Rational Numbers Manipulatives Guide
Tier 2/3 Intervention NC.7.NS.2 Rational Numbers
Hands-On Manipulatives Guide
Concrete-Representational-Abstract (CRA) instructional strategies for word problems 1 & 2
Grade 7 Math Intervention
Teacher Reference Guide
CRA Bridge: Negative decimals and debts can feel abstract. Move from physical directional displacement & fair sharing (Concrete) to number lines & bar models (Representational) before formulas (Abstract).
1
The Arctic Research Station
Expression: \(4 \times (-3.5) = -14^\circ\text{C}\)
"The temperature at a research station decreases by 3.5°C each hour for 4 hours. What is the total change?"
Recommended Concrete Tool
Vertical Thermometer / Slider Line
Calibrated from \(+5^\circ\) down to \(-20^\circ\). Use a clothes pin or magnetic slider to track physical descent.
Low-Cost / Tabletop Alternative
Two-Color Counters & Decimal Mats
Red chips = negative. Each of 4 hour plates receives 3 whole red chips and one cut half-chip (\(-3.5\)).
Intervention Procedure & Questioning:
- Step 1 (Anchor): Place slider at \(0^\circ\text{C}\). Ask: "Does 'decrease' move us up or down the column?"
- Step 2 (Iterate): Slide down 3.5 units four successive times: \(-3.5 \rightarrow -7.0 \rightarrow -10.5 \rightarrow -14.0\).
- Step 3 (Connect): Connect 4 jumps of downward 3.5 to the expression \(4 \times (-3.5) = -14\).
- Scaffold Check: "Why is the final answer negative even though the number of hours (4) is positive?"
2
Paying Off a Debt
Expression: \(-\$12.50 \div 5 = -\$2.50\)
"Jordan owes $12.50. He divides the debt into 5 equal weekly payments. Represent the debt as a negative number..."
Recommended Concrete Tool
Play Money in Red (or Red IOUs)
Print red play money: one $10 bill, two $1 bills, two quarters. 5 plastic sorting cups for the 5 weeks.
Low-Cost / Tabletop Alternative
Place Value Disks & Partition Mat
Red disks labeled \(-10\), \(-1\), and \(-0.10\). Students physically regroup across place values into 5 columns.
Intervention Procedure & Questioning:
- Step 1 (Physical Regrouping): Try to distribute the single \(-10\) bill across 5 cups. Ask: "What must we trade it for?" (Ten \(-1\) bills).
- Step 2 (Fair Sharing): Distribute twelve \(-1\) bills (2 per cup, with 2 bills left over). Trade remaining \(-\$2.00\) and \(-\$0.50\) for ten quarters (\(-0.25\) each). Distribute 2 quarters to each cup.
- Step 3 (Count & Interpret): Each cup holds \(-\$2.50\). Emphasize: "Each weekly payment reduces his debt by $2.50, signed as \(-2.50\) balance change."
- Common Pitfall: Students drop the negative sign. Remind: "Is each payment money Jordan gains or money he pays out?"
7th Grade Math Intervention Toolkit • NC.7.NS.2 Page 1 of 2
Tier 2/3 Intervention NC.7.NS.2 Rational Numbers
Hands-On Manipulatives Guide
Concrete-Representational-Abstract (CRA) instructional strategies for word problems 3 & 4
Grade 7 Math Intervention
Teacher Reference Guide
3
The Underwater Drone
Expression: \(7.5 \times (-2.4) = -18\text{ meters}\)
"An underwater drone descends at a rate of 2.4 meters per minute for 7.5 minutes. Change in position is negative..."
Recommended Concrete Tool
Vertical Depth Ribbon / Meter Tape
Anchor a scaled measuring tape vertically (table edge = 0 m). Attach 7 color clips for whole minutes plus a half-sized clip.
Semi-Concrete / Area Model
Base-Ten Blocks & Partial Products Mat
Decompose \(7.5 \times 2.4\) into \((7 \times 2.4) + (0.5 \times 2.4)\). Assign negative direction at the end.
Intervention Procedure & Questioning:
- Step 1 (Decompose Time): Break \(7.5\text{ min}\) into \(7\text{ full minutes}\) and \(0.5\text{ minute}\).
- Step 2 (Whole Minutes): Hop 7 times downward by \(2.4\text{ m}\): \(7 \times (-2.4) = -16.8\text{ m}\).
- Step 3 (The Half Minute): Ask: "If it travels 2.4 m in a full minute, how far in half a minute?" Fold a 2.4 strip in half to show \(1.2\text{ m}\) down (\(-1.2\)).
- Step 4 (Combine): Combine \(-16.8 + (-1.2) = -18.0\text{ m}\). Relate to the standard decimal algorithm.
4
Baking for the School Fair
Expression: \(2.5 \times \frac{3}{4} = \frac{15}{8} = 1\frac{7}{8}\text{ or }1.875\text{ cups}\)
"A recipe requires 3/4 cup sugar for 1 batch. The baking club makes 2.5 batches. How many cups needed altogether?"
Recommended Concrete Tool
Nesting Measuring Cups & Counters
Use physical \(\frac{1}{4}\)-cup scoops or transparent measuring cups with dried lentils/beans to build the recipe.
Semi-Concrete / Tiles
Fraction Strips / Tower Cubes
Align fourths and eighths strips side-by-side to visibly show \(\frac{1}{2} \text{ of } \frac{3}{4} = \frac{3}{8}\).
Intervention Procedure & Questioning:
- Step 1 (Whole Batches): Scoop \(\frac{3}{4}\) cup for Batch 1 and \(\frac{3}{4}\) cup for Batch 2: \(\frac{6}{4} = 1\frac{2}{4} = 1\frac{1}{2}\) cups (or six \(\frac{1}{4}\) scoops).
- Step 2 (The Half Batch): Ask: "What is half of 3 fourths?" Align three \(\frac{1}{4}\) tiles with six \(\frac{1}{8}\) tiles to reveal that half is \(\frac{3}{8}\) cup.
- Step 3 (Sum Total): \(\frac{6}{4} + \frac{3}{8} = \frac{12}{8} + \frac{3}{8} = \frac{15}{8} = 1\frac{7}{8}\) cups.
- Step 4 (Decimal Form): Connect to decimal conversion: \(1 + \frac{7}{8} = 1 + 0.875 = 1.875\) cups (or \(2.5 \times 0.75 = 1.875\)).
Manipulative Quick Reference NC.7.NS.2 Intervention
#1 Arctic Temp
Vertical Thermometer
Repeated descent
#2 Debt Split
Red Money / 5 Cups
Fair-share negative
#3 Drone Depth
Meter Tape / Area Mat
Partial products descent
#4 Cookie Sugar
Fraction Strips (1/4 & 1/8)
Dual fraction / decimal
7th Grade Math Intervention Toolkit • NC.7.NS.2 Page 2 of 2
