Download Grades K–5 - Barrington 220

Common Core State
Standards Overview
Grades K–5
World-Class Singapore Math
for Your Classrooms
“
Overall, the Common Core State Standards (CCSS)
are well aligned to Singapore’s Mathematics Syllabus.
Policymakers can be assured that in adopting the CCSS,
they will be setting learning expectations for students
that are similar to those set by Singapore in
terms of rigor, coherence and focus.
—Achieve* (achieve.org/CCSSandSingapore)
”
Table of Contents
Common Core State Standards Overview ...................2
Examples of support for the Common Core State Standards:
Standards for Mathematical Practice
1
2
3
4
5
6
7
8
* Achieve is a bipartisan, nonprofit educational reform organization that partnered with NGA and CCSSO on the Common Core State Standards Initiative.
Make sense of problems and
persevere in solving them. ..................................4
Reason abstractly and quantitatively. ................6
Construct viable arguments and critique
the reasoning of others. ......................................8
Model with mathematics...................................10
Use appropriate tools strategically. ..................12
Attend to precision. ...........................................14
Look for and make use of structure. .................16
Look for and express regularity in
repeated reasoning. ..........................................18
1
Teaching the Common Core State Standards
to mastery with Math in Focus
®
The Common Core State Standards for Mathematics is an initiative designed to implement more focused gradelevel standards. The research base used to guide the Common Core State Standards noted conclusions from TIMSS*, where
Singapore has been a top-scoring nation for over 15 years. Apparent in the TIMSS and other studies of high-performing
countries is a more coherent and focused curriculum. The Singapore math framework was one of the 15 national curricula
examined by the Common Core committee and had a particularly important impact on the Common Core writers and
contributors. Achieve, a bipartisan, nonprofit organization that partnered with NGA and CCSSO on the Common Core
State Standards Initiative, points out that “because of its quality, the Singapore Syllabus was an important resource
for the developers of the CCSS.”
Math in Focus emphasizes number and
operations in every grade, just as recommended
in the Common Core State Standards. The
textbook is divided into two books, roughly a
semester each. Approximately 75% of Book A is
devoted to number and operations and 60-70%
of Book B to geometry and measurement,
where the number concepts are practiced,
connected, and applied.
Focus on number, geometry, and
measurement in elementary grades
Common Core State Standards: Mathematics experiences
in early childhood settings should concentrate on (1) number
(which includes whole number, operations, and relations) and
(2) geometry, spatial relations, and measurement, with more
mathematics learning time devoted to number than to other topics.
(Common Core State Standards for Mathematics, 3)
Within the Common Core State Standards, several overarching initiatives are put forth, which parallel the framework
of the Singapore mathematics curriculum and Math in Focus. These initiatives include:
Organize content by big ideas, such as place value
Common Core State Standards: These Standards endeavor to follow such a
Curriculum must be focused and coherent
Math in Focus is organized to
teach fewer topics in each grade,
but to teach them thoroughly to
mastery. When a concept appears
in a subsequent grade level, it is
always at a higher level.
Common Core State Standards: For over a decade, research studies of
Le
mathematics education in high-performing countries have pointed to the
conclusion that the mathematics curriculum in the United States must become
substantially more focused and coherent in order to improve mathematics
achievement in this country. (Common Core State Standards for Mathematics, 3)
design, not only by stressing conceptual understanding of key ideas, but also by
continually returning to organizing principles such as place value or the properties of
operations to structure those ideas. (Common Core State Standards for Mathematics, 4)
arn You can use place-value charts to add numbers
with regrouping.
14 + 18 = ?
Look at the place-value chart.
14 = 1 ten 4 ones
18 = 1 ten 8 ones
Thousands Hundreds
Tens
Ones
Thousands Hundreds
Tens
Ones
7
0
1
6
0
1
6
8
0
0
1
8
0
70
Tens
Math in Focus is structured for
mastery learning. Rather than
repeating topics, students narrow
in on these critical areas and
master them. Then, in subsequent
grades they develop them to
more advanced levels. Moving
from addition and subtraction in
second grade to multiplication
and division in third grade is such
an example.
Teach to mastery
1
14
+
on four critical areas: (1) extending understanding of base ten notation; (2) building
fluency with addition and subtraction; (3) using standard units of measure; and
(4) describing and analyzing shapes. (Common Core State Standards for Mathematics, 17)
*The Trends in International Mathematics and Science Study (TIMSS) provides reliable
and timely data on the mathematics and science achievement of U.S. 4th- and 8th-grade
students compared to that of students in other countries.
70  10
Add the ones.
Skills Trace
Grade 2
Understand the concept of multiplication as
repeated addition and division as grouping or
sharing. Use objects and pictures to show the
concept of division as finding the number of
equal groups. (Chap. 5)
Grade 3
Multiply and divide 2-digit and 3-digit numbers
with and without regrouping. (Chaps. 6 to 9)
Grade 4
Multiply and divide multi-digit numbers using
place-value concepts. (Chap. 3)
1
1
160
160  10
4
8
2
1,800
1,800  10
4 ones + 8 ones = 12 ones
18
Regroup the ones.
12 ones = 1 ten 2 ones
70
Tens
In Grade 3, instructional time should focus on four critical areas: (1) developing
understanding of multiplication and division and strategies for multiplication and
division within 100; (2) developing understanding of fractions, especially unit fractions;
(3) developing understanding of the structure of rectangular arrays and of area; and
(4) describing and analyzing two-dimensional shapes.
(Common Core State Standards
for Mathematics, 21)
Step 1
Tens Ones
Common Core State Standards: In Grade 2, instructional time should focus
Common Core State Standards
Skills Traces in the Teacher’s
Edition help teachers understand
how concepts progress through
the grade levels.
2
Ones
What is the
pattern when
each number is
divided by 10?
Math in Focus is organized around
place value and the properties of
operations. The first chapter of
each grade level begins with place
value. In first grade, students learn
the teen numbers and math facts
through place value. In all the
grades, operations are taught with
place-value materials so students
understand how the standard
algorithms work.
Ones
Step 2
Add the tens.
70  10
Tens Ones
1
32
+
1
1
3
7
160
4
8
2
160  10
1,800
1 ten + 1 ten + 1 ten = 3 tens
1,800  10
1
Each digit moves
one place to the
right when the
number is divided
by 10.
So, 14 + 18 = 32.
98
Chapter 13
Addition and Subtraction to 40
This example, from Grade 1, shows how visual
place-value charts are used to reinforce
concepts early on to ensure that students
understand both how and why math works.
G1B_TB_Ch13(80-103).indd 98
12/31/08 9:17:22 AM
These visual representations are carried
Lesson 2.4 Dividing by Tens, Hundreds, or Thousands
throughout the program to reinforce the
underlying principle of place-value. Here, a
more complex place value chart is used in
Grade 5 to learn to divide by tens.
71
See how
Math in Focus
supports the
Full correlations are available at hmheducation.com/singaporemath
Common Core
State Standards.
3
Common Core State Standards: Standards for Mathematical Practice
1
Make sense of problems and persevere in solving them.
How Math in Focus Aligns:
Singapore Mathematics
Framework
Examples throughout the Math in Focus curriculum:
Grade 1
Notes
1
Grade 3
Put On Your Thinking Cap!
PRo B leM solVIN G
PRoBleM s olVIn G
Find the number of beads.
Use number bonds to help you.
solve.
1
Reasoning,
communication
and connections
Thinking skills
and heuristics
Applications and
modeling
Numerical, Algebraic, Geometrical
Statistical, Probabilistic, Analytical
Grade 4
Grade 5
Put On Your Thinking Cap!
Put On Your Thinking Cap!
PROBLEM SOLVING
PROBLEM SOLVING
Meena has 28 counters.
She puts some in a bag.
She puts the rest of the counters into 5 boxes.
If each box contains 5 counters, how many counters are in the bag?
There are 6 beads under the two cups.
CRITICAL THINKING SKILLS
CRITICAL THINKING SKILLS
CRITICAL THINKING SKILLS
Put On Your Thinking Cap!
Rita wrote three 4-digit numbers on a sheet of paper.
She accidentally spilled some ink on the paper.
Some digits were covered by the ink.
Using the clues given, help Rita find the digits covered by the ink.
PROBLEM SOLVING
Solve these problems.
2 Jessie had a whole graham cracker.
Minah had only part of another graham cracker.
1 Thenumberinthesquareistheproductofthenumbersinthe
twocirclesnexttoit.Findthenumbersinthecircles.
Jessie gave 1 of her graham cracker to Minah.
4
In the end, both girls had the same fractional part of
6
18
a graham cracker.
What fraction of a graham cracker did Minah have at first
How many mice will be hungry?
Jessie
2
2 3 boys have coats.
There are 8 beads under the two cups.
ES
CLU
Minah
The sum of all the ones is 17.
5.4
The ones digit of the first number is the greatest 1-digit number.
8
Here are 2 equal bars to
show that both of them had
an equal portion of a
graham cracker in the end.
The digit in the tens place of the second number is one more than
the digit in the tens place of the first number.
3 Thereare10beadsunderthethreecups.
There are 6 egg holders.
2.7
Work backward to find
the fraction of the graham
cracker Minah had at first.
The tens digit of the third number is 4 less than the tens digit of
the second number.
How many boys will be cold?
3
Mathematical
Problem
Solving
on
CRIT ICAl T HIn KIn G s KIlls
CRITIC Al T HIN KIN G s KIlls
There is cheese for 6 mice.
iti
From the Singapore Ministry of Education
Grade 2
Put On Your Thinking Cap!
Circle, count, and write.
gn
Concepts
Throughout the Math in Focus program, you will find problem solving at the heart of the curriculum. In addition
to solving problems in the Learn, Guided Practice, Let’s Practice, and independent practice portions
of each lesson, Put on your Thinking Cap! problems (Grades 1–5) and the Grade K Student Books challenge
students to put the skills they’ve learned to work, finding solutions in non-routine situations.
Grade K
Monitoring of
one’s own thinking
et
Self-regulation
ac
of learning
o
M
ls
”
Numerical calculation
Algebraic manipulation
Spatial visualization
Data analysis
Measurement
Use of mathematical tools
Estimation
Skil
Math in Focus is based on the premise that in
order for students to persevere and solve both
routine and non-routine problems, they need
to be given tools that they can use consistently
and successfully. They need to understand
both the how and the why of math so that they
can self-monitor and become empowered
problem solvers. This in turn spurs positive
attitudes that allow students to solidify their
learning and enjoy mathematics.
Beliefs
Interest
Appreciation
s
de
Confidence
u
t
ti
Perseverance
At
Pro
“
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for
entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about
the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They
consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into
its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on
the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator
to get the information they need. Mathematically proficient students can explain correspondences between equations,
verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for
regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a
problem. Mathematically proficient students check their answers to problems using a different method, and they continually
ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex
problems and identify correspondences between different approaches. —Common Core State Standards
cess
es
Math in Focus is built around the Singapore Ministry
of Education’s Mathematics Framework pentagon,
which places mathematical problem solving at the
core of the curriculum. Encircling the pentagon
are the skills and knowledge needed to develop
successful problem solvers, with concepts, skills,
and processes building a foundation for attitudes
and metacognition.
2 Simoneboughtatotalof10birthdayhatsandnoisemakers.
Eachbirthdayhatcost$1.50andeachnoisemakercost$2.50.
Thenoisemakerscost$13morethanthebirthdayhats.
Howmanyofeachitemdidshebuy?
10
There are
counters in the bag.
ON YOUR OWN
Go to Workbook A:
How many eggs are needed to fill all the egg holders?
58
lessonChapter
1 Making
2 Number Bonds
Kindi_SB1_Ch2.indd 58
Go to Workbook B:
Go to Workbook A:
Student Book A, Part 1, p. 58
1/6/11 4:09:10 PM
Kindergarten Book A, page 58: Kindergarten
students are introduced to problem solving in
a5.visual
way,
where
evaluate
a
Children
count
howthey
manylearn
objectstoare
needed to
situation
anda set.
determine the steps they need
complete
to take to get an answer.
6. For the first task, help children understand that the mice
being hungry implies that they do not have any cheese.
Then, have children circle the mice that do not have a
slice of cheese. Next, have them count the number of
mice they have circled and write this answer in the
4 blank provided.
7. For the second task, help children understand that the
boys being cold implies that they are not wearing a coat.
Then, have children circle the boys who are not wearing
Gr1 TB A_Ch 2.indd 37
37
8/19/08 4:34:24 PM
Student Book A, page 37: Students use number
bonds to determine unknowns, promoting early
algebraic thinking through relevant problem
solving.
lesson 3
G2B_TB_Ch 16.indd 217
Real-World Problems: Measurement and Money
Student Book B, page 217: In order to solve
problems like the one pictured here, students
cannot simply memorize. Rather, they need to
understand how math works and be able to
manipulate it to solve non-routine problems.
ON YOUR OWN
ON YOUR OWN
Go to Workbook B:
Go to Workbook A:
Put on Your Thinking Cap!
pages 51 – 54
Put on Your Thinking Cap!
pages 169–170
Put on Your Thinking Cap!
pages 173–174
Put on Your Thinking Cap!
pages 31–32
Chapter 2
Put on Your Thinking Cap!
pages 17–18
on YoUR oWn
oN YoUR oWN
217
12/17/08 6:17:47 PM
32
G3A_TB_Ch_01.indd 32
Chapter 6
Chapter 1 Numbers to 10,000
12/17/08 6:51:10 PM
Student Book A, page 32: Non-routine problems
like this one emphasize the necessity of
understanding. Math in Focus teaches students
to explore the meaning of operations so they can
go beyond simply identifying a symbol to determine
which operation to use. Instead, students are
challenged to think about the situation and choose
the operation based on reason and its application
to the problem.
G4_TB_Ch6.indd 269
Fractions and Mixed Numbers
269
12/11/08 4:54:21 PM
Student Book A, page 269: As students progress,
problems become increasingly complex, but
consistent problem-solving tools such as bar
modeling give students the tools they need to
persevere in solving them. Thought bubbles also
help students monitor their work and assess
whether or not they are on the right track and
whether or not their answers make sense.
Chapter 9MultiplyingandDividingDecimals81
Student Book B, page 81: By the time students
reach Grade 5, they have developed the confidence
and skills needed to become successful problem
solvers. Because they have consistently been
exposed to non-routine problems, they are ready
to handle the challenges of middle school
and enjoy mathematics.
Common Core State Standards for Mathematics: corestandards.org
© 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
5
Common Core State Standards: Standards for Mathematical Practice
2
Reason abstractly and quantitatively.
How Math in Focus Aligns:
Math in Focus’ concrete–pictorial–abstract progression helps students effectively contextualize and
decontextualize situations by developing a deep mastery of concepts. Each topic is approached with
the expectation that students will understand both how it works, and also why. Students start by
experiencing the concept through hands-on manipulative use. Then, they must translate what they
learned in the concrete stage into a visual representation of the concept. Finally, once they have gained
a strong understanding, they are able to represent the concept abstractly. Once students reach the
abstract stage, they have had enough exposure to the concept and they are able to manipulate it and
apply it in multiple contexts. They are also able to extend and make inferences; this prepares them for
success in more advanced levels of mathematics.
“
Mathematically proficient students make sense of quantities and their relationships in problem
situations. They bring two complementary abilities to bear on problems involving quantitative
relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and
manipulate the representing symbols as if they have a life of their own, without necessarily attending to their
referents—and the ability to contextualize, to pause as needed during the manipulation process in order to
probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent
representation of the problem at hand; considering the units involved; attending to the meaning of quantities,
not just how to compute them; and knowing and flexibly using different properties of operations
and objects. —Common Core State Standards
”
Le
Examples throughout the Math in Focus curriculum:
DAy
Grade 5
Grade 3
arn
Put On Your Thinking Cap!
Find 34 � 48.
2
Grade K
arn You
Grade
1 use number bonds to help you subtract.
can
How many beanbags are on the floor?
P ROB L E M S OLV ING
The 9 key on the calculator is not working.
48
50
Le
C RITIC A L THINKI N G SKI LLS
Add 2-digit numbers mentally using the ‘add the tens, then
subtract the extra ones’ strategy.
Explain how you can still use the calculator to find
1,234  79 in two ways.
2
Step 1 Add 50 to 34.
34 � 50 � 84
Step 2 Subtract 2 from the result.
84 � 2 � 82
1
79
Do you know why you add
50 and then subtract 2?
So, 34 � 48 � 82.
9−5=?
I can rewrite 79 as
1,234
to play three rounds of the game.
engage in the activity, end the day by
uestions such as:
know how many children should move?
?
1.
4. While children engage in the activity, ask check
questions such as:
• Are you sure?
• Can it be this number instead?
Step 2 Subtract
from the result.
arn Solve problems using the mean.
So, 35 � 57 �
.
part
GradePut
2
10 − 9 =
P R O B le M S O lV I N G
7. Explain to children that you are going to toss
your number cube and write a number on the
board. Have children read the number and let
each child build a tower using that number
of cubes.
?
42
whole
Chapter 2
Find the missing numbers in each part
box.
1
There
is
–
8
End the day by playing the game for
8–10 rounds.
74
Gr1 TB A_Ch 4.indd 74
C hapter 2: L esson 1
3
3
yellow2 bean.
G3A_TB_Ch2.indd 42
6
1
– 4
4
4
– 2
8
4
4
4
4
5
4
)
ON YOUR OWN
Go to Workbook A:
Chapter 2 Whole Number Multiplication and Division
109
�
�
�
MiF 5A PB U2 2.4-2.7.indd 109
12 lb
Mental Math and Estimation
Total
weight of the 2 tables � 16 � 2
� 32 lb
Weight of the other table � 32 � 12
� 20 lb
1/12/09 2:56:52 PM
Student Book A, page 109: Put On Your Thinking Cap!
problems like this one challenge students to consider
what quantities mean, how they are composed, and how
they can use model drawing to represent a solution.
Students are taught to think flexibly about numbers
so that they can be deconstructed if needed to solve
a problem. Here, while students may understand what
9 means, they must also understand how it can be
manipulated in order to solve.
3/12/10 2:22:01 PM
The weight of the other table is 20 pounds.
4
2
0
Chapter 4 Subtraction Facts to 10
Student Book A, page 88: After a lesson on subtraction up to 1,000,
Answerneed
the question.
students
to have a deep enough understanding in order to recognize
situations where the “-” sign doesn’t necessarily require taking away to solve.
4 Brian has a machine that changes numbers.
Even though these look like simple subtraction problems, students need to
He puts
one
number
into
machineinand
a different
understand
how
each
number
is the
functioning
order
to fill in the green squares.
3/2/11 10:41:01 AM
1,234
Put on Your Thinking Cap,
pages 75– 78
2 � 16 lb � 32 lb
On Your Thinking Cap!
1,234
1,234  79  (1,234 
C R I T I C A l T HI N K I N G S K I llS
6. Have children count 1 to 6 aloud and count their
cubes to be sure they each have 6.
)
groups
The mean weight of 2 tables is 16 pounds. The weight of one of the tables
is 12 pounds. What is the weight of the other table?
13 How many yellow beans are there?
5. Write the numeral 6 on the board. Explain to
children that this numeral represents the
number six.
1,234
Real-World Problems:
Data and Probability
Lesson Objective
Solve real-world
problems involving probability
to 35.
35 �
Step• 1 Add
and 4
measures of central tendency.
Grade
Student Book A, page 74: Students model subtraction concretely by
startingPractice
with physical objects. They then move on to the pictorial stage,
Guided
using number bonds to represent the action of taking away. Finally,
Use
number
bonds tosymbolically.
subtract. This concrete–pictorial–abstract
they
write subtraction
progression helps students make sense of quantities.
8. Then, have children hold up their towers and
check each other’s towers.
6
o
There are 4 beanbags on the floor.
3. Write the numerals 1 to 5 on the board. Ask:
What number is this?
1,234
1,234  79
Student Book A, page 42: Math in Focus teaches students ways to break
Add
mentally.
Use number
bonds
torequires
help you.
apart
numbers
to compute
mentally.
This
students to develop an
n of the meaning of quantities. In this example, students learn
understanding
2 Find 35 � 57.
that to add 34 + 48 is the same as adding 50 +34 and then subtracting 2.
A thought bubble
57reinforces this reasoning ability by asking students why
they would
60 manipulate the numbers like this.
3
4
part
Begin the day by dividing the class into
four groups.
2. Distribute the connecting cubes to the children.
Ensure that each child has six cubes.
9.
Guided Practice
9−5=4
whole
1,234
1,234  79  (1,234 
9
Math Focus: Make a connection between objects and
numerals from 1 to 6.
Materials: Connecting cubes, 6 per child
Number cube
Classroom Setup: Whole class
Less
Teacher’s Edition A,
Chapter 2: Math in
Focus Kindergarten
students participate
in Discover activities,
like the one pictured
here from Chapter 2.
These
ngineer to raise
his or Discover
her fingers activities
to
introduce
using
many train engines
he or sheconcepts
wants.
concrete, hands-on
nsure that thisactivities.
child does not
raise
more
This helps
students make sense of
quantities
and
numbers
class that if, for
example, the
train
so they
truly
understand
s four fingers, only
engines
1, 2,
3, and
o move. Engines
5 and
6 should
what
they
mean.
5
4
3 5 1
 1.
79 groups
Le
Discover
1 or
1,234 
part
Activity 2
1
Guided Practice
Student Book A, page 204: To solve problems like this one, students must
Solve.
be able
toShow
take your
their work.
understanding of mean and consider how it is used
to find
goes
beyond
simply
abstract and
1 weight.
Mr. Saco This
bought
chicken,
fish, and
shrimpcomputing
at a market. to
Theusing
mean weight
quantitative
consider
how itofrelates
operations.
of the reasoning
3 items was 7topounds.
The weight
chicken to
wasother
8 pounds
and the weight of fish was 4 pounds.
What was the weight of shrimp that Mr. Saco bought?
8/19/08 4:37:42 PM
number comes out.
When he puts 12 into the machine, the number 7 comes out.
When he puts 20 into the machine, the number 15 comes out.
The table on page 89 shows his results for 4 numbers.
Total weight of chicken, fish, and shrimp Mr. Saco bought �
�
Weight of chicken and fish Mr. Saco bought �
�
204
Chapter 5 Data and Probability
�
pounds
Common Core State Standards for Mathematics: corestandards.org
© 2010.
� National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
pounds
7
Activity 3
Common Core State Standards: Standards for Mathematical Practice
Explore
Construct viable arguments and critique
the reasoning of others.
“
How Math in Focus Aligns:
established
in constructing
arguments.
Begin results
1.
the day
by preparing
the cardsThey
for make conjectures and build a logical progression of
statements to
the truth of their conjectures. They are able to analyze situations by breaking them into Student Book A, Part 1, p. 26
theexplore
activity.
cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others,
2. Distribute
materials
to the children.
and respond
to the arguments
of others.
They reason inductively about data, making plausible arguments that
Activity 4
take into3.account
context
from
which‘same’,
the data
arose. Mathematically proficient students are also able to
Help the
them
read the
words
‘different’,
Apply
compare theand
effectiveness
‘color’. of two plausible arguments, distinguish correct logic or reasoning from that which is
Math
Focus:can
Apply
the concept
of counting up to 6 objects.
flawed, and—if there is a flaw in an argument—explain what it is. Elementary
students
construct
arguments
4. One child shuffles the cards and places them
Resource:
Student
Book
A,sense
Part 1, pp. 26 –29
using concrete
referents
such
as
objects,
drawings,
diagrams,
and
actions.
Such
arguments
can
make
face down. He or she then chooses
a card
and
Guided
Practice
Materials:
1 (TR01),learn
Numeral
and be correct,
even
not group
generalized
or made
grades.Numeral
Later, students
to 2 (TR02), Numeral 3 (TR03),
reads
thethough
task tothey
theare
other
members.
Forformal until later
Numeral 4 (TR05), Numeral 5 (TR06), and Numeral 6 (TR07)
determine domains
to4,
which
an argument
applies. Students at all grades can listen or read the arguments
example:
different,
color. Solve.
Paper, 1 sheet per child (optional)
of others, decide whether they make sense,
andbar
ask useful
questions
to clarify
or improve the arguments.
Use
models
to help
you.
Classroom
Setup: Children work independently.
5. The
other
group
members gather materials as
—Common
Core
State
Standards
per the task.
1
Carlos has 9 stickers. 1. Encourage children to return to their places and open
their Student Books to page 26.
Math
Talk
His
cousin
Encourage target vocabulary
by gives him 3 stickers.
6.
Kindi_SB1_Ch2.indd 26
1/6/11 4:07:19 PM
”
His chosen
sister buys him another
stickers.
asking group members why they have
2. 5Children
draw a line connecting the two boxes with the
their materials. Elicit replies such as:
I chose
4 stickers does Carlos
How
many
in all?
samehave
number
of wheels.
connecting cubes. 2 are red and 2 are yellow.
5
3. Remind
children to be sure they have counted each
Examples
throughout
thecolors.
Math in Focus curriculum:
Red and
yellow are different
wheel
and
say each counting word in order as they point
3
7.
After
introducing
the
activity,
let
children
to
each
wheel
in a set. Make sure children understand
Grade K
work independently.
that
the
last
number
they say is the total number in
Teacher’s Edition A, page 4: Check
9
questions throughout
the Kindergarten
the
set.
8. While children engage in the activity, ask check
Teacher’s Editions provide opportunities
for students
explain children
their thinking
and
4. toWhile
engage
construct viable arguments.
questions such as:
• How did you decide which cubes to use?
• Can I use this combination of cubes instead?
+
Grade 1
Carlos has
R eADI nG AnD WRITIn G M ATH
4
Math Journal
C hapter
2: L esson
1 pattern.
Tania completes
this number
32,
33,
34,
35,
36,
37,
38,
?
+
As seen on the Singapore Mathematics Framework pentagon,
metacognition is a foundational part of the Singapore curriculum.
Students are taught to self-monitor, so they can determine whether
or not their solutions make sense. Journal questions and other
opportunities to explain their thinking are found throughout the
program. Students are systematically taught to use visual diagrams
to represent mathematical relationships in such a way as to not
only accurately solve problems, but also to justify their answers.
Chapters conclude with a Put On Your Thinking Cap! problem. This is a
comprehensive opportunity for students to apply concepts and present
viable arguments. Games, explorations, and hands-on activities are also
strategically placed in chapters when students are learning concepts.
During these collaborative experiences, students interact with one
another to construct viable arguments and critique the reasoning of
others in a constructive manner.
Grade 2
BothAndyandRitathinkthat0.23isgreaterthan0.3.
23isgreaterthan3,
so0.23isgreaterthan0.3.
23tenthsisgreater
than3tenths,so0.23
isgreaterthan0.3.
Doyouagree?Whyorwhynot?Explainyouranswer.
REA DIN G A N D W RIT IN G MAT H
List the steps to arrange the numbers in order from least
to greatest.
Lesson 7.3 ComparingDecimals33
Example
Student Book B, page 33: This example presents students
with a mathematical statement and asks them whether
or not they agree. This prompts students to construct
an argument to support their answer and provides
opportunities for classroom discussion.
1,984 2,084 1,884
ST EP
1 I compare the thousands.
ST EP
2 I can see that 2,084 is the greatest.
ST EP
3 I compare the hundreds.
such as:
• Why have you chosen to match these two pictures?
• How do you know they match? Are you sure?
Student
BookWhy?
A, page
instead?
or104:
WhyThroughout
not?
= • Will this do
the program, students are taught to check
stickers in all.
Re a D iNG a ND WRiT iNG MaT H
Math Journal
Math Journal
in the activity, ask check questions
their answers and make sure their solutions
are reasonable. Look for the “Check!” icon
throughout the Student Books.
Grade 4
Grade 3
ST EP
4 I can see that 1,884 is the least number.
Arranged from least to greatest:
Grade 5
1,884 1,984 2,084
least
9,049 9,654 8,785
Le
3
Math Focus: Extend the concept of counting up to 6
objects.; Extend the concept of same and different.
Materials: Connecting cubes, 20 per group (10 yellow and
10 red)
Same and Different cards (TRAA–BB), 1 set per group
Classroom Setup: Children work in small groups at the
26
Chapter 2
math
center.
Mathematically
proficient students understand and use stated assumptions, definitions, and previously
Arranged from least to greatest:
arn
Some problems require two steps to solve.
The Fairfield Elementary School library is in the shape of a rectangle.
It measures 36 yards by 21 yards. The school’s principal, Mr. Jefferson,
wants to carpet the library floor. Find the cost of carpeting the library fully
if a 1-square-yard carpet tile costs $16.
List the steps to get your answer.
First, find the floor area of the library.
Check!
39
She explains how she found each number in the pattern.
3+5=8
1 to 32 to get 33.
Kindi_TE1_Chap2.inddI added
4
3/2/11 10:41:03 AM
+8=
I added 1 to 33 to get 34.
I just have to add 1 to get the next
number.
Estimate the answer.
36 rounds to 40.
21 rounds to 20.
40 20 800
756 is a reasonable answer.
Area length width
36 21
756 yd2
Lesson 1.3
Comparing and Ordering Numbers
The floor area of the library is 756 square yards.
31
Then, find the cost of carpeting.
G3A_TB_Ch_01.indd 31
12/17/08 6:51:08 PM
Cost of carpeting
Student Book A, page 31: Exercises that require students
to list the steps they take to get an answer help develop
the language students need to explain how they solved a
problem and justify their solutions.
Is the answer correct?
33 is 1 more than 32.
34 is 1 more than 33.
area cost of 1 yd2
756 $16
Estimate to check if the
answer is reasonable.
$12,096
32 + 1 = 33
It will cost $12,096 to carpet the library fully.
33 + 1 = 34
How do you find the missing numbers in this pattern?
40, 30,
Guided Practice
Student
Book A, page 98: Throughout Math in Focus, students are
Solve. Show your work.
asked to estimate in order to evaluate whether or not their answers
Rob fills 250-gallon fuel tanks at $3 per gallon at a gas station.
3
are reasonable.
develops
thetometacognitive
skills that are
HowThis
much money
does he need
pay for filling 9 such tanks?
highlighted in the Singapore mathematics framework, and promotes
the ultimate
developing
Total goal
amount of
of fuel
9 250 effective problem solvers.
, 10,
Student Book B, page 74: Students
learn how to explain their reasoning
through guided math journal exercises
and modeled thought processes.
In the pattern, is the next
number more or less?
Cost of fuel 74
8
G1B_TB_Ch12.indd 74
Chapter 12
Numbers to 40
104
Chapter 4
He needs to pay $
Using Bar Models: Addition and Subtraction
98
.
Chapter 2 Whole Number Multiplication and Division
Common Core State Standards for Mathematics: corestandards.org
© 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
12/29/08 4:25:54 PM
MiF 5A PB U2 2.4-2.7.indd 98
MS_Gr2A_unit04.indd 104
$3 $
8/26/08 4:38:56 PM
12/8/09 3:12:24 PM
9
Common Core State Standards: Standards for Mathematical Practice
Model with mathematics.
How Math in Focus Aligns:
”
ssoN
1
Making Number Bonds
Objectivescurriculum:
Examples throughout the Math 3inLesson
Focus
Vocabulary
DAy
• Useconnectingcubesoramathbalanceto
findnumberbonds.
• Finddifferentnumberbondsfornumbersto10.
Grade
1
le
Grade K
part
whole
number bond
a rn
You can make number bonds with
Grade 3
.
You can use a number train to make number bonds.
into two parts.
Sam put
part
Activity 4
Discover
Math Focus: Make a connection between the number of
objects and the terms one more and one less.
Materials: Counters, 10
Classroom Setup: Whole class with teacher direction.
1.
Begin the day by inviting children to stand
around a table.
2. Place one counter on the table. Ask: How many
are there? (1)
3. Then, add four more counters. Ask: How many
are there now? (5) Show the number with your
fingers.
4. Ask: Do I have more or less than I had before?
(More)
5. Remove two counters. Ask: How many are there
now? (3)
6. Ask: Do I have more or less than I had before?
(Less)
7.
Math Talk
Elicit from the children full
sentences such as:
•There are more.
•There are less.
8. Repeat steps 2 to 7 several times using different
numbers of counters.
9.
End the day by checking that children are
raising and putting down the correct number
of fingers.
Teacher’s Edition A, page 31: Students
use concrete manipulatives to model the
mathematics they are learning. This hands-on
approach helps students understand what the
numbers and concepts mean before they move
on to the abstract stage.
Explore
Math Focus: Extend the concept of pairing sets of objects
and dots to numerals.; Extend the concepts of one more,
one less, and the same number.
Materials: Counters, 10 per group
Student Numeral Cards 0–9, 1 set per group
Dot Cards 0–9, 1 set per group
part
Classroom Setup: Children work in small groups with
teacher direction.
How many are in each part?
1.
How many tickets did Sue sell?
b
How many tickets did they sell in all?
Nancy
a?
a
3 and 1 make 4.
4
3. Ask children
to then lay This
out the
sameshows
number
picture
a number bond.
1 dot card with the
of counters
whole and the
Part-Part-Whole in
1 Using
Addition
and Subtraction
corresponding number.
L
b
part
Math Talk
Give groups various instructions to
practice
the2 concepts
30
Chapter
Number Bonds of one more, one less, and
Lesson
Objectives
the
same number, such as:
•Usebarmodelstosolveadditionandsubtractionproblems.
•I
want the numeral to be one more.
•Applytheinverseoperationsofadditionandsubtraction.
Grade
•I want2
one less counter.
•I want the dots to be the same number as
n
r
atheYou
counters.
can use bar models to help you add.
4.
Gr1 TB A_Ch 2.indd 30
5. WhileMandy
children
engage
in the bars.
activity, ask check
makes
10 granola
Aida makes
12 granola bars.
questions
such as:
How
many
granola
bars
do
•How can you tell they match?they make in all?
•What number is one less than 2?
•What number is one less than 1?
10
12
Ones
Tens
Ones
Step 1
Dividethehundredsby3.
5hundreds31hundred
with2hundredsleftover
1
3 5 2 5
3 0 0
2 o
Less
Le
Addthetens.
20tens2tens22tens
3
1
5 2 5
3 0 0
2 2 5
Student Book A, page 96: Place-value charts are also used
consistently
throughout
the Math in Focus program. They
96
Chapter 3
WholeNumberMultiplicationandDivision
help students visualize and understand numbers so that they
understand why the standard algorithms work and can apply
them in non-routine situations.
12/11/08 5:11:20 PM
Grade 5
1,286 tickets
arn
Solve problems by drawing bar models.
Hector, Teddy, and Jim scored a total of 4,670 points playing a video game.
Teddy scored 316 points less than Hector. Teddy scored 3 times as many
points as Jim. How many points did Teddy score?
Sue sold fewer tickets
than Nancy. So, use a
comparison model.
316
Hector
3,450 1,286 2,164
Sue sold 2,164 concert tickets.
Teddy
4,670
Jim
3,450 2,164 5,614
They sold 5,614 concert
tickets in all.
Check!
First, subtract 316 points from Hector’s score so that he will have the
same number of points as Teddy.
This also means subtracting 316 points from the total number of points.
4,670 316 4,354
8/19/08 4:32:56 PM
122
Chapter 5
Using Bar Models: Addition and Subtraction
Student Book A, page 122: As students tackle increasingly complex
problems, they can use bar models to help them visualize, understand,
and solve.
G3A_TB_Ch5.indd 122
For Struggling Learners For children who are
having difficulty pairing objects to dots and numbers,
have them deal with numeral cards and dot cards
?
0–5 first, before moving on to 6–9.
Tens
Vocabulary
sum
difference
bar model
2,164 1,286 3,450
5,614 2,164 3,450
The answers are correct.
Student Book A,
page 96: Bar modeling
is introduced in Grade
2 and continued
throughout the
Math in Focus
curriculum.
6. Repeat steps 2 to 5 using different numbers.
5253?
G4_TB_Ch_03-1new.indd 96
b?
Sue
2. AskE Schildren
to lay3 out the numeral card ‘5’.
SON
Regroupthehundreds.
2hundreds20tens
3,450 tickets
Begin the day by distributing materials to
part
the children.
farmersellshiscropsto3restaurants.Hedivides
A
525headsoflettuceequallyamongthe3restaurants.
Howmanyheadsoflettucedoeseachrestaurantreceive?
Hundreds
Use bar models and addition or subtraction to solve
2-step real-world problems.
a
Vocabulary
regroup
Model division with regrouping in hundreds, tens, and ones.
Hundreds
Nancy and Sue sold tickets for a concert.
Nancy sold 3,450 tickets.
Sue sold 1,286 fewer tickets than Nancy.
Student Book A,
page 30: Students
use number bonds to
model part-part-whole
relationships.
Le
Activity 3
arn
arn
Real-World Problems:
Addition and Subtraction
n
Lesson Objective
• Use bar models to solve 2-step real-world problems on
addition and subtraction.
Le
2
DAy
Modeling Division
with Regrouping
n
Lesson Objectives
• Modelregroupingindivision.
• Dividea3-digitnumberbya1-digitnumberwithregrouping.
Singapore math is also known for its use of model drawing, often called
“bar modeling” in the U.S. Students are taught to use rectangular “bars”
to represent the relationship between known and unknown numerical
quantities and to solve problems related to these quantities. This gives
students the tools to develop mastery and tackle problems as they become
increasingly more complex.
Le
le
Grade 4
Math in Focus places a strong emphasis on number and number
relationships, using place-value manipulatives and place-value charts
to model concepts consistently throughout the program. In all grades,
operations are modeled with place-value materials so students understand
how the standard algorithms work.
o
“
Mathematically proficient students can apply the mathematics they know to solve problems arising
in everyday life, society, and the workplace. In early grades, this might be as simple as writing an
addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to
plan a school event or analyze a problem in the community. By high school, a student might use geometry
to solve a design problem or use a function to describe how one quantity of interest depends on another.
Mathematically proficient students who can apply what they know are comfortable making assumptions
and approximations to simplify a complicated situation, realizing that these may need revision later. They
are able to identify important quantities in a practical situation and map their relationships using such
tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships
mathematically to draw conclusions. They routinely interpret their mathematical results in the context
of the situation and reflect on whether the results make sense, possibly improving the model if
it has not served its purpose. —Common Core State Standards
Math in Focus follows a concrete–pictorial–abstract progression,
introducing concepts first with physical manipulatives or objects, then
moving to pictorial representation, and finally on to abstract symbols.
Less
4
The drawing shows there are 7 equal units after subtracting the 316 points.
Divide the remaining points by 7 to find the number of points that represent
one unit.
11/26/09 7:11:55 PM
Hector
Teddy
4,354
Jim
7 units
4,354 points
1 unit
4,354 7 622 points
3 units
3 622 1,866 points
Teddy scored 1,866 points.
10 + 12 = 22
They make 22 granola bars in all.
Check!
22 – 10 = 12
22 – 12 = 10
C hapter 2: L esson 5
The answer is correct.
10
Kindi_TE1_Chap2.indd 31
31
3/2/11 10:42:44 AM
96
Chapter 4 Using Bar Models: Addition and Subtraction
2.7 Real-World Problems:
Multiplication
Division
103
Student Book A, page 103: BarLesson
modeling
remains
a and
consistent
tool for students as they encounter new situations and need to
make sense of problems.
MiF 5A PB U2 2.4-2.7.indd 103
11/20/09 4:39:29 PM
Common Core State Standards for Mathematics: corestandards.org
© 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
11
Common Core State Standards: Standards for Mathematical Practice
Use appropriate tools strategically.
How Math in Focus Aligns:
Math in Focus helps students explore the different mathematical tools that are available to them.
New concepts are introduced using concrete objects, which help students break down concepts
to develop mastery. They learn how to use these manipulatives to attain a better understanding of
the problem and solve it appropriately. Math in Focus includes representative pictures and icons
as well as thought bubbles that model the thought processes students should use with the tools.
“
Mathematically proficient students consider the available tools when solving a mathematical
problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a
calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software.
Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound
decisions about when each of these tools might be helpful, recognizing both the insight to be gained
and their limitations. For example, mathematically proficient high school students analyze graphs of
functions and solutions generated using a graphing calculator. They detect possible errors by strategically
using estimation and other mathematical knowledge. When making mathematical models, they know
that technology can enable them to visualize the results of varying assumptions, explore consequences,
and compare predictions with data. Mathematically proficient students at various grade levels are able to
identify relevant external mathematical resources, such as digital content located on a website, and
use them to pose or solve problems. They are able to use technological tools to explore and
deepen their understanding of concepts. —Common Core State Standards
s
les o n
3
Measuring in Inches
lesson objectives
Vocabulary
• Use a ruler to measure length to the nearest inch.
le
Grade
2
• Draw parts of lines of given lengths.
Grade 4
inchB,
(in.)page 111
Student Book
arn You can use inches to measure the length of shorter objects.
Le
arn
Student Book B, page 88
Use a protractor to measure an angle in degrees.
Inches are marked on this ruler. There are 12 inches in one foot.
Ananglemeasureisafractionofafull
turn.Anangleismeasuredindegrees.
Forexample,arightanglehasa
measureof90degrees.Youcanwrite
thisas90°.
1 inch
”
Youcanuseaprotractortomeasureanangle.
C
C
What is inch?
50
11 0
80 70 120
60 130
50
15
40
180
0 170
0 16
10 0
20
30
A
100
0
vertex
B
80 90
70
100
60
11 0
120
0
13
14
It is a unit of
length like the
foot. You can use
it to measure
shorter objects.
0 10
20
180 170 16
30
0 1
5 0 40
14
0
5
B
A
center
base line
Step 1 Placethebaselineoftheprotractoron AB .
Step 2 Placethecenterofthebaselineoftheprotractoratthevertexof
theangle.
Examples throughout the Math in Focus curriculum:
sson
le
2
Grade K
Student Book B, Part 2, Chapter 15
Finding the Weight of Things
Step 3 Readtheouter scale. AC passesthroughthe45°mark.
So,themeasureoftheangleis45°.
The inch is a unit of length.
in. stands for inch.
Read 1 in. as one inch.
Inch is used to measure shorter lengths.
Lesson Objectives
• Use a non-standard object to find the weight
of things.
Lesson
lesson 3
• Compare weight using a non-standard object as
Grade
a unit 1
of measurement.
5 Finding Differences in Length
Using Non-standard Units
le
Count and write.
Student Book B, page 13
arn You can measure weight with objects.
Since AB passesthrough
thezeromarkofthe
outerscale,readthe
measureontheouter
scale.
e
g
Co
t pa
ntinued on nex
111
Measuring in Inches
G2B_TB_Ch 13.indd 111
88
12/17/08 5:57:47 PM
Chapter 9 Angles
Student Book B, page 179
Grade 3
glass
The caterpillar is
heavy as 8
longer than the ant.
Le
The glass is as
arn
Use yards to measure length.
.
1 yard
Student Book A, page 47
Grade 5
The weight of the glass is about 8
.
A yardstick is 3 times as long as a 12-inch ruler.
The pencil is
long.
A baseball bat is about 1 yard long.
A doorway is about 1 yard wide.
.
The crayon is
The glass is lighter than the cup.
long.
lesson 2
The pencil is
8
MSpring_StudentBk4_Unit15_B.indd 8
Finding the Weight of Things
G1B_TB_Ch10.indd 13
9/30/08 1:39:34 PM
12/30/08 2:23:02 PM
Using a Calculator
arn
Get to know your calculator.
Turnonyourcalculator.
Followthestepstoenternumbersonyourcalculator.
Toenter12,345,press: 1 2 3 4 5 13
longer than the crayon.
Chapter 15
Less
The height of a doorway is about
2 yards. The length of my garden
is about 10 yards. The distance
from my house to my neighbor’s
house is about 40 yards.
The cup is heavier than the glass.
n
Lesson Objective
• Useacalculatortoadd,subtract,multiply,anddividewholenumbers.
1 ft � 12 in.
3 ft � 12 � 3
� 36 in.
Le
The weight of the cup is about 15
o
The yard is another standard customary unit of length.
It is used for measuring long lengths and short distances.
yd stands for yard.
1 yard (yd) � 3 feet (ft)
1 yard (yd) � 36 inches (in.)
cup
Toclearthedisplayonyourcalculator,press: C
The boy is shorter than 1 yard.
The girl is taller than 1 yard.
This is a
yardstick.
Display
0
12345
0
The heights of both
the boy and the girl
are close to 1 yard.
So, they are about
1 yard tall.
e
g
Co
t pa
ntinued on nex
Lesson 15.1
12
3BTB_Chp15(163-185).indd 179
Measuring Length
179
Hands-On Activity
Common Core State Standards for Mathematics: corestandards.org
WORK IN PAIRS
© 2010. National Governors Association Center for
Best Practices and Council of Chief State School Officers. All rights reserved.
11/19/09 2:36:53 PM
Enterthesenumbersonyourcalculator.Clearthedisplayonyourcalculator
beforeenteringthenextnumber.
13
Common Core State Standards: Standards for Mathematical Practice
6
Attend to precision.
Workmat
“
3
Match me.
How Math in Focus Aligns:
As seen in the Singapore Mathematics Framework, metacognition, or the ability to monitor
one’s own thinking, is key in Singapore math. This is modeled for students throughout
Math in Focus through the use of thought bubbles, journal writing, and prompts to explain
reasoning. When students are taught to monitor their own thinking, they are better able
to attend to precision, as they consistently ask themselves, “does this make sense?” This
questioning requires students to be able to understand and explain their reasoning to others,
as well as catch mistakes early on and identify when incorrect labels or units have been used.
Precise language is an important aspect of Math in Focus. Students attend to the precision of
language with terms like factor, quotient,
arn difference, and capacity.
This one is
the same.
WORK MAT
Mathematically proficient students try to communicate precisely to others. They try to use clear
definitions in discussion with others and in their own reasoning. They state the meaning of the
symbols they choose, including using the equal sign consistently and appropriately. They are careful about
specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem.
They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate
for the problem context. In the elementary grades, students give carefully formulated explanations to
each other. By the time they reach high school they have learned to examine claims and make
explicit use of definitions. —Common Core State Standards
Examples throughout the Math in Focus curriculum:
Le
”
Grade 3
Add417and9,086.
Math Journal
6
er
M
Findthesumof$1,275and$876.
RE AD ING AND WRITING M ATH
ce
ath nt
Activity 3
Explore
Math Focus: Extend the concept of 8.; Extend the concept of same.
Resource: Student Book A, Part 1, Workmat 3
Materials: Counters, 4 per child (2 yellow and 2 green)
Connecting cubes, 4 per child (2 red and 2 blue)
Paper clips, 8 per child (optional)
Classroom Setup: Children work in pairs at the math center.
Begin the day by distributing the counters and
connecting cubes to the children.
7. After modelling the activity, let children work
independently.
8. While children engage in the activity, ask check questions
such as:
•Do both boxes have the same number of things?
•How do you know? Are you sure?
Math Journal
Teacher’s Edition A,
page 16: Math Talk
sections in the Teacher’s
Editions help teachers ask
the right questions, so
students begin expressing
mathematical concepts
accurately.
Display
0
C
Remembertowrite
These are the steps to find the factors of 12.
thecorrectunitin
1275
1275
1 Think of all the numbers that divide 12 exactly.
Grade 3, Student Book B, page 211:
youranswer.
CritiCAL tH ink inG Sk iLLS
12 4 1 5 12 12 4 4 5 3
876
Math
7 6 activities ask students
8 Journal
12 4 2 5 6
12 4 6 5 2
Think of
the Your Thinking
to consider
how they would find an
Put
On
Cap!
12 4 3 5 4
12 4 12 5 1
2
151
answer, requiring them to put their
multiplication tables.
ST EP
12 5 1 3 12
12 5 2 3 6
ST EP
reinforces the idea that the process
Thesumof$1,275and$876is$2,151.
sameiscapacity?
used to get an answer is just
1 Can containers of different shapes have thethat
Write the steps for finding the common
factors of 12 and 15.
Le
arn
Math Talk
5. Repeat the activity three times to allow children to
familiarize themselves with the concept of pairing.
68
C hapter 2: L esson 3
Grade 1
Kindi_TE1_Chap2.indd 16
3/2/11 10:41:35 AM
.
Order the numbers from greatest to least.
1
4 tens 8 ones – 5 ones = 4 tens
ones
2
7 tens 9 ones – 3 tens 2 ones =
tens 7 ones
Why is 95 greater
than
7 883?
4
On YOUr OWn
B:
1 7 Put
5thinking
Go3ontoYour
Workbook
8 Cap!
pages 135–136
 Lesson
6 15.3
9 Measuring
5 9Capacity
211

GradeThedifferenceis10,399.
5
Student Book A, page 65: Getting the correct answer is not always
the final goal. Students are also asked to explain why or why not an
answer is correct, and how to check to make sure. This kind of thinking
helps students establish the importance of precision and the need to
understand how they solve a problem in order to evaluate whether or
not the
answer is correct.
Subtract.
Student Book B, page 194: Thought bubbles throughout the
Student Books prompt students to explain their answers.
,
Press
Grade 4, Student Book A, page 55
let’s Practice
,
B
C
758 – 35 = 732
Is the answer correct?
Explain why you think so.
Show how you would check it.
The greatest number is
as important as the answer itself, and
that students must be able to explain
how they got a solution precisely
in order to ensure their result is
reasonable.
Use your calculator to subtract.
95
Math Journal
.
Why is it the
least number?
A
Re AD ING AND W RIT I NG M AT H
6. Ensure that the children exchange roles.
The least number is
Explain why or why not.
Subtract6,959from17,358.
Grade 2
83
thought process into words. This
2 The factors are 1, 2, 3, 4, 6, and 12.P rO B LeM SOLV
12 inG
5 3 3 4
3. Ask his or her partner to place identical objects on his
or her own workmat.
Encourage children to practice number
names. Ask: What do you have on your workmat? (I have
three counters and five cubes.)
Press
Example
For Advanced Learners For children who are more than
capable of counting up to 8 objects, add paper clips as
part of their material set. Having three different groups of
materials to make up 8 objects will challenge them.
2. Invite one child of each pair to place any number of
objects up to 8 on his or her workmat.
14

Thesumis9,503.
3/2/11 10:21:24 AM
DAy
16
417
You are given an empty container and a cup.
9 container.
0 8
explain how you would find the capacity
of the
Grade 4
Which2 is the least number?
Which is the greatest number?
4.
Display
0
417
9086
9503
C
Student Book A, Part 1, Workmat 3
Kindi_SB1_Ch2.indd 59
Grade K
1.
Press
re Ad inG And WritinG M AtH
Compare the numbers.
5
Use your calculator to add.
Display
0
17358
6959
10399
Findthedifferencebetween1,005poundsand248pounds.
Press
Student Book A,
page 48: Thought bubbles
also provide students with
reminders to consider units
and labels as they solve
problems.
Lesson 2.2 Factors
55
Rememberto
writepoundsin
youranswer.
G4_TB_Ch_02-1.indd 55
C
1005
 2 4 8 12/11/08 4:57:48 PM

Display
0
1005
248
757
Thedifferencebetween1,005poundsand248poundsis757pounds.
48
Chapter 2WholeNumberMultiplicationandDivision
Subtract.
3
– 2 4
9 8
– 5 6
MiF 5A PB U2 2.1-2.3.indd 48
Common Core State Standards for Mathematics: corestandards.org
© 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
15
1/12/09 2:51:20 PM
Le
arn Estimate the area of a figure.
7
Look for and make use of structure.
How Math in Focus Aligns:
The inherent pedagogy of Singapore math allows students to look for, and make use of, structure.
Math in Focus. Concepts in the program start
Place value is one of the underlying principles in
simple and grow in complexity throughout the chapter,
year,Region
and grade.
This of
helps
students
Shaded
Area
Shaded
Region
Approximate Area
master the structure of a given skill, see its utility, and advance to higher levels. Many of the
1squareunit
1squareunit
models in the program, particularly number bonds and bar models, allow students
to easily
1
1
see patterns within concepts and make inferences. As students progress through
grade levels,
squareunit
squareunit
2
2
this level of structure becomes more advanced.
greaterthan1squareunit
1squareunit
2
“
Mathematically proficient students look closely to discern a pattern or structure.
Young students, for example, might notice that three and seven more is the same amount as
seven and three more, or they may sort a collection of shapes according to how many sides the shapes have.
Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the
distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7.
They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing
an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They
can see complicated things, such as some algebraic expressions, as single objects or as being composed
of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a
square and use that to realize that its value cannot be more than 5 for any real numbers x and y.
Grade K
1, 2, 3, 4
1, 2, 3, 4
lessthan1squareunit
1, 2, 3, 4
Examples throughout
the Math in Focus curriculum:
Grade 2
Le
Grade 1
20
–4
7
3. Count the number of cubes with children,
starting at a different colored cube each time.
4. Remove all the cubes from the table.
5. Arrange eight cubes in a circle in this manner –
alternating yellow and blue.
11. Count the number of cubes with children,
starting at a different cube each time.
12. While children engage in the activity, end the
day by asking check questions such as:
•Will it still be the same number if we start from
another cube? Let’s count.
•Is it still the same?
•How else could we arrange eight cubes?
The answer is correct.
+
2 3
4
2 7
8
Hundreds
7
Tens
Ones
710
9
Whatisthepatternwhen
eachnumberismultiplied
Theareaofthecircleisabout12squareunits.
9
by10?
910
158
Chapter 12 AreaandPerimeter
10
×
1 2
1
2
3
4
5 60
10 3 6 
6
1
=23 12
4
5
6
7
8
9
9 3 6
A, page 160: Dot paper
12
and= thought
bubbles help
recognize structure
1 2 3 4 students
5 6
and how it can be used
1
to break down and solve
2
equations. These examples
demonstrate how students
 10 groups of 6
in Grades 2 and 3 learn that
2 1 group of 6
multiplication can be looked
 60 2 6
at in multiple ways.
 54
×
Hundreds
Tens
7
Ones
7
710
7
9
0
9
910
9
10
1
0
1
0
0
1
2
1
2
0
1010
12
1210
Eachdigitmovesone
placetotheleftwhenthe
numberismultipliedby10.
0
160
Chapter 6 Multiplication Tables of 2, 5, and 10
9 3 6 is the same as
10 3 6  6 3 10
subtracting 1 group of 6
 60
from 10 3 6.
36 – 3 = ?
Method 1
52
MS_Gr2A_unit06.indd 160
8/26/08 4:44:48 PM
Count back from the greater number.
Chapter 2WholeNumberMultiplicationandDivision
Student Book A, page 52: In this example, thought
bubbles prompt students to see the pattern when each
number is multiplied by 10. Place-value charts help them
visualize this pattern and solidify their learning.
MiF 5A PB U2 2.1-2.3.indd 52
Teacher’s Edition A, page 15: While participating
36,
,
,
in activities, teachers point out structure. In this
example from Kindergarten, students count out
connecting cubes and learn that no matter which
cube they start on, the total will be the same.
1/12/09 2:51:30 PM
Grade 3, Student Book A, page 154
154
G1B_TB_Ch13(80-103).indd 102
4
1010
1
25
3
4
5
6
7
8
9
10
Chapter 6 Multiplication Tables of 6, 7, 8, and 9
Addition and Subtraction to 40
G3_TB_Ch_06-1(New).indd 154
16
3
1210
Subtract.
Chapter 13
11
Lookattheplace-valuechart.
12
Student Book B, page 102: Grade 1 students learn to use
number bonds to demonstrate the structure of numbers and
Guided
Practice
understand properties.
102
4×2=2×4
These are related
10
multiplication facts .
9. Remove all the cubes from the table.
10. Arrange the eight blue cubes in a circle.
7
1 2 3 4 5 6
2 3 4 5 6
Use1 dot
paper to find the
missing numbers. Grade 2, Student Book
Remember, 7 – 4 = 3
3+4=7
If 27 – 4 = 23, then 23 + 4 should equal 27.
8. Repeat step 3.
2
Grade 5
9 3 6  ?
Check!
1
1
Theareaofthetriangleis
16squareunits.
Start with 10 groups of 6.
6. Repeat step 3.
7. Then, arrange the same eight cubes in a circle in
this manner – four yellow and four blue.
1
2
12
16
Guided Practice
So, 27 – 4 = 23.
2. Arrange eight connecting cubes in a circle in
this manner – red, blue, yellow, green, red, blue,
yellow, green.
=
=
=
=
=
=
Grade 3
Discover
1. Invite children to stand around a table.
1 2 3 4
1
2
3
4
Activity 2
Math Focus: Make a connection between number of objects
and number names up to 8.
Materials: Connecting cubes, 16 (8 blue, 4 yellow, 2 red, and
2 green)
Classroom Setup: Whole class
2
2
2
2
2
2
2×4=8
1 2
7–4=3
20 + 3 = 23
×
×
×
×
×
×
14 15
2
4 1
2
6
13
8
3
4
5
6
10
7
8
9 10 11 12
12
14
16
Lookatthecircle.Countthesquares.
18
20
5
6
arn You can multiply numbers in any order.
4×2=8
27
Lookatthetriangle.Countthesquares.
Student Book B, page 158:
Multiplication Table of 2
Students also learn to identify and
1
× dealing2 with =
use structure
when
geometry. In2this example,
students
×
2
=
must see 3how triangles
relate
×
2 to =
squares in order to determine the
4
×
2
=
area of the object.
5
6
7
8
9
10
0squareunit
2
Grade 4
”
—Common Core State Standards
Useroundingtoestimatetheareasofthefigures.
Common Core State Standards: Standards for Mathematical Practice
C hapter 2: L esson 3
15
12/29/08 4:52:59 PM
12/18/08 8:49:24 AM
Common Core State Standards for Mathematics: corestandards.org
© 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
17
Common Core State Standards: Standards for Mathematical Practice
Look for and express regularity in repeated reasoning.
A strong foundation in place value, combined with modeling
tools such as bar modeling and number bonds, gives students
the foundation they need to look for and express regularity in
repeated reasoning.
“
Mathematically proficient students notice if calculations are repeated, and look both for general
methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11
that they are repeating the same calculations over and over again, and conclude they have a repeating
decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on
the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3.
Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and
(x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they
work to solve a problem, mathematically proficient students maintain oversight of the process,
while attending to the details. They continually evaluate the reasonableness of their
intermediate results. —Common Core State Standards
sson
• Use equal groups and repeated
addition to multiply.
• Make multiplication stories about
pictures.
le
Grade
2
• Make multiplication
sentences.
times
repeated addition
equal
multiplication sentence
group
multiplication stories
le
90
ones
8
8
98 = 9 tens 8 ones
98 = 90 + 8
36
5
Chapter 2
Guided Practice
35
1/6/11 4:07:51 PM
Kindi_SB1_Ch2.indd 36
1/6/11 4:07:54 PM
1
d
als 5,
Use place value to findStudent
the missing
Booknumbers.
A, Part 1, p. 36
18
Tens
ones
•To trace the numeral 6, place the pencil at the top of
the numeral, follow the curve down, and then complete
the loop without lifting the pencil from the paper.
•To trace the numeral 7, place the pencil at the top left
87follow
=
tens line across
ones and then down to the
corner and
the
6ft
o
Less
Le
Length1 width 5 perimeter4 2
5184 2
59ft
?
rectangleA
Length1 width 59ft
61 width 5 9ft
width 592 6
5 3ft
ThewidthofrectangleAis3feet.
30
4
Lesson 12.2 RectanglesandSquares163
How to Multiply
Student Book B, page 163: The consistent use of models
throughout the program allows students to see connections
that allow them to simplify problems and understand how to
better solve them. Here, a bar model demonstrates that the
length + width of a rectangle is 1/2 the perimeter. Since they
are able to see this relationship, students can use it effectively.
G4_TB_Ch_12_Final.indd 163
Step 2 Subtract4onesfromtheresult.572 45 53
× is read as times.
It means to multiply, or to put
all the equal groups together.
127
3/18/09 8:53:49 AM
455
tens
ones
Step 1 Subtract
tensfrom79.792
Step 2 Subtract
onesfromtheresult.
So,792 455
2
5
5
2/24/09 10:50:31 AM
Comparing Numbers
to 10,000,000
n
Less
Guided Practice
Student
Book
A, page
Mental
math
is anyou.
important part of
Subtract
mentally.
Use44:
number
bonds
to help
the Math in Focus curriculum. Students develop a strong sense
45. understand the regularity of grouping by
1 Find792
of place
value, and
tens, hundreds, etc. This allows them to understand shortcuts
like this45one, where numbers can be broken apart by tens and
ones to facilitate mental subtraction.
o
So,872 345 53.
Student Book A, page 126:
Multiplication is introduced as
repeated addition. This helps
students understand how
multiplication works so they can
better apply their learning and
check their work.
Student Book B, page 184:
Early on, students consistently
think of numbers in terms of
place value. This allows them
to visually see the regularity
of grouping by tens, hundreds,
etc., so they understand how
operations work and can
evaluate the reasonableness
of their results.
Find one side of a rectangle given its perimeter and the other side.
TheperimeterofrectangleAis18feet.
Itslengthis6feet.Finditswidth.
Step 1 Subtract3tensfrom87.872 305 57
3 groups of 5 equal 15.
3 fives = 15
lesson 1
arn You can use place value to show numbers to 100.
arn
3453tens4ones
Co
x
ntinued on ne
MS_Gr2ATB_unit05.indd 127
width
length 1 width
to1ofitsperimeter.
Subtract 2-digit numbers mentally using the ‘subtract the
tens, then subtract the ones’ strategy.
ge
t pa
• Show 1
objects up to 100 as tens and ones.
Grade
9
arn
34
There are 15 horses in all.
3 × 5 = 15 is a multiplication sentence.
You read it as three times five is equal to fifteen.
• Use a place-value chart to show numbers up to 100.
length
So,thelength1 widthofarectangleisequal
Lesson Objectives
• Compareandordernumbersto10,000,000.
• Identifyandcompleteanumberpattern.
Grade
5
• Findaruleforanumberpattern.
Le
5 + 5 + 5 = 15
3 × 5 = 15
Youcanusea
modeltoshowthat
theperimeterofthe
rectangleisthesum
ofitstwolengths
andtwowidths.
2
First, count the number of equal groups.
There are 3 groups.
Then, count the number of horses in each group.
Place Value
Tens
Mental Subtraction
Lesson Objective
Grade
3
• Subtract2-digitnumbersmentallywithorwithoutregrouping.
arn You can multiply when you have equal groups.
There are 5 horses in each group.
Use repeated addition or multiply
to find the number of horses.
width
length 1 width
multiply
Le
2
length
Find872 34.
ess o n
Look and lsay.
width
n
There are two ways to find the number of horses.
Student Book A, Part 1: Starting in Kindergarten, students learn that
combining objects into groups can help determine quantity. Here, by
placing the bees in groups of 5, students can use repeated reasoning
to see that 6 lesson
is 5+1 objectives
and 7 is 5+2.
length
perimeter
How many horses are there?
6 6 6
7 7 7
arn Find the perimeter of a rectangle using a formula.
Perimeterofrectangle5 length1width1length1width
5totallengthofallfoursides
Vocabulary
Examples throughout the Math in Focus curriculum:
Trace.
Rectangles and Squares
Lesson Objective
• Solveproblemsinvolvingtheareaandperimeterofsquaresandrectangles.
Le
”
n
How to Multiply
Lesson Objectives
Grade K
Grade 4
Because students are given consistent tools for solving problems,
they have the opportunity to see the similarities in how different
problems are solved and understand efficient means for solving
them. Throughout the program, students see regularity with the
reasoning and patterns between the four key operations. Students
continually evaluate the reasonableness of solutions throughout
the program and the consistent models for solving, checking, and
self-regulation help them validate their answers.
o
1
Operations are taught with place-value materials so students
understand how the standard algorithms work.
Less
le
How Math in Focus Aligns:
arn
Vocabulary
greaterthan(>)
lessthan(<)
Compare numbers by using a place-value chart.
Whichnumberisless,
237,981or500,600?
Student Book A, page 21:
. consistent emphasis on place value
The
MentalSubtraction
allows students to seeLesson
the2.2structure
of
numbers and understand the reasoning
behind number comparisons. In this
example, students look at the placevalue chart and see how with every
number comparison, they can start with
the digits on the left. The visual image of
the place-value chart helps solidify this
process in the students’ minds.
Whencomparingnumbers,lookatthevalueofeachdigit
fromlefttoright.Remember,‘>’means‘greater than’
and‘<’means‘less than’.
Hundred
Ten
Thousands Hundreds
Thousands Thousands
Tens
Ones
2
3
7
9
8
1
5
0
0
6
0
0
Comparethevaluesofthedigitsstartingfromtheleft.
2hundredthousandsislessthan5hundredthousands.
So,237,981islessthan500,600.
Le
8
arn
237,981
<
500,600
Compare numbers greater than 1,000,000.
Whichnumberisless,3,506,017or5,306,007?
Hundred
Ten
Millions Thousands
Thousands
Hundreds
Tens
Onescorestandards.org
Common
Core State
Standards for
Mathematics:
Thousands
© 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
3
5
0
6
0
1
7
5
3
0
6
0
0
7
19
Notes
View full correlations to the Common Core
State Standards: Standards for Mathematical
Content and Mathematical Practice at
hmheducation.com/singaporemath
20
Introducing Math in Focus
Courses 1–3 for Grades 6–8.
Fully supports the Common Core State Standards!
To learn more, or to review
a virtual sample, visit:
hmheducation.com/singaporemath
800.289.4490
Math in Focus® is a registered trademark of Times Publishing Limited.
© Houghton Mifflin Harcourt Publishing Company. All rights reserved. Printed in the U.S.A. 07/12 MS52834