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NYC10 New York City Implementation Guide

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Lesson 1.2 1

Round to the Nearest Ten or HundredCommon Core Standard CC.3.NBT.1Use place value understanding to round whole numbers to the nearest 10 or 100.

Lesson Objective Round 2- and 3-digit numbers to the nearest ten or hundred.

Essential Question How can you round numbers?

Vocabulary round

Access Prior Knowledge Discuss with students situations where you do not need to know an exact number, but knowing about how much or about how many is suffi cient. For example, the length of a car is about 10 feet. There are about 100 seats in the auditorium. The height of a house is about 20 feet. • How are these numbers alike? They all have

zeros. They all tell about how many.

Unlock the Problem When would you round a number? Discuss the problem. Be sure that students understand that 32 is an exact number and they need to round 32 to the nearest ten.• What is an example of a rounded number?

Numbers with a zero at the end can be examples of rounded numbers, such as 10, 20, 30, and so on.

• What are the tens that are closest to the number 32? 30 and 40

One Way• Why is a number line a good way to think

about which numbers should be rounded? A number line can show how far apart numbers are from each other so they can be compared easily.

• Why does the fi rst number line include tens and not hundreds? Possible answer: because 32 is a 2-digit number and I am rounding 32 to the nearest 10

• Between which two tens is 32? Which ten is it closer to? Explain. 30 and 40. 32 is closer to 30. It is only 2 numbers away from 30 but 8 numbers away from 40.

Use Math Talk to focus students’ thinking on the fact that more than one number would round to 30. • What makes a number able to be rounded

to 30? It must be 25 or greater, or less than 35. • In which direction would you round 28 to

get to 30? up

• In which direction would you round 34 to get to 30? down

You might draw a number line on the board from 20 to 40 so students can see that the numbers 25–29 and 31–24 would round to 30. • Look at the second number line. How is

rounding to the nearest hundred similar to rounding to the nearest ten? Possible answer: when I round to the nearest ten, I look at the number line to see which ten the number is closer to. When I round to the nearest hundred, I look at the number line to see which hundred the number is closer to.

• What is 144 rounded to the nearest hundred? 100

1 ENGAGE

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AnimatedMath Models

MATHEMATICALPRACTICES

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Lesson 1.2

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New York City Implementation Guide NYC11

SOLUCIONA el problema

MÉTODOS MATEMÁTICOSMÉTODOS MATEMÁTICOS

Nombre

Capítulo 1 9

De una manera Usa una recta numérica para redondear.

A Redondea 32 a la decena más próxima.

100 20 30 40

32

Halla entre qué decenas está el número.

32 está entre _ y _.

32 está más cerca de _ que de _.

32 redondeado a la decena más próxima es _.

Entonces, la longitud del bate de Mara redondeada a la

decena de pulgadas más próxima es _ pulgadas.

B Redondea 174 a la centena más próxima.

0 100 200 300

174

Halla entre qué centenas está el número.

174 está entre _ y _.

174 está más cerca de _ que de _.

Entonces, 174 redondeado a la centena más próxima es _ .

Al redondear un número, hallas un número que te indica alrededor de cuánto o de cuántos.

El bate de béisbol de Mara mide 32 pulgadas de longitud. ¿Cuál es su longitud redondeada a la decena de pulgadas más próxima?

Lección 1.2Redondear a la decena o la centena más próxima Pregunta esencial ¿Cómo puedes redondear números?

Menciona otros tres números que se redondeen en 30 al redondearlos a la decena más próxima. Explícalo.

ESTÁNDAR COMÚN CC.3.NBT.1

Use place value understanding and properties of operations to perform multi-digit arithmetic.

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30

40

40

30

30

30

Respuesta posible: 28, 31, 34; Explicación posible: 28 está entre 20 y 30 pero está más cerca de 30; 31 y 34 están entre 30 y 40 pero están más cerca de 30.

200

200

100

100

200

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Lesson 1.2 3

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Try This! Discuss with students that there are different ways to round a number. Then, have students complete Parts A and B.• How is rounding a 3-digit number, like 718,

to the nearest ten similar to rounding the number to the nearest hundred? Possible answer: in each case, I fi nd the two tens or hundreds the number is between and then I decide which is closer.

• How is rounding a 3-digit number to the nearest ten different than rounding to the nearest hundred? Possible answers: the number lines I use are different. To round to the nearest ten, I need to fi nd the two tens that the number is between. To round to the nearest hundred, I need to fi nd the two hundreds that the number is between. I write a zero for the digit in the ones place when rounding to ten. I write a zero for the digits in the tens and ones places when rounding to hundred.

Another WayStudents should recognize that the result of rounding using place value is the same as rounding on a number line.• How might rounding using place value be

quicker? Possible answer: I don’t have to draw a number line to see the numbers.

• How is rounding to the nearest ten and rounding to the nearest hundred using place value the same? In each case, I look at the digit to the right of the place I am rounding to. If the digit is less than 5, the digit in the rounding place stays the same. If the digit is 5 or greater, the digit in the rounding place increases by 1. I write zeros for the digits to the right of the rounding place.

Use Math Talk to show students how using place value is similar to using a number line.• How can you look at the ones place to tell

if 54 should be rounded up or down? If the number in the ones place is 1, 2, 3, or 4, it should be rounded down. If the number in the ones place is 5, 6, 7, 8, or 9, it should be rounded up.

• Which place value digit should you look at to decide if 168 should be rounded to 200? Look at the tens place value.

• In which place value digit should you look at to decide if 81 should be rounded to 100? Look at the tens place value.

• In which place value digit should you look at to decide if 81 should be rounded to 80? Look at the ones place value.

c

COMMON ERRORS

COMMON ERRORS

Error Students may round numbers incorrectly because they do not look at the place to the immediate right.

Example To round 718 to the nearest hundred, students may look at the 8 and round to 800.

Springboard to Learning Have students circle the place to which they are rounding and underline the number to the immediate right before rounding.





700 710 720

718

700 750 800

718


10

A Decena más próxima B Centena más próxima

718 está más cerca de _

que de _.

Entonces, 718 se redondea en _.

718 está más cerca de _

que de _.

Entonces, 718 se redondea en _.

De otra manera Usa el valor posicional.

A Redondea 63 a la decena más próxima.

Piensa: El dígito en el lugar de las unidades indica si el número está más cerca de 60 o de 70.

3 5

Entonces, el dígito de las decenas queda igual. Escribe 6 en el lugar de las decenas.

Escribe cero en el lugar de las unidades.

Entonces, 63 redondeado a la decena más

próxima es _.

B Redondea 457 a la centena más próxima.

Piensa: El dígito en el lugar de las decenas indica si el número está más cerca de 400 o de 500.

5 5

Entonces, el dígito de las centenas aumenta en uno. Escribe 5 en el lugar de las centenas.

Escribe cero en el lugar de las decenas y de las unidades.

Entonces, 457 redondeado a la centena más

próxima es _.

457

¡Inténtalo! Redondea 718 a la decena y a la centena más próxima. Ubica y rotula 718 en las rectas numéricas.

Explica en qué se parecen usar el valor posicional y usar una recta numérica.

63

• Halla el lugar al que quieres redondear.• Observa el dígito de la derecha.• Si el dígito es menor que 5, el dígito en el lugar de redondeo queda igual.• Si el dígito es igual a o mayor que 5, el dígito en el lugar de redondeo aumenta en uno.• Escribe ceros en los dígitos a la derecha del lugar de redondeo.

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Charla matemática: Explicación posible: Cuando uso el valor posicional, observo el dígito a la derecha del lugar de redondeo para ver de qué decena o centena está más cerca. Si el dígito que sigue es igual a o mayor que 5, estará más cerca de la decena o la centena que sigue. Cuando uso una recta numérica, puedo ver de qué decena o centena está más cerca el número.

60

500

,

5

720 700

800

700720

710

Revise el trabajo de los estudiantes.

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Share and Show • Guided Practice

The fi rst problem connects to the learning model. Have students use the MathBoard to explain their thinking. Use Math Talk to focus on students’ understanding of rounding. Encourage students to explain their thinking. Use Exercises 6 and 7 for Quick Check. Students should show their answers for the Quick Check on the Math Board.

On Your Own • Independent

Practice

If students complete Exercises 6 and 7 correctly, they may continue with Independent Practice. Encourage students to complete the On Your Own section independently, but provide guidance as necessary. Ask questions to make sure students know what they need to fi nd out.• In Exercises 10–12, why is 550 not a

possible answer? I am asked to round to the nearest hundred, and 550 is not a hundred.

3 PRACTICE

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700 710 720

718

700 750 800

718

MATHEMATICAL PRACTICES

10

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A Nearest Ten B Nearest Hundred

718 is closer to _ than it is

to _.

So, 718 rounds to _.

718 is closer to _ than it is

to _.

So, 718 rounds to _.

Another Way Use place value.

A Round 63 to the nearest ten.

Think: The digit in the ones place tells if the number is closer to 60 or 70.

3 l 5

So, the tens digit stays the same. Write 6 as the tens digit.

Write zero as the ones digit.

So, 63 rounded to the nearest ten

is _.

B Round 457 to the nearest hundred.

Think: The digit in the tens place tells if the number is closer to 400 or 500.

5 l 5

So, the hundreds digit increases by one. Write 5 as the hundreds digit.

Write zeros as the tens and ones digits.

So, 457 rounded to the nearest hundred

is _.

457

• Find the place to which you want to round.• Look at the digit to the right.• If the digit is less than 5, the digit in the rounding place stays the same.• If the digit is 5 or greater, the digit in the rounding place increases by one.• Write zeros for the digits to the right of the rounding place.

Try This! Round 718 to the nearest ten and hundred. Locate and label 718 on the number lines.


Explain how using place value is similar to using a number line.

63

Math Talk: Possible explanation: when you use place value, you look at the digit to the right of the rounding place to see to which ten or hundred the digit is closer. If the next digit is 5 or greater, it will be closer to the next ten or hundred. When you use a number line, you can see to which ten or hundred the number is closer.

60

500

,

5

720 700

800

700720

710

Check students’ work.

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Lesson 1.2 5

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The fi rst problem connects to the learning model. Have students use the MathBoard to explain their thinking. Use Math Talk to focus on students’ understanding of rounding. Encourage students to explain their thinking. Use Exercises 6 and 7 for Quick Check. Students should show their answers for the Quick Check on the Math Board.

Quick Check

If

Rt I 1

2

3Quick Check

If

Rt I RR1

2

3

a student misses Exercises 6 and 7

Differentiate Instruction with • RtI Tier 1 Activity, p. 9B

• Reteach 1.2

Soar to Success Math 15.15, 15.17

Then


Practice

If students complete Exercises 6 and 7 correctly, they may continue with Independent Practice. Encourage students to complete the On Your Own section independently, but provide guidance as necessary. Ask questions to make sure students know what they need to fi nd out.• In Exercises 10–12, why is 550 not a

possible answer? I am asked to round to the nearest hundred, and 550 is not a hundred.

• Why are you being asked to fi nd which hundred 548 is closer to? That is how you fi gure out how to round the number.

• In Exercises 13–15, is it possible to have the same answer when rounding to the nearest ten and hundred? Explain. Yes, when a number is rounded to the closest ten, it is possible that the number rounds to a hundred, such as Exercise 14. The closest ten to 298 is 300. The closest hundred to 298 is also 300.

Go Deeper To extend their thinking, ask students to fi nd the greatest number that rounds to 500. Remind them that they are rounding to the hundreds place, not the tens place.• Would you be rounding up or down to fi nd

the greatest number that rounds to 500? down

• What is that number? 549

3 PRACTICE

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400 500

548

600

30 5040 60

46

Nombre

Capítulo 1 • Lección 2 11

Redondea a la decena más próxima.

Redondea a la centena más próxima.

Redondea a la decena y a la centena más próxima.

Comunicar y mostrarNUbica y rotula 46 en la recta numérica. Redondea a la decena más próxima.

4. 19 _ 5. 66 _ 6. 51 _

7. 463 _ 8. 202 _ 9. 658 _

13. 576 _

_

14. 298 _

_

15. 844 _

_

1. 46 está entre _ y _.

2. 46 está más cerca de _ que de _.

3. 46 redondeado a la decena más próxima

es _.

10. 548 está entre _ y _.

11. 548 está más cerca de _ que de _.

12. 548 redondeado a la centena más próxima es _.

¿Cuál es el mayor número que se redondea en 50 al redondearlo a la decena más próxima? ¿Cuál es el menor número? Explícalo.

Por tu cuentaUbica y rotula 548 en la recta numérica. Redondea a la centena más próxima.

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600

600

500

500

500

580

600 300

mayor número: 54; menor número: 45; Explicación posible: Como este grupo de números se redondea en 50 (45, 46, 47, 48, 49, 50, 51, 52, 53, 54), el mayor número es 54 y el menor número es 45.

300 840

800

50

5040

40

50

20

500 200 700

70 50



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MATHEMATICAL PRACTICESMATHEMATICAL PRACTICES

400 500

548

600

30 5040 60

46

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Chapter 1 • Lesson 2 11

Name

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Round to the nearest ten.

Round to the nearest hundred.

Round to the nearest ten and hundred.

Share and Show NLocate and label 46 on the number line. Round to the nearest ten.

4. 19 _ 5. 66 _ 6. 51 _

7. 463 _ 8. 202 _ 9. 658 _

13. 576 _

_

14. 298 _

_

15. 844 _

_

1. 46 is between _ and _.

2. 46 is closer to _ than it is to _.

3. 46 rounded to the nearest ten is _.

10. 548 is between _ and _.

11. 548 is closer to _ than it is to _.

12. 548 rounded to the nearest hundred is _.

What is the greatest number that rounds to 50 when rounded to the nearest ten? What is the least number? Explain.

On Your OwnNLocate and label 548 on the number line. Round to the nearest hundred.

600

600

500

500

500

580

600 300

greatest 54; least 45; Possibleexplanation: since this group of numbers rounds to 50 (45, 46, 47,48, 49, 50, 51, 52, 53, 54), the greatest number is 54 and the least number is 45.

300 840

800

50

5040

40

50

20

500 200 700

70 50



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Lesson 1.2 7

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Problem Solving For Exercises 16–18, students use information from a table and rounding to the nearest ten or hundred to solve problems. In Exercise 18, encourage students to discuss strategies they can use to determine which numbers round to 800.

ProblemTo solve Exercise 19, students will have to use higher order thinking skills. Remind them of when a number is rounded up to the nearest ten and when a number is rounded down to the nearest ten. • Does 351 round to 360 when rounding to

the nearest ten? Explain. No, 351 is closer to 350 than it is to 360, so it rounds to 350 instead of 360.

• Does 357 round to 360 when rounding to the nearest ten? Explain. Yes, the closest ten to 357 is 360.

• How can you use this thinking to fi nd other numbers that round to 360? I can think of which numbers are closest to 360 without rounding to another ten. For example, 356, 359, 361, and 364 all round to 360.

Test Prep CoachTest Prep Coach helps teachers to identify common errors that students can make. In Exercise 21, if students selected: • A or C, they rounded to the wrong ten.• D, they rounded to the nearest hundred.

Essential QuestionHow can you round numbers? I can use a number line or place value.

Math Journal Describe how to round 678 to the nearest hundred.


4 SUMMARIZE MATHEMATICALPRACTICES





Visitantes de laexposición de las jirafas

Día Cantidad de visitantes

Domingo

Lunes

Martes

Miércoles

Jueves

Viernes

894

793

438

362

839

725

598Sábado

Visitantes de laexposición de las jirafas

Día Cantidad de visitantes

Domingo

Lunes

Martes

Miércoles

Jueves

Viernes

894

793

438

362

839

725

598Sábado

Representar • Razonar • InterpretarMÉTODOSMATEMÁTICOS

12 PARA PRACTICAR MÁS:Cuaderno de práctica de los estándares, págs. P5 y P6

Resolución de problemasUsa la tabla para resolver los problemas 16 a 18.

16. ¿Qué día fueron alrededor de 900 visitantes a la exposición de las jirafas?

17. ¿Qué cantidad de visitantes redondeada a la decena más próxima fue el domingo a la exposición de las jirafas?

18. ¿Qué dos días fueron alrededor de 800 visitantes por día a la exposición de las jirafas?

19. Escribe cinco números que se redondeen en 360 al redondearlos a la decena más próxima.

20. ¿Cuál es el error? Camilo dijo que 555 redondeado a la decena más próxima es 600. ¿Cuál es el error de Camilo? Explícalo.

21. Preparación para la prueba ¿Cuánto es 438 redondeado a la decena más próxima?

A 450

B 440

C 430

D 400©

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Com

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PRÁCTICA ADICIONAL:Cuaderno de práctica de los estándares, pág. P27

domingo

lunes y jueves

Resultado posible: 356, 357, 359, 361, 364

El resultado debe ser 560. Explicación posible:

Camilo redondeó a la centena más próxima en lugar de a la decena más próxima.

890 visitantes

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Lesson 6.6 9

Investigate • Model with ArraysCommon Core Standard CC.3.OA.3Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quanti-ties, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.Also CC.3.OA.2

Lesson Objective Model division by using arrays.

Essential Question How can you use arrays to solve division problems?

Materials square tilesAccess Prior Knowledge Have students use square tiles to review making arrays for multiplication. Remind students that an array is a set of objects arranged in rows. There is the same number of tiles in each row.• What real-world examples can arrays

represent? Possible answers: marching bands or chairs set up in equal rows

Have students model 4 3 5 5 ______. • Why did you make 4 rows of 5? Possible

answer: 4 x 5 means 4 groups, or rows, of 5

• How did you fi nd the product? Possible answer: I skip counted by 5 four times to get 20.

Investigate Work together with students to complete the steps of the activity. Be sure students make equal rows of 5. Explain that making equal rows is necessary so that the array is easy to follow and analyze. Remind students of the difference between a row and the number of tiles in the row. • In which direction must you work to make

a row of tiles? across from left to right

• In which direction must you count to fi nd how many rows are in your completed array? from top to bottom

• What should you do if you put the array together and do not have an equal number of tiles in each row? I must count again to see if I made an error, because an array must have an equal number of tiles in each row.

Draw Conclusions • What number did you divide? Explain.

30; possible answer: I started with 30 tiles.

• What number did you divide by? Explain. 5; possible answer: I made rows of 5 tiles each.

ProblemExercise 3 requires students to generalize from the 5 by 6 array to fi nding a new array with 6 tiles in each row.• Use the same array to describe a strategy

for fi nding the number of rows of 6 tiles that are in 30. Possible answer: I knew that 30 4 5 5 6, so 30 4 6 5 5. Count the number of tiles in a row in the fi rst array; Think: What number times 6 equals 30?

• How is this array different from the one you put together in the Investigate section that shows how many rows of 5 there are in 30? The fi nal number, 30, is the same for both arrays. This one has 6 tiles in one row, while the fi rst array had 5 tiles in one row.

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Lesson 6.6





Investigar

Investigar

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Capítulo 6 231

Nombre

Representar con matricesPregunta esencial ¿Cómo puedes usar matrices para resolver problemas de división?

Lección 6.6

1. Explica cómo usaste las fichas cuadradas para hallar el número de hileras de 5 fichas que hay en 30.

2. ¿Qué ecuación de multiplicación podrías escribir para la matriz? Explícalo.

3. Aplica Indica cómo usar una matriz para hallar cuántas hileras de 6 hay en 30.

Sacar conclusionesN

Materiales ■ fichas cuadradas

Puedes usar matrices para representar la división y hallar grupos iguales.

A. Cuenta 30 fichas cuadradas. Forma una matriz para hallar cuántas hileras de 5 hay en 30.

B. Forma una hilera de 5 fichas cuadradas.

C. Sigue hasta que hayas formado todas las hileras de 5 fichas cuadradas posibles.

¿Cuántas hileras de 5 formaste? _________

ESTÁNDAR COMÚN CC.3.OA.3

Represent and solve problems involving multiplication and division.

Explicación posible: Dispuse las 30 � chas cuadradas en hileras de 5 � chas cuadradas.

Luego conté el número de hileras que formé. Había 6 hileras. Entonces, hay 6 hileras de 5 en 30.

6 3 5 5 30; Explicación posible: Hay 6 hileras de 5 � chas cuadradas cada una.

6 grupos de 5 son 30.

Respuesta posible: Formo hileras de 6 hasta usar las 30 � chas cuadradas. Hay

5 hileras de 6 en 30.

6 hileras

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Investigate

Investigate

Name

Chapter 6 231

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Model with ArraysEssential Question How can you use arrays to solve division problems?

Lesson 6.6

1. Explain how you used the tiles to find the number of rows of 5 in 30.

2. What multiplication equation could you write for the array? Explain.

3. Apply Tell how to use an array to find how many rows of 6 are in 30.

Draw ConclusionsN

Materials ■ square tiles

You can use arrays to model division and find equal groups.

A. Count out 30 tiles. Make an array to find how many rows of 5 are in 30.

B. Make a row of 5 tiles.

C. Continue to make as many rows of 5 tiles as you can.

How many rows of 5 did you make? _________

COMMON CORE STANDARD CC.3.OA.3

Represent and solve problems involving multiplication and division.

Possible explanation: I placed the 30 tiles in rows of 5 tiles. Then I counted

the number of rows I made. There were 6 rows. So, there are 6 rows of 5 in 30.

6 3 5 5 30; Possible explanation: there are 6 rows of 5 tiles each.

6 groups of 5 are 30.

Possible answer: make rows of 6 until all 30 tiles are used. There are 5 rows of 6 in 30.

6 rows

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Lesson 6.6 11

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Make ConnectionsHelp students connect the array to a division equation.• Why is 30 the dividend? because that is the

number of tiles being divided, or separated into equal groups

• What represents the quotient in this array? The number of rows is the quotient.

Try This!After students fi nd the quotient, have them share how they wrote the division equation that the array represents. Then check their quotients. • To divide, you have used drawing equal

groups or circling equal groups, repeated subtraction, counting back on a number line, and arrays. Which method do you think is the easiest? Explain.

Have several students explain their choices. Students’ explanations may include the following: • Drawing or circling the groups—it’s easier

when you can see the problem in a picture. • Repeated subtraction—its faster to keep

subtracting than to make a drawing. • Number line—it’s easy to count the jumps

to get the quotient. • Array—it’s easy to put the tiles in equal

rows and then count the number of rows. Use Math Talk to focus on students’ understanding of using an array to divide. Explain that students can count a whole row of an array as one number in a division problem. So, if an array has 4 rows, students can think: 24 4 4. The answer to that problem should be found by looking at the number of tiles in a row in the array, which is 6. Similarly, if there are 6 tiles in a row in the array, students can think: 24 4 6. They can fi nd the answer by looking at the number of rows, which is 4.• Why is it important to make neat rows

when making an array? If neat rows are not made, it is easy to make mistakes counting the rows and number of titles in a row, and therefore get the wrong answer.

c COMMON ERRORS

COMMON ERRORS

Error Students may make an incorrect array for a division problem.

Example How many rows of 3 are in 18?

Springboard to Learning Review with students that if the problem asks for rows of 3, you put that number in each row and count the number of rows to get the answer. If the problem gives the number of rows, you start by making that many rows with one tile in each row. You keep adding one tile to each row until all the tiles are used. Then you count the numbers of tiles in each row.





MÉTODOS MATEMÁTICOS

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Hacer conexionesN

Puedes escribir una ecuación de división para mostrar cuántas hileras de 5 hay en 30. Completa el siguiente dibujo para mostrar la matriz que formaste en la sección Investigar.

30 4 5 5 j

En 30, hay _ hileras de 5 fichas cuadradas.

Entonces, 30 4 5 5 _.

¡Inténtalo! Cuenta 24 fichas cuadradas. Forma una matriz que tenga el mismo número de fichas cuadradas en 4 hileras. Coloca 1 ficha cuadrada en cada una de las 4 hileras. Luego sigue colocando 1 ficha cuadrada en cada hilera hasta que hayas usado todas las fichas cuadradas. Dibuja tu matriz abajo.

• ¿Cuántas fichas cuadradas hay en cada hilera? ___

• ¿Qué ecuación de división puedes escribir para tu matriz? _____

Puedes dividir para hallar el número de hileras iguales o para hallar el número de objetos en cada hilera.

Explica cómo formar una matriz te ayudaa dividir.

Explicación posible: En la matriz se muestra cuántas chas cuadradas hay en cada hilera, y eso me ayuda a resolver el problema de división.

6

6

24 4 4 5 6 ó 24 4 6 5 4

6 chas cuadradas

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Exercises 5–8 are examples of partitive division. Before students complete the page, ask a volunteer to explain how he or she will fi nd the answer to Exercise 5. Separate 25 tiles into 5 groups by placing one tile in each of 5 rows. Place one tile at a time in each row until all tiles are used. Count the number of tiles in each row to fi nd the quotient.

Remind students to write the division equations their arrays represent.Use Exercises 2 and 6 for Quick Check. Students should show their answers for the Quick Check on the MathBoard. Use Math Talk to focus on students’ understanding of using arrays to model division. • How are the number of rows in an array

and the number of tiles in a row related? You must fi nd both to know how to divide using an array. If you know the total and one of the numbers, you can fi nd the other number.

Give students an extra example to make sure they understand the concept. • Look at Exercise 9 again. What does the

number 3 in the problem tell you? how many are in each row

• What does the number 5 in the problem tell you? the number of rows in the array

• What division problem can you write if you know that there is a total of 15 tiles and there are 5 tiles in a row? 15 4 5 5 3

• What does the 3 tell you about the array? It tells the number of rows.

3 PRACTICE







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Make ConnectionsN

You can write a division equation to show how many rows of 5 are in 30. Show the array you made in Investigate by completing the drawing below.

30 4 5 5 j

There are _ rows of 5 tiles in 30.

So, 30 4 5 5 _.

Try This! Count out 24 tiles. Make an array with the same number of tiles in 4 rows. Place 1 tile in each of the 4 rows. Then continue placing 1 tile in each row until you use all the tiles. Draw your array below.

• How many tiles are in each row? ___

• What division equation can you write for your array? _____

You can divide to find the number of equal rows or to find the number in each row.

Explain how making an array helps you divide.

Possible explanation: it helps me solve the division problem by showing how many tiles are in each row.

6

6

6 tiles

24 4 4 5 6 or 24 4 6 5 4

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Lesson 6.6 13

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Exercises 5–8 are examples of partitive division. Before students complete the page, ask a volunteer to explain how he or she will fi nd the answer to Exercise 5. Separate 25 tiles into 5 groups by placing one tile in each of 5 rows. Place one tile at a time in each row until all tiles are used. Count the number of tiles in each row to fi nd the quotient.

Remind students to write the division equations their arrays represent.Use Exercises 2 and 6 for Quick Check. Students should show their answers for the Quick Check on the MathBoard. Use Math Talk to focus on students’ understanding of using arrays to model division. • How are the number of rows in an array

and the number of tiles in a row related? You must fi nd both to know how to divide using an array. If you know the total and one of the numbers, you can fi nd the other number.

Give students an extra example to make sure they understand the concept. • Look at Exercise 9 again. What does the

number 3 in the problem tell you? how many are in each row

• What does the number 5 in the problem tell you? the number of rows in the array

• What division problem can you write if you know that there is a total of 15 tiles and there are 5 tiles in a row? 15 4 5 5 3

• What does the 3 tell you about the array? It tells the number of rows.

3 PRACTICE Quick Check

If

Rt I RR1

2

3



• Reteach 6.6

Soar to Success Math 13.17

Then






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Nombre

Comunicar y mostrarUsa fichas cuadradas para formar una matriz. Resuelve.

1. ¿Cuántas hileras de 3 hay en 18?

_____


_____


_____


_____

5. 25 fichas cuadradas en 5 hileras

_____


_____


_____


_____

11. Muestra dos manerasen que podrías formar una matriz confichas cuadradas para 18 4 6. Sombrea cuadrados en la cuadrícula para mostrar las matrices.

Forma una matriz. Luego escribe una ecuación de división.


_____


_____

Explica cuándo debes contar el número de hileras para hallar el resultado y cuándo debes contar el número de fichas cuadradas que hay en cada hilera para hallar el resultado.

6 hileras 2 hileras

4 hileras3 hileras

25 4 5 5 5 14 4 2 5 7

28 4 4 5 7 27 4 9 5 3

15 4 3 5 5 24 4 8 5 3

Explicación posible: Cuento el número de hileras cuando conozco el número que hay en cada hilera. Cuento el número de � chas cuadradas que hay en cada hilera cuando conozco el número de hileras.

Revise los dibujos de los estudiantes.

Revise las matrices de los estudiantes.

Revise las matrices de los estudiantes.

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MATHEMATICAL PRACTICESMATHEMATICAL PRACTICES

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Name


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Share and ShowUse square tiles to make an array. Solve.

1. How many rows of 3 are in 18?

_____


_____


_____


_____

5. 25 tiles in 5 rows

_____


_____


_____


_____

11. Show two ways you could make an array with tiles for 18 4 6. Shade squares on the grid to record the arrays.

Make an array. Then write a division equation. Check students’ arrays.


_____


_____

Explain when you count the number of rows to find the answer and when you count the number of tiles in each row to find the answer.

6 rows 2 rows

4 rows3 rows

25 4 5 5 5 14 4 2 5 7

28 4 4 5 7 27 4 9 5 3

15 4 3 5 5 24 4 8 5 3

Possible explanation: you count the number of rows when you know the number in each row. You count the number of tiles in each row when you know the number of rows.

Check students’ drawings.

Check students’ arrays.

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Lesson 6.6 15

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Unlock the Problem In Exercise 12, students draw an array to solve a problem in context. In Step d, have students share other strategies they could use to solve the problem. • Do you fi nd repeated subtraction to be

easier or harder than making arrays? Explain. Answers will vary.

Encourage students to express their thoughts about making arrays and why they fi nd them easier or harder than repeated subtraction. As students express their thoughts about arrays and what they fi nd diffi cult or easy about them, address their apprehensions . You may fi nd that students are unsure about whether they have made the correct number of rows or put the correct number of objects in each row.

Remind them that a row is the distance across from left to right. If they are showing that Thomas planted 4 seedlings in a row, they must draw 4 objects in one single row. Then they can fi nd the number of rows it takes to reach 28.

Test Prep CoachTest Prep Coach helps teachers to identify common errors that students can make. For Exercise 13, if students selected:

A They added 36 and 6.

B They subtracted 6 from 36.

C They incorrectly divided by 6.

Essential QuestionHow can you use arrays to solve division problems? Possible answer: I can fi nd how many equal groups by placing that number of tiles in each row of an array until all tiles are used. The number of rows is the answer. I can divide the tiles into a number of rows, placing 1 tile at a time in each row, until all the tiles are used. The number of tiles in each row is the answer.

Math Journal Draw an array to show how to arrange 20 chairs into 5 equal rows. Explain what each part of the array represents.


MATHEMATICALPRACTICES4 SUMMARIZE







PARA LA PRUEBA

PREPARACIÓN

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a. ¿Qué debes hallar?

b. ¿Qué operación podrías usar para resolver el problema?

c. Dibuja una matriz para hallar el número de hileras de plántulas de tomate.

d. ¿De qué otra manera podrías haber resuelto el problema?

e. Completa las oraciones.

Thomas tiene __ plántulas de tomate.

Quiere plantar __ plántulas

en cada __.

Entonces, Thomas plantará

__ hileras de plántulas de tomate.

f. Rellena el círculo del resultado correcto arriba.

13. Faith planta 36 flores en 6 hileras iguales. ¿Cuántas flores hay en cada hilera?

A 42 B 30 C 7 D 6

14. El sábado se vendieron 20 plantas en una tienda. Cada cliente compró 5 plantas. ¿Cuántos clientes compraron plantas?

A 3 B 4 C 5 D 6

12. Thomas tiene 28 plántulas de tomate para plantar en su jardín. Quiere plantar 4 plántulas en cada hilera. ¿Cuántas hileras de plántulas de tomate plantará Thomas?

A 5 B 6 C 7 D 8

PARA PRACTICAR MÁS:Cuaderno de práctica de los estándares, págs. P115 y P116

cuántas hileras de plántulas de tomate plantará Thomas

Respuesta posible: Podría haber

usado la resta repetida.

Revise los dibujos de los estudiantes. Debe haber una matriz con 7 hileras de 4 � chas cuadradas.

28

4

hilera

7


Respuesta posible:

la división

Untitled-5083 234 5/30/2011 5:56:31 AM





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Lesson 1.12 17

Problem Solving • Model Addition and SubtractionCommon Core Standard CC.3.OA.8Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Lesson Objective Solve addition and subtraction problems by using the strategy draw a diagram.

Essential Question How can you use the strategy draw a diagram to solve one- and two-step addition and subtraction problems?

Access Prior Knowledge Introduce the lesson by asking students: • Have you ever played a computer game

with another person? What kind of game was it? Did you keep score? How did you know who won?

Unlock the Problem Have students read the problem. Point out that a bar model is a diagram that can help them decide what operation to use to solve a problem.Guide students to read each question in the graphic organizer and answer it before solving the problem. • What question are you trying to answer?

What was Sami’s total score?

• What information do you know? He scored 84 points in the fi rst round and 21 more points in the second round.

• Did Sami score more points in the fi rst round of the game or the second round? He scored more points in the second round.

• How did you use the fi rst bar model? I used the fi rst bar model to fi gure out how many points Sami scored in the second round.

• How did you use the information from the problem to label the fi rst bar model? I labeled the longer bar 84 to show the points Sami scored in the fi rst round. I labeled the shorter bar 21 to show how many more points he scored in the second round. The number under the bars shows the total number of points Sami scored in the second round.

• What does the second bar model show? The second bar model shows Sami’s total score for the two rounds.

• How are the bar models related to the addition sentences that represent them? Possible answer: the addends are in the bars, and the sum is represented by the number under the bars.

Students may have diffi culty with the fact that this is a two-step problem. They may neglect to complete the second step of the problem and think that the answer is 105 points. Tell students that they should reread the problem carefully after they fi nd their answer to be sure that they have answered the question that is asked. • Why is it important to check the original

problem when you are fi nished to make sure you answered the right question? It is possible to get an answer that is mathematically correct, but answers the wrong question.

1 ENGAGE

2 TEACH and TALKc


Lesson 1.12





puntos21puntos84

puntos

puntos105puntos84

puntos


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Capítulo 1 51

Lee el problema

Sami anotó 84 puntos en la primera ronda de un videojuego nuevo. En la segunda ronda, anotó 21 puntos más que en la primera. ¿Cuál fue el puntaje total de Sami?

Puedes usar un modelo de barras para resolver el problema.

Resolución de problemas •Representar la suma y la restaPregunta esencial ¿Cómo puedes usar la estrategia hacer un diagrama para resolver problemas de suma y de resta de un paso y de dos pasos?

Resuelve el problema

¿Qué debo hallar?

Debo hallar

____.

¿Qué información debo usar?

Sami anotó _ puntos en la primera ronda.

Anotó _ puntos más en la segunda ronda.

¿Cómo usaré la información?

Dibujaré un modelo de barras para mostrar la cantidad de puntos que anotó Sami en cada ronda. Luego usaré el modelo para decidir qué operación usar.

Lección 1.12ESTÁNDAR COMÚN CC.3.OA.8

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

• Completa el modelo de barras para mostrar la cantidad de puntos que anotó Sami en la segunda ronda.

__ 1 __ 5 ■

__ 5 ■

• Completa otro modelo de barras para mostrar el puntaje total de Sami.

__ 1 __ 5 ▲

__ 5 ▲

Lección Lección 1.12RESOLUCIÓN DE PROBLEMAS

1. ¿Cuántos puntos anotó Sami en la segunda ronda?

2. ¿Cuál fue el puntaje total de Sami?

el puntaje total de Sami84

84 8421 105

105 puntos

105 puntos

105 189

21

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UNLOCK the Problem

points21points84

points

points105points84

points

Name

Chapter 1 51

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Read the Problem

Sami scored 84 points in the first round of a new computer game. He scored 21 more points in the second round than in the first round. What was Sami’s total score?

You can use a bar model to solve the problem.

Problem Solving •Model Addition and SubtractionEssential Question How can you use the strategy draw a diagram to solve one- and two-step addition and subtraction problems?

Solve the Problem

What do I need to find?

I need to find

____.

What information do I need to use?

Sami scored _points in the first round.

He scored _ more points than that in the second round.

How will I use the information?

I will draw a bar model to show the number of points Sami scored in each round. Then I will use the bar model to decide which operation to use.

PROBLEM SOLVINGLesson 1.12

COMMON CORE STANDARD CC.3.OA.8

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

• Complete the bar model to show the number of points Sami scored in the second round.

__ 1 __ 5 n

__ 5 n

• Complete another bar model to show Sami’s total score.

__ 1 __ 5 s

__ 5 s

1. How many points did Sami score in the second round? __

2. What was Sami’s total score? __

Sami’s total score

84

84 8421 105

105 189

21

105 points

189 points

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Lesson 1.12 19

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Try Another ProblemHave students read the problem and then answer the questions in the graphic organizer and solve the problem. Invite students to share their diagrams and explanations. • Who scored more points? How does the bar

model show this? Anna; the bar for Anna’s points is longer.

• How does the bar model help you solve the problem? Possible answers: it shows that the unknown part is the difference between Anna’s and Greg’s scores.

• What numbers do you need to subtract to solve the problem? I need to subtract Greg’s score from Anna’s score.

• How is the bar model related to the subtraction sentence that represents it? Possible answer: the number in the shorter bar is subtracted from the number in the longer bar to fi nd the difference, which is represented by the unknown quantity to the right of the shorter bar.

In problem 4, there are several ways to check for reasonableness, including estimation. Invite students to share their answers and discuss the different ways to estimate, such as rounding or using compatible numbers, to check for reasonableness.

Use Math Talk to focus on students’ understanding of how to use bar models to solve a problem. Ask students to look at the bar models again and compare the length of each bar model compared to Anna and Greg’s scores. • What would the bar model look like if Greg

scored more points than Anna? Greg’s bar model would be longer than Anna’s.

• Whose bar model would show the unknown part with the gray box? Greg’s bar model

You may suggest that students place completed Try Another Problem graphic organizers in their portfolios.

c

COMMON ERRORS COMMON ERRORS

Error Students may have diffi culty determin-ing how to label the bars in the bar model.

Example In Try Another Problem, students may draw a shorter bar for Anna’s points than for Greg’s points and, therefore, label it incorrectly.

Springboard to Learning Remind students that although the bars in a bar model do not have to be in exact proportion, longer bars should represent greater numbers. Have stu-dents fi rst determine whose score is greater before drawing their bar models.






Haz otro problema

Anna

Greg

puntos

puntos265

puntos142


52

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Anna anotó 265 puntos en un videojuego. Greg anotó 142 puntos. ¿Cuántos puntos más anotó Anna que Greg?

Puedes usar un modelo de barras para resolver el problema.

3. ¿Cuántos puntos más anotó Anna que Greg?

4. ¿Cómo sabes que tu resultado es razonable?

5. ¿Cómo te ayudó el dibujo a resolver el problema?

Lee el problema

Resuelve el problema

¿Qué debo hallar? ¿Qué información debo usar?

¿Cómo usaré la información?

Anota los pasos que seguiste para resolver el problema.

Explica cómo cambiaría la longitud de cada barra del modelo si Greg anotara más puntos que Anna, pero el puntaje total quedara igual.

Debo hallar cuántos puntos más anotó Anna que Greg.

Debo completar el modelo de barras para mostrar el puntaje de cada uno.

123 puntos

Respuesta posible: Puedo usar números amigos para

restar mentalmente. Resto 2 de 142 y obtengo 140.

Luego resto 2 de 265 y obtengo 263; 263 2 140 5 123.

Respuesta posible: El modelo de barras me ayudó a ver

Debo restar para hallar la parte desconocida.

265 2 142 5 ■

123 5 ■

Anna anotó 265 puntos. Greg anotó 142 puntos.

Usaré un modelo de barras para mostrar los puntos que anotó cada uno. Luego usaré ese modelo para decidir qué operación usar.

La barra de Anna sería la más corta y la barra de Greg sería la más larga.

que debía restar y qué números debía restar.

Untitled-34 52 5/9/2011 9:05:21 AM

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Be sure to point out to students that Exercise 1 is a two-step problem. Ask students to determine what they need to fi nd, what information they need to use, and how they can use the information.

ProblemExercise 2 requires that students use higher order thinking skills. The problem varies the scenario presented in Exercise 1. Students should connect the number and length of the bars to the numeric label of each bar. • What would be used to represent the third

student? A third bar model

• Is there just one answer for the number of votes that each student gets? Explain. No; There are many possibilities for the number of votes each student gets. The numbers can be any three numbers that add up to 121 votes.

Use Exercises 3 and 4 for Quick Check. Students should show their answers for the Quick Check on the MathBoard.

3 PRACTICE







Try Another Problem

Anna

Greg

points

points265

points142

52

Anna scored 265 points in a computer game. Greg scored 142 points. How many more points did Anna score than Greg?

You can use a bar model to solve the problem.

Explain how the length of each bar in the model would change if Greg scored more points than Anna but the totals remained the same.

3. How many more points did Anna score than Greg?

4. How do you know your answer is reasonable?

5. How did your drawing help you solve the problem?

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Read the Problem

Solve the Problem

What do I need to find?

What information do I need to use?

How will I use the information?

Record the steps you used to solve the problem.

I need to � nd how many more points Anna scored than Greg.

I need to complete the bar model to show each person’s score.

123 points

Possible answer: I can use friendly numbers to

subtract mentally. I subtract 2 from 142 to get 140. Then

I subtract 2 from 265 to get 263; 263 2 140 5 123.

Possible answer: the bar model helped me see that I had

to subtract and what numbers I needed to subtract.

I need to subtract to � nd the unknown part.

265 2 142 5 n

123 5 n

Anna scored 265 points.Greg scored 142 points.

I will use a bar model to show the points each person scored. Then I will use the bar model to decide which operation to use.

Anna’s bar would be the shorter bar and Greg’s bar would be the longer bar.

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Lesson 1.12 21

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Be sure to point out to students that Exercise 1 is a two-step problem. Ask students to determine what they need to fi nd, what information they need to use, and how they can use the information.

ProblemExercise 2 requires that students use higher order thinking skills. The problem varies the scenario presented in Exercise 1. Students should connect the number and length of the bars to the numeric label of each bar. • What would be used to represent the third

student? A third bar model

• Is there just one answer for the number of votes that each student gets? Explain. No; There are many possibilities for the number of votes each student gets. The numbers can be any three numbers that add up to 121 votes.

Use Exercises 3 and 4 for Quick Check. Students should show their answers for the Quick Check on the MathBoard.

Quick Check

If

Rt I RR1

2

3



• Reteach 1.2

Soar to Success Math 70.04

Then

Go Deeper To extend their thinking, have students write a problem and draw their own bar model to solve it. Explain that the problem can be about anything they wish, and can have two or more bar models. Remind students that they would need at least two bar models so that two quantities can be compared. • What kind of problem can you write that

would use a bar model to help solve it? The problem could be about comparing scores, lengths or weights of items, or any comparison in which one quantity is known, part of another quantity is known, and the total is unknown.

3 PRACTICEMATHEMATICAL

PRACTICES





SOLUCIONA el problemaSOLUCIONA el problemaPistas

Sara

Benji

votos73

votos25

votos

votos485

votos48votos73

votos

votos1215

89

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1. En la elección de la escuela, Sara obtuvo 73 votos. Benji obtuvo 25 votos menos que Sara. ¿Cuántos estudiantes votaron en total?

Primero, halla cuántos estudiantes votaron por Benji.

Piensa: 73 2 25 5 ■

Escribe los números en las barras.

Entonces, Benji obtuvo _ votos.

A continuación, halla la cantidad total de votos.

Piensa: 73 1 48 5 ▲

Escribe los números en las barras.

Entonces, __ estudiantes votaron en total.

2. ¿Qué pasaría si en otra elección hubiera que votar por 3 estudiantes y la cantidad total de votos fuera la misma? ¿Cómo sería el modelo de barras de la cantidad total de votos? ¿Cuántos votos podría obtener cada estudiante?

3. Plantea un problema Usa el modelo de barras que está a la derecha. Escribe un problema para él.

4. Resuelve tu problema. ¿Sumarás o restarás?

Comunicar y mostrarN√ Usa la pizarra de Resolución de

problemas.

√ Elige una estrategia que conozcas.

48

121

Respuesta posible: Habría 3 barras. Cantidad de

votos posibles: 55, 30 y 36

Problema posible: Russ y Juan coleccionan

estampillas. Entre ambos, coleccionaron 157 estampillas. Russ

coleccionó 89 estampillas. ¿Cuántas estampillas coleccionó Juan?

Respuesta posible: Restaré; 157 2 89 5 68.

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UNLOCK the ProblemUNLOCK the ProblemTips

Sara

Ben

votes73

votes25

votes

votes485

votes48votes73

votes

votes1215

89

157


Name

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1. Sara received 73 votes in the school election. Ben received 25 fewer votes than Sara. How many students voted in all?

First, find how many students voted for Ben.

Think: 73 2 25 5 n

Write the numbers in the bars.

So, Ben received _ votes.

Next, find the total number of votes.

Think: 73 1 48 5 s

Write the numbers in the bars.

So, __ students voted in all.

2. What if there were 3 students in another election and the total number of votes was the same? What would the bar model for the total number of votes look like? How many votes might each student get?

3. Pose a Problem Use the bar model at the right. Write a problem to match it.

4. Solve your problem. Will you add or subtract?

Share and ShowN

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√ Use the problem solving MathBoard.

√ Choose a strategy you know.

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Possible answer: there would be 3 bars. Possible

number of votes: 55, 30, and 36

Possible problem: Russ and Juan collect stamps.

Together they collected 157 stamps. Russ collected

89 stamps. How many stamps did Juan collect?

Possible answer: subtract; 157 2 89 5 68.

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Lesson 1.12 23

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Practice

If students complete Exercises 3 and 4 correctly, they may continue with Independent Practice. Encourage students to solve the problems independently, but provide assistance as needed. If students struggle with the On Your Own problems, ask them what they are trying to fi nd out, and what they already know about the problem. Encourage them to use the Show Your Work area of the page and then show it to you to see if their thinking and calculations are correct. Circle parts of their work that contains errors, and guide them through the process of making corrections.

ProblemExercise 8 requires that students use higher order thinking skills to solve the problem. • What can you do to fi nd out the greatest

number that could be rounded to 400? I must think of the greatest number that can be rounded down to 400.

• What number would be in the hundreds place? Explain. 4; I know I will be rounding down because I am looking for the greatest number that rounds to 400.

• What number would be in the tens place? Explain. 4; I know that any number that rounds to 400 must be less than 450.

• What number would be in the ones place? Explain. 9; If I have a 4 in the tens place, the greatest number that can be in the ones place and still round down to 400 would be 9. The answer is 449.

Test Prep CoachTest Prep Coach helps teachers to identify common errors that students can make. In Exercise 9, if students selected:

A They subtracted instead of added.

B They did not add the regrouped ten.

D They added incorrectly.

Essential QuestionHow can you use the strategy draw a diagram to solve one- and two-step addition and subtraction problems? Possible answer: I can draw a bar model to see if I need to add or subtract.

Math Journal Write an addition or subtraction problem and draw a diagram to solve it.

c

4 SUMMARIZE MATHEMATICALPRACTICES






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Por tu cuenta

5. En la Tienda Tecnológica de Tony hay una liquidación. Había 142 computadoras. En la liquidación se vendieron 91 computadoras. ¿Cuántas computadoras no se vendieron?

6. En la liquidación se vendieron 257 videojuegos. Esta cantidad es 162 videojuegos más que los que se vendieron la semana anterior a la liquidación. ¿Cuántos videojuegos se vendieron la semana anterior a la liquidación?

7. En una semana se vendieron 128 celulares. La semana siguiente se vendieron 37 celulares más que la semana anterior. ¿Cuántos celulares se vendieron en esas dos semanas?

8. El lunes la cantidad de clientes que fueron a la tienda redondeada a la centena más próxima fue 400. ¿Cuál es la mayor cantidad de clientes que pueden haber ido a la tienda? Explícalo.

9. Preparación para la prueba La cantidad de computadoras portátiles vendidas en un día fue 42. Esa cantidad es 18 menos que la cantidad de computadoras de escritorio vendidas. ¿Cuántas computadoras de escritorio se vendieron?

A 24 C 60

B 50 D 61

PARA PRACTICAR MÁS:Cuaderno de práctica de los estándares, págs. P25 y P26


51 computadoras

95 videojuegos

293 celulares

449 clientes; Explicación posible: Para redondear en

400, el número mayor debe tener un 4 en el lugar de las

decenas para que el lugar de las centenas quede igual.

El número mayor en el lugar de las unidades es 9.

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correctionkey=a do not edit--changes must be made through

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