Assessing Bar Model Solutions

It’s important in a Singapore style program for students to understand bar model drawing as a tool to help visualize relationships between the known and unknown in a word problem. It helps students see the algebraic structure in problems in a more concrete manner.  Developed by a Primary Mathematics Project team of the Curriculum Development Institute of Singapore in the 1980’s, the Model Method for problem solving (as it’s known in Singapore) was designed as a pictorial stage to help students learn abstract mathematics.

The model method has been incorporated into the Progressions documents for the Common Core State Standards for Math (CCSSM) as a “tape diagram”. The CCSSM glossary defines a tape diagram as:

“A drawing that looks like a segment of tape, used to illustrate number relationships. Also known as a strip diagram, bar model, fraction strip, or length model.”

As such, bar models are becoming ubiquitous in elementary schools. Books have been written about Singapore’s model method, my favorite being  The Singapore Model Method for Learning Mathematics:

Model Method


There’s plenty to be covered with bar models, and I thought I’d share answers to the most commonly asked questions I get from teachers on model drawing:

How do you get kids to draw models when they already know the math?
How do you assess them?

When models are introduced in Singapore textbooks, the problems are very simple and students will typically know what operation to use. The first part whole models in Primary Mathematics, Common Core Edition are in the 3A textbook:
Sum and difference

And the first part-whole model lessons in Math in Focus are in 2A:
MiF 2A models

Those are pretty easy word problems and most kids are waving their hands, with that “I KNOW the answer” look on their faces. It’s important to reiterate to students, “I’m sure you know the answer and even the equation to get there. What we’re learning today is a new way of drawing a model for our word problem so that we can work with more challenging problems in third grade. Problems like”:

A father shares $60 among his three sons and one daughter. If the daughter gets twice the amount that each son gets, how much money does his daughter receive?

or from 4A
Number of boy and girls in school

We’ll come back to that one.

When assessing word problems, the solution method is a significant part of the answer. Students should expect to show their work to get full credit for a problem. As a teacher, it’s important to keep in mind that there is no one-way to draw a bar model.

A word problem might consist of points for the method of solving as well as points for correct computation and answers. To encourage students to draw and become proficient with bar models, I have used a rubric for assessment:

  • 1 point for a representative diagram – Does the model make mathematical sense?
  • 1 point for correct labeling, including the “?”  to represent what is unknown.
  • 1 point for computation on the first step and, if more than one step…
  • 1 point for computation on the second or more steps
  • 1 point for a correct answer in a complete sentence

Let’s look at an example. This is a student-written two-step word problem from third grade:

Natasha and Analisa Meatballs

Does the bar model make sense? +1 point

Are the bars labeled? Are question marks in the correct places? +1 point

Is the computation to figure out the value of one unit correct? 384 ÷ 6 = 64 +1 point

How about the computation on the second step? 64 x 5 = 320 +1 point

Is the answer in a complete sentence? +1 point

So, what’s the deal with the 64 ÷ 6 equation? A byproduct of requiring students to show their work is that oftentimes, they will leave work on the paper, just in case. As a teacher, I love that this provides me with some insight into this student’s thought process:

I’m pretty sure I know what I’m doing, but I still am a little confused.

Which is exactly what the student said when I asked her about the extra equation. She realized her answer couldn’t be right because:

I ended up with 14 remainder 4 meatballs, and how could that happen?

Without a system to evaluate, you end up with:
Cassidy deposits moeny in the bank

I’m drawing boxes because I’m supposed to… I don’t get it.

This model doesn’t make sense. The student’s computation is slightly off, but she knows she needs to add. A teacher can also see from this model that more re-teaching and practice is necessary for this student and probably others in the class.

As students progress, the rubric may change. At the end of a unit/school year, the same problem might be worth three points; one apiece for diagram, computation, solution. I often tell students, “You don’t need to draw the model if you feel you understand the problem, however, you will be graded either all correct or all incorrect. Without the models I can’t give you partial credit for your thinking.”

Most students draw them anyway as a way to check their work. Some just start to see them in their heads. Here are a couple of examples from the end of a fractions unit in 5A

rice 3-001

rice 6-001

How would drawing a model have helped this student?

Number of boy and girls in school

 

 

 

 

 

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Fall 2015 dates for Model Drawing BER Seminar

Bar Modeling is my Force

Best. Seminar. Comment. Ever.

Here are the just-released Fall 2015 dates for my BER seminar “Boost Students’ Math Problem-Solving Skills Using Bar Models, Tape Diagrams and Strip Models (Grades 2-6)

Save the date!

A whole day of problem-solving with Bar Modeling, Tape Diagrams and Strip Models PLUS that handbook for your own home enjoyment! (- with the answers and fully worked solutions!)

——————————————————————————————————————————–

“How to Use the Best Strategies From Singapore Mathematics to Strengthen Your Math Instruction” (BER) will have 10 dates in spring of 2016 – I’ll update when available.

Your city not on the list? Contact me and I can bring my Singapore Math® workshop(s) to your school or district – email Cassy (at) singaporemathsource.com

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You Can Bar Model Anything!

nauty nice bar model

A fourth grader at a school I worked with this year included this on a Christmas card for her teacher.

Happy Holidays and Best Wishes for an Outstanding New Year!

 

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New Bar Modeling iPad Apps!

For many years, I’ve highly recommended Thinkingblocks.com to my students who hone their bar modeling skills while playing really fun games. These flash-based programs work great on a desktop or laptop, but required third-party solutions to work on mobile devices.

Tired of using Rover to run Thinkingblocks.com on your iPad?

Well, hop on over to the iTunes app store because Math Playground has just published  four new iPad apps based on the popular website Thinkingblocks.com that work perfectly, provide tracked progress and are FREE for a limited time:

Addition and Subtraction:

Thinking Blocks Addition and Subtraction

Multiplication and Division:

Thinking Blocks Multiplication & Division

Fractions:

Thinking Blocks Fractions

Ratios and Proportion:

Thinking Blocks Ratio & Proportion

 

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It Figures: Why U.S. Schools are Using Math from Singapore

Singapore’s Channel NewsAsia produces a show called “It Figures” described this way:

How do all the numbers and statistics on Singapore add up? IT FIGURES, figures it all out.

The premier episode of “It Figures Season 2” seeks “to find out why schools in the US are using our Math textbooks and adopting our way of teaching Math.”

Viewers were challenged to solve the Primary 5 Math problem below. Can you do it?

It Figures word problem

Most Singaporeans used Algebra.

Teacher Owen Lau uses the bar model method beginning at about 1:45:

[youtube]http://youtu.be/tblQg5dobIA[/youtube]From the Singapore TV Show It Figures.

Mr. Lau explains that “students can see better if a problem is presented in pictorial form.”  The following concepts are reinforced:

  • The Model Method: Drawing diagrams and bars to represent math problems visually.
  • The Model Method is not as abstract as Algebra, and helps students understand concepts instead of blindly apply math formulas.

The debut episode of “It Figures Season 2,” explaining the use of the Singapore curriculum in the US airs, on July 2, 2013.

Watch

Images and clip from the Channel NewsAsia website.

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