Number Models with Parentheses

Number Models
with Parentheses
Objective To introduce parentheses in number sentences.
www.everydaymathonline.com
ePresentations
eToolkit
Algorithms
Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Key Concepts and Skills
• Use basic and extended addition
and subtraction facts to solve
number sentences. [Operations and Computation Goal 1]
• Use multiplication facts to solve
number sentences. [Operations and Computation Goal 3]
• Write number models with parentheses
to match number stories. [Patterns, Functions, and Algebra Goal 3]
Key Activities
Children compare the use of commas in
word sentences to the use of parentheses
in number sentences. Children use
parentheses in writing number models
for number stories.
Family
Letters
Assessment
Management
Common
Core State
Standards
Curriculum
Focal Points
Ongoing Learning & Practice
Practicing with ×, ÷ Fact
Triangles
×, ÷ Fact Triangles
Children practice multiplication and
division with Fact Triangles.
Math Boxes 7 4
Math Journal 2, p. 165
Children practice and maintain skills
through Math Box problems.
Home Link 7 4
Math Masters, p. 215
Children practice and maintain skills
through Home Link activities.
Ongoing Assessment:
Informing Instruction See page 596.
Ongoing Assessment:
Recognizing Student Achievement
Use journal page 164. [Patterns, Functions, and Algebra Goal 3]
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Playing Name That Number
Math Masters, p. 451
Student Reference Book, pp. 299 and 300
per group: 4 each of number cards 0–10
and 1 each of number cards 11–20 (from
the Everything Math Deck, if available),
3" by 5" labeled index cards (see Advance
Preparation)
Children use at least three cards and two
operations to make target numbers.
ENRICHMENT
Describing Dot Patterns
with Number Models
Math Masters, p. 216
Children write number models to represent
the patterns in arrays.
ELL SUPPORT
Building a Math Word Bank
Differentiation Handbook, p. 132
Children add the term parentheses to their
Math Word Banks.
Key Vocabulary
parentheses
Materials
Math Journal 2, p. 164
Home Link 73 (teacher only)
Math Masters, p. 406 (optional)
slate counters (optional)
Advance Preparation
For the optional Readiness activity in Part 3, prepare 1 set of operation cards per partnership. Cut five
3" by 5" inch index cards in half. Write the following—one on each card: +, +, +, -, -, -, ×, ÷, =.
Teacher’s Reference Manual, Grades 1–3 pp. 82–84
594
Unit 7
Multiplication and Division
594_EMCS_T_TLG_G3_U07_L04_576892.indd 594
2/23/11 10:53 AM
Getting Started
Mental Math and Reflexes
Math Message
Have children practice quick recall of basic
multiplication facts. Suggestions:
Can you find more than one meaning for
each sentence?
3 × 3 9 4 × 4 16 5 × 5 25 6 × 6 36
2 × 9 18 3 × 9 27 4 × 9 36 5 × 9 45
7 × 7 49 8 × 8 64 9 × 9 81 9 × 8 72
Nancy fed Tom the big gray cat.
My sister Tess and Jimmy are going.
Discuss the meanings you found with a partner.
Home Link 7 3 Follow-Up
Remind children of the importance of frequent
practice with the facts. Ask them to name their
favorite activity for learning the facts. Do they set aside a certain
time for facts practice with someone at home?
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
Although the sentences illustrate the need for punctuation marks,
most children will need an explanation from you about the
meanings of the sentences.
One interpretation of the first sentence is that Nancy gave Tom a
big gray cat to eat! Another interpretation is that the cat’s name is
Tom, and Nancy gave Tom something to eat. If that is the
intended meaning, a comma needs to be inserted after Tom.
(Nancy fed Tom, the big gray cat.)
Without commas in the second sentence, two people are going
(Tess and Jimmy—Tess is my sister.). With commas, three people
are going (my sister, Tess, and Jimmy).
Comparing Punctuation Marks
to Parentheses
WHOLE-CLASS
ACTIVITY
ELL
Algebraic Thinking Explain that just as word sentences can often
be interpreted in more than one way, so can number sentences.
When a number sentence may be interpreted in more than one
way, parentheses are used to make the intended meaning clear.
Parentheses indicate which part of a number sentence should be
solved first. To support English language learners, write parentheses
on the board along with the number sentences listed below.
(25 - 8) + 7 = ?
25 - (8 + 7) = ?
Lesson 7 4
EM3cuG3TLG2_595-599_U07L04.indd 595
595
1/23/11 12:26 PM
In (25 - 8) + 7 = ?, the parentheses indicate that 25 - 8 is to be
solved first and that 7 is to be added to the result: 25 - 8 = 17
and 17 + 7 = 24. Replace the question mark with 24 in the
first number sentence.
In 25 - (8 + 7) = ?, the parentheses indicate that 8 + 7 is to
be solved first and that the sum is to be subtracted from 25:
8 + 7 = 15 and 25 - 15 = 10. Replace the question mark with
10 in the second number sentence.
Ask children to compare both number sentences and their answers.
Emphasize that the answer depends on where the parentheses
are placed.
Ongoing Assessment: Informing Instruction
Watch for children who have difficulty focusing their attention on the parentheses
in the number sentence. Cover the number outside the parentheses with a
stick-on note. After the computation inside the parentheses is completed, remove
the stick-on note so the remaining computation can be finished.
Write the following number sentences on the board. Have
children copy them onto their slates and solve them.
Adjusting the Activity
53 - (17 + 13) = 23
The problems to the right provide
children with practice in using mental math
strategies and parentheses. The problems
may be simplified so that children are able
to focus on the use of parentheses.
For example, instead of 53 - (17 + 13),
use 12 - (4 + 6).
AUDITORY
KINESTHETIC
TACTILE
27 = (17 - 8) × 3
48 = 68 - (4 × 5)
(12 + 38) - 15 = 35
VISUAL
Write the pairs of number sentences below on the board.
Have children copy them and insert parentheses to make
each answer correct.
NOTE Some calculators have parentheses
keys and evaluate expressions using a
different order of operations than four-function
calculators. Third Grade Everyday
Mathematics does not expect children to
have these more advanced calculators. If you
or one of your children has a calculator with
parentheses, however, you might make an
exploration of how it works using some of the
number sentences in this lesson. To further
explore order of operations without grouping
symbols, see Project 7, Order of Operations.
596
32 - (5 + 7) = 20
(38 - 14) - 9 = 15
40 - (5 × 2) = 30
(32 - 5) + 7 = 34
38 - (14 - 9) = 33
(40 - 5) × 2 = 70
Discuss solution strategies. Pose additional problems as needed.
Unit 7 Multiplication and Division
595-599_EMCS_T_TLG_G3_U07_L04_576892.indd 596
2/23/11 11:06 AM
Teaching Aid Master
Name
Links to the Future
Date
Time
Guide to Solving Number Stories
1. What do you understand from the story?
The activities in this lesson provide an introduction to using parentheses to write
number models for specific situations in number stories. This begins to teach
the concept that continues into algebra. Recognizing that parentheses affect
the order of operations is a Grade 3 Goal. Writing number sentences with
parentheses to fit specific situations is a Grade 4 Goal.
Read the story. What do you want to find out?
What do you know?
2. What will you do?
Add?
Subtract?
Multiply?
Divide?
Draw a picture?
Writing Number Models
Make tallies?
Use counters or base-10 blocks?
WHOLE-CLASS
ACTIVITY
Use a number grid or number line?
Make a table?
with Parentheses
Draw a diagram?
Write a number model?
(Math Journal 2, p. 164; Math Masters, p. 406)
3. Answer the question.
Solve the problem. Record your work.
Algebraic Thinking Explain to children that some number stories
have two steps. Number models for such stories can be written
with parentheses to show which step comes first.
Ask a child to read aloud Problem 1 on journal page 164: Alexis
scored 12 points and Nehemia scored 6 points. If their team scored
41 points altogether, how many points did the rest of the team
score?
Write the answer with the units.
4. Check.
Does your answer make sense? How do you know?
Math Masters, p. 406
EM3MM_G3_U02_036-063.indd 406
12/29/10 5:57 PM
Remind children of the Guide to Solving Number Stories on Math
Masters, page 406.
●
What do you want to find out? The number of points the rest of
the team scored
What do you know from reading the story? Alexis scored 12
points, Nehemia scored 6 points, and 41 points were scored by
the team altogether.
●
Explain that two calculations can be made to solve the story.
One strategy could be to first add to find the total points that
Alexis and Nehemia scored. Write 12 + 6 and tell children that
this expression models the first step in the solution. Add
parentheses around 12 + 6 to show that this calculation is to be
done first. The second calculation would then be to subtract
that sum from the 41 points the team scored altogether. For
step 2, write 41 - in front of (12 + 6). Remind children that a
letter can be used to represent what we want to find out. Since
we want to find the number of points the rest of the team
scored, write an R in the open sentence. 41 - (12 + 6) = R
Student Page
Date
Time
LESSON
74
Number Models with Parentheses
Write a number model using parentheses. Then, solve the number story.
1. Alexis scored 12 points, and Nehemie scored 6 points. If their team scored
41 points altogether, how many points did the rest of the team score?
Possible number models:
41 − (12 + 6) = R,
(12 + 6) + R = 41,
Answer: 23 points
(41 - 12) - 6 = R
Number model:
2. In a partner game, Quincy has 10 points, and Ellen has 14 points. They need
50 points to finish the game. How many more points are needed?
●
What is the answer? The rest of the team scored 23 points.
●
Does your answer make sense? yes How can you tell? Sample
answer: I knew that the rest of the team had to score fewer
points than the team scored altogether. Does your answer make
the open sentence true? yes
Write a summary number model on the board. 41 – (12 + 6) = 23
Ask children if they can think of a different way to solve the
number story. Some children may suggest adding 12 and 6 and
then counting up to 41. (12 + 6) + R = 41 Others may suggest
beginning with 41 and subtracting 12 and then 6. (41 – 12) – 6 = R
As children share their thinking, invite volunteers to write the
open sentences that reflect these strategies.
Possible number models:
50 − (10 + 14) = M,
10 + 14 + M = 50,
Answer: 26 points
(50 - 10) - 14 = M
Number model:
3. Quincy and Ellen earned 49 points but lost 14 points for a wrong move.
They gained 10 points back. What was their score at the end of the round?
Possible number models:
(49 − 14) + 10 = E,
(49 + 10) - 14 = E
Answer: 45 points
Number model:
Complete these number sentences.
4.
6.
4
35
= 18 − (9 + 5)
= 8 + (9 × 3)
5. (75 − 29) + 5 =
7. 36 + (15 ÷ 3) =
51
41
8. 20 −(10 + 4)= 6
11. (100 − 21)+ 10 = 89
9. (20 − 10) + 4 = 14
12. (27 − 8)+ 3 = 22
10. 100 −(21 + 10)= 69
13. 18 = 6 +(3 × 4)
Math Journal 2, p. 164
EM3MJ2_G3_U07_157-179.indd 164
1/21/11 3:30 PM
Lesson 7 4
595-599_EMCS_T_TLG_G3_U07_L04_576892.indd 597
Add parentheses to complete the number models.
597
3/17/11 3:28 PM
Student Page
Date
Time
LESSON
Math Boxes
74
1.
2.
Draw the lines of symmetry.
Draw a parallelogram.
Label the
_
_
vertices so AB CD. The
symbol means is parallel to.
A
There are
0
lines of symmetry.
B
D
C
108 109
122 123
3.
Solve.
4. Answer this riddle.
Unit
1,200
90,000 − 20,000 =
Then, have children complete the rest of the problems on the page.
I have four sides. My opposite
sides are equal in length. I have
two pairs of parallel sides. I do not
have any right angles.
= 400 + 800
3,000 + 7,000 =
What shape am I?
10,000
rhombus or
parallelogram
70,000
108 109
5.
6.
Complete the number-grid puzzle.
8,731 8,732 8,733
Divide the triangles into
three equal groups.
Ongoing Assessment:
Recognizing Student Achievement
Journal
Page 164
Problems 4 and 5
Use journal page 164, Problems 4 and 5 to assess children’s progress toward
recognizing that parentheses affect the order of operations. Children are making
adequate progress if they successfully complete Problems 4 and 5. Some
children may successfully complete the remaining problems on the page.
8,744
8,742
In the same manner, do the remaining two number stories on the
journal page with the class. In Problem 2, since we want to find
how many more points are needed, the letter M can be used in an
open sentence to represent what we want to find out. Likewise, for
Problem 3, since we want to find out the score at the end of the
game, the letter E can be used in an open sentence to represent
what we want to find out. (Of course, any letter may be used for
any variable.)
8,752
8,763
8,774
[Patterns, Functions, and Algebra Goal 3]
7 8
165
Math Journal 2, p. 165
EM3MJ2_G3_U07_157-179.indd 165
1/18/11 3:35 PM
2 Ongoing Learning & Practice
Practicing with ×, ÷
Fact Triangles
PARTNER
ACTIVITY
5–15 Min
Partners practice basic facts using the second set of Fact Triangles.
At first, children should limit themselves to finding products.
When children are well on their way to learning the products, they
can cover one of the other two numbers to practice finding missing
factors.
Home Link Master
Name
Date
HOME LINK
Time
Math Boxes 7 4
Parentheses Puzzles
74
Family
Note
Observe as your child adds parentheses and explains what to do first in the number sentence
puzzles in Problems 1 through 4. If needed, assist your child in writing a correct number
model for the Try This problem. You might ask how many gifts Dalia would need to fill
8 bags and how many she would need to also take care of Denise.
16 –17
Please return this Home Link to school tomorrow.
Show someone at home how to add parentheses to complete the number
sentences below. Remember that the parentheses are used to show what you
do first.
(
)
(
)
3 b. 24 - (17 - 6 )= 13
4 b. (3 × 6)+ 13 = 31
Make up other parentheses puzzles below. Sample answers:
5 a. 4 × (8 - 6) = 8
5 b. (4 × 8) - 6 = 26
(
)
(
)
3 a. (24 - 17)- 6 = 1
4 a. 3 × (6 + 13) = 57
1 a. 17 - 10 + 3 = 10
1 b. 17 - 10 + 3 = 4
2 a. 26 - 7 × 2 = 38
2 b. 26 - 7 × 2 = 12
6 a.
(7 + 3) × 4 = 40
6 b.
7 + (3 × 4) = 19
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 165)
Mixed Practice The Math Boxes in this lesson are paired
with the Math Boxes in Lesson 7-2. The skill in Problem
6 previews Unit 8 content.
Writing/Reasoning Have children write the answer to the
following: Write your own polygon riddle similar to the one
in Problem 4. Sample answer: I have 4 sides. My opposite
sides are equal in length. I have 4 right angles. What shape am I?
Rectangle.
Try This
7.
Dalia made 8 party bags for her birthday party. Each bag contained
4 small gifts for her friends. When Denise said that she could come,
Dalia had to make one more bag with 4 gifts. How many small gifts did
Dalia need to fill her bags?
Walter wrote this number model: 8 × (4 + 4) = 64
Explain Walter’s mistake.
Home Link 7 4
INDEPENDENT
ACTIVITY
(Math Masters, p. 215)
The parentheses are placed incorrectly.
The number model should be
(8 × 4) + 4 = 36.
Home Connection Children solve parentheses puzzles.
Math Masters, p. 215
206-236_EMCS_B_MM_G3_U07_576957.indd 215
598
3/29/11 11:13 AM
Unit 7 Multiplication and Division
595-599_EMCS_T_TLG_G3_U07_L04_576892.indd 598
3/30/11 11:27 AM
Game Master
Name
3 Differentiation Options
Date
Time
1 2
4 3
Name That Number Record Sheet
Target
Number
READINESS
Playing Name That Number
SMALL-GROUP
ACTIVITY
Number Sentence Solution
15–30 Min
Reminder: Write each step separately!
(Math Masters, p. 451; Student Reference Book,
pp. 299 and 300)
To provide experience with using basic facts to solve multistep
problems, have children play Name That Number. Encourage
children to work together to find solutions for the target numbers
using at least 3 cards and 2 operations. Children can use the
operations cards you prepared to physically model their thinking.
Have partners record their best round on Math Masters, page 451.
Have children read the number sentences they recorded.
ENRICHMENT
Describing Dot Patterns
INDEPENDENT
ACTIVITY
15–30 Min
Name
Date
Time
1 2
4 3
Name That Number Record Sheet
Target
Number
Number Sentence Solution
Reminder: Write each step separately!
Math Masters, p. 451
with Number Models
(Math Masters, p. 216)
To apply children’s understanding of number models, have
children write number models to represent the patterns in the
arrays on Math Masters, page 216.
ELL SUPPORT
Building a Math Word Bank
SMALL-GROUP
ACTIVITY
5–15 Min
To provide language support for number sentences, have children
use the Word Bank template found on Differentiation Handbook,
page 132. Ask the children to write the term parentheses, draw a
picture representing the term, and write other related words. See
the Differentiation Handbook for more information.
Teaching Master
Name
LESSON
74
Date
Time
Page
Dot Patterns
Title
Page Title
with Number Models
The total dots in this dot array can
be found by using patterns.
Here is one way to find the total:
9 + (4 × 4)
Find as many ways as you can to use patterns to find the total dots. Show each
pattern on the dot array and write a number model to describe the pattern.
Use parentheses in your number model if you can.
(5 × 4) + 5
(4 × 4) + 9
(4 × 4) + (3 × 3)
(5 × 2) + (7 × 2) + 1
py g
(9 × 2) + 7
g
p
(4 × 2) + (6 × 2) +
(1 × 2) + 3
Math Masters, p. 216
EM3MM_G3_U07_206-236.indd 216
1/18/11 1:03 PM
Lesson 7 4
595-599_EMCS_T_TLG_G3_U07_L04_576892.indd 599
599
2/23/11 11:06 AM