Lesson 46 covers:

• Elementary Level: Near Doubles (2+3, 3+4, etc.)
• Mid Level: Mixed Numbers and Improper Fractions
• High Level: Solving Two-Step Equations

Elementary Level (Kinder to Grade 2)

Subject: Near Doubles (2+3, 3+4, etc.)

1. Alignment with Standards:

• CCSS.MATH.CONTENT.1.OA.C.6
  • Add and subtract within 20, demonstrating fluency for addition within 10. Use strategies such as counting on; making ten; decomposing a number; and using the relationship between addition and subtraction.
• CCSS.MATH.CONTENT.1.OA.B.3
  • Apply properties of operations as strategies to add and subtract (e.g., commutative and associative properties).

2. Lesson Objectives:

By the end of the lesson, students will be able to:

• Identify near doubles (numbers that are one apart, like 3 and 4).
• Use doubles facts to solve near doubles problems (e.g., 3+4 = (3+3) +1 = 7).
• Demonstrate understanding through manipulatives, drawings, and verbal explanations.

3. Materials Needed:

• Counters (e.g., buttons, beads, linking cubes)
• Ten-frame mat (printed or drawn)
• Near Doubles flashcards (e.g., 2+3, 5+6, etc.)
• Whiteboard & markers
• Printable worksheet (with near doubles problems)

Lesson Procedure

1. Warm-Up (5-10 min) – Review Doubles Facts

• Activity: Sing a “Doubles Song” (e.g., *”1+1 is 2, 2+2 is 4…”*)
• Quick Practice:
  • Show flashcards with doubles (2+2, 3+3, etc.) and have the student answer orally.
  • Use counters to model doubles (e.g., place 3 counters + 3 counters and count).

2. Introduction to Near Doubles (10 min)

• Explain: *”Near doubles are numbers that are just 1 apart, like 4 and 5. If we know 4+4=8, then 4+5 is just one more!”*
• Demonstrate with Counters or Points (Spielgaben Set 10):
  • Place 3 counters and 4 counters side by side.
  • Ask: *”How is this like 3+3? What’s different?”*
  • Guide the student to see that 3+4 = (3+3) +1 = 7.

3. Guided Practice (15 min)

• Ten-Frame Activity:
  • Use a ten-frame to show near doubles (e.g., fill 3 spots + 4 spots, compare to 3+3).
• Flashcard Drills:
  • Hold up near doubles (2+3, 5+6) and have the student solve using the doubles strategy.
• Whiteboard Work:
  • Write problems like 4+5 and have the student break it into (4+4)+1.

4. Independent Practice (10 min)

• Worksheet: Simple near doubles problems with visuals (e.g., 2 dogs + 3 dogs = ?).
• Game: “Near Doubles Race” – Roll two dice (one regular, one modified to be +1) and solve quickly.

5. Wrap-Up & Assessment (5 min)

• Exit Question: *”If 6+6=12, what is 6+7?”*
• Verbal Check: Ask the student to explain how they would solve 5+6 using near doubles.

Extension Activities (Optional)

• Real-World Connection: Ask, “If you have 4 cookies and I have 5, how many do we have together?”
• Art Integration: Draw near doubles with stickers or stamps (e.g., 2 stars + 3 stars).

Assessment & Feedback

• Observation: Does the student use the doubles strategy correctly?
• Worksheet Accuracy: Check for correct application of near doubles.
• Verbal Explanation: Can they describe the strategy in their own words?

Mid Level (Grade 3 to 5)

Subject: Mixed Numbers and Improper Fractions

1. Alignment with Standards:

• CCSS.MATH.CONTENT.4.NF.B.3.B
  • Decompose a fraction into a sum of fractions with the same denominator and record as a mixed number (e.g., 5/4 = 1 + 1/4 = 1¼).
• CCSS.MATH.CONTENT.4.NF.B.3.C
  • Add and subtract mixed numbers with like denominators.
• CCSS.MATH.CONTENT.4.NF.A.1
  • Explain equivalence between fractions (including mixed numbers and improper fractions).

2. Lesson Objectives:

By the end of the lesson, students will be able to:

• Define mixed numbers (e.g., 2½) and improper fractions (e.g., 5/2).
• Convert mixed numbers to improper fractions and vice versa.
• Justify conversions using visual models (fraction circles, number lines).

3. Materials Needed:

• Fraction circles or strips (physical or printable)
• Graph paper or whiteboard for drawing models
• Index cards with mixed numbers/improper fractions
• Dice (for a conversion game)

Lesson Procedure

1. Warm-Up (5-10 min) – Review Fractions

• Quick Quiz:
  • *”What is 3/3 equal to? What about 5/1?”* (Reinforce whole numbers as fractions.)
• Visual Drill:
  • Show fraction circles for ¼, ½, ¾ and ask, “How many quarters make 1 whole?”

2. Introduction (10 min) – Mixed vs. Improper

• Define Terms:
  • Mixed Number: A whole number + fraction (e.g., 2⅓).
  • Improper Fraction: A fraction where numerator ≥ denominator (e.g., 7/3).
• Real-World Example:
  • *”If you eat 1 whole pizza and half of another, you ate 1½ pizzas (mixed). That’s the same as 3/2 pizzas (improper).”*
• Image Prompt: Display a pizza cut into halves with 3/2 shaded.

3. Guided Practice (20 min)

A. Converting Mixed → Improper (Step-by-Step)

1. Multiply the whole number by the denominator.
2. Add the numerator.
3. Keep the denominator the same.
   • Example: 2⅔ = (2×3) + 2 = 8 → 8/3

B. Converting Improper → Mixed (Step-by-Step)

1. Divide numerator by denominator (whole number = quotient).
2. Remainder = new numerator.
3. Denominator stays the same.
   • Example: 7/4 = 1 R3 → 1¾

C. Hands-On Activities:

• Fraction Circles: Have students build 2½, then count halves to make 5/2.
• Number Line: Plot 5/4 and show it equals 1¼.

4. Independent Practice (15 min)

• Worksheet: Conversions with visual aids (e.g., shade 11/4 as 2¾).
• Dice Game: Roll two dice to make an improper fraction (e.g., 5/3), then convert to mixed.

5. Wrap-Up & Assessment (10 min)

• Exit Ticket:
  • *”Convert 9/4 to a mixed number. Draw a model to prove it.”*
• Real-World Problem:
  • *”A recipe calls for 5/2 cups of flour. Is this the same as 2½ cups? Explain.”*

Extension Activities (Optional)

• Math in Baking: Adjust a recipe using mixed/improper fractions (e.g., “Triple 1⅓ cups of sugar”).
• Art Project: Create a “Fraction City” where buildings are built from mixed-number blocks.

Assessment & Feedback

• Accuracy: Check worksheet conversions.
• Verbal Explanation: Ask, *”How is 10/3 related to 3⅓?”*
• Error Analysis: If a student writes 5/2 = 1½, revisit division steps.

High Level (Grade 6 to 8)

Subject: Solving Two-Step Equations

1. Alignment with Standards:

• CCSS.MATH.CONTENT.7.EE.B.3
  • Solve multi-step real-life and mathematical problems with rational numbers.
• CCSS.MATH.CONTENT.7.EE.B.4.A
  • Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where *p*, *q*, and *r* are rational numbers.

2. Lesson Objectives:

By the end of the lesson, students will be able to:

• Solve two-step equations using inverse operations (e.g., *3x + 5 = 14*).
• Translate word problems into algebraic equations and solve them.
• Justify each step in the solving process and check solutions for accuracy.

3. Materials Needed:

• Whiteboard & markers
• Algebra tiles (or digital manipulatives like Desmos)
• Printed worksheets with word problems

Lesson Procedure

1. Warm-Up (5-10 min) – One-Step Equation Review

• Quick Practice: Solve 5 equations like *x + 7 = 12* or *2m = 16*.
• Think-Pair-Share:
  • *”How would you solve 2x + 1 = 5? What steps are different?”*

2. Direct Instruction (15 min) – Solving Two-Step Equations

A. Introduce the Process:

1. Identify the operations applied to the variable (e.g., in *3x − 4 = 11*, “×3” and “−4”).
2. Undo addition/subtraction first, then multiplication/division (inverse operations).
   • Example:
     • *3x − 4 = 11* → Add 4: *3x = 15* → Divide by 3: *x = 5*

B. Real-World Connection:

• Show a shopping scenario: “You buy 3 books for 5eachandpay5eachandpay2 tax. Total cost is $17. Write and solve the equation.”
  • Equation: *3x + 2 = 17* → Solution: *x = 5* (price per book).

3. Guided Practice (20 min)

A. Algebra Tile Activity:

• Model *2x + 3 = 7* with tiles to show removing constants first, then dividing.

B. Word Problem Breakdown:

• *”A gym charges 10/monthplusa10/monthplusa25 sign-up fee. You paid $65 total. How many months did you pay for?”*
  • Equation: *10m + 25 = 65* → Solve together.

4. Independent Practice (20 min)

A. Worksheet:

• 6 equations (e.g., *−4y + 6 = 18*) + 2 word problems.

**B. Real-Life Scenarios Task:

• Activity: Create your own word problem (e.g., cell phone plans, baking recipes) and solve.

5. Wrap-Up & Assessment (10 min)

• Exit Ticket: Solve *5(x − 2) = 30* and explain each step.
• Error Analysis: Present a solved equation with a mistake (e.g., *2x + 8 = 20 → x = 6*) and ask students to find the error.

Extension Activities (Optional)

• Math in Careers: Research how engineers/algebraists use equations (e.g., construction budgets).
• Digital Tool: Use PhET Interactive Simulations for equation balancing.

Assessment & Feedback

Real-World Application: Evaluate student-created word problems for relevance.

Accuracy: Grade worksheets for correct steps/solutions.

Verbal Justification: Ask, “Why do we undo addition before division?”