Algebra & functions

ILAW Daily Lesson Log (DLL)

Functions

Functions and Their Graphs · Term 1 · Weeks 9–10

Learning area: General Mathematics
Designed by: Jayous Mae A. De Leon
Grade level and section: Grade 11
Sessions / term: 5 sessions · Term 1
References: DepEd Grade 11 Three-Term Budget of Work (BOW) for General Mathematics
Galamay, D.O., & Albay, E.M. (2026). General Mathematics: A Strengthened Senior High School Textbook. Diwa Learning Systems Inc.
Digital tools: Desmos, GeoGebra, Spreadsheet, Presentation
Declaration of AI use (DO 3, s. 2026 Annex A): AI was used only as an auxiliary support tool in preparing and improving this lesson plan, in accordance with DepEd Order No. 003, s. 2026. The teacher retained human judgment and full responsibility for content selection, contextualization, assessment decisions, learner feedback, and final approval. AI-generated suggestions were reviewed, edited, and checked for accuracy, bias, appropriateness, data privacy, and alignment with the Learning Competencies. No learner personal data were entered into AI tools, and AI will not be used as the sole basis for grading, learner evaluation, disciplinary decisions, or other major academic decisions.

Intentions

Learning competency

Classify functions based on their graphs and/or equations to determine if they are linear, absolute value, quadratic, square root, or cube root.

gm11.t1.w9.c2

Learning objectives

At the end of the 45-minute period, the learners should be able to:

Classify functions as linear, absolute value, quadratic, square root, or cube root;
Justify their classification using mathematical evidence; and
Show appreciation on constructing an initial conceptual understanding of the relationship between an equation and its graph.

Learner context

Learners have prior experience with the Cartesian plane, ordered pairs, relations, and basic algebraic expressions. However, they may recognize familiar graph shapes without being able to explain why a particular equation produces that shape. They need opportunities to observe, compare, classify, and construct their own criteria before formal mathematical terms are introduced.

Learning experience

Pre-lesson

Hook

Engage

How can we determine what kind of function we are looking at without being told its name?

Activity: Notice and Wonder

Elicit

Display several graphs without labels. Learners individually record their observations before sharing.

Notice: What do you notice? What patterns do you see? Which graphs seem related?
Wonder: Why are some graphs straight? Why does one graph have a sharp corner? Why do some graphs have an endpoint? Why does one graph have an S-like shape?

Flow

Group Activity: Sort the Graphs

Explore

Give groups a collection of graphs. Learners sort them according to their own criteria. They may initially create categories such as: straight; curved; V-shaped; starts at a point; S-shaped.

"What rule did your group use to create your categories?"
"Could another group classify them differently?"
"What relationship do you notice between the equation and the shape of the graph?"

This encourages learners to recognize that mathematical classification requires defensible criteria, not simply visual guessing.

Defend Your Classification

Explain

Groups present their classification. Other groups are encouraged to challenge respectfully:

"What evidence supports your answer?"
"Would your rule work for this equation?"
"Can you find an example that contradicts your rule?"

This explicitly develops critical thinking and mathematical argumentation. Groups construct: Equation → Pattern → Graph → Characteristics → Possible Real-Life Situation.

Formal terminology

Explain

Only after learners have constructed their own categories does the teacher introduce the formal terminology. The teacher connects the students' discovered descriptions to the formal mathematical concepts. Say: "You discovered five different patterns. Mathematicians have names for these patterns..."

f(x)=x

f(x)=|x|

f(x)=x^{2}

Linear
Absolute Value
Quadratic
Square Root
Cube Root

Learning resources

Graph and equation cards
Graphing paper
Function classification worksheet
Desmos/GeoGebra/graphing technology
Learner's module/textbook

Opportunities for integration

ICT: Desmos for visualizing and comparing graphs.
Science: Relate graph patterns to relationships between measurable quantities.
Critical Thinking: Classification, justification, and evidence-based reasoning.

Assessment

Formative assessment

Exit Ticket: Prove Your Answer

Evaluate

Given an unfamiliar equation, learners must:

classify it;
identify evidence from the equation;
predict the shape of the graph; and
explain their reasoning.

Critical Thinking Focus: Analyze → Justify → Predict

Ways forward

Extended learning opportunities

Function Hunt: Where Do I See Functions?

Extend

Reinforce classification of linear, absolute value, quadratic, square root, and cube root functions.

Look around your home, community, textbook, or a teacher-provided online resource for 5 examples of graphs, equations, or situations that represent mathematical relationships.
For each example, identify the possible function type.
If you are unsure, use Desmos to graph the equation and examine its shape.
Complete the table below.
Choose one example and explain why you classified it that way.

Reflections

Think about what you need to change for the next session based on what happened today.

Day 1

Intentions

Learning competency

Classify functions based on their graphs and/or equations to determine if they are linear, absolute value, quadratic, square root, or cube root.

gm11.t1.w9.c2

Learning objectives

At the end of the 45-minute period, the learners should be able to:

Classify functions as linear, absolute value, quadratic, square root, or cube root;
Justify their classification using mathematical evidence; and
Show appreciation on constructing an initial conceptual understanding of the relationship between an equation and its graph.

Learner context

Learning experience

Pre-lesson

Hook

Engage

How can we determine what kind of function we are looking at without being told its name?

Activity: Notice and Wonder

Elicit

Display several graphs without labels. Learners individually record their observations before sharing.

Notice: What do you notice? What patterns do you see? Which graphs seem related?
Wonder: Why are some graphs straight? Why does one graph have a sharp corner? Why do some graphs have an endpoint? Why does one graph have an S-like shape?

Flow

Group Activity: Sort the Graphs

Explore

Give groups a collection of graphs. Learners sort them according to their own criteria. They may initially create categories such as: straight; curved; V-shaped; starts at a point; S-shaped.

"What rule did your group use to create your categories?"
"Could another group classify them differently?"
"What relationship do you notice between the equation and the shape of the graph?"

This encourages learners to recognize that mathematical classification requires defensible criteria, not simply visual guessing.

Defend Your Classification

Explain

Groups present their classification. Other groups are encouraged to challenge respectfully:

"What evidence supports your answer?"
"Would your rule work for this equation?"
"Can you find an example that contradicts your rule?"

This explicitly develops critical thinking and mathematical argumentation. Groups construct: Equation → Pattern → Graph → Characteristics → Possible Real-Life Situation.

Formal terminology

Explain

f(x)=x

f(x)=|x|

f(x)=x^{2}

Linear
Absolute Value
Quadratic
Square Root
Cube Root

Learning resources

Graph and equation cards
Graphing paper
Function classification worksheet
Desmos/GeoGebra/graphing technology
Learner's module/textbook

Opportunities for integration

ICT: Desmos for visualizing and comparing graphs.
Science: Relate graph patterns to relationships between measurable quantities.
Critical Thinking: Classification, justification, and evidence-based reasoning.

Assessment

Formative assessment

Exit Ticket: Prove Your Answer

Evaluate

Given an unfamiliar equation, learners must:

classify it;
identify evidence from the equation;
predict the shape of the graph; and
explain their reasoning.

Critical Thinking Focus: Analyze → Justify → Predict

Ways forward

Extended learning opportunities

Function Hunt: Where Do I See Functions?

Extend

Reinforce classification of linear, absolute value, quadratic, square root, and cube root functions.

Look around your home, community, textbook, or a teacher-provided online resource for 5 examples of graphs, equations, or situations that represent mathematical relationships.
For each example, identify the possible function type.
If you are unsure, use Desmos to graph the equation and examine its shape.
Complete the table below.
Choose one example and explain why you classified it that way.

Reflections

Think about what you need to change for the next session based on what happened today.

Day 2

Intentions

Learning competency

Sketch the graph of the following functions: linear, absolute value, quadratic, square root, or cube root with or without technology.

gm11.t1.w10.c1

Learning objectives

At the end of the 45-minute period, the learners should be able to:

sketch linear, absolute value, and quadratic functions;
explain how changes in an equation affect the graph; and
evaluate their predictions using mathematical evidence.

Learner context

Learners can plot points and work with tables of values from previous mathematics lessons. However, they may depend heavily on formulas or graphing technology and may find it difficult to visualize how changes in an equation affect a graph. They are challenged to predict a graph before using technology, construct it manually, and then verify or revise their thinking.

Learning experience

Pre-lesson

Hook

Engage

Can we predict the shape of a graph just by examining its equation?

Activity: Retrieve and Connect

Elicit

Learners receive the following equations. Without graphing, they draw what they think each graph looks like.

f(x)=x^{2}

f(x)=|x|

f(x)=|x-2|+1

Flow

Guided practice

Explore

Present the function. Ask the student to generate the table of values by substituting the assigned value of x. Then, identify the ordered pair and let the student plot it to see the graph. Learners discover that the graph forms a straight line.

f(x)=x

Investigation

Explore

Give the questions to the groups in advance. Present the function. Learners investigate rather than simply being told about symmetry.

f(x)=|x|

What happens when x changes from negative to positive?
Why do the two sides of the graph look symmetrical?
What feature of the equation might explain this?

Paired activity

Elaborate

Ask the learner to find a partner. Each student will compare their work and how to graph their function to each other and answer the following questions. To verify their answers, ask the learners to double check it with the graphing technology (Desmos/GeoGebra).

f(x)=3x

f(x)=\sqrt{x-3}+2

"What changed?"
"Where did the graph move?"
"How can you prove that?"

Learning resources

Tables-of-values worksheet
Coordinate/graphing paper
Guided graphing activity sheet
Calculator
Desmos/GeoGebra/graphing technology

Opportunities for integration

ICT: Desmos for visualizing and comparing graphs.
Science: Relate graph patterns to relationships between measurable quantities.
Critical Thinking: Classification, justification, and evidence-based reasoning.

Assessment

Formative assessment

Evaluate

Learners receive a new equation and must:

predict its graph;
sketch it;
explain their prediction;
verify with technology; and
explain any discrepancy.

Ways forward

Extended learning opportunities

Predict, Sketch, Check!

Extend

Develop the ability to sketch graphs without immediately relying on technology. For each function:

f(x)=2x+1

f(x)=-x+3

f(x)=|x-2|+1

f(x)=(x-2)^{2}-3

Look at the equation.
Predict what the graph will look like.
Identify at least 3 points.
Sketch the graph on graphing paper.
Use Desmos to check your answer.
Circle any differences between your prediction and the digital graph.
Explain what you learned from the comparison.

Reflections

Think about what you need to change for the next session based on what happened today.

Day 3

Intentions

Learning competency

Sketch the graph of the following functions: linear, absolute value, quadratic, square root, or cube root with or without technology.

gm11.t1.w10.c1

Learning objectives

At the end of the 45-minute period, the learners should be able to:

determine domain and range from representations;
explain why the graphs differ; and
compare the two functions using evidence.

Learner context

Learners have encountered radicals and coordinate graphs but may have limited conceptual understanding of why certain functions have restrictions in their domain or why different algebraic forms produce different graph behaviors. They investigate these functions through tables, graphs, and comparisons so that domain, range, endpoints, and graph behavior emerge from their own observations.

Learning experience

Pre-lesson

Hook

Engage

What does an equation tell us about where a graph can exist?

Flow

Activity: Function Detective

Explore

Learners investigate the following functions by constructing a table and graph.

f(x)=\sqrt{x}

f(x)=\sqrt[3]{x}

Why doesn't the graph continue to the left?
Does the same restriction occur?
What evidence do you have?
How can you distinguish a square-root function from a cube-root function?

Compare the two graphs

Explain

Group the learners to compare the two graphs and complete the table.

Question	Square Root	Cube Root
What does the graph look like?
Does it have an endpoint?
What is its domain?
What is its range?
Why?

Predict, then verify

Elaborate

Give learners an equation without its graph. Ask: "Predict where the graph begins." Learners sketch first, then verify digitally.

f(x)=\sqrt{x-3}+2

Peer explanation

Elaborate

Each learner explains to a partner: "How can you distinguish a square-root function from a cube-root function?" The partner asks for evidence.

Learning resources

Function investigation worksheet
Graphing paper
Domain and range organizer
Desmos/graphing technology
Calculator

Opportunities for integration

Science/Engineering: Connect square and cube roots to measurements involving area, volume, and physical quantities.
ICT: Interactive graphing through Desmos.
Communication: Peer explanation and mathematical reasoning.

Assessment

Formative assessment

Which Student Is Correct?

Evaluate

Present contradictory student explanations. Ask: Who is correct? Prove your answer.

Student A: "The square-root function can have any real number as its input."
Student B: "The square-root function has restrictions on its input."
Student C: "The cube-root function has a restricted domain like the square-root function."

Ways forward

Extended learning opportunities

What Happens to the Graph?

Extend

Strengthen understanding of square-root and cube-root functions, particularly domain, range, and graph behavior. Graph the functions using Desmos/GeoGebra.

f(x)=\sqrt{x+7}+4

f(x)=\sqrt[3]{x+5}-8

Before graphing, predict the possible values of x and y.
Record your observations.
Compare the square-root and cube-root graphs.
Explain why the two graphs behave differently.

Reflections

Think about what you need to change for the next session based on what happened today.

Day 4

Intentions

Learning competency

Sketch the graph of the following functions: linear, absolute value, quadratic, square root, or cube root with or without technology.

gm11.t1.w10.c1, gm11.t1.w10.c2

Learning objectives

At the end of the 45-minute period, the learners should be able to:

determine appropriate function types;
develop mathematical models;
evaluate whether a model reasonably represents a real situation.

Learner context

Learners encounter relationships involving rates, distance, cost, measurement, motion, and other quantities in everyday life and STEM subjects. However, they may be accustomed to solving equations provided by the teacher rather than deciding which mathematical model is appropriate. They are placed in authentic situations where they must identify variables, recognize patterns, select a function, construct a model, and justify their choice.

Learning experience

Pre-lesson

Hook

Engage

How do we decide which function best represents a real-life situation?

Elicit

Present: A delivery company charges a fixed fee plus a charge for every kilometer travelled. Ask: "How could we represent this situation mathematically?" Do not give the equation. Learners discuss:

What changes?
What stays constant?
What is the dependent variable?
What is the independent variable?

Flow

Group Activity

Explore

Groups receive different situations. Groups must determine: Situation → Variables → Pattern → Function → Equation → Graph.

Situation A: Taxi fare based on distance.
Situation B: Distance from a target value.
Situation C: Height of a ball over time.
Situation D: Side length of a square based on its area.
Situation E: Side length of a cube based on its volume.

Explain

Groups must answer: "Why is your chosen function appropriate?" They must provide at least three pieces of evidence. For example:

behavior of the variables;
shape of the graph;
equation structure;
behavior of the situation.

What-if challenge

Elaborate

Learners modify their models and predict the consequences. This promotes transfer and deeper conceptual understanding.

What if the fixed fee increases?
What if the rate changes?
What if the object is launched from a different height?
What if the dimensions change?

Learning resources

Function investigation worksheet
Graphing paper
Domain and range organizer
Desmos/graphing technology
Calculator

Opportunities for integration

STEM: Apply functions to motion, distance, measurement, finance, area, and volume.
ICT: Desmos or spreadsheets for mathematical modeling.
Financial Literacy: Use linear and quadratic functions in income, cost, or profit situations.
Critical Thinking: Evaluate the appropriateness and limitations of a mathematical model.

Assessment

Formative assessment

Mathematical Modeling Task

Evaluate

Learners submit:

situation;
variables;
selected function;
equation;
table;
graph;
solution;
interpretation; and
justification of the model.

Ways forward

Extended learning opportunities

Function in My World

Extend

Apply functions to authentic situations and develop mathematical modeling skills. Find one real-life situation at home or in your community involving a relationship between two quantities (e.g., transportation fare and distance; working hours and earnings; product quantity and total cost; distance and time; area and side length; volume and side length).

Identify the two variables.
Determine which function type best represents the situation.
Write an equation.
Create a table and draw the graph manually or create it using Desmos/GeoGebra.
Explain whether your model is reasonable.

Reflections

Think about what you need to change for the next session based on what happened today.

Day 5

Intentions

Learning competency

Apply the functions to model and solve real-life problems.

gm11.t1.w10.c2

Learning objectives

At the end of the 45-minute period, the learners should be able to:

defend mathematical decisions using evidence;
evaluate limitations of a mathematical model; and
reflect on how their understanding of functions developed.

Learner context

Learners have developed an initial understanding of the five function types and their graphical characteristics through the previous sessions. They are now ready to integrate these ideas in an unfamiliar or authentic context. Rather than simply obtaining a correct answer, learners are expected to create a model, defend their mathematical decisions, evaluate assumptions and limitations, respond to peer questions, and revise their thinking when necessary.

Learning experience

Pre-lesson

Hook

Engage

"How can functions help us explain, predict, and solve problems in the real world?"

Flow

Concept Reconstruction

Elicit

Ask the learners to reconstruct a Function Concept Map from memory: Five Functions → Equation → Graph → Characteristics → Real-Life Application.

Functions in our World

Elaborate

Learners choose an authentic situation involving mathematics (business; transportation; sports; science; engineering; construction; environmental issues; personal finance; school/community situations). They must answer:

What is the problem?
What quantities are involved?
What function could model the situation?
Why did you choose that function?
What equation represents it?
What does the graph tell us?
What solution does the model provide?
What assumptions did you make?
What are the limitations of your model?

Mathematical Defense

Evaluate

Groups present their models. The audience becomes a panel of mathematical critics using the rubric. Learners must defend, evaluate, and possibly revise their mathematical models. The panel may ask:

Why did you select this function?
Could another function work?
What evidence supports your model?
What happens if one variable changes?
Is your model realistic?
What are the limitations?
How would you improve the model?

Learning resources

Mathematical modeling/performance task sheet
Rubric and peer-assessment form
Graphing paper
Calculator
Desmos/GeoGebra/PowerPoint/Google Slides or other presentation tool

Opportunities for integration

STEM Integration: Develop a mathematical model based on an authentic problem.
ICT: Use Desmos, spreadsheets, or presentation tools.
Communication: Present and defend findings.
21st-Century Skills: Collaboration, problem-solving, creativity, critical thinking, and digital literacy.

Assessment

Formative assessment

Evaluate

Based on the feedback given by their classmates and the teacher, learners will also do a self-assessment using the rubric.

Ways forward

Extended learning opportunities

Remediation

For learners who have not yet achieved mastery: Concrete/Visual → Guided → Collaborative → Independent. Use:

graph cards;
equation cards;
partially completed tables;
guided questioning;
peer teaching;
technology-assisted exploration.

Enrichment

Extend

For advanced learners: "Can one real-life situation be modeled by more than one type of function?" Learners compare competing models and determine which is more appropriate. They must defend their choice using mathematical evidence.

Reflections

Think about what you need to change for the next session based on what happened today.

Rubric: Mathematical modeling and defense

Criteria	Weight
Correct identification/classification	15%
Accuracy of equation and graph	20%
Mathematical reasoning	20%
Quality of real-life model	20%
Evidence-based justification	15%
Communication and collaboration	10%
Total	100%

Teaching notes and common misconceptions

Learners often recognise graph shapes without being able to say why the equation produces that shape. Have them build classification criteria before naming the function types.
Many learners assume the cube-root function has a restricted domain like the square-root function. Use the Day 3 'Which Student Is Correct?' task to surface this.
Learners may over-rely on Desmos. Always predict and sketch by hand first, then verify digitally.

Worked examples

Example 1: Classify from the equation

Classify f(x) = |x − 2| + 1 and describe its graph.

f(x)=|x-2|+1

The absolute value of a linear expression gives a V shape.
The vertex is at (2, 1); the graph opens upward because the coefficient of |x − 2| is positive.

Example 2: Sketch from a table

Sketch f(x) = (x − 2)² − 3.

The values are symmetric about x = 2, the vertex is (2, −3), and the graph is a parabola opening upward.

x	0	1	2	3	4
f(x)	1	−2	−3	−2	1

Example 3: Domain of a square-root function

Find the domain and range of f(x) = √(x − 3) + 2.

x-3\ge 0 \Rightarrow x\ge 3

f(3)=2,\ \text{so } y\ge 2

Domain [3, ∞), range [2, ∞). The graph starts at (3, 2) and rises to the right.

Example 4: Choose a model

A tricycle ride costs ₱15 for the first kilometre and ₱5 for every kilometre after that. Model the fare for a ride of x kilometres (x ≥ 1).

F(x)=15+5(x-1)=5x+10

Constant change per kilometre means a linear function.
Check: a 4-km ride costs 5(4) + 10 = ₱30.