61
Step 1: Identify Outcomes to Address
Guiding Questions
•
What do I want my students to learn?
•
What can my students currently understand and do?
•
What do I want my students to understand and be able to do based on the Big Ideas and
specific outcomes in the program of studies?
Big Ideas
•
Addition and subtraction are related, with subtraction being the inverse of addition.
•
The order of the numbers does not matter when you add, but does when you subtract.
•
Traditional algorithms are often not the most efficient methods of computing. They are also
not naturally invented by students. If the traditional algorithms are not taught to students
early on, students will invent or adopt personal strategies that vary with the numbers and the
situation. What is important is that the methods used are understood by the user.
•
Personal strategies depend on taking apart and combining numbers in a variety of ways and
recognizing relationships between numbers.
•
Models, be they manipulatives or diagrams, can help a person recognize the operation
involved and make sense of a problem.
•
Examining any problems for the parts and the whole helps students make sense of the
problem and identify the operation required. (See
Categories of Addition and Subtraction
Problems Based on Structure
, p. 50.)
Principles and Standards for School Mathematics
states that computational fluency is a balance
between conceptual understanding and computational proficiency (NCTM 2000, p. 35).
Conceptual understanding requires flexibility in thinking about the structure of numbers (base-
ten system), the relationship among numbers and the connections between addition and
subtraction. The ability to generate equivalent representations of the same number provides a
foundation for using personal strategies to add and subtract, recognizing that for some problems
either operation may be used. Computational proficiency includes both efficiency and accuracy.
Personal strategies must be compared and evaluated so students adopt methods that are efficient,
as well as accurate.
Addition and subtraction problems include four main types:
•
join problems involving an initial amount, a change amount (the amount being added or
joined) and the resulting amount
•
separate problems also involve change as in join problems, but the whole is the result in the
join problems, whereas the whole is the initial amount in the separate problems
•
part–part–whole problems consider two static quantities either separately or combined
•
comparison problems determine how much two numbers differ in size.
Adapted from John A. Van de Walle, LouAnn H. Lovin,
Teaching Student-Centered Mathematics: Grades K–3
, 1e
(pp. 66, 67, 68, 69). Published by Allyn and Bacon, Boston, MA. Copyright © 2006 by Pearson Education.
Reprinted by permission of the publisher.
www.LearnAlberta.ca
Grade 2, Number (SO 8, 9)
© 2008 Alberta Education
Page 4 of 51
47
What is crucial is that students are familiar with the relationship between addition and
subtraction and all the possible forms these operations take in problems.
By using a variety of problems, students will construct their own meaning for the inverse
relationship between addition and subtraction and for the following properties:
•
commutative property of addition (numbers can be added in any order), which does not
function for subtraction
•
associative property of addition (grouping a set of numbers in different ways does not affect
the sum) and
•
the identity element for addition and subtraction, that is adding zero to or subtracting zero
from a number will result in the original or start number.
Students in Grade 2 generally do not know the names of these properties, but certainly learn to
recognize and describe them. Grade 2 teachers who have students using the traditional algorithm
will note that too many times their students subtract upside down or backwards, as if the
commutative property of addition could be equally applied to subtraction. For example, if the
equation to be solved is:
53
or
53 – 29 =
–29
students, upon noting that one cannot subtract 9 from 3, without even being aware they are
inverting the numbers, may subtract 3 from 9. This creates a difference that is untrue for the
equation given. This situation does not occur when students use invented personal strategies.
When using the traditional algorithm, the manipulatives will prevent students from making this
type of error. Some students will need to be made aware that they unconsciously do this when
solving problems without manipulatives, so they can guard against this error.
The use of manipulatives or models helps students understand the structure of the story problem
and also connects the meaning of the problem to the number sentence (Van de Walle 2001,
p. 108). To develop their understanding of the meaning of operations, students connect the story
problem to the manipulatives, connect it to the number sentence and then use personal strategies
to solve the problem.
Reflect upon the student who adds 28
+ 47
615
and prints this incorrect answer: 615 (as all too frequently happens with the traditional
algorithm). Errors of this magnitude do not happen when students use personal strategies.
Seldom will students use a personal strategy they do not understand. The number sense that
students are developing in Grade 2 is critical to their ability to progress to estimating the answer,
necessary in Grade 3. Students' understanding of addition and subtraction is enhanced as they
develop their own methods and share them with one another, explaining why their strategies
work and are efficient (NCTM 2000, p. 220).
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Grade 2, Number (SO 8, 9)
© 2008 Alberta Education
Page 5 of 51
136
Sequence of Outcomes from the Program of Studies
See http://education.alberta.ca/teachers/core/math/programs.aspx
for the complete program of
studies.
¨
¨
Grade 1
Grade 2
Grade 3
Specific Outcomes
8. Identify the number,
up to 20, that is:
•
one more
•
two more
•
one less
•
two less
than a given number.
9. Demonstrate an
understanding of
addition of numbers
with answers to 20 and
their corresponding
subtraction facts,
concretely, pictorially
and symbolically, by:
•
using familiar
mathematical
language to
describe additive
and subtractive
actions
•
creating and solving
problems in context
that involve
addition and
subtraction
•
modelling addition
and subtraction,
using a variety of
concrete and visual
representations, and
recording the
process
symbolically.
Specific Outcomes
Specific Outcomes
8. Demonstrate and explain
6. Describe and apply mental
mathematics strategies for
adding two 2-digit numerals,
such as:
the effect of adding zero
to, or subtracting zero
from, any number.
9. Demonstrate an
•
adding from left to right
understanding of addition
•
taking one addend to the
nearest multiple of ten and
then compensating
(limited to 1- and 2-digit
numerals) with answers to
100 and the corresponding
•
using doubles.
subtraction by:
7. Describe and apply mental
mathematics strategies for
subtracting two 2-digit
numerals, such as:
•
using personal
strategies for adding
and subtracting with
and without the support
•
taking the subtrahend to
the nearest multiple of ten
and then compensating
of manipulatives
•
creating and solving
problems that involve
•
thinking of addition
addition and subtraction
•
using doubles.
•
using the commutative
9. Demonstrate an
understanding of addition and
subtraction of numbers with
answers to 1000 (limited to
1-, 2- and 3-digit numerals),
concretely, pictorially and
symbolically, by:
property of addition
(the order in which
numbers are added does
not affect the sum)
•
using the associative
property of addition
(grouping a set of
•
using personal strategies
for adding and subtracting
with and without the
support of manipulatives
numbers in different
ways does not affect the
sum)
•
explaining that the
•
creating and solving
problems in context that
involve addition and
subtraction of numbers.
order in which numbers
are subtracted may
affect the difference.
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Grade 2, Number (SO 8, 9)
© 2008 Alberta Education
Page 6 of 51
64
Step 2: Determine Evidence of Student Learning
Guiding Questions
•
What evidence will I look for to know that learning has occurred?
•
What should students demonstrate to show their understanding of the mathematical concepts,
skills and Big Ideas?
Using Achievement Indicators
As you begin planning lessons and learning activities, keep in mind ongoing ways to monitor and
assess student learning. One starting point for this planning is to consider the achievement
indicators listed in the
Mathematics Kindergarten to Grade 9 Program of Studies with
Achievement Indicators
. You may also generate your own indicators and use them to guide your
observation of the students.
The following indicators may be used to determine whether or not students have met specific
outcomes 8 and 9. Can students:
•
add zero to a given number, and explain why the sum is the same as the given number?
•
subtract zero from a given number, and explain why the difference is the same as the given
number?
•
model addition and subtraction, using concrete or visual representations, and record the
process symbolically?
•
create a problem involving addition or subtraction given a number sentence?
•
create an addition and subtraction number sentence and corresponding story problem for a
given solution?
•
solve a given problem involving a missing addend, and describe the strategy used?
•
solve a given problem involving a missing minuend or subtrahend, and describe the strategy
used?
•
refine personal strategies to increase their efficiency?
•
match a number sentence to a given missing addend problem?
•
match a number sentence to a given missing subtrahend or minuend problem?
•
explain or demonstrate the commutative property for addition: the order of addition does not
affect the sum; e.g., 5 + 6 = 6 + 5?
•
add a given set of numbers, using the associative property of addition (grouping a set of
numbers in different ways does not affect the sum), and explain why the sum is the same;
e.g., 2 + 5 + 3 + 8 = (2 + 3) + 5 + 8 or 5 + 3 + (8 + 2)?
•
solve a given addition or subtraction computation in either horizontal or vertical formats?
•
identify what each number in the problem means in relation to a part or a whole?
•
recognize that some strategies are more efficient than others in particular cases?
•
recognize that subtraction is not commutative, and so do not subtract upside down or
backwards?
•
explain how a strategy for adding and subtracting works and apply it to another similar
problem (limited to 1- and 2-digit numerals)?
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Grade 2, Number (SO 8, 9)
© 2008 Alberta Education
Page 7 of 51
19
•
create a different personal strategy for adding and subtracting and decide which strategy is
more efficient when solving problems?
•
analyze a personal strategy created by another person and decide if it makes sense in solving
an addition or a subtraction problem?
•
solve problems that involve addition and/or subtraction of more than two numbers with a
sum or subtrahend of no more than 100?
Sample behaviours to look for related to these indicators are suggested for some of the activities
listed in
Step 3,
Section
C: Choosing Learning Activities
(p. 14).
www.LearnAlberta.ca
Grade 2, Number (SO 8, 9)
© 2008 Alberta Education
Page 8 of 51
68
Step 3: Plan for Instruction
Guiding Questions
What learning opportunities and experiences should I provide to promote learning of the
outcomes and to permit the students to demonstrate their learning?
•
What teaching strategies and resources should I use?
•
How will I meet the diverse learning needs of my students?
A. Assessing Prior Knowledge and Skills
Before introducing new material, consider ways to assess and build on students' knowledge and
skills related to counting. For example:
•
Can the student read number sentences such as 4 + 5 =
□
, 7 – 5 =
□
or variations such as
4 +
□
= 9,
□
+ 5 = 9,
□
– 5 = 2, 7 –
□
= 2? (Does the student use the terms: "plus,"
"minus," "add," "subtract", "equal" and, when appropriate, a term for the variable, such as
"something", "blank," "box," or "what/some number"?)
•
Can the student demonstrate the meaning of such number sentences to sums of 20 or with
minuends no larger than 20, by dramatizing a problem with manipulatives, with pictures or
with diagrams? If not, can the student do so to answers of 10? Clarify whether the problem is
with the size of the numbers, the concepts of addition and subtraction or the vocabulary. If
the student is not yet able to demonstrate competence with addition and subtraction facts to
ten, check for rote counting and one-to-one correspondence to 10 and then 20.
•
Model the addition of 12 and 4 using concrete or visual representations and record the
process symbolically. Can the student create addition models (to sums of 20) and write the
corresponding symbolic representations?
•
Model the subtraction of 8 from 13 using concrete or visual representations and record the
process symbolically. Can the student create subtraction models and record the process
symbolically (using numbers no larger than 20 for the minuend)?
•
Can the student create an addition or subtraction story problem for various number sentences
such as: 13 – 8 =
□
or 8 +
□
= 13?
•
How accurately and how rapidly does the student solve these problems?
•
Can the student verbalize the strategy used?
If a student appears to have difficulty with these tasks, consider further individual assessment,
such as a structured interview (see sample), to determine the student's level of skill and
understanding. The Kathy Richardson books listed in the bibliography contain excellent
structured interviews for investigating concept development. They also indicate how a teacher
can address the development of missing concepts. See
Sample Structured Interview: Assessing
Prior Knowledge and Skills
(p. 11).
If the student is very proficient in producing correct answers quickly and can verbalize the
strategies being employed, you may need to consider additional challenges for this student while
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Grade 2, Number (SO 8, 9)
© 2008 Alberta Education
Page 9 of 51
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