Educational Resource
Extensión del significado de las operaciones y sus relaciones inversas.
Grade 8 Mathematics Lesson Plan
Unit Title
Extending the Meaning of Operations and Their Inverse Relationships
Lesson Title
Divisibility, Prime Numbers, Greatest Common Factor, and Least Common Multiple
| Lesson information | Details |
|---|---|
| Grade | 8 |
| Subject | Mathematics |
| Lesson length | 60 minutes |
| Setting | Whole-class instruction with individual practice |
| Instructional approach | Explicit instruction, guided practice, and independent application |
| Essential question | How can divisibility rules and prime numbers help us find the greatest common factor or least common multiple and solve problems? |
| Learning target | I can use divisibility criteria and prime factorization to find the GCF or LCM, justify my choice, and solve a related problem. |
Standards Alignment
Because states adopt their own standards, use the following CCSS references as a flexible alignment guide and crosswalk them to the applicable state standards.
| Standard | Connection to the lesson |
|---|---|
| CCSS.Math.Practice.MP1: Make sense of problems and persevere in solving them. | Students identify what a word problem asks, select GCF or LCM, and check whether their answer is reasonable. |
| CCSS.Math.Practice.MP2: Reason abstractly and quantitatively. | Students connect quantities in a context to factors, multiples, and prime factorizations. |
| CCSS.Math.Practice.MP3: Construct viable arguments and critique the reasoning of others. | Students justify their method and explain why GCF or LCM fits the problem. |
| CCSS.Math.Practice.MP6: Attend to precision. | Students use accurate mathematical language, notation, and calculations. |
| CCSS.Math.Practice.MP7: Look for and make use of structure. | Students use prime factorization and divisibility patterns to identify common factors and multiples. |
| Related content connection: CCSS.Math.Content.6.NS.B.4 | Students apply the GCF and LCM concepts introduced in earlier grades in more complex reasoning and problem-solving contexts. This lesson reinforces and extends that content rather than presenting it as a new Grade 8 content standard. |
Measurable Objectives
By the end of the lesson, students will be able to:
1. Identify whether a number is divisible by 2, 3, 5, 9, or 10 by using the corresponding divisibility criteria, with at least 4 out of 5 items correct.
2. Write the prime factorization of a whole number using exponents when appropriate, with at least 4 out of 5 factorizations correct.
3. Calculate the GCF or LCM of two whole numbers using prime factorization, with at least 3 out of 4 problems correct.
4. Choose and justify whether a context calls for the GCF or LCM, using a written explanation that connects the operation to the problem.
Prerequisite Knowledge
Students should be familiar with:
- Multiplication facts and factor pairs
- Factors and multiples
- Whole-number multiplication and division
- Exponents as repeated multiplication
Materials
- Whiteboard or display
- Student notebooks or paper
- Pencils
- Divisibility and prime-factorization reference sheet
- Prime-factorization tree or factor-list graphic organizer
- Exit ticket
- Optional: counters, linking cubes, or number cards for visual modeling
Key Vocabulary
| Term | Student-friendly meaning |
|---|---|
| Divisible | A number can be divided by another number with no remainder. |
| Divisibility criterion | A shortcut for deciding whether one number divides evenly into another. |
| Prime number | A whole number greater than 1 with exactly two factors: 1 and itself. |
| Composite number | A whole number greater than 1 with more than two factors. |
| Prime factorization | Writing a number as a product of prime numbers. |
| Factor | A whole number that divides another whole number evenly. |
| Multiple | The product of a number and a whole number. |
| Greatest common factor (GCF) | The greatest factor shared by two or more numbers. |
| Least common multiple (LCM) | The smallest positive multiple shared by two or more numbers. |
Divisibility Criteria Reference
| Divisor | A number is divisible by it when… |
|---|---|
| 2 | Its last digit is 0, 2, 4, 6, or 8. |
| 3 | The sum of its digits is divisible by 3. |
| 5 | Its last digit is 0 or 5. |
| 9 | The sum of its digits is divisible by 9. |
| 10 | Its last digit is 0. |
Instructional Sequence — 60 Minutes
| Time | Lesson phase and teacher actions | Student actions and checks for understanding |
|---|---|---|
| 0–5 min | Launch and connect. Display: “Two school bells ring at different intervals. How could we figure out when they will ring together?” Introduce the essential question and learning target. Briefly review the meanings of factor and multiple. | Students write or share an initial idea. Teacher listens for whether students distinguish “shared groups” from “events happening together.” |
| 5–12 min | Explicit instruction: divisibility and primes. Model the criteria for 2, 3, 5, 9, and 10. Think aloud while testing 234: it is divisible by 2 because it ends in 4; by 3 because 2 + 3 + 4 = 9; and by 9 because its digit sum is 9. Explain that a prime number has exactly two positive factors. | Students use a response signal, such as thumbs up/down, to indicate whether sample numbers are divisible by a given divisor. Formative check: Ask students to explain which criterion they used, not just give an answer. |
| 12–22 min | Model prime factorization and GCF. Factor 36 and 60: 36 = 2² × 3²; 60 = 2² × 3 × 5. To find the GCF, select shared prime factors using the smaller exponent: 2² × 3 = 12. Explain that GCF is useful when dividing items into the largest possible equal groups with none left over. | Students copy the model and annotate why the smaller exponents are used. They answer: “Why is 12 a factor of both 36 and 60?” Formative check: Cold-call or invite volunteers to explain using factor language. |
| 22–32 min | Model LCM and compare choices. Use the same factorizations. To find the LCM, include every prime factor using the greater exponent: 2² × 3² × 5 = 180. Explain that LCM is useful for finding when repeating events occur together. Contrast a grouping situation with a repeating-event situation. | Students complete a two-column note: “GCF means…” and “LCM means…” Formative check: Present a short scenario and ask students to show GCF or LCM on a response card, then explain their choice. |
| 32–43 min | Guided practice. Work through two problems with the class. Problem A: Find the greatest number of identical gift bags that can be made from 24 pencils and 36 erasers, using all items. Problem B: One light flashes every 8 seconds and another every 12 seconds. If they flash together now, when will they next flash together? Prompt students to identify the context, select GCF or LCM, calculate, and label the answer. | Students solve each problem on paper or a whiteboard. For A, students identify GCF(24, 36) = 12 bags. For B, students identify LCM(8, 12) = 24 seconds. Formative check: Ask students to justify why the other operation would not answer the question. |
| 43–53 min | Independent application. Assign the performance task below. Circulate and use a checklist to note whether students choose the correct operation, show accurate prime factorization, and justify their reasoning. | Students complete the task independently. They may use the reference sheet or a factor-tree organizer. |
| 53–60 min | Closure and exit ticket. Revisit the essential question. Ask students to complete the exit ticket and rate their confidence from 1 to 4. Collect responses to plan the next lesson. | Students submit the exit ticket and complete the confidence rating. |
Performance Assessment
Student Task
A community center has 48 juice boxes and 60 granola bars.
1. What is the greatest number of identical snack packs the center can make if it uses every item and puts the same number of each item in every pack?
2. How many juice boxes and granola bars will be in each pack?
3. Show prime factorizations or another valid method.
4. Explain why this problem requires the GCF rather than the LCM.
Expected Reasoning
- 48 = 2⁴ × 3
- 60 = 2² × 3 × 5
- GCF(48, 60) = 2² × 3 = 12
- The center can make 12 identical packs.
- Each pack has 4 juice boxes and 5 granola bars.
- GCF is appropriate because the task asks for the greatest number of equal groups that use all items.
Scoring Guide
| Criterion | 2 points | 1 point | 0 points |
|---|---|---|---|
| Selects the operation | Correctly selects GCF and explains why | Selects GCF but gives an unclear explanation | Selects LCM or does not identify an operation |
| Shows mathematical method | Accurate prime factorization or valid equivalent method | Method has a minor error but shows partial understanding | No workable method shown |
| Calculates and labels answer | Correctly states 12 packs and items per pack | One answer or label is incorrect or missing | Answers are substantially incorrect |
| Justifies the result | Connects GCF to making the greatest number of equal groups using all items | Gives a partial or vague reason | Gives no relevant justification |
Proficiency target: At least 6 of 8 points, including a correct operation choice.
Exit Ticket
1. Is 378 divisible by 3 and by 9? Explain using the digit-sum criterion.
2. Find the GCF of 18 and 30 using prime factorization.
3. Two timers beep every 6 minutes and every 10 minutes. Which operation should you use to find when they next beep together: GCF or LCM? Explain briefly.
Answer guide:
1. Yes. The digit sum is 3 + 7 + 8 = 18, which is divisible by both 3 and 9.
2. 18 = 2 × 3²; 30 = 2 × 3 × 5; GCF = 2 × 3 = 6.
3. LCM, because the question asks when two repeating events happen together.
Differentiation and UDL Supports
| Learner need | Supports and instructional options |
|---|---|
| Students who need additional support | Provide a divisibility reference card, partially completed factor trees, a list of prime numbers, and a GCF/LCM decision chart. Reduce the number of practice items while preserving the same reasoning target. Check understanding after each modeled step. |
| Students developing academic English | Preteach vocabulary with examples, symbols, and visuals. Provide sentence frames: “I chose the GCF/LCM because the problem asks for ___.” Allow rehearsal with a partner before written explanation. |
| Students who benefit from concrete or visual models | Use counters or cubes to create equal groups; use factor trees and color-coded prime factors; show repeated events on a number line or schedule. |
| Students ready for extension | Ask them to compare the factorization method with listing factors or multiples, solve a three-number GCF/LCM problem, or create a context that requires GCF and defend their choice. |
| Whole-class access | State the learning target aloud and display it. Model worked examples, provide both oral and written directions, and allow students to show reasoning with words, equations, diagrams, or a combination. |
Accommodations
Follow each student’s IEP, 504 plan, and documented language supports. As appropriate:
- Provide extended time or a reduced-distraction setting.
- Offer large-print, high-contrast, or enlarged visual materials.
- Allow use of a multiplication chart, divisibility reference sheet, or calculator for checking calculations when calculation is not the assessed skill.
- Break multi-step directions into numbered steps and check for understanding between steps.
- Accept oral explanations, dictated responses, or assistive-technology responses when appropriate.
- Provide seating and response options that support access to instruction and participation.
Formative Assessment Plan
| Evidence | What the teacher looks for | Instructional response |
|---|---|---|
| Divisibility response signals | Correct use of a criterion and an explanation | Reteach the relevant criterion with examples if students rely on guessing. |
| GCF/LCM decision prompt | Students distinguish equal grouping from events coinciding | Revisit the context clues and model a comparison if choices are inconsistent. |
| Guided-practice work | Accurate factorization, selection of shared primes, and answer labels | Pull a brief small group or provide a worked example for students who need another model. |
| Independent task checklist | Correct operation, method, calculation, and justification | Use results to plan follow-up practice or extension. |
| Exit ticket | Individual understanding of divisibility, GCF, and LCM | Sort responses into ready for extension, needs targeted practice, and needs reteaching. |
Family and Community Connection
Invite students to notice situations involving equal groups or repeating schedules at home or in the community. Examples include arranging items
