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 is a bipartisan, nonprofit educational reform organization that partnered with NGA and CCSSO on the Common Core State Standards Initiative.
“
”
—Achieve
*(aChieve.Org/CCSSandSingapOre)
1
Common Core State Standards Overview ...2
Examples of support for the Common Core State Standards:
Standards for Mathematical Practice
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
Table of Contents
1
2
3
4
5
6
7
8
—Achieve
*(aChieve.Org/CCSSandSingapOre)
92_MS52834_MIF_CCbrochure_FilesOnly.indd 1 7/27/12 9:53 AM2
Teaching the
Common Core State Standards
to mastery with
Math in Focus
®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.
TeaCh TO MaSTery
Common Core State Standards:In Grade 2, instructional time should focus
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) 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)
CUrriCULUM MUST be FOCUSed and COherenT
Common Core State Standards:For over a decade, research studies of
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)
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.
Common Core State Standards Skills Traces in the Teacher’s Edition help teachers understand how concepts progress through the grade levels.
*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.
The Common Core State Standards for Mathematicsis an initiative designed to implement more focused grade-level 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.”
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:
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)
33
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)
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.
Organize COnTenT by big ideaS, SUCh aS pLaCe vaLUe
Common Core State Standards:These Standards endeavor to follow such a
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)
98 Chapter 13 Addition and Subtraction to 40
14 + 18 = ?
You can use place-value charts to add numbers with regrouping.
Learn
Step 1 Add the ones.
4 ones + 8 ones = 12 ones
Tens Ones
32
So, 14 + 18 = 32.
Step 2 Add the tens.
1 ten + 1 ten + 1 ten = 3 tens
Tens Ones
14
18
Regroup the ones. 12 ones = 1 ten 2 ones
1 1 Tens Ones 1 4 + 1 8 2 Tens Ones 1 4 + 1 8 3 2 14 = 1 ten 4 ones 18 = 1 ten 8 ones G1B_TB_Ch13(80-103).indd 98 12/31/08 9:17:22 AM
Look at the place-value chart.
Thousands Hundreds Tens Ones 70 70 10 160 160 10 1,800 1,800 10
Thousands Hundreds Tens Ones
70 7 0 70 10 7 160 1 6 0 160 10 1 6 1,800 1 8 0 0 1,800 10 1 8 0
Each digit moves one place to the right when the number is divided by 10. What is the pattern when each number is divided by 10?
Lesson 2.4 Dividing by Tens, Hundreds, or Thousands 71
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.
Full correlations are available at
hmheducation.com/singaporemath
These visual representations are carried 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.
See how
Math in Focus
supports the
Common Core
State Standards.
92_MS52834_MIF_CCbrochure_FilesOnly.indd 3 7/27/12 9:54 AM4
Make sense of problems and persevere in solving them.
1
Grade K Grade 1 Grade 2
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
exaMpLeS ThrOUghOUT The MAth in Focus CUrriCULUM:
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.
lesson 1 Making Number Bonds 37
CRITICAl THINKING sKIlls
Put On Your Thinking Cap!
PRoBleM solVING
Find the number of beads. Use number bonds to help you.
1 There are 6 beads under the two cups.
6
2 There are 8 beads under the two cups.
8
3 There are 10 beads under the three cups.
10
Chapter 2 Number Bonds 37 oN YoUR oWN
Go to Workbook A: Put on Your Thinking Cap!
pages 31–32
Gr1 TB A_Ch 2.indd 37 8/19/08 4:34:24 PM
PRoBleM solVInG
solve.
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 counters in the bag.
CRITICAl THInKInG sKIlls
Put On Your Thinking Cap!
on YoUR oWn
Go to Workbook B: Put on Your Thinking Cap!
pages 173–174 lesson 3 Real-World Problems: Measurement and Money 217
G2B_TB_Ch 16.indd 217 12/17/08 6:17:47 PM
Kindergarten Book A, page 58: Kindergarten
students are introduced to problem solving in a visual way, where they learn to evaluate a situation and determine the steps they need to take to get an answer.
Student Book A, page 37: Students use number
bonds to determine unknowns, promoting early algebraic thinking through relevant problem solving.
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.
Chapter 2: Lesson 6 37
Student Book A, Part 1, p. 58
58 Chapter 2
Circle, count, and write. 1 There is cheese for 6 mice.
How many mice will be hungry?
2 3 boys have coats.
How many boys will be cold?
3 There are 6 egg holders.
How many eggs are needed to fi ll all the egg holders?
Kindi_SB1_Ch2.indd 58 1/6/11 4:09:10 PM
5. Children count how many objects are needed to complete a set.
6. For the fi rst 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 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 a coat. Next, have them count the number of boys they have circled and write this answer in the blank provided. 8. For the third task, help children understand that eggs
are needed to fi ll only the empty egg holders. Then, have children circle the empty egg holders. Next, end the day by having them count the number of egg holders they have circled and write this answer in the blank provided.
Notes
Kindi_TE1_Chap2.indd 37 3/2/11 10:43:00 AM 4“
”
92_MS52834_MIF_CCbrochure_FilesOnly.indd 4 7/27/12 9:54 AMCommon 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.
Make sense of problems and persevere in solving them.
5
At
tit
udes
Metacognition
Pr
ocesses
Concepts
Skills
Mathematical
Problem
Solving
Numerical calculation Algebraic manipulation Spatial visualization Data analysis Measurement Use of mathematical toolsEstimation Beliefs Interest Appreciation Confidence Perseverance
Numerical, Algebraic, Geometrical Statistical, Probabilistic, Analytical
From the Singapore Ministry of Education
Reasoning, communication and connections Thinking skills and heuristics Applications and modeling Monitoring of
one’s own thinking Self-regulation
of learning
Singapore Mathematics
Framework
Grade 5
hOw MAth in Focus aLignS:
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.
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.
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.
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.
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.
Grade 3
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 fi nd the digits covered by the ink.
The sum of all the ones is 17.
The ones digit of the fi rst number is the greatest 1-digit number. The digit in the tens place of the second number is one more than the digit in the tens place of the fi rst number.
The tens digit of the third number is 4 less than the tens digit of the second number.
PROBLEM SOLVING
CLUES
ON YOUR OWN Go to Workbook A:
Put on Your Thinking Cap! pages 17–18
32 Chapter 1 Numbers to 10,000
G3A_TB_Ch_01.indd 32 12/17/08 6:51:10 PM
Grade 4
Here are 2 equal bars to show that both of them had an equal portion of a graham cracker in the end.
Work backward to fi nd the fraction of the graham cracker Minah had at fi rst.
CRITICAL THINKING SKILLS
Put On Your Thinking Cap!
2 Jessie had a whole graham cracker. Minah had only part of another graham cracker. Jessie gave 1
4 of her graham cracker to Minah.
In the end, both girls had the same fractional part of a graham cracker.
What fraction of a graham cracker did Minah have at fi rst
PROBLEM SOLVING
Jessie Minah
ON YOUR OWN Go to Workbook A:
Put on Your Thinking Cap! pages 169–170
Chapter 6 Fractions and Mixed Numbers 269
G4_TB_Ch6.indd 269 12/11/08 4:54:21 PM
CRITICAL THINKING SKILLS
Put On Your Thinking Cap! Solve these problems.
1 The number in the square is the product of the numbers in the two circles next to it. Find the numbers in the circles. 2 Simone bought a total of 10 birthday hats and noisemakers. Each birthday hat cost $1.50 and each noisemaker cost $2.50. The noisemakers cost $13 more than the birthday hats. How many of each item did she buy? PROBLEM SOLVING 18 2.7 5.4 Chapter 9 Multiplying and Dividing Decimals 81 ON YOUR OWN Go to Workbook B:
Put on Your Thinking Cap! pages 51 – 54
6
reason abstractly and quantitatively.
2
exaMpLeS ThrOUghOUT The MAth in Focus CUrriCULUM:
Teacher’s Edition A, Chapter 2: Math in
Focus Kindergarten
students participate in Discover activities, like the one pictured here from Chapter 2. These Discover activities introduce concepts using concrete, hands-on activities. This helps students make sense of quantities and numbers so they truly understand what they mean.
Grade K
Chapter 2: Lesson 1 3
12. Tell the train engineer to raise his or her fingers to indicate how many train engines he or she wants.
Best Practices Ensure that this child does not raise more than six fingers.
13. Explain to the class that if, for example, the train engineer raises four fingers, only engines 1, 2, 3, and 4 are allowed to move. Engines 5 and 6 should crouch down.
14. Allow children to play three rounds of the game. 15. While children engage in the activity, end the day by
asking check questions such as:
• How do you know how many children should move? • Are you sure?
Activity 2
Discover 3 54 1
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
1. 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. 3. Write the numerals 1 to 5 on the board. Ask:
What number is this?
4. While children engage in the activity, ask check questions such as:
• Are you sure?
• Can it be this number instead?
5. Write the numeral 6 on the board. Explain to children that this numeral represents the number six.
6. Have children count 1 to 6 aloud and count their cubes to be sure they each have 6.
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.
8. Then, have children hold up their towers and check each other’s towers.
9. End the day by playing the game for 8–10 rounds.
DAy 2
Kindi_TE1_Chap2.indd 3 3/2/11 10:41:01 AM
Student Book A, page 88: After a lesson on subtraction up to 1,000,
students need to have a deep enough understanding in order to recognize situations where the “-” sign doesn’t necessarily require taking away to solve. Even though these look like simple subtraction problems, students need to understand how each number is functioning in order to fill in the green squares.
Grade 2
CRITICAl THINKING SKIllS
Put On Your Thinking Cap!
Find the missing numbers in each box.
Answer the question.
4 Brian has a machine that changes numbers. He puts one number into the machine and a different 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.
– 4 4 4 4 4 4 2 – 1 8 8 1 6 5 4 – 2 4 4 2 0 3 PROBleM SOlVING 88 Chapter 3 Subtraction up to 1,000 MS_Gr2A_unit03.indd 88 8/26/08 12:23:40 PM
Student Book A, page 74: Students model subtraction concretely by
starting with physical objects. They then move on to the pictorial stage, using number bonds to represent the action of taking away. Finally, they write subtraction symbolically. This concrete–pictorial–abstract progression helps students make sense of quantities.
Grade 1
74 Chapter 4 Subtraction Facts to 10
You can use number bonds to help you subtract.
Learn
How many beanbags are on the floor? 9 − 5 = ?
There are 4 beanbags on the floor.
Use number bonds to subtract.
13 How many yellow beans are there?
part
part whole
There is yellow bean. Guided Practice 9 − 5 = 4 10 − 9 = 5 9 4 part part whole Gr1 TB A_Ch 4.indd 74 8/19/08 4:37:42 PM
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
“
”
7
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.
reason abstractly and quantitatively.
Grade 4 Grade 3
exaMpLeS ThrOUghOUT The MAth in Focus CUrriCULUM:
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.
Student Book A, page 42: Math in Focus teaches students ways to break
apart numbers to compute mentally. This requires students to develop an understanding of the meaning of quantities. In this example, students learn that to add 34 + 48 is the same as adding 50 +34 and then subtracting 2. A thought bubble reinforces this reasoning ability by asking students why they would manipulate the numbers like this.
Student Book A, page 204: To solve problems like this one, students must
be able to take their understanding of mean and consider how it is used to find weight. This goes beyond simply computing to using abstract and quantitative reasoning to consider how it relates to other operations.
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.
Grade 5
Add 2-digit numbers mentally using the ‘add the tens, then subtract the extra ones’ strategy.
Find 34 � 48.
50 48
2
Step 1 Add 50 to 34. 34 � 50 � 84 Step 2 Subtract 2 from the result. 84 � 2 � 82 So, 34 � 48 � 82.
Learn
Add mentally. Use number bonds to help you.
2 Find 35 � 57.
60 57
3
Step 1 Add to 35. 35 � � Step 2 Subtract from the result. � � So, 35 � 57 � .
Guided Practice
Do you know why you add 50 and then subtract 2?
42 Chapter 2 Mental Math and Estimation
G3A_TB_Ch2.indd 42 3/12/10 2:22:01 PM
Real-World Problems:
Data and Probability
Lesson Objective
• Solve real-world problems involving probability and measures of central tendency.
Solve problems using the mean.
Learn
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?
? 12 lb
2 � 16 lb � 32 lb
Total weight of the 2 tables � 16 � 2 � 32 lb Weight of the other table � 32 � 12
� 20 lb The weight of the other table is 20 pounds.
Solve. Show your work.
1 Mr. Saco bought chicken, fi sh, and shrimp at a market. The mean weight of the 3 items was 7 pounds. The weight of chicken was 8 pounds and the weight of fi sh was 4 pounds.
What was the weight of shrimp that Mr. Saco bought?
Total weight of chicken, fi sh, and shrimp Mr. Saco bought � �
� pounds
Weight of chicken and fi sh Mr. Saco bought � � � pounds Guided Practice Le sson
204 Chapter 5 Data and Probability
G4_TB_Ch5-2.indd 204 12/11/08 4:43:28 PM
CRITICAL THINKING SKILLS
Put On Your Thinking Cap!
The 9 key on the calculator is not working. PROBLEM SOLVING
Go to Workbook A:
Put on Your Thinking Cap, pages 75– 78 ON YOUR OWN 79 groups groups 1,234 1,234 79 I can rewrite 79 as 1 or 1. 1,234 79 (1,234 ) 1,234 79 (1,234 ) 1,234 1,234 1,234 1,234 1,234 1,234 79 1 1 Explain how you can still use the calculator to fi nd 1,234 79 in two ways.
Chapter 2 Whole Number Multiplication and Division 109
MiF 5A PB U2 2.4-2.7.indd 109 1/12/09 2:56:52 PM
8
Construct viable arguments and critique
the reasoning of others.
Grade 1 Grade K
Grade 2
exaMpLeS ThrOUghOUT The MAth in Focus CUrriCULUM:
104 Chapter 4 Using Bar Models: Addition and Subtraction
Guided Practice
Solve.
Use bar models to help you.
1
Carlos has 9 stickers.
His cousin gives him 3 stickers.
His sister buys him another 5 stickers.
How many stickers does Carlos have in all?
+
+
=
Carlos has
stickers in all.
9 ? 3 5
Check!
3 + 5 = 8 + 8 = Is the answer correct?MS_Gr2A_unit04.indd 104 8/26/08 4:38:56 PM
Tania completes this number pattern. 32, 33, 34, 35, 36, 37, 38, 39 She explains how she found each number in the pattern.
How do you fi nd the missing numbers in this pattern? 40, 30, , 10,
ReADInG AnD WRITInG MATH
Math Journal
I added 1 to 32 to get 33. I added 1 to 33 to get 34. I just have to add 1 to get the next number.
32 + 1 = 33 33 + 1 = 34
33 is 1 more than 32. 34 is 1 more than 33.
In the pattern, is the next number more or less?
74 Chapter 12 Numbers to 40
G1B_TB_Ch12.indd 74 12/29/08 4:25:54 PM
Student Book B, page 74: Students
learn how to explain their reasoning through guided math journal exercises and modeled thought processes.
Student Book A, page 104: Throughout
the program, students are taught to check their answers and make sure their solutions are reasonable. Look for the “Check!” icon throughout the Student Books.
Teacher’s Edition A, page 4: Check
questions throughout the Kindergarten Teacher’s Editions provide opportunities for students to explain their thinking and construct viable arguments.
4 Chapter 2: Lesson 1
Activity 4
Apply
Math Focus: Apply the concept of counting up to 6 objects.
Resource: Student Book A, Part 1, pp. 26 –29
Materials: Numeral 1 (TR01), Numeral 2 (TR02), Numeral 3 (TR03), Numeral 4 (TR05), Numeral 5 (TR06), and Numeral 6 (TR07) Paper, 1 sheet per child (optional)
Classroom Setup: Children work independently.
1. Encourage children to return to their places and open their Student Books to page 26.
2. Children draw a line connecting the two boxes with the same number of wheels.
3. Remind children to be sure they have counted each wheel and say each counting word in order as they point to each wheel in a set. Make sure children understand that the last number they say is the total number in the set.
4. While children engage in the activity, ask check questions such as:
• Why have you chosen to match these two pictures? • How do you know they match? Are you sure? • Will this do instead? Why? or Why not?
Student Book A, Part 1, p. 26
26
Chapter
Numbers to 9
Lesson 1 All About 6
Match.
2
Chapter 2 Kindi_SB1_Ch2.indd 26 1/6/11 4:07:19 PM Activity 3Explore
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
math center.
1. Begin the day by preparing the cards for
the activity.
2. Distribute materials to the children.
3. Help them read the words ‘same’, ‘different’, and ‘color’.
4. One child shuffles the cards and places them face down. He or she then chooses a card and reads the task to the other group members. For example: 4, different, color.
5. The other group members gather materials as per the task.
6. Math Talk Encourage target vocabulary by
asking group members why they have chosen their materials. Elicit replies such as: I chose 4 connecting cubes. 2 are red and 2 are yellow. Red and yellow are different colors.
7. After introducing the activity, let children work independently.
8. While children engage in the activity, ask check questions such as:
• How did you decide which cubes to use? • Can I use this combination of cubes instead?
DAy
3
2, same, color
Math center
Kindi_TE1_Chap2.indd 4 3/2/11 10:41:03 AM
3
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
—Common Core State Standards
“
”
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. 9
Construct viable arguments and critique
the reasoning of others.
Grade 3
Grade 4
Grade 5 READING AND WRITING MATH
Math Journal
List the steps to arrange the numbers in order from least to greatest.
Arranged from least to greatest:
List the steps to get your answer.
Example
1,984 2,084 1,884 STEP
1 I compare the thousands. STEP
2 I can see that 2,084 is the greatest. STEP
3 I compare the hundreds. STEP
4 I can see that 1,884 is the least number. Arranged from least to greatest: 1,884 1,984 2,084 least
9,049 9,654 8,785
Lesson 1.3 Comparing and Ordering Numbers 31
G3A_TB_Ch_01.indd 31 12/17/08 6:51:08 PM
Some problems require two steps to solve.
The Fairfi eld 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 fl oor. Find the cost of carpeting the library fully if a 1-square-yard carpet tile costs $16.
First, fi nd the fl oor area of the library. Area � length � width
� 36 � 21 � 756 yd2
The fl oor area of the library is 756 square yards. Then, fi nd the cost of carpeting.
Cost of carpeting � area � cost of 1 yd2
� 756 � $16 � $12,096
It will cost $12,096 to carpet the library fully.
Learn
Estimate the answer. 36 rounds to 40. 21 rounds to 20. 40 � 20 � 800 756 is a reasonable answer.
Estimate to check if the answer is reasonable.
Guided Practice
Solve. Show your work.
3 Rob fi lls 250-gallon fuel tanks at $3 per gallon at a gas station. How much money does he need to pay for fi lling 9 such tanks? Total amount of fuel � 9 � 250 �
Cost of fuel � � $3 � $ He needs to pay $ .
98 Chapter 2 Whole Number Multiplication and Division
MiF 5A PB U2 2.4-2.7.indd 98 12/8/09 3:12:24 PM
ReaDiNG aND WRiTiNG MaTH
Math Journal
Both Andy and Rita think that 0.23 is greater than 0.3. Do you agree? Why or why not? Explain your answer. 23 is greater than 3, so 0.23 is greater than 0.3. 23 tenths is greater than 3 tenths, so 0.23 is greater than 0.3. Lesson 7.3 Comparing Decimals 33hOw MAth in Focus aLignS:
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.
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.
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.
Student Book A, page 98: Throughout Math in Focus, students are
asked to estimate in order to evaluate whether or not their answers are reasonable. This develops the metacognitive skills that are highlighted in the Singapore mathematics framework, and promotes the ultimate goal of developing effective problem solvers.
10
4
Model with mathematics.
Grade 2 Grade 1 Grade K
exaMpLeS ThrOUghOUT The MAth in Focus CUrriCULUM:
Chapter 2: Lesson 5 31
Activity 3
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.
DAy 2
Activity 4
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
Classroom Setup: Children work in small groups with teacher direction.
1. Begin the day by distributing materials to the children.
2. Ask children to lay out the numeral card ‘5’. 3. Ask children to then lay out the same number
of counters and the dot card with the corresponding number.
4. Math Talk Give groups various instructions to practice the concepts of one more, one less, and
the same number, such as:
• I want the numeral to be one more. • I want one less counter.
• I want the dots to be the same number as the counters.
5. While children engage in the activity, ask check questions such as:
• How can you tell they match? • What number is one less than 2? • What number is one less than 1? 6. Repeat steps 2 to 5 using different numbers.
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.
DAy 3
Kindi_TE1_Chap2.indd 31 3/2/11 10:42:44 AM
30 Chapter 2 Number Bonds
Making Number Bonds
lessoN
1
Lesson Objectives • Use connecting cubes or a math balance to find number bonds. • Find different number bonds for numbers to 10. Vocabulary part whole number bondYou can make number bonds with .
learn
You can use a number train to make number bonds.
Sam put into two parts.
How many are in each part?
3 and 1 make 4.
This picture shows a number bond. 3 4 1 part whole part part part Gr1 TB A_Ch 2.indd 30 8/19/08 4:32:56 PM Student Book A, page 30: Students
use number bonds to model part-part-whole relationships.
Student Book A, page 96: Bar modeling
is introduced in Grade 2 and continued throughout the
Math in Focus
curriculum.
96 Chapter 4 Using Bar Models: Addition and Subtraction
Using Part-Part-Whole in
Addition and Subtraction
LESSON
1
Lesson Objectives
• Use bar models to solve addition and subtraction problems. • Apply the inverse operations of addition and subtraction.
LearnYou can use bar models to help you add.
Mandy makes 10 granola bars. Aida makes 12 granola bars.
How many granola bars do they make in all?
10 + 12 = 22
They make 22 granola bars in all.
10 12
?
Check!
22 – 10 = 12 22 – 12 = 10 The answer is correct.
MS_Gr2A_unit04.indd 96 8/26/08 4:37:43 PM
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.
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
“
”
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
Model with mathematics.
Grade 5
Grade 4
Grade 3
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?
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
The drawing shows there are 7 equal units after subtracting the 316 points. Divide the remaining points by 7 to fi nd the number of points that represent one unit.
7 units 4,354 points 1 unit 4,354 � 7 � 622 points 3 units 3 � 622 � 1,866 points Teddy scored 1,866 points.
Learn 4,670 316 Hector Teddy Jim 4,354 Hector Teddy Jim
Lesson 2.7 Real-World Problems: Multiplication and Division 103
MiF 5A PB U2 2.4-2.7.indd 103 11/20/09 4:39:29 PM
hOw MAth in Focus aLignS:
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.
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.
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.
Use bar models and addition or subtraction to solve 2-step real-world problems.
Nancy and Sue sold tickets for a concert. Nancy sold 3,450 tickets.
Sue sold 1,286 fewer tickets than Nancy.
a How many tickets did Sue sell?
b How many tickets did they sell in all?
a 3,450 � 1,286 � 2,164 Sue sold 2,164 concert tickets.
b 3,450 � 2,164 � 5,614 They sold 5,614 concert tickets in all.
Real-World Problems:
Addition and Subtraction
Less on Lesson Objective
• Use bar models to solve 2-step real-world problems on addition and subtraction.
Learn
Vocabulary
sum difference bar model
Sue sold fewer tickets than Nancy. So, use a comparison model.
Check! 2,164 � 1,286 � 3,450 5,614 � 2,164 � 3,450 The answers are correct.
a? 3,450 tickets Nancy Sue 1,286 tickets b?
122 Chapter 5 Using Bar Models: Addition and Subtraction
G3A_TB_Ch5.indd 122 11/26/09 7:11:55 PM
Modeling Division
with Regrouping
Less on Lesson Objectives • Model regrouping in division. • Divide a 3-digit number by a 1-digit number with regrouping. Vocabulary regroup Step 1 Divide the hundreds by 3. 5 hundreds 3 1 hundred with 2 hundreds left overModel division with regrouping in hundreds, tens, and ones.
Hundreds Tens Ones
Hundreds Tens Ones
Regroup the hundreds. 2 hundreds 20 tens Add the tens. 20 tens 2 tens 22 tens Learn 1 5 2 5 3 0 0 2 3 1 5 2 5 3 0 0 2 2 5 3 A farmer sells his crops to 3 restaurants. He divides 525 heads of lettuce equally among the 3 restaurants. How many heads of lettuce does each restaurant receive? 525 3 ? 96 Chapter 3 Whole Number Multiplication and Division G4_TB_Ch_03-1new.indd 96 12/11/08 5:11:20 PM
Student Book A, page 122: As students tackle increasingly complex
problems, they can use bar models to help them visualize, understand, and solve.
Student Book A, page 96: Place-value charts are also used
consistently throughout the Math in Focus program. They help students visualize and understand numbers so that they understand why the standard algorithms work and can apply them in non-routine situations.
Student Book A, page 103: Bar modeling remains a consistent
tool for students as they encounter new situations and need to make sense of problems.
12
Use appropriate tools strategically.
Grade 1
exaMpLeS ThrOUghOUT The MAth in Focus CUrriCULUM:
lesson 2 Finding the Weight of Things 13
Finding the Weight of Things
lesson
2
Lesson Objectives
• Use a non-standard object to fi nd the weight of things.
• Compare weight using a non-standard object as a unit of measurement.
You can measure weight with objects.
learn
The glass is as
heavy as 8 .
The weight of the glass is about 8 .
The weight of the cup is about 15 .
The cup is heavier than the glass. The glass is lighter than the cup.
glass
cup
G1B_TB_Ch10.indd 13 12/30/08 2:23:02 PM
Student Book B, Part 2, Chapter 15
Student Book B, page 13
Grade K
8 Chapter 15
Lesson5 Finding Differences in Length Using Non-standard Units
Count and write.
The pencil is long.
The crayon is long.
The pencil is longer than the crayon.
The caterpillar is longer than the ant.
MSpring_StudentBk4_Unit15_B.indd 8 9/30/08 1:39:34 PM
5
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
“
”
13
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.
Use appropriate tools strategically.
Student Book B, page 88
Grade 3
Grade 4
Grade 5 Grade 2
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.
Student Book B, page 111 lesson
3
Measuring in Inches
lesson objectives
• Use a ruler to measure length to the nearest inch. • Draw parts of lines of given lengths.
You can use inches to measure the length of shorter objects.
learn
Vocabulary
inch (in.)
Inches are marked on this ruler. There are 12 inches in one foot.
1 inch
It is a unit of length like the foot. You can use it to measure shorter objects. What is inch?
Continuedon next page 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 3 Measuring in Inches 111
G2B_TB_Ch 13.indd 111 12/17/08 5:57:47 PM
Student Book B, page 179
1 yard
Use yards to measure length.
A yardstick is 3 times as long as a 12-inch ruler. 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.) A baseball bat is about 1 yard long. A doorway is about 1 yard wide.
The boy is shorter than 1 yard. The girl is taller than 1 yard.
1 ft � 12 in. 3 ft � 12 � 3
� 36 in. 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.
This is a yardstick.
Continuedon next page The heights of both the boy and the girl are close to 1 yard. So, they are about 1 yard tall. Learn
Lesson 15.1 Measuring Length 179
3BTB_Chp15(163-185).indd 179 11/19/09 2:36:53 PM Learn An angle measure is a fraction of a full turn. An angle is measured in degrees. For example, a right angle has a measure of 90 degrees. You can write this as 90°. Since AB passes through the zero mark of the outer scale, read the measure on the outer scale.
Use a protractor to measure an angle in degrees.
You can use a protractor to measure an angle.
Step 1 Place the base line of the protractor on AB .
Step 2 Place the center of the base line of the protractor at the vertex of
the angle.
Step 3 Read the outer scale. AC passes through the 45° mark.
So, the measure of the angle is 45°. 88 Chapter 9 Angles vertex 90100 110120 130 140 150 16 0 170 180 80 70 60 50 40 30 20 10 0 0 10 20 30 40 50 60 70 80 100 110 120 130 140 150 160 170 18 0 base line center B A C B A C
Student Book A, page 47
Using a Calculator
Get to know your calculator. Turn on your calculator. Follow the steps to enter numbers on your calculator. To enter 12,345, press: 1 2 3 4 5 To clear the display on your calculator, press: C Le sson Lesson Objective • Use a calculator to add, subtract, multiply, and divide whole numbers. Learn Display 0 1 2 3 4 5 0 Hands-On Activity Enter these numbers on your calculator. Clear the display on your calculator before entering the next number. 1 735 2 9,038 3 23,104 4 505,602 Check each number on your calculator with your partner’s number. Do both calculators show the same number on the display screen? WORK IN PAIRS Lesson 2.1 Using a Calculator 47 MiF 5A PB U2 2.1-2.3.indd 47 1/12/09 2:51:16 PM 92_MS52834_MIF_CCbrochure_FilesOnly.indd 13 7/27/12 9:54 AM
14
attend to precision.
Grade K
exaMpLeS ThrOUghOUT The MAth in Focus CUrriCULUM:
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.
16 Chapter 2: Lesson 3
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?
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.
Student Book A, Part 1, Workmat 3
WORKMAT
WORKMAT
WORKMAT
Workmat 3 Match me. This one is the same.
Kindi_SB1_Ch2.indd 59 3/2/11 10:21:24 AM
DAy 2 Math center
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.
1. Begin the day by distributing the counters and connecting cubes to the children.
2. Invite one child of each pair to place any number of objects up to 8 on his or her workmat.
3. Ask his or her partner to place identical objects on his or her own workmat.
4. Math Talk Encourage children to practice number names. Ask: What do you have on your workmat? (I have three counters and five cubes.)
5. Repeat the activity three times to allow children to familiarize themselves with the concept of pairing. 6. Ensure that the children exchange roles.
Kindi_TE1_Chap2.indd 16 3/2/11 10:41:35 AM
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.
Grade 2
Student Book B, page 194: Thought bubbles throughout the
Student Books prompt students to explain their answers.
Grade 1
194
Chapter 16 Numbers to 100Compare the numbers.
5
Which is the least number?
Which is the greatest number?
68
83
95
The least number is
.
Why is it the
least number?
The greatest number is
.
Order the numbers from greatest to least.
,
,
greatest
Why is 95 greater
than 83?
G1B_TB_Ch16(186-203).indd 194 12/29/08 5:22:44 PM 758 – 35 = 732 Is the answer correct? Explain why you think so. Show how you would check it.ReADING AND WRITING MATH
Math Journal
let’s Practice
Subtract.
1 4 tens 8 ones – 5 ones = 4 tens ones 2 7 tens 9 ones – 3 tens 2 ones = tens 7 ones
Subtract. 3 7 8 – 2 4 4 9 8 – 5 6 Subtract. 5 38 – 15 = 6 77 – 24 = 7 97 – 3 =
lesson 1 Subtraction Without Regrouping 65
MS_Gr2A_unit03.indd 65 8/26/08 12:22:31 PM
6
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
“
”
15
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.
attend to precision.
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, difference, and capacity.
Grade 3, Student Book B, page 211:
Math Journal activities ask students to consider how they would find an answer, requiring them to put their thought process into words. This reinforces the idea that the process that is used to get an answer is just 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.
Student Book A, page 48: Thought bubbles
also provide students with reminders to consider units and labels as they solve problems.
Grade 4
Grade 3
Grade 5
reAdinG And WritinG MAtH
Math Journal
You are given an empty container and a cup.
explain how you would find the capacity of the container.
CritiCAL tHinkinG SkiLLS
Put On Your Thinking Cap!
PrOBLeM SOLVinG
1 Can containers of different shapes have the same capacity? Explain why or why not.
A B
On YOUr OWn
Go to Workbook B: Put on Your thinking Cap!
pages 135–136
Lesson 15.3 Measuring Capacity 211
READING AND WRITING MATH
Math Journal
ExampleThese are the steps to fi nd the factors of 12.
STEP
1 Think of all the numbers that divide 12 exactly. 12 4 1 5 12 12 4 4 5 3
12 4 2 5 6 12 4 6 5 2
12 4 3 5 4 12 4 12 5 1
STEP
2 The factors are 1, 2, 3, 4, 6, and 12.
Write the steps for fi nding the common factors of 12 and 15. Think of the multiplication tables. 12 5 1 3 12 12 5 2 3 6 12 5 3 3 4 Lesson 2.2 Factors 55 G4_TB_Ch_02-1.indd 55 12/11/08 4:57:48 PM
Use your calculator to add.
Add 417 and 9,086.
The sum is 9,503.
Find the sum of $1,275 and $876.
The sum of $1,275 and $876 is $2,151.
LearnPress
C 12
7
5
8
7
6
Display
0
1 2 7 5
8 7 6
2 1 5 1
Use your calculator to subtract.
Subtract 6,959 from 17,358.
The difference is 10,399.
Find the difference between 1,005 pounds and 248 pounds.
The difference between 1,005 pounds and 248 pounds is 757 pounds.
LearnPress
C 17
3
5
8
6
9
5
9
Display
0
1 7 3 5 8
6 9 5 9
1 0 3 9 9
Remember to write the correct unit in your answer.Press
C 10
0
5
2
4
8
Display
0
1 0 0 5
2 4 8
7 5 7
Press
C 41
7
9
0
8
6
Display
0
4 1 7
9 0 8 6
9 5 0 3
Remember to write pounds in your answer. 48 Chapter 2 Whole Number Multiplication and Division MiF 5A PB U2 2.1-2.3.indd 48 1/12/09 2:51:20 PMGrade 4, Student Book A, page 55
16
Look for and make use of structure.
exaMpLeS ThrOUghOUT
The MAth in Focus CUrriCULUM:
Student Book B, page 102: Grade 1 students learn to use
number bonds to demonstrate the structure of numbers and understand properties.
Teacher’s Edition A, page 15: While participating
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.
Grade 1 Grade K
Chapter 2: Lesson 3 15
Activity 2
Discover
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
1. Invite children to stand around a table. 2. Arrange eight connecting cubes in a circle in
this manner – red, blue, yellow, green, red, blue, yellow, green.
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. 6. Repeat step 3.
7. Then, arrange the same eight cubes in a circle in this manner – four yellow and four blue. 8. Repeat step 3.
9. Remove all the cubes from the table. 10. Arrange the eight blue cubes in a circle. 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? 1, 2, 3, 4
1, 2, 3, 4 1, 2, 3, 4
Kindi_TE1_Chap2.indd 15 3/2/11 10:41:35 AM
102 Chapter 13 Addition and Subtraction to 40
7 27 20 So, 27 – 4 = 23. Subtract. Guided Practice 1 36 – 3 = ?
Method 1 Count back from the greater number. 7 – 4 = 3
20 + 3 = 23 – 4
Check!
Remember, 7 – 4 = 3 3 + 4 = 7 If 27 – 4 = 23, then 23 + 4 should equal 27. The answer is correct.
2 3 + 4 2 7 36, , , G1B_TB_Ch13(80-103).indd 102 12/29/08 4:52:59 PM
7
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.
—Common Core State Standards