7th Grade Math ยท Unit 1
See how the same data look when the intervals change.
Same data, new groups
Changing the interval width changes how the shape looks. The values themselves do not change.
The total number of values stays the same. You regrouped them.
Detail and smooth shape
A narrow interval shows more detail. A wide interval looks smoother.
Neither display is always better. Use the width that shows the detail you need.
A wide bar can hide a split
A tall wide bar may join narrow bars that were not equal.
Do not say the wide bars prove the values are all the same.
Did the total change? Say no, and state the total.
These are the same quiz scores in two groupings.
Narrower intervals
Wider intervals
Did the total number of quiz scores change when the intervals got wider?
total = sum of the bar heights
narrow total = 3 + 3 + 4 + 4 + 3 + 1
narrow total = 18
wide total = 6 + 8 + 4
wide total = 18
The same values were regrouped. The total did not change.
Answer: No. The total is still 18.
These are the same practice sessions in two groupings.
Narrower intervals
Wider intervals
Did the total number of practice sessions change when the intervals got wider?
total = sum of the bar heights
narrow total = 3 + 3 + 2 + 3 + 3 + 2
narrow total = 16
wide total = 6 + 5 + 5
wide total = 16
The same values were regrouped. The total did not change.
Answer: No. The total is still 16.
These are the same plant heights in two groupings.
Narrower intervals
Wider intervals
Did the total number of plants change when the intervals got wider?
total = sum of the bar heights
narrow total = 3 + 3 + 3 + 4 + 2 + 2
narrow total = 17
wide total = 6 + 7 + 4
wide total = 17
The same values were regrouped. The total did not change.
Answer: No. The total is still 17.
These are the same noon temperatures in two groupings.
Narrower intervals
Wider intervals
Did the total number of days change when the intervals got wider?
total = sum of the bar heights
narrow total = 3 + 3 + 4 + 4 + 4 + 2
narrow total = 20
wide total = 6 + 8 + 6
wide total = 20
The same values were regrouped. The total did not change.
Answer: No. The total is still 20.
Look at the quiz scores again. The wide bars are taller.
Narrower intervals
Wider intervals
Did we add quiz scores when the bars got taller?
total = sum of the bar heights
narrow total = 3 + 3 + 4 + 4 + 3 + 1
narrow total = 18
wide total = 6 + 8 + 4
wide total = 18
The same values were regrouped. The total did not change.
Answer: No. The total is still 18.
Look at the practice minutes again.
Narrower intervals
Wider intervals
Are there more practice sessions in the wide display than in the narrow display?
total = sum of the bar heights
narrow total = 3 + 3 + 2 + 3 + 3 + 2
narrow total = 16
wide total = 6 + 5 + 5
wide total = 16
The same values were regrouped. The total did not change.
Answer: No. The total is still 16.
Look at the plant heights again.
Narrower intervals
Wider intervals
Did a plant get left out when the intervals got wider?
total = sum of the bar heights
narrow total = 3 + 3 + 3 + 4 + 2 + 2
narrow total = 17
wide total = 6 + 7 + 4
wide total = 17
The same values were regrouped. The total did not change.
Answer: No. The total is still 17.
Look at the temperatures again. The wide display has fewer bars.
Narrower intervals
Wider intervals
Do the two displays show different numbers of days?
total = sum of the bar heights
narrow total = 3 + 3 + 4 + 4 + 4 + 2
narrow total = 20
wide total = 6 + 8 + 6
wide total = 20
The same values were regrouped. The total did not change.
Answer: No. The total is still 20.
Add both quiz displays before you answer.
Narrower intervals
Wider intervals
Is the narrow total different from the wide total?
total = sum of the bar heights
narrow total = 3 + 3 + 4 + 4 + 3 + 1
narrow total = 18
wide total = 6 + 8 + 4
wide total = 18
The same values were regrouped. The total did not change.
Answer: No. The total is still 18.
Add both plant displays before you answer.
Narrower intervals
Wider intervals
Did the total change when the plant intervals got wider?
total = sum of the bar heights
narrow total = 3 + 3 + 3 + 4 + 2 + 2
narrow total = 17
wide total = 6 + 7 + 4
wide total = 17
The same values were regrouped. The total did not change.
Answer: No. The total is still 17.
Add both temperature displays before you answer.
Narrower intervals
Wider intervals
Did fewer days get counted in the wide temperature display?
total = sum of the bar heights
narrow total = 3 + 3 + 4 + 4 + 4 + 2
narrow total = 20
wide total = 6 + 8 + 6
wide total = 20
The same values were regrouped. The total did not change.
Answer: No. The total is still 20.
Say which width shows more detail, and use the bars in your reason.
Compare the two quiz score displays.
Narrower intervals
Wider intervals
Which interval width shows more detail, and why?
A narrower interval shows more detail. A wider interval looks smoother. Neither is always better.
Each narrow interval covers 10 possible numbers. Each wide interval covers 20 possible numbers.
The narrower display shows more detail because each bar covers fewer numbers.
The wide bar 60 to 79 has 8. Inside it, 60 to 69 has 4 and 70 to 79 has 4. The wide bar still hides that those are separate intervals.
The narrower display keeps 60 to 69 separate from 70 to 79.
Answer: The narrower intervals show more detail. Each one covers 10 possible numbers, while each wide interval covers 20. The wide bar 60 to 79 has 8. Inside it, 60 to 69 has 4 and 70 to 79 has 4. The wide bar still hides that those are separate intervals. The narrower display keeps 60 to 69 separate from 70 to 79.
Compare the two practice displays.
Narrower intervals
Wider intervals
Which interval width shows more detail? Use 10 to 19 in your reason.
A narrower interval shows more detail. A wider interval looks smoother. Neither is always better.
Each narrow interval covers 5 possible numbers. Each wide interval covers 10 possible numbers.
The narrower display shows more detail because each bar covers fewer numbers.
The wide bar 10 to 19 has 5. Inside it, 10 to 14 has 2 and 15 to 19 has 3. The wide bar hides that difference.
The wide bar 10 to 19 mixes a shorter narrow bar with a taller one.
Answer: The narrower intervals show more detail. Each one covers 5 possible numbers, while each wide interval covers 10. The wide bar 10 to 19 has 5. Inside it, 10 to 14 has 2 and 15 to 19 has 3. The wide bar hides that difference. The wide bar 10 to 19 mixes a shorter narrow bar with a taller one.
Compare the two height displays.
Narrower intervals
Wider intervals
Which display shows more detail? Use 10 to 19 in your reason.
A narrower interval shows more detail. A wider interval looks smoother. Neither is always better.
Each narrow interval covers 5 possible numbers. Each wide interval covers 10 possible numbers.
The narrower display shows more detail because each bar covers fewer numbers.
The wide bar 10 to 19 has 7. Inside it, 10 to 14 has 3 and 15 to 19 has 4. The wide bar hides that difference.
The narrow peak is 15 to 19. The wide peak is the wider bar 10 to 19.
Answer: The narrower intervals show more detail. Each one covers 5 possible numbers, while each wide interval covers 10. The wide bar 10 to 19 has 7. Inside it, 10 to 14 has 3 and 15 to 19 has 4. The wide bar hides that difference. The narrow peak is 15 to 19. The wide peak is the wider bar 10 to 19.
Compare the two temperature displays.
Narrower intervals
Wider intervals
Which display shows more detail? Use 60 to 69 in your reason.
A narrower interval shows more detail. A wider interval looks smoother. Neither is always better.
Each narrow interval covers 5 possible numbers. Each wide interval covers 10 possible numbers.
The narrower display shows more detail because each bar covers fewer numbers.
The wide bar 60 to 69 has 8. Inside it, 60 to 64 has 4 and 65 to 69 has 4. The wide bar still hides that those are separate intervals.
The narrow display has a three way tie. The wide display makes 60 to 69 the only peak.
Answer: The narrower intervals show more detail. Each one covers 5 possible numbers, while each wide interval covers 10. The wide bar 60 to 69 has 8. Inside it, 60 to 64 has 4 and 65 to 69 has 4. The wide bar still hides that those are separate intervals. The narrow display has a three way tie. The wide display makes 60 to 69 the only peak.
The quiz peak changes when the bars get wider.
Narrower intervals
Wider intervals
Did the peak move into a wider bar, and which display still shows more detail?
A narrower interval shows more detail. A wider interval looks smoother. Neither is always better.
Each narrow interval covers 10 possible numbers. Each wide interval covers 20 possible numbers.
The narrower display shows more detail because each bar covers fewer numbers.
The wide bar 60 to 79 has 8. Inside it, 60 to 69 has 4 and 70 to 79 has 4. The wide bar still hides that those are separate intervals.
Narrower peak: 60 to 69 and 70 to 79 with 4. Wider peak: 60 to 79 with 8. The narrower display still shows more detail.
Answer: The narrower intervals show more detail. Each one covers 10 possible numbers, while each wide interval covers 20. The wide bar 60 to 79 has 8. Inside it, 60 to 69 has 4 and 70 to 79 has 4. The wide bar still hides that those are separate intervals. Narrower peak: 60 to 69 and 70 to 79 with 4. Wider peak: 60 to 79 with 8. The narrower display still shows more detail.
On the practice graph, 10 to 19 and 20 to 29 are tied in the wide display.
Narrower intervals
Wider intervals
Which display shows more detail inside those two wide bars?
A narrower interval shows more detail. A wider interval looks smoother. Neither is always better.
Each narrow interval covers 5 possible numbers. Each wide interval covers 10 possible numbers.
The narrower display shows more detail because each bar covers fewer numbers.
The wide bar 20 to 29 has 5. Inside it, 20 to 24 has 3 and 25 to 29 has 2. The wide bar hides that difference.
The narrower bars inside 20 to 29 are not equal, and 10 to 19 is split as well.
Answer: The narrower intervals show more detail. Each one covers 5 possible numbers, while each wide interval covers 10. The wide bar 20 to 29 has 5. Inside it, 20 to 24 has 3 and 25 to 29 has 2. The wide bar hides that difference. The narrower bars inside 20 to 29 are not equal, and 10 to 19 is split as well.
The plant graph has a narrow peak and a wide peak.
Narrower intervals
Wider intervals
What happened to the narrow peak, and which width shows more detail?
A narrower interval shows more detail. A wider interval looks smoother. Neither is always better.
Each narrow interval covers 5 possible numbers. Each wide interval covers 10 possible numbers.
The narrower display shows more detail because each bar covers fewer numbers.
The wide bar 10 to 19 has 7. Inside it, 10 to 14 has 3 and 15 to 19 has 4. The wide bar hides that difference.
The peak moved into the wider bar 10 to 19. The narrow width still shows more detail.
Answer: The narrower intervals show more detail. Each one covers 5 possible numbers, while each wide interval covers 10. The wide bar 10 to 19 has 7. Inside it, 10 to 14 has 3 and 15 to 19 has 4. The wide bar hides that difference. The peak moved into the wider bar 10 to 19. The narrow width still shows more detail.
The temperature graph looks smoother with wide bars.
Narrower intervals
Wider intervals
Which width shows more detail, and what did the wide peak hide?
A narrower interval shows more detail. A wider interval looks smoother. Neither is always better.
Each narrow interval covers 5 possible numbers. Each wide interval covers 10 possible numbers.
The narrower display shows more detail because each bar covers fewer numbers.
The wide bar 60 to 69 has 8. Inside it, 60 to 64 has 4 and 65 to 69 has 4. The wide bar still hides that those are separate intervals.
70 to 74 was tied with the narrow bars inside 60 to 69. The wide display hides that tie.
Answer: The narrower intervals show more detail. Each one covers 5 possible numbers, while each wide interval covers 10. The wide bar 60 to 69 has 8. Inside it, 60 to 64 has 4 and 65 to 69 has 4. The wide bar still hides that those are separate intervals. 70 to 74 was tied with the narrow bars inside 60 to 69. The wide display hides that tie.
Someone likes the quiz graph with fewer bars.
Narrower intervals
Wider intervals
Are the wider intervals always better because the graph looks smoother?
A narrower interval shows more detail. A wider interval looks smoother. Neither is always better.
Each narrow interval covers 10 possible numbers. Each wide interval covers 20 possible numbers.
The narrower display shows more detail because each bar covers fewer numbers.
The wide bar 80 to 99 has 4. Inside it, 80 to 89 has 3 and 90 to 99 has 1. The wide bar hides that difference.
No. Wider looks smoother, and narrower shows more detail. Neither is always better.
Answer: The narrower intervals show more detail. Each one covers 10 possible numbers, while each wide interval covers 20. The wide bar 80 to 99 has 4. Inside it, 80 to 89 has 3 and 90 to 99 has 1. The wide bar hides that difference. No. Wider looks smoother, and narrower shows more detail. Neither is always better.
Someone likes the practice graph with more bars.
Narrower intervals
Wider intervals
Are the narrower intervals always better?
A narrower interval shows more detail. A wider interval looks smoother. Neither is always better.
Each narrow interval covers 5 possible numbers. Each wide interval covers 10 possible numbers.
The narrower display shows more detail because each bar covers fewer numbers.
The wide bar 0 to 9 has 6. Inside it, 0 to 4 has 3 and 5 to 9 has 3. The wide bar still hides that those are separate intervals.
No. Narrower shows more detail, and wider looks smoother. Neither is always better.
Answer: The narrower intervals show more detail. Each one covers 5 possible numbers, while each wide interval covers 10. The wide bar 0 to 9 has 6. Inside it, 0 to 4 has 3 and 5 to 9 has 3. The wide bar still hides that those are separate intervals. No. Narrower shows more detail, and wider looks smoother. Neither is always better.
The wide plant graph looks simple.
Narrower intervals
Wider intervals
Does a smoother histogram mean the heights are more accurate?
A narrower interval shows more detail. A wider interval looks smoother. Neither is always better.
Each narrow interval covers 5 possible numbers. Each wide interval covers 10 possible numbers.
The narrower display shows more detail because each bar covers fewer numbers.
The wide bar 0 to 9 has 6. Inside it, 0 to 4 has 3 and 5 to 9 has 3. The wide bar still hides that those are separate intervals.
No. Smoother only means the intervals are wider. The measurements did not become more accurate.
Answer: The narrower intervals show more detail. Each one covers 5 possible numbers, while each wide interval covers 10. The wide bar 0 to 9 has 6. Inside it, 0 to 4 has 3 and 5 to 9 has 3. The wide bar still hides that those are separate intervals. No. Smoother only means the intervals are wider. The measurements did not become more accurate.
Say what the wide bars hide.
Both quiz displays use the same scores.
Narrower intervals
Wider intervals
A classmate says, The wide intervals prove everyone got the same score. What is wrong with that conclusion?
A conclusion has to match the counts. Regrouping does not change the values or prove a cause.
The smallest value is 41 and the largest is 91.
Narrower peak: 60 to 69 and 70 to 79 with 4. Wider peak: 60 to 79 with 8.
The wide bar 60 to 79 has 8. Inside it, 60 to 69 has 4 and 70 to 79 has 4. The wide bar still hides that those are separate intervals.
The wide bars hide the spread. The scores are not all the same, and 60 to 79 still mixes 60 to 69 with 70 to 79.
Answer: The wide bars hide the spread. The scores are not all the same, and 60 to 79 still mixes 60 to 69 with 70 to 79.
Both practice displays use the same sessions.
Narrower intervals
Wider intervals
A classmate says, The wide intervals prove every practice took about the same time. What is wrong with that conclusion?
A conclusion has to match the counts. Regrouping does not change the values or prove a cause.
The smallest value is 1 and the largest is 28.
Narrower peak: 0 to 4, 5 to 9, 15 to 19, and 20 to 24 with 3. Wider peak: 0 to 9 with 6.
The wide bar 0 to 9 has 6. Inside it, 0 to 4 has 3 and 5 to 9 has 3. The wide bar still hides that those are separate intervals.
The wide bars hide the separate narrow intervals. The times are not all the same.
Answer: The wide bars hide the separate narrow intervals. The times are not all the same.
Both height displays use the same plants.
Narrower intervals
Wider intervals
A classmate says, The wide intervals prove the plants are all the same height. What is wrong with that conclusion?
A conclusion has to match the counts. Regrouping does not change the values or prove a cause.
The smallest value is 1 and the largest is 28.
Narrower peak: 15 to 19 with 4. Wider peak: 10 to 19 with 7.
The wide bar 10 to 19 has 7. Inside it, 10 to 14 has 3 and 15 to 19 has 4. The wide bar hides that difference.
The wide bars hide the heights inside them. The plants are not all the same height.
Answer: The wide bars hide the heights inside them. The plants are not all the same height.
Both temperature displays use the same days.
Narrower intervals
Wider intervals
A classmate says, The wide intervals prove the temperature did not vary. What is wrong with that conclusion?
A conclusion has to match the counts. Regrouping does not change the values or prove a cause.
The smallest value is 51 and the largest is 78.
Narrower peak: 60 to 64, 65 to 69, and 70 to 74 with 4. Wider peak: 60 to 69 with 8.
The wide bar 60 to 69 has 8. Inside it, 60 to 64 has 4 and 65 to 69 has 4. The wide bar still hides that those are separate intervals.
The wide bars hide the variation. The temperatures are not all the same, and the narrow bars inside 60 to 69 are separate.
Answer: The wide bars hide the variation. The temperatures are not all the same, and the narrow bars inside 60 to 69 are separate.
The tallest wide quiz bar is 60 to 79.
Narrower intervals
Wider intervals
A classmate says, Because that bar is tallest, every score is in the 60s. What is wrong with that conclusion?
A conclusion has to match the counts. Regrouping does not change the values or prove a cause.
The smallest value is 41 and the largest is 91.
Narrower peak: 60 to 69 and 70 to 79 with 4. Wider peak: 60 to 79 with 8.
The wide bar 60 to 79 has 8. Inside it, 60 to 69 has 4 and 70 to 79 has 4. The wide bar still hides that those are separate intervals.
60 to 79 includes 70 to 79, so those scores are not in the 60s. Other bars have scores too. The wide bar hides that split.
Answer: 60 to 79 includes 70 to 79, so those scores are not in the 60s. Other bars have scores too. The wide bar hides that split.
The wide practice display looks smoother.
Narrower intervals
Wider intervals
A classmate says, A smoother histogram means these practice times have no variability. What is wrong with that conclusion?
A conclusion has to match the counts. Regrouping does not change the values or prove a cause.
The smallest value is 1 and the largest is 28.
Narrower peak: 0 to 4, 5 to 9, 15 to 19, and 20 to 24 with 3. Wider peak: 0 to 9 with 6.
The wide bar 10 to 19 has 5. Inside it, 10 to 14 has 2 and 15 to 19 has 3. The wide bar hides that difference.
The times still vary. A smoother shape only means the intervals are wider. Inside 10 to 19, the narrow counts differ.
Answer: The times still vary. A smoother shape only means the intervals are wider. Inside 10 to 19, the narrow counts differ.
The tall wide bar for plant height is 10 to 19.
Narrower intervals
Wider intervals
A classmate says, That bar means most plants are exactly 14 inches. What is wrong with that conclusion?
A conclusion has to match the counts. Regrouping does not change the values or prove a cause.
The smallest value is 1 and the largest is 28.
Narrower peak: 15 to 19 with 4. Wider peak: 10 to 19 with 7.
The wide bar 10 to 19 has 7. Inside it, 10 to 14 has 3 and 15 to 19 has 4. The wide bar hides that difference.
The histogram does not show an exact height of 14. The wide bar hides the split between 10 to 14 and 15 to 19.
Answer: The histogram does not show an exact height of 14. The wide bar hides the split between 10 to 14 and 15 to 19.
The wide temperature bar 60 to 69 is the tallest.
Narrower intervals
Wider intervals
A classmate says, Those days were hotter because more days fell in that bar. What is wrong with that conclusion?
A conclusion has to match the counts. Regrouping does not change the values or prove a cause.
The smallest value is 51 and the largest is 78.
Narrower peak: 60 to 64, 65 to 69, and 70 to 74 with 4. Wider peak: 60 to 69 with 8.
The wide bar 60 to 69 has 8. Inside it, 60 to 64 has 4 and 65 to 69 has 4. The wide bar still hides that those are separate intervals.
A taller bar means more days, not a hotter temperature and not a cause. 70 to 79 is warmer and has fewer days. The wide bar also hides the narrow split.
Answer: A taller bar means more days, not a hotter temperature and not a cause. 70 to 79 is warmer and has fewer days. The wide bar also hides the narrow split.
The wide quiz display has fewer bars.
Narrower intervals
Wider intervals
A classmate says, We should use only wide intervals because they are always better. What is wrong with that conclusion?
A conclusion has to match the counts. Regrouping does not change the values or prove a cause.
The smallest value is 41 and the largest is 91.
Narrower peak: 60 to 69 and 70 to 79 with 4. Wider peak: 60 to 79 with 8.
The wide bar 40 to 59 has 6. Inside it, 40 to 49 has 3 and 50 to 59 has 3. The wide bar still hides that those are separate intervals.
Neither width is always better. The narrow display shows more detail, including the separate 90 to 99 bar. The wide display only looks smoother.
Answer: Neither width is always better. The narrow display shows more detail, including the separate 90 to 99 bar. The wide display only looks smoother.
The wide practice bar 10 to 19 has several sessions.
Narrower intervals
Wider intervals
A classmate says, That bar proves those practice sessions were better. What is wrong with that conclusion?
A conclusion has to match the counts. Regrouping does not change the values or prove a cause.
The smallest value is 1 and the largest is 28.
Narrower peak: 0 to 4, 5 to 9, 15 to 19, and 20 to 24 with 3. Wider peak: 0 to 9 with 6.
The wide bar 10 to 19 has 5. Inside it, 10 to 14 has 2 and 15 to 19 has 3. The wide bar hides that difference.
The bar is only a count of times. It does not show quality, and it hides the difference between 10 to 14 and 15 to 19.
Answer: The bar is only a count of times. It does not show quality, and it hides the difference between 10 to 14 and 15 to 19.
The wide plant bar 20 to 29 is the shortest.
Narrower intervals
Wider intervals
A classmate says, The short bar means those plants died. What is wrong with that conclusion?
A conclusion has to match the counts. Regrouping does not change the values or prove a cause.
The smallest value is 1 and the largest is 28.
Narrower peak: 15 to 19 with 4. Wider peak: 10 to 19 with 7.
The wide bar 20 to 29 has 4. Inside it, 20 to 24 has 2 and 25 to 29 has 2. The wide bar still hides that those are separate intervals.
A short bar is a count, not a cause. Nothing in the graph says a plant died. The wide bar also hides the split between 20 to 24 and 25 to 29.
Answer: A short bar is a count, not a cause. Nothing in the graph says a plant died. The wide bar also hides the split between 20 to 24 and 25 to 29.
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