Download Math 7 Unit 4 – Inferences (4 weeks) Unit Overview: In this unit

Math 7 Unit 4 – Inferences
(4 weeks)
Unit Overview: In this unit, students use data to make inferences about populations. In Grade 6, students build on the knowledge
and experiences in data analysis developed in earlier grades (see K-3 Categorical Data Progression and Grades 2-5 Measurement
Progression). They develop a deeper understanding of variability and more precise descriptions of data distributions, using numerical
measures of center and spread, and terms such as cluster, peak, gap, symmetry, skew, and outlier. They begin to use histograms
and box plots to represent and analyze data distributions. As in earlier grades, students view statistical reasoning as a four-step
investigative process:

Formulate questions that can be answered with data

Design and use a plan to collect relevant data

Analyze the data with appropriate methods

Interpret results and draw valid conclusions from the data that relate to the questions posed.
Such investigations involve making sense of practical problems by turning them into statistical investigations (MP1); moving from
context to abstraction and back to context (MP2); repeating the process of statistical reasoning in a variety of contexts (MP8).
In Grade 7, students move from concentrating on analysis of data to production of data, understanding that good answers to
statistical questions depend upon a good plan for collecting data relevant to the questions of interest. Because statistically sound
data production is based on random sampling, a probabilistic concept, students must develop some knowledge of probability before
launching into sampling. Their introduction to probability is based on seeing probabilities of chance events as long-run relative
frequencies of their occurrence, and many opportunities to develop the connection between theoretical probability models and
empirical probability approximations. This connection forms the basis of statistical inference.
With random sampling as the key to collecting good data, students begin to differentiate between the variability in a sample and the
variability inherent in a statistic computed from a sample when samples are repeatedly selected from the same population. This
understanding of variability allows them to make rational decisions, say, about how different a proportion of “successes” in a sample
is likely to be from the proportion of “successes” in the population or whether medians of samples from two populations provide
convincing evidence that the medians of the two populations also differ.
In grade 7, students’ statistical topics and investigations have dealt with univariate data, e.g., collections of counts or measurements
of one characteristic. These experiences will lay the foundation for them to study the association between two variables in linear
functions in grade 8, extending their understanding of “cluster” and “outlier” from univariate data to bivariate data.
Content Standards:
Use random sampling to draw inferences about a population.
MCC.7.SP.1 Understand that statistics can be used to gain information about a population by examining a sample of the population;
generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that
random sampling tends to produce representative samples and support valid inferences.
MCC.7.SP.2 Use data from a random sample to draw inferences about a population with an unknown characteristic of interest.
Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example,
estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based
on randomly sampled survey data. Gauge how far off the estimate or prediction might be. (ITBS)
Draw informal comparative inferences about two populations.
MCC.7.SP.3 Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring
the difference between the centers by expressing it as a multiple of a measure of variability. (ITBS)
MCC.7.SP.4 Use measures of center and measures of variability for numerical data from random samples to draw informal
comparative inferences about two populations. (ITBS)
Standards for Mathematical Practice:
3. Construct viable arguments and critique the reasoning of others.
5. Use appropriate tools strategically.
Diagnostic: Prerequisite Assessment
Standards for Mathematical Practice (3, 5)
EQ: How do you know if you have a convincing argument? (MP3) What makes a tool the “best” tool for the task? (MP5)
Learning Targets:
I can …
 understand and use stated assumptions, definitions, and previously established results in constructing arguments. (MP3)
 make conjectures and build a logical progression of statements to explore the truth of conjectures. (MP3)
 analyze situations by breaking them into cases, and can recognize and use counterexamples. (MP3)
 justify my conclusions, communicate them to others, and respond to the arguments of others. (MP3)
 reason inductively about data, making plausible arguments that take into account the context from which the data arose. (MP3)
 compare the effectiveness of two plausible arguments. (MP3)
 distinguish correct logic or reasoning from that which is flawed, and-if there is a flaw in an argument-explain what it is. (MP3)
 construct arguments using concrete referents such as objects, drawings, and diagrams. (MP3)
 listen or read the arguments of others. (MP3)
 decide whether an argument of another makes sense. (MP3)
 ask useful questions to clarify or improve the arguments. (MP3)
 consider the available tools when solving a mathematical problem. (MP5)
 make sound decisions about when each of these familiar tools might be helpful, recognizing both the insight to be gained and
their limitations. (MP5)
 detect possible errors by strategically using estimation and other mathematical knowledge. (MP5)
 explain how technology can enable me to visualize the results of varying assumptions, explore consequences, and compare
predictions with data after making mathematical models. (MP5)
 use technological tools to explore and deepen their understanding of concepts. (MP5)
Concept Overview:
MP3 Construct viable arguments and critique the reasoning of others.
In this unit, students construct arguments using verbal or written explanations accompanied by graphs, tables, and other data
displays (i.e. box plots, dot plots, histograms, etc.). They further refine their mathematical communication skills through
mathematical discussions in which they critically evaluate their own thinking and the thinking of other students. They pose questions
like, “How did you get that?”, “Why is that true?”, “Does that always work?” They explain their thinking to others and respond to
others’ thinking.
MP5 Use appropriate tools strategically.
Students consider available tools (including estimation and technology) when solving a mathematical problem and decide when
certain tools might be helpful. For instance, students may decide to represent similar data sets using dot plots with the same scale to
visually compare the center and variability of the data. Students might use physical objects or applets to generate probability data
and use graphing calculators or spreadsheets to manage and represent data in different forms.
Resources:
MP3 Inside Mathematics Website
MP5 Inside Mathematics Website
Random Sampling
Use random sampling to draw inferences about a population.
MCC.7.SP.1 Understand that statistics can be used to gain information about a population by examining a sample of the population;
generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that
random sampling tends to produce representative samples and support valid inferences.
MCC.7.SP.2 Use data from a random sample to draw inferences about a population with an unknown characteristic of interest.
Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example,
estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based
on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
EQ: How can random sampling methods be used to gain information about a population?
Learning Targets:
I can …
 make generalizations from statistical data about a population sample. (SP1)
 explain how a random sample increases the likelihood of obtaining a representative sample of a population. (SP1)
 answer questions related to sample size and validity―for example: How large is a large enough sample size? What makes a
sample valid? (SP1)
 make inferences about a population based on a sample. (SP2)
 make reasonable arguments about whether or not conclusions drawn from a sample are valid. (SP2)
Concept Overview: In earlier grades students have been using data, both categorical and measurement, to answer simple statistical
questions, but have paid little attention to how the data were selected.
 A primary focus for Grade 7 is the process of selecting a random sample, and the value of doing so. If three students are to
be selected from the class for a special project, students recognize that a fair way to make the selection is to put all the
student names in a box, mix them up, and draw out three names “at random.” Individual students realize that they may not
get selected, but that each student has the same chance of being selected. In other words, random sampling is a fair way to
select a subset (a sample) of the set of interest (the population).

A statistic computed from a random sample, such as the mean of the sample, can be used as an estimate of that same
characteristic of the population from which the sample was selected. This estimate must be viewed with some degree of
caution because of the variability in both the population and sample data.

A basic tenet of statistical reasoning, then, is that random sampling allows results from a sample to be generalized to a
much larger body of data, namely, the population from which the sample was selected.
 “What proportion of students in the seventh grade of your school choose football as their favorite sport?” Students realize
that they do not have the time and energy to interview all seventh graders, so the next best way to get an answer is to
select a random sample of seventh graders and interview them on this issue.
 The sample proportion is the best estimate of the population proportion, but students realize that the two are not the same
and a different sample will give a slightly different estimate.
 In short, students realize that conclusions drawn from random samples generalize beyond the sample to the population
from which the sample was selected, but a sample statistic is only an estimate of a corresponding population parameter
and there will be some discrepancy between the two. Understanding variability in sampling allows the investigator to gauge
the expected size of that discrepancy.
The variability in samples can be studied by means of simulation.
 Students are to take a random sample of 50 seventh graders from a large population of seventh graders to estimate the
proportion having football as their favorite sport. Suppose, for the moment, that the true proportion is 60%, or 0.60.
 How much variation can be expected among the sample proportions?
 The scenario of selecting samples from this population can be simulated by constructing a “population” that has 60% red
chips and 40% blue chips, taking a sample of 50 chips from that population, recording the number of red chips, replacing
the sample in the population, and repeating the sampling process. (This can be done by hand or with the aid of technology,
or by a combination of the two.) Record the proportion of red chips in each sample and plot the results.
The dot plots in the margin show results for 200 such random samples of size 50 each. Note that the sample proportions pile up
around 0.60, but it is not too rare to see a sample proportion down around 0.45 or up around .0.75. Thus, we might expect a
Vocabulary
population: The entire set of items from which data can be selected.
sample: A selection from a population.
representative sample: A sample that is similar to the entire population
random sample: A sample chosen at random; A sample chosen from a population such that each data unit in the population has an
equal chance of being chosen each time
systematic sample: A sample chosen according to a rule or formula (Example: Survey every 10 th person through the door.)
convenience sample: A sample that is easiest to reach (Example: Survey the first 10 people through the door.)
voluntary-response sample: Members volunteer to be in the sample.
Sample Problem(s): Solutions to Sample Problems
Sample Problem 1:
Find three examples in the media that demonstrate the use of samples to make a statement about the population. (SP1)
Sample Problem 2:
Design a method of gathering a random sample from the student body to determine the favorite NFL team. (SP1)
Sample Problem 3:
The school food service wants to increase the number of students who eat hot lunch in the cafeteria. The student council has been
asked to conduct a survey of the student body to determine the students’ preferences for hot lunch. They have determined two
ways to do the survey. The two methods are listed below. Identify the type of sampling used in each survey option. Which survey
option should the student council use and why? (SP1)
a. Write all of the students’ names on cards and pull them out in a draw to determine who will complete the survey.
b. Survey the first 20 students that enter the lunch room.
Sample Problem 4:
Below is the data collected from two random samples of 100 students regarding student’s school lunch preference. Make at least
two inferences based on the results. (SP2)
Sample Problem 5:
Given the first name of all students in your grade. Predict the most common name in the U.S. for 7th graders. How good an estimate
do you think your sample provides? Explain your reasoning. (SP2)
Standard
MCC.7.SP.1
Topic
Choosing random
sampling
methods that
produce
representative
samples
Resources
Teacher Notes
Holt 3
Samples and Surveys
Section 9 – 1 pgs. 462 –- 463
Model Lesson: Samples and Variability
Pearson 2
Random Samples and Surveys
Section 11 – 4 pgs. 550 - 551
Math 7 Unit 4- Differentiation Strategies
 How Black is a Zebra? (U)
 Reese’s Pieces Sampling
Activity (S, K)
Literacy Strategy: Word Sort. Introduction to unit terms that
can be used informally to determine how much the students
already know.
Math 7_Unit 4- Good Questions
Cooperative Learning Strategy: Sampling Candy
Surveys and sampling
MCC.7.SP.2
Making
inferences about
a population
based on random
samples
Holt 3
Samples and Surveys
Section 9 – 1 pgs. 462 –- 463
Pearson 2
Random Samples and Surveys
Section 11 – 4 pgs. 550 – 551
Student Misconceptions: Many words have very technical
meanings in statistics such as bias, sample, statistics,
accuracy, correlation, and random. Students can have
difficulty with the language. "Rather studying statistics is
akin to studying foreign language, for students need lots of
practice to become comfortable using these terms correctly"
(Rossman, Chance, & Medina, 2006, p. 10).
 TI Graphing Calculator
Math 7_Unit 4- Good Questions
Activity: Interrogating Data
from Random Sampling
Math 7 Unit 4- Differentiation Strategies
Literacy Strategy: Word Sort
Informal Comparative Inference
Draw informal comparative inferences about two populations.
MCC.7.SP.3 Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring
the difference between the centers by expressing it as a multiple of a measure of variability.
MCC.7.SP.4 Use measures of center and measures of variability for numerical data from random samples to draw informal
comparative inferences about two populations.
EQ: How can statistics be used to compare two populations?
Learning Targets:
I can …
 use visual representations to compare two populations using measures of center (median, mean) and measures of variation
(range, quartiles, interquartile range), visual overlap, and mean absolute deviation. (SP3)
 compare the degree of visual overlap of two data plots. (SP3)
 describe what that difference between two visual representations of data means. (SP3)
 use measures of center and measures of variability for numerical data from random samples to make informal inferences about
two populations. (SP4)
Skill Targets:
Students will be able to create a…
 Box and whisker plot (or boxplot) from a data set.
 Stem and leaf plot (or stemplot) from a data set.
 Histogram from a data set.
 Line plot (or dotplot).
 Graph with two data sets.
*In this transition year, be aware that students may not have mastery with all graph types.
Concept Overview: To estimate a population mean or median, the best practice is to select a random sample from that population
and use the sample mean or median as the estimate, just as with proportions. But, many of the practical problems dealing with
measures of center are comparative in nature, as in comparing average scores on the first and second exam or comparing average
salaries between female and male employees of a firm. Such comparisons may involve making conjectures about population
parameters and constructing arguments based on data to support the conjectures (MP3).
If all measurements in a population are known, no sampling is necessary and data comparisons involve the calculated measures of
center. Even then, students should consider variability. The figures below show the female life expectancies for countries of Africa
and Europe. It is clear that Europe tends to have the higher life expectancies and a much higher median, but some African countries
are comparable to some of those in Europe. The mean and MAD for Africa are 53.6 and 9.5 years, respectively, whereas those for
Europe are 79.3 and 2.8 years. In Africa, it would not be rare to see a country in which female life expectancy is about ten years away
from the mean for the continent, but in Europe the life expectancy in most countries is within three years of the mean.
For random samples, students should understand that medians and means computed from samples will vary from sample to sample
and that making informed decisions based on such sample statistics requires some knowledge of the amount of variation to expect.
Just as for proportions, a good way to gain this knowledge is through simulation, beginning with a population of known structure.
The following examples are based on data compiled from nearly 200 middle school students in the Washington, DC area
participating in the Census at Schools Project. Responses to the question, “How many hours per week do you usually spend on
homework?” from a random sample of 10 female students and another of 10 male students from this population gave the results
plotted in the margin.
Females have a slightly higher median, but students should realize that there is too much variation in the sample data to conclude
that, in this population, females have a higher median homework time. An idea of how much variation to expect in samples of size
10 is needed.
Simulation to the rescue! Students can take multiple samples of size 10 from the Census of Schools data to see how much the
sample medians themselves tend to vary. The sample medians for 100 random samples of size 10 each, with 100 samples of males
and 100 samples of females, is shown in the margin. This plot shows that the sample medians vary much less than the homework
hours themselves and provides more convincing evidence that the female median homework hours is larger than that for males. Half
of the female sample medians are within one hour of 4 while half of the male sample medians are within half hour of 3, although
there is still overlap between the two groups.
A similar analysis based on sample means gave the results seen in the margin. Here, the overlap of the two distributions is more
severe and the evidence weaker for declaring that the females have higher mean study hours than males.
This is the students’ first experience with comparing two data sets. Students build on their understanding of graphs, mean, median,
Mean Absolute Deviation (M.A.D.) and interquartile range from 6th grade. Students understand that
 a full understanding of the data requires consideration of the measures of variability as well as mean or median,
 variability is responsible for the overlap of two data sets and that an increase in variability can increase the overlap, and
 median is paired with the interquartile range and mean is paired with the mean absolute deviation.
Vocabulary:
centers (also measures of center): The location where roughly half the values are below it and the other half above it (mean and
median are numbers used to describe the center of a distribution).
interquartile range: A measure of variation in a set of numerical data, the interquartile range is the distance
between the first and third quartiles of the data set. Example: For the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120},
the interquartile range is 15 – 6 = 9. See also: first quartile, third quartile.
mean absolute deviation: A measure of variation in a set of numerical data, computed by adding the distances between each data
value and the mean, then dividing by the number of data values. Example: For the data set {2, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the
mean absolute deviation is 20.
variabilities (also measures of variability): A mathematical determination of how much the performance of group or data set as a
whole deviates from the mean or median. (range, mean absolute deviation, interquartile range are numbers used to describe the
variability of a data set)
cluster: When data seems to be "gathered" around a particular value.
peak: A high point in the data.
gap: A space in a graph with no data points.
symmetry: A graph that is balanced around a line of symmetry. The data is evenly distributed above and below the median.
skew: When data has a "long tail" on one side or the other, meaning it is asymmetrical.
outlier: A value much great or much less than the others in a data set.
Sample Problem(s): Solutions to Sample Problems
Sample Problem 6:
Jason wants to compare the mean height of the players on his favorite basketball and soccer teams. He thinks the mean height of
the players on the basketball team will be greater but doesn’t know how much greater. He also wonders if the variability of heights
of the athletes is related to the sport they play. He thinks that there will be a greater variability in the heights of soccer players as
compared to basketball players. He used the rosters and player statistics from the team websites to generate the following lists.
Basketball Team – Height of Players in inches for 2010-2011 Season
75, 73, 76, 78, 79, 78, 79, 81, 80, 82, 81, 84, 82, 84, 80, 84
Soccer Team – Height of Players in inches for 2010
73, 73, 73, 72, 69, 76, 72, 73, 74, 70, 65, 71, 74, 76, 70, 72, 71, 74, 71, 74, 73, 67, 70, 72, 69, 78, 73, 76, 69
Demonstrate how Jason can use visual representation of the data to verify his conjectures? (SP3)
Solution:
To compare the data sets, Jason creates a two dot plots on the same scale. The shortest player is 65 inches and the tallest players
are 84 inches.
In looking at the distribution of the data, Jason observes that there is some overlap between the two data sets. Some players on
both teams have players between 73 and 78 inches tall. Jason decides to use the mean and mean absolute deviation to compare the
data sets. Jason sets up a table for each data set to help him with the calculations.
The mean height of the basketball players is 79.75 inches as compared to the mean height of the soccer players at 72.07 inches, a
difference of 7.68 inches.
The mean absolute deviation (MAD) is calculated by taking the mean of the absolute deviations for each data point. The difference
between each data point and the mean is recorded in the second column of the table. Jason used rounded values (80 inches for the
mean height of basketball players and 72 inches for the mean height of soccer players) to find the differences. The absolute
deviation, absolute value of the deviation, is recorded in the third column. The absolute deviations are summed and divided by the
number of data points in the set.
The mean absolute deviation is 2.53 inches for the basketball players and 2.14 for the soccer players. These values indicate
moderate variation in both data sets. There is slightly more variability in the height of the soccer players. The difference between the
heights of the teams is approximately 3 times the variability of the data sets (7.68 ÷ 2.53 = 3.04).
Soccer Players (n = 29)
Basketball Players (n = 16)
Height (in)
Deviation from Mean Absolute
Height (in)
Deviation from
Absolute Deviation
(in)
Deviation (in)
Mean (in)
(in)
65
-7
7
73
-7
7
67
-5
5
75
-5
5
69
-3
3
76
-4
4
69
-3
3
78
-2
2
69
-3
3
78
-2
2
70
-2
2
79
-1
1
70
-2
2
79
-1
1
70
-2
2
80
0
0
71
-1
1
80
0
0
71
-1
1
81
1
1
71
-1
1
81
1
1
72
0
0
82
2
2
72
0
0
82
2
2
72
0
0
84
4
4
72
0
0
84
4
4
73
+1
1
84
4
4
73
+1
1
73
+1
1
73
+1
1
73
+1
1
73
+1
1
74
+2
2
74
+2
2
74
+2
2
74
+2
2
76
+4
4
76
+4
4
76
+4
4
78
+6
6
Σ = 2090
Σ = 62
Σ = 1276
Σ = 40
Mean = 2090 ÷ 29 =72 inches
Mean = 1276 ÷ 16 =80 inches
MAD = 62 ÷ 29 = 2.14 inches
MAD = 40 ÷ 16 = 2.53 inches
Sample Problem 7:
The average temperature in City 1 is 70 degrees and in City 2 it is 80 degrees. The mean absolute deviation of City 1 is 5 degrees and
in City 2 it is 5 degrees. If dot plots were made of the data for each city, what inferences can be made about the visual
representations of the data? (SP3)
Sample Problem 8:
Measure the heights of the girls versus boys in your class. Calculate the measures of center and measures of variability for each
group. Describe the similarities and differences. (SP3)
Sample Problem 9:
Decide whether girls or boys take longer to get ready for school in the morning. Justify your answer using measures of center and
spread. (SP4)
Sample Problem 10:
The two data sets below depict random samples of the housing prices sold in the King River and Toby Ranch areas of Arizona. Based
on the prices below, which measure of center will provide the most accurate estimation of housing prices in Arizona? Explain your
reasoning. (SP4)
 King River area {1.2 million, 242000, 265500, 140000, 281000, 265000, 211000}
 Toby Ranch homes {5million, 154000, 250000, 250000, 200000, 160000, 190000}
State Populations: East vs. West (SP3-4)
Standard
MCC.7.SP.3
Topic
Making
comparative
inferences based
upon visual
representations
Resources
Holt 3
Organizing Data
Section 9 – 2 pgs. 467 – 468
Variability
Section 9 – 4 pgs. 476 – 478
Displaying Data
Section 9 – 5 pgs. 485 – 486
Teacher Notes
Student Misconceptions: Students think that the middle of
a box and whisker plot is the mean. Often students do not
relate the five number summary (minimum, first quartile,
median, third quartile, and maximum) to the marks in a box
and whisker plot. When a student makes a plot without
scale they will make the quartiles the same length. This does
not allow the student to see the shape of the distribution
(skewed, normal, etc.)
Math 7_Unit 4- Good Questions
Pearson 2
Box and Whisker Plots
Activity Lab 1 – 10b. pg. 58
Reporting Frequency
Section 11 – 1 pgs. 532 – 534
Bar Graphs and Line Graphs
Activity Lab 11 – 2a. pg. 537
Data Display
Section 11 – 2 pg. 539
Math 7 Unit 4- Differentiation Strategies
Cooperative Learning Strategy:
Ruler Drop Activity
Literacy Strategy: Word Sort.
Words With Multiple Meanings Template with instructions
and examples.
Concept-Metaphor-Explanation with instructions and
examples.
Stem-and-Leaf Plots
Section 11 – 3 pgs. 544 - 545
**These tasks also cover
MCC.7.SP.4.
 Olympic Gold Times (U)
 Population Trends (U)
 Migraines (Box Plots) (S, K)
 Migraines (Histograms) (S,
K)
Teacher Notes on Variability
MCC.7.SP.4
Making
comparative
inferences based
upon numerical
data
Video on Variability
(PP for Students)
Holt 3
Measures of Central Tendency
Section 9 – 3 pgs. 472 – 473
Pearson 2
Mean, Median, Mode and
Range
Section 1 – 10 pgs. 53 – 55
Describing Data
Model Lesson: Using Measures of Center to Compare Data
Student Misconceptions:
Students think of variability only in terms of the range of
data.
The idea that statistics come from a sample taken from a
larger population and is only representative of the
population is a common issue. Often the term "statistic" is
used to describe the summary analysis from an entire
Guided Problem Solving pgs.
558 - 559
 MAD Learning Task:




Shakespeare and Harry
Potter (GaDOE
Frameworks) (K, U)
MAD Learning Task
example
MAD Learning Task
example MAD computation
Model Lesson Plan
Learning Task: Travel Times
to Work (GaDOE
Frameworks) (K, U)
population. When comparing the summary data from two
populations comparisons are definite and the difference can
be stated as such. When the data is a statistic from a sample
taken from a larger population the comparison introduce
some uncertainty. People often want to have definitive
comparisons from data sets and this is simply not
statistically possible when dealing with samples.
Math 7_Unit 4- Good Questions
Math 7 Unit 4- Differentiation Strategies
Literacy Strategy: Word Sort.
Words With Multiple Meanings Template with instructions
and examples.
Concept-Metaphor-Explanation with instructions and
examples.
Password Game. Can be used as a review activity. The
template is structured for easy adaption to your students
needs. Words can be added as necessary.
RAFT. This is a link that will assist in creating unique RAFTs .
Unit 4 Summative Assessment On-level Select appropriate questions from 18-28 for your students. Add lines or spaces as
necessary. You may use the other questions as preparation for the test
Unit 4 Summative Assessment Advanced Select appropriate questions from 20-35 for your students. Add lines or spaces as
necessary. You may use the other questions as preparation for the test.