Tackling word problems

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Tackling word problems

• NZAMT
• September 2005

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• Anne LawrenceAdviser in Numeracy, Mathematics
  NCEA
• Centre for Educational DevelopmentMassey
  University College of Education
• Palmerston North
• email a.lawrence_at_massey.ac.nz

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• Small Research Project
• with
• Marc Paterson
• HOD Mathematics
• Awatapu College
• Mathematics word problems and year 12
  students. Set Research information for teachers
  (NZCER,2005,1).

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Key questions

• What makes word problems difficult for many
  students?
• What can teachers do to help more students tackle
  word problems with more success?

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Definition of a Problem / context

• Solving a problem involves students in both
  choosing the mathematical skills or techniques to
  apply, and applying such skills or techniques
  accurately.
• Within the context of school mathematics a
  problem iseither a question set in a) a real
  context or b) a mathematical context.
  Common contexts will be proofs or the
  development of mathematical ideas
• A problem should not involve scaffolding or
  directions as to how to solve the problem, except
  in special situations.

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Wording problems marbles 1

• Peter, David and Jirka are playing marbles. They
  have 198 marbles altogether. Jirka has three
  times less than Peter, and Peter has six times
  more than David. How many marbles does each boy
  have?

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Wording problems marbles 2

• Peter, David and Jirka are playing marbles. They
  have 198 marbles altogether. Peter has 3 times
  more than Jirka, and Jirka has 2 times more than
  David. How many marbles does each boy have?

Nesher, P., Hershkovitz, S. and Novotna, J.
(2003). Situation model, text base and what else?
Factors affecting problem solving. Educational
Studies in Mathematics 52 151-176.
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Language factors

• Readability
• Technical words
• Connectives
• Order
• Omitted words
• Syntactic structure of sentences
• Length - sentences, whole problem
• Numbers - size, similarity, position, wording

Wiest, L. R. (2002). Aspects of word-problem
context that influence children's problem-solving
performance. FOCUS on Learning Problems in
Mathematics 24(2) 38-52.
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The Gift Box

• Sebastians holiday job is to wrap rectangular
  Christmas gift boxes with ribbon. Sebastian uses
  a single piece of ribbon that passes vertically
  around the middle of all four sides of the box
  and crosses itself at right angles on the top and
  on the bottom of the box. On the top is a bow
  that uses 30cm of ribbon. The most common box
  that Sebastian has to wrap is 20cm high, has a
  square base and uses 310 cm of ribbon. What are
  the lengths of the sides of this box?

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Contextual factors

• Interference
• Invisibility
• Avoidance
• Assumptions

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Doghouse task

• Julie wants to fence in an area in her yard for
  her dog. After paying for the materials to build
  her doghouse, she can afford only 36 feet of
  fencing. She is considering various different
  shapes for the enclosed area. However, she wants
  all of her shapes to have 4 sides that are whole
  number lengths and contain 4 right angles. All 4
  sides are to have fencing.
• What is the largest area that Julie can enclose
  with 36 feet of fencing?

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Apples for horses

• You have 18 apples that you want to have 3
  horses share fairly. You want to use up all the
  apples. How many apples does each horse get?

Kouba, V., Cezikturk, O., Sherwood, S. and Ho, C.
(1999). Setting the context for mathematics in
context. Mathematics Teaching in the Middle
School 2004 (24 June).
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Solvent task

• A paintbrush has just been used and the owner
  wishes to clean it. After the brush has been
  scraped against the side of the paint can, it
  still contains 4 fluid ounces of paint. The owner
  dips it into a quart of clean solvent and stirs
  well until the diluted paint solution is uniform.
  After draining, the brush stillholds 4 fluid
  ounces, part of which is paint and part solvent,
  since the diluted solution is uniform. The
  process is repeated with a fresh quart of
  solvent.
• Develop a mathematical model of the process.

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Pizza Task
PizzaStuff 6.25 6 slices per pizza
Sams Pizza 8.50 8 slices per pizza
Pizza Palace 10.25 10 slices per pizza

• If each person will eat two slices. If all the
  pizza slices are the same size, which is the best
  place to buy pizza for a class of 30 students?

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Teaching about context

• Students need to be context-wise
• Students need to understand why they are doing
  word problems
• Teachers need to be explicit about their
  expectations

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The problem solving cycle
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Nada has 15 apples. This is three times the
number of apples Kevin has. How many apples does
Kevin have?
Real problem
Step 1 Translation
Maths problem
Let x be the number of apples Kevin has. Then
3x15.
Step 2 Solving
Maths solution
If 3 times a number is 15, that number must be
15/35.
Step 3 Translation
Real solution
Kevin has 5 apples.
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Pat the Painter word problem

• Pat the Painter needs to know the lengths of the
  walls of the school gym. Unfortunately he loses
  the paper with all the measurements. However, he
  can remember three things. Firstly, the gym was
  rectangular, secondly, the area of the gym was
  375 m2 and lastly, the perimeter of the gym was
  80 m. Show how Pat is able to work out the
  lengths of the sides of the gym.

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Pat the Painter diagram
Length (L)
School Gym Area 375 m2 Perimeter 80m
Width (W)
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Pat the Painter equations

• 2L 2W 80
• L x W 375

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The Fair word problem

• Jenny is visiting the fair with two friends.
  When its her turn to buy the food she buys 2 hot
  dogs and one coffee. She pays with a 10 note and
  receives 3.50 change. When her friend buys the
  next lot of food she buys one hotdog and 2
  coffees and pays for them with a 20 note. She
  receives 14.50 change. How much is one hot dog
  worth?

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The Fair diagram
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The Fair equations

• 2H C 6.50
• H 2C 5.50

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What the students did
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Informal solution to the Fair problem
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Guess and check with some symbols used for the
Gym problem
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The problem-solving cycle
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The primitive problem-solving cycle
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The sophisticated problem-solving cycle
Translation
Algebraic problem
Formal solution
Algebraic solution
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Expert Blind spot

• Teachers familiarity with algebra tends to lead
  us to
• Overestimate student ease with the formal
  language of algebra
• Underestimate students ability to use informal
  strategies to solve problems

Koedinger, K. and Nathan, M. (2004). The real
story behind story problems effects of
representation on quantitative reasoning. The
Journal of the Learning Sciences 13(2) 129-164.
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The translation steps
Algebraic problem
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Suggestions for improving students use of
algebra to solve word problems

• Use materials or pictures
• Generate meaningful word problems
• Introduce the problem-solving cycle
• Practice translating between different
  representations

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• With nine more buckets, the tank on the left will
  be full
• With five more buckets, the tank on the right
  will be full.
• What can you say about this tank situation?

Lins (1994) cited by Nickson, M. (2000). Teaching
and learning mathematics A guide to recent
research and its applications London, Continuum.
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• Write a word problem to
• match the equation
• x 40 b y 15 b

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• Complete the diagram
• and write a word problem
• to match this equation
• x 12 b y - 15 b

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Matching activities

• Match each word problem with the equation that
  best fits.
• If no equation fits, write one
• If there is no matching word problem for an
  equation, write one

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• Ted earned 81.90 in one day, including 66 he
  received in tips. If he worked for 6 hours that
  day, how much does Ted make per hour?
• Julie borrowed some money from her mum for soccer
  gear. She needed 81.90 for a pair of soccer
  boots. Shin pads cost an extra 66. If she spread
  her payments to her mother evenly over 6 weeks,
  how much does she pay per week?
• Hari had 81.90 in his wallet. He bought 6 donuts
  at Wholey Donuts for 66 cents each. How much
  money did he have left?
• Rope is sold in coils of 66 metres in length.
  Sheree bought 6 coils of rope. She used 81.9
  metres. How much rope has she got left?

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• A
• B
• C
• D

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Cloze activities

• The word problem is missing some information.
  Fill the gaps so that the word problem fits the
  equation provided.
• The equation is missing some information. Finish
  the equation so that it matches the word problem
  provided.

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Winning Lotto

• Mum won 143.50 on lotto.
• She
• and then divided the remaining money evenly
  among her three sons. How much did each son get?

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Buying basketballs

• After buying a basketball with his four
  daughters, Mr Jordan wanted to find out how much
  each daughter had paid. He contributed of the
  68.39 cost of the ball. How much did each girl
  pay?

4 ? 25
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The helicopter task

• The Air Ways Helicopter Company is planning to
  produce its latest model, the Air Star. After
  extensive research, the company has determined
  that it will cost 11 million to produce the
  first one. This amount includes the cost of
  setting up the factory and the machinery. After
  that, each additional helicopter will cost 3
  million to produce.

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• Write an equation for this problem
• How much money will the company need to build
  the helicopters?
• Write a word problem for this equation
• 11 3x 47
• Write a word problem for this equation
• y 5.5 ? 10 (3 ? 10 1)
• Write an equation for this problem
• If the company charges 5.5 million for each
  helicopter, how many must it sell before making a
  profit?

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Tackling word problems

• Factors contributing to difficulty
• Linguistic
• Contextual
• Strategies for addressing difficulties
• Explicit problem-solving model
• Activities focussing on translation