1. Using Bar Models to Make Math Visual
GRADES 3-5
January 13, 2015
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2. Session goal and agenda
Explore the visual representation of bar models as a tool to solve word problems and support student learning.
Session Agenda:
Introduction to bar models
Modeling addition and subtraction
Modeling multiplication and division
Modeling 2-step problems
Modeling fraction problems
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3. A picture is worth a thousand words …
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4. Consider these problems …
Julie has three more apples than Lucy. Lucy has two apples. How many apples does Julie have?
Julie has three more apples than Lucy. Julie has five apples. How many apples does Lucy have?
A student who attacked these problems using key words would think that the same operation is appropriate for both problems.
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5. Introduction to bar models
Model drawing is a tool for representing relationships in problem situations.
Model drawing is often called …
Bar models
Bar diagrams
Strip models
Tape method
Tape diagrams
Singapore method
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6. Why use a bar model? Pictures are worth a thousand words.
Children find equations and abstract calculations difficult to understand. Bar models help to convert the numbers in a problem into pictorial images.
Use of bar models helps bridge students’ learning from elementary to secondary – from arithmetic to algebra.
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7. Making connections 9 + 6 = 15 Abstract Pictorial Concrete
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8. Benefits to bar models
Allows students to focus on comprehension of the problem’s situation rather than just finding numbers to crunch or just looking for a “key” word or phrase
Explicitly shows the problem structure along with the known and unknown quantities
Is a visual tool to help students determine the operation needed to solve the problem
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9. Developing conceptual understanding Concrete → Pictorial → Visualization → Abstract
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10. Bar model types
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11. Addition and subtraction
Part-whole
Comparison
part + part = whole
whole – part = part
larger quantity – smaller quantity = difference smaller quantity + difference = larger quantity
part
part
whole
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12. Part-whole model 62 29 ? Jorge Trevor
Jorge and Trevor collect baseball cards. Together they have 62 cards. If Jorge has 29 cards, how many cards does Trevor have? 62 29 ?
Jorge
Trevor
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13. Part-whole model 62 29 ? Jorge Trevor Another way to draw the model:
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14. Part-whole model . Ways to draw the model:
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15. Comparison model Girls 134 ? Boys 15
In the school’s art showcase, 134 girls entered their work which was 15 more than the number of boys. How many boys participated?
Girls
134
Boys ? 15
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16. Multiplication and division
Part-whole
Comparison
one part × number of parts = whole
whole ÷number of parts = one part
whole ÷ one part = number of parts
larger quantity ÷ smaller quantity = multiple
smaller quantity × multiple = larger quantity
larger quantity ÷ multiple = smaller quantity
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17. Part-whole model $8 $8 $8 $8 $8 ?
Devi saved $8 a week for 5 weeks. How much did she save altogether? $8 $8 $8 $8 $8 ?
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18. Comparison model White flowers 21 Red flowers 21 21 21 ? = 63
Mrs. Plant’s garden has 21 white flowers. Her garden has three times as many red flowers. How many red flowers are there?
White flowers 21
Red flowers 21 21 21 ?
= 63
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19. Two-step problems 32 8 8 8 8 Carla ? Abby 15
Carla has 4 packages of silly bands. Each package has 8 silly bands in it. Abby is supposed to get 15 fewer silly bands than Carla. How many silly bands should Abby get? 32 8 8 8 8
Carla ?
Abby 15
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20. Modeling fraction problems
Part-whole
Comparison
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21. Part-whole model
Kelly buys 24 flowers. Two-thirds of them are pink. How many pink flowers did Kelly buy? ? 24 ? ? 24
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22. Part-whole model ? 12 12 12 12 12 12 12 48
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23. Comparison model 75 Girls 15 15 15 15 15 Boys 15 15 15 75
Girls 15 15 15 15 15
Boys 15 15 15
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24. Summary Part-Whole Model
Also known as the ‘part-part-whole’ model, shows the various parts which make up a whole
Comparison Model
Shows the relationship between two quantities when they are compared
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25. Solving word problems Read the problem.
What are we trying to find? A total? A part? Are we comparing?
Draw a sketch representing the problem.
Use your diagram to solve the problem.
Write an answer sentence.
Check the answer to be sure it makes sense.
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26. Key points
When building proficiency with bar model skills, start with simple accessible situations and add complexities one at a time.
Develop habits of mind in students:
Continue to ask, “Is there anything else I can see in my model?” before moving on to the next sentence in the problem.
Reflect on the size of bars relative to one another by asking, “Who has more?” questions.
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27. Review …
Part-whole models are more helpful when modeling situations where you are given information relative to a whole.
Compare to models are best when comparing quantities.
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28. “Next step” ideas
Give the children a well-labeled bar model drawing and have them work in groups to write a story problem that goes with it.
Give the children a story problem, but without the question. Have the kids work with a partner to represent the situation and then write a question that could be answered.
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29. Create 2-3 story problems that involve the same names and numbers, but different situations and a bar model drawing of each. Cut them apart and have children work in groups to match each problem with its representation, explaining why it is a match.
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