1. Teaching Problem Solving Through Bar Modeling
Singapore Math
Teaching Problem Solving Through Bar Modeling

2. Please make a name tent

3. Some of the Tools used in Singapore Math
Place Value Strips
Place Value Disks
Place Value Charts
Number Bond Cards
Part-Whole Cards
Decimal Tiles
Decimal Strips

4. The most helpful part of the Singapore math tools is model drawing
When students learn to draw models, they have to master two stages.
They learn the process of using model drawing to solve story problems.
They learn how to apply the process independently when solving word problems.

5. Skills needed to solve story problems
Read
Understand
Strategize
Compute
Check their work
In Singapore the math is taught in English beginning in Kindergarten. *All students are ELL.
It is common that students make up stories to go with problems that do not have context.

6. A Step-By-step Approach to model drawing
Read the entire problem.
Rewrite the question in sentence form leaving a space for your calculation.
Determine who and/or what is involved in the problem.
Chunk the problem, adjust the unit bars, and mark what you need to find.
Compute and solve the problem.
Fill in the calculation in the sentence you wrote and make sure the answer makes sense.

7. The Modeling Method

8. Algebraic Word Problem
Jake is 3 years older than Kayla and 2 years younger than Larry. The total of their ages is 41 years. How old is Jake?
Initially show only all arguments on the model. Then go back and support the algebra.

9. It helps students derive algebraic expressions.
What advantages does this method give to students who are learning algebra?
It helps students derive algebraic expressions.
It helps students construct algebraic relations.
It helps students simplify algebraic relations.

10. Research in the area of word problem solving in mathematics has shown the following:
The ability to solve word problems requires more than procedural skills such as performing computations and conceptual understanding.
The ability to represent problems is critical. Representations of word problems often require a diagram.
The use of diagrams has been found to improve problem solving.
Useful representations allow students to reflect on them, modify them, and link them to a suitable strategy, computation, and procedure.
Representations that involve formal symbolic algebra have been found to cause difficulty in problem solving among younger students because they do not understand the meaning of the letter used and have difficulty translating information in text form into algebraic equations.

11. Important features in modeling
With students just learning algebra avoid using variables. Use a (?) to emphasize what is to be found.
The available information is recorded into the model to demonstrate the relationships between the parts and make the computations make sense.
Different types of dotted lines are used to model adding onto bars and taking away from bars.

12. The Part-Whole Model
By Part-Whole we mean a problem where two or more subsets (parts) make up a whole.
They may be drawn two ways.

13. There are 12 boys and 15 girls in a room
There are 12 boys and 15 girls in a room. How many children are there in the room altogether?
Draw a simple model.
Turn to a partner and explain your model to them.

14. Mrs. Lee baked a tray of 36 chocolate cookies and vanilla cookies
Mrs. Lee baked a tray of 36 chocolate cookies and vanilla cookies. There were 15 chocolate cookies. How many vanilla cookies were there?
Draw a simple model.
Turn to a partner and explain your model to them. But switch roles from before.

15. Help with statistics David made a bar graph but accidentally tore it.
Find the number of books David read in week 4.
With this example we want to also show how this can be used to set up an algebraic relation.

16. Enrico has traveled 11 miles of a 31-mile journey
Enrico has traveled 11 miles of a 31-mile journey. Find the distance Enrico still has to travel to complete the journey.
Here is my model. Explain my thinking. Why do you think I made this diagram a single bar broken into two adjoining pieces?
I want teachers to see the difference between a continuous model vs a discrete model.

17. Construct a bar model for this problem.
Make up a story problem to go with this problem.

18. Here is a good time for a break
Here is a good time for a break. Follow up with having someone display their answer.
Independent practice

19. What is different about this problem?
Two apples and a mango cost $4. Two apples and three mangoes cost $9. What is the cost of each type of fruit?
What is different about this problem?
How could we differentiate this problem for different grades and abilities?
Be sure to draw the Venn diagram showing the relationship. Draw as both a discrete model and a continuous model. MODEL BOTH WAYS.

20. Let’s develop some number sense

21. Now What
Break!

22. Comparison Models
Comparison models are used when one quantity is compared to another such as…
Juan has 3 more seashells than Kim. Juan and Kim have 15 seashells altogether. Find the number of seashells that Juan has.
Lanny has 3 times as much money as Ming. Lanny and Ming have $120 altogether. Find the amount of money Lanny has.

23. What is different about The two problems?
Additive comparisons contain one quantity that has a defined quantity that is more or less than the other one.
Multiplicative comparisons contain one quantity that is a defined factor of the other quantity.

24. What is the advantage of the first model over the second model?
Juan has 3 more seashells than Kim. Juan and Kim have 15 seashells altogether. Find the number of seashells that Juan has.
What is the advantage of the first model over the second model?
Does that mean that the second model is incorrect?
There are two set ups for this problem show both and discuss how one model is easier than the other, but both models are correct.

25. Lanny has 3 times as much money as Ming
Lanny has 3 times as much money as Ming. Lanny and Ming have $120 altogether. Find the amount of money Lanny has.
In this type of model, what are key characteristics to make it effective?
The focus of this model need to emphasize the importance that the blocks are all uniformly shapes rectangles to model equivalent values.

26. What does this example teach us?
Ali has $8 more than Sid. Trina has $6 less than Ali. The three of them together have $76 in all. Find the amount of money each of them have.
What does this example teach us?
Talking Point: Who has the most money, who has the least? (size of the bars).
Main idea is converting the bars to the same size as a defined unit bar.

27. Individual Practice If you teach grades K-2:
Ty has 9 fewer candy bars than Phoebe. They have 35 candy bars altogether. Find the number of candy bars Ty has.
If you teach grades 3-6:
Chris started saving money on Monday. Each day she saved $2 more than the day before. By Friday of the same week Chris had saved $35. Find the amount Chris saved on Wednesday.
If you teach grades 7 and up:
Mom is 28 years older than Zack. Mom is 4 years younger than Dad. Their total age is 84 years. What is mom’s age?
(7 and up, show the algebra demonstrated in your model and process)

28. Some Different types of Multiplicative Comparison models
There are four groups of students in the hall. There are twice as many boys as girls in the hall.
In Group A, there are 12 girls. How many boys are there in Group A?
In Group B, there are 12 boys. How many students are there in Group B?
In Group C, there are 12 students. How many girls are there in Group C?
In Group D, there are 12 more boys than girls. How many students are there in Group D?
Draw a model of each situation, one below the other.
Take 15 seconds to think to yourself how the models are different.
Take 1 minute and 28 seconds to discuss with each other your thinking.

29. A computer game costs twice as much as a soft toy
A computer game costs twice as much as a soft toy. The soft toy costs twice as much as a board game. If you were to purchase all three it would cost $224. Find the cost of the soft toy.
Why is it important to define what a single unit represents?

30. How are all four of these problems alike?
If the smaller number is 36, what is the sum of the two numbers?
If the larger number is 36, what is the sum of the two numbers?
If the difference between the two numbers is 36, what is the larger number?
If the sum of the two numbers is 36, what is the smaller number?
How are all four of these problems alike?

31. The sum of three numbers is 96
The sum of three numbers is 96. The largest number is 3 times as large as the smallest number. The middle number is twice as large as the smallest number. Find the value of the largest number.
What is a shortcut students with good number sense should see from the model?
Note that if you combine the block from the smallest number with the block from the middle number the largest number and the combo have the same number of tiles. To find the largest number you can divide the sum in half.

32. Choose one of the following problems
Mandy packs her clothes into a suitcase and it weighs 20 kilograms. Nat packs his clothes into an identical suitcase and it weighs 12 kilograms. Mandy’s clothes weighs twice as much as Nat’s clothes. Find the weight of the suitcase.
Package A is 4 times as heavy as Package B. How heavy is Package C?

33. To summarize thus far
Discuss differences in notation.

34. Additive and Multiplicative Comparison Models
Farid is 3 years younger than Erica. Don is 3 times as old as Erica. Together, Erica and Farid’s age is 15 years less than Don’s age. How old is Don?

35. Their suitcases weigh 61 kilograms altogether
Their suitcases weigh 61 kilograms altogether. Find the weight of the heaviest suitcase.

36. Choose One
There are 5 more boys than girls on a trip. There are twice as many teachers as girls on the trip. Not counting the girls, there are 29 persons on the trip. How many persons are on the trip?
The price of a sandwich is $.50 more than the price of a drink. The price of a pizza is twice the price of a drink. Two pizzas cost $1.90 more than a drink and a sandwich. Find the price of a drink.

37. Tasks for Kindergarten and Grade 1

38. Grades 1 and 2 type tasks

39. Grades 1 and 2 continued

40. Grades 1 and 2 continued

41. Tasks from Kindergarten trough Grade 3

42. Tasks from Kindergarten trough Grade 3

43. Tasks from Kindergarten trough Grade 3

44. Questions to ponder
Are problems from one of these categories more difficult than another? Within a category, is there one type of problem more difficult than another? For Kindergarten and Grade 1, present the problem to the students verbally.

45. For Grades 3 and 4
Ask students to solve comparison models and note which method they are using.
Is there a difference in student performance with respect to the type of problem or the methods used?

46. Arithmetic Problems

47. Algebraic Problems

48. Are Algebraic problems generally more difficult than arithmetic problems? Is any type of arithmetic comparison problem more difficult than another?

49. What time is it?

50. Modeling Fraction, Ratio, and Percent Problems
At a fair, there were half as many men as women. The number of adults to children was a third that of children. There were 3,600 people at the fair. How many children were there?

51. Guided Practice
¾ of the sales from onions is as much as 3/5 of the sales from carrots. The sales from carrots is $240 more than the sales from onions. Find the sales from the carrots and onions together.

52. At a sports festival, there were twice as many students who chose basketball as students who chose soccer. The number of students who chose basketball was one-fourth the number of students who chose baseball. There were 84 more students who chose baseball than soccer. How many students chose basketball?
How many things are we talking about?
Try solving it by using a single bar to represent the number of students who chose basketball.
Try it again by drawing a single bar that represents either of the other quantities.
Is there a difference in the answer?
Which quantity would be the most difficult to represent as a single unit bar and why?
Compare the operations and thinking involved in using the basketball students as a unit bar and the students who chose baseball as the unit bar. The thinking in dividing by a fraction becomes clear.

53. Comparisons using ratios
There are three boys in the Lee family. The ratio of the oldest boy’s age to that of the youngest is 4:1. the ratio of the middle boy’s age to that of the youngest is 2:1. the eldest boy is 4 years older than the middle boy. How old is the youngest boy?

54. How is the mathematics described by the modeling in this example?
The ratio of the number of red pebbles to the number of blue pebbles to the number of green pebbles in a container is 2 : 3 : 5. There are 80 pebbles in the container. How many red pebbles are there?
How is the mathematics described by the modeling in this example?
Participants should see that because we are given a total quantity that 2 tenths of the total is red.

55. There are 3 times as many Grade 6 students as Grade 2 students in Park Lane Elementary School. The ratio of the number of Grade 2 students to that of Grade 4 students is 1 : 4. There are 112 Grade 4 and Grade 6 students altogether. Find the number of Grade 4 students.
What do you think would be the most common error in this problem?

56. Percents
The number of tourists who visited a town this year is 75% of last year. There were 120,000 tourists last year. How many tourists visited the town this year?

57. How else could students think about the units in these problems?
Choose one
Pedro’s paycheck for Thursday is 90% of his paycheck for Friday. His total pay for Thursday and Friday is $76. Find Pedro’s pay for Friday.
Alvin’s salary is 20% more than Theodore’s salary. Alvin’s salary is $3,600. Find Theodore’s salary.
How else could students think about the units in these problems?

58. How could students think about the units in this problem?
Cindy’s salary is 25% less than Carla’s salary. Their total salary is $7,000. Find the difference in their salaries. What advantage does the thinking of units as 25% have over units as 1%?

60. It is interesting to note that by Grade 5, students with higher achievement scores used the model method more of ten than those with lower achievement scores in solving challenging before-after problems. Also 1 in 2 low achievers who used the model method were successful, compared to 1 in 3 who did not.
Goh, 2009
Before – After Models

61. Basic Before – After Models
What elements make a before - after problem?
There is an initial value of a quantity.
There is a change, which can be an increase or decrease.
There is a final value involved.

62. Progression through the development of the before – after model

63. Yani had $20. She donated $5 to her favorite charity
Yani had $20. She donated $5 to her favorite charity. How much money did yani have left?
How is the operation modeled by the bars? How else might a student model the problem using a single bar? How do the two models differ in the operation modeled?

64. How does the bar model show the 4 number sentences that can be written about this problem? Which number sentence is most beneficial to the student in solving this type of problem?

65. After Zack had added 19 songs to his playlist, there were 217 songs in his playlist. How many songs were in the playlist at first?

66. What is the difference in the two models for these problems?
After reading 19 pages, Edwin still has another 172 pages to read before he finishes a book. How many pages does the book have?
Donna deposited $75 into her bank account. She had $1,245 in her account after the deposit. How much money was in her account before she deposited the $75?

67. About to drop the F-Bomb
Before – After Models involving
Fractional changes

68. After giving away 2/5 of her rock collection, Amy still had 72 pieces
After giving away 2/5 of her rock collection, Amy still had 72 pieces. Find the number of pieces she gave away.
How is a problem like this similar to and different than the others?

69. Carmela has two brothers
Carmela has two brothers. She gave 1/6 of her candy sticks to one of them and 2/5 of the remainder to the other. She has 12 candy sticks left. How many candy sticks did Carmela have at first?
What does the model reveal about the fractional amount she gave away that might not be evident from the wording in the problem?

70. Bonita spent ¾ of her allowance on a new toy that cost $48
Bonita spent ¾ of her allowance on a new toy that cost $48. Find Bonita’s allowance.
How could is this problem similar to a previous strategy?

71. A store started selling a new gadget
A store started selling a new gadget. On the first day , it sold 3/7 of its stock. On the second day, it sold 3/4 of the remaining stock. On the third day, it sold the remaining 56 pieces. How many pieces did the store sell on each day?
What additional information does my model tell me? And… How does the picture help me with the problem?

72. Try to explain this model without drawing it.
Dana spent 1/3 of her money on a mathematics textbook and ½ of the remaining money on a science textbook. Both textbooks cost $72 altogether. How much money did she have at first?

73. Final Problem
The ratio of the amount of rice in Mr. Tan’s store to that in Mr. Lim’s store was 6 : 5. After they sold the same amount of rice, Mr. Tan had twice as much unsold rice as Mr. Lim, and the total amount of unsold rice was 240 kilograms altogether. Find the amount of rice they sold.

74. Evaluations
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75. This Concludes Our Broadcast Day