Smart Strategies for Basic Facts is a thinking approach to fact memorization! Children
are introduced to clever strategies for learning and internalizing the basic addition and
subraction facts, through a sequence of directed learning experiences with specific tools to
help them quickly recall the facts. Through guided introductory activities reinforced by
oral and written practice, they’ll quickly learn the mental tricks used by solid math students,
such as:
• Counting On
• Doubles
• Doubles Plus One
• Making Tens
• Tens and Extras
• Counting Back
• Counting Up
• And many more!
SMART STRATEGIES FOR BASIC FACTS
Mental Math Techniques for Mastering Addition and Subtraction Facts
The learning is then extended with mental math strategies for addressing problems beyond
just the basic facts. With emphasis on the decade numbers 10, 20, 30, etc., they apply their
skills to quickly skim and solve 2-digit facts (20 + 4, 54 + 1, etc.). Blackline masters of special
flash cards, a hundred number board, and a 10 frame are all included, as are assessments
all along the way and activities for the whole class as well as specific teaching suggestions
for each lesson.
Through these experiences, students learn to think flexibly about numbers, to use more than
one strategy to solve problems, and to truly understand number relationships.
Related Materials
Fast Facts Practice Book
Help children recall basic addition and subtraction facts with speed and accuracy. Students
complete basic exercises, then score their own work. 48 pages. Set of 5 or 20.
Visit our website to see all of our mathematics resources. www.primaryconcepts.com
#4598
Primary Concepts
P. O. Box 10043
Berkeley, CA 94709
COUNTING ON
DOUBLES
SMART STRATEGIES
DOUBLES
PLUS ONE
FOR TENS
MAKING
TENS AND EXTRAS
COUNTING BACK
COUNTING UP
COUNTING ON
DOUBLES
DOUBLES PLUS ONE
Mental Math Techniques
MAKING
TENS
for Mastering Addition
TENS
AND EXTRAS
and Subtraction
Facts
COUNTING BACK
COUNTING UP
COUNTING ON
DOUBLES
DOUBLES PLUS ONE
FF.part 1A_Add _ 2015_2015 10/2/15 5:48 PM Page ii
Working on a smartboard? Free digital download of this book at
https://primaryconcepts.exavault.com/share/view/8d08-ch63818k
Sample PDF for evaluation. Free digital download of entire book with
Purchase.
Written by Joan Westley, Heather McDonald
Design and Production: Candace Wesen
Editor: Kelly Stewart, Sarah Le Forge
©2016, 2009, 2000 Primary Concepts
All rights reserved.
Printed in the U.S.A.
Primary Concepts grants teachers permission to print or copy the reproducible pages from this
book. Permission is limited to the teacher for whom the book was purchased. A digital copy may be
downloaded for personal or work use but please do not share the digital file without the written
consent of the publisher.
Item no. 4598
ISBN 978-1-60184-313-5
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Contents
INTRODUCTION
vii
Section 1
Addition Basic Facts
Teaching Notes
3
Understanding Counting On
8
Think Big!
10
Just One More
12
Counting On 2 and 3
14
Counting On 1, 2, 3
16
Adding with Zero
18
Adding with Ten
20
Small Doubles
22
Double Trouble
24
Almost Doubles
26
On the Double
28
Numbers on the Ten Frame
30
Parts of Ten
32
Tens Facts
34
On Beyond Ten
36
Special Nines
38
Deconstructing Numbers
40
More with Tens and Extras
42
The Addition Chart
44
Individual Interview
46
Think Fast! #1 – #3
© Primary Concepts
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Contents
Section 2
Subtraction Basic Facts
Teaching Notes
53
Understanding Counting Back
58
Just One Less
60
Take Away 2 or 3
62
Counting Back Review
64
Take Away Zero
66
Take Away All
68
Understanding Counting Up
70
One Difference
72
Counting Up 2 and 3
74
Subtracting All the Ones
76
Subtracting Ten
78
Ten Take-Away
80
Spotting Doubles
82
Almost Doubles
84
Understanding Tens and Extras
86
Minus Nine
88
Minus 7 or 8
90
More with Tens and Extras
92
Difference of Nine
94
Down to Ten and More
96
Individual Interview
98
Think Fast! #1 – #3
iv
Smart Strategies for Basic Facts
100–102
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Contents
Sections 3–4
Addition Beyond the Basics
Teaching Notes
103
Decade Numbers Plus Extras
104
Decades Upon Decades
106
Two Digits and Counting
108
Decades and Two Digits
110
Spotting Doubles
112
Making a Decade
114
Looking for Tens
116
Decades and Extras
118
Freestyle Mental Math
120
Think Fast! #1 – #3
122–124
Subtraction Beyond the Basics
Teaching Notes
125
Tens from Tens
126
Take Away Tens
128
Down to a Decade
130
Subtracting Ones
132
Small Differences
134
Down from a Decade
136
Tens Combinations
138
Back Across the Decade
140
Bridging Up
142
Free Style Mental Math
144
Think Fast! #1 – #3
© Primary Concepts
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Smart Strategies for Basic Facts
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Contents
Section 5
Resources and Supplies
vi
149
Family Letters
150
Ten Frame and Ten Strip
154
Hundred Number Board
155
Flash Facts Addition Flash Cards
156
Flash Facts Subtraction Flash Cards
169
Answers
182
Addition Strategies List
188
Subtraction Strategies List
189
Smart Strategies for Basic Facts
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Introduction
Smart Strategies for Basic Facts
Learning the basic addition and subtraction facts becomes an
important part of the mathematics curriculum once students have a
strong foundation in the prerequisite skills of counting, one-to-one
correspondence, beginning number sense, and an awareness of the
meaning of addition as the joining of two or more sets. The time
at which this learning is undertaken varies from class to class, but
traditionally the focus has been on rote memorization and repetitive
practice as the “strategies” of choice.
This book, however, is based on the idea that thinking of the basic
facts in terms of classes of problems and specific strategies that can be
used to learn those facts swiftly and successfully is more efficient and
powerful than rote memorization of the same facts. When students
begin seeing networks of relationships between certain types of
problems, they become more active, engaged learners. Their instant
recognition of basic facts becomes rooted in powerful ways of thinking
about the problems rather than on memorizing each fact individually
—a decidedly less efficient method. Most powerful of all, students
begin using known facts to find the answers to new, forgotten, or
troublesome facts.
Using the Activities
The activity sequence outlined in this book reflects one logical way of
presenting basic facts families and strategies. In general, the flow is from
easiest to most difficult strategies. Don’t hesitate to act on teachable
moments, though. For example, if your students are really into doubles
and showing off their knowledge of these facts, you might jump ahead
to the lessons Small Doubles, Almost Doubles, and Double Trouble.
Each strategy is introduced and strengthened over a series of lessons.
Manipulatives are often used to introduce new strategies so that
students will understand why the strategies work. Once students
understand the strategies, they quickly move to using them mentally.
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The Flash Facts flash cards provide a set of basic facts to practice on—facts that lend
themselves specifically to the new strategy.
Each lesson includes four major components: a Warm-Up activity, the main lesson,
a practice worksheet, and a Flash Facts oral practice section. The Warm-Up activity
helps students review prerequisite skills essential to understanding the new strategy.
After the Warm-Ups, you may launch right into the main lesson, which introduces the
new strategy or a new part of the strategy that is being developed. Next, students are
ready for the worksheet that accompanies the lesson. This provides students with
immediate practice with the new strategy they have learned and helps them begin to
recognize when that strategy can be used. Finally, students are ready for Flash Facts.
This section of the teaching notes describes which Flash Facts flash cards (in the back
of the book) can now be used for oral review and practice. By this time, some of the
students may have learned the facts automatically. Others may still be learning them.
While doing the Flash Facts oral practice, be sure to ask students to explain the
strategies they are using to come up with the answers, either those strategies that have
been explicitly taught or their own personal strategies. Sharing ideas for strategies has
multiple benefits. Verbalizing strategies helps students work through their thought
processes. Hearing other students’s strategies gives students new ways of thinking about
a problem. And knowing how to solve something in multiple ways gives students a
way to check their work and helps them develop flexibility in their thinking.
Let’s think about the problem 8 + 6. Who knows the answer?
“Eight and two more is ten. Then it’s four more left, so 10…11, 12, 13, 14.”
“Eight and eight is sixteen, but this is two less, that’s 14.”
“Six and six is twelve. Plus two more I didn’t count yet—14.”
“Eight…9, 10, 11, 12, 13, 14. I just counted it up on my fingers.”
Materials and Manipulatives
Periodically, the activities use special manipulatives to introduce and help students
understand new strategies. You may purchase these materials, use those you already have
in your classroom, download the digital files (see copyright page), or make your own
with the blackline masters on pages 154 and 155. The following manipulatives are used:
• Counting Chips • Ten Frames
• Ten Strips
• Hundred Number Board
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Section 1
Addition
Basic Facts
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Teaching Notes
The Addition Strategies
The addition facts typically referred to as “the basic facts” include 0 + 0 through
10 + 10, sums from 0 to 20.
+
0
1
2
3
4
5
6
7
8
9
10
0
0
1
2
3
4
5
6
7
8
9
10
1
1
2
3
4
5
6
7
8
9
10
11
2
2
3
4
5
6
7
8
9
10
11
12
3
3
4
5
6
7
8
9
10
11
12
13
4
4
5
6
7
8
9
10
11
12
13
14
5
5
6
7
8
9
10
11
12
13
14
15
6
6
7
8
9
10
11
12
13
14
15
16
7
7
8
9
10
11
12
13
14
15
16
17
8
8
9
10
11
12
13
14
15
16
17
18
9
9
10
11
12
13
14
15
16
17
18
19
10
10
11
12
13
14
15
16
17
18
19
20
Among these addition facts, a number of categories, or families of related facts, can be
identified. This section helps students identify these categories and introduces logical,
efficient strategies for learning and internalizing each specific class of problem. Seven
strategies are presented: Counting On, Zeros, Ten, Doubles, Almost Doubles, Parts of
Ten, and Tens and Extras.
The blackline master on page 188 lists all 7 basic facts strategies, plus an icon to help
visually evoke each strategy. Use this list as a poster in your classroom, as a way to
discuss strategies as a class, or as a handy reference tool for individual students.
© Primary Concepts
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Addition Fact Strategies
+
0
1
2
3
4
5
6
7
8
9 10
0
0
1
2
3
4
5
6
7
8
9
1
1
2
3
4
5
6
7
8
9
10 11
2
2
3
4
5
6
7
8
9
10 11 12
3
3
4
5
6
7
8
9
10 11 12 13
4
4
5
6
7
8
9
10 11 12 13 14
5
5
6
7
8
9
10 11 12 13 14 15
6
6
7
8
9
10 11 12 13 14 15 16
7
7
8
9
10 11 12 13 14 15 16 17
8
8
9
10 11 12 13 14 15 16 17 18
9
9
10 11 12 13 14 15 16 17 18 19
10
Counting On
Here, students are introduced to the seemingly revolutionary idea
that you don’t need to count from zero to find a total. In the
problem 6 + 3, for instance, students might think, “6…7, 8, 9”
rather than counting from zero, “1, 2, 3, 4, 5, 6…7, 8, 9.”
Counting on works best when one of the addends is 1, 2, or 3.
10 10 11 12 13 14 15 16 17 18 19 20
+
0
1
2
3
4
5
6
7
8
9 10
0
0
1
2
3
4
5
6
7
8
9
1
1
2
3
4
5
6
7
8
9
10 11
2
2
3
4
5
6
7
8
9
10 11 12
3
3
4
5
6
7
8
9
10 11 12 13
4
4
5
6
7
8
9
10 11 12 13 14
5
5
6
7
8
9
10 11 12 13 14 15
6
6
7
8
9
10 11 12 13 14 15 16
7
7
8
9
10 11 12 13 14 15 16 17
8
8
9
10 11 12 13 14 15 16 17 18
9
9
10 11 12 13 14 15 16 17 18 19
10
Zeros
Some basic facts problems are just easier than others. Students
can be coached to look for problems featuring zero as one of the
addends. Once students understand the concept that adding
zero means adding nothing, these problems are a snap to learn!
10 10 11 12 13 14 15 16 17 18 19 20
+
0
1
2
3
4
5
6
7
8
9 10
0
0
1
2
3
4
5
6
7
8
9
1
1
2
3
4
5
6
7
8
9
10 11
2
2
3
4
5
6
7
8
9
10 11 12
3
3
4
5
6
7
8
9
10 11 12 13
4
4
5
6
7
8
9
10 11 12 13 14
5
5
6
7
8
9
10 11 12 13 14 15
6
6
7
8
9
10 11 12 13 14 15 16
7
7
8
9
10 11 12 13 14 15 16 17
8
8
9
10 11 12 13 14 15 16 17 18
9
9
10 11 12 13 14 15 16 17 18 19
10
Ten
Students can look for problems featuring ten as one of the addends
as well. Adding ten to a one-digit number is easy—the ones digit of
the sum is always the same as the one-digit number. For example,
10 + 6 = 16.
10 10 11 12 13 14 15 16 17 18 19 20
+
0
1
2
3
4
5
6
7
8
9 10
0
0
1
2
3
4
5
6
7
8
9
1
1
2
3
4
5
6
7
8
9
10 11
2
2
3
4
5
6
7
8
9
10 11 12
3
3
4
5
6
7
8
9
10 11 12 13
4
4
5
6
7
8
9
10 11 12 13 14
5
5
6
7
8
9
10 11 12 13 14 15
6
6
7
8
9
10 11 12 13 14 15 16
7
7
8
9
10 11 12 13 14 15 16 17
8
8
9
10 11 12 13 14 15 16 17 18
9
9
10 11 12 13 14 15 16 17 18 19
10
Doubles
This special group of problems is one most students take a very
early interest in. Pictures related to each fact can help give
students visual images to use as memory aids for these facts. For
instance, they may think of a pair of eyes for 1 + 1 = 2 or a
spider’s legs for 4 + 4 = 8.
10 10 11 12 13 14 15 16 17 18 19 20
4
Smart Strategies for Basic Facts
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+
0
1
2
3
4
5
6
7
8
0
0
1
2
3
4
5
6
7
8
9 10
9
1
1
2
3
4
5
6
7
8
9
10 11
2
2
3
4
5
6
7
8
9
10 11 12
3
3
4
5
6
7
8
9
10 11 12 13
4
4
5
6
7
8
9
10 11 12 13 14
5
5
6
7
8
9
10 11 12 13 14 15
6
6
7
8
9
10 11 12 13 14 15 16
7
7
8
9
10 11 12 13 14 15 16 17
8
8
9
10 11 12 13 14 15 16 17 18
9
9
10 11 12 13 14 15 16 17 18 19
10
Almost Doubles
If you know your doubles facts, it makes lots of other problems
easy, too. 5 + 6 is close to 5 + 5, so it’s a breeze to solve because
it’s just one more.
10 10 11 12 13 14 15 16 17 18 19 20
+
0
1
2
3
4
5
6
7
8
9 10
0
0
1
2
3
4
5
6
7
8
9
1
1
2
3
4
5
6
7
8
9
10 11
2
2
3
4
5
6
7
8
9
10 11 12
3
3
4
5
6
7
8
9
10 11 12 13
4
4
5
6
7
8
9
10 11 12 13 14
5
5
6
7
8
9
10 11 12 13 14 15
6
6
7
8
9
10 11 12 13 14 15 16
7
7
8
9
10 11 12 13 14 15 16 17
8
8
9
10 11 12 13 14 15 16 17 18
9
9
10 11 12 13 14 15 16 17 18 19
10
Parts of Ten
Helping students learn the number pairs for ten (6 + 4, 7 + 3 …)
inside and out increases their facility with another category of
basic facts. It also prepares them for tougher facts where ten is
a powerful benchmark.
10 10 11 12 13 14 15 16 17 18 19 20
+
0
1
2
3
4
5
6
7
8
9 10
0
0
1
2
3
4
5
6
7
8
9
1
1
2
3
4
5
6
7
8
9
10 11
2
2
3
4
5
6
7
8
9
10 11 12
3
3
4
5
6
7
8
9
10 11 12 13
4
4
5
6
7
8
9
10 11 12 13 14
5
5
6
7
8
9
10 11 12 13 14 15
6
6
7
8
9
10 11 12 13 14 15 16
7
7
8
9
10 11 12 13 14 15 16 17
8
8
9
10 11 12 13 14 15 16 17 18
9
9
10 11 12 13 14 15 16 17 18 19
10
Tens and Extras
Building on students’ knowledge of the parts of ten, we can begin to
help them visualize and reason out tricky basic facts with sums greater
than ten. For example, 8 + 4 is 8 + 2 (10) plus 2 more, 12 altogether.
10 10 11 12 13 14 15 16 17 18 19 20
© Primary Concepts
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Flash Facts Addition Flash Cards
The activities are designed to be used in conjunction with Flash Facts, a unique set of
flash cards highlighting the powerful categories and strategies that may be used to help
students learn and internalize the basic facts. The Flash Facts cards are organized in sets,
based on the strategy students might use.
Counting On
Set A1: One More
Set A2: Counting On 2 and 3
Almost Doubles
Set A7
Parts of Ten
Set A8
Zeros Set A3
Ten
Set A4
Tens and Extras
Set A9: Nine Pluses
Set A10: 7s and 8s
Doubles
Set A5: Small Doubles
Set A6: Large Doubles
You may make a set of Addition Flash Facts cards using the blackline masters on pages
156–168.
Using the Flash Cards
To organize your cards, use the number code (A1, for instance) in the upper left corner of
each card. Put all the cards with the same code together. Organize the sets in numerical
order to start—A1 through A10.
A1
2
+1
A1
7 8
4
6
3 +1 5 +1 +1 +
1
+1
+1
A1
A1
A1
A1
A1
A1
A1
1 1 1 1 1 1 1
+2 +3 +4 +5 +6 +7 +8
A1
A1
A1
A1
A1
As you go through the Addition activities, you will introduce one set of flash cards at a
time based on what strategy is being explored. After an initial introduction, the cards may
be joined with other previously introduced cards to form a larger practice set.
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Involving Families
A Family Letter is provided on pages 150 and 151. This letter explains the addition
strategies students have learned and suggests ways that families can extend and support
this work. The letter is intended to be sent home after the Section 1 activities have been
completed. Students should by then have the basic strategies under their belts and will
know many of their basic facts automatically. Families can be of great help assisting
students in staying familiar with the basic facts and in working on facts that are still
troublesome.
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S T R AT E G Y
Counting On
Understanding Counting On
Here, students are introduced to the seemingly revolutionary idea
that you don’t need to count from zero to find a total.
Manipulatives
• Counting Chips
Warm-Ups
I’ll name a number. You
count on from there to ten.
5… (6, 7, 8, 9, 10)
8…
3…
6…
2…
Introducing the Strategy
1
Display the problem 5 + 3 on the board. Use Counting Chips
to model the problem, setting out a group of five chips and a
group of three chips. Have students do likewise with their own
Counting Chips, and then ask them to count the total.
2
Let me show you an easier and faster way to count these
chips. Indicate the group of five chips. We know there are
five here, right? So why count them up again from zero? Why
don’t we just say “five,” and then count the other group of chips
from there? So we say, “5…6, 7, 8.” Place your hand over the
group of five when the class says “five,” and then point to each of
the remaining three chips in turn as they say “six, seven, eight.”
3
Practice this strategy several more times with problems
such as 4 + 2, 2 + 3, and 4 + 3. Each time have the students set
out two groups of counters and then count on from the first group
of counters to the total. Exaggerating the first number a bit and
drawing it out (5...6, 7, 8) helps some students transition to this
new way of counting, helping them hear the first number as a
group or running total before counting on from there.
4
Draw a simple drawstring bag for the students to see.
Label the bag 4. Next to the bag, draw 3 counters. Imagine
we have four counters in this bag, and we want to add three
more. Let’s count on from four. 4…5, 6, 7. Seven altogether. This
way of picturing the process of counting on gives students a handy
mental image to use. Have students practice the strategy of
counting on with the worksheet on the next page.
4
8
Smart Strategies for Basic Facts
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Name
Understanding Counting On
Solve the problems by counting on from the number in the bag.
4
4 + 3 = ___
5
5 + 3 = ___
5
5 + 2 = ___
6
6 + 2 = ___
4
4 + 1 = ___
3
3 + 2 = ___
7
7 + 2 = ___
8
8 + 1 = ___
© Primary Concepts
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S T R AT E G Y
Almost Doubles
Almost Doubles
Students use counters to help them understand and visualize
“almost doubles” facts such as 3 + 4.
Manipulatives
• Counting Chips
Flash Facts
• Set A7
Warm-Ups
Let’s freshen up on our
doubles facts. I’ll say a
fact; you say the answer
on the double!
5+5
2+2
4+4
8+8
9+9
0+0
7+7
1+1
3+3
6+6
10 + 10
Introducing the Strategy
1
Write 3 + 4 on the board. Show the problem with counters,
lining them up as shown below. This fact is called an
“almost double” fact because it is very nearly a double. What
double is this fact close to? Show how the fact is close to 3 + 3 by
removing one counting chip from the bottom row and saying 3 + 4
is one more than 3 + 3. One more than 6 is 7. Also show how the
fact is close to 4 + 4 by adding one counting chip to the top row.
3 + 4 is one less than 4 + 4. One less than 8 is 7.
3
+4
3
+3
4
+4
2
Have the students build the “almost doubles” facts 6 + 5 and
4 + 5 in a similar way, with counting chips lined up to show
the doubles. Ask students to name the close doubles facts. How
can you use the double fact you know to help you solve this fact?
For 4 + 5, for example, students might say that it is one more than
4 + 4 or one less than 5 + 5. Focus their attention on whether the
“almost double” fact is more or less than the related double fact.
3
Now give the students some more “almost doubles” facts like
those below and see if they can solve the problems using
counters.
7+8
5+4
6 +7
4+3
8+9
Flash Facts
How Do You Know?
Use Flash Facts Sets A1–A7.
Flash each card and have students say whether the fact
is an “almost double.” If it is, have them say the related
double fact.
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Name
Almost Doubles
Use the counters shown to help you solve these “almost doubles” facts.
4
+5
6
+5
3
+4
7
+6
7
+8
2
+3
9
+8
0
+1
10
+9
6
+7
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S T R AT E G Y
Tens and Extras
More with Tens and Extras
The idea of using “tens and extras” thinking is beginning to gel
for students. Now they work on moving away from manipulatives
toward more mental strategies.
Manipulatives
Introducing the Strategy
• Counting Chips
• Ten Frames
Flash Facts
• Set A10
Warm-Ups
1
Review with students the idea of tens and extras. Use the
problem 8 + 4 as an example. Place eight counters of one color
on the Ten Frame. Here’s eight. Now we want to add four more.
We’ll use the first few to fill up the Ten Frame, and then the rest
will be extras. How many more to get to ten? (2) So we use two
of the four to fill up the ten. How many of the four left over? (2).
So ten plus two extra is how many? (12)
I’ll say a number. You say
how many more to get to
ten.
8… (2)
6…
9…
10…
7…
5…
8+4
2
Repeat the procedure for the problems 4 + 7 and 8 + 6, but this
time distribute Ten Frames and Counting Chips to students and
have them model the problems as well. Tell the students that it’s best
to start with the larger addend on the Ten Frame so it’s easier to think
about how many more to fill up the ten and then how many extras.
Remind them of their recent work with deconstructing numbers. This
will help with the “how many to fill it up, how many extra” work.
3
Now let’s try a few problems without counters and Ten
Frames. Write the problem 5 + 8 on the board. What’s the
answer? How did you figure it out in your head? Students may
have explanations such as the following: “I thought of eight, and
it needs two more to get to ten. Then there’s three left, so it’s
thirteen altogether.” or, “Five and five is ten, and then three are
left from the eight. That’s thirteen.”
4
Repeat for the problems 7 + 5 and 8 + 3. Make sure to have
students share their thought processes.
Flash Facts
U
se Flash Facts Set A10 to give students more practice
with “tens and extras” thinking. Start by having
students say the answers “in their heads,” simply raising
their hands when they have the answer. This allows all
students to think through the problems without being
preempted by the fastest thinkers. Later, you may wish to
have students answer aloud altogether as they become more
familiar with the problems.
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Name
More with Tens and Extras
Solve the problems. Use “tens and extras” thinking to help.
8
+3
9
+6
7
+4
5
+9
7
+8
5
+8
9
+7
5
+7
4
+9
8
+4
6
+9
4
+8
9
+3
8
+9
9
+8
4
+7
2
+9
7
+9
8
+6
8
+5
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S T R AT E G Y
Counting Up
Manipulatives
Understanding Counting Up
“What’s the difference” is another way of thinking about
subtraction problems. In this lesson, students learn to count up
from the subtrahend to the starting number to determine the
difference between the two.
• Ten Strips
Flash Facts
• Sets S5 and S6
Warm-Ups
I’ll say a number, you
count up from there to ten.
5… (6, 7, 8, 9, 10)
2…
8…
3…
1…
Introducing the Strategy
1
Remind the students of their work with counting back. When
the number you were taking away wasn’t very big, it was
easy to count back to find the answer. Let the students know that
today they’ll be exploring a different strategy called counting up.
We’ll use counting up when the starting number and the number
to subtract are pretty close to each other.
2
Tell students that there are two ways to think about
subtraction problems. One way is “take away,” where you
have a starting number and a number to take away. The answer
tells how many are left. Another way to think about subtraction
is finding the difference between two numbers—how much
greater the starting number is than the other number. When the
two numbers are pretty close to each other, it’s easy to find the
difference between the two by counting up. Let’s try an example.
3
Distribute Ten Strips to students. Display the 7 – 5 flash card
from Set S6. Let’s see what the difference is between 7 and
5 — how much greater 7 is than 5. Show students how to start
with the smaller number (5), pointing to it on the Ten Strip and
then counting up to the larger number (7) by “hopping” their
finger. The number of hops you needed tells you how different the
two numbers are. How many hops between 5 and 7? (2). So the
difference between 7 and 5 is 2. Or we could say 7 – 5 = 2.
1 2 3 4 5 6 7 8 9 10
7–5
4
Repeat with one or two more cards from Sets S5 and S6,
always having students point to the subtrahend on the Ten
Strip and count up to the starting number. How many hops did it
take? What is the difference between the two numbers in the
problem?
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Name
Understanding Counting Up
Count up on the Ten Strip to solve each problem.
7 - 6 = __
1 2 3 4 5 6 7 8 9 10
8 - 5 = __
1 2 3 4 5 6 7 8 9 10
10 - 8 = __
1 2 3 4 5 6 7 8 9 10
4 - 2 = __
1 2 3 4 5 6 7 8 9 10
5 - 4 = __
1 2 3 4 5 6 7 8 9 10
6 - 4 = __
1 2 3 4 5 6 7 8 9 10
9 - 7 = __
1 2 3 4 5 6 7 8 9 10
7 - 4 = __
1 2 3 4 5 6 7 8 9 10
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S T R AT E G Y
Tens and Extras
Difference of Nine
Using “tens and extras” thinking is beginning to gel for students.
Now they explore a related set of problems and discover that all the
extras plus one more are taken away to leave a difference of nine.
Manipulatives
Introducing the Strategy
• Counting Chips
• Ten Frames
Flash Facts
• Set S14
Warm-ups
I’ll say a “tens and
extras” number. You say
how many extras. For
instance, the number 14
has four extras. Imagine it
on a Ten Frame—the
frame is filled, plus four
extras.
17… (7 extras)
12…
19…
13…
11…
15…
1
Display the Set S14 Flash Facts. What do you notice about all
of these facts? Give students time to look the set over. Some
students may notice that the answers are all 9. Now pay special
attention to the “extras” in the starting number as compared to
the number to be taken away. What do you notice? Students will
discover that the number to be subtracted is always one more than
the ones digit of the starting number.
2
Distribute Ten Frames and Counting Chips to pairs of
students. Together, model the problem 14 – 5. Fill the Ten
Frame with counters, and place the 4 extras off to the side. Now
the problem says to take away five. We’ll start with the extras
first. Take away the 4 extras. How many more do we still need
to take away? (1) Take 1 away from the Ten Frame. How many
left? (9)
14 – 5
3
Have pairs of students work each of the remaining Set S14
problems with their manipulatives, always taking away from
the extras first and then dipping into the Ten Frame. When
students have finished, ask them to describe what happened in all
of these problems. They should be able to explain that in each
problem all of the extras were taken away, plus 1 more, leaving 9
on the Ten Frame.
Flash Facts
U
se Flash Facts Set S14, mixed with a few other two-digit
minus one-digit facts from previously introduced sets.
Have students show thumbs up if the answer is 9 (all the
extras plus 1 more taken away), or thumbs down if the answer
is not 9. Remember, you can spot the “difference of nine”
problems easily because all of the extras and one more have
been taken away from the starting number.
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Name
Difference of Nine
Solve all the problems. Circle the ones where all the extras
plus one more were taken away.
14
-5
12
-6
11
-5
12
-3
15
-7
13
-4
17
-8
14
-8
15
-6
12
-8
11
-7
16
-7
11
-2
14
-7
17
-9
17
-8
10
-9
13
-6
18
-9
16
-9
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Individual Interview
About the Interview
Here’s an opportunity to sit with students one-on-one to check on their basic facts skills.
Because some students are just naturally quieter in group situations or don’t feel the need to
be first to say answers aloud or need more time to think about problems, it may be hard for
you to assess every student’s progress based solely on group participation. This interview
may be used with all students or just with those you feel a bit out of touch with. Use the
interview to get a more solid feel for their progress in learning and committing to memory
the basic subtraction facts.
As always in formal assessment situations, create a relaxed atmosphere during the interview.
The more comfortable students are, the more they show of their true abilities.
Conducting the Interview
Show the student the facts on the Interview Form on the facing page. Have the student
select one fact at a time, tell the answer, and attempt to tell you how he or she reached that
answer. If it seems the student knows the fact automatically, place a check mark near the
problem. For facts the student doesn’t know right away, encourage the student to talk aloud
while thinking through the problem: Tell me what you’re thinking inside your brain as you
do this problem. Write the student’s answer by each problem, along with a note about the
strategy the student used.
You may wish to have a copy of the Strategy List on page 189 on hand as you conduct the
interview. This provides easy reference for both you and the student as you discuss various
strategies the student is using.
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Name
Individual Interview
Look at the facts. Tell the answer for each one and
describe how you knew the answer.
10
-3
5
-5
18
-8
16
-8
9
-4
12
-7
9
-2
13
-4
3
-0
8
-6
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Name
Think Fast!
Solve as many of the problems as you can in one minute.
On your mark, get set, go!
15
- 10
8
-2
16
-9
11
-2
14
-8
12
-5
10
-4
7
-1
8
-4
6
-6
14
-6
7
-6
14
-4
4
-0
9
-4
11
-6
13
-9
15
-7
6
-4
11
-8
I finished ________ problems.
I solved _________ correctly.
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Section 3
Addition
Beyond the Basics
.
Teaching Notes
Beyond the Addition Basic Facts
The activities in Section 3 take students beyond the
addition basic facts, allowing them to see how previously
learned strategies may be used on more difficult problems.
Section 3 activities focus on the same addition strategies
students learned in relation to the basic facts but take
students beyond these basic facts to work with larger twodigit numbers. The major focus is on the decade numbers
(10, 20, 30, 40…) and on two-digit plus one-digit
problems. Although sums may “bridge the decade,” the
approach focuses on using mental math and efficient
strategies rather than on introducing the algorithm for
regrouping.
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S T R AT E G Y
Ten
Decade Numbers Plus Extras
Ten, twenty, thirty, forty… In this lesson, students explore adding
one-digit numbers to these special “decade” numbers.
Manipulatives
• Hundred Number Boards
Warm-Ups
10, 20, 30, 40, 50, 60, 70,
80, 90…We’ll call these
“decade” numbers. Decade
means “ten.” I’ll say a
decade number. You count
on by ones from there.
30… (31, 32, 33, 34, 35,
36, 37, 38, 39, 40)
Continue, naming a
different decade number
to start with each time.
Introducing the Strategy
1
Remind students of the strategy adding with tens. What
did you find out about the answers to those problems?
Students should recall that the two-digit answer always ends with
the number of ones they added to ten. For instance, 10 + 6 = 16.
The answer, 16, shows one ten and six ones.
2
Ask students what they can say about the answers to
problems like the following:
20 + 5
50 + 2
40 + 4
70 + 8
Let’s show some of these problems on the Hundred Number
Board. For each problem, mark the decade number
with your finger and count on from there the appropriate
number of spaces.
50 + 3
Think Fast!
T
he answers to decade numbers plus extras problems
should be easy. When you’re doing a page of addition
problems, look for problems like these. You can solve them
really fast. Let’s practice!
40 + 1
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Smart Strategies for Basic Facts
90 + 9
20 + 4
50 + 7
60 + 2
30 + 8
70 + 2
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Name
Decade Numbers Plus Extras
Circle and solve the “decades and extras” problems quickly.
Then go back and solve the rest of the problems.
30 + 9 = ____
8 + 8 = ____
80 + 6 = ____
8 + 9 = ____
3 + 30 = ____
2 + 70 = ____
6 + 10 = ____
90 + 5 = ____
40 + 7 = ____
20 + 3 = ____
90 + 1 = ____
10 + 4 = ____
60 + 7 = ____
2 + 80 = ____
7 + 3 = ____
60 + 0 = ____
5 + 80 = ____
6 + 5 = ____
8 + 4 = ____
40 + 9 = ____
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S T R AT E G Y
All
Freestyle Mental Math
In this lesson, students get an opportunity to solve a variety
of problems mentally and share their personal strategies.
Warm-Ups
Write one problem on the
board at a time and have
students answer as quickly
as possible.
38 + 3 (41)
26 + 4
25 + 40
Introducing the Strategy
1
Tell students that today they’ll be solving a variety of types
of problems and that they should rely on any strategy they’ve
learned or any of their own strategies that make sense as they
work the problems. Suggest, too, that you’re interested in having
students share their “brain talk” with classmates. As you solve each
problem today mentally, pay attention to what you’re saying
inside your head. This will help you share your thinking and
strategies with the rest of us.
2
As you present each problem below, have students raise their
hands to share their answers. Write down all answers
suggested for a particular problem, whether correct or not. Don’t
give any indication of correctness as you note each answer. When
several answers have been proposed, have students defend their
answers by explaining how they thought through the problem. In
the process, some students may discover their own errors and
adapt their answers.
37 + 4
40 + 29
43 + 27
48 + 2
31 + 53
55 + 36
3
At the end of the session, review the strategies used during
the session. Students may have referred to counting on,
doubles, ten as a benchmark, tens and extras, tens facts, wellknown basic facts, and more. Also review any personal strategies
students described that seemed to work well.
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Name
Freestyle Mental Math
Solve the problems. Use words, symbols, or diagrams to explain
how you solved each one.
17 + 5 = ____
26 + 44 = ____
38 + 28 = ____
54 + 15 = ____
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Name
Think Fast!
Solve as many of the problems as you can in one minute.
On your mark, get set, go!
33 + 7 = ___
20 + 5 = ___
57 + 3 = ___
48 + 3 = ___
40 + 40 = ___
22 + 0 = ___
54 + 30 = ___
50 + 7 = ___
74 + 2 = ___
20 + 6 = ___
79 + 3 = ___
21 + 9 = ___
20 + 63 = ___
38 + 1 = ___
60 + 30 = ___
45 + 5 = ___
68 + 4 = ___
40 + 3 = ___
I finished ________ problems.
I solved _________ correctly.
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S T R AT E G Y
Counting Up
Small Differences
Thinking of subtraction in terms of “what’s the difference” makes
problems like 47 – 44 easy. At a glance, students can see that the
difference is small.
Manipulatives
Introducing the Strategy
• Hundred Number Boards
Warm-Ups
I’ll say two numbers, you
tell me if they’re close
together or far apart.
57
38
49
21
58
and
and
and
and
and
23 (far apart)
35
46
7
53
1
Display the problem 47 – 44. Remind students that one way
to think about subtraction is finding the difference between
two numbers. This problem asks, What’s the difference between
47 and 44?
2
Tell students that when they see a problem like 47 – 44 where
the numbers are close together, they can simply count up
from the smaller number to the larger to find the difference. Since
the numbers are pretty close together, it’s not far to count up
from one to the other. Distribute Hundred Number Boards and
have students put their fingers on the number 44. Have the class
count up with you from 44 to 47. How many counts (finger hops)
did we need to get up to 47? (3)
3
Why wouldn’t we use counting up for a problem like 32 – 14?
Students should realize that that would be a long way to
count up. Display the following problems. Then point to one at a
time and have students show thumbs up if the problem lends itself
to counting up or thumbs down if it doesn’t.
75 – 73
43 – 13
4
84 – 21
18 – 15
55 – 54
66 – 63
Have pairs of students work with Hundred Number Boards
to solve the following problems:
66 – 64
77 – 77
49 – 46
37 – 34
19 – 17
Think Fast!
H
ow Do You Know?
Display the problems below
on the board. Review with students different strategies
for solving problems: counting up, counting back, tens and
extras thinking, and so on. For each problem, have students
share their answers and the strategies they used to think
about the problem. Some students will rely on strategies
that have been formally introduced. Others will describe
strategies that they themselves have come up with.
27 – 24
134
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80 – 30
54 – 4
54 – 2
37 – 20
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Name
Small Differences
Circle and solve the problems that have small differences.
Cross out the 5 problems that don’t have small differences!
34 - 32 = ____
35 - 32 = ____
65 - 64 = ____
91 - 49 = ____
25 - 3 = ____
97 - 96 = ____
73 - 71 = ____
59 - 57 = ____
47 - 44 = ____
43 - 17 = ____
78 - 23 = ____
36 - 33 = ____
69 - 67 = ____
25 - 24 = ____
44 - 44 = ____
84 - 56 = ____
86 - 84 = ____
16 - 14 = ____
19 - 18 = ____
75 - 74 = ____
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154
Smart Strategies for Basic Facts
Ten Strip
1 2 3 4 5 6 7 8 9 10
Ten Strip
Ten Frame
Ten Frame
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Hundred Number Board
Hundred Number Board
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
© Primary Concepts
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9 10
19 20
29 30
39 40
49 50
59 60
69 70
79 80
89 90
99 100
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A5
A6
Flash Facts – Addition
A5
A6
2 3 4 5 6
+2 +3 +4 +5 +6
7 8 9 10 4
+ 7 + 8 + 9 +10 + 5
A5
A6
A5
A6
A5
A7
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A7
A7
A7
A7
A7
A7
A7
A8
A7
Smart Strategies for Basic Facts
Flash Facts – Addition
A7
5 6 5 6 7
+6 +7 +4 +5 +6
7 8 8 9 1
+8 +9 +7 +8 +9
164
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A9
A9
A9
A9
A9
A9
A9
A10
A9
Smart Strategies for Basic Facts
Flash Facts – Addition
A9
9 9 9 2 3
+5 +6 +7 +9 +9
4 5 6 7 8
+9 +9 +9 +9 +3
166
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A10
A10
Flash Facts – Addition
A10
A10
8 8 8 7 7
+4 +5 +6 +4 +5
3 4 5 6 4
+8 +8 +8 +8 +7
A10
A10
A10
A10
A10
A10
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Flash Facts – Addition
A10
5
+7
168
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S4
S5
Flash Facts – Subtraction
S4
S5
9 10 5 6 7
- 9 -10 - 4 - 5 - 6
8 9 6 7 8
-7 -8 -4 -5 -6
S5
S6
S5
S6
S5
S6
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S6
S7
S6
S7
S6
S7
S6
S7
S7
Smart Strategies for Basic Facts
Flash Facts – Subtraction
S7
9 7 8 9 11
-7 -4 -5 -6 -1
12 13 14 15 16
-2 -3 -4 -5 -6
174
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S12
S13
Flash Facts – Subtraction
S12
S13
13 12 11 14 13
-9 -9 -9 -8 -8
12 11 12 11 11
-8 -8 -7 -7 -2
S12
S13
S13
S13
S13
S14
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S14
S15
S14
S15
S14
S15
S14
S15
S14
Smart Strategies for Basic Facts
Flash Facts – Subtraction
S15
12 13 14 15 16
-3 -4 -5 -6 -7
11 12 13 14 11
-3 -4 -5 -6 -4
180
© Primary Concepts