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APPENDIX L • For additional information on representing and interpreting
data in grades K–5, see the Progressions document available
CONNECTIONS TO THE COMMON CORE at: http://commoncoretools.files.wordpress.com/2011/06/ccss_
STATE STANDARDS FOR MATHEMATICS progression_md_k5_2011_06_20.pdf.
• For additional information on measurement in grades K–5,
see the Progressions document available at: http://
commoncoretools.files.wordpress.com/2012/07/ccss_progression_
gm_k5_2012_07_21.pdf.
CONSISTENCY WITH THE COMMON CORE STATE
STANDARDS FOR MATHEMATICS TABLE L-1 Key Topics Relevant to Science and the Grade at Which They Are First
Expected in the CCSSM
Science is a quantitative discipline, so it is important for educators Grade First
Number and Operations
to ensure that students’ science learning coheres well with their Expected
learning in mathematics.1,2 To achieve this alignment, the Next Multiplication and division of whole numbers 3
Generation Science Standards (NGSS) development team worked Concept of a fraction a/b 3
with the Common Core State Standards for Mathematics (CCSSM) Beginning fraction arithmetic 4
writing team to ensure the NGSS do not outpace or otherwise Coordinate plane 5
misalign to the grade-by-grade standards in the CCSSM. Every Ratios, rates (e.g., speed), proportional relationships 6
effort has been made to ensure consistency. It is essential that Simple percent problems 6
the NGSS always be interpreted, and implemented, in such a way
Rational number system/signed numbers—concepts 6
that the math does not outpace or misalign to the grade-by-grade
Rational number system/signed numbers—arithmetic 7
standards in the CCSSM (this includes the development of NGSS-
Grade First
aligned instructional materials and assessments). Measurement
Expected
For convenience, Table L-1 shows CCSSM grade placements for Standard length units (inch, centimeter, etc.) 2
key topics relevant to science. This table can help science educa- Area 3
tors ensure that students’ work in science does not require them Convert from a larger unit to a smaller in the same system 4
to meet the indicated CCSSM standards before the grade level in Convert units within a given measurement system 5
which they appear. Volume 5
Convert units across measurement systems (e.g., inches to centimeters) 6
Grade First
Statistics and Probability
Expected
Statistical distributions (including center, variation, clumping, outliers, 6–8
mean, median, mode, range, quartiles) and statistical associations or
1
For more on this point, see page 17 of the K–8 Publishers’ Criteria for the trends (including two-way tables, bivariate measurement data, scatter
Common Core State Standards for Mathematics and page 15 of the High plots, trend line, line of best fit, correlation)
School Publishers’ Criteria for the Common Core State Standards for Math-
ematics, both available at: www.corestandards.org. Probability, including chance, likely outcomes, probability models 7
2
For example, concepts of physical measurement are intertwined with students’
developing understanding of arithmetic in the elementary grades; see the NOTE: See the CCSSM for exact Statements of Expectations.
Progressions document available at: http://commoncoretools.files.wordpress.
com/2012/07/ccss_progression_gm_k5_2012_07_21.pdf and the brief essay
“Units, a Unifying Idea in Measurement, Fractions, and Base Ten” available at:
http://commoncoretools.me/2013/04/19/units-a-unifying-idea.
137

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During the middle school and high school years, students develop TABLE L-2 Middle School and High School Science and Engineering Practices
a number of powerful quantitative tools, from rates and pro- That Require Integrating the CCSSM Math/Statistics Tools into the NGSS-Aligned
portional relationships, to basic algebra and functions, to basic Instructional Materials and Assessments
statistics and probability. Such tools are applicable far beyond Science and 6–8 Condensed Practices 9–12 Condensed Practices
the mathematics classroom. Such tools can also be better under- Engineering (subset requiring (subset requiring
Practices integration) integration)
stood, and more securely mastered, by applying them in a variety
of contexts. Fortunately, the National Research Council (NRC) Analyzing and Apply concepts of Apply concepts of
report A Framework for K–12 Science Education (Framework) Intepreting Data statistics and probability statistics and probability
from the CCCSS (found in from the high school
makes clear in its science and engineering practices (Analyzing
grades 6-8.SP) CCCSS (found in S) to
and Interpreting Data, Using Mathematics and Computational to scientific and scientific and engineering
Thinking) that statistics and mathematics have a prominent role in engineering questions questions and problems,
science. The NGSS aim to give middle school and high school sci- and problems, using using digital tools when
digital tools when feasible.
ence educators a clear road map to prepare their students for the
feasible.
quantitative demands of college and careers, where students need
to apply quantitative tools in an applied or scientific context.3 For Using Apply concepts of ratio, Apply techniques of
all these reasons, the NGSS require key tools for grades 6−8 and Mathematics and rate, percent, basic algebra and functions
Computational operations, and simple to represent and solve
the high school Common Core State Standards (CCSS) to be inte-
Thinking algebra to scientific and scientific and engineering
grated into middle school and high school science instructional engineering questions problems. (See A and F in
materials and assessments. and problems. (See the CCSS.)
grades 6-7.RP, 6-8.NS,
For additional detail, see Table L-2, as well as the NGSS Condensed and 6-8.EE in the CCSS.) Apply key takeaways from
Practices (Appendix F) and the CCSS connections boxes that grades 6–8 mathematics,
appear throughout the NGSS. such as applying ratios,
rates, percentages, and
unit conversions (e.g., in
the context of complicated
CONNECTIONS TO CCSSM STANDARDS FOR measurement problems
MATHEMATICAL PRACTICE involving quantities with
derived or compound
Some general connections to the CCSSM can be found among units, such as mg/mL,
kg/m3, acre-feet, etc.).a
CCSSM’s Standards for Mathematical Practice. The three CCSSM
practice standards most directly relevant to science are: NOTE: Refer to the NGSS Science and Engineering Practices for context and to the CCSSM for
• MP.2. Reason abstractly and quantitatively. information about the standards notation: http://www.corestandards.org/Math.
• MP.4. Model with mathematics. a
See Table 1 of the High School Publishers’ Criteria for the Common Core State Standards for
• MP.5. Use appropriate tools strategically. Mathematics, available at: www.corestandards.org/resources.
Mathematical practice standards MP.2 and MP.4 are both about
using mathematics in context. The first practice standard, MP.2,
is about the back and forth between (1) manipulating symbols
abstractly and (2) attending to the meaning of those symbols
3
Table 1 of the High School Publishers’ Criteria for the Common Core State
Standards for Mathematics shows widely applicable prerequisites for college
and careers available at: www.corestandards.org.
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while doing so. For example, a kindergarten student might con- here. The difference is that scientific arguments are always based
nect a symbolic statement like “6 > 4” to the fact that there are on evidence, whereas mathematical arguments never are. It is this
more objects in one given set than in another. A middle school difference that renders the findings of science provisional and
student might rewrite the equation d = 65t for the motion of a car the findings of mathematics eternal. As Isaac Asimov wrote in the
in the equivalent form d/t = 65, recognizing that the new equation Foreword to A History of Mathematics, “Ptolemy may have devel-
abstractly expresses the steps in a computation of the car’s speed. oped an erroneous picture of the planetary system, but the system
A high school student might connect the 2 in the equation N = 2n of trigonometry he worked out to help him with his calculations
to the fact that each dividing cell gives rise to two daughter cells. remains correct forever” (Boyer and Merzbach, 1991, pp. vii–viii).
The second practice standard, MP.4, is also about applying mathe- Blurring the distinction between mathematical and scientific argu-
matics, but with more of a focus on results and less on the mental ments leads to a misunderstanding of what science is about. For
processes involved: more information about argumentation in science, see the NGSS
• In grades K–2, modeling with mathematics typically means science and engineering practice “Engaging in argument from
diagramming a situation mathematically, and/or solving a one- evidence.”
step addition/subtraction word problem. For more information on the standards for mathematical practice
• In grades 3–5, modeling with mathematics typically means rep- in general, see CCSSM, pp. 6–8. Also see pp. 72–73 for information
resenting and/or solving a one-step or multi-step word problem. on modeling in particular.
• In grades 6–8, modeling with mathematics typically means The rest of this appendix presents the remaining connections from
representing and/or solving a one-step or multi-step word the connections boxes. Illustrative science examples are provided
problem, possibly one in which certain assumptions necessary for a number of the connections, along with alignment notes in
to formulate the problem mathematically are not specified for select cases.
the student.
• In high school, modeling with mathematics typically includes the K-PS2 MOTION AND STABILITY: FORCES AND
kinds of problems seen in grades 6–8 as well as “full models”—
INTERACTIONS
that is, problems that include more of the steps of the modeling
cycle. (National Governors Association Center for Best Practices, As part of this work, teachers should give students opportunities
Council of Chief State School Officers, 2010, pp. 72–73) to use direct measurement:
Finally, the third practice standard, MP.5, refers not only to tech-
K.MD.A.1. Describe measurable attributes of objects, such as
nological tools, but also to such strategies as drawing diagrams
length or weight. Describe several measurable attributes of a
from kindergarten onward and, in later grades, using well-known
single object.
formulas and powerful representation schemes like the coordi-
K.MD.A.2. Directly compare two objects with a measurable attri-
nate plane. These tools, and the skill and judgment to use them
bute in common to see which object has more of/less of the
well, are important for quantitative work in science.
attribute and describe the difference. For example, directly
About CCSSM practice standard MP.3: None of the connections compare the heights of two children and describe one child as
boxes include a link to CCSSM practice standard MP.3, which reads, taller/shorter. Science examples: Students make a simple pulley
“Make viable arguments and critique the reasoning of others.” that uses one object to lift a second object. They describe one
The lack of a connection to MP.3 might appear surprising, given of the objects as heavier than the other. They try to predict
that science too involves making arguments and critiquing them. which will rise and which will fall. In consecutive trials that
However, there is a difference between mathematical arguments vary the weight of the first object (keeping the second object
and scientific arguments—a difference so fundamental that it the same), students conclude that a heavier object will lift a
would be misleading to connect any of the standards to MP.3 given target object faster.
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Alignment notes: (1) Data displays such as picture graphs and K.CC.A.4 Know number names and the count sequence. Science
bar graphs are not expected until grade 2. (2) Standard length example: Students write the number of sunny or rainy days in
units such as centimeters or inches are not expected until grade the previous month.
2. Informal units (e.g., a paper clip used as a length unit) are not K.MD.A.1. Describe measurable attributes of objects, such as
expected until grade 1. length or weight. Describe several measurable attributes of a
single object. Science example: Describe a beaker of water as
K-PS3 ENERGY being heavy and cold.
K.MD.B.3. Classify objects into given categories; count the number
As part of this work, teachers should give students opportunities of objects in each category and sort the categories by count.
to use direct measurement: Science example: Build a tally chart showing the number of
K.MD.A.2. Directly compare two objects with a measurable attri- rainy or sunny days as the month progresses. Count the num-
bute in common to see which object has more of/less of the ber of sunny or rainy days in the previous month (see K.CC.B).
attribute and describe the difference. For example, directly Were there more rainy days or sunny days (see K.CC.C)?
compare the heights of two children and describe one child as Alignment notes: (1) Data displays such as picture graphs and bar
taller/shorter. Science example: Directly compare a stone left in graphs are not expected until grade 2. (2) Standard length units
the sun with a stone left in the shade and describe one of the such as centimeters or inches are not expected until grade 2.
stones as warmer/cooler than the other.
K-ESS3 EARTH AND HUMAN ACTIVITY
K-LS1 FROM MOLECULES TO ORGANISMS: STRUCTURES
AND PROCESSES As part of this work, teachers should give students opportunities
to count and compare numbers (see K.CC). Science examples:
As part of this work, teachers should give students opportunities (1) Count the number of trees in each of two photographs. In
to use direct measurement: which photograph are there more trees? In which place might
you find more squirrels? (2) Keep a tally of the number of severe
K.MD.A.2. Directly compare two objects with a measurable attri-
weather days (forecast and actual). Count the number of severe
bute in common to see which object has more of/less of the
weather days at the end of the year.
attribute and describe the difference. For example, directly
compare the heights of two children and describe one child as
1-PS4 WAVES AND THEIR APPLICATIONS IN
taller/shorter. Science example: Directly compare a sunflower
TECHNOLOGIES FOR INFORMATION TRANSFER
grown in the shade with a sunflower grown in the sun. Which
flower is taller? Observe that these plants need light to thrive.
As part of this work, teachers should give students opportunities
Alignment notes: (1) Data displays such as picture graphs and bar to measure with non-standard units:
graphs are not expected until grade 2. (2) Standard length units
1.MD.A.1. Order three objects by length; compare the lengths of
such as centimeters or inches are not expected until grade 2.
two objects indirectly by using a third object. Science example:
The class makes string phones. Maria’s string is longer than
K-ESS2 EARTH’S SYSTEMS
Sue’s and Sue’s string is longer than Tia’s, so without measur-
As part of this work, teachers should give students opportunities ing directly we know that Maria’s string is longer than Tia’s.
to use numbers, counting, direct measurement, and classification:
4
The capital letter “A” in “K.CC.A” refers to the first cluster heading in domain
K.CC. See p. 11 of the CCSSM, available at: http://www.corestandards.org/assets/
CCSSI_Math%20Standards.pdf.
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1.MD.A.2. Express the length of an object as a whole number of (2) During the breeding season, a female cottontail rabbit has
length units, by layering multiple copies of a shorter object (the litters of five, six, five, and four bunnies. How many bunnies
length unit) end to end. Understand that the length measure- did the rabbit have during this time?
ment of an object is the number of same-size length units that
span it with no gaps or overlaps. Limit to contexts where the 1-LS3 HEREDITY: INHERITANCE AND VARIATION OF TRAITS
object being measured is spanned by a whole number of length
units with no gaps or overlaps. Science example: Using a shoe as As part of this work, teachers should give students opportu-
the length unit, the string for Sue’s string phone is 11 units long. nities to measure with non-standard units and use indirect
measurement:
Alignment note: Standard length units such as centimeters or
inches are not expected until grade 2. 1.MD.A.1. Order three objects by length and compare the lengths
of two objects indirectly by using a third object. Science exam-
1-LS1 FROM MOLECULES TO ORGANISMS: STRUCTURES ple: Every sunflower is taller than the ruler and every daisy
AND PROCESSES is shorter than the ruler, so without measuring directly we
know that every sunflower is taller than every daisy. The
As part of this work, teachers should give students opportunities sunflowers and daisies are not exactly like the plants from
to work with 2-digit numbers: which they grew, but they resemble the plants from which
they grew in being generally tall or generally short.
1.NBT.B.3. Compare two 2-digit numbers based on the meanings
of the tens and ones digits, recording the results of compari- Alignment note: Standard length units such as centimeters or
sons with the symbols >, =, and <. inches are not expected until grade 2.
1.NBT.C.4. Add within 100, including adding a 2-digit number and
a 1-digit number, and adding a 2-digit number and a multiple 1-ESS1 EARTH’S PLACE IN THE UNIVERSE
of 10, using concrete models or drawings and strategies based
on place value, properties of operations, and/or the relation- As part of this work, teachers should give students opportunities
ship between addition and subtraction. Relate the strategy to to practice addition and subtraction and represent and interpret
a written method and explain the reasoning uses. Understand data:
that in adding 2-digit numbers, one adds tens and tens, ones 1.OA.A.1. Use addition and subtraction within 20 to solve word
and ones; and sometimes it is necessary to compose a ten. problems involving situations of adding to, taking from, put-
1.NBT.C.5. Given a 2-digit number, mentally find 10 more or ting together, taking apart, and comparing, with unknowns in
10 less than the number, without having to count. Explain the all positions (e.g., by using objects, drawings, and equations to
reasoning used. represent the problem). Science example: There were 16 hours
1.NBT.C.6. Subtract multiples of 10 in the range 10–90 from mul- of daylight yesterday. On December 21, there were 8 hours
tiples of 10 in the range 10–90 (positive or zero differences), of daylight. How many more hours of daylight were there
using concrete models or drawings and strategies based on yesterday?
place value, properties of operations, and/or the relationship 1.MD.C.4. Organize, represent, and interpret data with up to three
between addition and subtraction. Relate the strategy to a categories. Ask and answer questions about the total number
written method and explain the reasoning used. of data points, how many are in each category, and how many
Science examples: (1) A mother wolf spider is carrying 40 baby more or less are in one category than in another. Science exam-
spiders on her back. There were 50 eggs in the egg sac. How ple: Based on the data collected so far and posted on the bul-
many of the hatchlings is the mother spider not caring for? letin board, which day has been the longest of the year so far?
Which day has been the shortest?
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Alignment notes: (1) Students in this grade are expected to be Make a bar graph with a single-unit scale showing the number
fluent in adding and subtracting within 10. (2) Picture graphs and of seedlings that sprout with and without watering.
bar graphs are not expected until grade 2. (3) Line plots are not Alignment notes: (1) Scaled bar graphs are not expected until
expected until grade 2. (4) The coordinate plane is not expected Grade 3. (2) Multiplication and division of whole numbers are not
until grade 5. expected until Grade 3.
2-PS1 MATTER AND ITS INTERACTIONS 2-LS4 BIOLOGICAL EVOLUTION: UNITY AND DIVERSITY
As part of this work, teachers should give students opportunities As part of this work, teachers should give students opportunities
to represent and interpret categorical data: to represent and interpret categorical data:
2.MD.D.10. Draw a picture graph and a bar graph (with a single- 2.MD.D.10. Draw a picture graph and a bar graph (with a single-
unit scale) to represent a data set with up to four categories. unit scale) to represent a data set with up to four categories.
Solve simple put-together, take-apart, and compare problems5 Solve simple put-together, take-apart, and compare problems7
using information presented in a bar graph. Science examples: using information presented in a bar graph. Science example:
(1) Make a bar graph with a single-unit scale showing how Make a picture graph with a single-unit scale showing the
many samples in a mineral collection are red, green, purple, or number of plant species, vertebrate animal species, and
various other colors. Based on the graph, how many samples invertebrate animal species observed during a field trip or
are represented in all? (2) As part of an investigation of which in a nature photograph. How many more plant species were
materials are best for different intended uses, make a picture observed than animal species?
graph with a single-unit scale showing how many tools in a
toolbox are made of metal, wood, rubber/plastic, or a combi- Alignment notes: (1) Scaled bar graphs are not expected until
nation. Based on the graph, how many tools are represented grade 3. (2) Multiplication and division of whole numbers are not
in all? expected until grade 3.
Alignment notes: (1) Scaled bar graphs are not expected until 2-ESS1 EARTH’S PLACE IN THE UNIVERSE
grade 3. (2) Multiplication and division of whole numbers are not
expected until grade 3. As part of this work, teachers should give students opportunities
to work with numbers to 1,000:
2-LS2 ECOSYSTEMS: INTERACTIONS, ENERGY, AND
DYNAMICS 2.NBT.A.8 Understand place value. Science example: As part of
comprehending media to identify the varying timescales on
As part of this work, teachers should give students opportunities which Earth events can occur, students understand that a
to represent and interpret categorical data: period of thousands of years is much longer than a period of
hundreds of years, which in turn is much longer than a period
2.MD.D.10. Draw a picture graph and a bar graph (with a single- of tens of years.
unit scale) to represent a data set with up to four categories.
Solve simple put-together, take-apart, and compare problems6 Alignment note: Rounding is not expected until grade 3.
using information presented in a bar graph. Science example:
7
Ibid.
5
See Glossary on p. 85 and Table 1 on p. 88 in the CCSSM, available at: http:// 8
The capital letter “A” in “2.NBT.A” refers to the first cluster heading in
www.corestandards.org/assets/CCSSI_Math%20Standards.pdf. domain 2.NBT. See p. 19 of the CCSSM, available at: http://www.corestandards.
6
Ibid. org/assets/CCSSI_Math%20Standards.pdf.
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2-ESS2 EARTH’S SYSTEMS Alignment notes: (1) Scaled bar graphs are not expected until
grade 3. (2) Multiplication and division of whole numbers are not
As part of this work, teachers should give students opportunities expected until grade 3.
to work with numbers to 1,000, to use standard units for length,
and to relate addition and subtraction to length: 3-PS2 MOTION AND STABILITY: FORCES AND
2.NBT.A.3. Read and write numbers to 1,000 using base-ten INTERACTIONS
numerals, number names, and expanded form. Science exam-
As part of this work, teachers should give students opportunities
ple: Students write about a lake that is 550 feet deep, a river
to work with continuous quantities:
that is 687 miles long, a forest that began growing about
200 years ago, and so on. 3.MD.A.2. Measure and estimate liquid volumes and masses of
2.MD.B.5. Use addition and subtraction within 100 to solve word objects using standard units such as grams (g), kilograms (kg),
problems involving lengths that are given in the same units and liters (l).10 Add, subtract, multiply, or divide to solve one-
(e.g., by using drawings [such as drawings of rulers] and equa- step word problems involving masses or volumes that are given
tions with a symbol for the unknown number to represent in the same units (e.g., by using drawings [such as a beaker
the problem). Science example: A gulley was 17 inches deep with a measurement scale] to represent the problem).11 Science
before a rainstorm and 42 inches deep after a rainstorm. How example: Estimate, then measure, the masses of two objects
much deeper did it get during the rainstorm? being used in an investigation of the effect of forces. Observe
that the change of motion due to an unbalanced force is larger
Alignment note: Students in this grade are expected to be fluent
for the smaller mass. (Students need not explain or quantify
in mentally adding and subtracting within 20, knowing single-
this observation in terms of Newton’s Laws of Motion.)
digit sums from memory by the end of grade 2; and are expected
to be fluent in adding and subtracting within 100 using strategies
3-LS1 FROM MOLECULES TO ORGANISMS: STRUCTURES AND
based on place value, properties of operations, and/or the rela-
PROCESSES
tionship between addition and subtraction.
As part of this work, teachers should give students opportunities
K-2-ETS1 ENGINEERING DESIGN to be quantitative in giving descriptions:
3.NF. Number and Operations—Fractions
As part of this work, teachers should give students opportunities
3.NBT. Number and Operations in Base Ten
to represent and interpret categorical data:
Science example: Be quantitative when describing the life cycles
2.MD.D.10. Draw a picture graph and a bar graph (with a single- of organisms, such as their varying life spans (e.g., ranging
unit scale) to represent a data set with up to four categories. from a fraction of a year up to thousands of years) and their
Solve simple put-together, take-apart, and compare problems9 varying reproductive capacity (e.g., ranging from a handful of
using information presented in a bar graph. Science example: offspring to thousands).
Make a bar graph with a single-unit scale showing the number
of seeds dispersed by two or three different design solutions
for seed dispersal.
10
Excludes compound units such as cubic centimeters (cm3) and finding the
geometric volume of a container. See p. 25 of the CCSSM, available at: http://
www.corestandards.org/assets/CCSSI_Math%20Standards.pdf.
11
Excludes multiplicative comparison problems (problems involving notions
of “times as much”). See Glossary on p. 85 and Table 2 on p. 89 in the CCSSM,
9
See Glossary on p. 85 and Table 1 on p. 88 in the CCSSM, available at: http:// available at: http://www.corestandards.org/assets/CCSSI_Math%20Standards.
www.corestandards.org/assets/CCSSI_Math%20Standards.pdf. pdf.
Connections to the Common Core State Standards for Mathematics 143

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3-LS2 ECOSYSTEMS: INTERACTIONS, ENERGY, AND many more of the individuals with the advantageous trait
DYNAMICS survived?
3.MD.B.4. Generate measurement data by measuring lengths
As part of this work, teachers should give students opportunities using rulers marked with halves and fourths of an inch. Show
to be quantitative in giving descriptions: the data by making a line plot, where the horizontal scale
is marked off in appropriate units—whole numbers, halves,
3.NBT. Number and Operations in Base Ten. Science example: Be
or quarters. Science example: Make a line plot to show the
quantitative when describing the group behaviors of animals
length of each fossil that is visible in a piece of shale. Do any
(e.g., describe groups ranging in size from a handful up to
of the fossils resemble modern organisms except for their
thousands of animals).
size?
3-LS3 HEREDITY: INHERITANCE AND VARIATION OF TRAITS
3-ESS2 EARTH’S SYSTEMS
As part of this work, teachers should give students opportunities
As part of this work, teachers should give students opportunities
to represent and interpret data:
to work with continuous quantities and represent and interpret
3.MD.B.4. Generate measurement data by measuring lengths categorical data:
using rulers marked with halves and fourths of an inch. Show
3.MD.A.2. Measure and estimate liquid volumes and masses of
the data by making a line plot, where the horizontal scale is
objects using standard units of grams (g), kilograms (kg), and
marked off in appropriate units—whole numbers, halves, or
liters (l).12 Add, subtract, multiply, or divide to solve one-step
quarters. Science examples: (1) Make a line plot to show the
word problems involving masses or volumes that are given in
height of each of a number of plants grown from a single par-
the same units (e.g., by using drawings [such as a beaker with
ent. Observe that not all of the offspring are the same size.
a measurement scale] to represent the problem).13 Science
Compare the sizes of the offspring to the size of the parent.
examples: (1) Estimate the mass of a large hailstone that dam-
(2) Make a similar plot for plants grown with insufficient water.
aged a car on a used-car lot. (2) Measure the volume of water
in liters collected during a rainstorm.
3-LS4 BIOLOGICAL EVOLUTION: UNITY AND DIVERSITY
3.MD.B.3. Draw a scaled picture graph and a scaled bar graph to
As part of this work, teachers should give students opportunities represent a data set with several categories. Solve one- and
to represent and interpret data: two-step “how many more” and “how many less” problems
using information presented in bar graphs. Science example:
3.MD.B.3. Draw a scaled picture or bar graph to represent a data Make a picture graph or bar graph to show the number of
set with several categories. Solve one- and two-step “how days with high temperatures below freezing in December,
many more” and “how many less” problems using informa- January, February, and March. How many days were below
tion presented in scaled bar graphs. For example, draw a bar freezing this winter?
graph in which each square in the bar graph might represent
five pets. Science examples: (1) Given a bar graph showing
the number of flower species found in several different habi-
tats, determine how many more flower species were found
12
Excludes compound units such as cubic centimeters (cm3) and finding the
geometric volume of a container. See p. 25 of the CCSSM, available at: http://
in a grassy meadow than in a dense forest. Would flower www.corestandards.org/assets/CCSSI_Math%20Standards.pdf.
species be affected if a forest were to spread into its habitat? 13
Excludes multiplicative comparison problems (problems involving notions
(2) Make a scaled bar graph to show the number of surviv- of “times as much”). See Glossary on p. 85 and Table 2 on p. 89 in the CCSSM,
available at: http://www.corestandards.org/assets/CCSSI_Math%20Standards.
ing individuals with and without an advantageous trait. How pdf.
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Alignment notes: (1) Students are not expected to understand sta- 4-PS3 ENERGY
tistical ideas such as average, mean, and median until grade 6.
(2) Graphing in the coordinate plane is not expected until grade 5. As part of this work, teachers should give students opportunities to
use the four operations with whole numbers to solve problems:
3-ESS3 EARTH AND HUMAN ACTIVITY 4.OA.A.3. Solve multi-step word problems posed with whole num-
bers and having whole-number answers using the four opera-
As part of this work, teachers should give students opportunities
tions, including problems in which remainders must be inter-
to work with continuous quantities, including area:
preted. Represent these problems using equations with a letter
3.MD.A.2. Measure and estimate liquid volumes and masses of standing for the unknown quantity. Assess the reasonableness
objects using standard units of grams (g), kilograms (kg), and of answers using mental computation and estimation strategies,
liters (l).14 Add, subtract, multiply, or divide to solve one-step including rounding. Science example: The class has 144 rubber
word problems involving masses or volumes that are given in bands with which to make rubber-band cars. If each car uses
the same units (e.g., by using drawings [such as a beaker with six rubber bands, how many cars can be made? If there are
a measurement scale] to represent the problem).15 28 students, at most how many rubber bands can each car
3.MD.C.5. Recognize area as an attribute of plane figures and have (if every car has the same number of rubber bands)?
understand concepts of area measurement.
Alignment note: Grade 4 students are expected to fluently add
a. square with a side length of one unit, called “a unit square,”
A
and subtract multi-digit whole numbers, multiply a number of
is said to have “one square unit” of area and can be used to
up to four digits by a 1-digit whole number, multiply two 2-digit
measure area.
numbers, and find whole-number quotients and remainders with
b. plane figure that can be covered without gaps or overlaps by
A
up to 4-digit dividends and 1-digit divisors.
n unit squares is said to have an area of n square units.
3.MD.C.6. Measure areas by counting unit squares (square centi-
4-PS4 WAVES AND THEIR APPLICATION IN TECHNOLOGIES
meters, square meters, square inches, square feet, and impro-
FOR INFORMATION TRANSFER
vised units).
Science example: In Hawaii some houses are raised on stilts to As part of this work, teachers should give students opportunities
reduce the impact of a tsunami. The force of a tsunami on to draw and identify lines and angles:
an object is greater if the object presents greater area to an
incoming wave. Based on a diagram of a stilt house, deter- 4.G.A.1. Draw points, lines, line segments, rays, angles (right,
mine how much area the stilts present to an incoming wave. acute, obtuse), and perpendicular and parallel lines. Identify these
How much area would the house present to an incoming wave in two-dimensional figures. Science example: Identify rays and
if it were not on stilts? angles in drawings of wave propagation.
4-LS1 FROM MOLECULES TO ORGANISMS: STRUCTURES
AND PROCESSES
As part of this work, teachers should give students opportunities
14
Excludes compound units such as cubic centimeters (cm3) and finding the to recognize symmetry:
geometric volume of a container. See p. 25 of the CCSSM, available at: http://
www.corestandards.org/assets/CCSSI_Math%20Standards.pdf. 4.G.A.3. Recognize a line of symmetry for a two-dimensional figure
15
Excludes multiplicative comparison problems (problems involving notions as a line across the figure such that the figure can be folded
of “times as much”). See Glossary on p. 85 and Table 2 on p. 89 in the CCSSM,
available at: http://www.corestandards.org/assets/CCSSI_Math%20Standards. across the line into matching parts. Identify line-symmetric
pdf. figures and draw lines of symmetry. Science example: Recognize
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symmetry, or lack of symmetry, in the internal and external (2, 24), (3, 36). Science example: One map shows that a par-
structures of plants and animals. Does the symmetry or lack ticular point in the ocean is 1,600 meters deep, while another
thereof contribute to the function of the organism? (For exam- map shows the same point as being 1.5 kilometers deep. Are
ple, bilateral symmetry is a signal of reproductive fitness in the two maps consistent?
many animals; the asymmetry in an owl’s face helps it pinpoint 4.MD.A.2. Use the four operations to solve word problems
the location of prey.) involving distances, intervals of time, liquid volumes, masses
of objects, and money, including problems involving simple
4-ESS1 EARTH’S PLACE IN THE UNIVERSE fractions or decimals and problems that require expressing
measurements given in a larger unit in terms of a smaller
As part of this work, teachers should give students opportunities unit. Represent measurement quantities using diagrams such
to solve problems involving measurement: as number line diagrams that feature a measurement scale.
4.MD.A.1. Know relative sizes of measurement units within one Science example: A coastline is reduced by an average of 4 feet
system of units, including km, m, and cm; kg and g; lb and oz; per year. In an 18-month period, approximately how much of
l, mL; and hr, min, and sec. Within a single system of measure- the coastline has been lost?
ment, express measurements in a larger unit in terms of a Alignment note: Expressing measurements in a smaller unit in
smaller unit. Record measurement equivalents in a two-column terms of a larger unit within the same system of measurement is
table. For example, know that 1 ft is 12 times as long as 1 in. not expected until grade 5.
Express the length of a 4 ft snake as 48 in. Generate a conver-
sion table for feet and inches listing the number pairs (1, 12), 4-ESS3 EARTH AND HUMAN ACTIVITY
(2, 24), (3, 36). Science example: A limestone layer with many
marine fossils is visible in the Grand Canyon. One reference As part of this work, teachers should give students opportunities
book lists this layer as being 300 feet thick. Another reference to be quantitative in descriptions:
book lists this layer as being 100 yards thick. Are the two ref- 4.OA.A.1. Interpret a multiplication equation as a comparison; for
erences consistent? example, interpret 35 = 5 × 7 as a statement that 35 is 5 times
Alignment note: Expressing measurements in a smaller unit in as many as 7 and 7 times as many as 5. Represent verbal state-
terms of a larger unit within the same system of measurement is ments of multiplicative comparisons as multiplication equations.
not expected until grade 5. Science example: Be quantitative when discussing environmen-
tal effects. For example, say not only that a particular oil spill
4-ESS2 EARTH’S SYSTEMS was “large,” but also that 5 million gallons was spilled or that
the oil spill was 40 times larger than the next-worst oil spill.
As part of this work, teachers should give students opportunities
to solve problems involving measurement: 5-PS1 MATTER AND ITS INTERACTIONS
4.MD.A.1. Know relative sizes of measurement units within one
As part of this work, teachers should give students opportunities
system of units, including km, m, and cm; kg and g; lb and oz;
to relate very large and very small quantities to place value and
l, mL; and hr, min, and sec. Within a single system of measure-
division, convert measurement units, and work with volume:
ment, express measurements in a larger unit in terms of a
smaller unit. Record measurement equivalents in a two-column 5.NBT.A.1. Explain patterns in the number of zeros of a product
table. For example, know that 1 ft is 12 times as long as 1 in. when multiplying a number by powers of 10, and explain pat-
Express the length of a 4 ft snake as 48 in. Generate a conver- terns in the placement of the decimal point when a decimal
sion table for feet and inches listing the number pairs (1, 12),
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is multiplied or divided by a power of 10. Use whole-number dissolved in 0.5 kg of water, what is the total weight of the
exponents to denote powers of 10. system? Answer in grams; then answer again in kilograms.
5.NF.B.7. Apply and extend previous understandings of division to After the water evaporates, see how much the sugar residue
divide unit fractions by whole numbers and whole numbers by weighs.
unit fractions.16 5.MD.C.3. Recognize volume as an attribute of solid figures and
a.
Interpret division of a unit fraction by a non-zero whole num- understand concepts of volume measurement.
ber and compute such quotients. For example, create a story a. cube with a side length of one unit, called a “unit cube,” is
A
context for (1/3) ÷ 4, and use a visual fraction model to show said to have “one cubic unit” of volume and can be used to
the quotient. Use the relationship between multiplication and measure volume.
division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 13. b. solid figure that can be packed without gaps or overlaps
A
b.
Interpret division of a whole number by a unit fraction and using n unit cubes is said to have a volume of n cubic units.
compute such quotients. For example, create a story context 5.MD.C.4. Measure volumes by counting unit cubes, using cubic
for 4 ÷ (1/5), and use a visual fraction model to show the quo- centimeters, cubic inches, cubic feet, and improvised units.
tient. Use the relationship between multiplication and division Science example: Compress the air in a cylinder to half its vol-
to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4. ume. Draw a picture of the volume before and after, and
c.
Solve real-world problems involving the division of unit frac- explain how you know that the new volume is half of the old
tions by non-zero whole numbers and the division of whole volume. Can you compress the volume by half again? Why is it
numbers by unit fractions (e.g., by using visual fraction models difficult to do?
and equations to represent the problem). For example, how Alignment notes: (1) Ratios are not expected until grade 6.
much chocolate will each person get if three people share 1/2 lb (2) Scientific notation is not expected until grade 8.
of chocolate equally? How many 1/3 cup servings are in 2 cups
of raisins? 5-PS2 MOTION AND STABILITY: FORCES AND
Science examples: (1) If you split a salt grain with a weight
INTERACTION
of 1 mg into 10 equal parts, find the weight of each part.
(Answer in milligrams.) If you next divide each of the parts N/A
into 10 equal parts, find the weight of one of the new parts.
(Answer in milligrams.) How many parts are there in the end? 5-PS3 ENERGY
(2) Suppose a salt grain with a weight of 1 mg is split into
10 equal parts, and each of those parts is split into 10 equal N/A
parts, and so on, until there are 108 parts. What is the weight
of one of these tiny parts? Write the number of these tiny 5-LS1 FROM MOLECULES TO ORGANISMS: STRUCTURES
parts as a whole number without using exponents. AND PROCESSES
5.MD.A.1. Convert among different-sized standard measurement
units within a given measurement system (e.g., convert 5 cm As part of this work, teachers should give students opportunities
to 0.05 m), and use these conversions in solving multi-step, to convert measurement units:
real-world problems. Science example: When 100 g of sugar is
5.MD.A.1. Convert among different-sized standard measurement
units within a given measurement system (e.g., convert 5 cm to
16
Students able to multiply fractions can generally develop strategies to divide 0.05 m), and use these conversions in solving multi-step, real-
fractions by reasoning about the relationship between multiplication and divi-
world problems. Science example: In an experiment to rule out
sion. But division of a fraction by a fraction is not a requirement at this grade.
See p. 36 in the CCSSM, available at: http://www.corestandards.org/Math/ soil as a source of plant food, Sue weighed the soil using units
Content/5/NF.
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of grams, but Katya weighed the plant using units of kilo- the year. What pattern is observed when the data are graphed
grams. The soil lost 4 grams, while the plant gained 0.1 kilo- on a coordinate plane? How can a model of the sun and Earth
grams. Did the plant gain much more than the soil lost? Much explain the pattern? (2) Students are given (x, y) coordinates
less? About the same? (A good way to begin is to express for the Earth at six equally spaced times during its orbit around
both figures in grams.) the sun (with the sun at the origin). Students graph the points
Alignment notes: (1) Converting between measurement systems to show snapshots of Earth’s motion through space.
(e.g., centimeters to inches) is not expected until grade 6. (2) Rate Alignment note: Scientific notation is not expected until grade 8.
quantities, such as annual rates of ecosystem production, are not
expected until grade 6. (3) Grade 5 students are expected to read, 5-ESS2 EARTH’S SYSTEMS
write, and compare decimals to thousandths and to perform deci-
mal arithmetic to hundredths. As part of this work, teachers should give students opportunities
to use the coordinate plane:
5-LS2 ECOSYSTEMS: INTERACTIONS, ENERGY, AND 5.G.A.2. Represent real-world and mathematical problems by
DYNAMICS graphing points in the first quadrant of the coordinate plane,
and interpret coordinate values of points in the context of the
As part of this work, teachers should give students opportunities
situation. Science example: Plot monthly data for high and
to be quantitative in giving descriptions. Science example: In a
low temperatures in two locations, one coastal and one inland
diagram showing matter flowing in a system, assign values to
(e.g., San Francisco County and Sacramento). What patterns
the arrows in a diagram to show the flows quantitatively.
are seen? How can the influence of the ocean be seen in the
observed patterns?
5-ESS1 EARTH’S PLACE IN THE UNIVERSE
Alignment notes: (1) Percentages are not expected until grade 6.
As part of this work, teachers should give students opportunities (2) Trends in scatterplots and patterns of association in two-way
to relate very large and very small quantities to place value and tables are not expected until grade 8.
to use the coordinate plane:
5-ESS3 EARTH AND HUMAN ACTIVITY
5.NBT.A.2. Explain patterns in the number of zeros of a product
when multiplying a number by powers of 10, and explain pat- As part of this work, teachers should give students opportuni-
terns in the placement of the decimal point when a decimal ties to be quantitative in giving descriptions. Science example: In
is multiplied or divided by a power of 10. Use whole-number describing ways that individual communities use science ideas to
exponents to denote powers of 10. Science example: The sun protect Earth’s resources and environment, provide quantitative
is about 1011 meters from Earth. Sirius, another star, is about information, such as the amount of energy saved and the cost of
1017 meters from Earth. Write these two numbers without the approach.
exponents; position the numbers directly below the other,
aligning on the 1. How many times farther away from Earth is 3-5-ETS1 ENGINEERING DESIGN
Sirius compared to the sun?
5.G.A.2. Represent real-world and mathematical problems by As part of this work, teachers should give students opportunities
graphing points in the first quadrant of the coordinate plane, to use the four operations to solve problems:
and interpret coordinate values of points in the context of the
situation. Science examples: (1) Over the course of a year, stu- OA: Operations and Algebraic Thinking (representing and solv-
dents compile data for the length of the day over the course of ing problems using the four operations; see each grade in the
CCSSM for detailed expectations). Science example: Analyze
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constraints on materials, time, or cost to draw implications for less than or much greater than 1. Also use convenient units
design solutions. For example, if a design calls for 20 screws such as unified atomic mass units.
and screws are sold in boxes of 150, how many copies of the Statistics and Probability (6-8.SP). Science example: Compile all of
design could be made? the boiling point measurements from the students into a line
plot and discuss the distribution in terms of clustering and outli-
MS-PS1 MATTER AND ITS INTERACTIONS ers. Why were not all of the measured values equal? How close
is the average value to the nominal/textbook value? Show the
As part of this work, teachers should give students opportunities average value and the nominal value on the line plot.
to work with ratios and proportional relationships, use signed
numbers, write and solve equations, and use order-of-magnitude MS-PS2 MOTION AND STABILITY: FORCES AND
thinking and basic statistics: INTERACTIONS
Ratios and Proportional Relationships (6-7.RP). Science examples:
As part of this work, teachers should give students opportunities
(1) A pile of salt has a mass of 100 mg. How much chlorine
to work with signed numbers and interpret expressions:
is in it? Answer in milligrams. What would the answer be
for a 500 mg pile of salt? (2) Twice as much water is twice The Number System (6-8.NS). Science examples: (1) Represent a
as heavy. Explain why twice as much water is not twice as third-law pair of forces as a +100 N force on one object and
dense. (3) Based on a model of a water molecule, recognize a −100 N force on the other object. (2) Represent balanced
that any sample of water has a 2:1 ratio of hydrogen atoms forces on a single object as equal and opposite numbers ±5 N.
to oxygen atoms. (4) Measure the mass and volume of a (3) Represent the net result of two or more forces as a sum of
sample of reactant and compute its density. (5) Compare a signed numbers. For example, given a large force and an oppo-
measured/computed density to a nominal/textbook value, sitely directed small force, represent the net force as (+100 N) +
converting units as necessary. Determine the percent differ- (−5 N) = +95 N. Relate the number sentence to the fact that the
ence between the two. net effect on the motion is approximately what it would have
The Number System (6-8.NS). Science examples: (1) Use positive been with only the large force.
and negative quantities to represent temperature changes in Expressions and Equations (6-8.EE). Science example: Interpret an
a chemical reaction (signs of energy released or absorbed). expression in terms of a physical context. For example, inter-
(2) For grade 7 or 8: Solve a simple equation for an unknown pret the expression F1 + F2 in a diagram as representing the net
signed number. For example, a solution was initially at room force on an object.
temperature. After the first reaction, the temperature change
was −8oC. After the second reaction, the temperature was 3oC MS-PS3 ENERGY
below room temperature. Find the temperature change during
the second reaction. Was energy released or absorbed in the As part of this work, teachers should give students opportuni-
second reaction? Show all of the given information on a num- ties to work with ratios and proportional relationships and basic
ber line/thermometer scale. Also represent the problem by an statistics:
equation.
Ratios and Proportional Relationships (6-7.RP) and Functions (8.F).
Expressions and Equations (6-8.EE). Science examples: (1) For
Science examples: (1) Analyze an idealized set of bivariate
grade 8: With substantial scaffolding, use algebra and quan-
measurement data for kinetic energy versus mass (holding
titative thinking to determine the interatomic spacing in a
speed constant). Decide whether the two quantities are in a
salt crystal. (2) For grade 8: Use scientific notation for atomic
proportional relationship (e.g., by testing for equivalent ratios
masses, large numbers of atoms, and other quantities much
or graphing on a coordinate plane and observing whether the
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graph is a straight line through the origin). (2) Do the same for Expressions and Equations (6-8.EE). Science examples: (1) Quantify
an idealized set of data for kinetic energy versus speed (hold- the sizes of cells and parts of cells, using convenient units such
ing mass constant). For grade 8 recognize from the data that as microns as well as (in grade 8) scientific notation.
the relationship is not proportional and that kinetic energy is (2) Appreciate the orders of magnitude that span the differ-
a non-linear function of speed. Draw conclusions such as that ence in size between cells, molecules, and atoms. (3) Write
doubling the speed more than doubles the kinetic energy. a number sentence that expresses the conservation of mass
What are some possible implications for driving safety? as food moves through an organism. Assign values to the
Statistics and Probability (6-8.SP). Science example: As part of carry- arrows in a diagram to show flows quantitatively. (4) Infer an
ing out a designed experiment, make a scatterplot showing the unknown mass by using the concept of conservation to write
temperature change of a sample of water versus the mass of ice and solve an equation with a variable.
added. For grade 8 if the data suggest a linear association, form Statistics and Probability (6-8.SP). Science examples: (1) For grade
a straight line, and informally assess the model fit by judging 8 use data in a two-way table as evidence to support an
the closeness of the data points to the line. Just for fun, com- explanation of how environmental and genetic factors affect
pute the slope of the line. What are the units of the answer? the growth of organisms. (2) For grade 8 use data in a two-
way table as evidence to support an explanation that different
MS-PS4 WAVES AND THEIR APPLICATIONS IN local environmental conditions impact growth in organisms.
TECHNOLOGIES FOR INFORMATION TRANSFER (3) For grade 7 or 8 use probability concepts and language to
describe and quantify the effects that characteristic animal
As part of this work, teachers should give students opportunities behaviors have on the likelihood of successful reproduction.
to use ratios and proportional relationships and functions:
Ratios and Proportional Relationships (6-7.RP) and Functions (8.F). MS-LS2 ECOSYSTEMS: INTERACTIONS, ENERGY, AND
Science examples: (1) Analyze an idealized set of bivariate DYNAMICS
measurement data for wave energy versus wave amplitude.
As part of this work, teachers should give students opportunities
Decide whether the two quantities are in a proportional
to work with ratios and proportional relationships, write and
relationship (e.g., by testing for equivalent ratios or graph-
solve equations, and use basic statistics:
ing on a coordinate plane and observing whether the graph
is a straight line through the origin). For grade 8 recognize Ratios and Proportional Relationships (6-7.RP). Science example:
that wave energy is a non-linear function of amplitude, and Use ratios and unit rates as inputs for evaluating plans for
draw conclusions such as that doubling the amplitude more maintaining biodiversity and ecosystem services (e.g., con-
than doubles the energy. Discuss possible implications for the sider the net cost or net value of developing a wetland, using
safety of wading in the ocean during a storm. (2) Interpret an inputs such as the value of various wetland services in dol-
idealized set of bivariate measurement data for wave energy lars per acre per year; and in analyzing urban biodiversity,
versus wave speed. rank world cities by the amount of green space as a fraction
of total land area; in analyzing social factors, determine the
MS-LS1 FROM MOLECULES TO ORGANISMS: STRUCTURES amount of green space per capita [square meters per person]).
AND PROCESSES Expressions and Equations (6-8.EE). Science examples: (1) Write
a number sentence that expresses the conservation of total
As part of this work, teachers should give students opportunities matter or energy in a system as matter or energy flows into,
to use order-of-magnitude thinking, write and solve equations, out of, and within it. Assign values to the arrows in a diagram
analyze data, and use concepts of probability: to show flows quantitatively. (2) Infer an unknown matter or
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energy flow in a system by using the concept of conservation Myr, Gyr, Ma, Ga). (3) Appreciate the spans of time involved in
to write and solve an equation with a variable. natural selection.
Statistics and Probability (6-8.SP). Science example: For grade 8 Alignment notes: (1) Exponential functions are not expected until
use data in a two-way table as evidence to support an expla- high school. (2) Laws of probability such as p(AB) = p(A)p(B A) are
nation of how social behaviors and group interactions benefit not expected until high school.
organisms’ abilities to survive and reproduce.
MS-ESS1 EARTH’S PLACE IN THE UNIVERSE
MS-LS3 HEREDITY: INHERITANCE AND VARIATION OF
TRAITS As part of this work, teachers should give students opportunities
to use ratios and proportional relationships and use order-of-
As part of this work, teachers should give students opportunities magnitude thinking:
to use concepts of probability:
Ratios and Proportional Relationships (6-7.RP). Science examples:
Statistics and Probability (6-8.SP). Science examples: (1) Recognize For grade 7: (1) Create a scale model or scale drawing of the
a Punnett square as a component of a probability model, solar system or Milky Way galaxy. (2) Create scale-preserving
and compute simple probabilities from the model. (2) Use a descriptions, such as “If the solar system were shrunk down
computer to simulate the variation that comes from sexual to the size of Earth, then Earth would shrink to the size of
reproduction, and determine probabilities of traits from the ______”; compute relevant scale factors and use them to deter-
simulation. mine a suitable object.
Expressions and Equations (6-8.EE). Science examples: For grade 8:
MS-LS4 BIOLOGICAL EVOLUTION: UNITY AND DIVERSITY (1) Use scientific notation for long intervals of time or for
dates in the distant past; also use convenient units (e.g., Myr,
As part of this work, teachers should give students opportunities
Gyr, Ma, Ga). Appreciate the spans of time involved in Earth’s
to work with ratios and proportional relationships, use concepts
history. (2) Are there more molecules of gas in a toy balloon or
of probability, and use order-of-magnitude thinking:
more stars in the Milky Way galaxy?
Ratios and Proportional Relationships (6-7.RP) and Statistics and
Probability (6-8.SP). Science examples: (1) Apply several ratios MS-ESS2 EARTH’S SYSTEMS
in combination to determine a net survival rate. For example,
if 50 animals in a population have trait A and 50 have trait As part of this work, teachers should give students opportunities
B, and each winter the survival rates are 80% for trait A and to work with positive and negative numbers and to use order-of-
60% for trait B, how many animals with each trait will be magnitude thinking:
alive after one winter? After two winters? Six winters? (2) Use The Number System (6-8.NS). Science examples: (1) Use posi-
scaled histograms to summarize the results of a simulation of tive and negative quantities to quantify changes in physical
natural selection over many generations. (3) For grade 7 or 8 quantities such as atmospheric pressure and temperature. For
use probability language and concepts when explaining how example, if the temperature drops from 24oC to 11oC, the tem-
variation in traits among a population leads to an increase in perature change is −13oC. (2) Solve word problems relating to
some traits in the population and a decrease in others. changes in signed physical quantities. For example, a shift in
Expressions and Equations (6-8.EE). Science examples: (1) Quantify the jet stream caused a 10oC temperature increase in a single
durations of time in interpreting the fossil record. (2) For day; if the temperature before the shift was −32oC, what was
grade 8 use scientific notation for long intervals of time or the temperature after?
for dates in the distant past; also use convenient units (e.g.,
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Expressions and Equations (6-8.EE). Science examples: For grade 8: the reasonableness of answers using mental computation and
(1) Use scientific notation for long intervals of time or for estimation strategies. For example: If a woman making $25 an
dates in the distant past; also use convenient units (e.g., Myr, hour gets a 10% raise, she will make an additional 1/10 of her
Gyr, Ma, Ga). (2) Use order-of-magnitude data on the rate old hourly wage, or $2.50, for a new hourly wage of $27.50. If
of seafloor spreading to estimate how long it has taken for you want to place a towel bar 9¾ inches long in the center of
two continents to separate. (3) Appreciate the spans of time a door that is 27½ inches wide, you will need to place the bar
involved in Earth’s history. Recognize that a period of time is about 9 inches from each edge; this estimate can be used as a
neither “long” nor “short” in itself, but only relatively long or check on the exact computation. Science example: Work with
relatively short compared to some other period of time. For tolerances, cost constraints, and other quantitative factors in
example, the Hawaiian islands have been forming for several evaluating competing design solutions.
million years, and this time period is neither long nor short. It Statistics and Probability (6-8.SP). Develop a probability model
is a long time in comparison to the duration of the last glacial and use it to find probabilities of events. Compare probabili-
period, but a short time in comparison to Earth’s entire history. ties from a model to observed frequencies; if the agreement is
not good, explain possible sources of the discrepancy. Science
MS-ESS3 EARTH AND HUMAN ACTIVITY example: For grade 7 use simulations to generate data that
As part of this work, teachers should give students opportunities can be used to modify a proposed object, tool, or process.
to use ratios and proportional relationships and use order-of-
magnitude thinking: HS-PS1 MATTER AND ITS INTERACTIONS
Ratios and Proportional Relationships (6-7.RP). Science example: As part of this work, teachers should give students opportunities
Work with measurement quantities that are formed through to reason quantitatively and use units to solve problems and to
division, such as atmospheric concentration of carbon dioxide, apply key takeaways from grades 6–8 mathematics:
extraction cost per barrel of oil in different forms, per-capita
Quantities (N-Q)/Reason quantitatively and use units to solve
consumption of given resources, and flow rates in freshwater
problems:
rivers.
Expressions and Equations (6-8.EE). Science example: For grade N-Q.1. Use units as a way to understand problems and to guide
8 use orders of magnitude and order-of-magnitude estimates the solution of multi-step problems; choose and interpret units
as part of oral and written arguments, evaluations of data consistently in formulas; choose and interpret the scale and ori-
from technical texts, design solutions, and explanations of the gin in graphs and data displays.
impact on Earth’s systems of increasing population and per- N-Q.2. Define appropriate quantities for the purpose of descrip-
capita consumption. tive modeling.
N-Q.3. Choose a level of accuracy appropriate to limitations on
MS-ETS1 ENGINEERING DESIGN measurement when reporting quantities.
Science examples: (1) Recognize the difference between intensive
As part of this work, teachers should give students opportunities and extensive quantities (e.g., a quantity with units of joules
to solve quantitative problems and use basic statistics: per kilogram is insensitive to the overall size of the sample in
question, unlike a quantity with units of joules). (2) Attend to
7.EE.3. Solve multi-step real-life and mathematical problems
units properly when using formulas such as density = mass/
posed with positive and negative rational numbers in any form
volume. (3) Carefully format data displays and graphs, attend-
(whole numbers, fractions, and decimals), using tools strategi-
ing to origin, scale, units, and other essential items.
cally. Apply properties of operations to calculate with numbers
in any form, convert between forms as appropriate, and assess
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Applying key takeaways from grades 6–8 mathematics. Science tity of interest. (2) Write and solve a linear equation to solve a
examples: (1) Convert a reference value of a quantity to match problem involving motion at a constant speed.
the units being used in a classroom experiment. (2) Interpret, Interpreting Functions (F-IF) and Interpreting Categorical and
write, or solve an equation that represents the conservation of Quantitative Data (S-ID). Science examples: (1) Informally fit a
energy or mass in a chemical reaction. quadratic function to the position-time data for a cart that rolls
up an incline (slowing as it climbs, then reversing direction and
HS-PS2 MOTION AND INSTABILITY: FORCES AND speeding up as it descends). Use the algebraic expression for the
INTERACTIONS fitted function to determine the magnitude of the cart’s accelera-
tion and initial speed. Over several trials, graph various quanti-
As part of this work, teachers should give students opportunities
ties (such as acceleration versus angle or peak displacement ver-
to model with mathematics, use basic algebra, reason quanti-
sus initial speed squared) and interpret the results. (2) Calculate
tatively, and use units to solve problems and to apply key take-
and interpret the average speed of a moving object by using data
aways from grades 6–8 mathematics:
from a distance-time graph.
Quantities (N-Q)/Reason quantitatively and use units to solve
Applying key takeaways from grades 6–8 mathematics. Science
problems:
examples: (1) Compute ratios of distances and times in order to
N-Q.1. Use units as a way to understand problems and to guide distinguish accelerated motion from motion with constant speed.
the solution of multi-step problems; choose and interpret units Reason qualitatively on that basis (e.g., a dropped stone falls far-
consistently in formulas; choose and interpret the scale and ori- ther between t = 1s and t = 2s than between t = 0s and t = 1s).
gin in graphs and data displays. (2) For an object moving at constant speed, compute the speed
N-Q.2. Define appropriate quantities for the purpose of descrip- by choosing a point from its distance-time graph.
tive modeling.
N-Q.3. Choose a level of accuracy appropriate to limitations on HS-PS3 ENERGY
measurement when reporting quantities.
Science examples: (1) Relate the units of acceleration (m/s2) to As part of this work, teachers should give students opportunities
the fact that acceleration refers to a change in velocity over to reason quantitatively and use units to solve problems and to
time. (2) Reconstruct the units of the universal gravitational apply key takeaways from grades 6–8 mathematics:
constant G by reference to the formula F = Gm1m2 / r 2, instead Quantities (N-Q)/Reason quantitatively and use units to solve
of having to memorize the units. (2) Attend to units properly problems:
when using formulas such as momentum = mass times veloc- N-Q.1. Use units as a way to understand problems and to guide
ity. (3) Carefully format data displays and graphs, attending to the solution of multi-step problems; choose and interpret units
origin, scale, units, and other essential items. consistently in formulas; choose and interpret the scale and ori-
Seeing Structure in Expressions (A-SSE). Science example: Draw con- gin in graphs and data displays.
clusions about gravitational or other forces by interpreting the alge- N-Q.2. Define appropriate quantities for the purpose of descrip-
braic structure of formulas. For example, conclude that the force on tive modeling.
an object in a gravitational field is proportional to its mass, by view- N-Q.3. Choose a level of accuracy appropriate to limitations on
ing the formula F = Gmsourcemobject /r 2 as F = (Gmsource/r 2) (mobject) measurement when reporting quantities.
and recognizing the same algebraic structure as in y = kx. Science examples: (1) Analyze units in expressions like mgh
and ½ mv2 to show that they both refer to forms of energy.
Creating Equations (A-CED). Science examples: (1) Rearrange a
(2) Observe in a range of situations within science that
formula (such as F = ma or p = mv) in order to highlight a quan-
quantities being added to one another or subtracted from
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one another are always quantities of the same general kind as long to reach the desired temperature as 1 lb of food? Why or
(energy, length, time, temperature); express such terms in the why not? How is this situation similar to or different from cook-
same units before adding or subtracting. (3) Carefully format ing in a conventional oven?
data displays and graphs, attending to origin, scale, units, and
other essential items. HS-LS1 FROM MOLECULES TO ORGANISMS: STRUCTURES
Applying key takeaways from grades 6–8 mathematics. Science AND PROCESSES
examples: (1) Fit a linear function to a data set showing the
As part of this work, teachers should give students opportunities
relationship between the change in temperature of an insulated
to model with mathematics:
sample of water and the number of identical hot ball bearings
dropped into it (all with the same initial temperature). Find the Interpreting Functions (F-IF) and Building Functions (F-BF).17
slope of the graph and use it to determine the specific heat of Science example: Use a spreadsheet or other technology to simu-
the metal. (2) Interpret, write, or solve an equation that repre- late the doubling in a process of cell division; graph the results;
sents the conservation of energy in a given process. write an expression to represent the number of cells after a divi-
sion in terms of the number of cells beforehand; express this in
HS-PS4 WAVES AND THEIR APPLICATIONS IN closed form as a population size in terms of time. Discuss real-
TECHNOLOGIES FOR INFORMATION TRANSFER world factors in the situation that lead to deviation from the
exponential model over time.
As part of this work, teachers should give students opportunities
to work with basic algebra and to apply key takeaways from HS-LS2 ECOSYSTEMS: INTERACTIONS, ENERGY, AND
grades 6–8 mathematics: DYNAMICS
Seeing Structure in Expressions (A-SSE). Science example: (1) Write
As part of this work, teachers should give students opportunities
expressions in equivalent forms to solve problems. For example,
to reason quantitatively and use units to solve problems, repre-
relate the formulas c = λf and c = λ/T by seeing that λ/T = λ(1/T)
sent quantitative data, and apply key takeaways from grades 6–8
= λf, instead of remembering both forms separately. (2) See the
mathematics:
conceptual and structural similarities between formulas such as
c = λ/T and v = d/t. How do these formulas relate to the formula Quantities (N-Q)/Reason quantitatively and use units to solve
(running speed) = (stride length) × (stride frequency), which is problems:
sometimes found in track-and-field coaching manuals? N-Q.1. Use units as a way to understand problems and to guide
the solution of multi-step problems; choose and interpret units
Creating Equations (A-CED). Science examples: (1) Rearrange a
consistently in formulas; choose and interpret the scale and ori-
formula in order to highlight a quantity of interest. (2) Write and
gin in graphs and data displays.
solve an equation in a problem involving wave motion. For exam-
N-Q.2. Define appropriate quantities for the purpose of descrip-
ple, as part of an activity to use seismographic data to locate the
tive modeling.
epicenter of an earthquake.
N-Q.3. Choose a level of accuracy appropriate to limitations on
Applying key takeaways from grades 6–8 mathematics. Apply sev- measurement when reporting quantities.
eral proportional relationships in combination. Science examples: Science examples: (1) Recognize the difference between intensive
(1) Estimate how long it would take a solar cell installation to and extensive quantities (e.g., a quantity with units of tons/
pay for itself by combining such factors as per-square-meter cost acre is insensitive to the overall size of the area in question,
of solar cells, collection efficiency, per-square-meter solar energy unlike a quantity with units of tons). (2) Carefully format data
flux, and per-kilowatt-hour cost of conventional electricity. (2) Will
4 lbs of food in a microwave oven take approximately four times 17
See also Linear, Quadratic, and Exponential Models (F-LE).
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displays and graphs, attending to origin, scale, units, and HS-ESS1 EARTH’S PLACE IN THE UNIVERSE
other essential items.
As part of this work, teachers should give students opportunities
Interpreting Categorical and Quantitative Data (S-ID) and Making
to model with mathematics, use basic algebra, reason quanti-
Inferences and Justifying Conclusions (S-IC). Science example: Use
tatively, and use units to solve problems and to apply key take-
a spreadsheet or other technology to analyze and display a his-
aways from grades 6–8 mathematics:
torical or simulated data set as part of an investigation of ecosys-
tem changes. Interpreting Functions (F-IF) and Interpreting Categorical and
Quantitative Data (S-ID).18 Science example: Work with exponential
Applying key takeaways from grades 6–8 mathematics. Science
models in connection with radiometric dating concepts and data.
examples: (1) Compute a percent change in a variable over the
period of a historical data set, as part of an explanation of ecosys- Creating Equations (A-CED). Science examples: (1) Rearrange a
tem changes (e.g., pesticide application, disease incidence, water formula (such as E = mc2) in order to highlight a quantity of inter-
temperature, invasive species population counts). (2) Merge two est. (2) Use Kepler’s Third Law to write and solve an equation in
data sets by converting values in one data set as necessary to order to solve a problem involving orbital motion.
match the units used in the other. (3) Interpret, write, or solve an Seeing Structure in Expressions (A-SSE). Science example: Draw
equation that represents the conservation of energy as it is trans- conclusions about astronomical phenomena by interpreting the
ferred from one trophic level to another. algebraic structure of formulas. For example, conclude from
λmaxT = b that cooler stars are redder and hotter stars are bluer;
HS-LS3 HEREDITY: INHERITANCE AND VARIATION OF conclude from Kepler’s Third Law that Earth and the moon take
TRAITS the same amount of time to revolve around the sun, even though
they have different masses (because only the mass of the sun
As part of this work, teachers should give students opportunities
appears in the law, not the mass of the orbiting body).
to apply key takeaways from grades 6–8:
Quantities (N-Q)/Reason quantitatively and use units to solve
Applying key takeaways from grades 6–8 mathematics (see espe-
problems:
cially 7.SP.B, 7.SP.C, and 8.SP.4). Science examples: (1) Use a prob-
N-Q.1. Use units as a way to understand problems and to guide
ability model to estimate the probability that a child will inherit a
the solution of multi-step problems; choose and interpret units
disease or other trait, given knowledge or hypotheses about the
consistently in formulas; choose and interpret the scale and ori-
parents’ traits. (2) Use observed or simulated frequencies to iden-
gin in graphs and data displays.
tify cases of non-Mendelian inheritance.
N-Q.2. Define appropriate quantities for the purpose of descrip-
tive modeling.
HS-LS4 BIOLOGICAL EVOLUTION: UNITY AND DIVERSITY N-Q.3. Choose a level of accuracy appropriate to limitations on
measurement when reporting quantities.
As part of this work, teachers should give students opportunities
Science examples: (1) Verify that mc2 has units of energy (e.g.,
to apply key takeaways from grades 6–8:
has the same units as mgh and mv2). (2) Use SI units as well as
Applying key takeaways from grades 6–8 mathematics (see espe- convenient units (e.g., mm/yr for sea floor spreading, Gya for
cially 6.SP). Science example: Assess differences between two dates in the early history of Earth). (3) Attend to units prop-
populations using measures of center and variation for each. erly when using formulas such as energy = mass times speed
Analyze a shift in the numerical distribution of traits and use of light squared. (4) Carefully format data displays and graphs,
these shifts as evidence to support an explanation. attending to origin, scale, units, and other essential items.
18
See also Linear, Quadratic, and Exponential Models (F-LE).
Connections to the Common Core State Standards for Mathematics 155

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Applying key takeaways from grades 6–8 mathematics. Science HS-ESS3 EARTH AND HUMAN ACTIVITY
examples: (1) Pinnacles National Monument is believed to have
been created when the San Andreas Fault moved part of a vol- As part of this work, teachers should give students opportunities
cano 195 miles northward over a period of 23 million years. On to reason quantitatively and use units to solve problems and to
average, how fast did this part of the volcano move, in centime- apply key takeaways from grades 6–8 mathematics:
ters per year? (2) Also express the answer in meters per second Quantities (N-Q)/Reason quantitatively and use units to solve
using scientific notation. (3) Use a spreadsheet to scatterplot the problems:
planets’ orbital periods against their orbital semimajor axes, and
explain how these data show non-linearity. Then scatterplot the N-Q.1. Use units as a way to understand problems and to guide
squares of the orbital periods against the cubes of their orbital the solution of multi-step problems; choose and interpret units
semimajor axes, and show that the relationship is linear. consistently in formulas; choose and interpret the scale and ori-
gin in graphs and data displays.
HS-ESS2 EARTH’S SYSTEMS N-Q.2. Define appropriate quantities for the purpose of descrip-
tive modeling.
As part of this work, teachers should give students opportunities N-Q.3. Choose a level of accuracy appropriate to limitations on
to reason quantitatively and use units to solve problems and to measurement when reporting quantities.
apply key takeaways from grades 6–8 mathematics: Science examples: (1) Quantify the impacts of human activities on
natural systems. For example, if a certain activity creates pol-
Quantities (N-Q)/Reason quantitatively and use units to solve
lution that in turn damages forests, go beyond a qualitative
problems:
statement by quantifying both the amount of pollution and
N-Q.1. Use units as a way to understand problems and to guide
the level of damage. (2) Carefully format data displays and
the solution of multi-step problems; choose and interpret units
graphs, attending to origin, scale, units, and other essential
consistently in formulas; choose and interpret the scale and ori-
items.
gin in graphs and data displays.
N-Q.2. Define appropriate quantities for the purpose of descrip- Applying key takeaways from grades 6–8 mathematics. Science
tive modeling. examples: (1) Use concepts of probability to describe risks of
N-Q.3. Choose a level of accuracy appropriate to limitations on natural hazards (e.g., volcanic eruptions, earthquakes, tsunamis,
measurement when reporting quantities. hurricanes, droughts) that impact human activity. (2) Use cost–
Science examples: (1) When coastal erosion is measured, what benefit ratios to evaluate competing design solutions. (3) Use
are its units? What does this say about what quantity is being per-capita measures such as consumption, cost, and resource
measured? (2) Use SI units as well as convenient units (e.g., needs.
My for durations of time or Gya for dates in the early his-
tory of Earth). (3) Carefully format data displays and graphs, HS-ETS1 ENGINEERING DESIGN
attending to origin, scale, units, and other essential items.
As part of this work, teachers should give students opportunities
Applying key takeaways from grades 6–8 mathematics. Science
to model with mathematics and to apply key takeaways from
example: Use order-of-magnitude thinking to appreciate rela-
grades 6–8 mathematics:
tive timescales in the co-evolution of Earth’s systems and life
on Earth. For example, when did photosynthetic life alter the Modeling. Science examples: (1) Identify variables and select
atmosphere? How long was it then until the appearance of land those that represent essential factors to understand, control, or
plants? optimize. (2) Use technology to vary assumptions, explore conse-
quences, and compare predictions with data.
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Applying key takeaways from grades 6–8 mathematics. Science http://www.teebweb.org/local-and-regional-policy-makers-report
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impacts, and other quantitative factors in evaluating competing Ts-5%20coastal%20erosion%20and%20mangrove%20progradation
design solutions. %20of%20southern%20thailand.pdf
http://www.uky.edu/Ag/CritterFiles/casefile/spiders/wolf/wolf.htm
REFERENCES
Boyer, C. B., and Merzbach, U. C. (1991). A history of mathematics,
Second Edition. New York: Wiley and Sons. Pp. vii–viii.
National Governors Association Center for Best Practices, Council of
Chief State School Officers. (2010). Common Core State Standards
Mathematics. Washington, DC: National Governors Association
Center for Best Practices, Council of Chief State School Officers.
Pp. 72–73.
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Connections to the Common Core State Standards for Mathematics 157