Research & Math Background Contents Planning Dr. Karen C. Fuson, Math Expressions Author RESEARCH—BEST PRACTICES Putting Research into Practice From Our Curriculum Research Project: Place Value In English, teen and 2-digit number words are complex and difficult to learn. In contrast, in some Asian languages the word for 13, for example, translates into “ten three.” In the English system, teen and 2-digit numbers look like two single-digit numbers written beside each other; nothing shows the ten value for the digit on the left. Young children need verbal and visual supports for understanding these number words and written numbers, a process we begin in Kindergarten. In this program, we provide this scaffolding by using tens and ones words, as well as standard number words, when working with teen and 2-digit numbers. We say 13 as thirteen and as 1 ten 3 ones and say 38 as thirty-eight and 3 tens 8 ones. These words are used interchangeably and help reinforce the embedded ten-based thinking and place value understanding. 273R | UNIT 4 | Overview
Transcript
Research & Math BackgroundContents Planning
Dr. Karen C. Fuson, Math Expressions Author
ReseaRch—Best PRactices
Putting Research into Practice
From Our Curriculum Research Project: Place Value
In English, teen and 2-digit number words are complex and difficult to learn. In contrast, in some Asian languages the word for 13, for example, translates into “ten three.” In the English system, teen and 2-digit numbers look like two single-digit numbers written beside each other; nothing shows the ten value for the digit on the left. Young children need verbal and visual supports for understanding these number words and written numbers, a process we begin in Kindergarten.
In this program, we provide this scaffolding by using tens and ones words, as well as standard number words, when working with teen and 2-digit numbers. We say 13 as thirteen and as 1 ten 3 ones and say 38 as thirty-eight and 3 tens 8 ones. These words are used interchangeably and help reinforce the embedded ten-based thinking and place value understanding.
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From Current Research: Place Value
It is absolutely essential that children develop a solid understanding of the base ten numeration system and place value concepts by the end of Grade 2. Children need many instructional experiences to develop their understanding of the system, including how numbers are written. They should recognize that the word ten may represent a single entity (1 ten) and, at the same time, 10 separate units (10 ones), and that these representations are interchangeable.
“Standards for Pre-K–2: Number and Operations.” Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics, 2000. 81.
Other Useful References: Place Value and Number
Miura, Irene T. “The Influence of Language on Mathematical Representations.” The Roles of Representation in School Mathematics. Reston, VA: National Council of Teachers of Mathematics, 2001. 53–61.
Copely, Juanita V. The Young Child and Mathematics, NAEYC, 2009
National Research Council. “Developing Proficiency with Whole Numbers.” Adding It Up: Helping Children Learn Mathematics. Washington: National Academy Press, 2001. 181–199.
Cross, Christopher T., Woods, Taniesha A., and Schweingruber, Heidi. Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity Editors; Committee on Early Childhood Mathematics; National Research Council, 2009
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ACTIVITY 1
ACTIVITY 2
Research & Math BackgroundContents Planning
Getting Ready to Teach Unit 4Using the Common Core Standards for Mathematical PracticeThe Common Core State Standards for Mathematical Content indicate what concepts, skills, and problem solving children should learn. The Common Core State Standards for Mathematical Practice indicate how children should demonstrate understanding. These Mathematical Practices are embedded directly into the Student and Teacher Editions for each unit in Math Expressions. As you use the teaching suggestions, you will automatically implement a teaching style that encourages children to demonstrate a thorough understanding of concepts, skills, and problems. In this program, Math Talk suggestions are a vehicle used to encourage discussion that supports all eight Mathematical Practices. See examples in Mathematical Practice 6.
Mathematical Practice 1Make sense of problems and persevere in solving them.
Children analyze and make conjectures about how to solve a problem. They plan, monitor, and check their solutions. They determine if their answers are reasonable and can justify their reasoning.
TeaCher ediTion: examples from Unit 4
Story Problems with Groups of Ten W H O L E C L A S S
MP.1, MP.4 Make Sense of Problems/Model with Mathematics Make a Math Drawing Present the story problem. Ask children what the problem means and have them solve it by making a drawing on their MathBoards or paper.
Each box in the bagel shop holds 10 bagels. There are 6 boxes and 8 extra bagels. How many bagels are there in the shop? 68 bagels
Lesson 10
MP.1 Make Sense of Problems Analyze the Problem Discuss the pictures in Exercises 1 and 2. Count the number of cars in the first row. 10 cars Explain that drivers may be directed to fill a row before parking in the next row of a parking lot. In the same way, people may be asked to fill a row of seats before sitting in the next row at a theater.
• How can a filled row help you count the number of cars or the number of people? Possible response: A filled row shows ten, so I can use the picture to count tens and extras.
Lesson 18
Mathematical Practice 1 is integrated into Unit 4 in the following ways:
Make Sense of ProblemsAnalyze the Problem
Make a Math Drawing
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Mathematical Practice 2reason abstractly and quantitatively.
Children make sense of quantities and their relationships in problem situations. They can connect diagrams and equations for a given situation. Quantitative reasoning entails attending to the meaning of quantities. In this unit, this involves decomposing numbers to add using doubles and to make a new ten.
Teacher edITIoN: examples from Unit 4
Doubles Plus 2 and Doubles Minus 2 W H O L E C L A S S
MP.2 reason abstractly and Quantitatively Write 4 + 5 = and 4 + 6 = on the board. Guide children to see that the numbers 4 and 5 are 1 apart, and the numbers 4 and 6 are 2 apart. Review with children how to add 4 + 5 using doubles plus 1. Then model how to add 4 + 6 using doubles plus 2. Invite a volunteer to name the hidden double in both equations, 4 + 4.
4 + 4 = 8 4 + 4 = 8
4 + 5 = 8 + 1 = 9 4 + 6 = 8 + 2 = 10
In a similar way, review with children how to add 8 + 7 using doubles minus 1, and apply that knowledge to add 8 + 6 using doubles minus 2.
8 + 8 = 16 8 + 8 = 16
8 + 7 = 16 - 1 = 15 8 + 6 = 16 - 2 = 14
Lesson 6
Model a New 10-Group W H O L E C L A S S
MP.2 reason Quantitatively Connect Drawings and Equations Now write on the board the addition equation 45 + 7 = . After the class solves it, show children how to group the loose ones into a new ten to make the total easier to see, if this has not been done by children already. Ringing the new ten or separating it in some other way is helpful. Two examples are shown below.
Lesson 15
Mathematical Practice 2 is integrated into Unit 4 in the following ways:
Reason Abstractly and Quantitatively
Reason Quantitatively
Connect Drawings and Equations
Connect Symbols and Models
Connect Symbols and Words
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Mathematical Practice 3Construct viable arguments and critique the reasoning of others.
Children use stated assumptions, definitions, and previously established results in constructing arguments. They are able to analyze situations and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others.
Children are also able to distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Children can listen to or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
Math Talk is a conversation tool by which children formulate ideas and analyze responses and engage in discourse. See also MP.6 Attend to Precision.
TeaCher ediTion: examples from Unit 4
Model Make a Ten W H O L E C L A S S
MP.3 Construct a Viable argument Justify Conclusions Discuss the equation 9 + 4 = 13, guiding children to recognize that the total, 13, is made up of a ten and some ones.
• Look at the total. How many tens and ones are in 13? 1 ten and 3 ones Does the total, 13, have a hidden 10 inside? yes What equation can you write that adds 1 ten and 3 ones? 10 + 3 = 13
• How do you know that 9 + 4 is 13? Is 9 + 4 the same as 10 + 3? Let’s make a drawing to prove it.
Lesson 4
What’s the Error? W H O L E C L A S S
MP.3, MP.6 Construct Viable arguments/Critique reasoning of others Puzzled Penguin Now discuss the Puzzled Penguin problem.
• Puzzled Penguin wrote 50 + 4 = 90. Is Puzzled Penguin correct? no What did Puzzled Penguin do wrong? Puzzled Penguin added 4 tens instead of 4 ones.
Guide a discussion to determine children’s understanding of the value of the tens and ones digits. Tell children to cross out the false equation and ask them how they could help Puzzled Penguin. Children can make a drawing to show 5 tens plus 4 ones, then write the correct total. 54
Lesson 14
Mathematical Practice 3 is integrated into Unit 4 in the following ways:
Construct a Viable ArgumentCritique Reasoning of Others
Justify ConclusionsPuzzled Penguin
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Mathematical Practice 4Model with mathematics.
Children can apply the mathematics they know to solve problems that arise in everyday life. This might be as simple as writing an equation to solve a problem. Children might draw diagrams to lead them to a solution for a problem. Children apply what they know and are comfortable making assumptions and approximations to simplify a complicated situation. They are able to identify important quantities in a practical situation and represent their relationships using such tools as diagrams, tables, graphs, and formulas.
Teacher edITIoN: examples from Unit 4
Solve a Teen Story Problem W H O L E C L A S S
MP.1, MP.4 Make Sense of Problems/Model with Mathematics Present a teen-grouping story problem to the class and have them solve it any way they can.
Sara has a bag of 10 tennis balls and 6 extra tennis balls. How many tennis balls does she have altogether? 16 tennis balls
• 16 means 1 ten and 6 extra ones. Let’s write an equation to show this: 10 + 6 = 16.
Use the Demonstration Secret Code Cards to show 16. Again, point out the ten “hiding” inside the number.
10
1 06
6 6
6 10
1 0
Lesson 2
MP.4 Model with Mathematics Write an Equation For each problem, children draw the numbers on the Dot Array and then write the equation. If they can solve these problems easily, continue with the more challenging packaging stories.
easy Packaging Stories
Molly has a bag of 10 peanuts and 3 extra peanuts. How many peanuts does she have altogether? 13 peanuts; 10 + 3 = 13
Wan buys 10 pieces of gum. He already has 9 pieces of gum. How many pieces of gum does Wan have now? 19 pieces of gum; 10 + 9 = 19
Lesson 3
Mathematical Practice 4 is integrated into Unit 4 in the following ways:
Model with MathematicsWrite an EquationMake a Math Drawing
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Mathematical Practice 5Use appropriate tools strategically.
Children consider the available tools and models when solving mathematical problems. Children make sound decisions about when each of these tools might be helpful. These tools might include paper and pencil, a straightedge, a ruler, or the MathBoard. They recognize both the insight to be gained from using the tool and the tool’s limitations. When making mathematical models, they are able to identify quantities in a practical situation and represent relationships using modeling tools such as diagrams, grid paper, tables, graphs, and equations.
Modeling numbers in problems and in computations is a central focus in Math Expressions lessons. Children learn and develop models to solve numerical problems and to model problem situations. Children continually use both kinds of modeling throughout the program.
Teacher ediTion: examples from Unit 4
MP.5 Use appropriate Tools MathBoard Modeling Explain that it is a good thing to “prove” that an answer is right. Guide children in using the MathBoard to prove that 9 + 4 is the same as 10 + 3. Ask the class to show 9 + 4 on the 10 × 10 Grid, using circles for the first partner and triangles for the second partner. Remind children to fill the entire first column before drawing in the second column.
• Make 9 circles and 4 triangles, and then add them together. You made a 10; now draw a 10-stick.
9 + 4
Lesson 4
Practice Finding Teen Totals IND IV IDUALS
MP.5 Use appropriate Tools Green Make-a-Ten Cards Children use the Green Make-a-Ten Cards to practice finding teen totals. Ask children to sort the cards according to the greater partner. They first find all the cards with 9 as the greater partner, then 8, then 7, then 6.
7+9=16
9 + 1 + 69
8+4=12
8 + 2 + 28
4+7=11
7 + 3 + 17
6+5=11
6 + 4 + 16
Have children practice first with the cards in the 9 pile. These all start in the same way (only 1 more to make 10). Then children practice with the cards in the 8 pile (2 more to make 10), the 7 pile (3 more to make 10), and the 6 pile (4 more to make 10). Tell the class that this method is still hard for many children at this time of the year, but it will get easier.
Lesson 5
Mathematical Practice 5 is integrated into Unit 4 in the following ways:
Use Appropriate ToolsMathBoard Modeling
Secret Code CardsStair Steps
Green Make-a-Ten CardsModel the Math
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Mathematical Practice 6attend to precision.
Children 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. They are careful about specifying units of measure to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, and express numerical answers with a degree of precision appropriate for the problem context. Children give carefully formulated explanations to each other.
Teacher edITIoN: examples from Unit 4
MP.6 attend to Precision Ask children to name the teen numbers from 11 to 19. Write them on the board as they count. Volunteers suggest different pairs of teen numbers, and children write to compare the numbers two ways. For example, 11 < 12 and 12 > 11. Suggest that they draw 10-sticks and circles to help them.MATH TALK Invite children to explain how
they remember which symbol to use.
Lesson 3
MP.6 attend to Precision Explain Solutions Select a card randomly from the Green Make-a-Ten Cards. Hold up the card or write the equation on the board. Invite volunteers to explain how to use the Make a Ten strategy to find the total. Repeat several times.
• This card shows the equation 5 + 6 = .How can we make a ten to find the total? Answers will vary. Possible response: I start with 6 because that is the greater number. Then, I take 4 from the 5 and add it to the 6 to make 10. There is 1 left over so the answer is 10 + 1 or 11.
Lesson 10
MATH TALK For each step of this activity, discuss how the representations of 47 are alike and different. You might wish to invite volunteers to suggest and describe different representations first.
Lesson 9
MATH TALK Discuss how the teen numbers are similar to and different from
the decade numbers. Encourage children to talk about place value when describing how the numbers are different.
•Howare18and80thesame?Answers will vary. Possible responses: Both have an eight; both names start with the same sound.
•Howare18and80different?Answers will vary. Possible responses: 18 has 8 ones, and 80 has 8 tens; the names have different ending sounds; when we say 18, we say eight-TEEN, the number of ones first and then the tens; when we say 80, we say the number of tens first, but there are no ones to say.
Lesson 8
MathematicalPractice6is integrated into Unit 4 in the following ways:
AttendtoPrecisionDescribeMethods
ExplainaSolutionDescribeaMethod
ExplainanExample
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Mathematical Practice 7Look for and make use of structure.
Children analyze problems to discern a pattern or structure. They draw conclusions about the structure of the relationships they have identified.
Teacher ediTion: examples from Unit 4
Identify Random Decade Numbers W H O L E C L A S S
MP.7 Use Structure Write the decade numbers in random order on the board. Have the class say each number and tell how many tens it has. Remind children that it is important to recognize that each number also has 0 ones, but it is not necessary to always say it.
Teacher writes: Children respond:
60 60 is 6 tens.
40 40 is 4 tens.
20 20 is 2 tens.
90 90 is 9 tens.
10 10 is 1 ten.
50 50 is 5 tens.
30 30 is 3 tens.
80 80 is 8 tens.
70 70 is 7 tens.
• How many tens and ones does 100 have? 100 is 10 tens and 0 ones.
Lesson 1
Model Tens to 100 W H O L E C L A S S
MP.7 Look for Structure Lead this activity using a MathBoard at the front of the room as children use their MathBoards at their seats.
Children draw a 10-stick through the first ten squares of the Number Path and then write a large 10 out to the right of the small 10 square.
Then they draw a 10-stick through each successive group of ten squares, writing the new 10 total out to the right (for 20 and 30) or under (for 40, 50, 60, 70) or to the left (for 80, 90, and 100).
Have children count by tens together pointing to the decade number they wrote (10, 20, 30, and so on) as you lead. You point with a sweeping motion across each ten, stopping at each new ten. (Children can just point to the numbers.)
Do this two more times so that all children see the structure of the groups of ten and how the numbers count to 100 in groups of ten. Discuss this structure.
Lesson 9
Mathematical Practice 7 is integrated into Unit 4 in the following ways:
Show tickets were sold on Friday, Saturday, and Sunday.
tens ones = tickets
tens ones = tickets
Friday
Saturday
Sunday
Compare the number of tickets sold. Use >, <, or =.
4. Friday Saturday
6. Saturday Sunday
5. Friday Sunday
7. Sunday Saturday
3. Write the number of tickets sold each day.
tens ones = tickets6 5 65
5 4 54
6 0 60Accept 5 tens 10 ones.
65
54606054
606554
< >
>>
128 UNIT 4 LESSON 18 Focus on Mathematical Practices
1_MNLESE813363_U04L18.indd 128 26/03/12 12:41 PM
STUdeNT edITIoN: LeSSoN 18, pageS 127–128
Mathematical practice 8Look for and express regularity in repeated reasoning.
Children use repeated reasoning as they analyze patterns, relationships, and calculations to generalize methods, rules, and shortcuts. As they work to solve a problem, children maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.
Teacher edITIoN: examples from Unit 4
Mp.8 Use repeated reasoning Generalize When the equations are completed, direct children’s attention to the totals. Help them see the pattern that is developing: as each number being doubled is one more, it results in a total that is two more. Perhaps they will be able to predict the next total, even though there are no more dots to ring.
Lesson 6
Mp.8 Use repeated reasoning Write these equations on the board and invite class discussion as you solve them together. Make sure children begin to notice and understand the regularity in the way they can start at any number and use counting on to find a total.
67 + 4 = 58 + 6 = 18 + 5 = Invite volunteers to make up more equations that add a 1-digit number to a 2-digit number and explain how to use counting on to find the answer.
Lesson 15
Mathematical Practice 8 is integrated into Unit 4 in the following ways:
Use Repeated ReasoningGeneralize
Focus on Mathematical practices Unit 4 includes a special lesson that involves solving real world problems and incorporates all eight Mathematical Practices. In this lesson children explore tens and ones in the real world context of a community theater.
71 64 23
Research & Math BackgroundContents Planning
Math Expressions VOCABULARY
As you teach this unit, emphasize
understanding of these terms.
• 10-group• 10-stick• teen number• decade number
See the Teacher Glossary.
Getting Ready to Teach Unit 4Learning Path in the Common Core StandardsThis unit builds on the work with teen numbers that began in Kindergarten. Children explore tens and ones using physical groupings and math drawings. Activities provide repeated experience in building 2-digit numbers with strong visual support. Children extend these place value concepts to adding with 1- and 2-digit numbers.
Help Children Avoid Common ErrorsMath Expressions gives children opportunities to analyze and correct errors, explaining why the reasoning was flawed.
In this unit we use Puzzled Penguin to show typical errors that children make. Children enjoy teaching Puzzled Penguin the correct way, why this way is correct, and why Puzzled Penguin made an error. Common errors are presented in the Puzzled Penguin feature in the following lessons:
→ Lesson 4: Correctly breaks apart a number to make a ten, but then adds the same part to 10 instead of the remaining part
→ Lesson 12: Compares the ones before comparing the tens
→ Lesson 14: Incorrectly adds tens instead of ones
In addition to Puzzled Penguin, there are other suggestions listed in the Teacher Edition to help you watch for situations that may lead to common errors. As a part of the Unit Test Teacher Edition pages, you will find a common error and prescription listed for each test item.
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from THE PROGRESSIONS FOR THE COMMON CORE STATE STANDARDS ON NUMBER AND OPERATIONS IN BASE TEN
Base-Ten Units Each place of a
base-ten numeral represents a
base-ten unit: ones, tens, tenths,
hundreds, hundredths, etc. The
digit in the ones place represents
0 to 9 of those units. Because ten
like units make a unit of the next
highest value, only ten digits are
needed to represent any quantity
in base ten. The basic unit is a
one (represented by the rightmost
place for whole numbers). In
learning about whole numbers,
children learn that ten ones
compose a new kind of unit called
a ten. They understand two-digit
numbers as composed of tens and
ones, and use this understanding in
computations, decomposing 1 ten
into 10 ones and composing a ten
from 10 ones.
Decade Numbers
Lesson
1
Using the MathBoard One of the first steps in mastering numbers to 100 involves learning the names of the decade numbers and making the association between these words and the corresponding tens they represent.
(10, 20, 30, . . .)
(1 ten, 2 tens, 3 tens, . . .)
With this knowledge, children are soon able to add tens mentally (30 + 40 = 3 tens + 4 tens). These skills are introduced and practiced with a number of physical supports.
Special configurations on the MathBoard, such as the Dot Array and the 10 × 10 Grid shown here, serve as concrete supports for counting by tens.
10 20
Using Their Fingers During the daily Quick Practice routines, children flash fingers as they practice counting to 100 by tens. Later in the unit, they will use number flashes to show a number with both tens and ones (34 = 3 tens and 4 ones).
20 10 30
31
32 33 34
34
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Using Secret Code Cards With the help of Demonstration Secret Code Cards, children practice saying and visualizing decade numbers in random order. The cards help children make the connection between these groupings and their numeric symbols.
60
6 90
90 060 is 6 tens. 90 is 9 tens.
Quick Practice student-led rhyming activities help children build fluency with adding a ten when decade numbers are given out of sequence. Other activities build fluency with numbers before and after these groups of ten.
40 lions in a den. Add a ten. (50)
70 lions in a den. Add a ten. (80)
39 tigers at the door. Here’s one more. (40)
90 tigers in a line. With 1 less, there’s 89.
Tens and Ones
Lessons
2 3 7 8
9 11 17 18
Teen Numbers After learning the decade numbers, children begin building an integrated concept of tens and ones starting with teen numbers. Integrating tens and ones into 2-digit numbers represents an enormous conceptual advance over simply counting by tens, and this skill takes practice. In Unit 4, practice is provided in a variety of ways as children repeatedly link tens groupings to concrete quantities, number words, and written numbers. In this way, they begin to construct the complex web of meanings and symbols that make up 2-digit numbers.
from THE PROGRESSIONS FOR THE COMMON CORE STATE STANDARDS ON NUMBER AND OPERATIONS IN BASE TEN
Teen Numbers Grade 1 students
take the important step of viewing
ten ones as a unit called a “ten.”
They learn to view the numbers
11 through 19 as composed of 1
ten and some ones. They learn to
view the decade numbers 10–90,
in written and in spoken form, as 1
ten to 9 tens. More generally, first
graders learn that the two digits
of a two-digit number represent
amounts of tens and ones, e.g., 67
represents 6 tens and 7 ones. The
number words continue to require
attention.
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10-Sticks and Circles Children learn to sort 2-digit numbers into tens and ones by drawing sticks (worth 10) and circles (worth 1). This system of representation allows them to visualize the meaning of the numbers and to understand the separate functions of the tens and ones in our number system. The system evolves gradually from the Dot Array on the MathBoard to freehand representations:
Development of 10-Sticks on Dot Array Freehand
10 + 4 = 14
Story Problems In this unit, children solve teen story problems by differentiating tens and ones.
Marco has a bag of 10 marbles and 6 extra marbles. How many marbles does he have?
Secret Code Cards A tens card and a ones card are used together to demonstrate the “invisible” zero in the tens place. These cards offer visual reinforcement of place value concepts.
40
4 0 40
4 03
33
343 = 40 + 3
The backs of the Secret Code Cards feature ten-sticks and circles that correspond to the numbers on the front. The back of the 70-card and the back of the 8-card are shown here.
Ones Tens
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Number Path The numbers to 100 are presented in order and in groups of ten around the edge of the MathBoard. Children draw tens and ones on the Number Path and relate them to the tens and ones they draw freehand.
73
70 + 3 = 73
7 tens and 3 ones
Math Games Math Expressions introduces several games to help children build concepts of tens and ones. In a game called One Hundred Ants, children draw numbers to represent ants and keep adding on new ants until they reach 100. The Number Path below shows 37 ants and 5 more just added. Children see that they made a new 10-group, and so they draw a 10-stick through the circles from 31 through 40. This prepares them for the regrouping they will do later.
28 34 37 422523
17
13
8
Unseen Numbers Children also solve problems with multiples of 10 in which the tens are unseen. They are represented as boxes, jars, or other containers labeled 10. Because it is not possible to count each object, children must apply ten-structured concepts to find the answer.
Each box has 10 muffins. How many muffins are there?
10 10
10 10
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from THE PROGRESSIONS FOR THE COMMON CORE STATE STANDARDS ON NUMBER AND OPERATIONS IN BASE TEN
Compare Numbers Grade 1
students use their base-ten work
to help them recognize that the
digit in the tens place is more
important for determining the size
of a two-digit number. They use
this understanding to compare
two two-digit numbers, indicating
the result with the symbols >, =,
and <. Correctly placing the > and < symbols is a challenge for early
learners. Accuracy can improve if
students think of putting the wide
part of the symbol next to the
larger number.
Comparing Numbers
Lessons
3 12 16
Secret Code Cards These place value cards help children focus on the place value of the numbers. Because they can pull the cards apart and put them together, they can always see the hidden ten. Children use familiar models to compare teen numbers and then 2-digit numbers. Children use tens and ones language to discuss which number is greater and why.
10
1 04
410
1 08
8
30
3 04
404
430
3 040
4 03
303
340
4 0
Finally, children write the comparison two ways.
Compare 53 and 54.
<
>
53
54
54
53
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Research & Math BackgroundContents Planning
Addition Strategies
Lessons
3 4 5 10
Counting On for Teen Numbers Children learn to solve problems such as 9 + 4 = using their fingers to count on. Children also solve teen story problems by counting on and then regrouping to emphasize the ten that is contained in the teen number.
10 11
12 13
9 Make a Ten for Teen Totals Children use the Green Make-a-Ten Cards to think through making a ten to add.
9+5=
back
front
9+5=14
9 + 1 + 49Green Make-a-Ten Cards
(Student Activity Bookpages 105–108)
Children also solve problems where they model making a ten to solve the problem.
9 alligators swim in the river.
Then 5 more alligators jump into the river.
How many alligators are in the river now?
from THE PROGRESSIONS FOR THE COMMON CORE STATE STANDARDS ON OPERATIONS AND ALGEBRAIC THINKING
Make a Ten
These make-a-ten methods have
three prerequisites reaching back
to Kindergarten:
a. knowing the partner that makes
10 for any number,
b. knowing all decompositions for
any number below 10 so that
you can find the partner for the
number that makes 10, and
c. knowing all teen numbers as
10 + n (e.g., 12 = 10 + 2,
15 = 10 + 5).
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from THE PROGRESSIONS FOR THE COMMON CORE STATE STANDARDS ON OPERATIONS AND ALGEBRAIC THINKING
Doubles Another Level 3 method
that works for certain numbers is a
doubles +1 or +2 method: 6 + 7 =
6 + (6 + 1) = (6 + 6) + 1 = 12 + 1 =
13. These methods do not connect
with place value the way make-a-
ten methods do.
Decomposing Numbers
Lessons
6 10 13 14
15 16
Doubles This strategy is based on understanding how to decompose one addend based on the other addend. This is a difficult strategy for young children, but with lots of practice it can help with mentally adding two numbers with a total greater than ten.
4 + 4 = 8 8 + 8 = 16
4 + 5 = 8 + 1 = 9 8 + 7 = 16 – 1 = 15
4 + 6 = 8 + 2 = 10 8 + 6 = 16 – 2 = 14
Adding Tens and Ones In Unit 4, children begin to prepare for multi-digit operations by adding decade numbers and by adding single-digit numbers to decade numbers. These exercises are designed to help children distinguish between tens and ones.
5 + 4
50 + 40
50 + 4
Adding a Two-Digit and One-Digit Number Later in the unit, children add single-digit numbers to any 2-digit number by counting on. Because most of these problems involve creating a new ten group, they help prepare children for later 2-digit addition with grouping.
UNIT 4 | Overview | 273II
Research & Math BackgroundContents Planning
Focus on Mathematical Practices
Lesson
18
The Standards for Mathematical Practice are included in every lesson of this unit. However, there is an additional lesson that focuses on all eight Mathematical Practices. In this lesson, children use what they know about addition to solve problems about a community theater.
