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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math III
(listing of included sources attached)
S.IC.1, IC.2
Understand and
evaluate random
processes underlying
statistical experiments.
Essential Understanding
Students need to know that data collection and the
analysis of the data influence most areas of our lives;
these analyses are what we call statistics and are
important to our health, wealth, and happiness,
when applied appropriately.
Understand statistics as a process for making inferences about
population parameters based on a random sample from that
population.
Students need to be able to identify whether a
particular statistical model is effective in a particular
context.
Decide if a specified model is consistent with results from a
given data-generating process, e.g., using simulation. For
example, a model says a spinning coin falls heads up with
probability 0.5. would a result of 5 tails in a row cause you to
question the model?
Students need to know that data can be distorted in
several ways; bad samples result from the use of
inappropriate methods to collect data and will bias
the results.
Extended Understanding
CCSSM Description
Decisions or predictions are often based on data—numbers in
context. These decisions or predictions would be easy if the data
always sent a clear message, but the message is often obscured
by variability. Statistics provides tools for describing variability in
data and for making informed decisions that take it into
account. This section will require students to determine the
correct mathematical model and to use the model appropriately
to solve problems.
I Can Statements
Columbus City Schools
Students should be provided opportunities to use
technology to make it possible to generate plots,
regression functions, and correlation coefficients, and
to simulate many possible outcomes in a short
amount of time.
Academic Vocabulary/
Language
biased, binomial distribution, bivariate,
categorical, cluster, confidence interval,
convenience, data, empirical rule, independence
test, inference, judgment, margin of error, mean,
measure of center, measures of spread, median,
mode, nonrandom samples, normal distribution,
null hypothesis, outlier, paired t-test, population,
population mean, population proportion,
purposive, p-value, qualitative, quantitative,
quota, random, sample, sample mean, sample
proportion, sample survey, significance,
simulation, simple, systematic, stratified, skew,
snowball, spread, standard deviation, treatment,
t-test, uniform, univariate, variance
Tier 2 Vocabulary
consistent, experiment, evaluate, process,
understand
Students should be provided opportunities to make
connections to functions and modeling. Functions
may be used to describe data; if the data suggest a
linear relationship, the relationship can be modeled
with a regression line, and its strength and direction
can be expressed through a correlation coefficient.
I can choose and use appropriate mathematics to analyze situations.
Clear Learning Targets Integrated Math III 2016-2017
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Instructional Strategies
Inferential statistics based on Normal probability models is a topic for Advanced Placement Statistics (e.g., t-tests). The idea here is that all students understand that
statistical decisions are made about populations (parameters in particular) based on a random sample taken from the population and the observed value of a sample
statistic (note that both words start with the letter “s”). A population parameter (note that both words start with the letter “p”) is a measure of some characteristic in the
population such as the population proportion of American voters who are in favor of some issue, or the population mean time it takes an Alka Seltzer tablet to dissolve. As
the statistical process is being mastered by students, it is instructive for them to investigate questions such as “If a coin spun five times produces five tails in a row, could
one conclude that the coin is biased toward tails?” One way a student might answer this is by building a model of 100 trials by experimentation or simulation of the number
of times a truly fair coin produces five tails in a row in five spins. If a truly fair coin produces five tails in five tosses 15 times out of 100 trials, then there is no reason to
doubt the fairness of the coin. If, however, getting five tails in five spins occurred only once in 100 trials, then one could conclude that the coin is biased toward tails (if the
coin in question actually landed five tails in five spins). A powerful tool for developing statistical models is the use of simulations. This allows the students to visualize the
model and apply their understanding of the statistical process. Provide opportunities for students to clearly distinguish between a population parameter which is a
constant, and a sample statistic which is a variable.
Common Misconceptions/Challenges
Students may believe: That population parameters and sample statistics are one in the same, e.g., that there is no difference between the population mean which is a
constant and the sample mean which is a variable. Making decisions is simply comparing the value of one observation of a sample statistic to the value of a population
parameter, not realizing that a distribution of the sample statistic needs to be created.
Common Core Support
Institute for Mathematics and Education Learning Progressions Narratives
High School Statistics and Probability
http://commoncoretools.me/wp-content/uploads/2012/06/ccss_progression_sp_hs_2012_04_21_bis.pdf
Ohio Department of Education Model Curriculum
http://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/High_School_Statistics-and-Probability_Model-Curriculum_March2015.pdf.aspx
Illustrative Mathematics
https://www.illustrativemathematics.org/blueprints/M3/5
Columbus City Schools
Clear Learning Targets Integrated Math III 2016-2017
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Textbook and Curriculum Resources
Integrated Math III, McGraw Hill
Chapter 10
Adventures with Mathematics: Statistics and Probability: Aligned With the Common Core State Standards, MCTM, 2012
Ants on the Loose, p. 14
Forecasting the Weather, p.20
When Bambi Hits the Blacktop, p. 22
Scrabble Express Vowel Bias, p. 28
The Magical Number 7, p. 36
The Definite Activity, p. 38
Which Gum Lasts Longer, p. 40
An A-MAZE-ING Comparison, p. 42
The Spelling Bee, p. 44
Archaeological Sampling, p. 46
Prior Knowledge
Future Learning
The four-step statistical process was introduced in Grade 6, with the recognition
of statistical questions. At the high school level, students need to become
proficient in all the steps of the statistical process. Using simulation to estimate
probabilities is a part of the Grade 7 curriculum as is initial understanding of
using random sampling to draw inferences about a population.
Next lessons will include making inference and justifying conclusions
from sample surveys, experiments, and observational studies.
Columbus City Schools
Clear Learning Targets Integrated Math III 2016-2017
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Performance Assessments/Tasks
Click on the links below to access performance tasks.
NRICH
http://nrich.maths.org/public/search.php?search=statistics
Illustrative Mathematics
Interpreting Data: Muddying the Waters
http://map.mathshell.org/download.php?fileid=1774
Career Connections
Students can explore the concepts of direct marketing, a marketing database, and a sales promotion as described in the High School Operations Research
Modules (http://hsor.org/modules.cfm?name=Gamz_Inc). Use the provided case studies to lead a discussion on how this content is critical to tasks
performed across various career fields (e.g., business, marketing, finance). Students will use the discussion to guide their research of related careers for
developing future career goals.
Columbus City Schools
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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math III
(listing of included sources attached)
S.IC.3, S.IC.4, S.IC.5,
S.IC.6
observational studies.
Make inferences and
justify conclusions from
sample surveys,
experiments, and
Recognize the purposes of and differences among sample surveys,
experiments, and observational studies; explain how randomization relates to
each other.
Use data from a sample survey to estimate a population mean or proportion:
develop a margin of error through the use of simulation models for random
sampling.
Use data from a randomized experiment to compare two treatments; use
simulations to decide if differences between parameters are significant.
Evaluate reports based on data.
CCSSM Description
Statistical inference refers simply to the process of drawing conclusions from
statistical data. Students need to be able to identify whether a particular
model is effective in a particular context. Data are gathered, displayed,
summarized, examined, and interpreted to discover patterns and deviations
from patterns. Which statistics to compare, which plots to use, and what the
results of a comparison might mean, depend on the question to be
investigated and the real-life actions to be taken. The conditions under which
data are collected are important in drawing conclusions from the data. In
critically reviewing uses of statistics in public media and other reports, it is
important to consider the study design, how the data were gathered, and the
analyses employed as well as the data summaries and the conclusions drawn.
I Can Statements
Essential Understanding
Students should know and understand the
different methods of data collection, specifically
the difference between an observational study
and a controlled experiment, and know the
appropriate use for each.
Students should be able to choose and use
appropriate mathematics to analyze situations;
students should be able to determine the correct
mathematical model and use the model to solve
problems.
Extended Understanding
While virtually all aspects of our human
experience have benefited from a responsible
use of statistics, data can be presented in ways
that are misleading. At times, this occurs
through carelessness or ignorance but other
times it is designed to be deceptive for the
purpose of obscuring unfavorable data or
accentuating data, which supports a certain
point of view. Provide an opportunity for
students to use their skills and identify
information found in magazines, newspapers, on
television, and via the Internet that consumers
should all be cautious of due to potential
misuses and abuses of statistical data.
Academic Vocabulary/
Language
biased, binomial distribution, bivariate,
categorical, cluster, confidence interval,
convenience, data, empirical rule,
independence test, inference, judgment,
margin of error, mean, measure of center,
measures of spread, median, mode,
nonrandom samples, normal distribution, null
hypothesis, outlier, paired t-test, population,
population mean, population proportion,
purposive, p-value, qualitative, quantitative,
quota, random, sample, sample mean, sample
proportion, sample survey, significance,
simulation, simple, systematic, stratified, skew,
snowball, spread, standard deviation,
treatment, t-test, uniform, univariate, variance
Tier 2 Vocabulary
consistent, experiment, evaluate, process,
understand
I can choose and use appropriate mathematics to analyze situations.
I can estimate a sample mean or sample proportion given data from a sample survey; I can
estimate the population value.
I can determine the correct mathematical model and use the model to solve problems.
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Clear Learning Targets Integrated Math III 2016-2017
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Instructional Strategies
This cluster is designed to bring the four-step statistical process (GAISE model) to life and help students understand how statistical decisions are made. The mastery of this cluster is
fundamental to the goal of creating a statistically literate citizenry. Students will need to use all of the data analysis, statistics, and probability concepts covered to date to develop a deeper
understanding of inferential reasoning. Students learn to devise plans for collecting data through the three primary methods of data production: surveys, observational studies, and
experiments. Randomization plays various key roles in these methods. Emphasize that randomization is not a haphazard procedure, and that it requires careful implementation to avoid
biasing the analysis. In surveys, the sample selected from a population needs to be representative; taking a random sample is generally what is done to satisfy this requirement. In
observational studies, the sample needs to be representative of the population as a whole to enable generalization from sample to population. The best way to satisfy this is to use random
selection in choosing the sample. In comparative experiments between two groups, random assignment of the treatments to the subjects is essential to avoid damaging problems when
separating the effects of the treatments from the effects of some other variable, called confounding. In many cases, it takes a lot of thought to be sure that the method of randomization
correctly produces data that will reflect that which is being analyzed. For example, in a two-treatment randomized experiment in which it is desired to have the same number of subjects in
each treatment group, having each subject toss a coin where Heads assigns the subject to treatment A and Tails assigned the subject to treatment B will not produce the desired random
assignment of equal-size groups. The advantage that experiments have over surveys and observational studies is that one can establish causality with experiments.
Also addressed with these standards estimation of the population proportion parameter and the population mean parameter. Data need not come from just a survey to cover this topic. A
margin-of-error formula cannot be developed through simulation, but students can discover that as the sample size is increased, the empirical distribution of the sample proportion and the
sample mean tend toward a certain shape (the Normal distribution), and the standard error of the statistics decreases (i.e. the variation) in the models becomes smaller. The actual formulas
will need to be stated.
Finally, this cluster of standards addresses testing whether some characteristic of two paired or independent groups is the same or different by the use of resampling techniques. Conclusions
are based on the concept of p-value. Resampling procedures can begin by hand but typically will require technology to gather enough observations for which a p-value calculation will be
meaningful. Use a variety of devices as appropriate to carry out simulations: number cubes, cards, random digit tables, graphing calculators, computer programs.
Common Misconceptions/Challenges
Students may believe:
That collecting data is easy; asking friends for their opinions is fine in determining what everyone thinks.
That causal effect can be drawn in surveys and observational studies, instead of understanding that causality is in fact a property of experiments.
That inference from sample to population can be done only in experiments. They should see that inference can be done in sampling and observational studies if data are
collected through a random process
Common Core Support
Institute for Mathematics and Education Learning Progressions Narratives
High School Statistics and Probability
http://commoncoretools.me/wp-content/uploads/2012/06/ccss_progression_sp_hs_2012_04_21_bis.pdf
Ohio Department of Education Model Curriculum
http://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/High_School_Statistics-and-Probability_Model-Curriculum_March2015.pdf.aspx
Illustrative Mathematics
https://www.illustrativemathematics.org/blueprints/M3/5
Columbus City Schools
Clear Learning Targets Integrated Math III 2016-2017
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Textbook and Curriculum Resources
http://www.virtualnerd.com/tutorials/?id=Alg1_9_2_17Integrated Math III, McGraw Hill
Chapter 10
Adventures with Mathematics: Statistics and Probability: Aligned With the Common Core State Standards, MCTM, 2012
Forecasting the Weather, p. 20
When Bambi Hits the Blacktop, p. 22
The Blob, p. 25
Scrabble Express Vowel Bias, p. 28
The State of Drunk Driving, p. 30
The Magical Number 7, p. 36
The Definite Activity, p. 38
Which Gum Lasts Longer?, p. 40
An A-MAZE_ING Comparison, p. 42
The Spelling Bee, p. 44
Archaeological Sampling, p. 46
Prior Knowledge
Future Learning
The four-step statistical process was introduced in middle school, with the first
step likely more often generated by teachers than students. At the high school
level, students need to become proficient in the first step of generating
meaningful questions, as well as designing a plan to collect their data using the
three primary methods: surveys, observational studies, and experiments. Using
simulation to estimate probabilities is a part of the Grade 7 curriculum, as is
introductory understanding of using random sampling to draw inferences about
a population
Next lessons will include interpreting categorical and quantitative data.
Columbus City Schools
Clear Learning Targets Integrated Math III 2016-2017
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Performance Assessments/Tasks
Click on the links below to access performance tasks.
NRICH
http://nrich.maths.org/public/search.php?search=statistics
Illustrative Mathematics
Interpreting Data: Muddying the Waters
http://map.mathshell.org/download.php?fileid=1774
Interpreting and Using Data: Testing a New Product
http://map.mathshell.org/download.php?fileid=1704
Career Connections
Statistics is the study of data organization to provide specific information and for measuring and determining uncertainty and probability. The discipline can
apply to problems in economics, engineering, education, biology and sports. Some of its uses include sports information for baseball players, calculations for
car insurance premiums and analyses of business efficiency.
Columbus City Schools
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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math III
(listing of included sources attached)
S.ID.4
measurement variable.
Summarize,
represent, and
interpret data on a
single count or
Use the mean and standard deviation of a data set to fit it to the
normal distribution and to estimate population percentages.
Recognize that there are data sets which such a procedure is not
appropriate. Use calculators, spreadsheets, and tables to
estimate areas under the normal curve.
CCSSM Description
Students will calculate and use summary statistics such as mean,
median, range, lower and upper quartile, interquartile range and
standard deviation to help describe the shape of data. The processes
by which mean and median are calculated have been previously taught;
however, students have not been introduced to standard deviation,
and must understand the process behind the calculation. Technology
should be used to calculate the standard deviation. Students will build
on their understanding of these calculations to comment on possible
outliers in a data set and to make well-informed decisions about the
best summary statistics to represent given data.
Essential Understanding
Students should be able to recognize that there are
data sets for which such a procedure is not
appropriate.
Academic Vocabulary/
Language
Students should use summary statistics and/or
graphical representations to write critical analyses of
a situation within the context of the given data.
biased, binomial distribution, bivariate,
categorical, cluster, confidence interval,
convenience, empirical rule, independence test,
inference, judgment, margin of error, mean,
measure of center, measures of spread, median,
mode, nonrandom samples, normal distribution,
null hypothesis, outlier, paired t-test, population,
population mean, population proportion,
purposive, p-value, qualitative, quantitative,
quota, random, sample, sample mean, sample
proportion, sample survey, significance,
simulation, simple, systematic, stratified, skew,
snowball, spread, standard deviation, treatment,
t-test, uniform, unimodal distribution, univariate,
variance
Extended Understanding
Tier 2 Vocabulary
Student should know that when data is notably
skewed or when meaningful outliers are present, the
median should be used to describe the distribution.
Students should know that the mean and standard
deviation should be used to describe unimodal and
symmetric data.
Opportunities should be provided for students to
work through the statistical process. Teachers and
students should make extensive use of resources in
order to perfect; make use global web resources for
projects.
consistent, experiment, evaluate, process,
understand
I can use the normal distribution to make estimates of frequencies (which can be expressed as
probabilities.
I can recognize that only some data are well described by a normal distribution.
I Can Statements
I can describe the characteristics of a normal distribution.
I can use a calculator, spreadsheet, and table to estimate areas under the normal curve
I can use the mean and standard deviation of a data set to fit it to a normal distribution.
I can use normal distribution to estimate population percentages.
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Clear Learning Targets Integrated Math III 2016-2017
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Instructional Strategies
It is helpful for students to understand that a statistical process is a problem-solving process consisting of four steps: formulating a question that can be answered by data;
designing and implementing a plan that collects appropriate data; analyzing the data by graphical and/or numerical methods; and interpreting the analysis in the context of
the original question. Opportunities should be provided for students to work through the statistical process. The richer the question formulated, the more interesting is the
process. Teachers and students should make extensive use of resources to perfect this very important first step. Global web resources can inspire projects. Although this
domain addresses both categorical and quantitative data, there is no reference to categorical data. This would be a good place to discuss graphs for one categorical variable
(bar graph, pie graph) and measure of center (mode). Have students practice their understanding of the different types of graphs for categorical and numerical variables by
constructing statistical posters. Note that a bar graph for categorical data may have frequency on the vertical (student’s pizza preferences) or measurement on the vertical
(radish root growth over time - days). Measures of center and spread for data sets without outliers are the mean and standard deviation, whereas median and interquartile
ranges are better measures for data sets with outliers. Introduce the formula of standard deviation by reviewing the previously learned MAD (mean absolute deviation).
The MAD is very intuitive and gives a solid foundation for developing the more complicated standard deviation measure. Informally observing the extent to which two
boxplots or two dotplots overlap begins the discussion of drawing inferential conclusions. Don’t shortcut this observation in comparing two data sets. As histograms for
various data sets are drawn, common shapes appear. To characterize the shapes, curves are sketched through the midpoints of the tops of the histogram’s rectangles. Of
particular importance is a symmetric unimodal curve that has specific areas within one, two, and three standard deviations of its mean. It is called the Normal distribution
and students need to be able to find areas (probabilities) for various events using tables or a graphing calculator.
Common Misconceptions/Challenges
Students may believe:
That a bar graph and a histogram are the same. A bar graph is appropriate when the horizontal axis has categories and the vertical axis is labeled by either frequency (e.g.,
book titles on the horizontal and number of students who like the respective books on the vertical) or measurement of some numerical variable (e.g., days of the week on
the horizontal and median length of root growth of radish seeds on the vertical). A histogram has units of measurement of a numerical variable on the horizontal (e.g., ages
with intervals of equal length).
That the lengths of the intervals of a boxplot (min, Q1), (Q1, Q2), (Q2, Q3), (Q3, max) are related to the number of subjects in each interval. Students should understand
that each interval theoretically contains one-fourth of the total number of subjects. Sketching an accompanying histogram and constructing a live boxplot may help in
alleviating this misconception.
That all bell-shaped curves are normal distributions. For a bell-shaped curve to be Normal, there needs to be 68% of the distribution within one standard deviation of the
mean, 95% within two, and 99.7% within three standard deviations.
Common Core Support
Institute for Mathematics and Education Learning Progressions Narratives
High School Statistics and Probability
http://commoncoretools.me/wp-content/uploads/2012/06/ccss_progression_sp_hs_2012_04_21_bis.pdf
Ohio Department of Education Model Curriculum
http://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/High_School_Statistics-and-Probability_Model-Curriculum_March2015.pdf.aspx
Illustrative Mathematics
https://www.illustrativemathematics.org/blueprints/M3/5
Columbus City Schools
Clear Learning Targets Integrated Math III 2016-2017
10
Textbook and Curriculum Resources
http://www.virtualnerd.com/tutorials/?id=Alg1_9_2_17Integrated Math III, McGraw Hill
Chapter 10
Adventures with Mathematics: Statistics and Probability: Aligned With the Common Core State Standards, MCTM, 2012
The Definite Activity, p. 38
The Spelling Bee, p. 44
Prior Knowledge
Future Learning
The four-step statistical process was introduced in Grade 6, with the recognition
of statistical questions. At the high school level, students need to become
proficient in the first step of generating meaningful questions.
The next focus of study includes interpreting the structure of expressions
in a variety of functions.
Columbus City Schools
Clear Learning Targets Integrated Math III 2016-2017
11
Performance Assessments/Tasks
Click on the links below to access performance tasks.
NRICH
http://nrich.maths.org/public/search.php?search=statistics
Illustrative Mathematics
Representing Data with Frequency Graphs
http://map.mathshell.org/download.php?fileid=1780
Representing Data with Box Plots
http://map.mathshell.org/download.php?fileid=1782
Career Connections
Statistics is the study of data organization to provide specific information and for measuring and determining uncertainty and probability. The discipline can
apply to problems in economics, engineering, education, biology and sports. Some of its uses include sports information for baseball players, calculations for
car insurance premiums and analyses of business efficiency. Other: Program Analyst, Data Specialist.
Columbus City Schools
Clear Learning Targets Integrated Math III 2016-2017
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Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics III
S.MD.6-7
6. (+) Use probabilities to
make fair decisions (e.g.,
drawing by lots, using a
random number generator).
7. (+) Analyze decisions and
strategies using probability
concepts (e.g., product
testing, medical testing,
pulling a hockey goalie at the
end of a game).
I Can Statements




Essential Understanding
- Students will understand the
concept of fairness as it applies to
probability.
Extended Understanding
- Students will be able to analyze
decisions and strategies using
concepts of probability.
Academic
Vocabulary/Language
- theoretical probability
- experimental probability
- random
Tier 2 Vocabulary
- use
- analyze
- fair
Common Misconceptions and Challenges
I can compute Theoretical and Experimental
Students may believe that probabilities and expected values are not
Probabilities.
useful in making decisions that affect one’s life. Students need to see
I can use probabilities to make fair decisions (e.g. drawing that these are not merely textbook exercises.
by lots, using random number generator).
I can recall prior understandings of probability.
I can analyze decision and strategies using probability
concepts (e.g., product testing, medical testing, pulling a
hockey goalie at the end of a game).
Columbus City Schools
Clear Learning Targets Integrated Math III 2016-2017
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Instructional Strategies
Columbus City Schools
Clear Learning Targets Integrated Math III 2016-2017
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Textbook and Curriculum Resources
McGraw-Hill: Integrated Math II
Chapter 13-3, 13-4


S-CP, S-MD But mango is my favorite…
S-MD Fred's Fun Factory
http://ccssmath.org/?s=md.6
http://ccssmath.org/?s=md.7
Career Connections
Computer and mathematical occupations
Actuaries
Computer programmers
Computer software engineers
Mathematicians
Statisticians
Architects, surveyors, and cartographers
Surveyors, cartographers, photogrammetrists, and surveying technicians
Food preparation and serving related occupations
Chefs, cooks, and food preparation workers
Personal care and service occupations
Animal care and service workers
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Clear Learning Targets Integrated Math III 2016-2017
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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math III
(listing of included sources attached)
A.SSE.2
Interpret the
structure of
expressions.
Use the structure of an
expression to identify ways to rewrite it. For example, see x4 –y4
as (x2 )2 - (y2 )2, thus recognizing it as difference of squares than be
be factored as (x 2 – y2) (x2 + y2).
CCSSM Description
Reading an expression with comprehension involves analysis of its
underlying structure. This may suggest a different but equivalent way of
writing the expression that exhibits some different aspect of its meaning.
Algebraic manipulations are governed by the properties of operations and
exponents, and the conventions of algebraic notation.
Essential Understanding
Students will need to be able to rewrite algebraic
expressions in different equivalent forms such as
factoring or combining like terms.
Students will need to be able to use factoring
techniques such as common factors, grouping, the
difference of two squares, the sum or difference of two
cubes, or a combination of methods to factor
completely.
Students will need to be able to simplify expressions
including combining like terms, using the distributive
property and other operations with polynomials.
Extended Understanding
Academic Vocabulary/
Language
combining like terms, common factors,
difference of squares, difference of two cubes,
equivalent, factoring, factor completely,
grouping, sum of two cubes
Tier 2 Vocabulary
analysis, manipulations, properties, rewrite,
structure
Students can use spreadsheets or a computer algebra
system (CAS) to experiment with algebraic expressions,
perform complicated algebraic manipulations, to better
understand how algebraic manipulations behave.
I can identify ways to rewrite expressions, such as difference of squares, factoring out a common
monomial, regrouping, etc.
I can identify ways to rewrite expressions based on the structure of the expression.
I Can Statements
I can use the structure of an expression to identify ways to rewrite it.
I can classify expression by structure and develop strategies to assist in classification.
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Clear Learning Targets Integrated Math III 2016-2017
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Instructional Strategies
Extending beyond simplifying an expression, this cluster addresses interpretation of the components in an algebraic expression. A student should recognize that in the
expression 2x + 1, “2” is the coefficient, “2” and “x” are factors, and “1” is a constant, as well as “2x” and “1” being terms of the binomial expression. Development and proper
use of mathematical language is an important building block for future content. Using real-world context examples, the nature of algebraic expressions can be explored. For
example, suppose the cost of cell phone service for a month is represented by the expression 0.40s + 12.95. Students can analyze how the coefficient of 0.40 represents the cost
of one minute (40¢), while the constant of 12.95 represents a fixed, monthly fee, and s stands for the number of cell phone minutes used in the month. Similar real-world
examples, such as tax rates, can also be used to explore the meaning of expressions. Factoring by grouping is another example of how students might analyze the structure of an
expression. To factor 3x(x – 5) + 2(x – 5), students should recognize that the “x – 5” is common to both expressions being added, so it simplifies to (3x + 2)(x – 5). Students
should become comfortable with rewriting expressions in a variety of ways until a structure emerges. Have students create their own expressions that meet specific criteria
(e.g., number of terms factorable, difference of two squares, etc.) and verbalize how they can be written and rewritten in different forms. Additionally, pair/group students to
share their expressions and rewrite one another’s expressions.
Common Misconceptions/Challenges
Students may believe that the use of algebraic expressions is merely the abstract manipulation of symbols. Use of real world context examples to demonstrate the meaning of
the parts of algebraic expressions is needed to counter this misconception. Students may also believe that an expression cannot be factored because it does not fit into a form
they recognize. They need help with reorganizing the terms until structures become evident.
Technology may be useful to help a student recognize that two different expressions represent the same relationship. For example, since (x – y)(x + y) can be rewritten as x2 –
y2 , they can put both expressions into a graphing calculator (or spreadsheet) and have it generate two tables (or two columns of one table), displaying the same output values
for each expression.
Common Core Support
Institute for Mathematics and Education Learning Progressions Narratives: Common Core Tools
http://commoncoretools.me/wp-content/uploads/2012/06/ccss_progression_sp_hs_2012_04_21_bis.pdf
http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_algebra_2013_07_03.pdf
Ohio Department of Education Model Curriculum
http://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/High_School_Algebra_Model_Curriculum_March2015.pdf.aspx
Ohio’s New Learning Standards
http://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/Math-Standards.pdf.aspx
Illustrative Mathematics
https://www.illustrativemathematics.org/blueprints/M3/5
Columbus City Schools
Clear Learning Targets Integrated Math III 2016-2017
17
Textbook and Curriculum Resources
http://www.virtualnerd.com/tutorials/?id=Alg1_9_2_17Integrated Math III, McGraw Hill
Chapter 0
Chapter 1
Problem-Based Tasks for Mathematics II, Walch Education, 2013
Factoring: On the Shelf, pp. 75-82
Completing the Square, pp. Curve Ball pp. 79-82
Solving Quadratic Inequalities: Dancing for Charity, pp. 86-89
Algebra I Station Activities for Common Core State Standards, Walch Education, 2013
Factoring Polynomials, pp. 1-6
Prior Knowledge
Future Learning
An introduction to the use of variable expressions and their meaning, as well as
the use of variables and expressions in real-life situations is included in the
Expressions and Equations Domain of Grade 7.
The next focus on learning will include creating equations that describe
numbers or relationships.
Columbus City Schools
Clear Learning Targets Integrated Math III 2016-2017
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Performance Assessments/Tasks
Click on the links below to access performance tasks.
Illustrative Mathematics
Mathematics Assessment Program College and Career Readiness Mathematics
http://map.mathshell.org/download.php?fileid=832
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 6: Algebraic Expressions—The Distributive Property Lesson 7: Algebraic Expressions—The Commutative and Associative Properties Lesson 8: Adding and
Subtracting Polynomials Lesson 9: Multiplying Polynomials
https://www.engageny.org/sites/default/files/resource/attachments/algebra-i-m1-teacher-materials.pdf
Career Connections
Students will evaluate cell phone plans across multiple providers to identify one that is the most cost effective for their expected use. They will consider the
fixed and variable costs to support their decision (e.g., unlimited plans, cost per unit, insurance protection, activation and cancellation fees). In collecting data
related to the cost of service, students will research the employment opportunities available across the telecommunication companies via website, phone, and
email.
Columbus City Schools
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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math III
(listing of included sources attached)
A.CED.1, A.CED.2
numbers or relationships.
Create
equations
that
describe
Create equations and inequalities in one variable and use
them to solve problems. Include equations arising from
linear and quadratics functions, and simple rational and
exponential functions.
Create equations in two or more variables to represent
relationships between quantities; graph equations on
coordinate axes with labels and scales.
CCSSM Description
An equation is a statement of equality between two
expressions, often viewed as a question asking for which
values of the variables the expressions on either side are in
fact equal. These values are the solutions to the equation.
Inequalities can be solved by reasoning about the properties
of inequality. Many, but not all, of the properties of equality
continue to hold for inequalities and can be useful in solving
them.
Essential Understanding
Students are expected to know how to solve all available types of
equations and inequalities, including root equations and
inequalities, in one variable.
Students are expected to know how: to describe the relationships
between the quantities in the problem (for example, how the
quantities are changing or growing with respect to each other);
express these relationships using mathematical operations to
create an appropriate equation or inequality to solve; compare
and contrast problems that can be solved by different types of
equations; Students are expected to know how to identify the
quantities in a mathematical problem or real world situation that
should be represented by distinct variables and describe what
quantities the variables represent; students will be expected to
know how to graph one or more created equations on a
coordinate axes with appropriate labels and scales.
Extended Understanding
Provide examples of real-world problems that can be solved by
writing an equation, and have students explore the graphs of the
equations on a graphing calculator to determine which parts of
the graph are relevant to the problem context.
Academic Vocabulary/ Language
absolute
value
function
coordinate
axes
cube root
function
equation
exponential
function
function
quadratic
inequality
graph
rational
function
relationship
linear
function
piecewise
function
quadratic
function
square root
function
variable
Tier 2 Vocabulary
appropriate
create
dependent
depict
describe
distinct
identify
independent
justify
label
represent
solve
I can solve quadratic equations in one variable; I can solve quadratic inequalities in one variable.
I can create quadratic equations and inequalities in one variable and use them to solve problems; I can create quadratic
equations and inequalities in one variable to model real-world situations.
I Can Statements
I can identify the quantities in a mathematical problem or real-world situation that should be represented by distinct
variables and describe what quantities the variables represent.
I can graph one or more created equation on a coordinate axes with appropriate labels and scales; I can determine
appropriate units for the labels and scale of a graph depicting the relationship between equations created in two or more
variables between equations created in two or more variables.
I can create at least two equations in two or more variables to represent relationships between quantities.
I can justify which quantities in a mathematical problem or real-world situation are dependent and independent of one
another and which operations represent those relationships.
Columbus City Schools
Clear Learning Targets Integrated Math III 2016-2017
I can determine appropriate units for the labels and scale of a graph depicting the relationship between
equations created in two or more variables
20
Instructional Strategies
Provide examples of real-world problems that can be modeled by writing an equation or inequality. Begin with simple equations and inequalities and build up to more complex equations in
two or more variables that may involve quadratic, exponential or rational functions. Discuss the importance of using appropriate labels and scales on the axes when representing functions
with graphs. Examine real-world graphs in terms of constraints that are necessary to balance a mathematical model with the real world context. For example, a student writing an equation to
model the maximum area when the perimeter of a rectangle is 12 inches should recognize that y = x(6 – x) only makes sense when 0 < x < 6. This restriction on the domain is necessary
because the side of a rectangle under these conditions cannot be less than or equal to 0, but must be less than 6. Students can discuss the difference between the parabola that models the
problem and the portion of the parabola that applies to the context. Explore examples illustrating when it is useful to rewrite a formula by solving for one of the variables in the formula. For
example, the formula for the area of a trapezoid (A = 1 2 h(b 1 + b2) ) can be solved for h if the area and lengths of the bases are known but the height needs to be calculated.
Use a graphing calculator to demonstrate how dramatically the shape of a curve can change when the scale of the graph is altered for one or both variables. Give students
formulas, such as area and volume (or from science or business), and have students solve the equations for each of the different variables in the formula.
Common Misconceptions/Challenges
Students may believe that equations of linear, quadratic and other functions are abstract and exist only “in a math book,” without seeing the usefulness of these functions
as modeling real-world phenomena. Additionally, they believe that the labels and scales on a graph are not important and can be assumed by a reader, and that it is always
necessary to use the entire graph of a function when solving a problem that uses that function as its model.
Common Core Support
Progressions for the Common Core State Standards in Mathematics (draft)
http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_algebra_2013_07_03.pdf
The Common Core in Ohio
https://www.ixl.com/standards/ohio/math
Ohio’s New Learning Standards
https://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/Math-Standards.pdf.aspx
Illustrative Mathematics
https://www.illustrativemathematics.org/blueprints/M3/5
Columbus City Schools
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Textbook and Curriculum Resources
http://www.virtualnerd.com/tutorials/?id=Alg1_9_2_17
Integrated Math III, McGraw Hill
Chapter 0
Chapter 1
Chapter 2
Common Core State Standards Station Activities for Mathematics I, Walch Education, 2014
Solving Inequalities, pp. 14-24
Solving Equations, pp. 25-50
Algebra 1 Station Activities for Common Core State Standards, Walch Education, 2013
Solving Linear Equations, pp. 54-79
Graphing Linear Equations/Solving Using Graphs, pp.28-45
Writing Linear Equations, pp. 46-53
Problem-Based Tasks for Mathematics I, Walch Education, 2013
Phone Card Fine Print, pp. 1-4
Investing Money, pp. 5-9
Rafting and Hiking Trip, pp. 10-13
Free Checking Accounts, pp. 22-24
Population Change, pp. 25-28
Problem-Based Tasks for Mathematics II, Walch Education, ,2013
Dancing for Charity, pp. 86-90
Prior Knowledge
Future Learning
Working with expressions and equations, including formulas, is an integral part
of the curriculum in Grades 7 and 8. In high school, students explore in more
depth the use of equations and inequalities to model real-world problems,
including restricting domains and ranges to fit the problem’s context, as well as
rewriting formulas for a variable of interest.
A.CED.1 and A.CED2 will be studied again when rational functions
become the topic in Grading Period 2.
Columbus City Schools
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Performance Assessments/Tasks
Click on the links below to access performance tasks.
NRICH
http://nrich.maths.org/public/search.php?search=statistics http://nrich.maths.org/public/search.php?search=creating equations&filters[ks4]=1
Illustrative Mathematics
Maximizing Profits: Selling Boomerangs
http://map.mathshell.org/download.php?fileid=1718
Triangular Frameworks
http://map.mathshell.org/download.php?fileid=814
Fearless Frames
http://map.mathshell.org/download.php?fileid=806
Pythagorean Triples
http://map.mathshell.org/download.php?fileid=812
Best Buy Tickets
http://map.mathshell.org/download.php?fileid=824
Skeleton Tower
http://map.mathshell.org/download.php?fileid=810
Printing Tickets
http://map.mathshell.org/download.php?fileid=772
Functions
http://map.mathshell.org/download.php?fileid=762
Career Connections
Occupations in Management: Computer, Engineering; Farmers Funeral; Industrial production managers; Medical and health services managers; Property,
real estate, and community association managers ; Purchasing managers, buyers, and purchasing agents--Business and financial operations occupations:
Insurance Computer and mathematical occupations: Actuaries, Computer programmers, Computer software engineers, Computer systems analysts,
Mathematicians, Statisticians
Engineers: Aerospace engineers , Computer hardware engineers, Environmental engineers , Industrial engineers, Nuclear engineers
http://www.xpmath.com/careers/topicsresult.php?subjectID=2&topicID=3
Columbus City Schools
Clear Learning Targets Integrated Math III 2016-2017
23
Ohio’s Learning Standards - Clear Learning Targets
Integrated Math III
(listing of included sources attached)
A.CED.3
Essential Understanding
Create equations that
describe numbers or
relationships.
Students should be able to create and solve
equations in one variable to answer
questions.
Represent constraints by equations or inequalities, and by systems of
equations and/or inequalities, and interpret solutions as viable or nonviable
options in a modeling context. For example, represent inequalities
describing nutritional and cost constraints on combinations of different
foods
Students should be able to interpret word
problems and form expressions, equations
and inequalities in order to solve a problem.
They must be able to translate a word
problem into an algebraic equation.
Students need to be able to identify when a
common formula is needed for the given
context.
Extended Understanding
-
CCSSM Description
Students need to be able to interpret results after translating words into
expressions, equations, and inequalities. They must be able to analyze an
equation and problem to see if they have followed all procedures correctly;
students must come up with the correct answer and determine if the
answer makes sense.. Finally, students must be able to manipulate
equations, following all the rules of Algebra, in order to solve for a given
variable (literal equation).
Students can start learning quadratic, rational,
and exponential functions to address all
aspects of this standard. Once students are
familiar with these operations individually,
they should be asked to distinguish them from
each other.



I Can Statements
Columbus City Schools

Academic Vocabulary/
Language
coefficient
equation
inequality
Linear
exponential
function
literal
polynomial
rational
system of
equation
variable
Tier 2 Vocabulary
describe
greater
than
interpret
reasoning
solve
solution
translate
unknown
I can recognize when a modeling context involves constraints
I can interpret solutions as viable or nonviable options in a modeling context
I can determine when a problem should be represented by equations,
inequalities, systems of equations and/or inequalities
I can represent constraints by equations or inequalities, and by systems of
equations and/or inequalities
Clear Learning Targets Integrated Math III 2016-2017
24
Instructional Strategies
Provide examples of real-world problems that can be modeled by writing an equation or inequality. Begin with simple equations and inequalities and build up to more
complex equations in two or more variables that may involve quadratic, exponential or rational functions.
Discuss the importance of using appropriate labels and scales on the axes when representing functions with graphs. Examine real-world graphs in terms of constraints that
are necessary to balance a mathematical model with the real world context. For example, a student writing an equation to model the maximum area when the perimeter
of a rectangle is 12 inches should recognize that y=x (6 – x) only makes sense when 0 < x < 6. This restriction on the domain is necessary because the side of a rectangle
under these conditions cannot be less than or equal to 0, but must be less than 6. Students can discuss the difference between the parabola that models the problem and
the portion of the parabola that applies to the context’
Provide examples of real-world problems that can be solved by writing an equation, and have students explore the graphs of the equations on a graphing calculator to
determine which parts of the graph are relevant to the problem context.
Use a graphing calculator to demonstrate how dramatically the shape of a curve can change when the scale of the graph is altered for one or both variables.
Give students formulas, such as area and volume (or from science or business), and have students solve the equations for each of the different variables in the formula
Common Misconceptions/Challenges
Students may believe that equations of linear, quadratic and other functions are abstract and exist only “in a math book,” without seeing the usefulness of these functions
as modeling real-world phenomena.
Students believe that the labels and scales on a graph are not important and can be assumed by a reader, and that it is always necessary to use the entire graph of a
function when solving a problem that uses that function as its model.
Common Core Support
Illustrative Mathematics
https://www.illustrativemathematics.org/content-standards/HSA/CED/A/3
Hunt Institute Video examples
http://www.ccsso.org/resources/digital_resources/common_core_implementation_video_series.htm
Columbus City Schools
Clear Learning Targets Integrated Math III 2016-2017
25
Textbook and Curriculum Resources
Integrated Math III McGraw Hill
Chapter 3
CCSS Math
http://ccssmath.org/?page_id=2121
Shmoop
http://www.shmoop.com/common-core-standards/ccss-hs-a-ced-3.html
Engage NY
https://www.engageny.org/ccls-math/aced3
Sophia
https://www.sophia.org/ccss-math-standard-9-12aced3-pathway
LearnZillion
https://learnzillion.com/resources/72824-represent-constraints-by-equations-or-inequalities-and-by-systems-of-equations-and-or-inequalities
Prior Knowledge
Future Learning
Working with expressions and equations, including formulas, is an integral part
of the curriculum in Grades 7 and 8. In high school, students explore in more
depth the use of equations and inequalities to model real-world problems,
including restricting domains and ranges to fit the problem’s context, as well as
rewriting formulas for a variable of interest.
Future learning will include the study of linear equations and inequalities
in two variables.
Columbus City Schools
Clear Learning Targets Integrated Math III 2016-2017
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Performance Assessments/Tasks
Click on the links below to access performance tasks.
Math Assessment Project
Maximizing Profits: Selling Boomerangs
http://map.mathshell.org/download.php?fileid=1718
Modeling Motion: Rolling Cups
http://map.mathshell.org/download.php?fileid=1746
Sorting Equations of Circle 1
http://map.mathshell.org/download.php?fileid=1766
Sorting Equations of Circle 2
http://map.mathshell.org/download.php?fileid=1768
Proving the Pythagorean Theorem
http://map.mathshell.org/download.php?fileid=1756
Inside Mathematics
Number Towers
http://www.insidemathematics.org/assets/common-core-math-tasks/number%20towers.pdf
Expressions
http://www.insidemathematics.org/assets/common-core-math-tasks/expressions.pdf
Sorting the Mix
http://www.insidemathematics.org/assets/problems-of-the-month/sorting%20the%20mix.pdf
Career/Everyday
Connections
https://www.illustrativemathematics.org/content-standards/HSF/LE/B/5/tasks/579
There
are many practical connections to creating equations describing numbers or relationships: deciding metered cab fares, mailing packages based upon
https://www.illustrativemathematics.org/content-standards/HSF/LE/B/5/tasks/758
weight, chemistry (preparing solutions mixing two given solutions; you will need to find how much of each given solution should be used to make your new
solution).business (determining inventory), etc.
Columbus City Schools
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27
Ohio’s Learning Standards - Clear Learning Targets
Integrated Math III
(listing of included sources attached)
A.REI.2, A.REI.11
Reasoning with
equations and
inequalities.
Solve simple rational and radical equations in one variable and give
examples showing how extraneous solutions may arise.
Explain why the x-coordinates of the points where the graphs of the
equations y = f(x) and y = g(x) intersect are the solutions of the
equation f(x) = g(x); find the solutions approximately, e.g., using
technology to graph the functions, make tables of values, or find
successive approximations. Include cases where f(x) and/or g(x) are
linear, polynomial, rational, absolute value, exponential, and
logarithmic functions.
CCSSM Description
Rational equations mean that fractions are involved. Radical equations
mean that square roots are involved. Students should know how to
deal with both separately and together. Also, students should
understand that an equation and its graph are just two different
representations of the same thing. The graph of the line or curve of a
two-variable equation shows in visual form all of the solutions (infinite
as they may be) to our equation in written form. When two equations
are set to equal one another, their solution is the point at which
graphically they intersect one another. Depending on the equations
(and the alignment of the planets), there might be one solution, or
more, or none at all.
Essential Understanding
Students are expected to be able to give examples
showing how extraneous solutions may
arise when solving rational and radical equations.
Students are expected to be able to determine the
domain of a rational function; students are
expected to know how to determine the domain
of a radical function.
Students are expected to know how to solve
radical equations in one variable; students are
expected to know how to solve rational equations
in one variable; students are expected to be able
to recognize and use function notation to
represent linear, polynomial, rational, absolute
value, exponential, and radical equations.
Academic Vocabulary/ Language
absolute
value
function
domain
exponential
function
function
function
notation
radical
equation
linear
function
logarithmic
function
polynomial
function
rational
equation
rational
functions
variable
Tier 2 Vocabulary
approximate
recognize
solve
successive
Extended Understanding
Provide visual examples of radical and rational
equations with technology so that students can see the
solution as the intersection of two functions and
further understand how extraneous solutions do not fit
the model. It is very important that students are able to
reason how and why extraneous solutions arise.
I can determine the domain of a rational function.
I can determine the domain of a radical function.
I can solve radical equations in one variable.
I Can Statements
I can solve rational equations in one variable.
I can give examples showing how extraneous solutions may arise when solving rational and radical equations.
I can approximate or find the solutions to a system.
I can explain why the solution to a system will occur at the point(s) of intersection.
Columbus City Schools
Clear Learning Targets Integrated Math III 2016-2017
28
Instructional Strategies
Challenge students to justify each step of solving an equation. Transforming 2x - 5 = 7 to 2x = 12 is possible because 5 = 5, so adding the same quantity to both sides of an equation makes the
resulting equation true as well. Each step of solving an equation can be defended, much like providing evidence for steps of a geometric proof. Provide examples for how the same equation
might be solved in a variety of ways as long as equivalent quantities are added or subtracted to both sides of the equation, the order of steps taken will not matter. Connect the idea of adding
two equations together as a means of justifying steps of solving a simple equation to the
process of solving a system of equations. A system consisting of two linear functions such as 2x + 3y = 8 and x - 3y = 1equation 2x - 4 = 5 can begin by adding the equation 4 = 4. Begin with
simple, one-step equations and require students to write out a justification for each step used to solve the equation. Ensure that students are proficient with solving simple rational and radical
equations that have no extraneous solutions before moving on to equations that result in quadratics and possible solutions that need to be eliminated.
Using technology, have students graph a function and use the trace function to move the cursor along the curve. Discuss the meaning of the ordered pairs that appear at the bottom of the
calculator, emphasizing that every point on the curve represents a solution to the equation. Begin with simple linear equations and how to solve them using the graphs and tables on a
graphing calculator. Then, advance students to nonlinear situations so they can see that even complex equations that might involve quadratics, absolute value, or rational functions can be
solved fairly easily using this same strategy. While a standard graphing calculator does not simply solve an equation for the user, it can be used as a tool to approximate solutions. Use the
table function on a graphing calculator to solve equations. For example, to solve the equation
x2 = x + 12, students can examine the equations y = x2 and y = x + 12 and determine that
they intersect when x = 4 and when x = -3 by examining the table to find where the y-values are the same.
Common Misconceptions/Challenges
Students may believe that the graph of a function is simply a line or curve “connecting the dots,” without recognizing that the graph represents all solutions to the equation.
Students may also believe that graphing linear and other functions is an isolated skill, not realizing that multiple graphs can be drawn to solve equations involving those
functions. Additionally, students may believe that two-variable inequalities have no application in the real world. Teachers can consider business related problems (e.g.,
linear programming applications) to engage students in discussions of how the inequalities are derived and how the feasible set includes all the points that satisfy the
conditions stated in the inequalities.
Common Core Support
Progressions for the Common Core State Standards in Mathematics (draft)
http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_algebra_2013_07_03.pdf
The Common Core in Ohio
https://www.ixl.com/standards/ohio/math
Ohio’s New Learning Standards
https://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/Math-Standards.pdf.aspx
Illustrative Mathematics
https://www.illustrativemathematics.org/blueprints/M3/5 https://www.illustrativemathematics.org/blueprints/M3
Columbus City Schools
Clear Learning Targets Integrated Math III 2016-2017
29
Textbook and Curriculum Resources
http://www.virtualnerd.com/tutorials/?id=Alg1_9_2_17Integrated Math III, McGraw Hill
Chapter 1
Chapter 4
Chapter 6
Chapter 7
Common Core State Standards Station Activities for Mathematics I, Walch Education, 2014
Comparing Linear Models, pp. 37-50
Using Systems in Applications, pp. 51-63
Problem-Based Tasks for Mathematics I, Walch Education,2013
Senior Trip, pp. 67-70
Prior Knowledge
Future Learning
Solving linear equations in one variable and analyzing pairs of simultaneous
linear equations is part of the Grade 8 curriculum. These ideas are extended in
high school, as students explore paper-and-pencil and graphical ways to solve
equations, as well as how to graph two variable inequalities and solve systems of
inequalities
These standards will be revisited when studying rational functions.
Columbus City Schools
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Performance Assessments/Tasks
Click on the links below to access performance tasks.
NRICH
http://nrich.maths.org/public/search.php?search=creating%20equations%20and%20inequalities&filters[ks3]=1
INISIDE Mathematics
Graphs 2006
http://www.insidemathematics.org/assets/common-core-math-tasks/graphs%20(2006).pdf
Hexagons
http://www.insidemathematics.org/assets/common-core-math-tasks/hexagons.pdf
Magic Squares
http://www.insidemathematics.org/assets/common-core-math-tasks/magic%20squares.pdf
Math Assessment Project
Evaluating Statements about Radicals
http://map.mathshell.org/download.php?fileid=1714
Building and Solving Complex Equations
http://map.mathshell.org/download.php?fileid=1722
Maximizing Profit: Selling Boomerangs
http://map.mathshell.org/download.php?fileid=1718
Representing Inequalities Graphically
http://map.mathshell.org/download.php?fileid=1742
Career Connections
Occupations using equations and inequalities: Management: Computer, Engineering; Farmers Funeral; Industrial production managers; Medical and health
services managers; Property, real estate, and community association managers ; Purchasing managers, buyers, and purchasing agents--Business and
financial operations occupations: Insurance Computer and mathematical occupations: Actuaries, Computer programmers, Computer software engineers,
Computer systems analysts, Mathematicians, Statisticians
Engineers: Aerospace engineers , Computer hardware engineers, Environmental engineers , Industrial engineers, Nuclear engineers
http://www.xpmath.com/careers/topicsresult.php?subjectID=2&topicID=3
Columbus City Schools
Clear Learning Targets Integrated Math III 2016-2017
31
Ohio’s Learning Standards - Clear Learning Targets
Integrated Math III
(listing of included sources attached)
F.IF.4, F.IF.6
Essential Understanding
Interpret funtions that
arise in applications in
terms of the context.
- Students must be able to interpret
functions that arise in applications in terms
of a specific context.
For a function that models a
relationship between two quantities, interpret key features of graphs and
tables in terms of the quantities, and sketch graphs showing key features
given a verbal description of the relationship. Key features include:
intercepts; intervals where the function is increasing, decreasing, positive,
or negative; relative maximums and minimums; symmetries; end
behavior; and periodicity.
Calculate and interpret the average rate of change of a function
(presented symbolically or as a table) over a specified interval. Estimate
the rate of change from a graph.
-
CCSSM Description
-Functions can be represented numerically,
graphically, algebraically (symbolically),
and/or verbally.
Extended Understanding
-Students will be expected to move flexibly
between the different representations of
the same function for comparison.
-Students will learn about independent and
dependent variables; an understanding of
these concepts provides the basis for later
work with functions
Because we continually make theories about dependencies between
quantities in nature and society, functions are important tools in the
construction of mathematical models. Students will use multiple
representations to represent functions in different contexts.
Academic Vocabulary/
Language
average
rate of
change
continuous
decreasing
domain
end
behavior
exponential
function
growth
periodicity
increasing
intercept
interval
linear
range
rate of change
representation
slope
maximum
minimum
symmetry
Tier 2 Vocabulary
algebraically
application
calculate
context
domain
end
behavior
features
graphically
identify
negative
numerically
positive
range
sketch
symbolically
verbally
I can, given a function, identify key features in graphs and tables including: intercepts,
intervals (increasing, decreasing, positive, negative), relative maximums and minimums,
symmetries, end behavior, and periodicity; I can, given the key features of a function, sketch
the graph.
I Can Statements
I can, given the graph of a function, determine the practical domain of the function as it
relates to the numerical relationship it describes.
I can calculate the average rate of change over a specified interval of a function presented
symbolically or in a table; I can estimate the average rate of change over a specified interval
of a function from the function’s graph; I can interpret,in context, the average rate of
change; I can demonstrate that the rate of change of a non-linear function is different for
different intervals.
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Instructional Strategies
Practice moving from examining a graph and describing its characteristics (e.g., intercepts, relative maximums, etc.) to using a set of given characteristics to sketch the graph
of a function.
Examine a table of related quantities and identify features in the table, such as intervals on which the function increases, decreases, or exhibits periodic behavior.
Begin with simple, linear functions to describe features and representations, and then move to more advanced functions, including non-linear situations.
Use various representations of the same function to emphasize different characteristics of that function.
Graphing utilities on a calculator and/or computer can be used to demonstrate the changes in behavior of a function as various parameters are varied.
Common Misconceptions/Challenges
Students may believe that all relationships having an input and an output are functions, and therefore, misuse the function terminology.
Students may experience challenges moving between the different representations.
Students may believe that it is reasonable to input any x-value into a function, so they will need to examine multiple situations in which there are various limitations to the
domains.
.
Common Core Support
Common Core Math: Computing Technology for Math Excellence: High School Functions Teaching and Learning Resources
http://www.ct4me.net/Common-Core/hsfunctions/hsf-interpreting-functions.htm
Institute for Mathematics and Education Learning Progressions Narratives
http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_functions_2013_07_02.pdf
Ohio Department of Education Model Curriculum
http://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/High_School_Functions_Model_Curriculum_March2015.pdf.aspx
Illustrative Mathematics: https://www.illustrativemathematics.org/blueprints/M1
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Textbook and Curriculum Resources
Problem-Based Tasks for Mathematics I Common Core State Standards, Walch Education, 2013
Identifying Key Features of Linear and Exponential Graphs, pp. 95-98
Proving Average Rate of Change, pp. 99-103
Recognizing Average Rate of Change, pp. 104-108.
Learnzillion
https://learnzillion.com/lessonsets/470
NCTM Illuminations:
Domain Representations: http://illuminations.nctm.org/Lesson.aspx?id=2071
Growth Rate: http://illuminations.nctm.org/Lesson.aspx?id=2265
Khan Academy
https://www.khanacademy.org/math/algebra/algebra-functions
Integrated Math I, McGraw Hill
Chapters 1,2,3
Prior Knowledge
Future Learning
Students have learned about correspondences between equations, verbal
descriptions, tables, and graphs and have studied regularity or trends.
Students will be expected to increase flexibility with moving between the
multiple representations. The Rule of 4 representing mathematical
functions--- visually (graphs, tables, charts), symbolically (algebraically),
numerically (concrete examples), and verbally (natural language) --- will
become increasingly prominent throughout students’ studies of
mathematics. Students will be expected to be proficient with modeling
and interpreting functions in terms of a context.
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Performance Assessments/Tasks
Click on the links below to access performance tasks where students are asked to demonstrate an understanding of functions.
Math Assessment Project
Linear Graphs
http://map.mathshell.org/download.php?fileid=1106
Interpreting Functions
http://map.mathshell.org/download.php?fileid=840
Inside Mathematics
Graphs
http://www.insidemathematics.org/assets/common-core-math-tasks/graphs%20(2004).pdf
Sorting Functions
http://www.insidemathematics.org/assets/common-core-math-tasks/sorting%20functions.pdf
Career Connections
Economics, Investment Brokers, Insurance, Actuarial Science, Architects
http://www.educationworld.com/a_curr/mathchat/mathchat010.shtml
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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math III
(listing of included sources attached)
F.IF.7B, F.IF.9
Analyze functions using
different representations.
Graph square root, cube root, and
piecewise functions, including step
functions and absolute value functions.
Compare properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions).
For example, given a graph of one quadratic function and an algebraic
expression for another, say which has the larger maximum.
Essential Understanding
Students will be expected to master flexible
movement between the multiple
representations.
Students are expected to increase comfort
level in understanding other
representations mentally even when only
one representation is given.
Academic Vocabulary/ Language
constant
exponential
linear
parent
function
rate of
change
sequence
slope
standard
form
x- intercept
y-intercept
Tier 2 Vocabulary
Extended Understanding
CCSSM Description
Students learn in different ways. The Rule of 4 ---visually (graphs,
tables, charts), symbolically (algebraically), numerically (concrete
examples), and verbally (natural language) facilitates and deepens
understanding by presenting the same concept in different modes.
Students can better understand the
characteristics of representations by
providing opportunities to study the eight
major families of functions.
algebraically
analyze
compare
contrast
domain
graphically
identify
numerically
sketch
symbolically
terms
variable
I can graph square root, cube root, and piecewise-defined functions, including step functions and
absolute value functions, by hand in simple cases or using technology for more complicated cases,
and show/label key features of the graph.
I can determine the difference between simple and complicated linear, quadratic, square root, cube
root, and piecewise-defined functions, including step functions and absolute value functions and
know when the use of technology is appropriate.
I can compare and contrast the domain and range of absolute vale, step and piece-wise defined
functions with linear, quadratic, and exponential.
I Can Statements
I can write a function in equivalent forms to show different properties of the function.
I can explain the different properties of a function that are revealed by writing a function in
equivalent forms.
I can identify types of functions based on verbal, numerical, algebraic, and graphical descriptions and
state key properties (e.g. intercepts, growth rates, average rates of change, and end behaviors).
I can differentiate between exponential and linear functions using a variety of descriptors
(graphically, verbally, numerically, and algebraically); I can use a variety of function representations
algebraically, graphically, numerically in tables, or by verbal descriptions) to compare and contrast
properties of two functions.
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Instructional Strategies
Graphing utilities on a calculator and/or computer can be used to demonstrate the changes in behavior of a function as various parameters are varied.
Add families of functions, one at a time, to the students’ knowledge base so they can see connections among behaviors of the various functions.
Provide numerous examples of real-world contexts such as exponential growth and decay situations (e.g., a population that declines by 10% per year) to help students
apply an understanding of functions in context.
Common Misconceptions/Challenges
Students may believe that each family of functions (e.g., quadratic, square root, etc.) is independent of the others, so they may not recognize commonalities among all
functions and their graphs.
Students may believe that the process of rewriting equations into various forms is simply an algebra symbol manipulation exercise rather than serving a purpose of allowing
different features of the function to be exhibited.
Common Core Support
Common Core Math: Computing Technology for Math Excellence: High School Functions Teaching and Learning Resources
http://www.ct4me.net/Common-Core/hsfunctions/hsf-interpreting-functions.htm
Institute for Mathematics and Education Learning Progressions Narratives
http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_functions_2013_07_02.pdf
Ohio Department of Education Model Curriculum
http://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/High_School_Functions_Model_Curriculum_March2015.pdf.aspx
Illustrative Mathematics
https://www.illustrativemathematics.org/blueprints/M1
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Textbook and Curriculum Resources
Problem-Based Tasks for Mathematics I Common Core State Standards, Walch Education, 2013
Fund-raising Concert, pp. 109113
Trout Pond, pp. 114-117
Supply and Demand, pp. 118-122
Analyzing Kidney Function, pp. 123-126
Station Activities for Mathematics I, Walch Education, 2014:
Interpreting Functions, pp. 94-117
Algebra I Station Activities for Common Core State Standards
Interpreting Functions, pp. 231-245
ORC (Ohio Resource Center: The Ohio State University)
http://www.ohiorc.org/search/results/?txtSearchText=functions
https://learnzillion.com/lessonsets/470
Virtual Nerd
http://www.virtualnerd.com/tutorials/?id=Alg1_9_2_17
Integrated Math I, McGraw Hill
Chapter 1
Extend Lesson 1-7: Graphing Technology Lab: Representing Functions
Lesson 1-8: Interpreting Graphs of Functions
Prior Knowledge
Future Learning
Students have been exposed to the idea that rewriting an expression can
provide more information on the expression. This idea is expanded upon as
students explore functions in high school and recognize how the form of the
equation can provide clues about zeros, asymptotes, etc.
Learning features of parent functions, the simplest form of a family of
function, and features of family functions can increase understanding of
functions.
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Performance Assessments/Tasks
Click on the links below to access performance tasks where students are asked to demonstrate an understanding of the concept of functions.
Inside Mathematics
Graphs 2004
http://www.insidemathematics.org/assets/common-core-math-tasks/graphs%20(2004).pdf
Graphs 2007
http://www.insidemathematics.org/assets/common-core-math-tasks/graphs%20(2007).pdf
NRICH
http://nrich.maths.org/773
http://nrich.maths.org/5872
Illustrative Mathematics
Throwing Baseballs
https://www.illustrativemathematics.org/content-standards/HSF/IF/C/9/tasks/1279
Modeling London's Population
https://www.illustrativemathematics.org/content-standards/HSF/IF/C/7/tasks/1595
Running Time
https://www.illustrativemathematics.org/content-standards/HSF/IF/C/7/tasks/1539
Career Connections
Any career that involves the need to articulate verbally the relationships between variables arising in everyday contexts can utilize the study of functions.
This include health care area, science, and any career involving sales.
Students can complete the following concept development activities (Representing Functions of Everyday Situations) where they are asked to :
• Translate between everyday situations and sketch graphs of relationships between variables.
• Interpret algebraic functions in terms of the contexts in which they arise.
• Reflect on the domains of everyday functions and in particular whether they should be discrete or Continuous
http://map.mathshell.org/download.php?fileid=1740
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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math III
(listing of included sources attached)
F.BF.1, F.BF.1B
Essential Understanding
Build a function that
models.
Examination of functions is extended to
include recursive and explicit
representations and sequences of
numbers that may not have a linear
relationship.
Write a linear function that
describes a relationship between
two quantities.
Combine standard function types using arithmetic operations. For
example, build a function that models the temperature of a cooling
body by adding a constant function to a decaying exponential, and
relate these functions to the model.
Academic Vocabulary/
Language
arithmetic
sequence
correspondence
Extended Understanding
-
CCSSM Description
Functions can be used to make predictions about future behaviors when
modeling real life situations. For students to recognize a functional
relationship, they need to recognize there is a correspondence and
see/understand the correspondence matches each element of the first set with an element of the second set. Once it is known that the
relationship is a function, students can determine the rule for the
function.
Using a variety of functions (e.g., linear,
exponential, constant, students can
increase understanding of the different
representations by representing functions
as a set of ordered pairs, a table, a graph,
and an equation.
direct variation
explicit
formula
function
geometric
sequence
inverse
function
inverse
relationship
recursive
quantities
Tier 2 Vocabulary
compare
construct
model
observe
prove
I can, from context, either write an explicit expression, define a recursive process, or
describe the calculations meeded to model a function between two quantities.
I can compose functioins; I can build standard functions to represent relevant
relationships/quantities.
I Can Statements
I can determine which arithmetic operation should be performed to build the appropriate
combined function; I can combine two functions using the operations of addition,
subtraction, multiplication, and division.
I can relate the combined function to the context of the problem; I can evaluate the domain
of the combined function.
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Instructional Strategies
Provide a real-world example (e.g., a table showing how far a car has driven after a given number of minutes, traveling at a uniform speed), and examine the
table by looking “down” the table to describe a recursive relationship, as well as “across” the table to determine an explicit formula to find the distance
traveled if the number of minutes is known.
Write out terms in a table in an expanded form to help students see what is happening. For example, if the y-values are 2, 4, 8, 16, they could be written as 2, 2(2), 2(2)
so that students recognize that 2 is being used multiple times as a factor.
Focus on one representation and its related language – recursive or explicit – at a time so that students are not confusing the formats.
Provide examples of when functions can be combined, such as determining a function describing the monthly cost for owning two vehicles when a function
for the cost of each (given the number of miles driven) is known.
Using visual approaches (e.g., folding a piece of paper in half multiple times), use the visual models to generate
Common Misconceptions/Challenges
sequences of numbers that can be explored and described with both recursive and explicit formulas. Emphasize that
Students may believe that the best (or only) way to generalize a table of data is by using a recursive formula. Students naturally tend to look “down” a table to find the
there
onethat
form
to describe
the
function
is preferred
over
the other.
patternare
buttimes
need when
to realize
finding
the 100th
term
requires
knowing the
99thterm
unless an explicit formula is developed.
Students may also believe that arithmetic and geometric sequences are the same. Students need experiences with both types of sequences to be able to recognize the
difference and more readily develop formulas to describe them.
Advanced students who study composition of functions may misunderstand function notation to represent multiplication (e.g., f(g(x)) means to multiply the f and g
function values).
When studying functions, students sometimes interchange the input and output values. This will lead to confusion about domain and range, and determining if a relation is
a function. This can also interfere with a student being able to find the appropriate inverse function, or the correct equation to model a relationship between two
quantities.
Common Core Support
Common Core Math: Computing Technology for Math Excellence: High School Functions Teaching and Learning Resources
http://www.ct4me.net/Common-Core/hsfunctions/hsf-interpreting-functions.htm
Institute for Mathematics and Education Learning Progressions Narratives
http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_functions_2013_07_02.pdf
Illustrative Mathematics: Practice and Content Standards: Toward Greater Focus and Coherence
https://www.illustrativemathematics.org/standards
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Textbook and Curriculum Resources
Problem-Based Tasks for Mathematics I Common Core State Standards, Walch Education, 2013
Texting for the Win, pp. 127-131
Jai’s Jeans, pp. 132-134
New Tablet, pp. 135-138
Glass Recycling, pp. 139-142
Problem-Based Tasks for Mathematics II Common Core State Standards, Walch Education, 2013
To Drill or Not to Drill?, pp. 138Pushing Envelopes, pp. 142-145
Common Core State Standards: Station Activities for Mathematics I
Relations Versus Functions/Domain and Range, pp. 85-93
Sequences, pp. 118-130
Real-World Situation Graphs pp. 194-208High School CCSS Mathematics I Curriculum Guide-Quarter 1 Curriculum Guide, 2013, pp. 161-204
Prior Knowledge
Future Learning
In Grade 8, students learn to compare functions by looking at
equations, tables and graphs, and focus primarily on linear
relationships.
Future learning will include working with inverse functions.
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Performance Assessments/Tasks
Click on the links below to access performance tasks.
Math Assessment Project
Generalizing Patterns: Table Tiles
http://map.mathshell.org/download.php?fileid=1716
Representing Linear and Exponential Growth
http://map.mathshell.org/download.php?fileid=1732
Modeling Motion: Rolling Cups
http://map.mathshell.org/download.php?fileid=1746
Inside Mathematics
Infinite Windows
http://www.insidemathematics.org/assets/problems-of-the-month/infinite%20windows.pdf
Slice and Dice
http://www.insidemathematics.org/assets/problems-of-the-month/slice%20and%20dice.pdf
Calculating Palindromes
http://www.insidemathematics.org/assets/problems-of-the-month/calculating palindromes.pdf
First Rate
http://www.insidemathematics.org/assets/problems-of-the-month/first%20rate.pdf
Cut It Out
http://www.insidemathematics.org/assets/problems-of-the-month/cut%20it%20out.pdf
Illutrative Mathematics
Summer Intern
https://www.illustrativemathematics.org/content-standards/HSF/BF/A/1/tasks/72
https://www.illustrativemathematics.org/content-standards/HSF/LE/B/5/tasks/579
https://www.illustrativemathematics.org/content-standards/HSF/LE/B/5/tasks/758
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Career Connections
Students can research and evaluate several options when purchasing a vehicle (e.g., new versus used, lease versus own, down payment, and interest rate).
They will examine the differences in gas mileage consumption by selecting two vehicles to evaluate (e.g., SUV versus compact hybrid). Once they choose a
vehicle, they will use their evaluations to show why they chose the vehicle. Their research will include interviewing automotive professionals, visiting
dealerships, and navigating company websites.
Applicable careers include business, finance, insurance and any career focused on making scholarly predictions.
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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math III
(listing of included sources attached)
F.BF.3, F.BF.4
Essential Understanding
Build a function
that models.
Identify the effect on
the graph of replacing
f(x) by f(x) + k, k(f(x), f(kx), and f(x + k) for specific values of k
(both positive and negative); find the value of k given the
graphs. Experiment with cases and illustrate an explanation
of the effects on the graph using technology. Include
recognizing even and odd functions from their graphs and
algebraic expressions for them.
Find the inverse functions.
-
CCSSM Description
Students identify appropriate types of functions to model a
situation. They adjust parameters to improve the model, and they
compare models by analyzing appropriateness of fit and making
judgments about the domain over which a model is a good fit.
Students should be able to identify appropriate types of
functions to model specific contexts.
Academic Vocabulary/
Language
Students should recognize that not all functions have
inverses.
algebraic
expression
arithmetic
sequence
correspondence
Extended Understanding
direct variation
Students should be able to find inverse functions.
Use real-world examples of functions and their inverses. For
example, students might determine that folding a piece of
paper in half 5 times results in 32 layers of paper, but that if
they are given that there are 32 layers of paper, they can
solve to find how many times the paper would have been
folded in half. Provide applied examples of exponential and
logarithmic functions, such as the use of a logarithm to
determine pH or the strength of an earthquake on the
Richter Scale. Both pH and Richter Scale values are powers
of 10 and are, therefore, logarithms. For example, the
magnitude of an earthquake, M, on the Richter Scale can be
calculated using the formula M = log10A, where A represents
the amplitude of measured seismic waves
even function
functions
function
geometric
sequence
inverse
function
inverse
relationship
logarithmic
function
odd
functions
parameters
quantities
value
Tier 2 Vocabulary
analyze
appropriate
compare
identify
illustrate
judgement
model
observe
prove
Identify, through experimenting with technology, the effect on the graph of a function by replacing f(x) with f(x) + k,
k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative).
Given the graphs of the original function and a transformation, determine the value of (k).
Recognize even and odd functions from their graphs and equations.
I Can Statements
Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the
inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠1; I can verify by composition that one function is the
inverse of another.
Read values of an inverse function from a graph or a table, given that the function has an inverse.
Produce an invertible function from a non-invertible f.
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Instructional Strategies
Use graphing calculators or computers to explore the effects of a constant in the graph of a function. For example, students should be able to distinguish between the graphs of y = x2 ,
y = 2x2 , y = x2 + 2, y = (2x) 2 , and y = (x + 2)2 . This can be accomplished by allowing students to work with a single parent function and examine numerous parameter changes to make
generalizations. Distinguish between even and odd functions by providing several examples and helping students to recognize that a function is even if f(-x) = f(x) and is odd if f(-x) = -f(x).
approaches to identifying the graphs of even and odd functions can be used as well. Provide examples of inverses that are not purely mathematical to introduce the idea.
Students should also recognize that not all functions have inverses. Again using a nonmathematical example, a function could assign a continent to a given country’s input, such as
g(Singapore) = Asia. However, g-1 (Asia) does not have to be Singapore (e.g., it could be China). Exchange the x and y values in a symbolic functional equation and solve for y to determine
the inverse function. Recognize that putting the output from the original function into the input of the inverse results in the original input value. Also, students need to recognize that
exponential and logarithmic functions are inverses of one another and use these functions to solve real-world problems. Nonmathematical examples of functions and their inverses can
help students to understand the concept of an inverse and determining whether a function is invertible.
Introduce finding the inverse of a function with the activity “Introduction to Inverse Functions” (included in the CCS Curriculum Guide, 2013, Math III, p. 112). In this activity, students will
intuitively attempt to find the inverse of functions, and then look at the actual inverse graphically and algebraically. Students will connect the domain and range of the original function to
the domain and range of the inverse relations. Students can do this in groups or individually. Have students work on the handout “Conversions – Applications of Inverses” (included in CCS
Curriculum Guide, 2013, Math III, p. 122) in order to see how inverse functions could be used in real life situations. They will use real life formulas to connect functions with their inverses
Common Misconceptions/Challenges
Students may believe that the graph of y = (x – 4)3 is the graph of y = x3 shifted 4 units to the left (due to the subtraction symbol). Examples should be explored by hand and
on a graphing calculator to overcome this misconception. Students may also believe that even and odd functions refer to the exponent of the variable, rather than the
sketch of the graph and the behavior of the function. Additionally, students may believe that all functions have inverses and need to see counter examples, as well
as examples in which a non-invertible function can be made into an invertible function by restricting the domain. For example, f(x) = x 2 has an inverse (f -1 (x) = square root
of x ) provided that the domain is restricted to x ≥ 0.
Common Core Support
Common Core Math: Computing Technology for Math Excellence: High School Functions Teaching and Learning Resources
http://www.ct4me.net/Common-Core/hsfunctions/hsf-building-functions.htm
Institute for Mathematics and Education Learning Progressions Narratives
http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_functions_2013_07_02.pdf
Illustrative Mathematics: Practice and Content Standards: Toward Greater Focus and Coherence
https://www.illustrativemathematics.org/standards
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Textbook and Curriculum Resources
http://www.virtualnerd.com/tutorials/?id=Alg1_9_2_17 Integrated Math III, McGraw Hill,
Chapter 5
Chapter 6
Chapter 7
Problem-Based Tasks for Mathematics I Common Core State Standards, Walch Education, 2013
Gym Fees, pp. 143-145
Problem-Based Tasks for Mathematics II Common Core State Standards, Walch Education, 2013
The Catch, pp. 146-149
Fewer Parabolas, Please, pp. 150-155
Falling Keys, pp. 156-159
Common Core State Standards: Station Activities for Mathematics II, Walch Education, 2010
Quadratics Transformations in Vertex Form, pp. 17-34
High School CCSS Mathematics IIII Curriculum Guide-Quarter 1 Curriculum Guide, 2013
Prior Knowledge
Future Learning
Understanding functional relationships as input and output values that have an
associated graph is introduced in Grade 8. In high school, changes in graphs is
explored in more depth, and the idea of functions having inverses is introduced.
Advanced students also expand their catalog of functions to include exponential
and logarithmic cases.
These standards will be revisited when studying rational functions.
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Performance Assessments/Tasks
Click on the links below to access performance tasks.
Math Assessment Project
Representing Trigonometric Functions
http://map.mathshell.org/download.php?fileid=1738
Representing Polynomials Graphically
http://map.mathshell.org/download.php?fileid=1744
NYS COMMON CORE MATHEMATICS CURRICULUM
Choosing a Model
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&ved=0ahUKEwjx483vxs3NAhUI6yYKHU3ODbkQFggrMAE&url=https%3A%2F%2Fwww.engageny.o
rg%2Ffile%2F109501%2Fdownload%2Falgebra-ii-m3-topic-c-lesson-22-teacher.pdf%3Ftoken%3DyFz_dJHn&usg=AFQjCNFk8Swf8dKGbZzhWu2MVVInJcrTsA&cad=rja
ILLUSTRATIVE MATHEMATICS
Transforming the Graph of a Function
https://www.illustrativemathematics.org/content-standards/tasks/742
https://www.illustrativemathematics.org/content-standards/HSF/LE/B/5/tasks/758
INSIDE MATHEMATICS
Tri-Triangles
http://www.insidemathematics.org/assets/problems-of-the-month/tri-triangles.pdf
Career Connections /Real World Applications
Engineers, Scientists
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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math III
(listing of included sources attached)
A.SSE.1A, A.SSE.1B
Interpret the structure
of expressions.
Interpret expressions that represent a
quantity in terms of its context.
Interpret parts of an expression, such as terms, factors, and coefficients.
Interpret complicated expression by viewing one or more of their parts as a
single entity. For example, see x4-y4 as (x2)2 – (y2)2, thus recognizing it as
difference of squares that can be factored as (x2-y2) (x2+y2).
Essential Understanding
Students should know that variable expressions are
used to communicate and model authentic
problems.
Students should know that math is a language and
has structures to ensure effective communication.
Students should know: how to write expressions
from descriptive words and from patterns in data.
Students should be able to describe in words an
expression in a given context.
Students should know how to explain the
difference between a variable and a constant.
CCSSM Description
Reading an expression with comprehension involves analysis of its
underlying structure. This may suggest a different but equivalent way of
writing the expression that exhibits some different aspect of its meaning.
Algebraic manipulations are governed by the properties of operations and
exponents, and the conventions of algebraic notation. Viewing an
expression as the result of operation on simpler expressions can sometimes
clarify its underlying structure.
Extended Understanding
Academic Vocabulary/
Language
average rate of change, binomial,
coefficient, constant, degree, difference
of squares, divisor, expression, factor,
end behavior, maximum, minimum,
monomial, polynomial, power, quotient,
rational, remainder, roots, terms,
trinomial, x-intercepts,
y-intercepts, zero product property,
zeros
Tier 2 Vocabulary
complex, context, identify, interpret,
recognizing, represent
Hands-on materials, such as algebra tiles, can
be used to establish a visual understanding of
algebraic expressions and the meaning of
terms, factors and coefficients.
I can identify the different parts of the expression and explain their meaning within the
context of a problem.
I Can Statements
I can decompose expressions and make sense of the multiple factors and terms by
explaining the meaning of the individual parts: terms, factors, and coefficients
I can interpret complex expressions by examining their variables
I can, for expressions that represent a contextual quantity, interpret complicated
expressions, in terms of the context, by viewing one or more of their parts as a single
entity.
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Instructional Strategies
Extending beyond simplifying an expression, this cluster addresses interpretation of the components in an algebraic expression. A student should recognize that in the
expression 2x + 1, “2” is the coefficient, “2” and “x” are factors, and “1” is a constant, as well as “2x” and “1” being terms of the binomial expression. Development and proper
use of mathematical language is an important building block for future content. Using real-world context examples, the nature of algebraic expressions can be explored. For
example, suppose the cost of cell phone service for a month is represented by the expression 0.40s + 12.95. Students can analyze how the coefficient of 0.40 represents the
cost of one minute (40¢), while the constant of 12.95 represents a fixed, monthly fee, and s stands for the number of cell phone minutes used in the month. Similar real-world
examples, such as tax rates, can also be used to explore the meaning of expressions. Factoring by grouping is another example of how students might analyze the structure of
an expression. To factor 3x(x – 5) + 2(x – 5), students should recognize that the “x – 5” is common to both expressions being added, so it simplifies to (3x + 2)(x – 5). Students
should become comfortable with rewriting expressions in a variety of ways until a structure emerges. Have students create their own expressions that meet specific criteria
(e.g., number of terms factorable, difference of two squares, etc.) and verbalize how they can be written and rewritten in different forms. Additionally, pair/group students to
share their expressions and rewrite one another’s expressions.
Common Misconceptions/Challenges
Students may believe that the use of algebraic expressions is merely the abstract manipulation of symbols. Use of real world context examples to demonstrate the meaning of
the parts of algebraic expressions is needed to counter this misconception. Students may also believe that an expression cannot be factored because it does not fit into a form
they recognize. They need help with reorganizing the terms until structures become evident.
.
Common Core Support
Institute for Mathematics and Education Learning Progressions Narratives
http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_algebra_2013_07_03.pdf
Ohio Learning Standards
http://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/Math-Standards.pdf.aspx
Illustrative Mathematics:
https://www.illustrativemathematics.org/blueprints/M1
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Textbook and Curriculum Resources
Integrated Math III, McGraw Hill
Chapter 1
Problem-Based Tasks for Mathematics I Common Core State Standards, Walch Education, 2013
Identifying Parts of an Expression in Context, pp. 14-16
Searching for a Greater Savings, pp. 17-21
Problem-Based Tasks for Mathematics II Common Core State Standards, Walch Education, 2013
Deck the Deck, pp. 27-29
Puppy Pen, pp. 30-33
Columbus City Schools Curriculum Guide, 2013, Math III, Quarter 2
https://www.khanacademy.org/math/algebra/algebra-functions
Prior Knowledge
Future Learning
An introduction to the use of variable expressions and their meaning, as well as
the use of variables and expressions in real-life situations is included in the
Expressions and Equations Domain of Grade 7.
Future learning will include arithmetic with polynomials and rational
expressions.
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Performance Assessments/Tasks
Click on the links below to access performance tasks where students are asked to demonstrate an understanding of functions.
The Physics Professor
https://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/23
Radius of a Cylinder
https://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/1366
Mixing Fertilizer
https://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/88
Increasing or Decreasing? Variation 1
https://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/89
Increasing or Decreasing? Variation 2
https://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/167
The Bank Account
https://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/390
Mixing Candies
https://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/389
Delivery Trucks
https://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/1343
Animal Populations
https://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/436
Modeling London’s Population
https://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/1595
Throwing Horseshoes
https://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/90
Seeing Dots
https://www.illustrativemathematics.org/content-standards/HSA/SSE/A/1/tasks/21
Career Connections
Students will evaluate cell phone plans across multiple providers to identify one that is the most cost effective for their expected use. They will consider the
fixed and variable costs to support their decision (e.g., unlimited plans, cost per unit, insurance protection, activation and cancellation fees). In collecting data
related to the cost of service, students will research the employment opportunities available across the telecommunication companies via website, phone,
and email.
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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math III
(listing of included sources attached)
A.APR.2, A.APR.3
Understand the relationship
between zeros and factors of
polynomials.
Essential Understanding
Academic Vocabulary/ Language
Students should be able to understand and
apply the Remainder Theorem.
average rate of change, binomial,
coefficient, constant, degree, divisor, factor,
end behavior, maximum, minimum,
monomial, polynomial, power, quotient,
rational, remainder, roots, terms, trinomial,
x-intercepts, y-intercepts, zero product
property, zeros
Know and apply the Remainder Theorem: For
a polynomial p(x) and a number a, the remainder on a division by x-a is p(a), so p(a)=0
if and only if (x-a) is a factor of p(x).
Students should know and understand that a is
a root of a polynomial function if and only if
x-a is a factor of the function.
Identify zeros of polynomials when suitable factorizations are available, and use the
zeros to construct a rough graph of the function defined by the polynomial
Students should be able to find the zeros of a
polynomial when the polynomial is factored.
Students should be able to use the zeros of a
function to sketch a graph of the function.
Extended Understanding
00
Tier 2 Vocabulary
apply, construct, identify, know, understand
Mathematical Practice 3:
Students can build a logical response,
providing examples, for the following essential
questions:
How are zeros and factors of a polynomial
related? How can a graph of a function be
estimated based on the zeros and factors of a
polynomial?
0
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CCSSM Description
A quick review…If we divide two integers, sometimes they make another integer
(6 ÷ 2 = 3), and other times they have remainders (13 ÷ 4 = 3 remainder 1). A
remainder of 0 means that the second number is a factor of first number. Polynomials
are the same way; if dividing polynomial p(x) by x – a has a remainder of 0, we’ll know
that x – a is a factor of p(x). In other words, p(x) = q(x) × (x – a) where q(x) is a
polynomial or an integer. Essentially, any polynomial p(x) can be written as a product
of (x – a) and some quotient q(x), plus the remainder p(a). The zeros of a polynomial
are the x values when we set the polynomial itself to equal zero. In other words, when
we plug in any of the zeros of a polynomial in for x, our answer should be 0. So the
zeros of x3 – 10x2 – 2x + 24 are the x values that make the equation x3 – 10x2– x+ 24= 0
true. Zero values are imporant because on the coordinate plane, zeros are the places
where the function crosses the x-axis. The zeros of the polynomial (also called the
solutions or “roots”) are the x-intercepts of the graph.
I can understand, define and apply the Remainder Theorem.
I can use the Remainder Theorem to show the relationship between a factor and a zero.
I can understand that a is a root of a polynomial function if and only if x-a is a factor of
the function.
I Can Statements
I can find the zeros of a polynomial when the polynomial is factored.
I can use the zeros of a function to sketch a graph of the function.
I can determine the domain of a rational function.
I can factor polynomials using any method.
I can sketch graphs of polynomials using zeroes and a sign chart.
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Instructional Strategies
By using technology to explore the graphs of many polynomial functions, and describing the shape, end behavior and number of zeros, students can begin to
make the following informal observations: The graphs of polynomial functions are continuous; an nth degree polynomial has at most n roots and at most n - 1
“changes of direction” (i.e., from increasing to decreasing or vice versa); an even-degree polynomial has the same end-behavior in both the positive and
negative directions: both heading to positive infinity, or both heading to negative infinity, depending upon the sign of the leading coefficient; an odd-degree
polynomial has opposite end-behavior in the positive versus the negative directions, depending upon the sign of the leading coefficient; an odd-degree
polynomial function must have at least one real root. Students can benefit from exploring the rational root theorem, which can be used to find all of the
possible rational roots (i.e., zeros) of a polynomial with integer coefficients. When the goal is to identify all roots of a polynomial, including irrational or
complex roots, it is useful to graph the polynomial function to determine the most likely candidates for the roots of the polynomial that are the x-intercepts of
the graph.
Common Misconceptions/Challenges
Difference between roots and zeros: the solution for a polynomial equation is called a root. The words root and zero are often used interchangeably, but technically, you find
the zero of a function and the root of an equation.
Common Core Support
Institute for Mathematics and Education Learning Progressions Narratives
http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_algebra_2013_07_03.pdf
Ohio Learning Standards
http://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/Math-Standards.pdf.aspx
Illustrative Mathematics:
https://www.illustrativemathematics.org/blueprints/A2/3
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Textbook and Curriculum Resources
Integrated Math III, McGraw Hill
Chapter 4
Common Core State Standards Station Activities for Mathematics II, Walch Education, 2014
Operations with Polynomials, pp. 44-56
Columbus City Schools Curriculum Guide, 2013, Math III, Quarter 2
https://www.khanacademy.org/math/algebra/algebra-functions
Prior Knowledge
Future Learning
Students have had exposure to expressions in equations in middle school which
should provide some comfort level with the understanding of roots and zeros.
Study of polynomial functions will continue with using polynomial
identities to solve problems.
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Performance Assessments/Tasks
Click on the links below to access performance tasks where students are asked to demonstrate an understanding of functions.
Math Assessment Project
Representing Polynomials Graphically
http://map.mathshell.org/download.php?fileid=1744
New York State Common Core Mathematics Curriculum: Engage New York: Math I
https://www.engageny.org/resource/algebra-ii-module-1-topic-b-lesson-12
Polynomials and Factoring
https://www.sophia.org/topics/polynomials-and-factoring
Career Connections/Everyday Applications
Polynomials can have real-world uses. Some careers require you to use complex math, including polynomials, to solve problems, draw conclusions and make
predictions. Economists: Economists use data and mathematical models and statistical techniques to conduct research, prepare reports, formulate plans and
interpret and forecast market trends. Their jobs often involve addressing economic problems related to the production and distribution of goods and services
and monetary and fiscal policies; Statisticians: Statisticians use mathematical techniques to analyze and interpret data and draw conclusions. Their work often
influences economic, social, political and military decisions, according to the BLS. Statisticians may work in government, education, health care and
manufacturing. Because the job requires the use of polynomials and other complex math, statisticians generally need at least a bachelor’s degree in statistics
or math with coursework in differential and integral equations, mathematical modeling and probability theory; Engineering Careers: Aerospace engineers,
chemical engineers, civil engineers, electrical engineers, environmental engineers, mechanical engineers and industrial engineers all need strong math skills.
Their jobs require them to make calculations using polynomial expressions and operations. For example, aerospace engineers may use polynomials to
determine acceleration of a rocket or jet, and mechanical engineers use polynomials to research and design engines and machines, according to
WeUseMath.org.; Science Careers: Physical and social scientists, including archaeologists, astronomers, meteorologists, chemists and physicists, need to use
polynomials in their jobs. Key scientific formulas, including gravity equations, feature polynomial expressions. These algebraic equations help scientists to
measure relationships between characteristics such as force, mass and acceleration. Astronomers use polynomials to help in finding new stars and planets and
calculating their distance from Earth, their temperature and other features, according to school-for-champions.com.
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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math III
(listing of included sources attached)
A.APR.4
Use polynomial
identities to solve
problems.
Prove polynomial identities and
use them to describe numerical relationships. For example, the
polynomial identity (x2 – y2)2=(x2 - y2)2 + (2xy)2 can be used to generate
Pythagorean triples.
Columbus City Schools
Essential Understanding
Students should be able to explain that an
identity shows a relationship between two
quantities or expressions, which is true for all
values of the variables, over a specified set.
Academic Vocabulary/
Language
Students should be able to prove polynomial
identities and use polynomial identities to
describe numerical relationships
average rate of change, binomial,
coefficient, constant, degree, divisor,
factor, end behavior, identity, maximum,
minimum, monomial, polynomial, power,
quotient, rational, remainder, roots,
terms, trinomial, x-intercepts, y-intercepts,
zero product property, zeros
Extended Understanding
Tier 2 Vocabulary
Some information below includes additional
mathematics that students should learn in order
to take advanced courses such as calculus,
advanced statistics, or discrete mathematics and
goes beyond the mathematics that all students
should study in order to be college- and careerready: Ask students to use the vertical
multiplication to write out term-by-term
multiplication to generate (x + y) 3 from the
expanded form of (x + y) 2. Then use that
expanded result to expand (x + y) 4, use that result
to expand (x + y) 5 , and so on. Students should
begin to see the arithmetic that generates the
entries in Pascal’s triangle.
generate, prove, triples
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CCSSM Description
The operations of addition, subtraction and multiplication can be
applied to polynomials. A polynomial identity is just a true equation,
often generalized so that it can apply to more than one situation.
Identities are proven by showing that one side of an equation is equal
to the other. This takes the same skills used to organize equations and
expressions.
I can prove polynomial identities
I Can Statements
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Instructional Strategies
This cluster is an opportunity to highlight polynomial identities that are commonly used in solving problems. To learn these identities, students need experience using them to
solve problems. Students should develop familiarity with the following special products. Students should be able to prove any of these identities. Furthermore, they should
develop sufficient fluency with the first four of these so that they can recognize expressions of the form on either side of these identities in order to replace that expression
with an equivalent expression in the form of the other side of the identity:
(x + y) 2 = x 2 + 2xy + y 2
(x - y) 2 = x 2 - 2xy + y 2
(x + y)(x - y) = x 2 - y 2
(x + a)(x + b) = x 2 + (a + b)x + ab
(x + y) 3 = x 3 + 3 x 2 y + 3xy 2 + y 3
(x - y) 3 = x 3 - 3x 2 y + 3xy 2 – y
With identities such as these, students can discover and explain facts about the number system. For example, in the multiplication table, the perfect squares appear on the
diagonal. Diagonally, next to the perfect squares are “near squares,” which are one less than the perfect square. Why? • Why is the sum of consecutive odd numbers
beginning with 1 always a perfect square? • Numbers that can be expressed as the sum of the counting numbers from 1 to n are called triangular numbers. What do you
notice about the sum of two consecutive triangular numbers? Explain why it works. • The sum and difference of cubes are also reasonable for students to prove. The focus of
this proof should be on generalizing the difference of cubes formula with an emphasis toward finite geometric series.
Common Misconceptions/Challenges
Students often look at a polynomial in a standard window on a grapher and do not investigate further properties that may be seen using different window settings. It is
important for students to use their knowledge of polynomials to predict what its graph may look like, then check their predictions on their grapher.
Common Core Support
Institute for Mathematics and Education Learning Progressions Narratives
http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_algebra_2013_07_03.pdf
Ohio Learning Standards
http://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/Math-Standards.pdf.aspx
Illustrative Mathematics:
https://www.illustrativemathematics.org/blueprints/M3/2
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Textbook and Curriculum Resources
Integrated Math III, McGraw Hill
Chapter 4
Common Core State Standards Station Activities for Mathematics II, Walch Education, 2014
Operations with Polynomials, pp. 44-56
Columbus City Schools Curriculum Guide, 2013, Math III, Quarter 2
https://www.khanacademy.org/math/algebra/algebra-functions
Prior Knowledge
Future Learning
In Grade 6, students began using the properties of operations to rewrite
expressions in equivalent forms.
The study of polynomial functions will continue with students learning to
rewrite rational expressions.
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Performance Assessments/Tasks
Click on the links below to access performance tasks where students are asked to demonstrate an understanding of functions.
New York State Common Core Mathematics Curriculum: Engage New York: Math I
https://www.engageny.org/resource/algebra-ii-module-1-topic-b-lesson-12
POLYNOMIALS
https://www.sophia.org/search/tutorials?q=polynomials
ILLUSTRATIVE MATHEMATICS
https://www.illustrativemathematics.org/blueprints/M3/2
Non Negative Polynomials
https://www.illustrativemathematics.org/content-standards/HSA/APR/A/1/tasks/1656
Powers of 11
https://www.illustrativemathematics.org/content-standards/HSA/APR/A/1/tasks/1654
Career Connections/Everyday Applications
Polynomials can have real-world uses. Some careers require you to use complex math, including polynomials, to solve problems, draw conclusions and make
predictions. Economists: Economists use data and mathematical models and statistical techniques to conduct research, prepare reports, formulate plans and
interpret and forecast market trends. Their jobs often involve addressing economic problems related to the production and distribution of goods and services
and monetary and fiscal policies; Statisticians: Statisticians use mathematical techniques to analyze and interpret data and draw conclusions. Their work often
influences economic, social, political and military decisions, according to the BLS. Statisticians may work in government, education, health care and
manufacturing. Because the job requires the use of polynomials and other complex math, statisticians generally need at least a bachelor’s degree in statistics
or math with coursework in differential and integral equations, mathematical modeling and probability theory; Engineering Careers: Aerospace engineers,
chemical engineers, civil engineers, electrical engineers, environmental engineers, mechanical engineers and industrial engineers all need strong math skills.
Their jobs require them to make calculations using polynomial expressions and operations. For example, aerospace engineers may use polynomials to
determine acceleration of a rocket or jet, and mechanical engineers use polynomials to research and design engines and machines, according to
WeUseMath.org.; Science Careers: Physical and social scientists, including archaeologists, astronomers, meteorologists, chemists and physicists, need to use
polynomials in their jobs. Key scientific formulas, including gravity equations, feature polynomial expressions. These algebraic equations help scientists to
measure relationships between characteristics such as force, mass and acceleration. Astronomers use polynomials to help in finding new stars and planets and
calculating their distance from Earth, their temperature and other features, according to school-for-champions.com.
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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math III
(listing of included sources attached)
A.APR.6
Rewrite rational
expressions.
Rewrite simple rational
expressions in different forms;
write a(x)/b(x) in the form q(x) +r(x)/b(x), where a(x), b(x), q(x), and
r(x) are polynomials with the degree of r(x) less than the degree of
b(x), using inspections, long division, or for more complicated
examples, a computer algebra system.
Essential Understanding
Students should be able to use inspection to
rewrite simple rational expressions in different
form.
Students should be able to use long division to
rewrite simple rational expressions in different
forms.
Students should be able to use a computer
algebra system to rewrite complicated rational
expressions in different form.
Academic Vocabulary/
Language
coefficient, CAS, constant, degree, divisor,
expression, factor, long division,
monomial, polynomial, power, quotient,
radical, rational, remainder, roots, terms,
synthetic division, trinomial, x-intercepts,
y-intercepts, zero product property, zeros
Tier 2 Vocabulary
inspection, rewrite, simple, strategy
CCSSM Description
Students will learn strategies for rewriting rational expressions in
different forms. In order to rewrite simple rational expressions in
different forms, students need to understand that the rules governing
the arithmetic of rational expressions are the same rules that govern
the arithmetic of rational numbers (i.e., fractions). To add fractions,
we use a common denominator: The operations of addition,
subtraction and multiplication can be applied to polynomials. This
cluster is the logical extension of the earlier standards on polynomials
and the connection to the integers. This takes the same skills used to
organize equations and expressions. Some students may need a
review with the arithmetic of simple rational expressions.
9
I Can Statements
Columbus City Schools
9
Extended Understanding
The use of synthetic division may be introduced
as a method but students should recognize its
limitations (division by a linear term). When
students use methods that have not been
developed conceptually, they often create
misconceptions and make procedural mistakes
due to a lack of understanding as to why the
method is valid. They also lack the understanding
to modify or adapt the method when faced with
new and unfamiliar situations. Suggested viewing
Synthetic Division: How to understand It by not
doing it. http://www.youtube.com/watch?v=VQ6jBYn3Oc
I can rewrite rational expressions using different strategies: inspection, long or synthetic
division, computer algebra systems.
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Instructional Strategies
This cluster is the logical extension of the earlier standards on polynomials and the connection to the integers. Now, the arithmetic of rational functions is governed by the
same rules as the arithmetic of fractions, based first on division. This cluster is the logical extension of the earlier standards on polynomials and the connection to the integers.
Now, the arithmetic of rational functions is governed by the same rules as the arithmetic of fractions, based first on division. In order to rewrite simple rational expressions in
different forms, students need to understand that the rules governing the arithmetic of rational expressions are the same rules that govern the arithmetic of rational numbers
(i.e., fractions). To add fractions, we use a common denominator. Suggested resources/tools include: graphing calculators, graphing software (including dynamic geometry
software), Computer Algebra Systems.
Common Misconceptions/Challenges
Students with only procedural understanding of fractions are likely to cancel terms (rather than factors of) in the numerator and denominator of a fraction. Emphasize the
structure of the rational expression: that the whole numerator is divided by the whole denominator. In fact, the word “cancel” likely promotes this misconception. It would be
more accurate to talk about dividing the numerator and denominator by a common factor.
Common Core Support
Institute for Mathematics and Education Learning Progressions Narratives
http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_algebra_2013_07_03.pdf
Ohio Learning Standards
http://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/Math-Standards.pdf.aspx
Illustrative Mathematics:
https://www.illustrativemathematics.org/blueprints/M3/2
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Textbook and Curriculum Resources
Integrated Math III, McGraw Hill
Chapter 4
Common Core State Standards Station Activities for Mathematics II, Walch Education, 2014
Operations with Polynomials, pp. 44-56
Columbus City Schools Curriculum Guide, 2013, Math III, Quarter 2
https://www.khanacademy.org/math/algebra/algebra-functions
Prior Knowledge
Future Learning
In Grade 6, students began using the properties of operations to rewrite
expressions in equivalent forms.
The study of polynomial functions will continue with students learning to
representing and solving equations and inequalities graphically.
Columbus City Schools
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Performance Assessments/Tasks
Click on the links below to access performance tasks where students are asked to demonstrate an understanding of functions.
New York State Common Core Mathematics Curriculum: Engage New York: Math I
https://www.engageny.org/resource/algebra-ii-module-1-topic-b-lesson-12
POLYNOMIALS
https://www.sophia.org/search/tutorials?q=polynomials
ILLUSTRATIVE MATHEMATICS
https://www.illustrativemathematics.org/blueprints/M3/2
Non Negative Polynomials
https://www.illustrativemathematics.org/content-standards/HSA/APR/A/1/tasks/1656
Powers of 11
https://www.illustrativemathematics.org/content-standards/HSA/APR/A/1/tasks/1654
Career Connections/Everyday Applications
Polynomials can have real-world uses. Some careers require you to use complex math, including polynomials, to solve problems, draw conclusions and make
predictions. Economists: Economists use data and mathematical models and statistical techniques to conduct research, prepare reports, formulate plans and
interpret and forecast market trends. Their jobs often involve addressing economic problems related to the production and distribution of goods and services
and monetary and fiscal policies; Statisticians: Statisticians use mathematical techniques to analyze and interpret data and draw conclusions. Their work often
influences economic, social, political and military decisions, according to the BLS. Statisticians may work in government, education, health care and
manufacturing. Because the job requires the use of polynomials and other complex math, statisticians generally need at least a bachelor’s degree in statistics
or math with coursework in differential and integral equations, mathematical modeling and probability theory; Engineering Careers: Aerospace engineers,
chemical engineers, civil engineers, electrical engineers, environmental engineers, mechanical engineers and industrial engineers all need strong math skills.
Their jobs require them to make calculations using polynomial expressions and operations. For example, aerospace engineers may use polynomials to
determine acceleration of a rocket or jet, and mechanical engineers use polynomials to research and design engines and machines, according to
WeUseMath.org.; Science Careers: Physical and social scientists, including archaeologists, astronomers, meteorologists, chemists and physicists, need to use
polynomials in their jobs. Key scientific formulas, including gravity equations, feature polynomial expressions. These algebraic equations help scientists to
measure relationships between characteristics such as force, mass and acceleration. Astronomers use polynomials to help in finding new stars and planets and
calculating their distance from Earth, their temperature and other features, according to school-for-champions.com.
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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math III
(listing of included sources attached)
Graph polynomial
F.IF.7C
functions, identifying
zeros when suitable
factorizations are
avaialble and showing end behavior.
Graph polynomial functions, identifying zeros when suitable
factorizations are available, and showing end behavior.
CCSSM Description
A function can be described in various ways, such as by a graph.
The graph of a function is often a useful way of visualizing the
relationship of the function models. Manipulating a
mathematical expression for a function can throw light on the
function’s properties. A graphing utility or a computer algebra
system can be used to experiment with properties of these
functions and their graphs .
Essential Understanding
Students should be able to accurately graph
polynomial functions.
Academic Vocabulary/
Language
constant
Extended Understanding
Add families of functions, one at a time, to
the students’ knowledge base so they can
see connections among behaviors of the
various functions. Provide numerous
examples of real-world contexts, such as
exponential growth and decay situations
(e.g., a population that declines by 10%
per year) to help students apply an
understanding of functions in context.
Examine rational functions on a graphing
calculator and discuss why, for example,
the tabular representation shows an
“Error” message for some values of y.
Students need to be able to verbalize why
a function has asymptotes and distinguish
between asymptotes and holes.
end
behavior
exponential
factorization
linear
parent
function
polynomial
function
rate of
change
sequence
slope
standard
form
x- intercept
y-intercept
zeros
Tier 2 Vocabulary
algebraically
analyze
compare
contrast
domain
graph
identify
numerically
sketch
symbolically
terms
variable
I can graph polynomial functions accurately.
I can graph functions expressed symbolically and show key features of the graph.
I Can Statements
I can graph simple cases by hand and use technology to show more complicated cases
I can identify zeros when factorable and show end behavior.
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Instructional Strategies
Graphing utilities on a calculator and/or computer can be used to demonstrate the changes in behavior of a function as various parameters are varied. Real-world
problems, such as maximizing the area of a region bound by a fixed perimeter fence, can help to illustrate applied uses of families of functions
Common Misconceptions/Challenges
Students may believe that each family of functions (e.g., quadratic, square root, etc.) is independent of the others, so they may not recognize commonalities among all
functions and their graphs.
Students may believe that each family of functions (e.g., quadratic, square root, etc.) is independent of the others, so they may not recognize commonalities among all
functions and their graphs. Students may also believe that skills such as factoring a trinomial or completing the square are isolated within a unit on polynomials, and that
they will come to understand the usefulness of these skills in the context of examining characteristics of functions.
Common Core Support
Common Core Math: Computing Technology for Math Excellence: High School Functions Teaching and Learning Resources
http://www.ct4me.net/Common-Core/hsfunctions/hsf-interpreting-functions.htm
Institute for Mathematics and Education Learning Progressions Narratives
http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_functions_2013_07_02.pdf
Ohio Department of Education Model Curriculum
http://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/High_School_Functions_Model_Curriculum_March2015.pdf.aspx
Illustrative Mathematics
https://www.illustrativemathematics.org/blueprints/M1
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Textbook and Curriculum Resources
ORC (Ohio Resource Center: The Ohio State University)
http://www.ohiorc.org/search/results/?txtSearchText=functions
https://learnzillion.com/lessonsets/470
Virtual Nerd
Graphing Polynomials
http://www.virtualnerd.com/search/search.php?query=graphing+polynomials&search=1
Integrated Math I, McGraw Hill
Chapter 4
Prior Knowledge
Future Learning
In Grade 7, students are exposed to the idea that rewriting an expression can
shed light on the meaning of the expression. This idea is expanded upon as
students explore functions in high school and recognize how the form of the
equation can provide clues about zeros, asymptotes, etc.
Learning features of parent functions, the simplest form of a family of
function, and features of family functions can increase understanding of
functions. These skills will be needed when students study trigonometric
functions later in the year.
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Performance Assessments/Tasks
Click on the links below to access performance tasks where students are asked to demonstrate an understanding of the concept of functions.
Inside Mathematics
Sorting Functions
http://www.insidemathematics.org/assets/common-core-math-tasks/sorting%20functions.pdf
NRICH
http://nrich.maths.org/773
Illustrative Mathematics
Modeling London's Population
https://www.illustrativemathematics.org/content-standards/HSF/IF/C/7/tasks/1595
Career Connections
Any career that involves the need to articulate verbally the relationships between variables arising in everyday contexts can utilize the study of functions.
This include health care area, science, and any career involving sales.
Students can complete the following concept development activities (Representing Functions of Everyday Situations) where they are asked to :
• Translate between everyday situations and sketch graphs of relationships between variables.
• Interpret algebraic functions in terms of the contexts in which they arise.
• Reflect on the domains of everyday functions and in particular whether they should be discrete or Continuous
http://map.mathshell.org/download.php?fileid=1740
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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math III
(listing of included sources attached)
A.SSE.3C, ASSE.4
Write expressions in
equivalent forms to solve
problems.
Use the properties of exponents to transform expressions for exponential
functions. For example the expression 1.15t can be rewritten as (1.151/12)12t
to reveal thee approximate equivalent monthly interest rate if the annual
rate is 15%.
Essential Understanding
Students should be able to use the
properties of exponents to transform
simple expressions for exponential
functions.
Students should know the difference
between an arithmetic sequence and a
geometric sequence.
Students should be able to use a formula
to solve real world problems.
Derive the formula for the sum of a finite geometric series (when the
common ratio is not 1), and use the formula to solve problems. For example,
calculate mortgage payments.
Academic Vocabulary/ Language
arithmetic
expression
coefficient
completing
the square
derive
finite
geometric
sequence
linear
numerical
expression
sequence
Tier 2 Vocabulary
equivalent
factors
interpret
phenomena
variable
term
properties
sum
CCSSM Description
The ability to interpret and create expressions to model mathematical phenomena is
one of the most important skills an education in mathematics can offer. The different
expressions can tell us about the quantities they represent; being able to rewrite
expressions in another form leads to efficiency when solving a problems. Changing
the forms of expressions, such as factoring or completing the square, or transforming
expressions from one exponential form to another, are processes that are guided by
goals (e.g., investigating properties of families of functions and solving contextual
problems).
Extended Understanding
Provide opportunities for students to use
graphing utilities to explore the effects of
parameter changes on a graph.
I can identify the different parts of the expression and explain their meaning within the context
of a problem.
I Can Statements
I can use properties of exponents (such as power of a power, product of powers, power of a
product, and rational exponents, etc.) to write an equivalent form of an exponential function
to reveal and explain specific information about its approximate rate of growth or decay.
I can develop the formula for the sum of a finite geometric series when the ratio is not 1.
I can use the formula to solve real world problems such as calculating the height of a tree after
n years given the initial height of the tree and the rate the tree grows each year.
I can calculate mortgage payments
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Instructional Strategies
This cluster focuses on linking expressions and functions, i.e., creating connections between multiple representations of functional relations – the dependence
between a quadratic expression and a graph of the quadratic function it defines, and the dependence between different symbolic representations of exponential
functions. Teachers need to foster the idea that changing the forms of expressions, such as factoring or completing the square, or transforming expressions from one
exponential form to another, are not independent algorithms that are learned for the sake of symbol manipulations. They are processes that are guided by goals (e.g.,
investigating properties of families of functions and solving contextual problems).
Factoring methods that are typically introduced in elementary algebra and the method of completing the square reveals attributes of the graphs of quadratic
functions, represented by quadratic equations.
• The solutions of quadratic equations solved by factoring are the x – intercepts of the parabola or zeros of quadratic functions.
• A pair of coordinates (h, k) from the general form f(x) = a(x – h) 2 +k represents the vertex of the parabola, where h represents a horizontal shift and k represents a
vertical shift of the parabola y = x2 from its original position at the origin.
• A vertex (h, k) is the minimum point of the graph of the quadratic function if a › 0 and is the maximum point of the graph of the quadratic function if a ‹ 0.
Understanding an algorithm of completing the square provides a solid foundation for deriving a quadratic formula.
Translating among different forms of expressions, equations and graphs helps students to understand some key connections among arithmetic, algebra and geometry.
The reverse thinking technique (a process that allows working backwards from the answer to the starting point) can be very effective. Have students derive
information about a function’s equation, represented in standard, factored or general form, by investigating its graph. Offer multiple real-world examples of
exponential functions.
Common Misconceptions/Challenges
Some students may believe that factoring and completing the square are isolated techniques within a unit of quadratic equations. Teachers should help students to see the
value of these skills in the context of solving higher degree equations and examining different families of functions. Students may think that the minimum (the vertex) of the
graph of y = (x + 5)2 is shifted to the right of the minimum (the vertex) of the graph y = x 2 due to the addition sign. Students should explore examples both analytically and
graphically to overcome this misconception. Some students may believe that the minimum of the graph of a quadratic function always occur at the y-intercept. Some
students cannot distinguish between arithmetic and geometric sequences, or between sequences and series. To avoid this confusion, students need to experience both
types of sequences and series
Common Core Support
Illustrative Mathematics: Learning Progressions
http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_algebra_2013_07_03.pdf
Ohio Department of Education Model Curriculum
https://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/High_School_Algebra_Model_Curriculum_March2015.pdf.aspx
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Textbook and Curriculum Resources
Problem-Based Tasks for Mathematics I Common Core State Standards, Walch Education, 2013
Options of Interest, pp. 45-48
Columbus City Schools Curriculum Guide, Math III Quarter 3, 2013
https://learnzillion.com/lessonsets/470http://www.virtualnerd.com/tutorials/?id=Alg1_9_2_17Integrated Math I, McGraw Hill
Chapter 6
Prior Knowledge
Future Learning
In Grade 8, students compare tables, graphs, expressions and equations of linear
relationships.
Future learning will continue study of exponential functions with
creating equations that describe numbers or relationships.
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Performance Assessments/Tasks
Click on the links below to access performance tasks.
Illustrative Mathematics
Forms of exponential expressions
https://www.illustrativemathematics.org/content-standards/HSA/SSE/B/3/tasks/1305
A Lifetime of Savings
https://www.illustrativemathematics.org/content-standards/HSA/SSE/B/4/tasks/1283
Triangle Series
https://www.illustrativemathematics.org/content-standards/HSA/SSE/B/4/tasks/442
Cantor Set
https://www.illustrativemathematics.org/content-standards/HSA/SSE/B/4/tasks/929
Course of Antibiotics
https://www.illustrativemathematics.org/content-standards/HSA/SSE/B/4/tasks/805
YouTube Explosion
https://www.illustrativemathematics.org/content-standards/HSA/SSE/B/4/tasks/1797
Career/ Everyday Connections
Insurance, Real Estate, Sales, Science & Engineering
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Ohio’s Learning Standards – Clear Learning Targets
Integrated Mathematics III
Essential Understanding
A.APR.1
Understand that polynomials
form a system analogous to the
integers, namely, they are
closed under the operations of addition,
subtraction, and multiplication; add, subtract, and
multiply polynomials.
- Students can add, subtract, and
multiply polynomials.
Extended Understanding
- Students can divide polynomials.
- Students can factor polynomials.
Academic
Vocabulary/Language
- Polynomial
- Monomial
- Binomial
- Trinomial
- distribute
- like terms
Tier 2 Vocabulary
- understand
- analogous
I Can Statements



I can identify that the sum, difference, or product of two polynomials will always be a polynomial, which means that polynomials are
closed under the operations of addition, subtraction, and multiplication.
I can define “closure”.
I can apply arithmetic operations of addition, subtraction, and multiplication to polynomials.
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Instructional Strategies
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Common Misconceptions and Challenges
Textbook and Curriculum Resources
McGraw-Hill: Integrated Math II
Chapter 1-1, 1-2, 1-3, 1-3 (explore), 1-4
http://ccssmath.org/?s=apr.1
Polynomial Addition and Subtraction- APR.1
Polynomial Multiplication- APR.1
Adding, Multiplying, and Subtracting Monomials- APR.1
https://www.sophia.org/ccss-math-standard-9-12aapr1-pathway


A-APR Non-Negative Polynomials
A-APR Powers of 11
https://sites.google.com/site/commoncorewarwick/home/unit-ofstudies/algebra-2/a-apr-1
Career Connections
Social scientists and related occupations
Economists
Education, training, library, and museum occupations
Teachers-adult literacy and remedial and self-enrichment education
Teachers-postsecondary
Teachers-preschool, kindergarten, elementary, middle, and secondary
Teachers-special education
Health diagnosing and treating occupations
Registered nurses
Columbus City Schools
Aerospace engineers
Chemical engineers
Civil engineers
Electrical engineers
Environmental engineers
Industrial engineers
Materials engineers
Mechanical engineers
Nuclear engineers
Petroleum engineers
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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math III
(listing of included sources attached)
F.IF.7E, F.IF.8B
Analyze functions using
different
representations.
Graph exponential and logarithmic
functions, showing intercepts and end behavior.
Use the properties of exponents to interpret expressions for exponential
functions. For example, identify percent rate of change functions such as
y-(1.02)t , y=(0.97)t , y=(1.01)12t , y=(1.2)t/10 , and classify them as representing
exponential growth or decay.
CCSSM Description
A function can be described in various ways, such as by a graph (e.g.,
the trace of a seismograph); by a verbal rule, as in, “I’ll give you a
state, you give me the capital city;” by an algebraic expression like f(x)
= a + bx; or by a recursive rule. The graph of a function is often a
useful way of visualizing the relationship of the function models, and
manipulating a mathematical expression for a function and can throw
light on the function’s properties. A graphing utility or a computer
algebra system can be used to experiment with properties of these
functions and their graphs and to build computational models of
functions.
9
Essential Understanding
Students will be expected to be able to graph
functions expressed symbolically and show key
features of the graph, by hand in simple cases and
using technology for more complicated
cases.*(Modeling standard).
Students will be expected to be able to graph
exponential and logarithmic functions, showing
intercepts and end behavior, and trigonometric
functions, showing period, midline, and
amplitude.
Students will be expected to write a function
defined by an expression in different but
equivalent forms to reveal and explain different
properties of the function
Academic Vocabulary/
Language
arithmetic, axes, base, constant, decay,
differences, equation, explicit,
exponential, expression, factors, formula,
function, geometric sequence, graph,
growth, input, intervals, inverse, linear,
logarithm, model, ordered pair, output,
parameters, percent, polynomial,
quadratic, quantity, rate, recursive,
relation, scale, sequence, table, unit
Tier 2 Vocabulary
graph, identify, interpret,
Extended Understanding
Involve students in activities that include
collection and analysis of data to generalize
function behaviors. For example, they can take a
cup filled with pennies, spill them onto a table,
count how many came up “heads,” put only those
pennies back in the cup, and repeat this process
several times. In the end, they will generate a
table of values that will model an exponential
decay function with a base of ½.
I can graph exponential, logarithmic, and trigonometric functions.
I can describe key features of exponential, logarithmic, and trigonometric functions.
I Can Statements
I can classify the exponential function as exponential growth or decay by examining the base.
I can use the properties of exponents to interpret expressions for exponential functions in a
real-world context.
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Instructional Strategies
Explore various families of functions and help students to make connections in terms of general features. Use various representations of the same function to emphasize
different characteristics of that function. For example, the y-intercept of the function y = x2 -4x – 12 is easy to recognize as (0, -12). However, rewriting the function as
y = (x – 6)(x + 2) reveals zeros at (6, 0) and at ( -2, 0). Furthermore, completing the square allows the equation to be written as y = (x – 2)2 – 16, which shows that the
vertex (and minimum point) of the parabola is at (2, -16).
Hands-on materials (e.g., paper folding, building progressively larger shapes using pattern blocks, etc.) can be used as a visual source to build numerical tables
for examination.
Common Misconceptions/Challenges
Students oversimplify rules of exponents. For example, a student might think/claim 𝑒 𝑎+𝑏 = 𝑒 𝑎 + 𝑒 𝑏. This may be the result of students failing to attribute meaning to
exponential symbols. Students interpret negative exponents incorrectly or fail to connect the negative symbol back to the idea of inverses. Students make the assumption
that a correctly followed algorithm will only ever give correct answers. For example, in solving 𝑙𝑜𝑔2 (𝑥 − 4) = 3 − 𝑙𝑜𝑔2(𝑥 + 3) a student might correctly follow the solution
algorithm and claim the answer is 𝑥 = −4, 5 without noting the fact that -4 is an invalid solution since plugging it in to 𝑙𝑜𝑔2 (𝑥 − 4) results in an input value outside the domain
of the logarithmic function.
Common Core Support
Institute for Mathematics and Education Learning Progressions Narratives
http://commoncoretools.me/wp-content/uploads/2012/12/ccss_progression_functions_2012_12_04.pdf
Ohio Learning Standards
http://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/Math-Standards.pdf.aspx
Illustrative Mathematics:
https://www.illustrativemathematics.org/blueprints/M3
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Textbook and Curriculum Resources
Integrated Math III, McGraw Hill
Chapter 6
Common Core State Standards Station Activities for Mathematics I, Walch Education, 2014
Interpreting Exponential Functions, pp. 110-117
Columbus City Schools Curriculum Guide, 2013, Math III, Quarter 3
https://www.khanacademy.org/math/algebra/algebra-functions
Prior Knowledge
Future Learning
In Grade 8, students compare functions by looking at equations, tables and
graphs, and focus primarily on linear relationships. In high school, examination of
functions is extended to include recursive and explicit representations and
sequences of numbers that may not have a linear relationship.
Future learning will include analyzing functions—comparing properties of
two functions each represented in a different way.
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Performance Assessments/Tasks
Click on the links below to access performance tasks where students are asked to demonstrate an understanding of functions.
POLYNOMIALS
https://www.sophia.org/search/tutorials?q=polynomials
ILLUSTRATIVE MATHEMATICS
Classifying Equations of Parallel and Perpendicular Lines
https://www.illustrativemathematics.org/blueprints/M3/2 http://map.mathshell.org/download.php?fileid=1724
Representing Quadratic Functions Graphically
http://map.mathshell.org/download.php?fileid=1734
Representing Functions of Everyday Situations
http://map.mathshell.org/download.php?fileid=1740
Representing Polynomials Graphically
http://map.mathshell.org/download.php?fileid=1744
Career Connections/Everyday Applications
Logarithms (graphing/analyzing), the inverses of exponential functions, are used in many occupations. Perhaps the most well-known use of logarithms is in the
Richter scale, which determines the intensity and magnitude of earthquakes. Yet, there are many other professionals who use logarithms in their careers.
Anyone who calculates the quantity of things that increase or decrease exponentially uses logarithms. This includes engineers, coroners, financiers, computer
programmers, mathematicians, medical researchers, farmers, physicists and archaeologists. Because there is no definitive list of careers that require the use
of logarithms, below is a brief sampling of how some careers employ these log
Read more : http://www.ehow.com/info_8649362_careers-use-logarithms.html
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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math I
(listing of included sources attached)
F.BF.1, F.BF.1A,
F.BF.2
Build a function that
models.
Write a linear function that
describes a relationship between
two quantities.
Determine an explicit expression, a recursive process, or steps for
calculation from a context.
Write arithmetic sequences both recursively and with an explicit
formula use them to model situations, and translate between the two forms.
CCSSM Description
Functions can be used to make predictions about future behaviors when
modeling real life situations. For students to recognize a functional
relationship, they need to recognize there is a correspondence and
see/understand the correspondence matches each element of the first set with an element of the second set. Once it is known that the
relationship is a function, students can determine the rule for the
function.
Essential Understanding
Examination of functions is extended to
include recursive and explicit
representations and sequences of
numbers that may not have a linear
relationship.
Academic Vocabulary/
Language
arithmetic
sequence
correspondence
Extended Understanding
Using a variety of functions (e.g., linear,
exponential, constant, students can
increase understanding of the different
representations by representing functions
as a set of ordered pairs, a table, a graph,
and an equation.
direct variation
explicit
formula
function
geometric
sequence
inverse
function
inverse
relationship
recursive
quantities
Tier 2 Vocabulary
compare
construct
model
observe
prove
I can, from context, either write an explicit expression, define a recursive process, or
describe the calculations meeded to model a function between two quantities.
I can combine standard function types, such as linear and exponential, using arithmetic
operations. I can compose functioins.
I Can Statements
I can write arithmetic sequences recursively and explicityly, use the two forms to model a
sitation and translate between the two forms.
I can write geometric sequences recursively and expliciitly, use the tow forms to model a
sistuation, and translate between the two forms.
I can understand that linear functions are the explicity form of recursievely-defined
arithmetic sequences and that exponential functions are the explicit form of recursivelydefined geometric sequences.
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Instructional Strategies
Provide a real-world example (e.g., a table showing how far a car has driven after a given number of minutes, traveling at a uniform speed), and examine the
table by looking “down” the table to describe a recursive relationship, as well as “across” the table to determine an explicit formula to find the distance
traveled if the number of minutes is known.
Write out terms in a table in an expanded form to help students see what is happening. For example, if the y-values are 2, 4, 8, 16, they could be written as 2, 2(2), 2(2)
so that students recognize that 2 is being used multiple times as a factor.
Focus on one representation and its related language – recursive or explicit – at a time so that students are not confusing the formats.
Provide examples of when functions can be combined, such as determining a function describing the monthly cost for owning two vehicles when a function
for the cost of each (given the number of miles driven) is known.
Using visual approaches (e.g., folding a piece of paper in half multiple times), use the visual models to generate sequences of numbers that can be explored and describ
recursive and explicit formulas. Emphasize that there are times when one form to describe the function is preferred over the other.
Common Misconceptions/Challenges
Students may believe that the best (or only) way to generalize a table of data is by using a recursive formula. Students naturally tend to look “down” a table to find the
pattern but need to realize that finding the 100th term requires knowing the 99thterm unless an explicit formula is developed.
Students may also believe that arithmetic and geometric sequences are the same. Students need experiences with both types of sequences to be able to recognize the
difference and more readily develop formulas to describe them. Advanced students who study composition of functions may misunderstand function notation to represent
multiplication (e.g., f(g(x)) means to multiply the f and g function values).
When studying functions, students sometimes interchange the input and output values. This will lead to confusion about domain and range, and determining if a relation is
a function. This can also interfere with a student being able to find the appropriate inverse function, or the correct equation to model a relationship between two
quantities.
Common Core Support
Common Core Math: Computing Technology for Math Excellence: High School Functions Teaching and Learning Resources
http://www.ct4me.net/Common-Core/hsfunctions/hsf-interpreting-functions.htm
Institute for Mathematics and Education Learning Progressions Narratives
http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_functions_2013_07_02.pdf
Illustrative Mathematics: Practice and Content Standards: Toward Greater Focus and Coherence
https://www.illustrativemathematics.org/standards
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Textbook and Curriculum Resources
Problem-Based Tasks for Mathematics I Common Core State Standards, Walch Education, 2013
Texting for the Win, pp. 127-131
Jai’s Jeans, pp. 132-134
New Tablet, pp. 135-138
Glass Recycling, pp. 139-142
Problem-Based Tasks for Mathematics II Common Core State Standards, Walch Education, 2013
To Drill or Not to Drill?, pp. 138Pushing Envelopes, pp. 142-145
Common Core State Standards: Station Activities for Mathematics I
Relations Versus Functions/Domain and Range, pp. 85-93
Sequences, pp. 118-130
Real-World Situation Graphs pp. 194-208High School CCSS Mathematics I Curriculum Guide-Quarter 1 Curriculum Guide, 2013, pp. 161-204
Prior Knowledge
Future Learning
In Grade 8, students learn to compare functions by looking at
equations, tables and graphs, and focus primarily on linear
relationships.
Future learning will include working with inverse functions.
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Performance Assessments/Tasks
Click on the links below to access performance tasks.
Math Assessment Project
Generalizing Patterns: Table Tiles
http://map.mathshell.org/download.php?fileid=1716
Representing Linear and Exponential Growth
http://map.mathshell.org/download.php?fileid=1732
Modeling Motion: Rolling Cups
http://map.mathshell.org/download.php?fileid=1746
Inside Mathematics
Infinite Windows
http://www.insidemathematics.org/assets/problems-of-the-month/infinite%20windows.pdf
Slice and Dice
http://www.insidemathematics.org/assets/problems-of-the-month/slice%20and%20dice.pdf
First Rate
http://www.insidemathematics.org/assets/problems-of-the-month/calculating palindromes.pdf
http://www.insidemathematics.org/assets/problems-of-the-month/first%20rate.pdf
Cut It Out
http://www.insidemathematics.org/assets/problems-of-the-month/cut%20it%20out.pdf
Illustrative Mathematics
Summer Intern
Career
Connections
https://www.illustrativemathematics.org/content-standards/HSF/BF/A/1/tasks/72
Students can research and evaluate several options when purchasing a vehicle (e.g., new versus used, lease versus own, down payment, and interest rate).
They will examine the differences in gas mileage consumption by selecting two vehicles to evaluate (e.g., SUV versus compact hybrid). Once they choose a
vehicle,
they will use their evaluations to show why they chose the vehicle. Their research will include interviewing automotive professionals, visiting
https://www.illustrativemathematics.org/content-standards/HSF/LE/B/5/tasks/579
dealerships,
and navigating company websites.
https://www.illustrativemathematics.org/content-standards/HSF/LE/B/5/tasks/758
Applicable careers include business, finance, insurance and any career focused on making scholarly predictions.
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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math I
(listing of included sources attached)
F.BF.4, F.BF.4A
Essential Understanding
Build new functions from
existing functions.
Find inverse functions.
Solve a linear equation of the form
f(x)=c for a simple function f that has an inverse and write an expression for
the inverse.
Students should understand that an
inverse function does the reverse of a
given function. Square and square root
functions are examples of an inverse
function within the domain of
nonnegative numbers
.
Extended Understanding
Advanced students can expand their catalog
of functions to include exponential and
logarithmic cases. Students can learn to
contrast an invertible and non-invertible
function which is mentioned in the Functions
Progressions document as a reasonable
extension of the standard.
CCSSM Description
In simple terms, an inverse function undoes what the original function
does. Continued studies with parent functions can facilitate deeper
understanding of functions.
Academic Vocabulary/
Language
dependent
variable
direct variation
function
independent
variable
inverse
inverse function
invertible
non-invertible
Tier 2 Vocabulary
build
interchanging
-
I Can Statements
Columbus City Schools
I can solve a function for the dependent variable and write the inverse of a function by
interchanging the values of th dependent and independent variables.
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Instructional Strategies
Provide examples of inverses that are not purely mathematical to introduce the idea. For example, given a function that names the capital of a state,
f(Ohio) = Columbus, the inverse would be to input the capital city and have the state be the output such that f—1 (Denver) = Colorado.
Allow students to initially make tables of values by hand for some simple examples, such as y = x + 3 to examine the effects of changing the constant,
including the existence of inverses. Students can then examine additional effects and more complicated functions with technology.
Use real-world examples of functions and their inverses. For example, students might determine that folding a piece of paper in half 5 times results in 32
layers of paper, but that if they are given that there are 32 layers of paper, they can solve to find how many times the paper would have been folded in half.
Common Misconceptions/Challenges
Students may believe that the graph of y = (x – 4)3 is the graph of y = x 3 shifted 4 units to the left (due to the subtraction symbol). Examples should be explored by hand and
on a graphing calculator to overcome this misconception.
Students may also believe that even and odd functions refer to the exponent of the variable, rather than the sketch of the graph and the behavior of the function.
In f -1 (x) =3x+3, students may think -1 is an exponent.
Common Core Support
Institute for Mathematics and Education Learning Progressions Narratives
http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_functions_2013_07_02.pdf
Illustrative Mathematics
High School: Functions
https://www.illustrativemathematics.org/content-standards/HSF/BF
Integrated Math I
https://www.illustrativemathematics.org/blueprints/M1
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Textbook and Curriculum Resources
Problem-Based Tasks for Mathematics II Common Core State Standards, Walch Education, 2013
Finding Inverse Functions, pp. 156-159
LearnZillion
https://learnzillion.com/resources/31835#fndtn-resource__content
YouTube
Khan Academy: Finding Inverse Functions
https://www.youtube.com/watch?v=W84lObmOp8M
Inverse Functions
https://www.youtube.com/watch?v=Y-wxZdMMcYc
Inverse Functions- The Basics!:patrickjmt
https://www.youtube.com/watch?v=nSmFzOpxhbY
Integrated Math I, McGraw Hill
Chapter 4
Prior Knowledge
Future Learning
Understanding functional relationships as input and output values that have an
associated graph is introduced in Grade 8. In high school, changes in graphs is explored
in more depth, and the idea of functions having inverses is introduced.
Students will begin studies of linear equations and inequalities in one
variable and exponentials in future studies.
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Performance/Assessment Tasks
Click on the links below to access performance tasks where students are asked to demonstrate an understanding of functions.
Inside Mathematics
Digging Dinosaurs
http://www.insidemathematics.org/assets/problems-of-the-month/digging%20dinosaurs.pdf
Illustrative Mathematics
https://www.illustrativemathematics.org/content-standards/HSF/LE/B/5/tasks/579 Invertible or Not?
https://www.illustrativemathematics.org/content-standards/HSF/LE/B/5/tasks/758
https://www.illustrativemathematics.org/content-standards/HSF/BF/B/4/tasks/1374
Households
https://www.illustrativemathematics.org/content-standards/HSF/BF/B/4/tasks/234
Temperature Conversions
https://www.illustrativemathematics.org/content-standards/HSF/BF/B/4/tasks/364
Temperature in Degrees Fahrenheit and Celsius
https://www.illustrativemathematics.org/content-standards/HSF/BF/B/4/tasks/501
Career Connections
Population Studies: choosing a linear function to model the given data, and then use the inverse function to interpolate a data point (see Households task
above).
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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math III
(listing of included sources attached)
F.LE.4
Construct and compare
linear, quadratics, and
exponential models and
solve problems.
For exponential models, express as logarithm the solution to abct =d
where a,c, and d are numbers and the base b is 2, 10, or e; evaluate
the logarithm using technology.
Essential Understanding
Students will be expected to know how to express
logarithms as solutions to exponential functions
using bases 2, 10, and e.
Students will be expected to know how to use
technology to evaluate a logarithm.
Extended Understanding
Provide opportunities where students care given
examples of real-world situations that apply linear
and exponential functions to compare their
behaviors.
Academic Vocabulary/
Language
arithmetic, axes, base, constant, decay,
differences, equation, explicit,
exponential, expression, factors, formula,
function, geometric sequence, graph,
growth, input, intervals, inverse, linear,
logarithm, model, ordered pair, output,
parameters, percent, polynomial,
quadratic, quantity, recursive, relation,
scale, sequence, solution, table, unit
Tier 2 Vocabulary
CCSSM Description
express, properties, technology
Given sufficient information, e.g., a table of values together with
information about the type of relationship represented, students will
learn how to construct/evaluate the appropriate model. Technology
will be used.
9
I can use the properties of logs.
I Can Statements
I can describe the key features of logs.
I can use logarithmic form to solve exponential models.
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Instructional Strategies
Use technology to solve exponential equations such as 3*10x = 450. (In this case, students can determine the approximate power of 10 that would generate a value of 150.)
Students can also take the logarithm of both sides of the equation to solve for the variable, making use of the inverse operation to solve.
Instructional Resource Tools: Examples of real-world situations that apply linear and exponential functions to compare their behaviors; graphing calculators or computer
software that generates graphs and tables of functions; a graphing tool such as the one found at nlvm.usu.edu is one option.
Common Misconceptions/Challenges
Students may believe that all functions have a first common difference and need to explore to realize that, for example, a quadratic function will have equal second common
differences in a table. Students may also believe that the end behavior of all functions depends on the situation and not the fact that exponential function values will
eventually get larger than those of any other polynomial functions.
Common Core Support
Institute for Mathematics and Education Learning Progressions Narratives
http://commoncoretools.me/wp-content/uploads/2012/12/ccss_progression_functions_2012_12_04.pdf
Ohio Learning Standards
http://education.ohio.gov/getattachment/Topics/Ohio-s-New-Learning-Standards/Mathematics/Math-Standards.pdf.aspx
Illustrative Mathematics:
https://www.illustrativemathematics.org/blueprints/M3/2
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Textbook and Curriculum Resources
Integrated Math III, McGraw Hill
Chapters 2-7
Columbus City Schools Curriculum Guide, 2013, Math III, Quarter 3
https://www.khanacademy.org/math/algebra/algebra-functions
Prior Knowledge
Future Learning
While students in Grade 8 examine some nonlinear situations, most of the
functions explored are linear. Students will build on the understanding of
exponents that began in Grade 8.8.EE.1.
Next studies will include trigonometric functions---interpreting
functions.
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Performance Assessments/Tasks
Click on the links below to access performance tasks where students are asked to demonstrate an understanding of functions.
Carbon 14 Dating
https://www.illustrativemathematics.org/content-standards/HSF/LE/A/4/tasks/369
Bacteria Populations
https://www.illustrativemathematics.org/content-standards/HSF/LE/A/4/tasks/370
Comparing Exponentials
https://www.illustrativemathematics.org/content-standards/HSF/LE/A/tasks/213
Newton’s Law of Cooling
https://www.illustrativemathematics.org/content-standards/HSF/LE/A/4/tasks/382
Exponential Kiss
https://www.illustrativemathematics.org/content-standards/HSF/LE/A/4/tasks/1824
Other Tasks
https://www.illustrativemathematics.org/content-standards/HSF/LE/A/4
Career Connections/Everyday Applications
Jobs using quadratics/exponential models include: Military and Law Enforcement; Engineering , Science, Management and Clerical Work, Agriculture:
http://www.ehow.com/info_8711999_careers-use-quadratic-equations.html?%20%20%20utm_source=eHowMobileShare%26utm_medium=email
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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math I
(listing of included sources attached)
F.TF.1, F.TF.2
Extend domain of
trigonometric
function using the
unit circle.
Understand radian measure of an angle as the length of the
arc on the unit circle subtended by the angle.
Explain how the unit circle in the coordinate plane enables
the extension of trigonometric functions to all real numbers,
interpreted as radian measures of angles traversed
counterclockwise around the unit circle.
-
Essential Understanding
Students are expected to know that the radian
measure of an angle is the length of the arc on
the unit circle subtended by the angle.
Students are expected to be able to explain how
the unit circle in the coordinate plane enables
the extension of trigonometric functions to all
real numbers interpreted as radian measures of
angles traversed counterclockwise around the
unit circle
Academic Vocabulary/
Language
amplitude, angle, arc, arccosine(arccos),
arcsine (arcsin), arctangent (arctan), axes,
circle, clockwise, constant, coordinate, cosine
(cos), counterclockwise, degree, differences,
equation, expression, formula, Frequency,
function, graph, identity, input, intervals,
inverse, midline, model, ordered pair, output,
period, quadrant, quantity, radian, relation,
sine (sin), subtend, table, tangent (tan),
trigonometric, unit, unit circle
Tier 2 Vocabulary
explain, understand
CCSM Description
A unit circle is a circle with a radius of one. In trigonometry,
the unit circle is the circle of radius one centered at the origin
(0, 0) in the Cartesian coordinate system in the Euclidean
plane. Because the radius of the unit circle is one, the
trigonometric functions sine and cosine have special
relevance for the unit circle.
I can understand and explain that if the length of an arc subtended by an angle is
length as the radius of the circle, then the measure of the angle is 1 radian.
I Can Statements
the s
I can understand and explain that the graph of the function, f, is the graph of the equation y=f(x).
I can explain how radian measures of angles rotated counterclockwise in a unit circle are in a one-to-one
correspondence with the nonnegative real numbers, and that angles rotated clockwise in a unit circle are in a onto-one correspondence with the non-positive real numbers.
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Instructional Strategies
Use a compass and straightedge to explore a unit circle with a fixed radius of 1. Help students to recognize that the circumference of the circle is 2π, which
represents the number of radians for one complete revolution around the circle. Students can determine that, for example, π/4 radians would represent
a revolution of 1/8 of the circle or 45°. Having a circle of radius 1, the cosine, for example, is simply the x-value for any ordered pair on the circle
(adjacent/hypotenuse where adjacent is the x-length and hypotenuse is 1). Students can examine how a counterclockwise rotation determines a coordinate
of a particular point in the unit circle from which sine, cosine, and tangent can be determined. Some information below includes additional mathematics
that students should learn in order to take advanced courses such as calculus, advanced statistics, or discrete mathematics and goes beyond the mathematics that all s
study in order to be college- and career-ready: Some students can use what they know about 30-60-90 triangles and right isosceles triangles to determine
the values for sine, cosine, and tangent for π/3, π/4, and π/6. In turn, they can determine the relationships between, for example, the sine of π/6, 7π/6,
and 11π/6, as all of these use the same reference angle and knowledge of a 30-60-90 triangle. Provide students with real-world examples of periodic
functions. One good example is the average high (or low) temperature in a city in Ohio for each of the 12 months. These values are easily located at
weather.com and can be graphed to show a periodic change that provides a context for exploration of these functions. Allow plenty of time for students
to draw – by hand and with technology – graphs of the three trigonometric functions to examine the curves and gain a graphical understanding of why, for
example, cos (π/2) = 0 and whether the function is even (e.g., cos(-x) = cos(x)) or odd (e.g., sin(-x) = -sin(x)). Similarly, students can generalize how function
values repeat one another, as illustrated by the behavior of the curves.
Common Misconceptions/Challenges
Students may believe that there is no need for radians if one already knows how to use degrees. Students need to be shown a rationale for how radians are unique in terms
of finding function values in trigonometry since the radius of the unit circle is 1. Students may also believe that all angles having the same reference values have identical
sine, cosine and tangent values. They will need to explore in which quadrants these values are positive and negative.
Other challenges include: failure to identify the advantages of radian measurements over degree measurement.; overgeneralization; e.g. assuming all trigonometry
functions have a range of -1 to 1; confusion regarding domain restrictions when defining inverses.; confusion over inverse notation; failure to connect the Pythagorean
Theorem to other aspects of trigonometry; failure to identify the relationship between various trig functions such as sine and cosine; algorithmic oversimplification; e.g.
assuming sin(𝑎 + 𝑏) = sin(𝑎) + sin(𝑏).
Common Core Support
Common Core Math: Computing Technology for Math Excellence: High School Functions Teaching and Learning Resources
http://www.ct4me.net/Common-Core/hsfunctions/hsf-interpreting-functions.htm
Institute for Mathematics and Education Learning Progressions Narratives
http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_functions_2013_07_02.pdf
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Textbook and Curriculum Resources
Integrated Math III, McGraw Hill
Chapters 11-12
High School CCSS Mathematics I Curriculum Guide-Quarter 3 Curriculum Guide, 2013
Prior Knowledge
Future Learning
Students begin studying right triangles using the Pythagorean Theorem in Grade
8.
Future learning will include modeling periodic phenomena with
trigonometric functions.
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Performance Assessments/Tasks
Click on the links below to access performance tasks.
Illustrative Mathematics
Bicycle Wheel
https://www.illustrativemathematics.org/content-standards/HSF/TF/A/1/tasks/1873
What Exactly Is A Radian?
https://www.illustrativemathematics.org/content-standards/HSF/TF/A/1/tasks/1874
Trigonometric functions for arbitrary angles
https://www.illustrativemathematics.org/content-standards/HSF/TF/A/2/tasks/1692
Trigonometric Identities and Rigid Motions
https://www.illustrativemathematics.org/content-standards/HSF/TF/A/2/tasks/1698
Trig Functions and the Unit Circle
https://www.illustrativemathematics.org/content-standards/HSF/TF/A/2/tasks/1820
Properties of Trigonometric Functions
https://www.illustrativemathematics.org/content-standards/HSF/TF/A/2/tasks/1704
https://www.illustrativemathematics.org/content-standards/HSF/LE/B/5/tasks/579
Career
Connections
https://www.illustrativemathematics.org/content-standards/HSF/LE/B/5/tasks/758
Management Occupations, Administrative Support, Construction, Production, Professional, Farming, Installation
www.xpmath.com/careers
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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math III
(listing of included sources attached)
Essential Understanding
F.TF.5
Students are expected to be able to define the
parameters of trigonometric functions.
Model periodic
phenomena with
trigonometric
functions.
Students are expected to be able to interpret
trigonometric functions in context.
Choose trigonometric functions to model periodic
phenomena with specified amplitude, frequency, and midline.
Students are expected to identify and model
periodic phenomena in real world situations.
Extended Understanding
-
CCSM Description
The study of trigonometry is reserved for high school
students. In the Geometry conceptual category, students
explore right triangle trigonometry, with advanced students
working with laws of sines and cosines. In the conceptual
category of Functions, students connect the idea of functions
with trigonometry and explore the effects of parameter
changes on the amplitude, frequency and midline of
trigonometric graphs.
Provide students with a list of real-world applications
of periodic situations that can be modeled by using
trigonometric functions for students to explore.
Utilize graphing calculators or computer software to
generate the graphs of trigonometric functions.
Academic Vocabulary/
Language
amplitude, angle, arc, arccosine(arccos),
arcsine (arcsin), arctangent (arctan), axes,
circle, clockwise, constant, coordinate, cosine
(cos), counterclockwise, degree, differences,
equation, expression, formula, Frequency,
function, graph, identity, input, intervals,
inverse, midline, model, ordered pair, output,
period, quadrant, quantity, radian, relation,
sine (sin), subtend, table, tangent (tan),
trigonometric, unit, unit circle
Tier 2 Vocabulary
explain, understand
I can define and recognize the parameters of trigonometric functions.
I can interpret trig functions in real-world situations.
I Can Statements
Columbus City Schools
I can identify and model periodic phenomena in real-world situations.
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Instructional Strategies
Allow students to explore real-world examples of periodic functions. Examples include average high (or low) temperatures throughout the year, the height
of ocean tides as they advance and recede, and the fractional part of the moon that one can see on each day of the month. Graphing some real-world
examples can allow students to express the amplitude, frequency, and midline of each. Help students to understand what the value of the sine (cosine, or
tangent) means (e.g., that the number represents the ratio of two sides of a right triangle having that angle measure). Using graphing calculators or
computer software, as well as graphing simple examples by hand, have students graph a variety of trigonometric functions in which the amplitude,
frequency, and/or midline is changed. Students should be able to generalize about parameter changes, such as what happens to the graph of y = cos(x)
when the equation is changed to y = 3cos(x) + 5. Some information below includes additional mathematics that students should learn in order to take
advanced courses such as calculus, advanced statistics, or discrete mathematics and goes beyond the mathematics that all students should study in order to
be college- and career-ready: Some students can explore the inverse trigonometric functions, recognizing that the periodic nature of the functions depends
on restricting the domain. These inverse functions can then be used to solve real-world problems involving trigonometry with the assistance of technology
Common Misconceptions/Challenges
Students may believe that all trigonometric functions have a range of 1 to -1. Students need to see examples of how coefficients can change the range and the look of the
graphs. Students may also believe that restrictions to the domain of trigonometric functions are not necessary for defining inverse functions. Students may also believe that
sin-1 A = 1/sin A, thus confusing the ideas of inverse and reciprocal functions. Additionally, students may not understand that when sin A = 0.4, the value of A represents an
angle measure and that the function sin-1 (0.4) can be used to find the angle measure.
Common Core Support
Common Core Math: Computing Technology for Math Excellence: High School Functions Teaching and Learning Resources
http://www.ct4me.net/Common-Core/hsfunctions/hsf-interpreting-functions.htm
Institute for Mathematics and Education Learning Progressions Narratives
http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_functions_2013_07_02.pdf
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Textbook and Curriculum Resources
Math III, McGraw Hill
Chapters 11-12
High School CCSS Mathematics I Curriculum Guide-Quarter 3 Curriculum Guide, 2013
Prior Knowledge
Future Learning
Students begin studying right triangles using the Pythagorean Theorem in Grade 8.
Future learning will include proving and applying trigonometric
identities.
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Performance Assessments/Tasks
Click on the links below to access performance tasks.
Illustrative Mathematics
As the Wheel Turns
https://www.illustrativemathematics.org/content-standards/HSF/TF/B/5/tasks/595
Foxes and Rabbits 2
https://www.illustrativemathematics.org/content-standards/HSF/TF/B/5/tasks/816
Foxes and Rabbits 3
https://www.illustrativemathematics.org/content-standards/HSF/TF/B/5/tasks/817
Hours of Daylight 1
https://www.illustrativemathematics.org/content-standards/HSF/TF/B/5/tasks/1832
https://www.illustrativemathematics.org/content-standards/HSF/LE/B/5/tasks/579
https://www.illustrativemathematics.org/content-standards/HSF/LE/B/5/tasks/758
Career Connections
Management Occupations, Administrative Support, Construction, Production, Professional, Farming, Installation
http://www.xpmath.com/careers/math_jobs.php
http://www.xpmath.com/careers
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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math III
(listing of included sources attached)
Essential Understanding
F.TF.8
Prove and apply
trigonometric
identities.
Students are expected to be able to define the
trigonometric ratios.
Students are expected to prove the Pythagorean
identity.
Students are expected to be able to use the
Pythagorean identity to find sin (Ө), cos (Ө), or
tan (Ө), given sin(Ө), cos(Ө), or tan(Ө), and the
quadrant of the angle.
Prove the Pythagorean identity sin2 (Ө) + cos2 (Ө)
=1 and use it to find sin(Ө), cos(Ө), or tan(Ө) and
the quadrant of the angle.
-
Academic Vocabulary/
Language
amplitude, angle, arc, arccosine(arccos),
arcsine (arcsin), arctangent (arctan), axes,
circle, clockwise, constant, coordinate, cosine
(cos), counterclockwise, degree, differences,
equation, expression, formula, Frequency,
function, graph, identity, input, intervals,
inverse, midline, model, ordered pair, output,
period, quadrant, quantity, radian, relation,
sine (sin), subtend, table, tangent (tan),
trigonometric, unit, unit circle
Extended Understanding
Tier 2 Vocabulary
Provide students opportunity to draw the unit circle;
drawings can be useful in showing why the
Pythagorean relationship must be true. Dynamic
geometry software, such as Geometer’s Sketchpad or
Geogebra, can be used to demonstrate that,
regardless of the angle measure, the Pythagorean
relationship always holds in the unit circle.
define, explain, prove, understand, use
I can define trigonometric ratios as related to the unit circle.
I can prove the Pythagorean identity sin2 (Ө) + cos2 (Ө) =1.
I Can Statements
Columbus City Schools
I can use the Pythagorean identity, sin2 (Ө) + cos2 (Ө) =1, to find sin (Ө), cos (Ө), or tan (Ө), given
sin(Ө), cos(Ө), or tan(Ө), and the quadrant of the angle.
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Instructional Strategies
In the unit circle, the cosine is the x-value, while the sine is the y-value. Since the hypotenuse is always 1, the Pythagorean relationship sin2 (θ) + cos2 (θ) = 1
is always true. Students can make a connection between the Pythagorean Theorem in geometry and the study of trigonometry by proving this relationship.
In turn, the relationship can be used to find the cosine when the sine is known, and vice-versa. Provide a context in which students can practice and apply
skills of simplifying radicals. Some information below includes additional mathematics that students should learn in order to take advanced courses such
as calculus, advanced statistics, or discrete mathematics and goes beyond the mathematics that all students should study in order to be college- and
career-ready: Some students can explore other trigonometric identities, such as the half-angle, double-angle, and addition/subtraction formulas to extend
on the Pythagorean relationship. Formulas should be proven and then used to determine exact values when given an angle measure, to prove identities, and
to solve trigonometric equations. For example, by dividing the formula sin2 (θ) + cos2 (θ) = 1 by cos2 (θ), a new formula is generated ( tan2 (θ) +1= sec 2 (θ) ).
Common Misconceptions/Challenges
Students may believe that there is no connection between the Pythagorean Theorem and the study of trigonometry. Students may also believe that there is no relationship
between the sine and cosine values for a particular angle. The fact that the sum of the squares of these values always equals 1 provides a unique way to view trigonometry
through the lens of geometry. Additionally, students may believe that sin(A +B) = sinA + sinB and need specific examples to disprove this assumption.
Common Core Support
Common Core Math: Computing Technology for Math Excellence: High School Functions Teaching and Learning Resources
http://www.ct4me.net/Common-Core/hsfunctions/hsf-interpreting-functions.htm
Institute for Mathematics and Education Learning Progressions Narratives
http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_functions_2013_07_02.pdf
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Textbook and Curriculum Resources
Math III, McGraw Hill
Chapters 11-12
High School CCSS Mathematics I Curriculum Guide-Quarter 3 Curriculum Guide, 2013
Prior Knowledge
Future Learning
Students in Grade 8 grade learn to use the Pythagorean Theorem, while high school
students in a geometry unit study right triangle trigonometry. This cluster allows high
school students to connect these ideas as they derive a Pythagorean relationship for the
trigonometric functions.
Future learning will include study of coordinated geometry.
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Performance Assessments/Tasks
Click on the links below to access performance tasks.
Illustrative Mathematics
Trigonometric Ratios and the Pythagorean Theorem
https://www.illustrativemathematics.org/content-standards/HSF/LE/B/5/tasks/579
https://www.illustrativemathematics.org/content-standards/HSF/TF/C/8/tasks/1693
https://www.illustrativemathematics.org/content-standards/HSF/LE/B/5/tasks/758
Finding Trig Values
https://www.illustrativemathematics.org/content-standards/HSF/TF/C/8/tasks/1835
Calculations with sine and cosine
https://www.illustrativemathematics.org/content-standards/HSF/TF/C/8/tasks/1868
Career Connections
Management Occupations, Administrative Support, Construction, Production, Professional, Farming, Installation
http://www.xpmath.com/careers/math_jobs.php
http://www.xpmath.com/careers
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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math III
(listing of included sources attached)
Essential Understanding
G.GPE.4, G.GPE.5,
G.GPE.6, G.GPE.7
Use coordinates to
prove simple
geometric theorems
algebraically
Use coordinates to prove simple geometric theorems algebraically. For
example, prove or disprove that a figure defined by four given point in
the coordinate plane is a rectangle; prove or disprove that the point (1,
√3 ) lies on the circle centered at the origin and containing the point (0,
2).
Prove the slope criteria for parallel and perpendicular lines and use them
to solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point
Find the point on a directed line segment between two given points that
partitions the segment in a given ratio.
Use coordinates to compute perimeters of polygons and areas of
triangles and rectangles, e.g., using the distance formula.
CCSM Description
Students are expected to prove simple geometric
theorems algebraically; students are expected to prove
the slope criteria for parallel and perpendicular lines and
sue them to solve problems; students are expected to
know how to use coordinates to compute perimeters of
polygons and areas of triangles and rectangles.
Extended Understanding
Provide students opportunities to: find pictures of real
world examples of parallel lines. They can use magazines,
clip art, internet pictures or take pictures themselves.
Overlaying graph paper on their picture, instruct them to
prove the lines are parallel. ; use Google Earth to find a
real-world shape (i.e, a metro park, their yard, the stadium
at OSU). Ask the students to determine the perimeter and
area of their diagram using coordinate geometry. Discuss
scale factor with students, reminding them to use a
realistic scale to determine the perimeter and area. (You
might also have several students use the same picture so
they can compare their perimeters and areas)
Academic Vocabulary/
Language
altitude, area, centroid, diagonal, directed segment,
distance formula, intersecting lines, line segment,
median, midpoint, ordered pair, parallel, parallelogram,
partitioning a segment, perimeter, perpendicular,
perpendicular bisector, polygon, Pythagorean Theorem,
quadrilateral, ratio, reciprocal, segment bisector,
segment partition, slope
Tier 2 Vocabulary
define, explain, find, prove, understand, use
The correspondence between numerical coordinates and geometric
points allows methods from algebra to be applied to geometry and vice
versa. The solution set of an equation becomes a geometric curve,
making visualization a tool for doing and understanding algebra.
Geometric shapes can be described by equations, making algebraic
manipulation into a tool for geometric understanding, modeling, and
proof.
I Can Statements
Columbus City Schools
I can use coordinate geometry to prove geometric theorems algebraically; I can, using slope, prove lines are parallel
or perpendicular ; I can find equations of lines based on certain slope criteria such as; finding the equation of a line
parallel or perpendicular to a given line that passes through a given point;I can, given two points, find the point on
the line segment between the two points that divides the segment into a given ratio; I can use coordinate geometry
and the distance formula to find the perimeters of polygons and the areas of triangles and rectangles.
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Instructional Strategies
Review the concept of slope as the rate of change of the y-coordinate with respect to the x-coordinate for a point moving along a line, and derive the slope
formula. Use similar triangles to show that every nonvertical line has a constant slope. Review the point-slope, slope-intercept and standard forms for
equations of lines. Investigate pairs of lines that are known to be parallel or perpendicular to each other and discover that their slopes are either equal or
have a product of –1, respectively. Pay special attention to the slope of a line and its applications in analyzing properties of lines. Allow adequate time for
students to become familiar with slopes and equations of lines and methods of computing them. Use slopes and the Euclidean distance formula to solve
problems about figures in the coordinate plane such as: Given three points, are they vertices of an isosceles, equilateral, or right triangle? Given four points,
are they vertices of a parallelogram, a rectangle, a rhombus, or a square? Given the equation of a circle and a point, does the point lie outside, inside, or on
the circle? Given the equation of a circle and a point on it, find an equation of the line tangent to the circle at that point. Given a line and a point not on it,
find an equation of the line through the point that is parallel to the given line. Given a line and a point not on it, find an equation of the line through the
point that is perpendicular to the given line. Given the equations of two non-parallel lines, find their point of intersection. Given two points, use the distance
formula to find the coordinates of the point halfway between them. Generalize this for two arbitrary points to derive the midpoint formula. Use linear
interpolation to generalize the midpoint formula and find the point that partitions a line segment in any specified ratio. Use the distance formula to find the
length of each side of a polygon whose vertices are known, and compute the perimeter of that figure.
Common Misconceptions/Challenges
Students may claim that a vertical line has infinite slopes. This suggests that infinity is a number. Since applying the slope formula to a vertical line leads to division by zero,
we say that the slope of a vertical line is undefined. Also, the slope of a horizontal line is 0. Students often say that the slope of vertical and/or horizontal lines is “no slope,”
which is incorrect.
Common Core Support
Common Core State Standards: Geometry
http://www.corestandards.org/Math/Content/HSG/introduction/
Illustrative Mathematics: Practice and Content Standards: Toward Greater Focus and Coherence
https://www.illustrativemathematics.org/standards
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Textbook and Curriculum Resources
Math III, McGraw Hill
Chapters 14, 15
Problem Based Tasks for Mathematics I, Walch Education, 2013
Field of Dreams, pp. 228-236
Building Fences, pp. 242-246
Problem Based Tasks for Mathematics II, Walch Education, 2013
A Circle Graph for Lunch, pp. 308-311
Points of Shade, pp. 312-316
Geometry Station Activities for Common Core State Standards, Walch Education, 2013
Similarity, Right Triangles, and Trigonometry, pp. 109-135
Common Core State Standards Station Activities for Mathematics I, Walch Education, 2014
Parallel Lines, Slopes, and Equations, pp. 159-168
Perpendicular Lines, pp. 169-180
Coordinate Proof with Quadrilaterals, pp. 181-190
Prior Knowledge
Future Learning
Rates of change and graphs of linear equations were studied in Grade 8 and generalized in
The next area of study will be geometric constructions and
the Functions and Geometry Conceptual Categories in high school. Therefore, an
measurement.
alternative
way
to define
the slope IofCurriculum
a line is to call
it the tangent 4ofCurriculum
an angle of Guide, 2013
High
School
CCSS
Mathematics
Guide-Quarter
inclination of the line. In calculus, the concept of slope will be extended again to the slope
of a curve at a particular point
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Performance Assessments/Tasks
Click on the links below to access performance tasks.
Illustrative Mathematics
Midpoint Miracle
https://www.illustrativemathematics.org/content-standards/HSG/GPE/B/4/tasks/605
SRT Unit Squares and Triangles
https://www.illustrativemathematics.org/content-standards/HSG/GPE/B/4/tasks/918
Parallel Lines in the Coordinate Plane
https://www.illustrativemathematics.org/content-standards/HSG/GPE/B/5/tasks/1880
SRT Slope Criterion for Perpendicular Lines
https://www.illustrativemathematics.org/content-standards/HSG/GPE/B/5/tasks/1876
Triangles inscribed in a Circle
https://www.illustrativemathematics.org/content-standards/HSG/GPE/B/5/tasks/1332
Equal Area Triangles on the Same Base I
https://www.illustrativemathematics.org/content-standards/HSG/GPE/B/5/tasks/1347
Equal Area Triangles on the Same Base II
https://www.illustrativemathematics.org/content-standards/HSG/GPE/B/5/tasks/1348
Career Connections
Management Occupations, Administrative Support, Construction, Production, Professional, Farming, Installation, Computer and mathematical Occupations,
Architects/Surveyors/Cartographers, Engineering, Business/Finance, Scientists, Pilots,
http://www.xpmath.com/careers/topicsresult.php?subjectID=2&topicID=7
http://www.xpmath.com/careers
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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math I
(listing of included sources attached)
G.CO.12, G.CO.13
Essential Understanding
Make geometric
constructions.
Students should be able to apply definitions,
properties, theorems about line segments,
rays, and angles to support geometric
constructions.
Make formal geometric
constructions with a variety of
tools and methods (compass and straightedge, string, reflective
devices, paper folding, dynamic geometric software, etc.). Copying a
segment: copying an angle; bisecting a segment; constructing
perpendicular lines, including the perpendicular bisector of a line
segment and constructing a line parallel to a given line through a
point not on the line.
Student should be able to apply properties,
theorems about parallel and perpendicular
lines to support geometric constructions.
Students should be able to construct a square
equilateral triangle, regular hexagon inscribed
in a circle.
Construct an equilateral triangle, a square, and a regular hexagon
inscribe in a circle.
Academic Vocabulary/ Language
arc
equilateral
bisector
triangle
circle
circumscribe
congruent
diameter
inscribe
parallel
perpendicular
radius
regular
hexagon
regular
polygon
square
straightedge
triangle
Tier 2 Vocabulary
Extended Understanding
CCSSM Description
-
Students should be able to formalize and explain the construction of
geometric figures using a variety of tools and methods.
Students can create drawings using nothing
more than a compass and straightedge: e.g.,
stars inside of a circle, dodecagons; students
can then calculate the each inscribed image.
compass
construct
draw
explain
sketch
I can copy: a segment, an angle.
I can bisect: a segment, an angle.
I can construct perpendicular lines, including the perpendicular bisector of a line segment.
I Can Statements
I can construct a line parallel to a given line to a point not on the line.
I can consturct an equilateral triangle so that each vertex of the triangel is on the circle.
I can construct a square so that each vertex of the squate is on the circle.
I can construct a regular hexagon so that each vertex of the regular hexagon is on the circle.
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Instructional Strategies
Students should analyze each listed construction in terms of what simpler constructions are involved (e.g., constructing parallel lines can be done with two
different construction of perpendicular lines).
Challenge students to perform the same construction using a compass and a string. Use paper folding to produce a reflection; use bisections to produce
reflections.
Ask students to produce “how to” manuals, giving verbal instructions for particular constructions.
Provide meaningful opportunities (constructing the centroid or the incenter of a triangle) to offer students practice in executing basic constructions.
Compare dynamic geometry commands to sequences of compass-and- straightedge steps.
Utilize technology in construction activities.
To ensure that students are correctly making instructions and not just estimating a parallel line or the bisector of an angle, remind students that you will be
looking for the marks made by the sharp points of th3e compass and that there should be arcs made of the drawing; it should be clear where the arcs cross
each other.
Common Misconceptions/Challenges
Some students believe that construction is the same as sketching or drawing.
Teachers should emphasize the need for precision and accuracy when doing constructions. Stress the ideas that a compass and straightedge are identical to
a protractor and ruler. Explain the definition of measurement and construction.
If not using safety compasses, make certain that students know to use tool in a cautious, safe manner.
Remind students to keep compass opened at the same setting throughout the entire construction unless they are told to readjust the tool.
Common Core Support
Illustrative Mathematics
Constructions and Rigid Motions
https://www.illustrativemathematics.org/blueprints/G/1
Achieve the Core Modules, Resources
http://achievethecore.org/category/416/mathematics-tasks?&g%5B%5D=9&g%5B%5D=10&g%5B%5D=11&g%5B%5D=12&sort=name
The Common Core in Ohio
http://www.ccsso.org/resources/digital_resources/common_core_implementation_video_series.htm
https://www.ixl.com/standards/ohio/math
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Textbook and Curriculum Resources
Integrated Math I, McGraw Hill
Chapters 10,11,12,13, 14
Problem-Based Tasks for Mathematics I Common Core State Standards, Walch Education, 2013
Copying Segments and Angles, pp. 199-204
Bisecting Segments and Angles, pp. 205-208
Constructing Perpendicular and Parallel Lines, pp. 209-212
Problem-Based Tasks for Mathematics II Common Core State Standards, Walch Education, 2013
Sensing Distance, pp. 177-180
Calibrating Consoles, pp. 181-184
Life-Size Support, pp. 185-188
Sailing Centroid, pp. 189-195
Common Core Standards Station Activities for Mathematics II, Welch Education, 2014
Circumcenter, Incenter, Orthocenter, and Centroid, pp. 94-107
Geometry Station Activities Common Core State Standards: Welch Education, 2013
Classifying Triangles and Angle Theorems, pp. 13-27
Bisectors, Medians, and Altitudes, pp. 50-63
Triangle Inequalities, pp. 64-75
Ratio Segments, pp. 123-135
Prior
Knowledge
Future Learning
High
School
CCSS Mathematics I Curriculum Guide-Quarter 2 Curriculum Guide, 2013,
Drawing geometric shapes with rulers, protractors, and technology is developed
Future learning will include study of basic geometric definitions and rigid
in Grade 7. In high school, students perform formal geometric constructions
motions, geometric relationships and properties.
using a variety of tools. Students will utilize proofs to justify validity of their
constructions.
Patrickjmt
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Performance Assessments/Tasks
Click on the links below to access performance tasks.
Math Assessment Project
Inscribing and Circumscribing Right Triangles
http://map.mathshell.org/download.php?fileid=1758
Transforming 2D Figures
http://map.mathshell.org/download.php?fileid=1772
Evaluating Statements About Length and Area
http://map.mathshell.org/download.php?fileid=1750
Inside Mathematics
Circles in Triangles
http://www.insidemathematics.org/assets/common-core-math-tasks/circles%20in%20triangles.pdf
What’s My Angle?
http://www.insidemathematics.org/assets/problems-of-the-month/what's%20your%20angle.pdf
The Shape of Things
http://www.insidemathematics.org/assets/problems-of-the-month/the%20shape%20of%20things.pdf
Polly Gone
http://www.insidemathematics.org/assets/problems-of-the-month/polly%20gone.pdf
Once Upon a Time
http://www.insidemathematics.org/assets/problems-of-the-month/once%20upon%20a%20time.pdf
Career/Everyday Connections
With regard to constructing perpendicular lines and bisectors, engage students in a discussion where students will identify the applications of
concepts/skills in career areas such as: landscaping, agriculture, construction, architecture, logistics, and engineering.
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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math III
(listing of included sources attached)
G.GMD.4
three-dimensional objects.
Essential Understanding
Visualize
relationships
between twodimensional and
When given a three-dimensional object, students will be
expected to identify the shape made when the object is cut into
cross sections.
Students are expected to know the three-dimensional figure
that is generated when a two dimensional figure is rotating.
Identify the shapes of two-dimensional cross-sections
of three-dimensional objects, and identify threedimensional objects generated by rotations of twodimensional objects.
Students are expected to know that a cross section of a solid is
an intersection of a plane (two dimensional) and a solid (threedimensional).
Academic Vocabulary/
Language
area, base, bisect, circle,
circumference, construct,
coplanar, cone, cross section,
cutting plane, cube, cylinder,
diameter, dimension, equilateral,
line, parallel, perpendicular, pi,
plane, radius, regular, rotation,
slid, solid of revolution, volume
Extended Understanding
Provide opportunities such as the following, for students
to engage in experiences using skills learned in this
sections:
CCSSM Description
Students will learn to analyze characteristics and
properties of two- and three-dimensional geometric
shapes and develop mathematical arguments about
geometric relationships.
-
Tennis Balls in a Can
http://www.illustrativemathematics.org/illustrations/512
a real life situation using a can of tennis balls and an x-ray
machine at the airport to see the cross sections of the can,
and to determine what the cross section would look like in
different circumstances.
Tier 2 Vocabulary
identify
I I can, given a three- dimensional object, identify the shape made when the object is cut into
cross-sections.
I Can Statements
Columbus City Schools
I can, when rotating a two- dimensional figure, such as a square, know the three-dimensional
figure that is generated, such as a cylinder. Understand that a cross section of a solid is an
intersection of a plane (twodimensional) and a solid (three-dimensional).
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Instructional Strategies
Review vocabulary for names of solids (e.g., right prism, cylinder, cone, sphere, etc.). Slice various solids to illustrate their cross sections. For example, cross
sections of a cube can be triangles, quadrilaterals or hexagons. Rubber bands may also be stretched around a solid to show a cross section. Cut a half-inch
slit in the end of a drinking straw, and insert a cardboard cutout shape. Rotate the straw and observe the three-dimensional solid of revolution generated by
the two-dimensional cutout. Java applets on some web sites can also be used to illustrate cross sections or solids of revolution. Encourage students to
create three-dimensional models to be sliced and cardboard cutouts to be rotated. Students can also make three-dimensional models out of modeling clay
and slice through them with a plastic knife.
Common Misconceptions/Challenges
Some cross sections are more difficult to visualize than others. For example, it is often easier to visualize a rectangular cross section of a cube than a
hexagonal cross section. Generating solids of revolution involves motion and is difficult to visualize by merely looking at drawings.
Common Core Support
Illustrative Mathematics
Constructions and Rigid Motions
https://www.illustrativemathematics.org/blueprints/G/1
The Common Core in Ohio
http://www.ccsso.org/resources/digital_resources/common_core_implementation_video_series.htm
https://www.ixl.com/standards/ohio/math
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Textbook and Curriculum Resources
Integrated Math I McGraw Hill
Chapter 15
High School CCSS Mathematics III Curriculum Guide-Quarter4 Curriculum Guide, 2013,
Prior Knowledge
Future Learning
Students have had experiences with visualizing two and three dimensional
figures in middle school: 7.G.3.( Describe the two-dimensional figures that
result from slicing three-dimensional figures, as in plane sections of right
rectangular prisms and right rectangular pyramids).
Future learning will include study of circles and conics.
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Performance Assessments/Tasks
Click on the links below to access performance tasks.
Math Assessment Project
Modeling Motion: Rolling Cups
http://map.mathshell.org/download.php?fileid=1746
Representing 3D Objects in 2D
http://map.mathshell.org/download.php?fileid=1762
Calculating Volumes of Compound Objects
http://map.mathshell.org/download.php?fileid=1764
Inside Mathematics
Piece It Together
http://www.insidemathematics.org/assets/problems-of-the-month/piece%20it%20together.pdf
Cutting a Cube
http://www.insidemathematics.org/assets/problems-of-the-month/cutting%20a%20cube.pdf
Global Positioning System I
https://www.illustrativemathematics.org/content-standards/HSG/GMD/B/4/tasks/1215
Global Positioning System II
https://www.illustrativemathematics.org/content-standards/HSG/GMD/B
Career/Everyday Connections
Transportation, Art, Architecture, Medicine, Engineering, Event Planner
Careers using geometry: http://work.chron.com/careers-require-geometry-10361.html
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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math III
(listing of included sources attached)
G.C.1, G.C.2
Understand
and apply
theorems
about circles.
Students should know that unlike polygons that have dimensions
independent of one another (base and height, for instance), a
circle's size depends only on one measurement: the radius r.
Students should know that since all aspects of a circle's size
depend on r; the size can be changed of any circle simply by
dilating the radius by a constant scale factor.
Prove that all circles are similar
Identify and describe relationships among inscribed
angles, radii, and chords. Include the relationship
between central, inscribed, and circumscribed angles;
inscribed angles on a diameter are right angles; the
radius of a circle is perpendicular to the tangent where
the radius intersects the circle.
CCSSM Description
Students should already know that dilations, whether they're
expansions or contractions, are similarity transformations; the
size off the circle is changing but not its shape.
Academic Vocabulary/
Language
center, central angle, centroid, chord, circle,
circumcenter, circumference, , circumscribed
angle, cyclic, diameter, dilations, equidistant,
focus , incenter, inscribed angle, latus rectum,
proportions, quadrilateral, Radian, radius,
scalersimilar, translations
Tier 2 Language
construct
draw
sketch
Extended Understanding
Provide opportunities for students to engage in activities that will
allow them to enhance understanding such as:
ttp://learnzillion.com/lessonsets/427-prove-that-all-circles-are-similar
Learning to recall, understand, apply, prove and
extend theorems about circles is useful because it
leads to being able to find angles in and around
circles; it becomes a functional (real-life)
application skill used in occupations such as
engineering and design and, this leads to
developing skills at geometric proof and geometric
reasoning.
I Can Statements
Essential Understanding
This is an all in one unit to prove all circles are similar. It includes talk
about using translations and dilations as well as triangles to prove that
all circles are similar.
I can, using the fact that the ratio of diameter to circumference is the same for circles, prove that all circles
are similar.
I can, using definitions, properties, theorems, identiry and describe relationships among inscribed angles,
radii, and chords. Include central, inscribed, and circumscribed angles.
I can understand that inscribed angles on a diameter are right angles.
I can understand that the radius of a circle is perpendicular to the tangent where the radius intersects the
circle
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Instructional Strategies
Given any two circles in a plane, show that they are related by dilation. Guide students to discover the center and scale factor of this dilation and make a
conjecture about all dilations of circles. Starting with the special case of an angle inscribed in a semicircle, use the fact that the angle sum of a triangle is
180° to show that this angle is a right angle. Using dynamic geometry, students can grab a point on a circle and move it to see that the measure of the
inscribed angle passing through the endpoints of a diameter is always 90°. Then extend the result to any inscribed angles. For inscribed angles, proofs can be
based on the fact that the measure of an exterior angle of a triangle equals the sum of the measures of the nonadjacent angles. Consider cases of acute or
obtuse inscribed angles. Use properties of congruent triangles and perpendicular lines to prove theorems about diameters, radii, chords, and tangent lines.
Use formal geometric constructions to construct perpendicular bisectors of the sides and angle bisectors of a given triangle. Their intersections are the
centers of the circumscribed and inscribed circles, respectively.
Common Misconceptions/Challenges
Students sometimes confuse inscribed angles and central angles. Students may think they can tell by inspection whether a line intersects a circle in exactly one point. It
may be beneficial to formally define a tangent line as the line perpendicular to a radius at the point where the radius intersects the circle
Common Core Support
Illustrative Mathematics
Circles
https://www.illustrativemathematics.org/blueprints/G/6
https://www.illustrativemathematics.org/blueprints/G/3
The Common Core in Ohio
http://www.ccsso.org/resources/digital_resources/common_core_implementation_video_series.htm
https://www.ixl.com/standards/ohio/math
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Textbook and Curriculum Resources
Integrated Math III, McGraw Hill
High School CCSS Mathematics I Curriculum Guide -Quarter 4- Columbus City Schools, 2013, pp. 83-207
Geometry Station Activities for Common Core State Standards, Walch Education, 2013
Circumference, Angles, Arcs, Chord, and Inscribed Angles, pp. pp. 147-160
Special Segments, Angle Measurements, and Equations of Circles, pp. 161-172
Common Core State Standards: Problem-Based Tasks for Mathematics II, Walch Education, 2013
Following in Arhimedes’ Footsteps, pp. 265-267
Masking the Problem, pp. 268-270
The Circus Is In Town, Is It Safe?, pp. pp. 271-274
Prior Knowledge
Future Learning
Middle school experiences with circles in Geometry in 7th grade is when they
are expected to draw, construct and describe geometrical figures and describe
the relationships between them and solve real-life and mathematical problems
involving angle measure, area, surface area, and volume. In 8th grade students
begin work with volume of cylinders, cones and spheres.
Future learning includes constructing inscribed and circumscribed circles
of a triangle, and proving properties of angles for a quadrilateral
inscribed in a circle.
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Performance Assessments/Tasks
Click on the links below to access performance tasks.
Math Assessment Project
Inscribing and Circumscribing Right Triangles
http://map.mathshell.org/download.php?fileid=1758
Solving Problems with Circles and Triangles
http://map.mathshell.org/download.php?fileid=1760
http://schools.nyc.gov/NR/rdonlyres/49162FEC-37E2-4A96-93C1-6671664FACD5/0/NYCDOEHSMathCompanyLogo_Final.pdf
INSIDE Mathematics
Circles In Squares
http://www.insidemathematics.org/assets/common-core-math-tasks/circle%20and%20squares.pdf
Circles In Triangles
http://www.insidemathematics.org/assets/common-core-math-tasks/circles%20in%20triangles.pdf
Similar Circles
https://www.illustrativemathematics.org/content-standards/HSG/C/A/1/tasks/1368
Right triangles inscribed in Circles I
https://www.illustrativemathematics.org/content-standards/HSG/C/A/2/tasks/1091
Right triangles inscribed in Circles II
https://www.illustrativemathematics.org/content-standards/HSG/C/A/2/tasks/1093
Career/Everyday Connections
Architectural Engineering
Construction Engineering
Forensics
Landscaping Engineering
Crop circles are an interesting and controversial phenomenon that can best be described as a pattern in a field where the crop (usually wheat) has been flattened. Many
believe that the circles are made using a string and a piece of wood to flatten the crops.
The geometry of a basketball court, food engineering and efficiency, and landscaping a great are great studies of circle applications.
http://algebralab.org/practice/practice.aspx?file=Word_WP-CircleApplications.xml
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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math III
(listing of included sources attached)
G.C.3
Essential Understanding
Understand and apply
theorems about circles.
Construct the inscribed and circumscribed
circles of a triangle, and prove
properties of angles for a quadrilateral inscribed in a circle,
Students need to understand that when
a circle is inscribed in a polygon, then the
polygon is circumscribed about the circle
and when a circle is circumscribed about
a polygon, then the polygon is inscribed
in the circle.
Students need to understand that when
a circle is inscribed in a polygon, then the
polygon is circumscribed about the circle
and when a circle is circumscribed about
a polygon, then the polygon is inscribed
in the circle.
Extended Understanding
-
CCSSM Description
A circle is inscribed in a polygon if each side of the polygon is tangent to the
circle, so an inscribed circle touches each side of the polygon at exactly one
point.
Challenge students to generalize the
results about angle sums of triangles and
quadrilaterals to a corresponding result
for n-gons.
A circle is circumscribed about a polygon if each vertex of the polygon lies on
the circle. A circumscribed circle passes through each vertex of the polygon.
Academic Vocabulary/
Language
acute triangle, angles, alternate interior angles,
alternate exterior angles, base, base angles,
bisect, bisector, centroid, circumcenter,
circumscribe, concurrent, consecutive interior
angles, corresponding angles, diagonal,
equiangular triangle, equidistant, equilateral
triangle, exterior angle, hypotenuse, incenter,
inscribe, inscribed arc, inscribed angle, inscribed
quadrilateral, interior angle, isosceles triangle,
leg, linear pair, lines, midsegment, obtuse
triangle, orthocenter, parallel lines,
parallelogram, perpendicular, perpendicular
bisector, quadrilateral, rectangle, remote angle,
rhombus, right angles, scalene triangle, square, ,
transversal line, vertex angle, vertical angles
Tier 2 Language
construct
draw
sketch
Tier 2 Vocabulary
construct
draw
sketch
I can construct inscribed circles of a triangle.
I Can Statements
I can construct circumscribed circles of a triangle.
I can, using definitions, properties, and theorems, prove properties of angles for a
quadrilateral inscribed in a circle
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Instructional Strategies
Given any two circles in a plane, show that they are related by dilation. Guide students to discover the center and scale factor of this dilation and make a
conjecture about all dilations of circles.
Starting with the special case of an angle inscribed in a semicircle, use the fact that the angle sum of a triangle is 180° to show that this angle is a right angle.
Using dynamic geometry, students can grab a point on a circle and move it to see that the measure of the inscribed angle passing through the endpoints of a
diameter is always 90°. Then extend the result to any inscribed angles. For inscribed angles, proofs can be based on the fact that the measure of an exterior
angle of a triangle equals the sum of the measures of the nonadjacent angles. Consider cases of acute or obtuse inscribed angles.
Use properties of congruent triangles and perpendicular lines to prove theorems about diameters, radii, chords, and tangent lines. Use formal geometric
constructions to construct perpendicular bisectors of the sides and angle bisectors of a given triangle. Their intersections are the centers of the
circumscribed and inscribed circles, respectively.
Dissect an inscribed quadrilateral into triangles, and use theorems about triangles to prove properties of these quadrilaterals and their angles.
Common Misconceptions/Challenges
Students sometimes confuse inscribed angles and central angles.
Students may think they can tell by inspection whether a line intersects a circle in exactly one point. It may be beneficial to formally define a tangent line as the line
perpendicular to a radius at the point where the radius intersects the circle.
Remembering which point of concurrency is created by the four special triangle segments. The medians make the centroid, the perpendicular bisectors make the
circumcenter, the angle bisectors make the incenter, and the altitudes make the orthocenter.
Common Core Support
Illustrative Mathematics
Circles
https://www.illustrativemathematics.org/blueprints/G/6
https://www.illustrativemathematics.org/blueprints/G/3
The Common Core in Ohio
http://www.ccsso.org/resources/digital_resources/common_core_implementation_video_series.htm
https://www.ixl.com/standards/ohio/math
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Textbook and Curriculum Resources
Integrated Math I, McGraw Hill
Chapter 15
Integrated Math II, McGraw Hill
Chapter 11
High School CCSS Mathematics I Curriculum Guide -Quarter 3- Columbus City Schools, 2013, pp. 83-207
Geometry Station Activities for Common Core State Standards, Walch Education, 2013
Circumcenter, Incenter, Orthocenter, and Centroid, pp. 173-186
Common Core State Standards: Problem-Based Tasks for Mathematics II, Walch Education, 2013
First Aid Station, pp. 275-278
Building a New Radio Station, pp. 279-282
King Arthur and His Round Table, pp. 283-285
Prior Knowledge
Future Learning
Constructing inscribed and circumscribed circles of a triangle is an application of
the formal constructions studied in G – CO.12
Statistics will be topic covered in next lessons.
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Performance Assessments/Tasks
Click on the links below to access performance tasks.
NYC Department of Education G.C.3: Circles: Understand And Apply Theorems About Circles: Understand And Apply Theorems About Circles
https://www.engageny.org/ccls-math/gc3
Math Assessment Project
Inscribing and Circumscribing Right Triangles
http://map.mathshell.org/download.php?fileid=1758
Solving Problems with Circles and Triangles
http://map.mathshell.org/download.php?fileid=1760
http://schools.nyc.gov/NR/rdonlyres/49162FEC-37E2-4A96-93C1-6671664FACD5/0/NYCDOEHSMathCompanyLogo_Final.pdf
INSIDE Mathematics
Circles In Squares
http://www.insidemathematics.org/assets/common-core-math-tasks/circle%20and%20squares.pdf
Circles In Triangles
http://www.insidemathematics.org/assets/common-core-math-tasks/circles%20in%20triangles.pdf
What’s My Angle
http://www.insidemathematics.org/assets/problems-of-the-month/what's%20your%20angle.pdf
Career/Everyday Connections
Architectural Engineering
Construction Engineering
Forensics
Landscaping Engineering
Crop circles are an interesting and controversial phenomenon that can best be described as a pattern in a field where the crop (usually wheat) has been flattened. Many
believe that the circles are made using a string and a piece of wood to flatten the crops.
The geometry of a basketball court, food engineering and efficiency, and landscaping a great are great studies of circle applications.
http://algebralab.org/practice/practice.aspx?file=Word_WP-CircleApplications.xml
29.30
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Ohio’s Learning Standards - Clear Learning Targets
Integrated Math III
(listing of included sources attached)
G.MG.1, G.MG.2,
G.MG.3
Essential Understanding
Apply geometric
concepts in modeling
situations in modeling.
Students are expected to apply and model
geometric concepts.
Use geometric shapes, their measures, and their properties to
describe objects (e.g., modeling a tree trunk or a human torso as a
cylinder).
Apply concepts of density based on area and volume in modeling
situations (e.g., persons per square mile, BTUs per cubic foot).
Academic Vocabulary/
Language
gometric concepts, geometric methods,
properties
Extended Understanding
-
Apply geometric methods to solve design problems (e.g., designing
an object or structure to satisfy physical constraints or minimize
cost; working with typographic grid systems based on ratios).
CCSSM Description
An understanding of the attributes and relationships of geometric
objects can be applied in diverse contexts—interpreting a schematic
drawing, estimating the amount of wood needed to frame a sloping
roof, rendering computer graphics, or designing a sewing pattern for
the most efficient use of material. Modeling activities are a good
way to show connections among various branches of mathematics
and science. Dynamic geometry environments provide students with
experimental and modeling tools that allow them to investigate
geometric phenomena.
Encourage students to engage in a project(s)
using real-world applications of geometry.
Resources: A Sourcebook of Applications of School
Mathematics, compiled by a Joint Committee of
the Mathematical Association of America and the
National Council of Teachers of Mathematics
(1980); Mathematics: Modeling our World, Course
1 and Course 2, by the Consortium for
Mathematics and its Applications (COMAP);
Geometry & its Applications (GeoMAP) -- an
exciting National Science Foundation project to
introduce new discoveries and real-world
applications of geometry to high school students.
Produced by COMAP; Measurement in School
Mathematics, NCTM 1976 Yearbook.
Tier 2 Vocabulary
analyze, describe, design, model, solve
I can use geometric shapes, their measures, and their properties to describe objects (e.g.,
modeling a tree trunk or a human torso as a cylinder).
I Can Statements
I can use the concept of density when referring to situations involving area and volume models,
such as persons per square mile.
I can solve design problems by designing an object or structure that satisfies certain constraints,
such as minimizing cost or working with a grid system based on ratios (i.e., The enlargement of a
picture using a grid and ratios and proportions)
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Instructional Strategies
Genuine mathematical modeling typically involves more than one conceptual category. For example, modeling a herd of wild animals may involve
geometry, measurement, proportional reasoning, estimation, probability and statistics, functions, and algebra. It would be somewhat misleading to try to
teach a unit with the title of “modeling with geometry.” Instead, these standards can be woven into other content clusters. A challenge for teaching
modeling is finding problems that are interesting and relevant to high school students and, at the same time, solvable with the mathematical tools at the
students’ disposal. The resources listed below are a beginning for addressing this difficulty.
Common Misconceptions/Challenges
When students ask to see “useful” mathematics, what they often mean is, “Show me how to use this mathematical concept or skill to solve the homework problems.”
Mathematical modeling, on the other hand, involves solving problems in which the path to the solution is not obvious. Geometry may be one of several tools that can be
used.
Common Core Support
https://www.illustrativemathematics.org/blueprints/G/3
The Common Core in Ohio
http://www.ccsso.org/resources/digital_resources/common_core_implementation_video_series.htm
https://www.ixl.com/standards/ohio/math
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Textbook and Curriculum Resources
Integrated Math III, McGraw Hill
High School CCSS Mathematics I Curriculum Guide –Quarter - Columbus City Schools, 2013
Common Core State Standards: Problem-Based Tasks for Mathematics II, Walch Education, 2013
Designing a Tablecloth, pp. pp. 317-320
Cylinders of Sand, pp. 321-324
Prior Knowledge
Future Learning
Students have acquired knowledge to empower them to experience modeling
geometric concepts/skills.
Precalculus
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Performance Assessments/Tasks
Click on the links below to access performance tasks.
Math Assessment Project
Solving Quadratic Equations
http://map.mathshell.org/download.php?fileid=1736
Modeling Motion: Rolling Cups
http://map.mathshell.org/lessons.php?unit=9300&collection=8&redir=1
http://schools.nyc.gov/NR/rdonlyres/49162FEC-37E2-4A96-93C1-6671664FACD5/0/NYCDOEHSMathCompanyLogo_Final.pdf
Career/Everyday Connections
Architectural Engineering
Construction Engineering
Forensics
Landscaping Engineering
Crop circles are an interesting and controversial phenomenon that can best be described as a pattern in a field where the crop (usually wheat) has been flattened. Many
believe that the circles are made using a string and a piece of wood to flatten the crops.
The geometry of a basketball court, food engineering and efficiency, and landscaping a great are great studies of circle applications.
http://algebralab.org/practice/practice.aspx?file=Word_WP-CircleApplications.xml
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