Download Lesson Plan: Z-Tables - Howard County Public School System

Lesson Title: Z-Tables
Date: _____________
Teacher(s): ____________________
Course: Common Core Algebra II, Unit 6
Start/end times: _________________________
Lesson Standards/Objective(s): What mathematical skill(s) and understanding(s) will be developed? Which
Mathematical Practices do you expect students to engage in during the lesson?
S.ID.A.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate
population percentages. Recognize that there are data sets for which such a procedure is not appropriate.
Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
(This lesson is designed to take 2 or more days)
MP1:
MP3:
MP4:
MP5:
MP6:
MP7:
Make sense of problems and persevere in solving them.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Common Core Algebra II, Unit 6
Lesson Launch Notes: Exactly how will you use the
first five minutes of the lesson?
Ask how many students will be taking the SAT or have
taken it. Ask what the scoring range is (200-800 on
each part, for a total of 600-2400).
Ask students what they think the mean score is for
math (in 2012, mean was 514). Tell them the standard
deviation was 114 in 2012.
Ask them what this means. What would a lower SD
mean? What would a higher SD mean? ( The larger the
SD the greater the variability or spread of the data
about the mean.) (Look for evidence of MP2 and MP3.)
Lesson Closure Notes: Exactly what summary activity,
questions, and discussion will close the lesson and
connect big ideas? List the questions. Provide a
foreshadowing of tomorrow.
Say, “Let us assume we curve a test. That is, we assume
the test scores are normally distributed. What would be a
good scale of Z-scores for the letter grades A.B.C.D.E?”
Allow for discussion, either whole group or in groups.
Encourage consideration of percentages of students who
get each letter grade. (Look for evidence of MP3 and
MP7.)
Lesson Tasks, Problems, and Activities (attach resource sheets):
PART I (empirical rule and z-scores)
1. Use the data set, “50 Students’ Scores on SAT Math.” The data is created so it has the approximate mean and
standard deviation referenced in the lesson launch (mean of 505 and SD of 108.4). Group students in groups
of 3 to 4. Ask students to find the mean and standard deviation of the data using the TI-84 (TI-84 Instructions
on Wiki) or other device. Have groups create a frequency histogram of the data. This can be done on the TI- 84
(TI-84 Instructions on Wiki), on this site: http://www.shodor.org/interactivate/activities/Histogram/ (select
“my data” on the drop down menu) or by hand on graph paper. (Look for evidence of MP4 and MP5.)
2. Ask groups to sketch a curve over the histogram that touches the tops of the bars. Ask students to
comment on the shape of the histogram and the curve. Highlight concepts like symmetric, balanced,
high point at the mean. Some students may even know it as a “bell curve” or a testing curve. Tell them
this is known as a normal curve and the data is called normally distributed data. This type of
distribution was investigated previously with the “Random Rectangle” activity. (At this point, a
discussion of what it really means to “curve” a test, vs. merely inflating grades, may be appropriate.)
(Look for evidence of MP4 and MP5.)
3. Explain that many sets of data will fall into this normal curve pattern (some examples are test scores,
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: Z-Tables
Date: _____________
4.
5
6.
6.
7.
Teacher(s): ____________________
Course: Common Core Algebra II, Unit 6
Start/end times: _________________________
peoples’ heights, IQ) and this curve has specific properties. Ask groups to find the percentage of the
scores that fall between the Mean  one Standard Deviation, the Mean  two Standard Deviations,
and the Mean  three Standard Deviations (answers should be approximately 68%, 95%, and 99.7%).
These approximate the empirical rule. Explain to students that for large sets of normally distributed
data, the percentages of the empirical rule will generally hold. That is, approximately 68% of the data
will fall within one Standard Deviation of the mean,
approximately 95% of the data will fall within two Standard Deviations of the mean, and
approximately 99.7% (or virtually all) of the data will fall within three Standard Deviations of the
mean. Show illustration 1 from the “Empirical Rule Illustrations” document on the wiki. (Look for
evidence of MP1 and MP6.)
Ask students if they have ever gotten a percentile score (vs. a percentage score) for a standardized exam.
Lead a discussion of what the score means. (Guide responses to the concept that a percentile indicates
the percentage of other scores at or below the given score.) Return to the mean and standard deviation
referenced in the lesson launch. Ask groups to answer this question: “If a student scored a 628, what
would the percentile score be?” (Answer: 84%). It may be necessary to point out that 628 is one
Standard Deviation above the mean. It also may be necessary to address the symmetry of the
percentages about the mean. Illustration 2 on the “Empirical Rule Illustrations” document on the wiki
will help with this. Ask randomly selected groups to present and defend their answers. Allow for
discussion and critique with the whole group. Ask groups to find the percentile scores for scores of 742
(answer: 97.5) and 400 (answer: 16). (These scores are exactly two standard deviations above and one
standard deviation below the mean respectively.) (Look for evidence of MP1 and MP4.)
Ask groups to complete a chart of the percentile scores for each level of integer standard deviations
above and below the mean ( (Z) Table of Percentile Scores on Wiki with answers on the second page).
Tell them to skip the half standard deviations above and below for now. Check groups’ answers.
Students may ask about the middle column labeled Z-scores. If they do not, draw attention to them. Ask
groups to conjecture what a Z-score means. Randomly select groups to answer. (It is the number of
standard deviations above (positive) or below (negative) the mean for a score). Allow for discussion
and critique. Display Illustration 3 from the “Empirical Rule Illustrations.” Ask groups to correct any
errors on their chart and to complete the chart with the half Z-values. (Look for evidence of MP1, MP3,
and MP5.)
Ask groups to find the scores on the SAT Math 2012 that would align with each Z-score. (Answers are
in orange on the (Z) Table of Percentile Scores on Wiki with answers on the third page)
(The top and bottom scores are outside the given range. They could arguably be given as the max of
800 and the min of 200 respectively. A rich discussion about real results vs. computed results can
ensue. It also stands to reason since such a small percentage may fall beyond these scores.) Have
randomly selected groups present their answers and methods. Allow for discussion and critique. (Look
for evidence of MP1, MP3, and MP6.)
Ask groups to discuss how they would find a Z-score for a score that did not fall on one of the table
values. After they have formulated a strategy, ask them to find the Z –score for this SAT score,
rounded to the nearest hundredth: 550 (answer is (560-514)/114  .40). Circulate and help groups that
do not have a correct answer. Some groups may use linear extrapolation to estimate their value. This
will produce a less accurate answer as the normal curve is not linear. This will then need to be
addressed. Display the following scores and ask groups to find their Z-scores (answers are here in red.)
SAT SCORE 700
514
300
420
750
Z-SCORE 1.63
0
-1.88
-.82
2.07
Offer more practice if the answers do not show good comprehension.
(Look for evidence of MP6 and MP7.)
Explain that there is a table of percentages that go with Z-scores that covers MANY more Z-scores than
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: Z-Tables
Date: _____________
Teacher(s): ____________________
Course: Common Core Algebra II, Unit 6
Start/end times: _________________________
the thirteen Z-scores table the groups have completed. It is a two page table with positive Z-scores on
one page and negative Z-scores on the other. Hand out tables to the groups (Z-table on Wiki). Have
groups find the percentile score of each SAT score and subsequent Z-score found in part 6, rounded to a
whole number.
Percentile Scores 94
50
3
21
98
Offer more practice if the answers do not show good comprehension.
(Look for evidence of MP6 and MP7.)
Evidence of Success: What exactly do I expect students to be able to do by the end of the lesson, and how will I
measure student success? That is, deliberate consideration of what performances will convince you (and any outside
observer) that your students have developed a deepened and conceptual understanding.
Given a set of normally distributed data, students will be able to construct a frequency histogram.
Given a set of normally distributed data, students will discover and state the empirical rule about the data.
Given a mean and a standard deviation, students will move between Z-scores, raw scores, percentile scores.
Notes and Nuances: Vocabulary, connections, anticipated misconceptions (and how they will be addressed), etc.
normal curve
frequency histogram
mean
Standard deviation
Z-score
percentile score
Resources: What materials or resources are essential
for students to successfully complete the lesson tasks or
activities?
Homework: Exactly what follow-up homework tasks,
problems, and/or exercises will be assigned upon the
completion of the lesson?
TI-84 Calculator
Resource sheets on wiki:
(Z) Table of Percentile Scores
50 Students’ SAT scores
Empirical Rule Illustrations
Z-table
Students may need more practice with converting raw
scores to Z-scores and Z-scores to percentile scores. This
type of practice and be found online or in statistics texts
Lesson Reflections: How do you know that you were effective? What questions, connected to the lesson
standards/objectives and evidence of success, will you use to reflect on the effectiveness of this lesson?
Can students fluently move between percentile scores, Z-scores and raw scores.
Do students have a sense of the meaning of the relationship between the three scores?
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product
under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.