Math: Common Core Information Session
Mathematical Practices & Math Content Must InteractThe
Standards for Mathematical Practice describe ways in which
developing student practitioners of the discipline of mathematics
increasingly ought to engage with the subject matter as they grow
in mathematical maturity and expertise throughout the elementary,
middle and high school years. The Standards for Mathematical
Content are a balanced combination of procedure and understanding.
Expectations that begin with the word understand are often
especially good opportunities to connect the practices to the
content. Students who lack understanding of a topic may rely on
procedures too heavily which may effectively prevent a student from
engaging in the mathematical practices.The content standards which
set an expectation of understanding are potential points of
intersection between the Standards for Mathematical Content and the
Standards for Mathematical Practice. Designers of curricula,
assessments, and professional development should all attend to the
need to connect the mathematical practices to mathematical content
in mathematics instruction. Without a flexible base from which to
work, they may be less likely to consider analogous problems,
represent problems coherently, justify conclusions, apply the
mathematics to practical situations, use technology mindfully to
work with the mathematics, explain the mathematics accurately to
other students, step back for an overview, or deviate from a known
procedure to find a shortcut. These points of intersection are
intended to be weighted toward central and generative concepts in
the school mathematics curriculum that most merit the time,
resources, innovative energies, and focus necessary to
qualitatively improve the curriculum, instruction, assessment,
professional development, and student achievement in
mathematics.34
Assessments in mathematics aligned to the Common Core State
Standards, so that students leave high school prepared for
postsecondary success in college or a career through increased
student learning and improved teaching.2Bases of the Standards for
Mathematical PracticeDesired expertise mathematics educators at all
levels should seek to develop in their students. These practices
rest on important processes and proficiencies.The first of these
are the NCTM process standards of problem solving, reasoning and
proof, communication, representation, and connections. The second
are the strands of mathematical proficiency specified in the
National Research Councils reportAdding It Up: adaptive reasoning,
strategic competence, conceptual understanding (comprehension of
mathematical concepts, operations and relations), procedural
fluency (skill in carrying out procedures flexibly, accurately,
efficiently and appropriately), and productive disposition
(habitual inclination to see mathematics as sensible, useful, and
worthwhile, coupled with a belief in diligence and ones own
efficacy).Overview of the Standards for Mathematical Practice1 -
Make sense of problems and persevere in solving them.2 - Reason
abstractly and quantitatively.3 - Construct viable arguments and
critique the reasoning of others.4 - Model with mathematics.5 - Use
appropriate tools strategically.6 - Attend to precision.7 - Look
for and make use of structure.8 - Look for and express regularity
in repeated reasoning.How can we get students over their fear of
word problems?
1 - Make sense of problems and persevere in solving them.Start
by explaining to themselves the meaning of a problem and looking
for entry points to its solution. Analyze givens, constraints,
relationships, and goals. Make conjectures about the form and
meaning of the solution and plan a solution pathway rather than
simply jumping into a solution attempt. Consider analogous
problems, and try special cases and simpler forms of the original
problem in order to gain insight into its solution. Monitor and
evaluate their progress and change course if necessary. Explain
correspondences between equations, verbal descriptions, tables, and
graphs or draw diagrams of important features and relationships,
graph data, and search for regularity or trends. Check their
answers to problems using a different method, and they continually
ask themselves, Does this make sense? Understand the approaches of
others to solving complex problems and identify correspondences
between different approaches.Older students might, depending on the
context of the problem, transform algebraic expressions or change
the viewing window on their graphing calculator to get the
information they need. Younger students might rely on using
concrete objects or pictures to help conceptualize and solve a
problem. 62 - Reason abstractly and quantitatively.Make sense of
quantities and their relationships in problem situations. The
ability todecontextualizeto abstract a given situation and
represent it symbolically and manipulate the representing
symbolsThe ability tocontextualize, to pause as needed during the
manipulation process in order to probe into the referents for the
symbols involved. Quantitative reasoning entails habits of creating
a coherent representation of the problem at hand.
They bring two complementary abilities to bear on problems
involving quantitative relationships: as if they have a life of
their own, without necessarily attending to their referents
considering the units involved; attending to the meaning of
quantities, not just how to compute them; and Knowing and flexibly
using different properties of operations and objects.73 - Construct
viable arguments and critique the reasoning of others.Understand
and use stated assumptions, definitions, and previously established
results in constructing arguments. Make conjectures and build a
logical progression of statements to explore the truth of their
conjectures. Analyze situations by breaking them into cases, and
can recognize and use counterexamples. Justify their conclusions,
communicate them to others, and respond to the arguments of others.
Reason inductively about data, making plausible arguments that take
into account the context from which the data arose. Compare the
effectiveness of two plausible arguments, distinguish correct logic
or reasoning from that which is flawed, andif there is a flaw in an
argumentexplain what it is. Determinedomains to which an argument
applies. Listen or read the arguments of others, decide whether
they make sense, and ask useful questions to clarify or improve the
arguments.
Elementary students can construct arguments using concrete
referents such as objects, drawings, diagrams, and actions. Such
arguments can make sense and be correct, even though they are not
generalized or made formal until later grades.84 - Model with
mathematics.Apply the mathematics they know to solve problems
arising in everyday life, society, and the workplace. Comfortable
making assumptions and approximations to simplify a complicated
situation, realizing that these may need revision later. Identify
important quantities in a practical situation and map their
relationships using such tools as diagrams, two-way tables, graphs,
flowcharts and formulas. Analyze those relationships mathematically
to draw conclusions. Routinely interpret their mathematical results
in the context of the situation and reflect on whether the results
make sense, possibly improving the modelIn early grades, this might
be as simple as writing an addition equation to describe a
situation. In middle grades, a student might apply proportional
reasoning to plan a school event or analyze a problem in the
community. By high school, a student might use geometry to solve a
design problem or use a function to describe how one quantity of
interest depends on another. 9While we are still teaching
mathematics, how students engage in learning the concepts MUST be
different!
Boy, some things change with time, but math sure doesnt!5 - Use
appropriate tools strategicallyConsider the available tools when
solving a mathematical problem - tools might include pencil and
paper, concrete models, a ruler, a protractor, a calculator, a
spreadsheet, a computer algebra system, a statistical package, or
dynamic geometry software. Sufficiently familiar with tools
appropriate for their grade or course to make sound decisions about
when each of these tools might be helpful, recognizing both the
insight to be gained and their limitations.Detect possible errors
by strategically using estimation and other mathematical knowledge.
Know that technology can enable them to visualize the results of
varying assumptions, explore consequences, and compare predictions
with data. Identify relevant external mathematical resources, such
as digital content located on a website, and use them to pose or
solve problems. Use technological tools to explore and deepen their
understanding of concepts.
For example, mathematically proficient high school students
analyze graphs of functions and solutions generated using a
graphing calculator. 116 - Attend to precision.Try to communicate
precisely to others. Use clear definitions in discussion with
others and in their own reasoning. State the meaning of the symbols
they choose, including using the equal sign consistently and
appropriately. Carefully specify units of measure, and labeling
axes to clarify the correspondence with quantities in a problem.
Calculate accurately and efficiently, express numerical answers
with a degree of precision appropriate for the problem context.
In the elementary grades, students give carefully formulated
explanations to each other. By the time they reach high school they
have learned to examine claims and make explicit use of
definitions.12
7 - Look for and make use of structure.Look closely to discern a
pattern or structure. Young students group figures with same number
of sides. Students see 9 X 8 equals the remembered 5 X 8 + 4 X 8
when learning the distributive propertyOlder students recognize 9
as 7 + 2 and 14 as 7 X 2 in the expression x2+ 9x+ 14, when
learning to factor trinomials Recognize the significance of an
existing line in a geometric figure and can use the strategy of
drawing an auxiliary line for solving problems. Can step back for
an overview and shift perspective. See complicated things, such as
some algebraic expressions, as single objects or as being composed
of several objects. Young students, for example, might notice that
three and seven more is the same amount as seven and three more, or
they may sort a collection of shapes according to how many sides
the shapes have. Later, students will see 7 8 equals the well
remembered 7 5 + 7 3, in preparation for learning about the
distributive property. In the expressionx2+ 9x+ 14, older students
can see the 14 as 2 7 and the 9 as 2 + 7. For example, they can see
5 3(xy)2as 5 minus a positive number times a square and use that to
realize that its value cannot be more than 5 for any real
numbersxandy
148 - Look for and express regularity in repeated
reasoning.Notice if calculations are repeated, and look both for
general methods and for shortcuts. Upper elementary students might
notice when dividing 25 by 11 that they are repeating the same
calculations over and over again, and conclude they have a
repeating decimal. By paying attention to the calculation of slope
as they repeatedly check whether points are on the line through (1,
2) with slope 3, middle school students might abstract the equation
(y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel
when expanding (x 1)(x+ 1), (x 1)(x2+x+ 1), and (x 1)(x3+x2 +x+ 1)
might lead them to the general formula for the sum of a geometric
series. Maintain oversight of the process, while attending to the
details as they work to solve a problem. Continually evaluate the
reasonableness of their intermediate results.A variety of
activities and/or problems are on the tables for you to reviewRead
each problem or activityConsider a possible solution to the
problem(s)Discuss which standard(s) for mathematical practice may
be reinforced as students work through the problem/activityUse a
sticky note to record the number of the standard on the solution
chartWe will discuss overall approaches to developing these skills
in studentsCommon Core ActivityYou own a square pool that is 6
meters by 6 meters. You want to put in a tile surrounding. Each
tile is 1 x 1 meter. How many tiles will you need to surround your
pool?If your friends square pool is x by x, how many tiles would he
need to surround his pool?Another friend has a rectangular pool, x
by y. How many tiles would she need to surround her pool?
Class DiscussionWhat solution did you get for Question #1?
What solution did you get for Question #2?
Is there more than one correct solution?
If yes, are these expressions equivalent?
What solution did you get for Question #3?
Again, is there more than one correct solution?
Lastly, are these expressions equivalent?Warm-up (6 min.)The
expression 82/3 can be written as (8 1/3)2 or as (82)1/3Write these
expressions in radical form. How would you confirm that these forms
are equivalent?Considering that Which form would be easier to
simplify without a calculator? Why?When calculating , Kyle entered
9^3/2 into his calculator. Karen entered it as . They came out with
different results. Which was right and how could the other student
modify their input?
Think-Pair-Shair (5 min.)Your class is given the quadratic
equation ax2+ bx + c = 0, knowing that one solution is 2 + 3i. The
class came up with several different possibilities for the second
solution; (3 + 2i), (2 3i), (-2 3i), and (-2 + 3i). Are either of
these proposed solutions correct? Why, or why not?ExamplesAir is
being blown into a sphere at a rate of 6 cubic inches per minute.
How fast is the radius changing when the radius of the sphere is 2
inches? Create a list of steps that you may use in general for a
related rates problem.Use the steps created above for the problems
below:The edge of a cube is increasing at a rate of 2 inches per
minute. At the instant the edge is 3 inches, how fast is the volume
of the cube changing?The diagonal of a square is increasing at a
rate of 3 inches per minute. When the area of the square is 18
square inches, how fast (in inches per minute) is the perimeter
increasing? (Give exact value)Group WorkThe region bounded by the
curves y = x and y = is rotated about the x axis.- Sketch the
region and the solid formed by the rotation.- Find the volume of
the solid formed.The base of a solid is a region in the first
quadrant bounded by the x-axis, the y-axis, and the line y = 1 x.If
cross sections of the solid perpendicular to the x-axis are
semicircles make a sketch of the cross section in the region in the
first quadrantWhat is the volume of the solid described
above?Partner WorkIn a womans professional tennis tournament, the
money a player wins depends on her finishing place in the
standings. The first-place finisher wins half of $1,500,000 in
total prize money. The second-place finisher wins half of what is
left; then the third-place finisher wins half of that, and so on.a.
Write a rule to calculate the actual prize money in dollars won by
the player finishing in nth place, for any positive integer n.b.
Graph the relationship that exists between the first 10 finishers
and the prize money in dollars. c. What pattern do you notice in
the graph? What type of relationship exists between the two
variables?Group ActivityConsider parallelogram ABCD with
coordinates A(2,-2), B(4,4), C(12,4) and D(10,-2). Perform the
following transformations. Make predictions about how the lengths,
perimeter, area and angle measures will change under each
transformation. a. A reflection over the x-axis.b. A rotation of
270 about the origin.c. A dilation of scale factor 3 about the
origin.d. A translation to the right 5 and down 3. Verify your
predictions. Compare and contrast which transformations preserved
the size and/or shape with those that did not preserve size and/or
shape. Generalize, how could you determine if a transformation
would maintain congruency from the preimage to the
image?Think-Pair-Share (10 min.)Donna is building a swing set. She
wants to countersink the nuts by drilling a hole that will
perfectly circumscribe the hexagonal nuts that she is using. Each
side of the regular hexagonal nut is 8mm.
Construct a diagram that shows what size drill she should use to
countersink the nut.How could you determine the amount of the hole
that is not filled by the nut?Warm-up (6 min.)At the left is a
table that represents the relationship between daily income, I, for
an amusement park and the number of paying visitors in thousands,
n.a. What are the x-intercepts and y-intercepts and explain them in
the context of the problem?b. Identify any maximums or minimums and
explain their meaning in the context of the problem.c. What pattern
of change do these ordered pairs develop? Explain. d. Is the
pattern and/or graph of the data symmetrical? How do you know? e.
Describe the intervals of increase and decrease and explain them in
the context of the problemnI 00152839485560ExampleKierra is a
saleswoman whose pay depends on the number of diamonds she sells.
She earns a base salary plus a commission on each sale. Using the
table below, determine the rate of change in earnings as sales
increase.
What part of Kierras pay does this represent? Determine an
equation that could represent Kierras pay.What is her base salary
for a week when she doesnt sale any diamonds? Number of
Diamonds2468Weekly Earnings60096013201680Group WorkTen bacteria are
placed in a test tube and each one splits in two after one minute.
After 1 minute, the resulting 10 bacteria each split in two,
creating 20 bacteria. This process continues for one hour until
test tube is full.
a. How many bacteria are in the test tube after 5 minutes? 15
minutes? b. Describe how to take any current number of bacteria to
find the number of bacteria at the next minute (this is writing a
NOW - NEXT rule).c. Write an equation that will determine the
number of bacteria after any number of minutes. d. How many
bacteria are in the test tube after one hour?e. For further
research, Dr. Bland removes 5 bacteria after each minute from the
original test tube to start a new cell culture. Write the resulting
equation and describe how this affected your rule in parts b. and
c.?Think-Pair-ShareTwo students are discussing moving between
degrees and radians. Critique and use their arguments in the
questions that follow their discussion.JamesI remember that 360o =
2p radiansIn finding the radian measure of 120o, I divided 120 by
360 to get 1/3.Since my angle was 1/3 of the whole circle and that
there are 2p radians in a circle I think that the angle will be 1/3
of 2p which is 2p/3JimIf 3600 = 2p radians then if I divide both
sides by 360, then 1 degree is equal to p/180. Therefore, 120
degrees times p/180 would be radian measure, which reduces to
2p/3.a. Using James method, find the radian measure of 300.b. Using
Jims method, find the radian measure of 300.c. Which method is your
preferred method? Why?Warm-up (5 min.)You are planning to take on a
part time job as a waiter at a local restaurant. During your
interview, the boss told you that their best waitress, Jenni, made
an average of $70 a night in tips last week. However, when you
asked Jenni about this, she said that she made an average of only
$50 per night last week. She provides you with a copy of her
nightly tip amounts from last week (see below). Calculate the mean
and the median tip amount. DayTip
AmountSunday$50Monday$45Wednesday$48Friday$125Saturday$85a. Which
value is Jennis boss using to describe the average tip? Why do you
think he chose this value?b. Which value is Jenni using? Why do you
think she chose this value?c. Which value best describes the
typical amount of tips per night? Explain why.Partner WorkDescribe
the following subset of outcomes for choosing one card from a
standard deck of cards as the intersection of two events: {queen of
hearts, queen of diamonds}.
Create and prove two events are independent from drawing a card
from a standard deck
Create and prove two events are dependent from drawing a card
from a standard deck
Example of an Extended Response Item1. Make a sketch of the
pieces you will need to cut out for the body of the bag. Your
sketch does not have to be to scale. On your sketch, show all the
measurements you will need.2. You are going to make one of these
bags from a roll of cloth 1 meter wide. What is the shortest length
that you need to cut from the roll for the bag? Describe, using
words and sketches, how you arrive at your answer.You have been
asked to design a sports bag.The length of the bag will be 60
cm.The bag will have circular ends of diameter 25 cm.The main body
of the bag will be made from 3 pieces of material; a piece for the
curved body, and the two circular end pieces.Each piece will need
to have an extra 2 cm all around it for a seam, so that the pieces
may be stitched together.
Another Example of an Extended Response ItemA square metal sheet
(6 feet x 6 feet) is to be made into an open-topped water tank by
cutting squares from the four corners of the sheet, and bending the
four remaining rectangular pieces up, to form the sides of the
tank. These edges will then be welded together.A. How will the
final volume of the tank depend upon the size of the squares cut
from the corners?Describe your answer by:i) Sketching a rough
graphii) Explaining the shape of your graph in wordsiii) Writing an
algebraic formula for the volumeB. How large should the four
corners be cut, so that the resulting volume of the tank is as
large as possible?
Resourceshttp://www.dpi.state.nc.us/acre/standards/common-core/http://maccss.ncdpi.wikispaces.nethttp://www.corestandards.org/Math/Practice
