Chapter 3 · Lesson 1 of 6 · Grade 6 · The Number System · MAT-06-NS-007
Use Positive and Negative Numbers
Represent and solve use positive and negative numbers problems using precise mathematical notation, models, and reasoning.
Math V4 · Course chapter
Chapter 3: Rational Numbers and the Coordinate Plane
Direction, distance, order, and location
Essential question: How do signs, absolute value, and coordinates describe quantities and locations that extend beyond positive whole numbers?
Where this lesson fits
Lesson 1 of 6. Each lesson builds on earlier chapter representations and ideas rather than resetting the topic.
Chapter destination
Create a coordinate map with elevations, temperatures, gains/losses, and route distances. Explain the meaning of each signed value.
Learning objectives
What you should be able to do
- Represent and solve use positive and negative numbers problems using precise mathematical notation, models, and reasoning.
- Explain why a method for use positive and negative numbers works, interpret the result in context, and verify it independently.
Prerequisite check
Make sure the foundation is ready.
The new lesson depends on this prerequisite as active knowledge.
Learn
Build the idea from meaning, not memorization.
Chapter 3: Rational Numbers and the Coordinate Plane
Direction, distance, order, and location. This lesson opens Chapter 3. Begin with the chapter question: How do signs, absolute value, and coordinates describe quantities and locations that extend beyond positive whole numbers? The goal is to build a reusable idea that later lessons will extend.
Connection to the course
Cumulative knowledge used here includes number lines, fraction and decimal comparison, Chapter 2 arithmetic fluency. As you work, connect today's idea to earlier lessons in the chapter and keep those earlier representations available for comparison.
Mathematical meaning
Use Positive and Negative Numbers extends arithmetic to signed, fractional, decimal, or coordinate quantities. Operations must preserve both magnitude and direction.
Represent the relationship
The Math Box uses rational number line to expose the structure. Change one input, predict the result, then connect the visual change to an equation, table, graph, number line, or geometric model.
Calculate, interpret, and verify
Verify with an inverse operation, equivalent representation, estimate, or second method so both the calculation and reasoning are auditable.
Math Box · Interactive lesson
Touch the math. Change it. See what stays true.
Use the interactive model before and after the worked examples. Change the inputs, make a prediction, then use the model to test whether your reasoning holds.
Rational Number Line
Move through positive and negative halves and compare distance from zero.
- Model one example of use positive and negative numbers in the Math Box.
- Change one input, predict the effect, and test the prediction.
- Connect the model to another representation used earlier in this chapter.
- State the relationship or invariant that explains what stayed mathematically consistent.
Worked examples
Twenty different ways to see the concept work.
Use -10 as a signed number and identify its opposite.
opposite of -10 is 10Opposites are the same distance from zero on opposite sides.
Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Use -9 as a signed number and identify its opposite.
opposite of -9 is 9The representation should show the same mathematical relationship as the calculation. Opposites are the same distance from zero on opposite sides.
Explain why a valid method works, then solve: Use -8 as a signed number and identify its opposite.
opposite of -8 is 8A complete explanation names the relationship or property being preserved. Opposites are the same distance from zero on opposite sides.
Estimate or predict first, then calculate and decide whether the result is reasonable: Use -7 as a signed number and identify its opposite.
opposite of -7 is 7The estimate is a reasonableness check, not a replacement for the exact result. Opposites are the same distance from zero on opposite sides.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Use -6 as a signed number and identify its opposite.
opposite of -6 is 6Verification should independently support the result. Opposites are the same distance from zero on opposite sides.
Interpret the answer in context after solving. What does the result mean here? Use -5 as a signed number and identify its opposite.
opposite of -5 is 5State the result with its meaning, units, direction, or comparison—not only a number. Opposites are the same distance from zero on opposite sides.
Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Use -4 as a signed number and identify its opposite.
Use a number line, context, or inverse operation. Correct solution: opposite of -4 is 4Error analysis requires identifying the broken idea, not merely replacing the final answer. Opposites are the same distance from zero on opposite sides.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Use -3 as a signed number and identify its opposite. Case B: Use 6 as a signed number and identify its opposite.
Case A: opposite of -3 is 3 Case B: opposite of 6 is -6Compare the structure, representation, units, and result rather than only the numbers. Opposites are the same distance from zero on opposite sides.
Use -2 as a signed number and identify its opposite.
opposite of -2 is 2Opposites are the same distance from zero on opposite sides.
Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Use -1 as a signed number and identify its opposite.
opposite of -1 is 1The representation should show the same mathematical relationship as the calculation. Opposites are the same distance from zero on opposite sides.
Explain why a valid method works, then solve: Use 0 as a signed number and identify its opposite.
opposite of 0 is 0A complete explanation names the relationship or property being preserved. Opposites are the same distance from zero on opposite sides.
Estimate or predict first, then calculate and decide whether the result is reasonable: Use 1 as a signed number and identify its opposite.
opposite of 1 is -1The estimate is a reasonableness check, not a replacement for the exact result. Opposites are the same distance from zero on opposite sides.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Use 2 as a signed number and identify its opposite.
opposite of 2 is -2Verification should independently support the result. Opposites are the same distance from zero on opposite sides.
Interpret the answer in context after solving. What does the result mean here? Use 3 as a signed number and identify its opposite.
opposite of 3 is -3State the result with its meaning, units, direction, or comparison—not only a number. Opposites are the same distance from zero on opposite sides.
Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Use 4 as a signed number and identify its opposite.
Use a number line, context, or inverse operation. Correct solution: opposite of 4 is -4Error analysis requires identifying the broken idea, not merely replacing the final answer. Opposites are the same distance from zero on opposite sides.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Use 5 as a signed number and identify its opposite. Case B: Use 14 as a signed number and identify its opposite.
Case A: opposite of 5 is -5 Case B: opposite of 14 is -14Compare the structure, representation, units, and result rather than only the numbers. Opposites are the same distance from zero on opposite sides.
Use 6 as a signed number and identify its opposite.
opposite of 6 is -6Opposites are the same distance from zero on opposite sides.
Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Use 7 as a signed number and identify its opposite.
opposite of 7 is -7The representation should show the same mathematical relationship as the calculation. Opposites are the same distance from zero on opposite sides.
Explain why a valid method works, then solve: Use 8 as a signed number and identify its opposite.
opposite of 8 is -8A complete explanation names the relationship or property being preserved. Opposites are the same distance from zero on opposite sides.
Estimate or predict first, then calculate and decide whether the result is reasonable: Use 9 as a signed number and identify its opposite.
opposite of 9 is -9The estimate is a reasonableness check, not a replacement for the exact result. Opposites are the same distance from zero on opposite sides.
Practice
Guided practice with hints
01Solve and verify the result with a second method, inverse operation, or equivalent representation: Use 10 as a signed number and identify its opposite.
Hint: Choose a representation before computing.
Answer: opposite of 10 is -10
02Interpret the answer in context after solving. What does the result mean here? Use 11 as a signed number and identify its opposite.
Hint: Explain what relationship or property makes your method valid.
Answer: opposite of 11 is -11
03Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Use 12 as a signed number and identify its opposite.
Hint: Choose a representation before computing.
Answer: Use a number line, context, or inverse operation. Correct solution: opposite of 12 is -12
04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Use 13 as a signed number and identify its opposite. Case B: Use 22 as a signed number and identify its opposite.
Hint: Explain what relationship or property makes your method valid.
Answer: Case A: opposite of 13 is -13 Case B: opposite of 22 is -22
05Use 14 as a signed number and identify its opposite.
Hint: Choose a representation before computing.
Answer: opposite of 14 is -14
06Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Use 15 as a signed number and identify its opposite.
Hint: Explain what relationship or property makes your method valid.
Answer: opposite of 15 is -15
Practice
Independent practice
01Explain why a valid method works, then solve: Use 16 as a signed number and identify its opposite.
Answer: opposite of 16 is -16
02Estimate or predict first, then calculate and decide whether the result is reasonable: Use 17 as a signed number and identify its opposite.
Answer: opposite of 17 is -17
03Solve and verify the result with a second method, inverse operation, or equivalent representation: Use 18 as a signed number and identify its opposite.
Answer: opposite of 18 is -18
04Interpret the answer in context after solving. What does the result mean here? Use 19 as a signed number and identify its opposite.
Answer: opposite of 19 is -19
05Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Use 20 as a signed number and identify its opposite.
Answer: Use a number line, context, or inverse operation. Correct solution: opposite of 20 is -20
06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Use 21 as a signed number and identify its opposite. Case B: Use 30 as a signed number and identify its opposite.
Answer: Case A: opposite of 21 is -21 Case B: opposite of 30 is -30
07Use 22 as a signed number and identify its opposite.
Answer: opposite of 22 is -22
08Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Use 23 as a signed number and identify its opposite.
Answer: opposite of 23 is -23
Common mistakes
Learn to catch the error, not just the answer.
Use a number line, context, or inverse operation.
For negatives, values closer to zero are greater.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Model temperature, elevation, debt, gains and losses, coordinates, and measurements beyond whole numbers.
- Compute accurately with fractions, decimals, and signed quantities while checking magnitude and sign.
Challenge problems
Explain why a valid method works, then solve: Use 24 as a signed number and identify its opposite.
opposite of 24 is -24
Estimate or predict first, then calculate and decide whether the result is reasonable: Use 25 as a signed number and identify its opposite.
opposite of 25 is -25
Solve and verify the result with a second method, inverse operation, or equivalent representation: Use 26 as a signed number and identify its opposite.
opposite of 26 is -26
Interpret the answer in context after solving. What does the result mean here? Use 27 as a signed number and identify its opposite.
opposite of 27 is -27
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 6 Chapter 3 Reasoning Lab
Solve use positive and negative numbers problems accurately, choose an appropriate representation, and justify each result with a check.
Practice
Mastery check
01Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Use 28 as a signed number and identify its opposite.
Answer: Use a number line, context, or inverse operation. Correct solution: opposite of 28 is -28
02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Use 29 as a signed number and identify its opposite. Case B: Use 38 as a signed number and identify its opposite.
Answer: Case A: opposite of 29 is -29 Case B: opposite of 38 is -38
03Use 30 as a signed number and identify its opposite.
Answer: opposite of 30 is -30
04Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Use 31 as a signed number and identify its opposite.
Answer: opposite of 31 is -31
05Explain why a valid method works, then solve: Use 32 as a signed number and identify its opposite.
Answer: opposite of 32 is -32
06Estimate or predict first, then calculate and decide whether the result is reasonable: Use 33 as a signed number and identify its opposite.
Answer: opposite of 33 is -33
07Solve and verify the result with a second method, inverse operation, or equivalent representation: Use 34 as a signed number and identify its opposite.
Answer: opposite of 34 is -34
08Interpret the answer in context after solving. What does the result mean here? Use 35 as a signed number and identify its opposite.
Answer: opposite of 35 is -35
Terminology
Words to know
- rational number
- A number expressible as a ratio of two integers with nonzero denominator.
- absolute value
- Distance from zero on a number line.
- opposite
- A number the same distance from zero on the other side.
- coordinate
- A number locating a point relative to an axis.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
Grade 6 scope, sequence, standards, and public task structure
Reference for coherent sequencing, representations, dependency-aware progression, and reasoning-rich task types.Eureka Math² Grade 6 program structure
Reference for module/topic coherence, recap, mixed-practice, and cumulative-learning patterns.enVision Mathematics Grade 6 instructional model
Reference for problem-based entry points, visual learning, modeling, and application patterns.Common Core State Standards for Mathematics — Grade 6
Grade-level content and mathematical-practice expectations.