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Common Core State Standards Overview Grades K–5

World-Class Singapore Math for Your Classrooms

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“

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)

”

* Achieve is a bipartisan, nonprofit educational reform organization that partnered with NGA and CCSSO on the Common Core State Standards Initiative.

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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

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

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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.” 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:

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 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.

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)

Common Core State Standards Skills Traces in the Teacher’s Edition help teachers understand how concepts progress through the grade levels.

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*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.

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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)

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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)

Organize content by big ideas, such as place value Common Core State Standards: These Standards endeavor to follow such a

Le

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

Ones

70 10

Step 1 Add the ones.

160

Tens Ones 1

14

+

1 1

160 10

4 8 2

1,800 1,800 10

4 ones + 8 ones = 12 ones

18

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.

Regroup the ones. 12 ones = 1 ten 2 ones

70

Tens

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.

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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

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Common Core State Standards. 3

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1

Make sense of problems and persevere in solving them.

“

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. Grade K

Grade 1

Notes

CRIT ICAl T HIn KIn G s KIlls

CRIT ICAl T HIN KIN G s KIlls

Put On Your Thinking Cap!

Put On Your Thinking Cap!

Circle, count, and write.

1

Grade 2

PRoBleM s olVIN G

PRoBleM s olVIn G

Find the number of beads. Use number bonds to help you.

solve.

There is cheese for 6 mice.

1

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.

6

How many mice will be hungry? 2

2 3 boys have coats.

There are 8 beads under the two cups.

8

How many boys will be cold?

3

3 There are 10 beads under the three cups.

There are 6 egg holders.

10 There are

counters in the bag. on YoUR oWn

oN YoUR oWN

How many eggs are needed to ﬁll all the egg holders? 58

Chapter 2

lessonChapter 1 Making 2 Number Bonds

Kindi_SB1_Ch2.indd 58

Student Book A, Part 1, p. 58

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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 a coat. Next, have them count the number of boys they have circled and write this answer in the blank provided.

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Go to Workbook B:

Go to Workbook A:

Put on Your Thinking Cap! pages 173–174

Put on Your Thinking Cap! pages 31–32

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Student Book A, page 37: Students use number bonds to determine unknowns, promoting early algebraic thinking through relevant problem solving.

lesson 3

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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.

217

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Common Core State Standards: Standards for Mathematical Practice

. How Math in Focus Aligns:

Singapore Mathematics Framework Monitoring of one’s own thinking et Self-regulation ac of learning o

M

gn

iti

Mathematical Problem Solving

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

on

Pro 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.

Reasoning, communication and connections Thinking skills and heuristics Applications and modeling

Concepts Numerical, Algebraic, Geometrical Statistical, Probabilistic, Analytical From the Singapore Ministry of Education

Grade 3

Grade 4

Grade 5 CRITICAL THIN KIN G SKILL S

CRITICAL THIN KIN G SKILLS

C RIT IC A L T H INKING SK I LLS

Put On Your Thinking Cap!

Put On Your Thinking Cap!

Put On Your Thinking Cap!

PROBLEM SOLVIN G

P RO BL E M S O LV ING

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.

P ROBLEM SOLVIN G

Solve these problems.

2 Jessie had a whole graham cracker. Minah had only part of another graham cracker.

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.

Jessie gave 1 of her graham cracker to Minah. 4 In the end, both girls had the same fractional part of

18

a graham cracker. What fraction of a graham cracker did Minah have at ﬁrst Jessie

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.

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.

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.

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?

ON YOUR OWN

Go to Workbook A:

Put on Your Thinking Cap! pages 17–18

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 32

Chapter 6 Fractions and Mixed Numbers

Chapter 1 Numbers to 10,000

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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.

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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 9 Multiplying and Dividing Decimals 81

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

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2

Reason abstractly and quantitatively.

“

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

”

Examples throughout the Math in Focus curriculum: 2

DAy

Le

Grade K arn You Grade 1 use number bonds to help you subtract. can

How many beanbags are on the floor? 9−5=? part Activity 2

Discover

Teacher’s Edition A, Chapter 2: Math in Focus Kindergarten students participate in Discover activities, like the one pictured here from Chapter 2. These he train engineer to raise his or Discover her fingers activities to introduce using te how many train engines he or sheconcepts wants. concrete, hands-on ctices Ensure that this child does not raise more activities. This helps ngers. students make sense of quantities and numbers in to the class that if, for example, the train so they truly understand eer raises four fingers, only engines 1, 2, 3, and allowed to move. Engines 5 and 6 should what they mean.

h down. children to play three rounds of the game.

children engage in the activity, end the day by g check questions such as: w do you know how many children should move? you sure?

5

4 3 5 1

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

1.

part

Begin the day by dividing the class into four groups.

There are 4 beanbags on the floor.

2. Distribute the connecting cubes to the children. Ensure that each child has six cubes.

Student Book A, page 74: Students model subtraction concretely by

3. Write the numerals 1 to 5 on the board. Ask: What number is this?

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

4. While children engage in the activity, ask check questions such as: • Are you sure? • Can it be this number instead?

progression helps students make sense of quantities.

13 How many yellow beans are there?

part

5. Write the numeral 6 on the board. Explain to children that this numeral represents the number six.

C R I T I C A l T HI N K I N G S K I llS

GradePut 2

6. Have children count 1 to 6 aloud and count their cubes to be sure they each have 6.

1 There is

8

End the day by playing the game for 8–10 rounds.

74

C hapter 2: L esson 1

3

3

yellow2 bean.

–

Gr1 TB A_Ch 4.indd 74

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10 − 9 =

whole

Find the missing numbers in each part box.

8. Then, have children hold up their towers and check each other’s towers.

6

On Your Thinking Cap!

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.

9.

9−5=4 4

whole

6

1

– 4

4

4

– 2

8

4

4

4

4

5

4 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.

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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.

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Common Core State Standards: Standards for Mathematical Practice

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.

Grade 5

Le

Grade 3 arn

CR I T I CA L T H I N K I N G S K I L L S

Add 2-digit numbers mentally using the ‘add the tens, then subtract the extra ones’ strategy.

Put On Your Thinking Cap!

Find 34 � 48.

P RO BL E M S O LVI N G

The 9 key on the calculator is not working.

48

50

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

1 79

Do you know why you add 50 and then subtract 2?

So, 34 � 48 � 82.

I can rewrite 79 as

1 or

1.

1,234 1,234

1,234

1,234

79 groups

)

1,234 79 (1,234

Guided Practice

1,234 79

Less

o

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

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

1,234 79 (1,234

ON YOUR OWN

Chapter 2 Whole Number Multiplication and Division

?

G3A_TB_Ch2.indd 42

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

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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.

2 � 16 lb � 32 lb

Chapter 2

)

Go to Workbook A:

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?

42

1,234

Put on Your Thinking Cap, pages 75– 78

Le

Step 2 Subtract from the result. arn Solve problems using the mean. So, 35 � 57 � .

1,234 groups

3/12/10 2:22:01 PM

The weight of the other table is 20 pounds.

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? Total weight of chicken, ﬁsh, and shrimp Mr. Saco bought � � Weight of chicken and ﬁsh Mr. Saco bought � � 204

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.

7

pounds

Chapter 5 Data and Probability

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� pounds

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Activity 3

Explore

Construct viable arguments and critique the reasoning of others.

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

“

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

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”

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 eADIn G An D 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,

?

+

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? Student BookWhy? A, page instead? or104: WhyThroughout not? = • Will this do the program, students are taught to check

stickers in all.

Grade 2

their answers and make sure their solutions are reasonable. Look for the “Check!” icon throughout the Student Books.

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

3/2/1

+8=

I added 1 to 33 to get 34. I just have to add 1 to get the next number.

Is the answer correct?

33 is 1 more than 32. 34 is 1 more than 33.

32 + 1 = 33 33 + 1 = 34 How do you find the missing numbers in this pattern? 40, 30,

, 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?

74

8

Chapter 12

Numbers to 40

G1B_TB_Ch12.indd 74

104

Chapter 4

MS_Gr2A_unit04.indd 104

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Using Bar Models: Addition and Subtraction

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Common Core State Standards: Standards for Mathematical Practice

How 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.

Grade 4 R e aDi N G aN D W R iTi N G M aTH

Math Journal

Both Andy and Rita think that 0.23 is greater than 0.3.

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.

Grade 3

Do you agree? Why or why not? Explain your answer.

R E A D I N G A N D W RITING MATH

Math Journal

List the steps to arrange the numbers in order from least to greatest.

Lesson 7.3 Comparing Decimals 33

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.

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

Le

9,049 9,654 8,785 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. 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

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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.

� area � cost of 1 yd2 � 756 � $16

Estimate to check if the answer is reasonable.

� $12,096

It will cost $12,096 to carpet the library fully.

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. Cost of fuel � He needs to pay $ 98

� $3 � $ .

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. MiF 5A PB U2 2.4-2.7.indd 98

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4

Model with mathematics.

“

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 le

”

ssoN

1

Making Number Bonds Objectivescurriculum: Examples throughout the Math 3inLesson Focus Vocabulary DAy

• Use connecting cubes or a math balance to find number bonds.

2

DAy

• Find different number bonds for numbers to 10. Grade 1 le

Grade K

part whole number bond

arn You can make number bonds with

.

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.

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.

Begin the day by distributing materials to part the children.

2. AskE Schildren to lay3 out the numeral card ‘5’. SON L

and 1 make 4. Using 3Part-Part-Whole in Addition and Subtraction

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 corresponding number. part

1

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: • Use bar models to solve addition and subtraction problems. • I want the numeral to be one more. • Apply the inverse operations of addition and subtraction. 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?

8/19/08 4:32:56 PM

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. 10

12

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. 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

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31

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96

MS_Gr2A_unit04.indd 96

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Student Book A, page 30: Students use number bonds to model part-part-whole relationships.

Explore

Le

Activity 3

Chapter 4 Using Bar Models: Addition and Subtraction

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Common Core State Standards: Standards for Mathematical Practice

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.

Grade 4

Le

Lesson Objectives • Model regrouping in division. • Divide a 3-digit number by a 1-digit number with regrouping.

Lesson Objective • Use bar models to solve 2-step real-world problems on addition and subtraction.

Le

Grade 3

farmer sells his crops to 3 restaurants. He divides A 525 heads of lettuce equally among the 3 restaurants. How many heads of lettuce does each restaurant receive?

525 3 ?

Hundreds

How many tickets did Sue sell?

b

How many tickets did they sell in all?

a? a b

Ones

5 hundreds 3 1 hundred with 2 hundreds left over 1 3 5 2 5 3 0 0 2

Add the tens. 20 tens 2 tens 22 tens

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 Whole Number Multiplication and Division 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 Le

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

2,164 � 1,286 � 3,450 5,614 � 2,164 � 3,450 The answers are correct. 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

Tens

Step 1 Divide the hundreds by 3.

G4_TB_Ch_03-1new.indd 96

b?

Sue

Ones

Vocabulary sum difference bar model

3,450 tickets Nancy

Tens

Regroup the hundreds. 2 hundreds 20 tens

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.

122

arn

Real-World Problems: Addition and Subtraction

n

Less

o

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.

arn

Modeling Division with Regrouping

n

Less

o

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.

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.

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.

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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.

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5

Use appropriate tools strategically.

“

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

”

Examples throughout the Math in Focus curriculum: sson le

2

Grade K

Student Book B, Part 2, Chapter 15

Finding the Weight of Things

Lesson Objectives • Use a non-standard object to ﬁnd the weight of things.

Lesson

• 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.

glass The glass is as The caterpillar is

heavy as 8

longer than the ant.

The weight of the glass is about 8

.

.

cup

The pencil is

long.

The weight of the cup is about 15

.

The cup is heavier than the glass.

The crayon is

The glass is lighter than the cup.

long.

lesson 2

The pencil is 8

G1B_TB_Ch10.indd 13

Chapter 15

MSpring_StudentBk4_Unit15_B.indd 8

Finding the Weight of Things

13

longer than the crayon. 12/30/08 2:23:02 PM

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12

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Common Core State Standards: Standards for Mathematical Practice

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. 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

Le

arn You can use inches to measure the length of shorter objects.

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.

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°.

1 inch

You can use a protractor to measure an angle. C

C

What is inch?

15

40

180 0 170 0 16 10 0 20

30

A

11 0 80 70 120 60 130 50

0

vertex

B

100

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

50

B

A

center

base line

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°.

The inch is a unit of length. in. stands for inch. Read 1 in. as one inch. Inch is used to measure shorter lengths.

Since AB passes through the zero mark of the outer scale, read the measure on the outer scale.

e

g Co t pa ntinued on nex

lesson 3

111

Measuring in Inches

G2B_TB_Ch 13.indd 111

88

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Le

arn

Chapter 9 Angles

Student Book B, page 179

Grade 3 Use yards to measure length. 1 yard

Student Book A, page 47

Grade 5 A yardstick is 3 times as long as a 12-inch ruler. Less

n

Using a Calculator

Lesson Objective • Use a calculator to add, subtract, multiply, and divide whole numbers.

1 ft � 12 in. 3 ft � 12 � 3 � 36 in.

Le

A baseball bat is about 1 yard long. A doorway is about 1 yard wide.

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.)

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.

arn

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

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

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

13

Enter these numbers on your calculator. Clear the display on your calculator before entering the next number.

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1 735

2 9,038

3 23,104

4 505,602

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6

Attend to precision. Workmat

“

3

Match me.

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:

Compare the numbers.

Student Book A, Part 1, Workmat 3

Kindi_SB1_Ch2.indd 59

Which2 is the least number? Which is the greatest number? M

DAy

ce ath nt

er

5

3/2/11 10:21:24 AM

Grade K

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.

1.

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?

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.

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. 3. Ask his or her partner to place identical objects on his or her own workmat. Math Talk Encourage children to practice number names. Ask: What do you have on your workmat? (I have three counters and five cubes.)

4.

5. Repeat the activity three times to allow children to familiarize themselves with the concept of pairing.

68

Grade 2

83

95 Math Journal

Re AD ING AND W RIT I NG M AT H

6. Ensure that the children exchange roles.

The least number is 16

C hapter 2: L esson 3

Grade 1

.

758 – 35 = 732 Is the answer correct? Explain why you think so. Show how you would check it.

Kindi_TE1_Chap2.indd 16

Why is it the least number?

3/2/11 10:41:35 AM

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.

let’s Practice

Student Book B, page 194: Thought bubbles throughout the Student Books prompt students to explain their answers.

The greatest number is 14

.

Order the numbers from greatest to least. ,

greatest 92_MS52834_MIF_CCbrochure_FilesOnly.indd

, 14

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

Subtract. 3

– 2 4

9 8 – 5 6

Subtract. 5

38 – 15 =

6

77 – 24 =

7

97 – 3 =

7/27/12 9:54 AM

Common Core State Standards: Standards for Mathematical Practice

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.

Le

Use your calculator to add. Grade 3 Add 417 and 9,086.

Press

re A d i nG A nd Wri t i nG MAtH

Math Journal

Display 0 417 9086 9503

C 4 1 7

You are given an empty container and a cup. 9 container. 0 8 explain how you would find the capacity of the

6

The sum is 9,503.

Grade 4

Find the sum of $1,275 and $876.

RE A DING A ND WRITING MAT H

Math Journal

Press

Display 0 C Remember to write These are the steps to find the factors of 12. the correct unit in 1275 1 2 7 5 1 Think of all the numbers that divide 12 exactly. Grade 3, Student Book B, page 211: your answer. Cri t i CA L t Hi nki nG S ki LLS 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. Example ST EP

12 5 1 3 12 12 5 2 3 6

ST EP

thought process into words. This

2 The factors are 1, 2, 3, 4, 6, and 12.P rOBLe M S O 12 LVi5 nG3 3 4

reinforces the idea that the process The sum of $1,275 and $876 is $2,151. sameiscapacity? used to get an answer is just 1 Can containers of different shapes have thethat

Le

Write the steps for finding the common factors of 12 and 15.

arn

Explain why or why not. A

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. B

Subtract 6,959 from 17,358.

Press C

Grade 4, Student Book A, page 55

On YOUr OWn

B: 1 7 Put 5thinking Go3ontoYour Workbook 8 Cap! pages 135–136

Lesson 6 15.3 9 Measuring 5 9Capacity

211

GradeThe difference is 10,399. 5

Display 0 17358 6959 10399

Find the difference between 1,005 pounds and 248 pounds. 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 Remember to write pounds in your answer.

G4_TB_Ch_02-1.indd 55

C 1 0 0 5 2 4 8

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Display 0 1005 248 757

The difference between 1,005 pounds and 248 pounds is 757 pounds. 48

Chapter 2 Whole Number Multiplication and Division

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7

Look for and make use of structure.

“

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

Grade K 1, 2, 3, 4

1, 2, 3, 4

1, 2, 3, 4

Examples throughout the Math in Focus curriculum: Grade 1

27

20

–4

7

7–4=3 20 + 3 = 23

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

So, 27 – 4 = 23.

Check! Remember, 7 – 4 = 3 3+4=7 If 27 – 4 = 23, then 23 + 4 should equal 27.

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.

+

The answer is correct.

2 3 4 2 7

Student Book B, page 102: Grade 1 students learn to use number bonds to demonstrate the structure of numbers and Guided Practice understand properties.

Subtract.

6. Repeat step 3. 7. Then, arrange the same eight cubes in a circle in this manner – four yellow and four blue.

1

36 – 3 = ?

Method 1

8. Repeat step 3.

Count back from the greater number.

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?

16

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.

102

Chapter 13

G1B_TB_Ch13(80-103).indd 102

Kindi_TE1_Chap2.indd 15

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Addition and Subtraction to 40

C hapter 2: L esson 3

15

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Le

arn Estimate the area of a figure.

Use rounding to estimate the areas of the figures. Common Core State Standards: Standards for Mathematical Practice

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 1 square unit 1 square unit 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, square unit square unit 2 2 this level of structure becomes more advanced. greater than 1 square unit 1 square unit 2 less than 1 square unit

Look at the triangle. Count the squares.

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

Le

Grade 2

× × × × × ×

2 2 2 2 2 2

14 15 2 4 1 2 6 13 8 3 4 5 6 10 7 8 9 10 11 12 12 14 16 Look at the circle. Count the squares. 18 20 5 6

= = = = = =

arn You can multiply numbers in any order. 4×2=8

2×4=8

1 2

1 2 3 4

1 2 3 4

1 2

0 square unit

2

Grade 4

12

1

2

7

11

3

4

8

4×2=2×4 These are related 10 multiplication facts .

16 The area of the triangle is 16 square units.

Grade 5 Look at the place-value chart. Hundreds

7

Tens

Ones

7 10

9

What is the pattern when

each number is multiplied The area of the circle is about 12 square units. 9 by 10?

9 10

158

Chapter 12 Area and Perimeter

10

Grade 3

10 10

9 3 6 ?

12

Start with 10 groups of 6.

12 10

Guided Practice 1 2 3 4 5 6 2 3 4 5 6 Use1 dot paper to find the missing numbers. Grade 2, Student Book 1 25 3 4 5 6 7 8 9 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

Each digit moves one place to the left when the number is multiplied by 10.

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.

52

MS_Gr2A_unit06.indd 160

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Chapter 2 Whole Number Multiplication and Division

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.

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Grade 3, Student Book A, page 154 154

Chapter 6

Multiplication Tables of 6, 7, 8, and 9

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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.

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17

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8

“

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 le

sson

1

”

How to Multiply

Lesson Objectives

Vocabulary

Examples throughout the Math in Focus curriculum:

• Use equal groups and repeated addition to multiply. • Make multiplication stories about pictures.

Grade 2 • Make multiplication sentences. le

Grade K

6 6 6 7 7 7 2

There are 5 horses in each group. Use repeated addition or multiply to find the number of horses. 5 + 5 + 5 = 15 3 × 5 = 15

Place Value

ge

lesson 1

le

arn You can use place value to show numbers to 100.

90

ones

9

8

8

98 = 9 tens 8 ones 98 = 90 + 8 Chapter 2

Guided Practice

07, and 1 to 4 eed

numerals 5,

Kindi_SB1_Ch2.indd 36

Use place value to findStudent the missing Booknumbers. A, Part 1, p. 36

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1

18

Tens

× is read as times. It means to multiply, or to put all the equal groups together.

a Co xt p ntinued on ne

MS_Gr2ATB_unit05.indd 127

Tens

3 groups of 5 equal 15. 3 fives = 15

There are 15 horses in all. 3 × 5 = 15 is a multiplication sentence. You read it as three times five is equal to fifteen.

• Show 1 objects up to 100 as tens and ones. Grade

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multiplication stories

multiply

arn You can multiply when you have equal groups.

• Use a place-value chart to show numbers up to 100.

t 1, p. 35

multiplication sentence

group

First, count the number of equal groups. There are 3 groups. Then, count the number of horses in each group.

ess o n

Look and lsay.

36

repeated addition

equal

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.

35

times

How many horses are there?

Trace.

1l 2 3 4 5

n2

Look for and express regularity in repeated reasoning.

ones

How to Multiply

127

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.

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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.

• 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 bottom, without lifting the pencil from the paper. 184

Chapter 16

Numbers to 100

18 numerals 5, 6, and 7. rst, place92_MS52834_MIF_CCbrochure_FilesOnly.indd the 11. Children trace the

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Common Core State Standards: Standards for Mathematical Practice

How Math in Focus Aligns: 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. Operations are taught with place-value materials so students understand how the standard algorithms work. Grade 4 n

Rectangles and Squares

Less

o

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.

Le

Lesson Objective • Solve problems involving the area and perimeter of squares and rectangles. arn Find the perimeter of a rectangle using a formula. length

width

Perimeter of rectangle 5 length 1 width 1 length 1 width 5 total length of all four sides

You can use a model to show that the perimeter of the rectangle is the sum of its two lengths and two widths.

length

n

Less

o

perimeter

width

length 1 width

Mental Subtraction

length

width

length 1 width

So, the length 1 width of a rectangle is equal to 1 of its perimeter. 2

Le

arn

Le

Lesson Objective Grade 3 • Subtract 2-digit numbers mentally with or without regrouping.

6 ft

Length 1 width 5 perimeter 4 2 5 18 4 2 5 9 ft

Find 87 2 34.

Find one side of a rectangle given its perimeter and the other side. The perimeter of rectangle A is 18 feet. Its length is 6 feet. Find its width.

Subtract 2-digit numbers mentally using the ‘subtract the tens, then subtract the ones’ strategy.

34

arn

?

rectangle A

Length 1 width 5 9 ft 6 1 width 5 9 ft width 5 9 2 6 5 3 ft The width of rectangle A is 3 feet.

30 4

Lesson 12.2 Rectangles and Squares 163

34 5 3 tens 4 ones Step 1 Subtract 3 tens from 87. 87 2 30 5 57

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 Subtract 4 ones from the result. 57 2 4 5 53

45 5

tens

ones

Step 1 Subtract

tens from 79. 79 2

Step 2 Subtract

ones from the result.

So, 79 2 45 5

2

5 5

Comparing Numbers to 10,000,000

Less

n

Lesson Objectives • Compare and order numbers to 10,000,000. • Identify and complete a number pattern. Grade 5 • Find a rule for a number pattern. Le

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 Find 79 2 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, 87 2 34 5 53.

2/24/09 10:50:31 AM

arn

Compare numbers by using a place-value chart. Which number is less, 237,981 or 500,600?

Student Book A, page 21: . consistent emphasis on place value The Mental Subtraction allows students to seeLesson the2.2 structure 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.

Vocabulary greater than (>) less than (’ means ‘greater than’ and ‘

View more...
World-Class Singapore Math for Your Classrooms

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“

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)

”

* Achieve is a bipartisan, nonprofit educational reform organization that partnered with NGA and CCSSO on the Common Core State Standards Initiative.

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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

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

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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.” 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:

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 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.

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)

Common Core State Standards Skills Traces in the Teacher’s Edition help teachers understand how concepts progress through the grade levels.

2

*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.

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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)

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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)

Organize content by big ideas, such as place value Common Core State Standards: These Standards endeavor to follow such a

Le

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

Ones

70 10

Step 1 Add the ones.

160

Tens Ones 1

14

+

1 1

160 10

4 8 2

1,800 1,800 10

4 ones + 8 ones = 12 ones

18

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.

Regroup the ones. 12 ones = 1 ten 2 ones

70

Tens

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.

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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

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Common Core State Standards. 3

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1

Make sense of problems and persevere in solving them.

“

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. Grade K

Grade 1

Notes

CRIT ICAl T HIn KIn G s KIlls

CRIT ICAl T HIN KIN G s KIlls

Put On Your Thinking Cap!

Put On Your Thinking Cap!

Circle, count, and write.

1

Grade 2

PRoBleM s olVIN G

PRoBleM s olVIn G

Find the number of beads. Use number bonds to help you.

solve.

There is cheese for 6 mice.

1

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.

6

How many mice will be hungry? 2

2 3 boys have coats.

There are 8 beads under the two cups.

8

How many boys will be cold?

3

3 There are 10 beads under the three cups.

There are 6 egg holders.

10 There are

counters in the bag. on YoUR oWn

oN YoUR oWN

How many eggs are needed to ﬁll all the egg holders? 58

Chapter 2

lessonChapter 1 Making 2 Number Bonds

Kindi_SB1_Ch2.indd 58

Student Book A, Part 1, p. 58

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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 a coat. Next, have them count the number of boys they have circled and write this answer in the blank provided.

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Go to Workbook B:

Go to Workbook A:

Put on Your Thinking Cap! pages 173–174

Put on Your Thinking Cap! pages 31–32

Gr1 TB A_Ch 2.indd 37

37

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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.

217

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Common Core State Standards: Standards for Mathematical Practice

. How Math in Focus Aligns:

Singapore Mathematics Framework Monitoring of one’s own thinking et Self-regulation ac of learning o

M

gn

iti

Mathematical Problem Solving

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

on

Pro 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.

Reasoning, communication and connections Thinking skills and heuristics Applications and modeling

Concepts Numerical, Algebraic, Geometrical Statistical, Probabilistic, Analytical From the Singapore Ministry of Education

Grade 3

Grade 4

Grade 5 CRITICAL THIN KIN G SKILL S

CRITICAL THIN KIN G SKILLS

C RIT IC A L T H INKING SK I LLS

Put On Your Thinking Cap!

Put On Your Thinking Cap!

Put On Your Thinking Cap!

PROBLEM SOLVIN G

P RO BL E M S O LV ING

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.

P ROBLEM SOLVIN G

Solve these problems.

2 Jessie had a whole graham cracker. Minah had only part of another graham cracker.

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.

Jessie gave 1 of her graham cracker to Minah. 4 In the end, both girls had the same fractional part of

18

a graham cracker. What fraction of a graham cracker did Minah have at ﬁrst Jessie

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.

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.

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.

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?

ON YOUR OWN

Go to Workbook A:

Put on Your Thinking Cap! pages 17–18

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 32

Chapter 6 Fractions and Mixed Numbers

Chapter 1 Numbers to 10,000

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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.

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G4_TB_Ch6.indd 269

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 9 Multiplying and Dividing Decimals 81

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

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2

Reason abstractly and quantitatively.

“

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

”

Examples throughout the Math in Focus curriculum: 2

DAy

Le

Grade K arn You Grade 1 use number bonds to help you subtract. can

How many beanbags are on the floor? 9−5=? part Activity 2

Discover

Teacher’s Edition A, Chapter 2: Math in Focus Kindergarten students participate in Discover activities, like the one pictured here from Chapter 2. These he train engineer to raise his or Discover her fingers activities to introduce using te how many train engines he or sheconcepts wants. concrete, hands-on ctices Ensure that this child does not raise more activities. This helps ngers. students make sense of quantities and numbers in to the class that if, for example, the train so they truly understand eer raises four fingers, only engines 1, 2, 3, and allowed to move. Engines 5 and 6 should what they mean.

h down. children to play three rounds of the game.

children engage in the activity, end the day by g check questions such as: w do you know how many children should move? you sure?

5

4 3 5 1

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

1.

part

Begin the day by dividing the class into four groups.

There are 4 beanbags on the floor.

2. Distribute the connecting cubes to the children. Ensure that each child has six cubes.

Student Book A, page 74: Students model subtraction concretely by

3. Write the numerals 1 to 5 on the board. Ask: What number is this?

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

4. While children engage in the activity, ask check questions such as: • Are you sure? • Can it be this number instead?

progression helps students make sense of quantities.

13 How many yellow beans are there?

part

5. Write the numeral 6 on the board. Explain to children that this numeral represents the number six.

C R I T I C A l T HI N K I N G S K I llS

GradePut 2

6. Have children count 1 to 6 aloud and count their cubes to be sure they each have 6.

1 There is

8

End the day by playing the game for 8–10 rounds.

74

C hapter 2: L esson 1

3

3

yellow2 bean.

–

Gr1 TB A_Ch 4.indd 74

92_MS52834_MIF_CCbrochure_FilesOnly.indd 6

10 − 9 =

whole

Find the missing numbers in each part box.

8. Then, have children hold up their towers and check each other’s towers.

6

On Your Thinking Cap!

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.

9.

9−5=4 4

whole

6

1

– 4

4

4

– 2

8

4

4

4

4

5

4 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

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.

7/27/12 9:54 AM

Common Core State Standards: Standards for Mathematical Practice

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.

Grade 5

Le

Grade 3 arn

CR I T I CA L T H I N K I N G S K I L L S

Add 2-digit numbers mentally using the ‘add the tens, then subtract the extra ones’ strategy.

Put On Your Thinking Cap!

Find 34 � 48.

P RO BL E M S O LVI N G

The 9 key on the calculator is not working.

48

50

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

1 79

Do you know why you add 50 and then subtract 2?

So, 34 � 48 � 82.

I can rewrite 79 as

1 or

1.

1,234 1,234

1,234

1,234

79 groups

)

1,234 79 (1,234

Guided Practice

1,234 79

Less

o

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

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

1,234 79 (1,234

ON YOUR OWN

Chapter 2 Whole Number Multiplication and Division

?

G3A_TB_Ch2.indd 42

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.

2 � 16 lb � 32 lb

Chapter 2

)

Go to Workbook A:

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?

42

1,234

Put on Your Thinking Cap, pages 75– 78

Le

Step 2 Subtract from the result. arn Solve problems using the mean. So, 35 � 57 � .

1,234 groups

3/12/10 2:22:01 PM

The weight of the other table is 20 pounds.

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? Total weight of chicken, ﬁsh, and shrimp Mr. Saco bought � � Weight of chicken and ﬁsh Mr. Saco bought � � 204

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.

7

pounds

Chapter 5 Data and Probability

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� pounds

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Activity 3

Explore

Construct viable arguments and critique the reasoning of others.

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

“

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 eADIn G An D 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,

?

+

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? Student BookWhy? A, page instead? or104: WhyThroughout not? = • Will this do the program, students are taught to check

stickers in all.

Grade 2

their answers and make sure their solutions are reasonable. Look for the “Check!” icon throughout the Student Books.

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

3/2/1

+8=

I added 1 to 33 to get 34. I just have to add 1 to get the next number.

Is the answer correct?

33 is 1 more than 32. 34 is 1 more than 33.

32 + 1 = 33 33 + 1 = 34 How do you find the missing numbers in this pattern? 40, 30,

, 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?

74

8

Chapter 12

Numbers to 40

G1B_TB_Ch12.indd 74

104

Chapter 4

MS_Gr2A_unit04.indd 104

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Using Bar Models: Addition and Subtraction

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Common Core State Standards: Standards for Mathematical Practice

How 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.

Grade 4 R e aDi N G aN D W R iTi N G M aTH

Math Journal

Both Andy and Rita think that 0.23 is greater than 0.3.

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.

Grade 3

Do you agree? Why or why not? Explain your answer.

R E A D I N G A N D W RITING MATH

Math Journal

List the steps to arrange the numbers in order from least to greatest.

Lesson 7.3 Comparing Decimals 33

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.

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

Le

9,049 9,654 8,785 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. 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.

� area � cost of 1 yd2 � 756 � $16

Estimate to check if the answer is reasonable.

� $12,096

It will cost $12,096 to carpet the library fully.

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. Cost of fuel � He needs to pay $ 98

� $3 � $ .

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. MiF 5A PB U2 2.4-2.7.indd 98

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9

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4

Model with mathematics.

“

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 le

”

ssoN

1

Making Number Bonds Objectivescurriculum: Examples throughout the Math 3inLesson Focus Vocabulary DAy

• Use connecting cubes or a math balance to find number bonds.

2

DAy

• Find different number bonds for numbers to 10. Grade 1 le

Grade K

part whole number bond

arn You can make number bonds with

.

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.

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.

Begin the day by distributing materials to part the children.

2. AskE Schildren to lay3 out the numeral card ‘5’. SON L

and 1 make 4. Using 3Part-Part-Whole in Addition and Subtraction

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 corresponding number. part

1

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: • Use bar models to solve addition and subtraction problems. • I want the numeral to be one more. • Apply the inverse operations of addition and subtraction. 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?

8/19/08 4:32:56 PM

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. 10

12

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. 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

MS_Gr2A_unit04.indd 96

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Student Book A, page 30: Students use number bonds to model part-part-whole relationships.

Explore

Le

Activity 3

Chapter 4 Using Bar Models: Addition and Subtraction

8/26/08 4:37:43 PM

7/27/12 9:54 AM

Common Core State Standards: Standards for Mathematical Practice

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.

Grade 4

Le

Lesson Objectives • Model regrouping in division. • Divide a 3-digit number by a 1-digit number with regrouping.

Lesson Objective • Use bar models to solve 2-step real-world problems on addition and subtraction.

Le

Grade 3

farmer sells his crops to 3 restaurants. He divides A 525 heads of lettuce equally among the 3 restaurants. How many heads of lettuce does each restaurant receive?

525 3 ?

Hundreds

How many tickets did Sue sell?

b

How many tickets did they sell in all?

a? a b

Ones

5 hundreds 3 1 hundred with 2 hundreds left over 1 3 5 2 5 3 0 0 2

Add the tens. 20 tens 2 tens 22 tens

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 Whole Number Multiplication and Division 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 Le

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

2,164 � 1,286 � 3,450 5,614 � 2,164 � 3,450 The answers are correct. 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

Tens

Step 1 Divide the hundreds by 3.

G4_TB_Ch_03-1new.indd 96

b?

Sue

Ones

Vocabulary sum difference bar model

3,450 tickets Nancy

Tens

Regroup the hundreds. 2 hundreds 20 tens

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.

122

arn

Real-World Problems: Addition and Subtraction

n

Less

o

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.

arn

Modeling Division with Regrouping

n

Less

o

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.

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.

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.

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11

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5

Use appropriate tools strategically.

“

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

”

Examples throughout the Math in Focus curriculum: sson le

2

Grade K

Student Book B, Part 2, Chapter 15

Finding the Weight of Things

Lesson Objectives • Use a non-standard object to ﬁnd the weight of things.

Lesson

• 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.

glass The glass is as The caterpillar is

heavy as 8

longer than the ant.

The weight of the glass is about 8

.

.

cup

The pencil is

long.

The weight of the cup is about 15

.

The cup is heavier than the glass.

The crayon is

The glass is lighter than the cup.

long.

lesson 2

The pencil is 8

G1B_TB_Ch10.indd 13

Chapter 15

MSpring_StudentBk4_Unit15_B.indd 8

Finding the Weight of Things

13

longer than the crayon. 12/30/08 2:23:02 PM

9/30/08 1:39:34 PM

12

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Common Core State Standards: Standards for Mathematical Practice

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. 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

Le

arn You can use inches to measure the length of shorter objects.

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.

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°.

1 inch

You can use a protractor to measure an angle. C

C

What is inch?

15

40

180 0 170 0 16 10 0 20

30

A

11 0 80 70 120 60 130 50

0

vertex

B

100

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

50

B

A

center

base line

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°.

The inch is a unit of length. in. stands for inch. Read 1 in. as one inch. Inch is used to measure shorter lengths.

Since AB passes through the zero mark of the outer scale, read the measure on the outer scale.

e

g Co t pa ntinued on nex

lesson 3

111

Measuring in Inches

G2B_TB_Ch 13.indd 111

88

12/17/08 5:57:47 PM

Le

arn

Chapter 9 Angles

Student Book B, page 179

Grade 3 Use yards to measure length. 1 yard

Student Book A, page 47

Grade 5 A yardstick is 3 times as long as a 12-inch ruler. Less

n

Using a Calculator

Lesson Objective • Use a calculator to add, subtract, multiply, and divide whole numbers.

1 ft � 12 in. 3 ft � 12 � 3 � 36 in.

Le

A baseball bat is about 1 yard long. A doorway is about 1 yard wide.

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.)

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.

arn

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

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

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

13

Enter these numbers on your calculator. Clear the display on your calculator before entering the next number.

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1 735

2 9,038

3 23,104

4 505,602

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6

Attend to precision. Workmat

“

3

Match me.

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:

Compare the numbers.

Student Book A, Part 1, Workmat 3

Kindi_SB1_Ch2.indd 59

Which2 is the least number? Which is the greatest number? M

DAy

ce ath nt

er

5

3/2/11 10:21:24 AM

Grade K

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.

1.

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?

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.

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. 3. Ask his or her partner to place identical objects on his or her own workmat. Math Talk Encourage children to practice number names. Ask: What do you have on your workmat? (I have three counters and five cubes.)

4.

5. Repeat the activity three times to allow children to familiarize themselves with the concept of pairing.

68

Grade 2

83

95 Math Journal

Re AD ING AND W RIT I NG M AT H

6. Ensure that the children exchange roles.

The least number is 16

C hapter 2: L esson 3

Grade 1

.

758 – 35 = 732 Is the answer correct? Explain why you think so. Show how you would check it.

Kindi_TE1_Chap2.indd 16

Why is it the least number?

3/2/11 10:41:35 AM

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.

let’s Practice

Student Book B, page 194: Thought bubbles throughout the Student Books prompt students to explain their answers.

The greatest number is 14

.

Order the numbers from greatest to least. ,

greatest 92_MS52834_MIF_CCbrochure_FilesOnly.indd

, 14

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

Subtract. 3

– 2 4

9 8 – 5 6

Subtract. 5

38 – 15 =

6

77 – 24 =

7

97 – 3 =

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Common Core State Standards: Standards for Mathematical Practice

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.

Le

Use your calculator to add. Grade 3 Add 417 and 9,086.

Press

re A d i nG A nd Wri t i nG MAtH

Math Journal

Display 0 417 9086 9503

C 4 1 7

You are given an empty container and a cup. 9 container. 0 8 explain how you would find the capacity of the

6

The sum is 9,503.

Grade 4

Find the sum of $1,275 and $876.

RE A DING A ND WRITING MAT H

Math Journal

Press

Display 0 C Remember to write These are the steps to find the factors of 12. the correct unit in 1275 1 2 7 5 1 Think of all the numbers that divide 12 exactly. Grade 3, Student Book B, page 211: your answer. Cri t i CA L t Hi nki nG S ki LLS 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. Example ST EP

12 5 1 3 12 12 5 2 3 6

ST EP

thought process into words. This

2 The factors are 1, 2, 3, 4, 6, and 12.P rOBLe M S O 12 LVi5 nG3 3 4

reinforces the idea that the process The sum of $1,275 and $876 is $2,151. sameiscapacity? used to get an answer is just 1 Can containers of different shapes have thethat

Le

Write the steps for finding the common factors of 12 and 15.

arn

Explain why or why not. A

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. B

Subtract 6,959 from 17,358.

Press C

Grade 4, Student Book A, page 55

On YOUr OWn

B: 1 7 Put 5thinking Go3ontoYour Workbook 8 Cap! pages 135–136

Lesson 6 15.3 9 Measuring 5 9Capacity

211

GradeThe difference is 10,399. 5

Display 0 17358 6959 10399

Find the difference between 1,005 pounds and 248 pounds. 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 Remember to write pounds in your answer.

G4_TB_Ch_02-1.indd 55

C 1 0 0 5 2 4 8

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Display 0 1005 248 757

The difference between 1,005 pounds and 248 pounds is 757 pounds. 48

Chapter 2 Whole Number Multiplication and Division

MiF 5A PB U2 2.1-2.3.indd 48

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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

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7

Look for and make use of structure.

“

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

Grade K 1, 2, 3, 4

1, 2, 3, 4

1, 2, 3, 4

Examples throughout the Math in Focus curriculum: Grade 1

27

20

–4

7

7–4=3 20 + 3 = 23

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

So, 27 – 4 = 23.

Check! Remember, 7 – 4 = 3 3+4=7 If 27 – 4 = 23, then 23 + 4 should equal 27.

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.

+

The answer is correct.

2 3 4 2 7

Student Book B, page 102: Grade 1 students learn to use number bonds to demonstrate the structure of numbers and Guided Practice understand properties.

Subtract.

6. Repeat step 3. 7. Then, arrange the same eight cubes in a circle in this manner – four yellow and four blue.

1

36 – 3 = ?

Method 1

8. Repeat step 3.

Count back from the greater number.

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?

16

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.

102

Chapter 13

G1B_TB_Ch13(80-103).indd 102

Kindi_TE1_Chap2.indd 15

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Addition and Subtraction to 40

C hapter 2: L esson 3

15

12/29/08 4:52:59 PM

3/2/11 10:41:35 AM

7/27/12 9:54 AM

Le

arn Estimate the area of a figure.

Use rounding to estimate the areas of the figures. Common Core State Standards: Standards for Mathematical Practice

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 1 square unit 1 square unit 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, square unit square unit 2 2 this level of structure becomes more advanced. greater than 1 square unit 1 square unit 2 less than 1 square unit

Look at the triangle. Count the squares.

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

Le

Grade 2

× × × × × ×

2 2 2 2 2 2

14 15 2 4 1 2 6 13 8 3 4 5 6 10 7 8 9 10 11 12 12 14 16 Look at the circle. Count the squares. 18 20 5 6

= = = = = =

arn You can multiply numbers in any order. 4×2=8

2×4=8

1 2

1 2 3 4

1 2 3 4

1 2

0 square unit

2

Grade 4

12

1

2

7

11

3

4

8

4×2=2×4 These are related 10 multiplication facts .

16 The area of the triangle is 16 square units.

Grade 5 Look at the place-value chart. Hundreds

7

Tens

Ones

7 10

9

What is the pattern when

each number is multiplied The area of the circle is about 12 square units. 9 by 10?

9 10

158

Chapter 12 Area and Perimeter

10

Grade 3

10 10

9 3 6 ?

12

Start with 10 groups of 6.

12 10

Guided Practice 1 2 3 4 5 6 2 3 4 5 6 Use1 dot paper to find the missing numbers. Grade 2, Student Book 1 25 3 4 5 6 7 8 9 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

Each digit moves one place to the left when the number is multiplied by 10.

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.

52

MS_Gr2A_unit06.indd 160

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Chapter 2 Whole Number Multiplication and Division

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

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Grade 3, Student Book A, page 154 154

Chapter 6

Multiplication Tables of 6, 7, 8, and 9

G3_TB_Ch_06-1(New).indd 154

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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.

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17

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8

“

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 le

sson

1

”

How to Multiply

Lesson Objectives

Vocabulary

Examples throughout the Math in Focus curriculum:

• Use equal groups and repeated addition to multiply. • Make multiplication stories about pictures.

Grade 2 • Make multiplication sentences. le

Grade K

6 6 6 7 7 7 2

There are 5 horses in each group. Use repeated addition or multiply to find the number of horses. 5 + 5 + 5 = 15 3 × 5 = 15

Place Value

ge

lesson 1

le

arn You can use place value to show numbers to 100.

90

ones

9

8

8

98 = 9 tens 8 ones 98 = 90 + 8 Chapter 2

Guided Practice

07, and 1 to 4 eed

numerals 5,

Kindi_SB1_Ch2.indd 36

Use place value to findStudent the missing Booknumbers. A, Part 1, p. 36

1/6/11 4:07:54 PM

1

18

Tens

× is read as times. It means to multiply, or to put all the equal groups together.

a Co xt p ntinued on ne

MS_Gr2ATB_unit05.indd 127

Tens

3 groups of 5 equal 15. 3 fives = 15

There are 15 horses in all. 3 × 5 = 15 is a multiplication sentence. You read it as three times five is equal to fifteen.

• Show 1 objects up to 100 as tens and ones. Grade

1/6/11 4:07:51 PM

multiplication stories

multiply

arn You can multiply when you have equal groups.

• Use a place-value chart to show numbers up to 100.

t 1, p. 35

multiplication sentence

group

First, count the number of equal groups. There are 3 groups. Then, count the number of horses in each group.

ess o n

Look and lsay.

36

repeated addition

equal

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.

35

times

How many horses are there?

Trace.

1l 2 3 4 5

n2

Look for and express regularity in repeated reasoning.

ones

How to Multiply

127

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.

3/18/09 8:53:49 AM

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.

• 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 bottom, without lifting the pencil from the paper. 184

Chapter 16

Numbers to 100

18 numerals 5, 6, and 7. rst, place92_MS52834_MIF_CCbrochure_FilesOnly.indd the 11. Children trace the

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Common Core State Standards: Standards for Mathematical Practice

How Math in Focus Aligns: 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. Operations are taught with place-value materials so students understand how the standard algorithms work. Grade 4 n

Rectangles and Squares

Less

o

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.

Le

Lesson Objective • Solve problems involving the area and perimeter of squares and rectangles. arn Find the perimeter of a rectangle using a formula. length

width

Perimeter of rectangle 5 length 1 width 1 length 1 width 5 total length of all four sides

You can use a model to show that the perimeter of the rectangle is the sum of its two lengths and two widths.

length

n

Less

o

perimeter

width

length 1 width

Mental Subtraction

length

width

length 1 width

So, the length 1 width of a rectangle is equal to 1 of its perimeter. 2

Le

arn

Le

Lesson Objective Grade 3 • Subtract 2-digit numbers mentally with or without regrouping.

6 ft

Length 1 width 5 perimeter 4 2 5 18 4 2 5 9 ft

Find 87 2 34.

Find one side of a rectangle given its perimeter and the other side. The perimeter of rectangle A is 18 feet. Its length is 6 feet. Find its width.

Subtract 2-digit numbers mentally using the ‘subtract the tens, then subtract the ones’ strategy.

34

arn

?

rectangle A

Length 1 width 5 9 ft 6 1 width 5 9 ft width 5 9 2 6 5 3 ft The width of rectangle A is 3 feet.

30 4

Lesson 12.2 Rectangles and Squares 163

34 5 3 tens 4 ones Step 1 Subtract 3 tens from 87. 87 2 30 5 57

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 Subtract 4 ones from the result. 57 2 4 5 53

45 5

tens

ones

Step 1 Subtract

tens from 79. 79 2

Step 2 Subtract

ones from the result.

So, 79 2 45 5

2

5 5

Comparing Numbers to 10,000,000

Less

n

Lesson Objectives • Compare and order numbers to 10,000,000. • Identify and complete a number pattern. Grade 5 • Find a rule for a number pattern. Le

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 Find 79 2 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, 87 2 34 5 53.

2/24/09 10:50:31 AM

arn

Compare numbers by using a place-value chart. Which number is less, 237,981 or 500,600?

Student Book A, page 21: . consistent emphasis on place value The Mental Subtraction allows students to seeLesson the2.2 structure 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.

Vocabulary greater than (>) less than (’ means ‘greater than’ and ‘

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