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The content that follows is what the teacher sees on the Lesson Overview page for each lesson in the teacher course. The 5E Lesson Plan is composed of individual tabs for each phase of the 5E lesson. For easier viewing as you review this sample lesson, they are combined into one page here.
Lesson Overview
Focusing Questions: How can you use finite differences to construct a quadratic model for a data set?
Learning Outcome:
- I can use finite differences to write a quadratic function that describes a data set.
Textbook Alignment: Chapter 1, Section 7
Sec 1.7 Algebraic Reasoning TWE 2024
| Texas Essential Knowledge and Skills (TEKS) |
| AR.2D Patterns and structure. The student applies mathematical processes to connect finite differences or common ratios to attributes of functions. The student is expected to determine a function that models real-world data and mathematical contexts using finite differences such as the age of a tree and its circumference, figurative numbers, average velocity, and average acceleration. |
| AR.1A Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to apply mathematics to problems arising in everyday life, society, and the workplace; |
| English Language Proficiency Standards (ELPS) |
| 5.B Cross-curricular second language acquisition/writing. The ELL writes in a variety of forms with increasing accuracy to effectively address a specific purpose and audience in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in writing. In order for the ELL to meet grade-level learning expectations across foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student’s level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations do not apply until the student has reached the stage of generating original written text using a standard writing system. The student is expected to write using newly acquired basic vocabulary and content-based grade-level vocabulary; |
Pacing Guide
- Engage (5-10 minutes)
- Explore (25-30 minutes)
- Explain (20 minutes)
- Elaborate (40-45 minutes)
- Evaluate (10 minutes)
Teacher Materials
- Color tiles
- Square-inch grid paper or chart paper.
- Teacher Wraparound Edition (TWE) of Algebraic Reasoning textbook
Student Materials
- Pencil
- Paper
- Graphing calculator, spreadsheet or graphic application
- Student Edition (SE) of Algebraic Reasoning textbook
Vocabulary
- quadratic function
- finite differences
- maximum
- minimum
Overview of Process Standard(s)
In this lesson, students will apply mathematics to real-world situations in everyday life, society, and the workplace in the following ways:
- Students can analyze tables and graphs of real-world data to determine quadratic functions to model the situation and then use those quadratic functions to make predictions about the situation.
- Students can analyze a verbal description of the real-world situation to determine mathematics in the situation that may be modeled by a quadratic function.
5E Lesson Plan
Engage (5-10 minutes)
- Present the task on page 88 about Mrs. Hernandez’s sandbox. Sec 1.7 Algebraic Reasoning SE 2024
- Facilitate students’ mathematical reasoning as they discuss what dimensions are possible for a rectangular sandbox.
- As needed, support student reasoning and productive struggle through questioning.
- Clarifying Question(s)
- When you say __________, what do you mean?
- Focusing Question(s)
- What is this problem about?
- Advancing Question(s) [ask and walk away]
- How could you cut the wood to make the sides of the sandbox?
- What patterns do you notice?
- Assessing Question(s)
- How can you determine whether dimensions are incorrect for this situation?
- Clarifying Question(s)
Explore (25 – 30 minutes)
- Group students in a visibly random way (e.g. playing cards, drawing straws, spinner, etc.) into groups no larger than three students per group.
- Provide each group of no more than three students with color tiles.
- Facilitate student group exploration and reflection on pages 88-90. Sec 1.7 Algebraic Reasoning SE 2024
- Support for Emergent Bilinguals:
- Writing with familiar English language words helps students both deepen their understanding of the mathematical content as well as become more comfortable with the English language. Having students use the reflection questions for a journal entry in their interactive math notebooks provides an opportunity to reinforce this language proficiency skill.
- As needed, support student reasoning and productive struggle through questioning.
- Clarifying Question(s)
- Can you explain that in a different way?
- Did I paraphrase you correctly?
- Focusing Question(s)
- How is area different from perimeter?
- What do you think the problem means by, “second finite difference”?
- Advancing Question(s) [ask and walk away]
- What pattern(s) do you notice?
- What do you notice about the finite differences?
- What do you think the pattern in the finite differences might mean about the pattern in the second finite differences?
- Assessing Question(s)
- How can you verify that your equation represents the data in the table accurately?
- Clarifying Question(s)
Explain (20 minutes)
- Student-Centered Learning Option
- Allow time for students to read the Explain section on pages 90 – 92 and , including worked out examples 2 and 3. Sec 1.7 Algebraic Reasoning SE 2024
- Have students try the You Try It problems 2 and 3.
- As needed, support student reasoning and productive struggle through questioning.
- Guide students to watch the associated videos linked beneath the QR code on page 91 as needed.
- Explicit Instruction Option
- Prior to teaching the lesson, read Section 1.7 in the Algebraic Reasoning TWE. Sec 1.7 Algebraic Reasoning TWE 2024
- Before teaching the lesson, watch only the lesson videos for Section 1.7 that are indicated below. Other learning in this section was addressed in Unit 1.
- Section 1.7 Explain Video (5:08)
- Section 1.7, You Try It, Question 2 Video (1:49)
- Section 1.7, You Try It, Question 3 Video (4:51)
- Utilize slides 2 and 3 in the 1.7 Examples.pptx PowerPoints to directly teach this portion of the lesson using the provided examples.
- Guide students to try the corresponding You Try It problem between your presentation of each example.
- Instructional Hints:
- Use technology such as a graphing calculator or spreadsheet application on a display screen to show students how, no matter the numbers present in the quadratic function, the second differences will always be constant.
- Encourage students to annotate the flowchart they created during Unit 1 with abbreviated directions for writing function rules. Summarizing learning in one flowchart will help students study and process the content.
- Justification of why data represents a specific type of function is important for students to demonstrate mastery of the concept. Have students write complete sentences with explanations of their answers.
- Support for Emergent Bilinguals:
- Asking students to write using newly acquired basic vocabulary and content-based grade level vocabulary helps emergent bilinguals make connections among vocabulary terms and important mathematical ideas. Students encountered quadratic functions in Algebra 1 and are using finite differences to deepen their understanding of how to use quadratic functions to model real-world phenomena. Writing with previous and new vocabulary terms helps students merge these ideas while they are learning the English language.
- Clarifying Question(s)
- What does the problem ask?
- Can you explain your thinking in another way?
- Focusing Question(s)
- Which value(s) represent the y-intercept?
- How are the parameters of the function related to the data in the table or the finite differences?
- How does this data look on a scatterplot?
- Advancing Question(s) [ask and walk away]
- How can you tell from looking at a table that a function is quadratic?
- How do finite differences help you determine a quadratic function?
- Assessing Question(s)
- How would you explain to someone who has never seen this math lesson how to write a quadratic function based on a table of values?
Elaborate (40 – 45 minutes)
- Differentiated elaboration assignments based on instructional assessments in the previous portions of the lesson
- Not Yet Proficient: p. 100-102 #9, 11-14, 18-20
- Students may work in a small group or 1:1 with the teacher initially
- Somewhat Proficient: p. 100-102 #7, 11-14, 18-20
- Proficient: p. 100-102 #8, 10, 11, 13-17
- Highly Proficient: p. 100-102 #7-9, 13-17
- Not Yet Proficient: p. 100-102 #9, 11-14, 18-20
- As needed, support student reasoning and productive struggle through questioning.
- Clarifying Question(s)
- What does the problem ask?
- Focusing Question(s)
- Which value(s) represent the y-intercept?
- How does this data look on a scatterplot?
- Advancing Question(s) [ask and walk away]
- How can you tell from looking at a table that a function is quadratic?
- How do finite differences help you determine a function that models the data?
- How can you identify the y-intercept from a table of values?
- Assessing Question(s)
- Explain how you use a table to determine a quadratic function.
- How can you tell whether the data in a table can be modeled with a quadratic function when the data is not perfectly quadratic?
- Clarifying Question(s)
Evaluate (10 minutes)
- Paired Problem Solving
- Display #5 on page 131 with a table of data. Ch 1 Chapter Review AR SE 2024
- Ask students to pair up to discuss the table, determine whether it can be represented by a linear function or a quadratic function, and then write a function rule that models the situation.
- As needed, support student reasoning and productive struggle through questioning.
- Clarifying Question(s)
- When you say __________, what do you mean?
- Focusing Question(s)
- What is this problem about?
- Advancing Question(s) [ask and walk away]
- How do you find finite differences?
- I wonder how a scatterplot and a graph could be helpful?
- Assessing Question(s)
- How did you determine that your answer was reasonable?
- Clarifying Question(s)
