Elementary Level: Patterns and Sequences

Mid Level: Mental Math Strategies

High Level: Review of Ratios, Integers, and Rational Numbers

Elementary Level (Kinder to Grade 2)

Subject: Patterns and Sequences

Alignment with Standards:

• Common Core State Standards (CCSS):
  • CCSS.MATH.CONTENT.1.OA.C.5: Relate counting to addition and subtraction (e.g., by counting on 2 to add 2) (links to number sequences).
  • CCSS.MATH.CONTENT.1.OA.D.8: Determine the unknown whole number in an addition or subtraction equation relating three whole numbers (extends to filling in patterns).
  • CCSS.MATH.CONTENT.K.OA.A.3: Decompose numbers less than or equal to 10 into pairs in more than one way (builds foundation for pattern recognition).

Objectives

1. Understand that patterns are repeating or growing sequences of shapes, colors, or numbers.
2. Identify and create patterns using objects and extend them logically.
3. Explore patterns with blocks and number sequences through hands-on activities.
4. Build observation, prediction, and early algebraic thinking skills.

Materials

• Pattern blocks (e.g., triangles, squares, hexagons in different colors) or household items (e.g., buttons, cereal)
• Paper or notebook for drawing/writing
• Crayons, markers, or pencils
• Number sequence cards (e.g., 2, 4, 6, __ or 5, 10, 15, __)
• Optional: Counting bears, a number line (0-20), or a pattern book (e.g., Pattern Fish by Trudy Harris)

Activities

Day 1: Introduction to Patterns with Blocks (30-45 minutes)

1. Warm-Up (5-10 minutes):
   • Ask: “Look at my shirt—stripes go red, blue, red, blue. That’s a pattern! What’s a pattern you see?”
   • Say: “Patterns repeat or grow—let’s make some!”
2. Direct Instruction (10 minutes):
   • Show blocks: “Red triangle, blue square, red triangle, blue square—what’s next? (red triangle!)”
   • Explain: “Patterns follow a rule—like ‘red, blue’ over and over.”
   • Try another: “Yellow circle, yellow circle, green hexagon—rule? (two yellows, one green)”
3. Practice (15-20 minutes):
   • Build: Start “red, blue, red, blue” with blocks—they add 3 more.
   • Create: “Make your own pattern!” (e.g., “circle, square, circle, square”). Say it aloud.
   • Read Pattern Fish (if available): Spot patterns (e.g., “stripe, dot, stripe”).
4. Wrap-Up (5 minutes):
   • Ask: “What’s your pattern rule? Show me what’s next!”

Day 2: Number Sequences (45 minutes)

1. Review (10 minutes):
   • Recap: “What’s a pattern? Show me ‘big, small, big, small’ with your hands!”
   • Quick block pattern: “Red, red, blue—what’s next? (red!)”
2. Hands-On Activity: Number Patterns (25 minutes):
   • Show: “2, 4, 6—what’s next? (8) Rule? (add 2 each time)”
   • Count aloud: “5, 10, 15—next? (20)” Use fingers or blocks to see “add 5.”
   • Cards: Lay out “3, 6, 9, __” and “1, 3, 5, __”—they finish (12, 7).
   • Make one: “Start with 4, add 3 each time—go!” (4, 7, 10, 13)
3. Wrap-Up (10 minutes):
   • Ask: “What’s the rule for 10, 20, 30? (add 10) How do numbers pattern?”

Day 3: Extending and Assessing Patterns (45 minutes)

1. Warm-Up (10 minutes):
   • Say: “Clap a pattern: loud, soft, loud, soft—what’s next?”
   • Quickie: “2, 5, 8—next? (11)”
2. Hands-On Activity: Mixed Patterns (20 minutes):
   • Blocks: “Red, yellow, red, yellow, red—extend it!” (yellow, red, yellow).
   • Numbers: “1, 2, 3, 4—next 3!” (5, 6, 7) or “10, 8, 6—next!” (4).
   • Draw: “Make a shape pattern—circle, square, triangle, circle—keep going!”
3. Assessment: Fill in Missing Elements (15 minutes):
   • Give patterns:
     • Blocks: “Blue, green, blue, __” (green).
     • Numbers: “5, 10, __, 20” (15).
     • Mixed: “Star, heart, star, __, star” (heart).
   • Check: “What’s the rule? How do you know?” (e.g., “Add 5!”)
   • Draw one missing: “2, 4, __, 8” (6).
4. Wrap-Up (5 minutes):
   • Celebrate: “You’re a pattern pro! What’s a pattern you’d make tomorrow?”

Assessment

• Informal Observation: Note their ability to spot and extend patterns in activities.
• Block Patterns: Check if they add 2-3 correct steps (e.g., red, blue, red, blue → red, blue).
• Assessment Task: Ensure they fill in 4/5 missing elements correctly with a rule explanation.

Extensions

• Pattern hunt: Find patterns outside (e.g., “leaf, no leaf” on a plant).
• Sound patterns: Clap or tap “loud, loud, soft”—extend it!
• Challenge: “3, 6, 9, 12—backwards!” (9, 6, 3).

Mid Level (Grade 3 to 5)

Subject: Mental Math Strategies

Alignment with Standards:

• Common Core State Standards (CCSS):
  • CCSS.MATH.CONTENT.4.NBT.B.4: Fluently add and subtract multi-digit whole numbers using the standard algorithm (mental strategies enhance fluency).
  • CCSS.MATH.CONTENT.4.OA.A.1: Interpret a multiplication equation as a comparison (supports mental multiplication).
  • CCSS.MATH.CONTENT.4.NBT.B.5: Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations.
  • CCSS.MATH.CONTENT.4.NBT.B.6: Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors (mental division builds on this).

Objectives

1. Understand and apply mental math strategies for addition, subtraction, multiplication, and division.
2. Perform calculations quickly and accurately without paper or pencil.
3. Practice mental math skills through timed challenges to build fluency and confidence.
4. Develop number sense and problem-solving abilities through strategic thinking.

Materials

• Flashcards with math problems (e.g., 45 + 27, 63 – 19, 7 × 8, 36 ÷ 4)
• Timer or stopwatch (manual or phone app)
• Whiteboard or paper (for explanations, not solving during challenges)
• Small counters or blocks (optional for visualizing)
• Mental math worksheet (sample provided below)
• Optional: Dice, a number chart (0-100)

Activities

Day 1: Mental Math Strategies for Addition and Subtraction (45 minutes)

1. Warm-Up (10 minutes):
   • Ask: “What’s 10 + 7? (17) Now 17 – 5? (12) How did you do it in your head?”
   • Say: “We’re learning tricks to solve math fast—without writing!”
2. Direct Instruction (15 minutes):
   • Teach strategies:
     • Addition:
       • Break apart: 45 + 27 = 40 + 20 = 60, 5 + 7 = 12, 60 + 12 = 72.
       • Round and adjust: 48 + 19 = 48 + 20 = 68, 68 – 1 = 67.
     • Subtraction:
       • Count up: 63 – 19 = 19 + 1 = 20, 20 + 40 = 60, 60 + 3 = 63, so 44.
       • Adjust: 50 – 28 = 50 – 30 = 20, 20 + 2 = 22.
   • Model: “35 + 18 = 35 + 20 = 55, 55 – 2 = 53.”
3. Practice (15-20 minutes):
   • Try: 26 + 39 (65), 72 – 47 (25). “Tell me your trick!”
   • Flashcards: 5 quick ones (e.g., 54 + 29, 81 – 36)—verbal answers.
4. Wrap-Up (5 minutes):
   • Ask: “Which strategy was easiest? Where could you use this?”

Day 2: Mental Math for Multiplication and Division (45 minutes)

1. Review (10 minutes):
   • Recap: “What’s 43 + 19? (62) 65 – 27? (38) How’d you do it?”
2. Hands-On Activity: Multiplication and Division Strategies (25 minutes):
   • Teach strategies:
     • Multiplication:
       • Doubles/Halves: 6 × 8 = 3 × 16 = 48 (double 6, half 8).
       • Break apart: 14 × 3 = 10 × 3 = 30, 4 × 3 = 12, 30 + 12 = 42.
     • Division:
       • Related facts: 36 ÷ 4 = think 4 × ? = 36 (9).
       • Chunking: 48 ÷ 6 = 6 × 8 = 48, so 8.
   • Model: “7 × 9 = 7 × 10 = 70, 70 – 7 = 63.” “24 ÷ 3 = 3 × 8 = 24, so 8.”
   • Practice: 5 × 12 (60), 45 ÷ 5 (9). “What’s your method?”
   • Flashcards: 5 quick ones (e.g., 8 × 7, 32 ÷ 4).
3. Wrap-Up (10 minutes):
   • Discuss: “Which was faster—multiplying or dividing? Why?”

Day 3: Timed Math Challenges (45 minutes)

1. Warm-Up (10 minutes):
   • Quickie: “9 + 8? (17) 15 × 2? (30) 20 ÷ 4? (5)”
   • Say: “Let’s race our brains with timed challenges!”
2. Hands-On Activity: Challenges (25 minutes):
   • Set timer (1-2 min per round):
     • Round 1 (Addition): 5 problems (e.g., 34 + 28, 67 + 19). Goal: 4/5 right.
     • Round 2 (Subtraction): 5 problems (e.g., 83 – 47, 55 – 29).
     • Round 3 (Multiplication): 5 problems (e.g., 6 × 9, 13 × 4).
     • Round 4 (Division): 5 problems (e.g., 40 ÷ 5, 72 ÷ 9).
   • Worksheet sample:
     • 25 + 37 = __ (62)
     • 91 – 46 = __ (45)
     • 7 × 8 = __ (56)
     • 63 ÷ 7 = __ (9)
   • Say answers aloud, track score: “How many in 1 minute?”
   • Retry: Beat your time or score!
3. Wrap-Up (10 minutes):
   • Reflect: “What strategy helped most? What’s your fastest operation?”

Assessment

• Informal Observation: Note their strategy use and speed during practice.
• Flashcards: Check accuracy (e.g., 80% correct verbally).
• Timed Challenges: Ensure 4/5 correct per round with a clear method explained.

Extensions

• Math duel: Parent vs. kid—5 problems, who’s faster?
• Real life: “Estimate grocery cost—$12 + $8, then split by 2 people.”
• Mix it: “15 + 6 × 2—what’s first?” (27).

High Level (Grade 6 to 8)

Subject: Review of Ratios, Integers, and Rational Numbers

Alignment with Standards:

• Common Core State Standards (CCSS):
  • CCSS.MATH.CONTENT.7.RP.A.1: Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units.
  • CCSS.MATH.CONTENT.7.NS.A.1: Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
  • CCSS.MATH.CONTENT.7.NS.A.2: Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
  • CCSS.MATH.CONTENT.7.NS.A.3: Solve real-world and mathematical problems involving the four operations with rational numbers.

Objectives

1. Review and reinforce understanding of ratios, integers, and rational numbers.
2. Apply operations (addition, subtraction, multiplication, division) to integers and rational numbers accurately.
3. Consolidate skills through engaging games and quizzes that test ratios and number concepts.
4. Build confidence and fluency in working with proportional relationships and rational number operations.

Materials

• Index cards or paper slips for game problems (e.g., “2:3 ratio, 6 total—what’s each part?”)
• Dice (2-3) or a spinner for random numbers
• Number line (drawn or printed, -10 to 10)
• Quiz worksheet (sample provided below)
• Notebook or paper for tracking scores/answers
• Optional: Counters, a timer, digital quiz tool (e.g., Kahoot if available)

Activities

Day 1: Review and Ratio Games (45 minutes)

1. Warm-Up (10 minutes):
   • Ask: “What’s 3:5 if there’s 8 total? (3 and 5) What’s -4 + 6? (2)”
   • Say: “We’re brushing up on ratios, integers, and rational numbers with fun games!”
2. Direct Instruction (15 minutes):
   • Recap:
     • Ratios: Compare parts (e.g., 2:3 = 2 to 3, or 2/5 total). Unit rate: 12 miles in 3 hours = 4 mph.
     • Integers: Whole numbers, positive/negative (e.g., -5, 0, 7). Add: -3 + 8 = 5. Subtract: 4 – (-2) = 6.
     • Rational Numbers: Fractions/decimals (e.g., 1/2, -0.75). Multiply: 1/3 × 6 = 2. Divide: -4 ÷ 1/2 = -8.
   • Model: “Ratio 4:5, 9 total. 4 parts = 4, 5 parts = 5, 1 part = 1, so 4 and 5.”
3. Hands-On Activity: Ratio Game (20 minutes):
   • Game: Ratio Roll
     • Roll 2 dice (e.g., 2, 3). Make a ratio (2:3). “15 total—how many each?” (6 and 9).
     • 5 rounds, track correct answers.
   • Discuss: “How’d you split it? What’s the unit rate?” (e.g., 2/5 = 0.4 per part).
4. Wrap-Up (5 minutes):
   • Ask: “What’s a ratio you made? How do integers fit in?”

Day 2: Integer and Rational Number Games (45 minutes)

1. Review (10 minutes):
   • Recap: “What’s -7 + 3? (-4) 2:4 ratio, 10 total? (4 and 6)”
2. Hands-On Activity: Number Line Race (25 minutes):
   • Game: Integer Dash
     • Use number line. Start at 0. Draw card (e.g., “+5,” “-3”). Move and say new spot (5, then 2).
     • Add rationals: “+1/2,” “-3/4.” 5 rounds, aim for a target (e.g., 4).
   • Game: Rational Match
     • Cards: 1/2 × 4, -6 + 2, 8 ÷ 1/4. Match to answers (2, -4, 32). 5 pairs, time it.
   • Discuss: “What’s trickier—adding negatives or dividing fractions?”
3. Wrap-Up (10 minutes):
   • Reflect: “Which move was fastest? How do rationals change the game?”

Day 3: Quiz and Mixed Review (45 minutes)

1. Warm-Up (10 minutes):
   • Quickie: “3:7, 20 total? (6 and 14) -5 × 2? (-10) 1/3 + 2/3? (1)”
2. Hands-On Activity: Quiz Challenge (25 minutes):
   • Worksheet quiz:
     • Ratios: “3:2, 15 total—what’s each?” (9, 6)
     • Integers: “-8 + 12 = __” (4), “6 – (-3) = __” (9)
     • Rationals: “1/4 × 8 = __” (2), “-6 ÷ 1/2 = __” (-12)
     • Mixed: “Ratio 1:3, 1 part is -4—what’s total?” (-16)
   • Time it (10 min): 8-10 questions, check together.
   • Bonus game: “Roll 3 dice (2, 4, 5). Make a problem!” (e.g., 2:4, 5 is 2 parts—total 12.5).
3. Wrap-Up (10 minutes):
   • Discuss: “What’s your quiz score? Which skill was strongest?”

Assessment

• Informal Observation: Note accuracy and strategy use in games.
• Game Scores: Check 70-80% correct answers (e.g., 4/5 per round).
• Quiz: Ensure 80% accuracy (e.g., 8/10 right) across ratios, integers, rationals.

Extensions

• Real-world mix: “2:3 juice ratio, 10 cups total, 20% spills—left?” (8 cups).
• Number line art: Plot 5 moves (e.g., -2, +3/4), draw the path.
• Speed round: 10 rapid-fire questions—beat your time!