Name of Task: Pictorial to Abstract
Task:
In the following picture,
represents x
represents –x
represents 1
represents –1
=
Option 1: Write an equation that models the pictorial representation shown above. Show how your
equation relates to the picture. Solve your equation. Show how your solution relates to the picture.
Write a different equation that models the pictorial representation shown above. Show how your second
equation relates to the picture. Solve your second equation. Show how your solution relates to the picture.
Option 2:
Carlos writes the following equation to match the pictorial representation:
4𝑥 + −𝑥 + 8 + −2 = −2𝑥 + 4
Sam disagrees with Carlos and says that the following equation matches the pictorial representation:
4(𝑥 + 2) + 1(−𝑥 − 2) = 2(−𝑥 + 2)
Edward disagrees with both of them and writes the following equation:
3𝑥 + 6 = −2𝑥 + 4
1.
Who is correct? Draw connections between the equation(s) and the picture to prove your answer.
2. Solve each equation. What do you notice?
3. If you were writing an equation to match the picture, whose would you write and why?
4. Can you think of a different equation to match the picture?
Alignment to Content Standards:
Alignment to Practice Standards:
Mathematical Goals:
What do you want students to know and understand about mathematics as a result of this lesson?
Prerequisite Knowledge:
What definitions, concepts, or ideas do students
need to know to begin to work on the task?
Accessing Prior Knowledge:
What questions will you ask to help students
access their prior knowledge and relevant life and
cultural experiences?
Task Solutions:
What are all the ways the task can be solved?
Anticipated Student Solutions:
Which of these methods do you think your students
will use?
Anticipating Misconceptions:
What misconceptions might students have? What errors might students make?
Resources/Tools:
What resources or tools will students have to use in their work that will give them entry into, and help
them reason through, the task?
Structure:
How will the students work—independently, in small groups, or in pairs—to explore this task? How long
will they work individually or in small groups or pairs? Will students be partnered in a specific way? If
so, in what way?
Questions that Assess and Advance Student Learning:
How will you help a group get started on the task? How will you focus student thinking on the key
mathematical ideas in the task? How will you assess student understanding of key mathematical ideas,
problem-solving strategies, or the representations? How will you advance student understanding of the
mathematical ideas? How will you encourage all students to share their thinking with others or to
assess their understanding of their peers’ ideas?
Reminder: The questions you ask should keep the task at a high level, not push kids toward
a particular solution path, turning the task into an opportunity to practice procedures.
Selection:
Sequencing:
What solution paths do you want to have shared
In what order will the solutions be presented?
during the class discussion?
Why?
Connect:
What specific questions will you ask so that students: Make sense of the mathematical ideas you want
them to learn? Expand on, debate, question, the solutions being shared? Make connections among the
different strategies that are presented? Look for patterns? Begin to form generalizations?
Extensions of the Task:
What will you do tomorrow that will build on this lesson?
Task: Pictorial to Abstract – Option 1
In the following picture,
represents x
represents –x
represents 1
represents –1
=
Write an equation that models the pictorial representation shown above.
Show or explain how your equation relates to the picture.
Can you think of another equation that models the pictorial representation shown above?
Show or explain how your equation relates to the picture.
If you came up with two equations, explain how it is possible to have two different equations that represent the same
picture.
Solve your equations.
Task: Pictorial to Abstract – Option 2
*recommended that this is printed single-sided
In the following picture,
represents x
represents –x
represents 1
represents –1
=
Carlos writes the following equation to match the pictorial representation:
4𝑥 + −𝑥 + 8 + −2 = −2𝑥 + 4
Sam disagrees with Carlos and says that the following equation matches the pictorial representation:
4(𝑥 + 2) + 1(−𝑥 − 2) = 2(−𝑥 + 2)
Edward disagrees with both of them and writes the following equation:
3𝑥 + 6 = −2𝑥 + 4
1.
Who is correct? Draw connections between the equation(s) and the picture to prove your answer.
2. Solve each equation. What do you notice?
3. If you were writing an equation to match the picture, whose would you write and why?
4. Can you think of a different equation to match the picture?