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 Factorization, Greatest Common Factor, and Least Common Multiple
Grade: 8
Instructional setting: Whole-class instruction
Primary approach: Explicit instruction with guided and independent practice
Suggested duration: Two 50-minute class periods
Essential question:
How can divisibility rules and prime numbers help us solve problems involving the greatest common factor (GCF) and least common multiple (LCM), and how can we justify which operation to use?
Learning Objectives
By the end of the lesson, students will be able to:
1. Apply divisibility criteria for 2, 3, 5, 9, and 10 to determine whether a whole number is divisible by each divisor, with at least 80% accuracy.
2. Identify prime and composite numbers and express composite numbers as products of prime factors.
3. Calculate the GCF and LCM of two or more whole numbers using prime factorization, showing a valid strategy and accurate work.
4. Choose whether a context calls for the GCF or LCM, explain why, and justify the answer using mathematical reasoning.
5. Check whether a solution is reasonable by relating the GCF and LCM to the original numbers and the context.
Standards Alignment
CCSS-Math is a widely used reference framework; states may adopt different versions or standards. Teachers should verify the exact wording and grade placement in their state’s standards.
| Standard | Connection to the lesson |
|---|---|
| CCSS.MATH.CONTENT.6.NS.B.4 | Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers with a common factor. This is a foundational standard that may be reviewed or extended in Grade 8. |
| CCSS.MATH.PRACTICE.MP1: Make sense of problems and persevere in solving them | Students interpret context, decide whether GCF or LCM is appropriate, and check whether their answer makes sense. |
| CCSS.MATH.PRACTICE.MP2: Reason abstractly and quantitatively | Students connect numerical calculations with quantities and relationships in real-world situations. |
| CCSS.MATH.PRACTICE.MP3: Construct viable arguments and critique the reasoning of others | Students explain and defend their choice of GCF or LCM and evaluate a partner’s reasoning. |
| CCSS.MATH.PRACTICE.MP6: Attend to precision | Students use terms such as factor, multiple, prime, composite, GCF, and LCM accurately and show clear calculations. |
| State or local standards | Adapt the alignment to the state’s grade-level expectations for number theory, factors and multiples, mathematical practices, and problem solving. |
Alignment note: The selected Mathematical Practices are the primary emphasis. The listed Grade 6 content standard is an appropriate prerequisite or review connection; teachers should confirm how their state places this content across grades.
Materials
- Whiteboard or display
- Student notebooks or paper
- Pencils and highlighters
- Prime factorization chart or factor-tree template
- Divisibility rules reference card
- Practice problems and exit ticket
- Optional: number cards, counters, or digital visualizer
- Optional: calculators for checking work after students show their reasoning
Key Vocabulary
| Term | Student-friendly meaning |
|---|---|
| Divisible | A number can be divided by another number with no remainder. |
| Factor | A whole number that divides another whole number evenly. |
| Multiple | A product of a number and a whole number. |
| 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. |
| 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 criterion | A rule that helps determine whether one number divides another evenly. |
| Remainder | The amount left after division when the division is not even. |
Divisibility Criteria Reference
| Divisor | A whole number is divisible by… |
|---|---|
| 2 | …2 if its last digit is 0, 2, 4, 6, or 8. |
| 3 | …3 if the sum of its digits is divisible by 3. |
| 5 | …5 if its last digit is 0 or 5. |
| 9 | …9 if the sum of its digits is divisible by 9. |
| 10 | …10 if its last digit is 0. |
Lesson Sequence
Day 1: Divisibility, Prime Numbers, and Prime Factorization
| Time | Instructional sequence | Teacher actions and prompts | Student actions and formative checks |
|---|---|---|---|
| 0–5 min | Launch: Notice and wonder | Display 36 and 45. Ask: “What do you notice about how these numbers can be divided?” “What might they have in common?” | Students write one observation and one question. Listen for factors, divisibility, and shared factors. |
| 5–12 min | State the target and connect prior knowledge | Read the essential question and objectives. Review factor, multiple, prime, and composite. Ask students to classify 2, 7, 9, and 15. | Students use a quick response signal or mini-whiteboard. Correct misconceptions immediately, especially that 1 is not prime. |
| 12–22 min | Explicit instruction: Divisibility criteria | Model the rules for 2, 3, 5, 9, and 10 using 270, 315, and 428. Think aloud: “The digit sum of 315 is 9, so 315 is divisible by 3 and 9.” | Students record the rules and answer brief checks: “Is 1,245 divisible by 3? By 5? How do you know?” |
| 22–32 min | Explicit instruction: Prime factorization | Model a factor tree for 60: \(60=6\times10=2\times3\times2\times5=2^2\times3\times5\). Emphasize that the prime factors must multiply back to the original number. | Students complete a guided factor tree for 84. Teacher checks that students continue until every factor is prime. |
| 32–42 min | Guided practice: Factor and verify | Assign 72 and 90. Model the first step for one number, then prompt students to complete the factorization. Ask: “How can you verify your factorization?” | Students factor both numbers and check by multiplication. Use cold-call or partner explanation to sample reasoning. |
| 42–48 min | Quick application | Present: “A student says 1 is prime because it has one factor. Is the student correct? Explain.” | Students write a brief claim and justification. Look for the definition of prime as having exactly two factors. |
| 48–50 min | Closure | Ask students to complete: “One divisibility rule I can explain is…” | Collect responses to plan the next lesson’s review. |
Day 2: GCF, LCM, and Problem Solving
| Time | Instructional sequence | Teacher actions and prompts | Student actions and formative checks |
|---|---|---|---|
| 0–5 min | Retrieval warm-up | Display 48 and 75. Ask students to state one applicable divisibility rule and identify whether each number is prime or composite. | Students respond on paper or mini-whiteboards. Review errors before moving on. |
| 5–15 min | Explicit instruction: GCF using prime factorization | Use 36 and 60: \(36=2^2\times3^2\); \(60=2^2\times3\times5\). Explain that the GCF uses shared prime factors with the smallest exponents: \(2^2\times3=12\). | Students annotate which factors are shared and why the smaller exponents are used. |
| 15–25 min | Explicit instruction: LCM using prime factorization | Use the same numbers. Explain that the LCM includes every prime factor needed, using the greatest exponent: \(2^2\times3^2\times5=180\). Check that 180 is divisible by both 36 and 60. | Students compare the GCF and LCM methods and explain why the exponent choices differ. |
| 25–33 min | Model choosing GCF or LCM | Model two situations: dividing 24 markers and 36 pencils into identical groups with none left over (GCF); two events repeating every 6 and 8 days (LCM). Think aloud about the meaning of “largest equal groups” versus “first time together again.” | Students identify the operation and justify the choice before calculating. |
| 33–43 min | Guided practice: Context problems | Give students two problems (see below). Prompt them to underline context clues, select GCF or LCM, calculate, and label the answer. | Students solve independently, then compare reasoning with a partner. Teacher uses a checklist for operation choice, calculation, and explanation. |
| 43–48 min | Independent performance assessment | Assign the summative task below. Remind students to show their factorization and explain their choice. | Students complete the task independently. |
| 48–50 min | Closure and reflection | Ask: “What clue helps you decide between GCF and LCM?” Collect an exit response. | Students submit a one- or two-sentence explanation. |
Guided Practice Problems
1. GCF context: A teacher has 48 pencils and 60 erasers. She wants to make the greatest possible number of identical supply kits, using all the items. How many kits can she make? What will each kit contain?
Expected reasoning: Find \(\text{GCF}(48,60)=12\). She can make 12 kits. Each kit has 4 pencils and 5 erasers.
2. LCM context: One school announcement repeats every 6 minutes, and another repeats every 8 minutes. If both repeat now, how many minutes will pass before they repeat together again?
Expected reasoning: Find \(\text{LCM}(6,8)=24\). They repeat together again in 24 minutes.
Formative Checks
Use these checks throughout the lesson:
- Mini-whiteboard responses: Students apply a divisibility criterion and show how they know.
- Factorization scan: Check whether students break composite factors down until all factors are prime.
- Verbal reasoning prompts: “Why does this situation call for the GCF?” “Why do we use the greatest exponents for the LCM?”
- Partner explanation: Students explain a solution and listen for accurate vocabulary and reasoning.
- Exit response: “Describe one clue that tells you to use the GCF and one clue that tells you to use the LCM.”
- Teacher checklist: Record whether each student can:
- Apply a divisibility rule.
- Write an accurate prime factorization.
- Calculate GCF and LCM.
- Select the appropriate operation in context.
- Explain and check the answer.
Summative Performance Assessment
Task
A community center has 72 juice boxes and 90 granola bars for an event.
1. The staff wants to make the greatest possible number of identical snack bags, using all the items. How many bags can they make? How many juice boxes and granola bars will be in each bag?
2. At a later event, a reminder announcement plays every 12 minutes and a music cue plays every 18 minutes. If both play at the same time now, when will they next play at the same time?
3. Show prime factorizations, identify whether each question requires the GCF or LCM, and explain why.
Answer Key
- \(72=2^3\times3^2\)
- \(90=2\times3^2\times5\)
- \(\text{GCF}(72,90)=2\times3^2=18\)
The staff can make 18 snack bags. Each bag contains 4 juice boxes and 5 granola bars.
- \(12=2^2\times3\)
- \(18=2\times3^2\)
- \(\text{LCM}(12,18)=2^2\times3^2=36\)
The announcement and music cue will next play together in 36 minutes.
Scoring Rubric
| Criteria | 4 – Strong | 3 – Meets | 2 – Developing | 1 – Beginning |
|---|---|---|---|---|
| Operation choice | Correctly selects GCF and LCM and clearly explains each choice. | Selects both correctly with a reasonable explanation. | Selects one correctly or gives limited explanation. | Does not identify an appropriate operation. |
| Prime factorization and calculation | Factorizations and answers are accurate and clearly shown. | Minor notation error, but method and answers are substantially correct. | Several errors or incomplete work. | Work does not show a usable method. |
| Contextual interpretation | Labels answers and correctly explains what each means. | Gives answers with mostly clear labels. | Labels or interpretations are incomplete. | Answers are not connected to the context. |
| Reasoning and verification
