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Lesson Plan

What If We Changed the Book? Problem-Posing with Sixteen Cows

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What If We Changed the Book? Problem-Posing with Sixteen Cows

Grades 3 – 5
Lesson Plan Type Standard Lesson
Estimated Time Two 50-minute sessions
Lesson Author

David Whitin

David Whitin

Grosse Pointe Woods, Michigan

Phyllis Whitin

Phyllis Whitin

Grosse Pointe Woods, Michigan


National Council of Teachers of English


Student Objectives

Session One

Session Two


Student Assessment/Reflections



Students will:

  • identify attributes of a story.

  • pose a problem based on these attributes.

  • use pictures, numbers and words to solve a mathematical extension of the story.

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Session One

  1. Introduce Sixteen Cows to the class or other selected book for problem posing.

  2. Invite a few students to respond to each page as you read it aloud.

  3. After reading the story, invite the class to respond further by asking:
    • What did you find interesting about this story?
    • What else did you notice?
  4. As the students are responding to the story, fill in the chart with their observations.

  5. If the students do not mention some of the mathematical features of the story, then ask them to do so directly:

    • What do you notice about the numbers in this story?

    • How do the numbers change?
  6. Add these observations to the left hand side of the chart as well.

  7. Encourage many different answers, even though they may seem redundant. The language of each observation can offer different possibilities for extensions. Possible observations include the following:

    • “Both numbers are 8.”

    • “They each have the same number of cows.”

    • “They each have an even number of cows.”

    • “The cows didn’t look the same but the people each had 8.”
  8. After the students have shared about 10 observations (including at least 6-7 mathematical ones) and they have been recorded, demonstrate how to create some problem extensions. If needed, follow the Teacher Talk handout for problem-posing examples.

  9. Brainstorm some possible “what-ifs” for four or five of the students’ observations.

  10. Decide on one “what-if” extension (based on the various abilities of your students) and ask each student to solve it, using pictures, numbers and words to explain their answer.

  11. After the children have had time to solve this problem, invite different children to share their solutions with the whole class.

  12. Emphasize the different ways that children used pictures, numbers and words in their explanations. For instance, you might say, “I appreciate how you used arrows to show how you added those numbers together” or “I appreciate how you used the word ‘double’ to explain what you did first.”

  13. Begin a class chart that lists effective ways that children use pictures, numbers and words to make their thinking visible.

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Session Two

  1. Return to the chart of observations and “what-if” extensions.

  2. Read the list over together, emphasizing how the extensions were derived from the initial observations.

  3. Select two other problems to solve, and invite students to solve at least one of these. Again, encourage students to use pictures, numbers and words in their explanation.

  4. Provide another sharing time when the class can discuss the solutions as well as their different ways of representing their thinking.

  5. Invite students to revise and edit one of their solutions to publish. Some options are as a class book or bulletin board display on problem-posing. Some of their work might be included in a family newsletter as well.

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  • Invite students to develop an alternative story that is based on some of their “what-if” extensions. The following alternative provides an example:
If there are now 4 people sharing 16 cows, how might this mathematical variation change the story line? What will be the central problem of this story? Will there now be two couples marrying at the end of the tale?

  • Read other books that lend themselves to problem-posing extensions. Before reading the next book aloud to the children, you might create a chart yourself (or with other teachers) of your own observations and “what-if” extensions. In this way, you can envision more of the mathematical possibilities ahead of time; then, you can share the story with students and brainstorm together some observations and possible extensions.
  • Extend your study of Sixteen Cows with the additional teaching ideas on the author's Website.

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Feedback for this lesson should be ongoing, integrated with the students' problem-posing and problem-solving process. It's not something that occurs after the project is completed, but while students are working. Since the focus is on the strategies and processes students use, feedback and reflection takes the form of kidwatching and specific commentary that helps students expand and extend their language strategies. Use this information to create formal or informal checklists, which you can refer to as you observe students at work.

Whole group and small group participation

In your observations of student groups, these questions can guide your observations:

  • Are students engaged in the group work?
  • Do they listen to the suggestions of others?
  • Do students work cooperatively together throughout the lesson?
  • Do students provide useful ideas when participating in the group?
  • Do the members actively look for solutions for one another’s problem-posing?
  • Do students listen to, share with, and support the efforts of others?

Student explanations of their strategies

As you read or listen to students’ explanations, these questions can guide your observations:

  • Are the pictures, numbers, and words clear?
  • Do they add to the reader’s or listener’s understanding of the strategies?
  • Does the explanation show a complete understanding of the mathematical concepts used to solve the problem?

Quality of story problem written extensions and individual student follow-up work, including clarity of ideas and details in written work

As you read students’ story problems, these questions can guide your observations:

  • Is the problem and solution correct? creative?
  • Is the solution easy-to-understand and logical?
  • Are the details clear and complete?
  • Does the story follow a clear sequence? Are the ideas clear and organized?
  • Does the story problem include creative details and/or descriptions?
  • Is the story is readable, clean, neat and attractive?

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