STANDARD 8 - NUMERICAL OPERATIONS
K-12 Overview
All students will understand, select, and apply various methods of
performing numerical operations.
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Descriptive Statement
Numerical operations are an essential part of the mathematics
curriculum. Students must be able to select and apply various
computational methods, including mental math, estimation,
paper-and-pencil techniques, and the use of calculators. Students
must understand how to add, subtract, multiply, and divide whole
numbers, fractions, and other kinds of numbers. With calculators that
perform these operations quickly and accurately, however, the
instructional emphasis now should be on understanding the meanings and
uses of the operations, and on estimation and mental skills, rather
than solely on developing paper-and-pencil skills.
Meaning and Importance
The wide availability of computing and calculating technology has
given us the opportunity to significantly reconceive the role of
computation and numerical operations in our school mathematics
programs. Up until this point in our history, the mathematics program
has called for the expenditure of tremendous amounts of time in
helping children to develop proficiency with paper-and-pencil
computational procedures. Most people defined proficiency as a
combination of speed and accuracy with the standard algorithms. Now,
however, adults who need to perform calculations quickly and
accurately have electronic tools that are more accurate and more
efficient than any human being. It is time to re-examine the reasons
to teach paper-and-pencil computational algorithms to children and to
revise the curriculum in light of that re-examination. Mental
mathematics, however, should continue to be stressed; students should
be able to carry out simple computations without resort to either
paper-and-pencil or calculators. Fourth-graders must know the basic
facts of the multiplication table, and seventh-graders must be able to
evaluate in their heads simple fractions, such as
What's two-thirds of 5 tablespoons?
K-12 Development and Emphases
At the same time that technology has made the traditional focus on
paper-and-pencil skills less important, it has also presented us with
a situation where some numerical operations, skills, and concepts are
much more important than they have ever been. Estimation
skills, for example, are critically important if one is to be a
competent user of calculating technology. People must know the range
in which the answer to a given problem should lie before doing any
calculation, they must be able to assess the reasonableness of the
results of a string of computations, and they should be able to be
satisfied with the results of an estimation when an exact answer is
unnecessary. They should also be able to work quickly and easily with
changes in order of magnitude, using powers of ten and their
multiples. Mental mathematics skills also play a more
importantrole in a highly technological world. Simple two-digit
computations or operations that involve powers of ten should be
performed mentally by a mathematically literate adult. Students
should have enough confidence in their ability with such computations
to do them mentally rather than using either a calculator or paper and
pencil. Most importantly, a student's knowledge of the
meanings and uses of the various arithmetic operations is
essential. Even with the best of computing devices, it is still the
human who must decide which operations need to be performed and in
what order to answer the question at hand. The construction of
solutions to life's everyday problems, and to society's
larger ones, will require students to be thoroughly familiar with when
and how the mathematical operations are used.
The major shift in this area of the curriculum, then, is one away
from drill and practice of paper-and-pencil procedures and toward
real-world applications of operations, wise choices of appropriate
computational strategies, and integration of the numerical operations
with other components of the mathematics curriculum. So what is
the role of paper-and-pencil computation in a mathematics program
for the year 2000? Should children be able to perform any
calculations by hand? Are those procedures worth any time in
the school day? Of course they should and of course they are.
Most simple paper-and-pencil procedures should still be
taught and one-digit basic facts should still be committed to
memory. We want students to be proficient with two- and three-digit
addition and subtraction and with multiplication and division
involving two-digit factors or divisors, but there should be changes
both in the way we teach those processes and in where we go from
there. The focus on the learning of those procedures should be on
understanding the procedures themselves and on the development of
accuracy. There is no longer any need to concentrate on the
development of speed. To serve the needs of understanding and
accuracy, non-traditional paper-and-pencil algorithms, or algorithms
devised by the children themselves, may well be better choices than
the standard algorithms. The extensive use of drill in multi-digit
operations, necessary in the past to enable people to perform
calculations rapidly and automatically, is no longer necessary and
should play a much smaller role in today's curriculum.
For procedures involving larger numbers, or numbers with a greater
number of digits, the intent ought to be to bring students to the
point where they understand a paper-and-pencil procedure well enough
to be able to extend it to as many places as needed, but certainly not
to develop an old-fashioned kind of proficiency with such problems.
In almost every instance where the student is confronted with such
numbers in school, technology should be available to aid in the
computation, and students should understand how to use it effectively.
Calculators are the tools that real people in the real world use when
they have to deal with similar situations and they should not be
withheld from students in an effort to further an unreasonable and
antiquated educational goal.
In summary, numerical operations continue to be a critical
piece of the school mathematics curriculum and, indeed, a very
important part of mathematics. But, there is perhaps a greater need
for us to rethink our approach here than to do so for any other
component. An enlightened mathematics program for today's
children will empower them to use all of today's tools rather
than require them to meet yesterday's expectations.
note: Although each content standard is discussed
in a separate chapter, it is not the intention that each be
treated separately in the classroom. Indeed, as noted the
Introduction to this Framework, an effective curriculum is one
that successfully integrates these areas to present students with rich
and meaningful cross-strand experiences.
Standard 8 - Numerical Operations - Grades K-2
Overview
The wide availability of computing and calculating technology has
given us the opportunity to significantly reconceive the role of
computation and numerical operations in our elementary mathematics
programs, but, in kindergarten through second grade, the effects will
not be as evident as they will be in all of the other grade ranges.
This is because the numerical operations content taught in these
grades is so basic, so fundamental, and so critical to further
progress in mathematics that much of it will remain the same. The
approach to teaching that content, however, must still be changed to
help achieve the goals expressed in the New Jersey
Mathematics Standards.
Learning the meanings of addition and subtraction, gaining
facility with basic facts, and mastering some computational
procedures for multi-digit addition and subtraction are still the
topics on which most of the instructional time in this area will be
spent. There will be an increased conceptual and developmental focus
to these aspects of the curriculum, though, away from a traditional
drill-and-practice approach, as described in the K-12 Overview;
nevertheless, students will be expected to be able to respond quickly
and easily when asked to recall basic facts.
By the time they enter school, most young children can use counters
to act out a mathematical story problem involving addition or
subtraction and find a solution which makes sense. Their experiences
in school need to build upon that ability and deepen the
children's understanding of the meanings of the
operations. School experiences also need to strengthen the
children's sense that modeling such situations as a way to
understand them is the right thing to do. It is important that they
be exposed to a variety of different situations involving addition and
subtraction. Researchers have separated problems into categories
based on the kind of relationships involved (Van de Walle, 1990,
pp. 75-6); students should be familiar with problems in all of the
following categories:
Join problems
- Mary has 8 cookies. Joe gives her 2 more. How
many cookies does Mary have in all?
- Mary has some cookies. Joe gives her 2 more. Now
she has 8. How many cookies did Mary have to begin
with? (Missing addend)
- Mary has 8 cookies. Joe gives her some more. Now
Mary has 10. How many cookies did Joe give Mary?
(Missing addend)
Separate problems
- Mary has 8 cookies. She eats 2. How many are
left? (Take away)
- Mary has some cookies. She eats 2. She has 6
left. How many cookies did Mary have to begin
with?
- Mary has 8 cookies. She eats some. She has 6
left. How many cookies did Mary eat? (Missing
addend)
Part-part-whole problems
- Mary has 2 nickels and 3 pennies. How many
coins does she have?
- Mary has 8 coins. Three are pennies, the rest
nickels. How many nickels does Mary have?
Compare problems
- Mary has 6 books. Joe has 4. How many more
books does Mary have than Joe?
- Mary has 2 more books than Joe. Mary has 6 books.
How many books does Joe have?
- Joe has 2 fewer books than Mary. He has 4 books.
How many books does Mary have?
Basic facts in addition and subtraction continue to be very
important. Students should be able to quickly and easily recall
one-digit sums and differences. The most effective way to accomplish
this has been shown to be the focused and explicit use of basic fact
strategies-conceptual techniques that make use of the
child's understanding of number parts and relationships to help
recover the appropriate sum or difference. By the end of second
grade, students should not only be able to use counting on,
counting back, make ten, and doubles and near
doubles strategies, but also explain why these strategies work by
modeling them with counters. Building on their facility with learning
doubles like 7 + 7 = 14, children recast 7 + 8 as 7 +
7 + 1, which they then recognize as 15 (near doubles).
Make ten involves realizing that in adding 8 + 5, you
need two to make ten, and recasting the sum as 8 + 2 + 3 which
is 10 + 3 or 13. Counting on involves starting
with the large number and counting on the smaller number so that
adding 9 + 3, for example, would involve counting on 10,
11, and then 12. Counting back is used for
subtraction, so that finding 12 - 4, the child
might count 11, 10, 9, and then 8.
Students must still be able to perform multi-digit addition and
subtraction with paper and pencil, but the widespread availability
of calculators has made the particular procedure used to perform the
calculations less important. It need no longer be the single fastest,
most efficient algorithm chosen without respect to the degree to which
children understand it. Rather, the teaching of multi-digit
computation should take on more of a problem solving approach, a more
conceptual, developmental approach. Students should first use the
models of multi-digit number that they are most comfortable with (base
ten blocks, popsicle sticks, bean sticks) to explore the new class of
problems. Students who have never formally done two-digit addition
might be asked to use their materials to help figure out how many
second graders there are in all in the two second grade classes in the
school. Other similar real-world problems should follow, some
involving regrouping and others not. After initial exploration,
students share with each other all of the strategies they've
developed, the best ways they've found for working with the tens
and ones in the problems, and their own approaches (and names!) for
regrouping. Most students can, with direction, take the results of
those discussions and create their own paper-and-pencil procedures for
addition and subtraction. The discussions can, of course, include the
traditional approaches, but these ought not to be seen as the only
right way to do these operations.
Kindergarten through second grade teachers are also responsible for
setting up an atmosphere where estimation and mental
math are seen as reasonable ways to do mathematics. Of course
students at these grade levels do almost exclusively mental math until
they reach multi-digit operations, but estimation should also comprise
a good part of the activity. Students regularly involved in
real-world problem solving should begin to develop a sense of when
estimation is appropriate and when an exact answer is necessary.
Technology should also be an important part of the
environment in primary classrooms. Calculators provide a valuable
teaching tool when used to do student-programmed skip counting, to
offer estimation and mental math practice with target games,
and to explore operations and number types that the students have not
formally encountered yet. They should also be used routinely to
perform computation in problem solving situations that the students
may not be able to perform otherwise. This use prevents the need to
artificially contrive the numbers in real-world problems so that their
answers are numbers with which the students are already
comfortable.
The topics that should comprise the numerical operations focus of
the kindergarten through second grade mathematics program are:
- addition and subtraction basic facts
- multi-digit addition and subtraction
Standard 8 - Numerical Operations - Grades K-2
Indicators and Activities
The cumulative progress indicators for grade 4 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in kindergarten
and grades 1 and 2.
Experiences will be such that all students in grades K-2:
1. Develop meaning for the four basic arithmetic
operations by modeling and discussing a variety of
problems.
- Students use unifix cube towers of two colors to show all
the ways to make "7" (for example: 3+4,
2+5, 0+7, and so on). This activity focuses more on developing a
sense of "sevenness" than on addition
concepts, but a good sense of each individual number makes the
standard operations much easier to understand.
- Kindergartners and first graders use workmats depicting
various settings in which activity takes place to make up and act out
story problems. On a mat showing a vacant playground, for instance,
students place counters to show 3 kids on the swings and 2 more in the
sandbox. How many kids are there in all? How many more are on the
swings than in the sandbox? What are all of the possibilities
for how many are boys and how many are girls?
- Students work through the Sharing a Snack lesson
that is described in the Introduction to this Framework. It
challenges students to find a way to share a large number of cookies
fairly among the members of the class, promoting discussion of early
division, fraction, and probability ideas.
- Students learn about addition as they read Too
Many Eggs by M. Christina Butler. They place eggs in different
bowls as they read and then make up addition number sentences to find
out how many eggs were used in all.
- Kindergarteners count animals and learn about
addition as they read Adding Animals by Colin Hawkins. This
book uses addends from one through four and shows the number sentences
that go along with the story.
- Students are introduced to the take-away meaning
for subtraction by reading Take Away Monsters by Colin
Hawkins. Students see the partial number sentence (e.g., 5
- 1 = ), count to find the answer, and then pull the
tab to see the result.
- Students explore subtraction involving missing
addend situations as they read The Great Take-Away by
Louise Mathews. This book tells the story of one lazy hog who decides
to make easy money by robbing the other pigs in town. The answers to
five subtraction mysteries are revealed when the thief is
captured.
- Students make booklets containing original word
problems that illustrate different addition or subtraction situations.
These may be included in a portfolio or evaluated
independently.
2. Develop proficiency with and memorize basic
number facts using a variety of fact strategies (such as
"counting on" and
"doubles").
- Students play one more than dominoes by changing the
regular rules so that a domino can be placed next to another only if
it has dots showing one more than the other. Dominoes of any
number can be played next to others that show 6 (or 9 in
a set of double nines). One less than dominoes is also
popular.
- Students work through the Elevens Alive lesson that
is described in the Introduction to this Framework. It asks
them to consider the parts of eleven and the natural, random,
occurrence of different pairs of addends when tossing eleven
two-colored counters.
- Second graders regularly use the doubles and near
doubles, the make ten, and the counting on
and counting back strategies for addition and subtraction.
Practice sets of problems are structured so that use of all of these
strategies is encouraged and the students are regularly asked to
explain the procedures they are using.
- Students play games like addition war to practice
their basic facts. Each of two children has half of a deck of playing
cards with the face cards removed. They each turn up a card and the
person who wins the trick is the first to say the sum (or difference)
of the two numbers showing. Calculators may be used to check answers,
if necessary.
- Students use the calculator to count one more
than by pressing + 1= = =. The display will increase by
one every time the student presses the = key. Any number can replace
the 1 key.
- Students use two dice to play board games
(Chutes and Ladders or home-made games). These situations
encourage rapid recall of addition facts in a natural way. In order
to extend practice to larger numbers, students may use 10-sided
dice.
- Students use computer games such as Math
Blaster Plus or Math Rabbit to practice basic
facts.
3. Construct, use, and explain procedures for
performing whole number calculations in the various
methods of computation.
- Second graders use popsicle sticks bundled as tens and
ones to try to find a solution to the first two-digit addition problem
they have formally seen: Our class has 27 children and
Mrs. Johnson's class has 26. How many cupcakes will we
need for our joint party? Solution strategies are shared and
discussed with diversity and originality praised. Other problems,
some requiring regrouping and others not, are similarly solved using
the student-developed strategies.
- Students use calculators to help with the computation
involved in a first-grade class project: to see how many books are
read by the students in the class in one month. Every Monday morning,
student reports contribute to a weekly total which is then added to
the monthly total.
- Students look forward to the hundredth day of school, on
which there will be a big celebration. On each day preceding it, the
students use a variety of procedures to determine how many days are
left before day 100.
- As part of their assessment, students explain how
to find the answer to an addition or subtraction problem (such as
18 + 17) using pictures and words.
- Students find the answer to an addition or
subtraction problem in as many different ways as they can. For
example, they might solve 28 + 35 in the following
ways:
- 8 + 5 = 13 and 20 + 30 = 50, so 13 + 50 = 63
- 28 + 30 = 58. Two more is 60, and 3 more is 63
- 25 + 35 = 60 and 3 more is 63.
- Students use estimation to find out whether a
package of 40 balloons is enough for everyone in the class of 26 to
have two balloons. They discuss the strategies they use to solve this
problem and decide if they should buy more packages.
4. Use models to explore operations with
fractions and decimals.
- Kindergartners explore part/whole relations with pattern
blocks by seeing which shapes can be created using other blocks. You
might ask: Can you make a shape that is the same as the
yellow hexagon with 2 blocks of some other color? with 3 blocks of
some other color? with 6 blocks of some other color? and
so on.
- Students use paper folding to begin to identify and name
common fractions. You might ask: If you fold this rectangular
piece of paper in half and then again and then again, how many
equal parts are there when you open it up? Similarly folded
papers, each representing a different unit fraction, allow for early
comparison activities.
- Second graders use fraction circles to model
situations involving fractions of a pizza. For example: A pizza
is divided into six pieces. Mary eats two pieces. What
fraction of the pizza did Mary eat? What fraction is
left?
- Students use manipulatives such as pattern blocks
or Cuisenaire rods to model fractions. For example: If the red rod
is one whole, then what number is represented by the yellow
rod?
5. Use a variety of mental computation and
estimation techniques.
- Students regularly practice a variety of oral counting
skills, both forward and backward, by various steps. For instance,
you might instruct your students to: Count by ones -
start at 1, at 6, at 12, from 16 to 23; Count by tens
- start at 10, at 30, at 110, at 43, at 67, from 54
to 84, and so on.
- Students estimate sums and differences both before doing
either paper-and-pencil computation or calculator computation and
after so doing to confirm the reasonableness of their answers.
- Students are given a set of index cards on each of which
is printed a two-digit addition pair (23+45, 54+76, 12+87, and
so on). As quickly as they can they sort the set into three piles:
more than 100, less than 100, and equal to 100.
- Students play "Target 50" with
their calculator. One student enters a two-digit number and the other
must add a number that will get as close as possible to
50.
6. Select and use appropriate computational
methods from mental math, estimation, paper-and-pencil, and
calculator methods, and check the reasonableness of results.
- The daily calendar routine provides the students
with many opportunities for computation. Questions like these arise
almost every day: There are 27 children in our class.
Twenty-four are here today. How many are absent? Fourteen are buying
lunch; how many brought their lunch? or It's now 9:12.
How long until we go to gym at 10:30? The students are
encouraged to choose a computation method with which they feel
comfortable; they are frequently asked why they chose their method and
whether it was important to get an exact answer. Different solutions
are acknowledged and praised.
- Students regularly have human vs. calculator races.
Given a list of addition and subtraction basic facts, one student uses
mental math strategies and another uses a calculator. They quickly
come to realize that the human has the advantage.
- Students regularly answer multiple choice questions like
these with their best guesses of the most reasonable answer: A
regular school bus can hold: 20 people, 60 people, 120 people?
The classroom is: 5 feet high, 7 feet high, 10 feet high?
- As part of an assessment, students tell how they
would solve a particular problem and why. They might circle a picture
of a calculator, a head (for mental math), or paper-and-pencil for
each problem.
7. Understand and use relationships among
operations and properties of operations.
- Students explore three-addend problems like 4 + 5 + 6
=. First they check to see if adding the numbers in different
orders produces different results and, later, they look for pairs of
compatible addends (like 4 and 6) to make the addition
easier.
- Students make up humorous stories about adding and
subtracting zero. I had 27 cookies. My mean brother took
away zero. How many did I have left?
- Second graders, exploring multiplication arrays, make a
4 x 5 array of counters on a piece of construction
paper and label it: 4 rows, 5 in each row = 20. Then they
rotate the array 90 degrees and label the new array, 5
rows, 4 in each row = 20. Discussions follow which lead to
intuitive understandings of commutativity.
References
-
Butler, M. Christina. Too Many Eggs. Boston: David
R. Godine Publisher, 1988.
Hawkins, Colin. Adding Animals. New York:
G. P. Putnam's Sons, 1984.
Hawkins, Colin. Take Away Monsters. New York:
G. P. Putnam's Sons, l984.
Mathews, Louise. The Great Take-Away. New York: Dodd,
Mead, & Co., 1980.
Van de Walle, J. A. Elementary School Mathematics: Teaching
Developmentally. New York: Longman, 1990.
Software
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Math Blaster Plus. Davidson.
Math Rabbit. The Learning Company.
On-Line Resources
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http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/
The Framework will be available at this site during
Spring 1997. In time, we hope to post additional resources
relating to this standard, such as grade-specific activities submitted
by New Jersey teachers, and to provide a forum to discuss the
Mathematics Standards.
Standard 8 - Numerical Operations - Grades 3-4
Overview
The widespread availability of computing and calculating technology
has given us the opportunity to reconceive the role of computation and
numerical operations in our third and fourth grade mathematics
programs. Traditionally, tremendous amounts of time were spent at
these levels helping children to develop proficiency and accuracy with
paper-and-pencil procedures. Now, adults needing to perform
calculations quickly and accurately have electronic tools that are
both more accurate and more efficient than those procedures. At the
same time, though, the new technology has presented us with a
situation where some numerical operations, skills, and concepts are
much more important than they used to be. As described in the K-12
Overview, estimation, mental computation, and understanding
the meanings of the standard arithmetic operations all play
a more significant role than ever in the everyday life of a
mathematically literate adult.
The major shift in the curriculum that will take place in this
realm, therefore, is one away from drill and practice of
paper-and-pencil procedures with symbols and toward real-world
applications of operations, wise choices of appropriate
computational strategies, and integration of the numerical
operations with other components of the mathematics curriculum.
Third and fourth graders are primarily concerned with cementing
their understanding of addition and subtraction and developing new
meanings for multiplication and division. They should be in an
environment where they can do so by modeling and otherwise
representing a variety of real-world situations in which these
operations are appropriately used. It is important that the variety
of situations to which they are exposed include all the different
scenarios in which multiplication and division are used. There are
several slightly different taxonomies of these types of problems, but
minimally students at this level should be exposed to repeated
addition and subtraction, array, area, and expansion
problems. Students need to recognize and model each of these problem
types for both multiplication and division.
Basic facts in multiplication and division continue to be
very important. Students should be able to quickly and easily recall
quotients and products of one-digit numbers. The most effective
approach to enabling them to acquire this ability has been shown to be
the focused and explicit use of basic fact strategies-conceptual
techniques that make use of the child's understanding of the
operations and number relationships to help recover the appropriate
product or quotient. Doubles and near doubles are
useful strategies, as are discussions and understandings regarding the
regularity in the nines multiplication facts, the roles
of one and zero in these operations, and the roles of
commutativity and distributivity.
Students must still be able to perform two-digit multiplication
and division with paper and pencil, but the widespread
availability of calculators has made the particular procedure used to
perform the calculations less important. It need no longer be the
single fastest, most efficient algorithm chosen without respect to the
degree to which children understand it. Rather, the teaching of
two-digit computation should take on more of a problem solving
approach, a more conceptual, developmental approach. Students should
first use the models of multi-digit numbers that they are most
comfortable with (base ten blocks, money) to explore this new class of
problems. Students who have never formally done two-digit
multiplication might be asked to use their materials to help figure
out how many pencils are packed in the case just received in the
school office. There are 24 boxes with a dozen pencils in each box.
Are there enough for everystudent in the school to have
one? Other, similar, real-world problems would follow, some
involving regrouping and others not.
After initial exploration, students share with each other all of
the strategies they've developed, the best ways they've
found for working with the tens and ones in the problem, and their own
approaches to dealing with the place value issues involved. Most
students can, with direction, take the results of those discussions
and create their own paper-and-pencil procedures for multiplication
and division. The discussions can, of course, include the traditional
approaches, but these ought not to be seen as the only right
way to perform these operations.
Estimation and mental math become critically
important in these grade levels as students are inclined to use
calculators for more and more of their work. In order to use that
technology effectively, third and fourth graders must be able to use
estimation to know the range in which the answer to a given problem
should lie before doing any calculation. They also must be able to
assess the reasonableness of the results of a computation and be
satisfied with the results of an estimation when an exact answer is
unnecessary. Mental mathematics skills, too, play a more important
role in third and fourth grade. Simple two-digit addition and
subtraction problems and those involving powers of ten should be
performed mentally. Students should have enough confidence in their
ability with these types of computations to do them mentally instead
of relying on either a calculator or paper and pencil.
Technology should be an important part of the environment in
third and fourth grade classrooms. Calculators provide a valuable
teaching tool when used to do student-programmed repeated addition or
subtraction, to offer estimation and mental math practice with
target games, and to explore operations and number types that
the students have not yet formally encountered. Students should also
use calculators routinely to find answers to problems that they might
not be able to find otherwise. This use prevents the need to
artificially contrive real-world problems so that their answers are
numbers with which the students are already comfortable.
The topics that should comprise the numerical operations focus of
the third and fourth grade mathematics program are:
- multiplication and division basic facts
- multi-digit whole number addition and subtraction
- two-digit whole number multiplication and division
- decimal addition and subtraction
- explorations with fraction operations
Standard 8 - Numerical Operation - Grades 3-4
Indicators and Activities
The cumulative progress indicators for grade 4 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in grades 3 and
4.
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 3-4 will be such that all students:
1. Develop meaning for the four basic arithmetic
operations by modeling and discussing a variety of
problems.
- Students broaden their initial understanding of
multiplication as repeated addition by dealing with situations
involving arrays, expansions, and combinations. Questions of these
types are not easily explained through repeated addition: How many
stamps are on this 7 by 8 sheet? How big would this painting
be if it was 3 times as big? How many outfits can you make
with 2 pairs of pants and 3 shirts?
- Students use counters to model both repeated subtraction
(There are 12 cookies. How many bags of 3?) and
sharing (There are 12 cookies and 3 friends. How many
cookies each?) meanings for division and write about the
difference in their journals.
- Students work through the Sharing Cookies
lesson that is described in the First Four Standards of this
Framework. They investigate division by using 8 cookies to be
shared equally among 5 people, and discuss the problem of simplifying
the number sentence which describes the amount of each person's
share.
- From the beginning of their work with division, children
are asked to make sense out of remainders in problem situations. The
answers to these three problems are different even though the division
is the same: How many cars will we need to transport 19 people if
each car holds 5? How many more packages of 5 ping-pong balls
can be made if there are 19 balls left in the bin? How much
does each of 5 children have to contribute to the cost of a $19
gift?
- Students explore division by reading The
Doorbell Rang by Pat Hutchins. In this story, Victoria and Sam
must share 12 cookies with increasing numbers of friends. Students
can use counters to show how many cookies each person
gets.
- Students learn about multiplication as an array
by reading One Hundred Hungry Ants by Elinor Pinczes, Lucy
and Tom's 1, 2, 3 by Shirley Hughes or Number
Families by Jane Srivastava.
- Students make books showing things that come in
3's, 4's, 5's, 6's, or 12's.
2. Develop proficiency with and memorize
basic number facts using a variety of fact strategies
(such as "counting on" and
"doubles").
- Students use streets and alleys as both a mental
model of multiplication and a useful way to recover facts when needed.
It simply involves drawing a series of horizontal lines(streets) to
represent one factor and a series of vertical lines (alleys) crossing
them to represent the other. The number of intersections of the
streets and alleys is the product!
- Students use a double maker on a calculator for
practice with doubles. They enter x 2 = on the calculator.
Any number pressed then, followed by the equal sign, will show the
number's double. Students work together to try to say the double
for each number before the calculator shows it.
- Students regularly use doubles, near doubles, and
use a related fact strategies for multiplication; they are
using the near doubles strategy when they calculate a sum like
15 + 17 by recognizing that it is 2 more than double
15. More generally, they are using the use a related
fact strategy when they use any fact they happen to remember, like
8 + 4 = 12, to make a related calculation like 8 + 5
=12 + 1 = 13. They also recover facts by knowledge of the role of
zero and one in multiplication, of commutativity, and of the regular
patterned behavior of multiples of nines. Practice sets of problems
are structured so that use of all these strategies is encouraged and
the students are regularly asked to explain the procedures they are
using.
- Pairs of students play Circles and Stars
(Burns, 1991). Each student rolls a die and draws as many circles as
the number shown, then rolls again and puts that number of stars in
every circle, and then writes a multiplication number sentence and
records how many stars there are all together. Each student takes
seven turns, and adds the total. The winner is the student with the
most stars.
- Students use color tiles to show how a given
number of candies can be arranged in a rectangular box.
- Students play multiplication war, using a
deck of cards with kings and queens removed. All of the cards are
dealt out. Each player turns up two cards and multiplies their values
(Jacks count as 0; aces count as 1). The
"general" draws a target number from a hat. The player
closest to the target wins a point. The first player to get 10
points wins the game.
- Students use computer programs such as Math
Workshop to practice multiplication facts.
3. Construct, use, and explain procedures for
performing whole number calculations in the various methods of
computation.
- Students work through the Product and Process
lesson that is described in the Introduction to this Framework.
It challenges students to use calculators and four of the five digits
1, 3, 5, 7, and 9 to discover the multiplication
problem that gives the largest product.
-
Students explore lattice multiplication and try to figure out
how it works. For example, the figure at the right shows 14
× 23 = 322.
- Students use the skills they've developed with arrow
puzzles (See Standard 6-Number Sense-Grades
3-4-Indicator 3) to practice mental addition and subtraction of
2- and 3-digit numbers. To add 23 to 65, for instance,
they start at 65 on their "mental hundred number
chart," go down twice and to the right three times.
- Students use base ten blocks to help them decide
how many blocks there would be in eachgroup if they divided 123 blocks
among 3 people. The students describe how they used the blocks to
help them solve the problem and compare their solutions and solution
strategies.
4. Use models to explore operations with fractions and
decimals.
- Students use fraction circle pieces (each unit
fraction a different color) to begin to explore addition of fractions.
Questions like: Which of these sums are greater than 1? and
How do you know? are frequent.
- Students use the base ten models that they are most
familiar with for whole numbers and relabel the components with
decimal values. Base ten blocks represent 1 whole, 1 tenth, 1
hundredth, and 1 thousandth. Coins, which had represented a whole
number of cents, now represent hundredths of dollars.
- Students operate a school store with school supplies
available for sale. Other students, using play money, decide on
purchases, pay for them, receive and check on the amount of
change.
- In groups, students each roll a number cube and
use dimes to represent the decimal rolled. For example, a student
rolling a 4 would take 4 dimes to represent 4 tenths of a
dollar. When a student gets 10 dimes, he turns them in for a dollar.
The first student to get $5 wins the game.
- Students use money to represent fractions. For
example, a quarter and a quarter equals half a dollar.
- Students demonstrate equivalent fractions using
pattern blocks. For example, if a yellow hexagon is one whole, then
three green triangles (3/6) is the same size as one red
trapezoid (1/2). Pattern blocks may also be used to represent
addition and subtraction of fractions.
5. Use a variety of mental computation and estimation
techniques.
- Students frequently do warm-up drills that enhance their
mental math skills. Problems like: 3,000 x 7 = , 200
x 6 = , and 5,000 x 5 + 5 = are put on the
board as individual children write the answers without doing any
paper-and-pencil computation.
- Students make appropriate choices from among
front-end, rounding, and compatible numbers
strategies in their estimation work depending on the real-world
situation and the numbers involved. Front end strategies
involve using the first digits of the largest numbers to get an
estimate, which of course is too low, and then adjusting
up. Compatible numbers involves finding some numbers which can
be combined mentally, so that, for example, 762 + 2,444 + 248
is about (750 + 250) + 2,500, or 3,500.
- Students use money and shopping situations to practice
estimation and mental math skills. Is $20.00 enough to buy items
priced at $12.97, $4.95, and 3.95? About how much would 4 cans
of beans cost if each costs $0.79?
- Students explore estimation involving division as
they read The Greatest Guessing Game: A Book about Dividing
by Robert Froman. A little girl and her three friends solve a
variety of problems, estimating first and discussing what to do with
remainders.
6. Select and use appropriate computational methods from
mental math, estimation, paper-and-pencil, and calculator
methods, and check the reasonableness of results.
- Students play addition max out. Each
student has a 2 x 3 array of blanks (in standard 3-digit addition
form) into each of which will be written a digit. One student rolls a
die and everyone must write the number showing into one of their
blanks. Once the number is written in, it can not be changed.
Another roll - another number written, and so on. The object is
to be the player with the largest sum when all six digits have been
written. If a player has the largest possible sum that can be made
from the six digits rolled, there is a bonus for maxing
out.
- Students discuss this problem from the NCTM Standards
(p. 45): Three fourth grade teachers decided to take their
classes on a picnic. Mr. Clark spent $26.94 for refreshments.
He used his calculator to see how much the other two teachers should
pay him so that all three could share the cost equally. He
figured they each owed him $13.47. Is his answer reasonable?
As a follow-up individual assessment, they write about how they
might find an answer.
7. Understand and use relationships among
operations and properties of operations.
- Students take 7x8 block rectangular grids printed
on pieces of paper. They each cut along any one of the 7
block-long segments to produce two new rectangles, for example, a
7x6 and a 7x2 rectangle. They then discuss all of the
different rectangle pairs they produced and how they are all related
to the original one.
- Students write a letter to a second grader explaining why
2+5 equals 5+2 to demonstrate their understanding of
commutativity.
- Students explore modular, or clock, addition as an
operation that behaves differently from the addition they know how to
do. For example: 6 hours after 10 o'clock in the
morning is 4 o'clock in the afternoon, so 10 + 6 =
4 on a 12-hour clock. How is clock addition different
from regular addition? How is it the same? How would modular
subtraction and multiplication work?
References
-
Burns, Marilyn. Math By All Means: Multiplication, Grade 3.
New Rochelle, New York: Cuisenaire, 1991.
Froman, Robert. The Greatest Guessing Game: A Book About
Dividing. New York: Thomas Y. Crowell Publishers, 1978.
Hughes, Shirley. Lucy and Tom's 1, 2, 3. New
York: Viking Kestrel, 1987.
Hutchins, Pat. The Doorbell Rang. New York: Greenwillow
Books, 1986.
National Council of Teachers of Mathematics. Curriculum and
Evaluation Standards for School
Mathematics. Reston, VA, 1989.
Pinczes, Elinor J. One Hundred Hungry Ants. Boston:
Houghton Mifflin Company, 1993.
Srivastava, Jane. Number Families. New York: Thomas
Y. Crowell, 1979.
Software
-
Math Workshop. Broderbund.
On-Line Resources
-
http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/
The Framework will be available at this site during
Spring 1997. In time, we hope to post additional resources
relating to this standard, such as grade-specific activities submitted
by New Jersey teachers, and to provide a forum to discuss the
Mathematics Standards.
Standard 8 - Numerical Operations - Grades 5-6
Overview
As indicated in the K-12 Overview, the widespread availability of
computing and calculating technology has given us the opportunity to
significantly reconceive the role of computation and numerical
operations in our fifth and sixth grade mathematics programs. Some
skills are less important while others, such as estimation, mental
computation, and understanding the meanings of the standard
arithmetic operations, all play a more significant role
than ever in the everyday life of a mathematically literate adult.
The major shift in the curriculum that will take place in grades 5
and 6, therefore, is one away from drill and practice of
paper-and-pencil symbolic procedures and toward real-world
applications of operations, wise choices of appropriate
computational strategies, and integration of the numerical
operations with other components of the mathematics curriculum. At
these grade levels, students are consolidating their understanding of
whole number operations (especially multiplication and division) and
beginning to develop computational skills with fractions and decimals.
A sample unit on fractions for the sixth-grade level can be found in
Chapter 17 of this Framework.
Much research in the past decade has focused on students'
understandings of operations with large whole numbers and work with
fractions and decimals. Each of these areas requires students to
restructure their simple conceptions of number that were adequate for
understanding whole number addition and subtraction.
Multiplication requires students to think about different meanings
for the two factors. The first factor in a multiplication problem is
a "multiplier." It tells how many groups one has of a size
specified by the second factor. Thus, students need different
understandings of the roles of the two numbers in the operation of
multiplication than their earlier understandings of addition, in which
both addends meant the same thing.
A similar restructuring is necessary for dealing appropriately with
operations involving fractions and decimals. This restructuring
revolves around the role of the "unit" in these numbers. In
earlier grades, students thought about 5 or 498 as
numbers that represented that many things. The understood
unit, one, is the number which was used to count a group of objects.
With fractions and decimals, though, the unit, still one and still
understood, is a harder concept to deal with because its essential use
is to help define the fraction or decimal rather than as a counter.
When we speak of 5 poker chips or 35 students, our
message is reasonably clear to elementary students. But when we speak
of 2/3 of the class or 0.45 of the price of the
sweater, the meaning is significantly less clear and we must be
much more explicit about the role being played by the unit.
The topics that should comprise the numerical operations focus of
the fifth and sixth grade mathematics program are:
- multi-digit whole number multiplication and division
- decimal multiplication and division
- fraction operations
- integer operations
Standard 8 - Numerical Operations - Grades 5-6
Indicators and Activities
The cumulative progress indicators for grade 8 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in grades 5 and
6.
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 5-6 will be such that all students:
6*. Select and use appropriate
computational methods from mental math, estimation,
paper-and-pencil, and calculator methods, and check the
reasonableness of results.
- Fifth and sixth grade students have calculators available
to them at all times, but frequently engage in competitions to see
whether it is faster to do a given set of computations with the
calculators or with the mental math techniques they've learned.
- Fifth graders make rectangular arrays with base-ten blocks
to try to figure out how to predict how many square foot tiles they
will need to tile a 17' by 23' kitchen floor.
- Students are challenged to answer this
question and then discuss the appropriate use of estimation when an
exact answer is almost certain to be wrong: The Florida's Best
Orange Grove has 15 rows of 21 orange trees. Last year's yield
was an average of 208.3 oranges per tree. How many oranges
might they expect to grow this year? What factors might affect
that number?
- Students play multiplication max out. Each student
has a 2 x 2 array of blanks (in the standard form of a
2-digit multiplication problem) into each of which a digit will be
written. One student rolls a die and everyone must write the number
showing into one of the blanks. Once a number is written, it cannot
be moved. Another roll-another number written, and so on. The
object is to be the player with the largest product when all four
digits have been written. If a player has the largest possible
product that can be made from the four digits rolled, there is a bonus
for maxing out.
8. Extend their understanding and use of arithmetic
operations to fractions, decimals, integers, and rational
numbers.
9. Extend their understanding of basic arithmetic
operations on whole numbers to include powers and
roots.
- Students explore the exponent key, the x2
key, and the square root key on their calculators. The groups are
challenged to define the function of each key, to tell how each works,
and to create a keypress sequence using these keys, the result of
which they predict before they key it in.
- Students work through The Powers of the Knight
lesson that is described in the Introduction to this Framework.
It introduces a classic problem of geometric growth which engages them
as they encounter notions of exponential notation.
- Students work through the Pizza Possibities
lesson that is described in the First FourStandards of this
Framework. In it, students discover that the number of pizzas
possible doubles every time another choice of topping is added. They
work through the Two-Toned Towers lesson that is also
described in the First Four Standards and note the similarities in the
problems and in their solutions.
- Students join the midpoints of the sides of a 2 x
2 square on a geoboard to form a smaller square. They determine the
area of the smaller square and explore the lengths of its four
sides.
10. Develop, apply, and explain procedures for
computation and estimation with whole numbers, fractions,
decimals, integers, and rational numbers.
- Students working in groups develop a method to estimate the
products of two-digit whole numbers and decimals by using the kinds of
base-ten block arrays described in Indicator 6 above. Usually just
focusing on the "flats" results in a reasonable
estimate.
- Students follow up a good deal of experience with concrete
models of fraction operations using materials such as fraction
bars or fraction squares by developing and defending their
own paper-and-pencil procedures for completing those operations.
- Students develop rules for integer operations by using
postman stories, as described in Robert Davis' Discovery in
Mathematics. The teacher plays the role of a postman who delivers
mail to the students. Sometimes the mail delivered contains money
(positive integers) and sometimes bills (negative integers).
Sometimes they are delivered to the students (addition) and sometimes
picked up from them (subtraction).
- Students model subtraction with two-color chips
by adding pairs of red and yellow chips. (First, they must agree that
an equal number of red and yellow chips has a value of 0.) For
example, to show 4 - (-2), they lay out
4 red chips, add 2 pairs of red and yellow chips (whose value is
0), and then take away 2 yellow chips. They note that 4
- (-2) and 4 + 2 give the
same answer and try to explain why this is so.
11. Develop, apply, and explain methods for solving
problems involving proportions and percents.
- Students develop an estimate of pi by carefully
measuring the diameter and circumference of a variety of circular
objects (cans, bicycle tires, clocks, wooden blocks). They list the
measures in a table and discuss observations and possible
relationships. After the estimate is made, pi is used to solve a
variety of real-world circle problems.
- Students use holiday circulars advertising big sales on
games and toys to comparison shop for specific items between different
stores. Is the new Nintendo game, Action Galore, cheaper at
Sears where it is 20% off their regular price of $49.95 or at
Macy's where it's specially priced at
$41.97?
- One morning, as the students arrive at school, they see a
giant handprint left on the blackboard overnight. They measure it and
find it to be almost exactly one meter long. How big was the person
who left the print? Could she have fit in the room to make the
print, or did she have to reach in through the window? How
could you decide how much she weighs?
- Students read Jim and the Giant Beanstalk
by Raymond Briggs. Jim helps the aging giantby measuring his head and
getting giant eyeglasses, false teeth and a wig. The students use the
measures given in the book to find the size of the giant's hand
and then his height.
- Students develop a sampling strategy and use
proportions to determine the population of Bean City (NCTM
Addenda Booklet, Grade 6), whose inhabitants consist of three
types of beans.
- Students discuss different ways of finding
"easy" percents, such as 50% of
30 or 15% of 25. They then generate percent
exercises that can be solved mentally and share them with their
classmates.
12. Understand and apply the standard algebraic order
of operations.
- Students bring in calculators from home to examine their
differences. Among other activities, they each key in " 6 + 2
x 4 = " and then compare their calculator
displays. Some of the displays show 32 and others show
14. Why? Which is right? Are the other calculators
broken?
- Students play rolling numbers. They use four white
dice and one red one to generate four working numbers and one target
number. They must combine all of the working numbers using any
operations they know to formulate an expression that equals the target
number. For example, for 2, 3, 4, 5 with target number
1, the following expression works: (2+5)/(3+4)=1.
Questions about order of operations and about appropriate use of
parentheses frequently arise.
References
-
Briggs, Raymond. Jim and the Giant Beanstalk.
Coward-McCann, Inc., 1970.
Ciardi, John. "Little Bits," in You Read to Me,
I'll Read to You. New York: Lippincott, 1962.
Davis, Robert. Discovery in Mathematics. New Rochelle,
NY: Cuisenaire, 1980.
Hiebert, J., and Behr, M. (Eds.) Number Concepts and
Operations in the Middle Grades. Reston, VA: National Council
of Teachers of Mathematics, 1988.
National Council of Teachers of Mathematics.
Addenda Booklet, Grade 6. Reston, VA, 1992.
Schwartz, David. If You Made a Million. New York:
Lothrop, Lee, & Shepard Books, 1989.
Silverstein, Shel. A Giraffe and a Half. New York:
Harper and Row, 1964.
Tahan, Malba. "Beasts of Burden," in The Man Who
Counted: A Collection of Mathematical
Adventures. W. W. Norton, 1993.
On-Line Resources
-
http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/
The Framework will be available at this site during
Spring 1997. In time, we hope to post additional resources
relating to this standard, such as grade-specific activities submitted
by New Jersey teachers, and to provide a forum to discuss the
Mathematics Standards.
Standard 8 - Numerical Operations - Grades 7-8
Overview
Traditionally, tremendous amounts of time were spent at these grade
levels helping students to finish their development of complex
paper-and-pencil procedures for the four basic operations with whole
numbers, fractions, and decimals. While some competency with
paper-and-pencil computation is necessary, estimation,
mental computation, and understanding the meanings of
the standard arithmetic operations all play a more significant
role than ever in the everyday life of a mathematically literate
adult.
As indicated in the K-12 Overview, then, the major shift in the
curriculum that will take place at these grade levels is one away from
drill and practice of paper-and-pencil symbolic procedures and toward
real-world applications of operations, wise choices
of appropriate computational strategies, and integration of the
numerical operations with other components of the mathematics
curriculum.
Seventh- and eighth-graders are relatively comfortable with the
unit shift discussed in this standard's Grades 5-6 Overview.
Operations on fractions and decimals, as well as whole numbers, should
be relatively well developed by this point, allowing the focus to
shift to a more holistic look at operations. "Numerical
operations" becomes less a specific object of study and more a
process, a set of tools for problem setting. It is critical that
teachers spend less time focused on numerical operations, per se, so
that the other areas of the Standards-based curriculum receive
adequate attention.
One important set of related topics that needs to receive some
significant attention here, however, is ratio, proportion,
and percent. Seventh and eighth graders are cognitively
ready for a serious study of these topics and to begin to incorporate
proportional reasoning into their set of problem solving tools. Work
in this area should start out informally, progressing to the student
formulation of procedures that make proportions and percents the
powerful tools they are.
Two other topics that receive greater attention here, even though
they have been informally introduced earlier, are integer
operations and powers and roots. Both of these types of
operations further expand the students' knowledge of the types of
numbers that are used and the ways in which they are used.
Estimation, mental math, and technology use
begin to mature in seventh and eighth grades as students use these
strategies in much the same way that they will as adults. If earlier
instruction in these skills has been successful, students will be able
to make appropriate choices about which computational strategies to
use in given situations and will feel confident in using any of these
in addition to paper-and-pencil procedures. For example, students
should evaluate simple problems involving fractions, such as
what's two-thirds of 5 tablespoons? using mental
math. Students need to continue to develop alternatives to
paper-and-pencil as they learn more about operations on other types of
numbers, but the work here is primarily on the continuing use of all
of the strategies in rich real-world problem solving settings.
The topics that should comprise the numerical operations focus of
the seventh and eighth grade mathematics program are:
-
rational number operations |
|
powers and roots |
integer operations |
|
proportion and percent |
Standard 8 - Numerical Operations - Grades 7-8
Indicators and Activities
The cumulative progress indicators for grade 8 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in grades 7 and
8.
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 7-8 will be such that all students:
6*. Select and use appropriate
computational methods from mental math, estimation,
paper-and-pencil, and calculator methods, and check the
reasonableness of results.
- Students choose a stock from the New York Stock Exchange
and estimate and then compute the net gain or loss each week for a
$1,000 investment in the company.
- Students use spreadsheets to "program" a set of
regular, repeated, calculations. They might, for example, create a
prototype on-line order blank for a school supply company that lists
each of the ten items available, the individual price, a cell for each
item in which to place the quantity ordered, the total computed price
for each item, and the total price for the order.
- Students regularly have human vs. calculator races.
Given a list of specially selected computation exercises (e.g., 53
x 20, 40 x 10, 95 + 17 +5 ), one student uses mental
math strategies and another uses a calculator. They quickly come to
realize that the human has the advantage in many situations.
8. Extend their understanding and use of arithmetic
operations to fractions, decimals, integers, and rational
numbers.
- Given a decimal or a fractional value for a piece of a
tangram puzzle, the students determine a value for each of the other
pieces and a value for the whole puzzle.
- Students use fraction squares to show why the
multiplication of two fractions less than one results in a product
that is less than either.
- Students demonstrate their understanding of operations on
rational numbers by formulating their own reasonable word problems to
accompany given number sentences such as
3/4 divided by 1/2 = 1 1/2.
9. Extend their understanding of basic arithmetic
operations on whole numbers to include powers and
roots.
- Students play powers max out. Each student
has a set of 5 blanks, into each of which will be written a digit.
They are in the form VWX + YZ. One student rolls a die and
everyone must write the number showing into one of their
blanks. Once written, anumber can not be moved. Another roll -
another number written, and so on. The object is to be the player
with the largest-valued expression when all five digits have been
written. If a player has the largest possible value that can be made
from the five digits rolled, there is a bonus for maxing
out.
- Students develop their own "rules" for
operations on numbers raised to powers by rewriting the expressions
without exponents. For example, 72 x
74 = (7 x 7) x
(7 x 7 x 7 x 7) = 7 x 7 x 7
x 7 x 7 x 7 = 76. You
just add the exponents!
- Students read The
King's Chessboard, The Rajah's Rice: A
Mathematical Folktale from India, or A Grain of
Rice. All of these stories involve a situation in which a
quantity is doubled each day. Students use the story to discuss
powers of 2 and to look for patterns in the sums of the powers
of 2.
- Students use the relationship between the area
of a square and the length of one of its sides to begin their study of
roots. Starting with squares on a geoboard with areas of 1,
4, 9, and 16, they then are asked to find squares
whose areas are 2, 5, and 13.
- Students work through the Rod Dogs lesson
that is described in the First Four Standards of this
Framework. They investigate how the surface area and volume of
an object changes as it is enlarged by various scale
factors.
10. Develop, apply, and explain procedures for
computation and estimation with whole numbers, fractions,
decimals, integers, and rational numbers.
- Students use a videotape of a youngster walking forward
and backward as a model for multiplication of integers. The
"product" of running the tape forward (+) with the student
walking forward (+) is walking forward (+). The "product"
of running it backward (-) with the student walking forward (+)
is walking backwards (-). The other two combinations also work
out correctly.
- Students use base ten blocks laid out in an array to show
decimal multiplication. How could the values of the blocks
be changed to allow it to work? What new insights do we gain
from the use of this model?
- Students judge the reasonableness of the results of
fraction addition and subtraction by "rounding off" the
fractions involved to 0, 1/2, or 1.
- Students explore the equivalence between
fractions and repeated decimals by finding the decimal representations
of various fractions and using the resulting patterns to find the
fractional equivalents of some repeated decimals.
11. Develop, apply, and explain methods for solving
problems involving proportions and percents.
- Students use The Geometer's Sketchpad
software to draw a geometric figure on a computer screen, scale it
larger or smaller, and then compare the lengths of the sides of the
original with those of the scaled image. They also compare the areas
of the two.
- Students are comfortable using a variety of approaches to
the solution of proportion problems. Example: If 8 pencils
cost 40 cents, how much do 10 pencils cost? This problem
can be solved by:
-
unit-rate method |
8 pencils for 40 cents means 5 cents per pencil or 10 x 5=50 cents for 10 |
factor-of-change method |
10 pencils is 10/8 of 8 pencils, so cost is (10/8)x40=50 cents |
cross multiplication method |
8/40 = 10/x, 8x = 400, so x=50 cents. |
- Students set up a part/whole proportion as one method of
solving percent problems.
- Students spend $100 by selecting items from a
catalog. They must compute sales tax and consider it in deciding what
they will buy.
12. Understand and apply the standard algebraic order
of operations.
- Students bring in calculators from home to examine their
differences. Among other activities, they each key in "3 + 15
÷ 3" and then compare their calculator
displays. Some of the displays show 6 and others show 8.
Why? Which is right? Are the other calculators broken?
Students decide what key sequence would work for the calculators
that do not use order of operations.
- Students play with the software How the West was One +
Three x Four, which requires them to construct numerical
expressions that use the standard order of operations.
- Students use the digits 1, 2, 3, and 4 to
find expressions for each of the numbers between 0 and
50. For example, 7 = (3x4)/2 + 1.
References
-
Barry, David. The Rajah's Rice: A
Mathematical Folktale from India. San Francisco, CA:
W. J. Freeman, 1995.
Birch, David. The King's Chessboard. Puffin
Pied Piper Books, 1988.
Pittman, Helena Clare. A Grain of Rice. Bantam
Skylark, 1986.
Software
-
Geometer's Sketchpad. Key
Curriculum Press.
How the West Was One + Three x Four.
Sunburst Communications.
On-Line Resources
-
http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/
The Framework will be available at this site during
Spring 1997. In time, we hope to post additional resources
relating to this standard, such as grade-specific activities submitted
by New Jersey teachers, and to provide a forum to discuss the
Mathematics Standards.
Standard 8 - Numerical Operations - Grades 9-12
Overview
In the ninth through twelfth grades, the themes described in the
K-12 Overview - estimation, mental computation, and
appropriate calculator and computer use-become the focus
of this standard. What is different about this standard at this level
when compared to the traditional curriculum is its mere presence. In
the traditional academic mathematics curriculum, work on numerical
operations was basically finished by eighth grade and focus then
shifted exclusively to the more abstract work in algebra and geometry.
But, in the highly technological and data-driven world in which
today's students will live and work, strong skills in numerical
operations have perhaps even more importance than they once did. By
giving older students a variety of approaches and strategies for the
computation that they encounter in everyday life, approaches with
which they can confidently approach numerical problems, they will be
better prepared for their future.
The major work in this area, then, that will take place in the high
school grades, is continued opportunity for real-world
applications of operations, wise choices of appropriate
computational strategies, and integration of the numerical
operations with other components of the mathematics
curriculum.
The new topics to be introduced in this standard for these grade
levels involve factorials, matrices, operations with polynomials, and
operations with irrational numbers as useful tools in problem solving
situations.
Estimation, mental math, and technology use
should fully mature in the high school years as students use these
strategies in much the same way that they will as adults. If earlier
instruction in these skills has been successful, students will be able
to make appropriate choices about which computational strategies to
use in given situations and will feel confident in using any of these
in addition to paper-and-pencil techniques. Students need to continue
to develop alternatives to paper-and-pencil as they learn about
operations with matrices and other types of number, but the work here
is almost exclusively on the continuing use of all of the strategies
in rich, real-world, problem solving settings.
The topics that should comprise the numerical operations focus of
the ninth through twelfth grade mathematics program are:
- operations on real numbers
- translation of arithmetic skills to algebraic operations
- operations with factorials, exponents, and matrices
Standard 8 - Numerical Operations - Grades 9-12
Indicators and Activities
The cumulative progress indicators for grade 12 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in grades 9, 10,
11 and 12.
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 9-12 will be such that all students:
6*. Select and use appropriate
computational methods from mental math, estimation,
paper-and-pencil, and calculator methods, and check the
reasonableness of results.
- Students frequently use all of these computational
strategies in their ongoing mathematics work. Inclinations to
over-use the calculator, in situations where other strategies would be
more appropriate, are overcome with five minute "contests,"
speed drills, and warm-up exercises that keep the other skills sharp
and point out their superiority in given situations.
- Numerical problems in class are almost always worked out
in "rough" form before any precise calculation takes place
so that everyone understands the "ballpark" in which the
computed answer should lie and which answers would be considered
unreasonable.
- Students use estimation in their work with
irrational numbers, approximating the results of operations such as
sqrt(15) + sqrt(17) or sqrt(32)
sqrt(8), and developing general rules.
- Students discuss the advantages and disadvantages
of using graphing calculators or computers to perform computations
with matrices.
- Students demonstrate their ability to select and
use appropriate computational methods by generating examples of
situations in which they would choose to use a calculator, to
estimate, or to use mental math.
- Students solve given computational problems using
an assigned strategy and discuss the advantages and disadvantages of
using that particular strategy with that particular
problem.
13. Extend their understanding and use of operations
to real numbers and algebraic procedures.
- Students work on the painted cube problem to enhance their
skill in writing algebraic expressions: A 3-inch cube is painted
red. It is then cut into 1-inch cubes. How many of them have
3 red faces? 2 red faces? 1-red face? No red faces? Repeat the
problem using an original 4-inch cube, then a five-inch cube,
then an n-inch cube.
- Students develop a procedure for binomial multiplication
as an extension of their work with 2-digit whole number multiplication
arrays. Using algebra tiles, they uncover the parallels
between 23 x 14 (which can be thought of as
(20+3)(10+4)) and (2x+3)(x+4).
- Students work through the Ice Cones lesson
that is described in the First Four Standards ofthis Framework.
They discover that in order to graph the equation to determine the
maximum volume of the cones, they need to use algebraic procedures to
solve for h in terms of r.
- Students devise their own procedures and
"rules" for operations on variables with exponents by
performing trials of equivalent computations on whole numbers.
- Students use algebra tiles to develop
procedures for adding and subtracting polynomials.
- Students use compasses and straightedges to
construct a Golden Rectangle and find the ratio of the length to the
width (1 + sqrt(5))/2.
- Students consider the ratios of successive terms
of the Fibonacci sequence (1, 1, 2, 3, 5, 8, ...), where
each term after the first two is the sum of the two preceding terms,
finding that the ratios get closer and closer to the Golden Ratio
(1 + û5)/2.
14. Develop, apply, and explain methods for solving
problems involving factorials, exponents, and matrices.
- Students work through the Breaking the Mold lesson
that is described in the Introduction to this Framework. It
uses a science experiment with growing mold to involve students in
discussions and explorations of exponential growth.
- Students use their graphing calculators to find a curve
that best fits the data from the population growth in the state of New
Jersey over the past 200 years.
- Students discover the need for a factorial notation and
later incorporate it into their problem solving strategies when
solving simple combinatorics problems like: How many
different five card poker hands are there? In how many
different orders can four students make their class
presentations? In how many different orders can six packages be
delivered by the letter carrier?
- Students compare
2101,
(250) 2, and
3x2100 to decide which is largest.
They explain their reasoning.
- Students read "John Jones's
Dollar" by Harry Keeler and discuss how it demonstrates
exponential growth. They check the computations in the story,
determining their accuracy.
References
-
Keeler, Harry Stephen. "John Jones's Dollar," in
Clifton Fadiman, Ed. Fantasia Mathematica. New York:
Simon and Schuster, 1958.
On-Line Resources
-
http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/
The Framework will be available at this site during
Spring 1997. In time, we hope to post additional resources
relating to this standard, such as grade-specific activities submitted
by New Jersey teachers, and to provide a forum to discuss the
Mathematics Standards.
* Activities are included here for Indicator 6,
which is also listed for grade 4, since the Standards specify that
students demonstrate continued progress in this indicator.
* Activities are included here for Indicator 6,
which is also listed for grade 8, since the Standards specify that
students demonstrate continued progress in this indicator.
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