Lesson 1.2
Round to the Nearest Ten or
Hundred
• Between which two tens is 32? Which ten is
it closer to? Explain. 30 and 40. 32 is closer to 30.
Common Core Standard CC.3.NBT.1
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:
It is only 2 numbers away from 30 but 8 numbers away
from 40.
Use 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
1 ENGAGE
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 sufficient. 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.
2 TEACH and TALK
Unlock the Problem
GO
Online
Animated
Math Models
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
MATHEMATICAL
PRACTICES
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?
© Houghton Mifflin Harcourt Publishing Company
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
c
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 first 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
Lesson 1.2
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Lección 1.2
Nombre
Redondear a la decena o la centena
más próxima
ESTÁNDAR COMÚN CC.3.NBT.1
Use place value understanding and properties of
operations to perform multi-digit arithmetic.
Pregunta esencial ¿Cómo puedes redondear números?
SOLUCIONA el problema
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?
De una manera Usa una recta numérica
para redondear.
A Redondea 32 a la decena más próxima.
32
0
10
30
20
40
Halla entre qué decenas está el número.
MÉTODOS MATEMÁTICOS
30
40 .
32 está entre _
y_
30
40 .
que de _
32 está más cerca de _
30 .
32 redondeado a la decena más próxima es _
Entonces, la longitud del bate de Mara redondeada a la
30
pulgadas.
decena de pulgadas más próxima es _
© Houghton Mifflin Harcourt Publishing Company
B Redondea 174 a la centena más próxima.
174
0
100
200
Menciona otros
tres números que se redondeen
en 30 al redondearlos a la
decena más próxima. Explícalo.
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.
300
Halla entre qué centenas está el número.
200 .
100 y _
174 está entre _
100 .
200 que de _
174 está más cerca de _
200 .
Entonces, 174 redondeado a la centena más próxima es _
Capítulo 1
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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 find 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 find the two tens that the number is between.
To round to the nearest hundred, I need to find 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.
c
Another Way
Students 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
• 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.
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.
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.
© Houghton Mifflin Harcourt Publishing Company
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.
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¡Inténtalo! Redondea 718 a la decena y a la centena más
próxima. Ubica y rotula 718 en las rectas numéricas.
A Decena más próxima
B Centena más próxima
718
700
Revise el trabajo de
los estudiantes.
710
718
700
720
750
800
720
718 está más cerca de _
700
718 está más cerca de _
710 .
que de _
800 .
que de _
720 .
Entonces, 718 se redondea en _
700 .
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.
• 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.
63
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.
60 .
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.
457
5 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
500 .
próxima es _
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.
MÉTODOS MATEMÁTICOS
Explica en qué se parecen usar
el valor posicional y usar una recta numérica.
© Houghton Mifflin Harcourt Publishing Company
Entonces, 63 redondeado a la decena más
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• Why are you being asked to find which
hundred 548 is closer to? That is how you figure
3 PRACTICE
out how to round the number.
Share and Show •
Guided Practice
The first 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
Quick Check
c
3
2
3
1
2
1
Rt I
R
Rt I
If
If
a student misses Exercises 6 and 7
Then
Differentiate Instruction with
• RtI Tier 1 Activity, p. 9B
• Reteach 1.2
Soar to Success Math 15.15, 15.17
On Your Own •
• 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
MATHEMATICAL
PRACTICES
To extend their thinking, ask students to
find 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 find
the greatest number that rounds to 500?
down
• What is that number? 549
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 find
out.
• In Exercises 10–12, why is 550 not a
possible answer? I am asked to round to the
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
nearest hundred, and 550 is not a hundred.
Lesson 1.2
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Nombre
Comunicar y mostrarN
Ubica y rotula 46 en la recta numérica.
Redondea a la decena más próxima. Revise el trabajo de los estudiantes.
MÉTODOS MATEMÁTICOS
¿Cuál es el mayor
número que se redondea en 50 al
redondearlo a la decena más
próxima? ¿Cuál es el menor número?
Explícalo.
46
30
40
60
50
1.
50 .
40
46 está entre _
y_
2.
40 .
50
46 está más cerca de _
que de _
3.
46 redondeado a la decena más próxima
50 .
es _
Redondea a la decena más próxima.
4.
20
19 _
5.
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.
70
66 _
6.
50
51 _
9.
700
658 _
Redondea a la centena más próxima.
7.
500
463 _
8.
200
202 _
Por tu cuenta
Ubica y rotula 548 en la recta numérica.
Redondea a la centena más próxima. Revise el trabajo de los estudiantes.
548
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
400
600
500
10.
600 .
500 y _
548 está entre _
11.
500 que de _
600 .
548 está más cerca de _
12.
500 .
548 redondeado a la centena más próxima es _
Redondea a la decena y a la centena más próxima.
13.
580
576 _
600
_
14.
300
298 _
300
_
15.
840
844 _
800
_
Capítulo 1 • Lección 2
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Problem Solving
MATHEMATICAL
PRACTICES
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.
Problem
To 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
4 SUMMARIZE
MATHEMATICAL
PRACTICES
Essential Question
How can you round numbers? I can use a number
line or place value.
Math Journal
Describe how to round 678 to the nearest
hundred.
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 find 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 Coach
© Houghton Mifflin Harcourt Publishing Company
Test 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.
Lesson 1.2
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MÉTODOS
MATEMÁTICOS
Representar • Razonar • Interpretar
Resolución de problemas
Usa la tabla para resolver los problemas
16 a 18.
16. ¿Qué día fueron alrededor de
900 visitantes
a la exposición de las jirafas?
domingo
17. ¿Qué cantidad de visitantes redondeada a
la decena más próxima fue el domingo a la
exposición de las jirafas?
890 visitantes
18. ¿Qué dos días fueron alrededor de
800
visitantes por día a la exposición de las
jirafas?
Visitantes de la
exposición de las jirafas
Día
Cantidad de visitantes
Domingo
894
Lunes
793
Martes
438
Miércoles
362
Jueves
839
Viernes
725
Sábado
598
lunes y jueves
19.
Escribe cinco números que se
redondeen en 360 al redondearlos a la
decena más próxima.
Resultado posible: 356, 357, 359, 361, 364
20.
¿Cuál es el error? Camilo
dijo que 555 redondeado a la decena más
próxima es 600. ¿Cuál es el error de Camilo?
Explícalo.
El resultado debe ser 560. Explicación posible:
21.
Preparación para la prueba ¿Cuánto es
438 redondeado a la decena más próxima?
A 450
B 440
C 430
D 400
12
PARA PRACTICAR MÁS:
Cuaderno de práctica de los estándares, págs. P5 y P6
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© Houghton Mifflin Harcourt Publishing Company
Camilo redondeó a la centena más próxima en
lugar de a la decena más próxima.
PRÁCTICA ADICIONAL:
Cuaderno de práctica de los estándares, pág. P27
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New York City Implementation Guide NYC17
Lesson 6.6
Investigate • Model with Arrays
Common Core Standard CC.3.OA.3
Use multiplication and division within 100 to solve word problems in
situations involving equal groups, arrays, and measurement quantities, 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?
• 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.
c 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.
Problem
1 ENGAGE
Materials square tiles
Access 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 find the product? Possible
Exercise 3 requires students to generalize from
the 5 by 6 array to finding a new array with 6
tiles in each row.
• Use the same array to describe a strategy
for finding 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 first 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 final number, 30, is the same for both
arrays. This one has 6 tiles in one row, while the first
array had 5 tiles in one row.
answer: I skip counted by 5 four times to get 20.
2 TEACH and TALK
© Houghton Mifflin Harcourt Publishing Company
c
Investigate
GO
iTools
Online
MATHEMATICAL
PRACTICES
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 find
how many rows are in your completed
array? from top to bottom
Lesson 6.6
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Lección 6.6
Nombre
Representar con matrices
ESTÁNDAR COMÚN CC.3.OA.3
Pregunta esencial ¿Cómo puedes usar matrices para resolver problemas de
división?
Represent and solve problems involving
multiplication and division.
Investigar
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.
6 hileras
¿Cuántas hileras de 5 formaste? _________
Sacar conclusionesN
1. Explica cómo usaste las fichas cuadradas para hallar
el número de hileras de 5 fichas que hay en 30.
Explicación posible: Dispuse las 30 fichas cuadradas en hileras de 5 fichas cuadradas.
Luego conté el número de hileras que formé. Había 6 hileras. Entonces, hay 6 hileras
de 5 en 30.
© Houghton Mifflin Harcourt Publishing Company
2. ¿Qué ecuación de multiplicación podrías escribir para la
matriz? Explícalo.
6 3 5 5 30; Explicación posible: Hay 6 hileras de 5 fichas cuadradas cada una.
6 grupos de 5 son 30.
3.
Aplica Indica cómo usar una matriz para hallar
cuántas hileras de 6 hay en 30.
Respuesta posible: Formo hileras de 6 hasta usar las 30 fichas cuadradas. Hay
5 hileras de 6 en 30.
Capítulo 6
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c Make Connections
Help 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
COMMON ERRORS
Error Students may make an incorrect array
for a division problem.
Example How many rows of 3 are in 18?
• What represents the quotient in this array?
The number of rows is the quotient.
© Houghton Mifflin Harcourt Publishing Company
Try This!
After students find 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
find 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
COMMON ERRORS
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.
made, it is easy to make mistakes counting the rows
and number of titles in a row, and therefore get the
wrong answer.
Lesson 6.6
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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.
Puedes dividir para
hallar el número de
hileras iguales o para
hallar el número de
objetos en cada hilera.
30 4 5 5 j
6
En 30, hay _
hileras de 5 fichas cuadradas.
6 .
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
Explicación posible: En la
matriz se muestra cuántas
fichas cuadradas hay en
cada hilera, y eso me ayuda
a resolver el problema de
división.
MÉTODOS MATEMÁTICOS
Explica cómo
formar una matriz te ayuda
a dividir.
6 fichas cuadradas
• ¿Cuántas fichas cuadradas hay en cada hilera? ___
24 4 4 5 6 ó 24 4 6 5 4
• ¿Qué ecuación de división puedes escribir para tu matriz? _____
© Houghton Mifflin Harcourt Publishing Company
fichas cuadradas. Dibuja tu matriz abajo.
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3 PRACTICE
Share and Show •
Quick Check
Guided Practice
Exercises 5–8 are examples of partitive
division. Before students complete the page,
ask a volunteer to explain how he or she will
find 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 find the quotient.
3
2
1
Rt I
R
If
a student misses Exercises 2 and 6
Then
Differentiate Instruction with
• RtI Tier 1 Activity, p. 231B
• Reteach 6.6
Soar to Success Math 13.17
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 find both to know how to divide using an
array. If you know the total and one of the numbers,
you can find 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?
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
It tells the number of rows.
Lesson 6.6
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Nombre
Comunicar y mostrar
Usa fichas cuadradas para formar una matriz. Resuelve.
1. ¿Cuántas hileras de
3 hay en 18?
2. ¿Cuántas hileras de
6 hileras
_____
3. ¿Cuántas hileras de
Revise las matrices de los
estudiantes.
6 hay en 12?
2 hileras
_____
7 hay en 21?
4. ¿Cuántas hileras de
3 hileras
_____
8 hay en 32?
4 hileras
_____
Revise las matrices
Forma una matriz. Luego escribe una ecuación de división. de los estudiantes.
5. 25 fichas cuadradas en 5 hileras
6. 14 fichas cuadradas en 2 hileras
25 4 5 5 5
_____
7.
14 4 2 5 7
_____
28 fichas cuadradas en 4 hileras
8.
28 4 4 5 7
_____
9. ¿Cuántas hileras de
27 4 9 5 3
_____
3 hay en 15?
10. ¿Cuántas hileras de
15 4 3 5 5
_____
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
8 hay en 24?
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 fichas
cuadradas que hay en cada hilera cuando
conozco el número de hileras.
11.
27 fichas cuadradas en 9 hileras
MÉTODOS MATEMÁTICOS
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.
Muestra dos maneras
en que podrías formar una matriz con
fichas cuadradas para 18 4 6. Sombrea
cuadrados en la cuadrícula para mostrar
las matrices.
Revise los dibujos de los estudiantes.
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Unlock the Problem
MATHEMATICAL
PRACTICES
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 find 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 find them
easier or harder than repeated subtraction. As
students express their thoughts about arrays
and what they find difficult or easy about
them, address their apprehensions. You may
find that students are unsure about whether
they have made the correct number of rows or
put the correct number of objects in each row.
4 SUMMARIZE
MATHEMATICAL
PRACTICES
Essential Question
How can you use arrays to solve division
problems? Possible answer: I can find 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.
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 find the number of rows it takes to
reach 28.
Test Prep Coach
Test 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.
© Houghton Mifflin Harcourt Publishing Company
C They incorrectly divided by 6.
Lesson 6.6
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MÉTODOS
MATEMÁTICOS
Representar • Razonar • Interpretar
EPARACIÓN
PR PARA
LA PRUEBA
SOLUCIONA el problema
28 plántulas de tomate para plantar en su
jardín. Quiere plantar 4 plántulas en cada hilera. ¿Cuántas
12. Thomas tiene
hileras de plántulas de tomate plantará Thomas?
A 5
B 6
a. ¿Qué debes hallar?
C 7
D 8
cuántas hileras de plántulas de tomate plantará Thomas
b. ¿Qué operación podrías usar para resolver el problema? Respuesta posible:
la división
c. Dibuja una matriz para hallar el número
de hileras de plántulas de tomate.
d. ¿De qué otra manera podrías haber
resuelto el problema?
Revise los dibujos de los
estudiantes. Debe haber una matriz
con 7 hileras de 4 fichas cuadradas.
Respuesta posible: Podría haber
usado la resta repetida.
e. Completa las oraciones.
28
plántulas
Thomas tiene __
de tomate.
4
plántulas
Quiere plantar __
hilera
.
en cada __
Entonces, Thomas plantará
7
__
hileras de plántulas de
© Houghton Mifflin Harcourt Publishing Company
tomate.
f. Rellena el círculo del resultado
correcto arriba.
13. Faith planta
234
36 flores en 6 hileras
14. El sábado se vendieron
20 plantas
iguales. ¿Cuántas flores hay en cada
hilera?
en una tienda. Cada cliente compró
5 plantas. ¿Cuántos clientes
compraron plantas?
A 42
A 3
B 30
C 7
D 6
PARA PRACTICAR MÁS:
Cuaderno de práctica de los estándares, págs. P115 y P116
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B 4
C 5
D 6
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New York City Implementation Guide NYC25
Lesson 1.12
Problem Solving •
Model Addition and Subtraction
Common Core Standard CC.3.OA.8
Solve 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?
1 ENGAGE
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?
2 TEACH and TALK
c
Unlock the Problem
• How did you use the information from the
problem to label the first bar model? I labeled
the longer bar 84 to show the points Sami scored in
the first 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 difficulty 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 find 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 finished to make
sure you answered the right question? It is
possible to get an answer that is mathematically correct,
but answers the wrong question.
MATHEMATICAL
PRACTICES
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?
© Houghton Mifflin Harcourt Publishing Company
What was Sami’s total score?
• What information do you know? He scored
84 points in the first round and 21 more points in the
second round.
• Did Sami score more points in the first
round of the game or the second round?
He scored more points in the second round.
• How did you use the first bar model? I used
the first bar model to figure out how many points Sami
scored in the second round.
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RESOLUCIÓN DE PROBLEMAS
Nombre
Lección 1.12
Resolución de problemas •
Representar la suma y la resta
ESTÁNDAR COMÚN CC.3.OA.8
Solve problems involving the four operations, and
identify and explain patterns in arithmetic.
Pregunta esencial ¿Cómo puedes usar la estrategia hacer un diagrama
para resolver problemas de suma y de resta de un paso y de dos pasos?
SOLUCIONA 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.
Lee el problema
¿Qué debo hallar?
Debo hallar
el
puntaje total de Sami .
____
¿Cómo usaré la
información?
¿Qué información
debo usar?
84 puntos
Sami anotó _
en la primera ronda.
Dibujaré un modelo de
barras para mostrar la
cantidad de puntos que
21 puntos más en
Anotó _
anotó Sami en cada ronda.
la segunda ronda.
Luego usaré el modelo para
decidir qué operación usar.
Resuelve el problema
• Completa el modelo de barras para
• Completa otro modelo de barras para
© Houghton Mifflin Harcourt Publishing Company
mostrar la cantidad de puntos que anotó
Sami en la segunda ronda.
84
puntos
21
mostrar el puntaje total de Sami.
84
puntos
105
puntos
puntos
puntos
puntos
84
21
__
1 __
5■
84
105
__
1 __
5▲
105
__
5■
189
__
5▲
1. ¿Cuántos puntos anotó Sami en la segunda ronda? 105 puntos
2. ¿Cuál fue el puntaje total de Sami? 105 puntos
Capítulo 1
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c
Try Another Problem
Have 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 find
the difference, which is represented by the unknown
quantity to the right of the shorter bar.
You may suggest that students place
completed Try Another Problem graphic
organizers in their portfolios.
COMMON
COMMON ERRORS
ERRORS
Error Students may have difficulty 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 students first determine whose score is greater
before drawing their bar models.
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?
© Houghton Mifflin Harcourt Publishing Company
Greg’s bar model
Lesson 1.12
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Haz otro problema
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.
Lee el problema
¿Qué debo hallar?
¿Qué información
debo usar?
Debo hallar cuántos
puntos más anotó Anna
que Greg.
Anna anotó 265 puntos.
Greg anotó 142 puntos.
¿Cómo usaré la
información?
Usaré un modelo de barras
para mostrar los puntos
que anotó cada uno. Luego
usaré ese modelo para
decidir qué operación usar.
Resuelve el problema
Anota los pasos que seguiste para resolver el problema.
Debo completar el modelo de barras para
mostrar el puntaje de cada uno.
Debo restar para hallar la parte
desconocida.
265 2 142 5 ■
Anna
Greg
265
142
puntos
123
5■
puntos
puntos
123 puntos
4. ¿Cómo sabes que tu resultado es razonable?
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.
5. ¿Cómo te ayudó el dibujo a resolver el problema?
Respuesta posible: El modelo de barras me ayudó a ver
que debía restar y qué números debía restar.
MÉTODOS MATEMÁTICOS
Explica cómo
cambiaría la longitud de cada
barra del modelo si Greg
anotara más puntos que
Anna, pero el puntaje total
quedara igual.
La barra de Anna sería
la más corta y la barra
de Greg sería la más
larga.
© Houghton Mifflin Harcourt Publishing Company
3. ¿Cuántos puntos más anotó Anna que Greg?
52
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Go Deeper
3 PRACTICE
Share and Show •
Guided Practice
Be sure to point out to students that Exercise
1 is a two-step problem. Ask students to
determine what they need to find, what
information they need to use, and how they
can use the information.
Problem
Exercise 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.
MATHEMATICAL
PRACTICES
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.
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
3
2
1
Rt I
R
a student misses Exercises 3 and 4
Then
Differentiate Instruction with
• RtI Tier 1 Activity, p. 51B
• Reteach 1.2
Soar to Success Math 70.04
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
If
Lesson 1.12
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Pista
s
SOLUCIONA el problema
Nombre
√ Usa la pizarra de Resolución de
Comunicar y mostrarN
problemas.
1. En la elección de la escuela, Sara obtuvo
√ Elige una estrategia que conozcas.
73 votos. Benji obtuvo 25 votos menos que
Sara. ¿Cuántos estudiantes votaron en total?
Primero, halla cuántos estudiantes votaron
por Benji.
Sara
votos
73
Benji
Piensa: 73 2 25 5 ■
25
votos
votos
Escribe los números en las barras.
48 votos.
Entonces, Benji obtuvo _
48
5
votos
A continuación, halla la cantidad total de votos.
73
Piensa: 73 1 48 5 ▲
votos
Escribe los números en las barras.
votos
votos
121
Entonces, __
estudiantes votaron
en total.
2.
48
5 121 votos
¿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?
Respuesta posible: Habría 3 barras. Cantidad de
votos posibles: 55, 30 y 36
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
3. Plantea un problema Usa el modelo de barras
que está a la derecha. Escribe un problema
para él.
89
157
Problema posible: Russ y Juan coleccionan
estampillas. Entre ambos, coleccionaron 157 estampillas. Russ
coleccionó 89 estampillas. ¿Cuántas estampillas coleccionó Juan?
4. Resuelve tu problema. ¿Sumarás o restarás?
Respuesta posible: Restaré; 157 2 89 5 68.
Capítulo 1 • Lección 12
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c
On Your Own •
Independent
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
find 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.
Problem
Exercise 8 requires that students use higher
order thinking skills to solve the problem.
• What can you do to find out the greatest
number that could be rounded to 400?
I must think of the greatest number that can be
rounded down to 400.
Test Prep Coach
Test 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.
4 SUMMARIZE
MATHEMATICAL
PRACTICES
Essential Question
How 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.
• 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
© Houghton Mifflin Harcourt Publishing Company
number that can be in the ones place and still round
down to 400 would be 9. The answer is 449.
Lesson 1.12
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MÉTODOS
MATEMÁTICOS
Por tu cuenta
Representar • Razonar • Interpretar
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?
51 computadoras
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?
95 videojuegos
128 celulares. La
semana siguiente se vendieron 37 celulares más
7. En una semana se vendieron
que la semana anterior. ¿Cuántos celulares se
vendieron en esas dos semanas?
293 celulares
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.
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.
9.
54
© Houghton Mifflin Harcourt Publishing Company
El número mayor en el lugar de las unidades es 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
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New York City Implementation Guide NYC33