The Mystery of 1: Number Patterns and Problem-Solving
Materials Needed
- Paper or a math notebook
- Pencil and colored pencils
- 20 small objects, such as buttons, LEGO bricks, coins, or cereal pieces
- Index cards or small pieces of paper
- A calculator, optional
- A timer, optional
Learning Objectives
By the end of the lesson, the learner will be able to:
- Explain how multiplying by 1 and dividing by 1 affect a number.
- Identify and create number patterns involving 1.
- Use multiplication and division facts to solve real-world problems.
- Explain their mathematical thinking using words, numbers, pictures, or objects.
Success Criteria
I can:
- Correctly solve at least 8 out of 10 practice problems.
- Explain why a number multiplied by 1 stays the same.
- Create a number-pattern puzzle and explain its rule.
- Apply the idea of 1 to a practical situation.
Introduction: The One-Thing Challenge
Hook: Imagine that every object in your home suddenly became one object. One book, one shoe, one snack, or one LEGO brick. What would change? What would stay the same?
- Place 5 small objects in front of the learner.
- Ask: “What happens if we make 1 group of 5?”
- Repeat with 1 group of 8 and 1 group of 12.
- Invite the learner to describe what they notice.
Think-pair-share alternative: The learner thinks independently, explains the answer to a parent or partner, and then shares one observation.
Tell the learner: “Today we will investigate the special role of the number 1. We will discover patterns, solve puzzles, and design our own math challenge.”
Body
Part 1: I Do — Discovering the Identity Rule
Explain that when a number is multiplied by 1, the number keeps its identity. This is called the identity property of multiplication.
Model the following examples using objects, drawings, or equations:
- 1 × 4 = 4: One group of four has four objects.
- 1 × 9 = 9: One group of nine has nine objects.
- 1 × 25 = 25: One group of twenty-five has twenty-five objects.
Then show the related division facts:
- 4 ÷ 1 = 4: Four objects placed into one group leaves four objects in that group.
- 9 ÷ 1 = 9
- 25 ÷ 1 = 25
Say: “Multiplying by 1 or dividing by 1 does not change the number. The number remains itself.”
Quick Check
Ask the learner to answer mentally and explain one answer:
- 1 × 7 = ?
- 1 × 36 = ?
- 18 ÷ 1 = ?
- 52 ÷ 1 = ?
Part 2: We Do — The Number Detective Game
Work together to solve each mystery. The learner may use objects, drawings, or equations.
-
Snack Mystery: One bag contains 6 crackers. How many crackers are in 1 bag?
Equation: 1 × 6 = 6 -
Book Mystery: A shelf has 24 books. If the books are placed into 1 group, how many books are in the group?
Equation: 24 ÷ 1 = 24 -
Sticker Mystery: One sheet has 15 stickers. How many stickers are on 1 sheet?
Equation: 1 × 15 = 15
After each problem, ask:
- What information did you notice?
- Which operation did you use?
- Why did the answer stay the same?
Pattern Practice
Complete each pattern and describe the rule:
- 1, 4, 7, 10, ___, ___
- 1, 5, 25, 125, ___
- 36, 36, 36, 36, ___
- 1 × 3 = 3, 1 × 6 = 6, 1 × 9 = 9, 1 × 12 = ___
Discuss how the number 1 can appear in different kinds of patterns. Emphasize that the learner must look for the rule rather than simply guess the next number.
Part 3: You Do — Create a Number-Pattern Puzzle
The learner creates a puzzle for someone else to solve.
- Choose a pattern involving 1.
- Write at least five numbers or equations in the pattern.
- Remove two or three parts of the pattern.
- Write a clue explaining the rule.
- Have a parent, sibling, friend, or teacher solve it.
Example:
1 × 8 = 8, 1 × 11 = 11, 1 × 14 = ___
Clue: “The number 1 is multiplied by numbers that increase by 3.”
Part 4: Real-World Application — Plan a One-Group Event
Imagine you are planning a small event with exactly one group of guests. Choose one scenario:
- A one-table picnic
- A one-team game
- A one-shelf book display
- A one-box art kit
The learner decides:
- How many people or objects are in the group
- How many items each person receives, if applicable
- Which multiplication or division equations describe the situation
- How to show the solution with a drawing, diagram, or written explanation
Example: “My one art kit contains 8 crayons, 3 markers, and 1 notebook. I can write 1 × 8 = 8 to show that one group of 8 crayons has 8 crayons.”
Formative Assessment
- Listen for accurate explanations during the object demonstration.
- Ask the learner to explain why 1 × 42 equals 42 without using a calculator.
- Check the learner’s pattern rule, not just the missing answers.
- Ask the learner to show one answer in two ways, such as an equation and a drawing.
Independent Check
Solve the following problems. Show work for at least two.
- 1 × 13 = _____
- 1 × 48 = _____
- 27 ÷ 1 = _____
- 91 ÷ 1 = _____
- 1 × _____ = 64
- _____ ÷ 1 = 37
- Complete the pattern: 1, 2, 4, 8, _____, _____
- Write a real-world problem that can be solved with 1 × 16.
- Explain in your own words why 1 × 73 = 73.
- Create a number puzzle involving multiplication by 1.
Answer Guide
- 13
- 48
- 27
- 91
- 64
- 37
- 16, 32
- Answers will vary. Example: “One box has 16 pencils. How many pencils are in one box?”
- Answers should explain that one group of 73 contains 73.
- Answers will vary, but the pattern should be accurate and solvable.
Differentiation and Adaptations
Support for Learners Who Need More Help
- Use physical objects to model every equation.
- Begin with numbers from 1 to 10 before using larger numbers.
- Provide sentence frames: “When I multiply by 1, the answer stays the same because _____.”
- Allow the learner to draw groups instead of writing equations first.
Extensions for Learners Ready for More Challenge
- Compare multiplication by 1 with multiplication by 0 and explain the difference.
- Investigate why dividing by 1 is different from dividing 1 by a number.
- Create a puzzle with two operations, such as 1 × 24 + 6.
- Write a short explanation of the identity property using a real-world example.
Flexible Learning Options
- Hands-on: Use coins, blocks, snacks, or household objects.
- Digital: Create the puzzle in a document, presentation, or spreadsheet.
- Verbal: Explain the patterns aloud and record the explanation.
- Creative: Design a “Number 1 Detective” comic or poster.
Conclusion: Tell What You Learned
Ask the learner to complete these statements:
- “When I multiply a number by 1, _____.”
- “When I divide a number by 1, _____.”
- “One pattern I discovered was _____.”
- “I can use this idea in real life when _____.”
Review the main idea: Multiplying or dividing a number by 1 leaves that number unchanged. The learner should be able to recognize this rule, explain why it works, and use it to solve or create problems.
Optional Follow-Up Challenge
For one day, look for the number 1 in real life. Record five examples, such as one door, one serving, one dollar, one group, or one item per person. For each example, write a matching multiplication or division equation.