Chapter 6 · Lesson 4 of 12 · Grade 6 · Statistics & Probability · MAT-06-SP-004
Build and Interpret Box Plots
Represent and solve build and interpret box plots problems using precise mathematical notation, models, and reasoning.
Math V4 · Course chapter
Chapter 6: Statistics, Variability, and Mathematical Modeling
Using data and reasoning to support claims
Essential question: How can data, models, and mathematical arguments be used responsibly to describe variation and evaluate conclusions?
Where this lesson fits
Lesson 4 of 12. Each lesson builds on earlier chapter representations and ideas rather than resetting the topic.
Chapter destination
Collect a small data set, choose appropriate displays and summary measures, critique a misleading claim, and present a defensible conclusion.
Learning objectives
What you should be able to do
- Represent and solve build and interpret box plots problems using precise mathematical notation, models, and reasoning.
- Explain why a method for build and interpret box plots 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 6: Statistics, Variability, and Mathematical Modeling
Using data and reasoning to support claims. This is Lesson 4 of 12 in Chapter 6. It builds on earlier chapter ideas instead of restarting the topic from scratch.
Connection to the course
Cumulative knowledge used here includes all previous chapters. As you work, connect today's idea to earlier lessons in the chapter and keep those earlier representations available for comparison.
Mathematical meaning
Build and Interpret Box Plots describes data or chance while accounting for variability. Calculations must be interpreted in the context of data collection or a probability model.
Represent the relationship
The Math Box uses statistics plot 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 calculations from the raw values, then decide whether the sample, display, center, spread, or probability model actually supports the claim.
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.
Statistics Lab
Change the data and watch center and spread respond.
- Model one example of build and interpret box plots 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.
For 1, 3, 5, 7, 9, identify the median.
5With five ordered values, the middle value is the median.
Represent before calculating. Use a data display, ordered data set, summary measure, or written statistical claim for this task, then solve: For 2, 4, 6, 8, 10, identify the median.
6The representation should show the same mathematical relationship as the calculation. With five ordered values, the middle value is the median.
Explain why a valid method works, then solve: For 3, 5, 7, 9, 11, identify the median.
7A complete explanation names the relationship or property being preserved. With five ordered values, the middle value is the median.
Estimate or predict first, then calculate and decide whether the result is reasonable: For 4, 6, 8, 10, 12, identify the median.
8The estimate is a reasonableness check, not a replacement for the exact result. With five ordered values, the middle value is the median.
Solve and verify the result with a second method, inverse operation, or equivalent representation: For 5, 7, 9, 11, 13, identify the median.
9Verification should independently support the result. With five ordered values, the middle value is the median.
Interpret the answer in context after solving. What does the result mean here? For 6, 8, 10, 12, 14, identify the median.
10State the result with its meaning, units, direction, or comparison—not only a number. With five ordered values, the middle value is the median.
Error analysis: a student says, "Reporting one statistic without considering variability." Explain the mistake, then solve this related task correctly: For 7, 9, 11, 13, 15, identify the median.
Interpret center with spread, shape, and sampling method. Correct solution: 11Error analysis requires identifying the broken idea, not merely replacing the final answer. With five ordered values, the middle value is the median.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: For 8, 10, 12, 14, 16, identify the median. Case B: For 17, 19, 21, 23, 25, identify the median.
Case A: 12 Case B: 21Compare the structure, representation, units, and result rather than only the numbers. With five ordered values, the middle value is the median.
For 9, 11, 13, 15, 17, identify the median.
13With five ordered values, the middle value is the median.
Represent before calculating. Use a data display, ordered data set, summary measure, or written statistical claim for this task, then solve: For 10, 12, 14, 16, 18, identify the median.
14The representation should show the same mathematical relationship as the calculation. With five ordered values, the middle value is the median.
Explain why a valid method works, then solve: For 11, 13, 15, 17, 19, identify the median.
15A complete explanation names the relationship or property being preserved. With five ordered values, the middle value is the median.
Estimate or predict first, then calculate and decide whether the result is reasonable: For 12, 14, 16, 18, 20, identify the median.
16The estimate is a reasonableness check, not a replacement for the exact result. With five ordered values, the middle value is the median.
Solve and verify the result with a second method, inverse operation, or equivalent representation: For 13, 15, 17, 19, 21, identify the median.
17Verification should independently support the result. With five ordered values, the middle value is the median.
Interpret the answer in context after solving. What does the result mean here? For 14, 16, 18, 20, 22, identify the median.
18State the result with its meaning, units, direction, or comparison—not only a number. With five ordered values, the middle value is the median.
Error analysis: a student says, "Reporting one statistic without considering variability." Explain the mistake, then solve this related task correctly: For 15, 17, 19, 21, 23, identify the median.
Interpret center with spread, shape, and sampling method. Correct solution: 19Error analysis requires identifying the broken idea, not merely replacing the final answer. With five ordered values, the middle value is the median.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: For 16, 18, 20, 22, 24, identify the median. Case B: For 25, 27, 29, 31, 33, identify the median.
Case A: 20 Case B: 29Compare the structure, representation, units, and result rather than only the numbers. With five ordered values, the middle value is the median.
For 17, 19, 21, 23, 25, identify the median.
21With five ordered values, the middle value is the median.
Represent before calculating. Use a data display, ordered data set, summary measure, or written statistical claim for this task, then solve: For 18, 20, 22, 24, 26, identify the median.
22The representation should show the same mathematical relationship as the calculation. With five ordered values, the middle value is the median.
Explain why a valid method works, then solve: For 19, 21, 23, 25, 27, identify the median.
23A complete explanation names the relationship or property being preserved. With five ordered values, the middle value is the median.
Estimate or predict first, then calculate and decide whether the result is reasonable: For 20, 22, 24, 26, 28, identify the median.
24The estimate is a reasonableness check, not a replacement for the exact result. With five ordered values, the middle value is the median.
Practice
Guided practice with hints
01Solve and verify the result with a second method, inverse operation, or equivalent representation: For 21, 23, 25, 27, 29, identify the median.
Hint: Choose a representation before computing.
Answer: 25
02Interpret the answer in context after solving. What does the result mean here? For 22, 24, 26, 28, 30, identify the median.
Hint: Explain what relationship or property makes your method valid.
Answer: 26
03Error analysis: a student says, "Reporting one statistic without considering variability." Explain the mistake, then solve this related task correctly: For 23, 25, 27, 29, 31, identify the median.
Hint: Choose a representation before computing.
Answer: Interpret center with spread, shape, and sampling method. Correct solution: 27
04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: For 24, 26, 28, 30, 32, identify the median. Case B: For 33, 35, 37, 39, 41, identify the median.
Hint: Explain what relationship or property makes your method valid.
Answer: Case A: 28 Case B: 37
05For 25, 27, 29, 31, 33, identify the median.
Hint: Choose a representation before computing.
Answer: 29
06Represent before calculating. Use a data display, ordered data set, summary measure, or written statistical claim for this task, then solve: For 26, 28, 30, 32, 34, identify the median.
Hint: Explain what relationship or property makes your method valid.
Answer: 30
Practice
Independent practice
01Explain why a valid method works, then solve: For 27, 29, 31, 33, 35, identify the median.
Answer: 31
02Estimate or predict first, then calculate and decide whether the result is reasonable: For 28, 30, 32, 34, 36, identify the median.
Answer: 32
03Solve and verify the result with a second method, inverse operation, or equivalent representation: For 29, 31, 33, 35, 37, identify the median.
Answer: 33
04Interpret the answer in context after solving. What does the result mean here? For 30, 32, 34, 36, 38, identify the median.
Answer: 34
05Error analysis: a student says, "Reporting one statistic without considering variability." Explain the mistake, then solve this related task correctly: For 31, 33, 35, 37, 39, identify the median.
Answer: Interpret center with spread, shape, and sampling method. Correct solution: 35
06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: For 32, 34, 36, 38, 40, identify the median. Case B: For 41, 43, 45, 47, 49, identify the median.
Answer: Case A: 36 Case B: 45
07For 33, 35, 37, 39, 41, identify the median.
Answer: 37
08Represent before calculating. Use a data display, ordered data set, summary measure, or written statistical claim for this task, then solve: For 34, 36, 38, 40, 42, identify the median.
Answer: 38
Common mistakes
Learn to catch the error, not just the answer.
Interpret center with spread, shape, and sampling method.
Finite experimental frequencies vary even when a model is correct.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Summarize data, compare groups, evaluate survey claims, and describe uncertainty.
- Build probability models and simulations for repeated random processes.
Challenge problems
Explain why a valid method works, then solve: For 35, 37, 39, 41, 43, identify the median.
39
Estimate or predict first, then calculate and decide whether the result is reasonable: For 36, 38, 40, 42, 44, identify the median.
40
Solve and verify the result with a second method, inverse operation, or equivalent representation: For 37, 39, 41, 43, 45, identify the median.
41
Interpret the answer in context after solving. What does the result mean here? For 38, 40, 42, 44, 46, identify the median.
42
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 6 Chapter 6 Reasoning Lab
Solve build and interpret box plots problems accurately, choose an appropriate representation, and justify each result with a check.
Practice
Mastery check
01Error analysis: a student says, "Reporting one statistic without considering variability." Explain the mistake, then solve this related task correctly: For 39, 41, 43, 45, 47, identify the median.
Answer: Interpret center with spread, shape, and sampling method. Correct solution: 43
02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: For 40, 42, 44, 46, 48, identify the median. Case B: For 49, 51, 53, 55, 57, identify the median.
Answer: Case A: 44 Case B: 53
03For 41, 43, 45, 47, 49, identify the median.
Answer: 45
04Represent before calculating. Use a data display, ordered data set, summary measure, or written statistical claim for this task, then solve: For 42, 44, 46, 48, 50, identify the median.
Answer: 46
05Explain why a valid method works, then solve: For 43, 45, 47, 49, 51, identify the median.
Answer: 47
06Estimate or predict first, then calculate and decide whether the result is reasonable: For 44, 46, 48, 50, 52, identify the median.
Answer: 48
07Solve and verify the result with a second method, inverse operation, or equivalent representation: For 45, 47, 49, 51, 53, identify the median.
Answer: 49
08Interpret the answer in context after solving. What does the result mean here? For 46, 48, 50, 52, 54, identify the median.
Answer: 50
Terminology
Words to know
- distribution
- The pattern of values including center, spread, shape, and unusual features.
- variability
- The degree to which observations differ.
- sample
- A subset of a population used to collect information.
- probability
- A number from 0 to 1 describing likelihood.
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.