey, Rod (ed.). «Turing machines to word problems» (PDF). Turing’s Legacy: 330. doi:10.1017/CBO9781107338579.010. hdl:11343/51723. ISBN 9781107338579. Retrieved 6 December 2021.
  4. ^ Stillwell, John (1982). «The word problem and the isomorphism problem for groups». Bulletin of the American Mathematical Society. 6 (1): 33–56. doi:10.1090/S0273-0979-1982-14963-1.
  5. ^ Müller-Stach, Stefan (12 September 2021). «Max Dehn, Axel Thue, and the Undecidable». p. 13. arXiv:1703.09750 [math.HO].
  6. ^ Steinby, Magnus; Thomas, Wolfgang (2000). «Trees and term rewriting in 1910: on a paper by Axel Thue». Bulletin of the European Association for Theoretical Computer Science. 72: 256–269. CiteSeerX 10.1.1.32.8993. MR 1798015.
  7. ^ Dehn, Max (1911). «Über unendliche diskontinuierliche Gruppen». Mathematische Annalen. 71 (1): 116–144. doi:10.1007/BF01456932. ISSN 0025-5831. MR 1511645. S2CID 123478582.
  8. ^ Dehn, Max (1912). «Transformation der Kurven auf zweiseitigen Flächen». Mathematische Annalen. 72 (3): 413–421. doi:10.1007/BF01456725. ISSN 0025-5831. MR 1511705. S2CID 122988176.
  9. ^ Greendlinger, Martin (June 1959). «Dehn’s algorithm for the word problem». Communications on Pure and Applied Mathematics. 13 (1): 67–83. doi:10.1002/cpa.3160130108.
  10. ^ Lyndon, Roger C. (September 1966). «On Dehn’s algorithm». Mathematische Annalen. 166 (3): 208–228. doi:10.1007/BF01361168. hdl:2027.42/46211. S2CID 36469569.
  11. ^ Schupp, Paul E. (June 1968). «On Dehn’s algorithm and the conjugacy problem». Mathematische Annalen. 178 (2): 119–130. doi:10.1007/BF01350654. S2CID 120429853.
  12. ^ Power, James F. (27 August 2013). «Thue’s 1914 paper: a translation». arXiv:1308.5858 [cs.FL].
  13. ^ See History of the Church–Turing thesis. The dates are based on On Formally Undecidable Propositions of Principia Mathematica and Related Systems and Systems of Logic Based on Ordinals.
  14. ^ Post, Emil L. (March 1947). «Recursive Unsolvability of a problem of Thue» (PDF). Journal of Symbolic Logic. 12 (1): 1–11. doi:10.2307/2267170. JSTOR 2267170. S2CID 30320278. Retrieved 6 December 2021.
  15. ^ Mostowski, Andrzej (September 1951). «A. Markov. Névožmoinost’ nékotoryh algoritmov v téorii associativnyh sistém (Impossibility of certain algorithms in the theory of associative systems). Doklady Akadémii Nauk SSSR, vol. 77 (1951), pp. 19–20». Journal of Symbolic Logic. 16 (3): 215. doi:10.2307/2266407. JSTOR 2266407.
  16. ^ Turing, A. M. (September 1950). «The Word Problem in Semi-Groups With Cancellation». The Annals of Mathematics. 52 (2): 491–505. doi:10.2307/1969481. JSTOR 1969481.
  17. ^ Novikov, P. S. (1955). «On the algorithmic unsolvability of the word problem in group theory». Proceedings of the Steklov Institute of Mathematics (in Russian). 44: 1–143. Zbl 0068.01301.
  18. ^ Boone, William W. (1954). «Certain Simple, Unsolvable Problems of Group Theory. I». Indagationes Mathematicae (Proceedings). 57: 231–237. doi:10.1016/S1385-7258(54)50033-8.
  19. ^ Boone, William W. (1957). «Certain Simple, Unsolvable Problems of Group Theory. VI». Indagationes Mathematicae (Proceedings). 60: 227–232. doi:10.1016/S1385-7258(57)50030-9.
  20. ^ Britton, J. L. (October 1958). «The Word Problem for Groups». Proceedings of the London Mathematical Society. s3-8 (4): 493–506. doi:10.1112/plms/s3-8.4.493.
  21. ^ Boone, William W. (1958). «The word problem» (PDF). Proceedings of the National Academy of Sciences. 44 (10): 1061–1065. Bibcode:1958PNAS…44.1061B. doi:10.1073/pnas.44.10.1061. PMC 528693. PMID 16590307. Zbl 0086.24701.
  22. ^ Boone, William W. (September 1959). «The Word Problem». The Annals of Mathematics. 70 (2): 207–265. doi:10.2307/1970103. JSTOR 1970103.
  23. ^ Higman, G. (8 August 1961). «Subgroups of finitely presented groups». Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 262 (1311): 455–475. Bibcode:1961RSPSA.262..455H. doi:10.1098/rspa.1961.0132. S2CID 120100270.
  24. ^ Britton, John L. (January 1963). «The Word Problem». The Annals of Mathematics. 77 (1): 16–32. doi:10.2307/1970200. JSTOR 1970200.
  25. ^ Simpson, Stephen G. (18 May 2005). «A Slick Proof of the Unsolvability of the Word Problem for Finitely Presented Groups» (PDF). Retrieved 6 December 2021.
  26. ^ «Subgroups of finitely presented groups». Mathematics of the USSR-Sbornik. 103 (145): 147–236. 13 February 1977. doi:10.1070/SM1977v032n02ABEH002376.
  27. ^ a b Matiyasevich, Yuri; Sénizergues, Géraud (January 2005). «Decision problems for semi-Thue systems with a few rules». Theoretical Computer Science. 330 (1): 145–169. doi:10.1016/j.tcs.2004.09.016.
  28. ^ Davis, Martin (1978). «What is a Computation?» (PDF). Mathematics Today Twelve Informal Essays: 257–259. doi:10.1007/978-1-4613-9435-8_10. ISBN 978-1-4613-9437-2. Retrieved 5 December 2021.
  29. ^ a b Baader, Franz; Nipkow, Tobias (5 August 1999). Term Rewriting and All That. Cambridge University Press. pp. 59–60. ISBN 978-0-521-77920-3.
  30. ^
    • Matiyasevich, Yu. V. (1967). «Простые примеры неразрешимых ассоциативных исчислений» [Simple examples of undecidable associative calculi]. Doklady Akademii Nauk SSSR (in Russian). 173 (6): 1264–1266. ISSN 0869-5652.
    • Matiyasevich, Yu. V. (1967). «Simple examples of undecidable associative calculi». Soviet Mathematics. 8 (2): 555–557. ISSN 0197-6788.

  31. ^ Novikov, P. S. (1955). «On the algorithmic unsolvability of the word problem in group theory». Trudy Mat. Inst. Steklov (in Russian). 44: 1–143.
  32. ^ Statman, Rick (2000). «On the Word Problem for Combinators». Rewriting Techniques and Applications. Lecture Notes in Computer Science. 1833: 203–213. doi:10.1007/10721975_14. ISBN 978-3-540-67778-9.
  33. ^ Beke, Tibor (May 2011). «Categorification, term rewriting and the Knuth–Bendix procedure». Journal of Pure and Applied Algebra. 215 (5): 730. doi:10.1016/j.jpaa.2010.06.019.
  34. ^ Peter T. Johnstone, Stone Spaces, (1982) Cambridge University Press, Cambridge, ISBN 0-521-23893-5. (See chapter 1, paragraph 4.11)
  35. ^ Whitman, Philip M. (January 1941). «Free Lattices». The Annals of Mathematics. 42 (1): 325–329. doi:10.2307/1969001. JSTOR 1969001.
  36. ^ Whitman, Philip M. (1942). «Free Lattices II». Annals of Mathematics. 43 (1): 104–115. doi:10.2307/1968883. JSTOR 1968883.
  37. ^ K. H. Bläsius and H.-J. Bürckert, ed. (1992). Deduktionsssysteme. Oldenbourg. p. 291.; here: p.126, 134
  38. ^ Apply rules in any order to a term, as long as possible; the result doesn’t depend on the order; it is the term’s normal form.
Источник

From Wikipedia, the free encyclopedia

Not to be confused with word problem (mathematics), the problem of deciding whether two given expressions are equivalent in rewriting.

Not to be confused with an essay question, another type of exam question that also requires word use.

In science education, a word problem is a mathematical exercise (such as in a textbook, worksheet, or exam) where significant background information on the problem is presented in ordinary language rather than in mathematical notation. As most word problems involve a narrative of some sort, they are sometimes referred to as story problems and may vary in the amount of technical language used.

Example[edit]

A typical word problem:

Tess paints two boards of a fence every four minutes, but Allie can paint three boards every two minutes. If there are 240 boards total, how many hours will it take them to paint the fence, working together?

Solution process[edit]

Word problems such as the above can be examined through five stages:

  • 1. Problem Comprehension
  • 2. Situational Solution Visualization
  • 3. Mathematical Solution Planning
  • 4. Solving for Solution
  • 5. Situational Solution Visualization

The linguistic properties of a word problem need to be addressed first. To begin the solution process, one must first understand what the problem is asking and what type of solution the answer will be. In the problem above, the words «minutes», «total», «hours», and «together» need to be examined.

The next step is to visualize what the solution to this problem might mean. For our stated problem, the solution might be visualized by examining if the total number of hours will be greater or smaller than if it were stated in minutes. Also, it must be determined whether or not the two girls will finish at a faster or slower rate if they are working together.

After this, one must plan a solution method using mathematical terms. One scheme to analyze the mathematical properties is to classify the numerical quantities in the problem into known quantities (values given in the text), wanted quantities (values to be found), and auxiliary quantities (values found as intermediate stages of the problem). This is found in the «Variables» and «Equations» sections above.

Next, the mathematical processes must be applied to the formulated solution process. This is done solely in the mathematical context for now.

Finally, one must again visualize the proposed solution and determine if the solution seems to make sense for the realistic context of the problem. After visualizing if it is reasonable, one can then work to further analyze and draw connections between mathematical concepts and realistic problems.[1]

The importance of these five steps in teacher education is discussed at the end of the following section.

Purpose and skill development[edit]

Word problems commonly include mathematical modelling questions, where data and information about a certain system is given and a student is required to develop a model. For example:

  1. Jane had $5.00, then spent $2.00. How much does she have now?
  2. In a cylindrical barrel with radius 2 m, the water is rising at a rate of 3 cm/s. What is the rate of increase of the volume of water?

As the developmental skills of students across grade levels varies, the relevance to students and application of word problems also varies. The first example is accessible to primary school students, and may be used to teach the concept of subtraction. The second example can only be solved using geometric knowledge, specifically that of the formula for the volume of a cylinder with a given radius and height, and requires an understanding of the concept of «rate».

There are numerous skills that can be developed to increase a students’ understanding and fluency in solving word problems. The two major stems of these skills are cognitive skills and related academic skills. The cognitive domain consists of skills such as nonverbal reasoning and processing speed. Both of these skills work to strengthen numerous other fields of thought. Other cognitive skills include language comprehension, working memory, and attention. While these are not solely for the purpose of solving word problems, each one of them affects one’s ability to solve such mathematical problems. For instance, if the one solving the math word problem has a limited understanding of the language (English, Spanish, etc.) they are more likely to not understand what the problem is even asking. In Example 1 (above), if one does not comprehend the definition of the word «spent,» they will misunderstand the entire purpose of the word problem. This alludes to how the cognitive skills lead to the development of the mathematical concepts. Some of the related mathematical skills necessary for solving word problems are mathematical vocabulary and reading comprehension. This can again be connected to the example above. With an understanding of the word «spent» and the concept of subtraction, it can be deduced that this word problem is relating the two.[2] This leads to the conclusion that word problems are beneficial at each level of development, despite the fact that these domains will vary across developmental and academic stages.

The discussion in this section and the previous one urge the examination of how these research findings can affect teacher education. One of the first ways is that when a teacher understands the solution structure of word problems, they are likely to have an increased understanding of their students’ comprehension levels. Each of these research studies supported the finding that, in many cases, students do not often struggle with executing the mathematical procedures. Rather, the comprehension gap comes from not having a firm understanding of the connections between the math concepts and the semantics of the realistic problems. As a teacher examines a student’s solution process, understanding each of the steps will help them understand how to best accommodate their specific learning needs. Another thing to address is the importance of teaching and promoting multiple solution processes. Procedural fluency is often times taught without an emphasis on conceptual and applicable comprehension. This leaves students with a gap between their mathematical understanding and their realistic problem solving skills. The ways in which teachers can best prepare for and promote this type of learning will not be discussed here.[1][3]

History and culture[edit]

The modern notation that enables mathematical ideas to be expressed symbolically was developed in Europe from the sixteenth century onwards. Prior to this, all mathematical problems and solutions were written out in words; the more complicated the problem, the more laborious and convoluted the verbal explanation.

Examples of word problems can be found dating back to Babylonian times. Apart from a few procedure texts for finding things like square roots, most Old Babylonian problems are couched in a language of measurement of everyday objects and activities. Students had to find lengths of canals dug, weights of stones, lengths of broken reeds, areas of fields, numbers of bricks used in a construction, and so on.

Ancient Egyptian mathematics also has examples of word problems. The Rhind Mathematical Papyrus includes a problem that can be translated as:

There are seven houses; in each house there are seven cats; each cat kills seven mice; each mouse has eaten seven grains of barley; each grain would have produced seven hekat. What is the sum of all the enumerated things?

In more modern times the sometimes confusing and arbitrary nature of word problems has been the subject of satire. Gustave Flaubert wrote this nonsensical problem, now known as the Age of the captain:

Since you are now studying geometry and trigonometry, I will give you a problem. A ship sails the ocean. It left Boston with a cargo of wool. It grosses 200 tons. It is bound for Le Havre. The mainmast is broken, the cabin boy is on deck, there are 12 passengers aboard, the wind is blowing East-North-East, the clock points to a quarter past three in the afternoon. It is the month of May. How old is the captain?

Word problems have also been satirised in The Simpsons, when a lengthy word problem («An express train traveling 60 miles per hour leaves Santa Fe bound for Phoenix, 520 miles away. At the same time, a local train traveling 30 miles an hour carrying 40 passengers leaves Phoenix bound for Santa Fe…») trails off with a schoolboy character instead imagining that he is on the train.

Both the original British and American versions of the game show Winning Lines involve word problems. However, the problems are worded so as to not give away obvious numerical information and thus, allow the contestants to figure out the numerical parts of the questions to come up with the answers.

See also[edit]

  • Cut-the-knot

References[edit]

  1. ^ a b Rich, Kathryn M.; Yadav, Aman (2020-05-01). «Applying Levels of Abstraction to Mathematics Word Problems». TechTrends. 64 (3): 395–403. doi:10.1007/s11528-020-00479-3. ISSN 1559-7075. S2CID 255311095.
  2. ^ Lin, Xin (2021-09-01). «Investigating the Unique Predictors of Word-Problem Solving Using Meta-Analytic Structural Equation Modeling». Educational Psychology Review. 33 (3): 1097–1124. doi:10.1007/s10648-020-09554-w. ISSN 1573-336X. S2CID 225195843.
  3. ^ Scheibling-Sève, Calliste; Pasquinelli, Elena; Sander, Emmanuel (March 2020). «Assessing conceptual knowledge through solving arithmetic word problems». Educational Studies in Mathematics. 103 (3): 293–311. doi:10.1007/s10649-020-09938-3. ISSN 0013-1954. S2CID 216314124.

Further reading[edit]

  • L Verschaffel, B Greer, E De Corte (2000) Making Sense of Word Problems, Taylor & Francis
  • John C. Moyer; Margaret B. Moyer; Larry Sowder; Judith Threadgill-Sowder (1984) Story Problem Formats: Verbal versus Telegraphic Journal for Research in Mathematics Education, Vol. 15, No. 1. (Jan., 1984), pp. 64–68. JSTOR 748989
  • Perla Nesher Eva Teubal (1975)Verbal Cues as an Interfering Factor in Verbal Problem Solving Educational Studies in Mathematics, Vol. 6, No. 1. (Mar., 1975), pp. 41–51. JSTOR 3482158
  • Madis Lepik (1990) Algebraic Word Problems: Role of Linguistic and Structural Variables, Educational Studies in Mathematics, Vol. 21, No. 1. (Feb., 1990), pp. 83–90., JSTOR 3482220
  • Duncan J Melville (1999) Old Babylonian Mathematics http://it.stlawu.edu/%7Edmelvill/mesomath/obsummary.html
  • Egyptian Algebra — Mathematicians of the African Diaspora
  • Mathematical Quotations — F
  • Andrew Nestler’s Guide to Mathematics and Mathematicians on The Simpsons
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What are word problems?

We explain what a word problem is and give examples of the types of word problems your child might be challenged with in each primary-school maths year group, from Year 1 to Year 6.

What is a word problem?

A word problem is a few sentences describing a ‘real-life’ scenario where a problem needs to be solved by way of a mathematical calculation.

Word problems are seen as a crucial part of learning in the primary curriculum, because they require children to apply their knowledge of various different concepts to ‘real-life’ scenarios. 

Word problems also help children to familiarise themselves with mathematical language (vocabulary like fewer, altogether, difference, more, share, multiply, subtract, equal, reduced, etc.).

Teachers tend to try and include word problems in their maths lessons at least twice a week.
 

What is RUCSAC?

In the classroom children might be taught the acronym RUCSAC (Read, Understand, Choose, Solve, Answer, Check) to help them complete word problems.

By following the acronym step by step children learn to apply a structured, analytical strategy to their calculations. They will need to understand what the problem is asking them to find out by reading the question carefully, choosing the correct mathematical operation to help them solve the query and finally checking their answer by using the inverse operation.

Word problem examples for Years 1 to 6

The following are example word problems that apply to each primary year group.

Year 1

In Year 1 a child would usually been given apparatus to help them with a problem (counters, plastic coins, number cards, number lines or picture cards).

Sarah wants to buy a teddy bear costing 30p. How many 10p coins will she need?

Brian has 3 sweets. Tom has double this number of sweets. How many sweets does Tom have?

Year 2

In Year 2, children continue to use apparatus to help them with problem-solving.

Faye has 12 marbles. Her friend Louise has 9 marbles. How many marbles do they both have altogether?

Three children are each given 5 teddy bears. How many teddy bears do they have altogether?

Year 3

In Year 3, some children may use apparatus, but on the whole children will tend to work out word problems without physical aids. Teachers will usually demonstrate written methods for the four operations (addition, subtraction, multiplication and division) to support children in their working out of the problems.

A jumper costs £23. How much will 4 jumpers cost?

Sarah has 24 balloons. She gives a quarter of them away to her friend. How many balloons does she give away?

Children will also start to do two-step problems in Year 3. This is a problem where finding the answer requires two separate calculations, for example:

I have £34. I am given another £26. I divide this money equally into four different bank accounts. How much money do I put in each bank account?

  • In this case, the first step would be to add £34 and £26 to make £60.
  • The second step would be to divide £60 by 4 to make £15.

Year 4

Children should feel confident in an efficient written method for each operation at this stage. They will continue to be given a variety of problems and have to work out which operation and method is appropriate for each. They will also be given two-step problems. 

I have 98 marbles. I share them equally between 6 friends. How many marbles does each friend get? How many marbles are left over?

Year 5

Children will continue to do one-step and two-step problems. They will start to carry out problem-solving involving decimals. 

My chest of drawers is 80cm wide and my table is 1.3m wide. How much wall space do they take up when put side by side?

There are 24 floors of a car park. Each floor has room for 45 cars. How many cars can the car park fit altogether?

Year 6

In Year 6 children solve ‘multi-step problems’ and problems involving fractions, decimals and percentages. 

Sarah sees the same jumper in two different sales:
In the first sale, the original price of the jumper is £36.15, but has been reduced by a third.
In the second sale, the jumper was priced at £45, but now has 40% off.
How much does each jumper cost and which one is the cheapest?

In the past, calculators were sometimes used for solving two-step problems like the one above, but the new curriculum does not include the use of calculators at any time during primary school.

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Table of Contents

  1. How do you solve 5th grade word problems?
  2. What is guess and problem-solving?
  3. When should I look for patterns?
  4. How do you know if a pattern is good?
  5. Why have we become so good at Recognising patterns?
  6. Why do some people see patterns?
  7. Do human brains look for patterns?
  8. Is Pareidolia a disorder?
  9. Is Pareidolia good or bad?
  10. Is Pareidolia a psychosis?
  11. What is Pareidolia caused by?

Word problems commonly include mathematical modelling questions, where data and information about a certain system is given and a student is required to develop a model. For example: Jane had $5.00, then spent $2.00. How much does she have now?

How do you solve 5th grade word problems?

Here are the seven strategies I use to help students solve word problems.

  1. Read the Entire Word Problem.
  2. Think About the Word Problem.
  3. Write on the Word Problem.
  4. Draw a Simple Picture and Label It.
  5. Estimate the Answer Before Solving.
  6. Check Your Work When Done.
  7. Practice Word Problems Often.

What is guess and problem-solving?

“Guess and Check” is a problem-solving strategy that students can use to solve mathematical problems by guessing the answer and then checking that the guess fits the conditions of the problem.

When should I look for patterns?

Finding patterns is extremely important. Patterns make our task simpler. Problems are easier to solve when they share patterns, because we can use the same problem-solving solution wherever the pattern exists. The more patterns we can find, the easier and quicker our overall task of problem solving will be.

How do you know if a pattern is good?

There are two really easy ways to develop pattern recognition skills:

  1. Be born with them.
  2. Put in your 10,000 hours.
  3. Study nature, art and math.
  4. Study (good) architecture.
  5. Study across disciplines.
  6. Find a left-brain hobby.
  7. Don’t read (much) in your own discipline.
  8. Listen for echoes and watch for shadows.

Why have we become so good at Recognising patterns?

Recognizing patterns allows us to predict and expect what is coming. The development of neural networks in the outer layer of the brain in humans has allowed for better processing of visual and auditory patterns.

Why do some people see patterns?

Seeing familiar objects or patterns in otherwise random or unrelated objects or patterns is called pareidolia. It’s a form of apophenia, which is a more general term for the human tendency to seek patterns in random information. The ability to experience pareidolia is more developed in some people and less in others.

Do human brains look for patterns?

The brain looks for patterns and fills in the blanks. It uses patterns to understand the relationship between things—putting them in context. “The human brain is a pattern-recognition machine. They are learning patterns and developing rules that guide their decision and make them faster and more accurate.”

Is Pareidolia a disorder?

Pareidolia is a type of complex visual illusion that occurs in health but rarely reported in patients with Depression. We present a unique case of treatment-resistant Major Depressive Disorder with co-occurring complex visual disturbance that responded to augmentation of treatment with an anxiolytic.

Is Pareidolia good or bad?

While pareidolia was at one time thought to be related to psychosis, it’s now generally recognized as a perfectly healthy tendency.

Is Pareidolia a psychosis?

Pareidolia was at one time considered a symptom of psychosis, but it is now seen as a normal human tendency.

What is Pareidolia caused by?

Studies show that neurotic people, and people in negative moods, are more likely to experience pareidolia. The reason for this seems to be that these people are on higher alert for danger, so are more likely to spot something that isn’t there.

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