The students will be able to draw a model to help them write
expressions that can be use to solve one‐ and two‐step problems.
- About Teaching Mathematics: A K‐8 Resource, by Marilyn Burns; ISBN 0‐941355‐25‐X
- 8‐Step Model Drawing: Singapore's Best Problem‐Solving MATH Strategies, by Bob
Hogan and Char Forsten; ISBN 13‐973‐1‐88458‐95‐6
- Word Problems for Model Drawing Practice, by Catherine Jones Kuhns; ISBN 978‐1‐934026‐53‐3
- Marjorie Montague. Math Problem Solving for Upper Elementary Students With Disabilities.
The Access Center: Improving Outcomes for All Students K‐8.
Background for Teachers
The skills and strategies needed for successful mathematical problem solving start developing in the preschool years, when children acquire a basic conceptual understanding of the base 10 numerical system
Students in grades three, four, and five continue to apply
and refine the skills and strategies necessary to solve real life mathematics problems
However, many students, especially students with learning disabilities, have difficulty solving math word problems because they often cannot decide what to do to solve the
problem. Most textbooks are not very helpful when it comes to teaching students how to solve math problems.
Teaching mathematical problem solving is a challenge for many teachers, many of whom rely almost exclusively on mathematics textbooks to guide instruction. Most mathematics
textbooks simply instruct students to draw a picture or make a diagram
Students who have difficulty solving math word problems usually draw a picture of the problem without considering the relationship among the problem components and, as a
result, still do not understand the problem and therefore cannot make a plan to solve it. So, it is not simply a matter of drawing a picture or making a diagram; rather, it is the type of picture or diagram that is important. Effective visual representations, whether with
manipulatives, with paper and pencil, or in ones imagination, show the relationship among the problem parts.
For upper elementary students, math problem solving instruction should start with onestep problems involving only whole numbers. When students have mastered the problems at this level, they can progress to one‐step problems with decimals. They can then progress
to two‐step problems with whole numbers, and so on.
(From Math Problem Solving for
Upper Elementary Students with Disabilities; Marjorie Montague, Ph.D., University of Miami;
This lesson will explore a variety of strategies, methods, and classroom structures that can
be used in helping students to become better problem solvers, but the main focus of the
lesson will be on using a whole/part model that can be used as a tool to help students
understand and solve one and two operation problems that are required by the fifth Grade
Math Core. This whole/part model also relates to the fifth Grade Science Core, Standard 1,
which deals with physical and chemical changes in matter. In this science standard
students learn that matter cannot be created or destroyed, and that the sum of the parts is
equal to the whole. Often this is a hard concept for students to understand, but, by using
the whole/part model in math and science, it can help to deepen students understanding
of this fundamental concept.
When students approach a math problem, they need to be armed with strategies that will
allow them to be successful. Since students face a wide variety of problems, they need a
variety of strategies. The whole/part approach is just one approach to problem solving
and is not applicable to all types of problems. It works well for many one step choose an
operation type problems and many multiple‐step problems.
Intended Learning Outcomes
- Become effective problem solvers by selecting appropriate methods, employing a variety
of strategies, and exploring alternative approaches to solving problems.
Invitation to Learn:
Explain to the class that today they will be discussing problem solving, and to get everyone into the spirit of problem solving you are going to give them some problems to solve. Have
students find the Problem Solving Quiz (this quiz has trick questions and riddles) and work the problems. Give participants time to complete the quiz. Discuss the answers to the questions and have students share strategies they used to find the answers.
Problem Solving Quiz Answers
- Some months have 30 days; some months have 31 days. How many months have 28 days?
12, they all have at least 28 days
- How many birthdays does the average person have?
- A farmer had 17 sheep. All but 9 died. How many does the farmer have left?
He has 9; the rest died
- How much dirt may be removed from a hole that is 3 feet deep, 2 feet wide, and 10 feet long?
None; you cannot take dirt out of a hole.
- Take 2 apples from 3 apples and what do you have?
You have 2 apples
- Divide 30 by ½ and add 10. What's the answer?
70 because 30 / ½ = 30 x 2/1 = 60; 60 + 10 = 70
- I've got 2 U.S. coins that total 55 cents. One of the coins is not a nickel. What are the 2 coins?
A half‐dollar and a nickel. One's not a nickel; the other one is.
- There are 12 1‐cent stamps in a dozen, but how many 2‐cent stamps are there in a dozen?
12 is always a dozen
- How is the moon like a dollar?
They both have 4 quarters
- What is the difference between a new penny and an old quarter?
- Where do you buy a ruler that is 3 feet long?
At a yard sale
- How many times can you subtract 6 from 30?
Only once; after that you no longer have 30
- Students should be seated in cooperative teams of 4. Explain that they will be doing numbered heads together so everyone needs a number between 1‐4. When the team is given a question or problem, they should discuss it and be prepared to share the team's responses with the class.
- Begin by posing the question: What is the most important aspect of teaching
students how to be successful problem solvers? Give the teams time to
discuss the question.
- Call a number. Members of each team with that number should stand and
share with the whole class their team's comments about the question.
- Use the Problem Solving PowerPoint with ideas from About Teaching Mathematics by Marilyn Burns on "Managing the Classroom for Problem Solving" and from Marjorie Montague on what research says about problem solving, to continue the discussion (see Resources).
- Introduce the "whole/part" model by modeling the drawing process using
the problems on the Practice Problems worksheet.
- Continue with guided practice. Have students work the next problems on
their worksheets as a team. Discuss each question and have teams share
- Have participants work the final problems on the worksheet individually.
Give help where needed.
- Make the following points:
- The "whole/part" model does not work with all types of problems but is
effective with many "choose an operation" and multiple‐step problems.
- With practice, students can learn to identify when to use "whole/part."
- Some students will not need the help of this strategy, especially with
simple one‐step problems, but if they learn how to use it they will have it
as a tool when they encounter more difficult problems.
- Singapore Math uses similar model drawing strategies that expand to
many different types of problems (see the resource section for more on
Singapore model drawing).
- The "whole/part" model can be used to help students understand the
relationship of the parts to the whole in the fifth Grade Science Standard
dealing with physical and chemical changes of matter.
- Use numbered heads together to discuss the question: What is the most
effective way to teach students to use the "whole/part" model to solve
problems? Bring out the following points:
- Doing a problem a day can be effective, because students can focus on one problem at a time and grow their abilities by working on increasingly
difficult problems over time.
- This helps create a classroom culture where students are unafraid to try new approaches to problems and share their successes and failures.
- Teaching students to write math problems can help them to become better
problems‐solvers as they look closely at how different types of problems are
structured. (See the Activity section below.)
- Discuss ways students can be guided to write grade‐level‐appropriate
problems, such as:
- Suggesting themes or situations they can use to build problems around.
- Providing students with partial problems that need some information
added to make them complete.
- Providing students with a drawing and having them write a problem to
- Having students act out a problem situation.
- Providing data sets to use in the problem.
Lesson and Activity Time Schedule:
- Invitation to Learn: 10 minutes
- Lesson procedures: 50 minutes
- Problem writing activity: 30 minutes
- Total lesson and activity time 90 minutes.
Activity Connected to Lesson:
Problem Writing Activity:
- Divide the cooperative teams of 4‐6 in half to make teams of 2‐3.
- Show examples of student‐written problems.
- Explain that each team will need to write 4 different problems. One problem should use the "whole/part" model drawing provided on a Drawing Card, one problem using information from the U.S. Census data set, one about life at Core Academy, and one free choice.
- Problems should be multiple step problems and apply to the fifth Grade Core.
- Each problem should be clearly written in large letters on an 8 ½" by 11" sheet of paper. All problems should be displayed on one piece of chart paper. An illustration to decorate the problems and humor to make us laugh is appreciated.
- The solution to each problem should be clearly written in large letters on an 8½" by 11" sheet of paper and pasted to the back side of the chart paper.
- Allow time for teams to work.
- When problems have been completed, have each team share. Have the class solve problems as time permits.
Assign the class a Problem of the Day. Have students work in pairs, teams or individually
to solve the problem. Have students share their solutions with the class and discuss.
Encourage students to share or teach what they have learned about model drawing with
parents and family members.
Use student problem solutions to determine their understanding of the process and what
type of problems they need to be assigned to continue to progress.