STANDARD 8 - NUMERICAL OPERATIONS
All students will understand, select, and apply various methods of
performing numerical operations.
Standard 8 - Numerical Operations - Grades K-2
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
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
- 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?
- 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
- Mary has 8 cookies. She eats some. She has 6
left. How many cookies did Mary eat? (Missing
- 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?
- 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
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
- 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
- Students make booklets containing original word
problems that illustrate different addition or subtraction situations.
These may be included in a portfolio or evaluated
2. Develop proficiency with and memorize basic
number facts using a variety of fact strategies (such as
"counting on" and
- 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
- 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
- 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,
- 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
- Students use computer games such as Math
Blaster Plus or Math Rabbit to practice basic
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
- 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
- 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
- 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
- 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
5. Use a variety of mental computation and
- 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
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
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
- 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.
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.
Math Blaster Plus. Davidson.
Math Rabbit. The Learning Company.
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