Bar Model Solutions – by Students

After the post on Assessing Bar Model Solutions went up, Beth Curran sent a message: “We just did that problem!” She agreed to share some student work:


boys and girls 2

boys and girls 3

boys and girls 5

And when the students didn’t draw a model:

boys and girls 4

I see this as a comparison problem:

thinking blocks

5 units -> 125 students
1 unit -> 125 ÷ 5 = 25
7 units for boys -> 7 x 25 = 175 boys in all

(That’s the Thinking Blocks Model Drawing tool that allows you to insert your own word problems and solve – or you can use the pre-made questions!)

 

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

 

 

 

 

 

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

Who to see at National Math Educators Conferences this year?

Me!

As a full-time trainer and instructional coach, I know the importance of professional development and continuing education. Since embarking on this career path seven years ago, I’ve attended annual meetings of both the National Council of Teachers of Mathematics and NCSM (formerly the National Council of Supervisors of Mathematics). Attending NCTM, NCSM and other regional, national and international math conferences and educational events fulfills my personal ongoing learning goals.

In 2012, I was thrilled to have a proposal to present at NCTM accepted! (Here are two posts on that presentation: tools and thoughts).

NCTM 2015 header  2015BostonAM_590x90_WebBanner_p1For 2015, I submitted similar proposals to both NCTM and NCTM…and BOTH were accepted. Woot! And good news, my co presenter will be Lauri Susi of Conceptua Math.

Here’s the description:

Strip Models, Tape Diagrams, and Bar Models, Oh My!

Lead Speaker: Cassandra Turner
Co-Speaker: Lauri Susi


These visual components sit at the intersection of Common Core, Singapore Math®, and now technology! Learn why this visual model for word problems is so powerful, try some problems from the simple to the complex, and investigate web-based programs and iPad apps that will help anyone incorporate this effective strategy into their classrooms.

The NCSM Presentation is on Tuesday, April 14, 2015, 10:00 AM – 11:00 AM.

The NCTM Presentation is on Thursday, April 16, 2015, from 11:00 AM – 12:00 PM (details here).

But that’s not all!

This year, Singapore Math Inc. asked me to present an NCTM workshop to address some practical considerations schools face when implementing the curriculum.

Filling Knowledge Gaps with Critical Singapore Math® Approach Across Grade Levels (Gr. 3-5)


One of the biggest challenges during a Singapore Math® implementation can be the gaps in student knowledge. Upper elementary students frequently lack the foundation provided by the 1st-3rd grade Singapore curriculum. In this direct from the classroom session, you’ll learn the critical lessons and concepts students must master before jumping into their grade-level content.

This NCTM Workshop is on Friday, April 16, 2015, from 12:30 – 1:30 PM

When the complete conference schedules for both NCTM and NCSM are available, I’ll publish an overview of Singapore Math® presentations along with recommendations for session at each conference as I have done every year since 2010.

If you are headed to Boston in April, please consider attending one of these sessions.  I’d love to connect with you!

NCSM Boston2015BostonAM_200x2002015 NCSM Annual Conference – April 13-15, 2015

SHINING THE LIGHT ON LEARNING:
A Vision for Mathematics Leaders


2015 NCTM Annual Meeting & Exposition – April 15-18, 2015

Effective Teaching to Ensure Mathematical Success for All

Boston, Massachusetts
Boston Convention & Exhibition Center

[Note: Conference organizers include a “Times Subject to Change” disclaimer. I’ll update this information if it changes.]

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!