www.everydaymathonline.com eToolkit ePresentations Interactive Teacher’s Lesson Guide Algorithms Practice EM Facts Workshop Game™ Assessment Management Family Letters Curriculum Focal Points 798 Unit 9 More about Variables, Formulas, and Graphs Key Concepts and Skills • Add and subtract signed numbers. [Operations and Computation Goal 1] • Add and subtract decimals. [Operations and Computation Goal 1] • Add and subtract fractions. [Operations and Computation Goal 3] • Substitute values for a variable to check simplified expressions. [Patterns, Functions, and Algebra Goal 2] • Use distributive strategies to simplify algebraic expressions. [Patterns, Functions, and Algebra Goal 4] Key Activities Students use the distributive property to combine like terms. They simplify expressions containing like terms. Key Vocabulary like terms combine like terms simplify an expression Materials Math Journal 2, pp. 330 and 331 Study Link 9 2 slate Estimating and Measuring in Millimeters Math Journal 2, p. 332 metric ruler Students measure line segments to the nearest millimeter. Math Boxes 9 3 Math Journal 2, p. 333 Geometry Template Students practice and maintain skills through Math Box problems. Ongoing Assessment: Recognizing Student Achievement Use Math Boxes, Problems 2a and 2b. [Patterns, Functions, and Algebra Goal 2] Study Link 9 3 Math Masters, p. 288 Students practice and maintain skills through Study Link activities. READINESS Playing Algebra Election Student Reference Book, pp. 304 and 305 Math Journal 1, Activity Sheets 3 and 4 Math Masters, pp. 434 and 435 per group: 4 counters or pennies, 1 six-sided die, calculator, index cards (optional) Students practice substituting values for variables and solving equations. ENRICHMENT Simplifying and Evaluating Expressions Math Masters, p. 289 Students find missing terms in expressions. They also simplify and evaluate expressions. EXTRA PRACTICE Generating Equivalent Expressions Differentiation Handbook, p. 135 Students record equivalent expressions in name-collection boxes. They substitute values for variables to check that expressions are equivalent. ELL SUPPORT Building a Math Word Bank Differentiation Handbook, p. 130 Students add the phrases combine like terms and simplify an expression to their Math Word Banks. Teaching the Lesson Ongoing Learning & Practice Differentiation Options Simplifying Expressions: Combining Like Terms Objective To simplify algebraic expressions by combining like terms. Advance Preparation For the optional Readiness activity in Part 3, students will need Algebra Election Cards and the Electoral Vote Map from Lesson 6 11. Teacher’s Reference Manual, Grades 4–6 pp. 289–291 Common Core State Standards
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www.everydaymathonline.com
eToolkitePresentations Interactive Teacher’s
Lesson Guide
Algorithms Practice
EM FactsWorkshop Game™
AssessmentManagement
Family Letters
CurriculumFocal Points
798 Unit 9 More about Variables, Formulas, and Graphs
Key Concepts and Skills• Add and subtract signed numbers.
[Operations and Computation Goal 1]
• Add and subtract decimals. [Operations and Computation Goal 1]
• Add and subtract fractions. [Operations and Computation Goal 3]
• Substitute values for a variable to check simplified expressions. [Patterns, Functions, and Algebra Goal 2]
• Use distributive strategies to simplify algebraic expressions. [Patterns, Functions, and Algebra Goal 4]
Key ActivitiesStudents use the distributive property to combine like terms. They simplify expressions containing like terms.
Key Vocabularylike terms � combine like terms � simplify an expression
MaterialsMath Journal 2, pp. 330 and 331Study Link 9�2slate
Estimating and Measuring in MillimetersMath Journal 2, p. 332metric rulerStudents measure line segments to the nearest millimeter.
Math Boxes 9�3Math Journal 2, p. 333Geometry Template Students practice and maintain skillsthrough Math Box problems.
Ongoing Assessment: Recognizing Student Achievement Use Math Boxes, Problems 2a and 2b. [Patterns, Functions, and Algebra Goal 2]
Study Link 9�3Math Masters, p. 288 Students practice and maintain skillsthrough Study Link activities.
READINESS
Playing Algebra ElectionStudent Reference Book, pp. 304 and 305Math Journal 1, Activity Sheets 3 and 4Math Masters, pp. 434 and 435per group: 4 counters or pennies, 1 six-sided die, calculator, index cards (optional)Students practice substituting values for variables and solving equations.
ENRICHMENTSimplifying and Evaluating ExpressionsMath Masters, p. 289Students find missing terms in expressions. They also simplify and evaluate expressions.
EXTRA PRACTICE Generating Equivalent ExpressionsDifferentiation Handbook, p. 135Students record equivalent expressions in name-collection boxes. They substitute values for variables to check that expressions are equivalent.
ELL SUPPORT Building a Math Word BankDifferentiation Handbook, p. 130Students add the phrases combine like terms and simplify an expression to their Math Word Banks.
Teaching the Lesson Ongoing Learning & Practice Differentiation Options
Simplifying Expressions:Combining Like Terms
Objective To simplify algebraic expressions by combininglike terms.
������
Advance PreparationFor the optional Readiness activity in Part 3, students will need Algebra Election Cards and the Electoral Vote Map from Lesson 6�11.
Teacher’s Reference Manual, Grades 4–6 pp. 289–291
Common Core State Standards
798_EMCS_T_TLG2_G6_U09_L03_576922.indd 798 2/22/11 11:35 AM
Combining Like TermsLESSON
9�3
Date Time
Algebraic expressions contain terms. For example, the expression 4y + 2x - 7y contains the terms 4y, 2x, and 7y. The terms 4y and 7y are called like terms because they are multiples of the same variable, y. To combine like terms means to rewrite the sum or difference of like terms as a single term. In the case of 4y + 2x - 7y, the like terms 4y and -7y can be combined and rewritten as -3y.
To simplify an expression means to write the expression in a simpler form. Combining like terms is one way to do that. Reminder: The multiplication symbol (∗) is often not written. For example, 4 ∗ y is often written as 4y, and (x + 3) ∗ 5 as (x + 3)5.
Example 1: Simplify the expression 5x - (-8)x. Use the distributive property. 5x - (-8)x = (5 ∗ x) - (-8 ∗ x ) = (5 - (-8)) ∗ x = (5 + 8) ∗ x = 13 ∗ x, or 13x
Check your answer by substituting several values for the variable.
Math MessageThese expressions have two terms. Rewrite each expression as a single term.
4y + 7y 4y - 7y
Study Link 9�2 Follow-UpBriefly review answers.
Mental Math and Reflexes Students write the value of each expression when x = 2.
3x + 7 13 4x + 8x 24
x + 2x + 3x + 4x 20 x ∗ 2x - 9 -1
6 + x - ( 1 _ 2 )x 7 -(2x) - (-9) 5
1 Teaching the Lesson
▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION
Algebraic Thinking Draw two name-collection boxes on the board, one for each expression. Record students’ answers in the name-collection boxes. (See margin for example.) Discuss their answers and explanations. Some students might use objects or amounts to justify their answers. For example:
● Think of 4y + 7y as 4 oranges + 7 oranges. That’s 11 oranges, so 4y + 7y = 11y.
● 4 dollars minus 7 dollars is 3 dollars owed, so 4y - 7y is -3y.
Remind students that the multiplication symbol (∗ or ×) is frequently omitted. For example, 7y means 7 ∗ y, and 5(2y + 3) means 5 ∗ (2y + 3).
Ask students to name other expressions that are equivalent to the expressions in the Math Message. Record these in the name-collection boxes. Sample answers for 4y + 7y: 12y – 1y; 88y ÷ 8; 3y ∗ 3 + 2y
▶ Combining Like Terms
WHOLE-CLASS ACTIVITY
(Math Journal 2, pp. 330 and 331)
Algebraic Thinking Read the first two paragraphs on journal page 330 as a class. Terms are the items that are added and subtracted in expressions. Like terms are either number terms (constant terms) or terms that contain the same variables raised to the same power (variable terms). To support English language learners, discuss the meaning of like in this context. Remind students that the number part of a variable term is called the coefficient.
Example 1 shows how to use the distributive property to combine like terms, which means to rewrite the sum or difference of like terms as a single term.
An expression such as 2y � 6 � 4y � 8 � 9y � (�3) is difficult to work with because itis made up of 6 different terms that are added and subtracted.
There are 2 sets of like terms in the expression. The terms 2y, 4y, and 9y are 1 set oflike terms. The constant terms 6, 8, and (�3) are a second set of like terms. Each setof like terms can be combined into a single term. To simplify an expression that hasmore than one set of like terms, combine each set into a single term.
Example 3: Simplify 2y � 6 � 4y � 8 � 9y � (�3) by combining like terms.
Step 1 Combine the y terms. 2y � 4y � 9y � 6y � 9y � �3y
Simplify each expression by combining like terms. Check each answer by substitutingseveral values for the variable.
11. 4 � 7y � 20
12. 5x � 3x � 8
13. 5n � 6 � 8n � 2 � 3n
14. n � π � 2n � �12�π
15. �2.5x � 9 � 1.4x � 0.6
16. 9d � 2a � (�6a) � 3d � 15d
7y � 24
�6n � 43n � �
12��, or 3n � �
�2�
�1.1x � 9.6�3d � 8a
2x � 8
252
Math Journal 2, p. 331
Student Page
Adjusting the Activity
800 Unit 9 More about Variables, Formulas, and Graphs
Redo the Math Message problems using the distributive property to combine like terms.
� To simplify 4y + 7y, think:
4y + 7y = (4 ∗ y) + (7 ∗ y)
= (4 + 7) ∗ y
= 11 ∗ y = 11y
So, 4y + 7y = (4 + 7)y = 11y.
� To simplify 4y - 7y, think:
4y - 7y = (4 ∗ y) - (7 ∗ y)
= (4 - 7) ∗ y
= -3 ∗ y = -3y
So, 4y - 7y = (4 - 7)y = -3y.
To simplify an expression means to write the expression in a simpler form. To support English language learners, explain that even though one expression is in simpler form, the expressions are equivalent. Combining like terms is one way to simplify an expression. Removing parentheses is another way. This method will be the focus of Lesson 9-4.
Example 2 shows how to extend the method in Example 1 to combine more than two like terms. The first two terms are combined. The result is then combined with the next term and so on.
At the board, demonstrate a factoring procedure for combining like terms. Write the expression 2n - 7n + 3n - 4n. Enclose the expression in parentheses and cross out (or erase) each occurrence of the variable n. Then write the variable n to the right of the parentheses, saying that you have undistributed the variable. Evaluate the numerical expression within parentheses. The simplified expression is -6n.
2n - 7n + 3n - 4n = (2n/ - 7n/ + 3n/ - 4n/ )n
= (2 - 7 + 3 - 4)n
= -6n
A U D I T O R Y � K I N E S T H E T I C � T A C T I L E � V I S U A L
Provide additional practice by writing several more expressions containing only one kind of term on the board or overhead. Then have students simplify them on their slates. Suggestions:
2b - (-6b) + 4b 12b
4p - 7p - (-4p) 1p or p
1 _ 2 m + 3 _ 4 m - (- 5 _ 4 m) 10 _ 4 m, 5 _ 2 m, or 2 1 _ 2 m
-2.7w + 9.6w - 7.5w –0.6w
Partners complete the problems on journal page 330. When most students have finished, briefly go over the answers.
799-803_EMCS_T_TLG2_G6_U09_L03_576922.indd 800 2/16/11 7:05 AM
Estimating and Measuring in MillimetersLESSON
9�3
Date Time
All measurements are approximations. The pencil shown below measures about 4 inches.A more precise measurement is about 3�
1106� inches. An even more precise measurement
is about 93 millimeters.
The smaller the unit you use to measure an object, the more precise the measurementwill be.
1. Explain why 93 millimeters is a more precise measurement than 3�1106� inches for the
length of the pencil.
Measure each line segment below to the nearest millimeter.
2. Length of A�B� � mm
3. Length of D�E� � mm
4. Measure P�Q�. Then, in the space provided, draw a line segment that is �58� the length
of P�Q�. Label the segment R�S�.
Length of P œQ œ � mm
Length of RœS œ � mm
5. Measure F�G�. Then, in the space provided, draw a line segment that is 125% thelength of F�G�. Label the segment J�K�.
Length of F�G� � mm
Length of J�K� � mmJ K
60F G
R S
56P Q
73D E
22A B
43210 INCHES
0 1 2 3 4 5 6 7 8 9 10 11CM
35
75
Sample answer: Millimeters are smaller units than sixteenths of an inch.
Math Journal 2, p. 332
Student Page
4. I am a regular polygon with all obtuseangles. I have the smallest number ofsides of any polygon with obtuse angles.How many sides do I have?
Use your Geometry Template to draw this polygon below.
5
1. The area of the rectangle is 238 units2.
Write a number sentence to find the value of x.
Number sentence
Solve for x.
x � units
238 � 7(9 � x)
3. Triangles ABC and DEF are similar.
a. Length of A�B� � units
b. Length of D�F� � units
c. � units �87�Perimeter of triangle ABC
���Perimeter of triangle DEF
3532
5. Use the diagonals to find the sum of the interior angle measures of the octagon below.
Sum of the interior angle measures �
�1,080165 233
2. Solve.
a. �100 � 12w � 4
Solution
b. �24 � p � 8 � 4
Solution
c. 4j � 24 � 6j � 120
Solution
x 9
7 w � �8
p � 20
j � �72
24
40
A
C B
21
28
D
E F
�
248 249 250 251
179
Math BoxesLESSON
9�3
Date Time
25
Math Journal 2, p. 333
Student Page
Lesson 9�3 801
Read the paragraphs at the top of journal page 331 and go over Example 3, which shows how to combine like terms when there is more than one kind of term. Demonstrate Example 3 on the board as shown below. Remind students that they identified variable terms and constant terms in their work with equations (Lesson 6-11).
In the expression 2y + 6 + 4y - 8 - 9y + (-3),
� the variable terms 2y, 4y, and 9y are one set of like terms.
� the constant terms 6, 8, and -3 are a second set of like terms.
� the variable terms are combined into one single term.
� the constant terms are combined into a second single term.
variable termsvariable terms
constant terms
variable terms constant terms
Assign the problems on journal page 331. Go over the answers. Note that an instruction to check answers is included with the problem set. Remind students to check their answers by substituting at least two different values for the variable in each expression.
2 Ongoing Learning & Practice
▶ Estimating and Measuring
SMALL-GROUP ACTIVITY
in Millimeters(Math Journal 2, p. 332)
Students measure line segments to the nearest millimeter.
▶ Math Boxes 9�3
INDEPENDENT ACTIVITY
(Math Journal 2, p. 333)
Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lessons 9-1 and 9-5. The skills in Problems 4 and 5 preview Unit 10 content.
Writing/Reasoning Have students write a response to the following: Explain how the ratio in Problem 3c is related to the ratio of the corresponding sides. Sample answer: The ratio of the perimeters is equivalent to the ratios of the corresponding sides.
799-803_EMCS_T_TLG2_G6_U09_L03_576922.indd 801 2/16/11 7:05 AM
STUDY LINK
9�3
Name Date Time
Combining Like Terms
252
Simplify each expression by rewriting it as a single term.
1. 3x � 12x � 2. (1�35�)y � (1�1
30�)y �
3. �(5t) � 6t � 4. 4d � (�3d) �
Complete each equation.
5. 15k � (9 � )k 6. 3.6p � p � � 0.4p
7. (8 � ) � m � 5m 8. j � 4.5j � 3.8j
Simplify each expression by combining like terms. Check your answers bysubstituting the given values for the variables. Show your work on the back of this sheet.
Example: 18 � 6m � 2m � 26Combine the m terms. 6m � 2m � 8mCombine the number, or constant, terms. 18 � 26 � 44So, 18 � 6m � 2m � 26 � 8m � 44.
802 Unit 9 More about Variables, Formulas, and Graphs
Ongoing Assessment: Math Boxes Problems �2a and 2bRecognizing Student Achievement
Use Math Boxes, Problems 2a and 2b to assess students’ ability to solve equations. Students are making adequate progress if they can use a method (trial and error or equivalent equations) to find a solution and then check the solution by substituting it for the variable. Some students might be able to combine like terms to solve Problem 2c. [Patterns, Functions, and Algebra Goal 2]
▶ Study Link 9�3
INDEPENDENT ACTIVITY
(Math Masters, p. 288)
Home Connection Students simplify algebraic expressions by combining like terms.
3 Differentiation Options
READINESS PARTNER ACTIVITY
▶ Playing Algebra Election 15–30 Min
(Student Reference Book, pp. 304 and 305; Math Journal 1, Activity Sheets 3 and 4; Math Masters, pp. 434 and 435)
Algebraic Thinking Students practice substituting for variables in equations by playing Algebra Election, which was introduced in Lesson 6-11. In this game, students travel through the United States capturing electoral votes by solving a variety of problems. They may play a shorter version of the game by going through the 32 cards only one time. You can tailor the game to address any skills you wish to review by writing additional problems on index cards. The team with the most votes after going through the cards once wins the game.
NOTE: Alaska and Hawaii are not drawn to scale.
Math Masters, pages 434 and 435 show the electoral votes for each state based on the 2000 census.
NOTE A state’s electoral votes equal the total number of its representatives and senators in the United States Congress. The number of representatives depends on the state’s population as determined by the decennial Census. The vote distribution on this map is based on the 2000 Census. Following the 2010 Census, it is likely that some states will lose votes and some will gain votes because of changes in the population.
799-803_EMCS_T_TLG2_G6_U09_L03_576922.indd 802 2/22/11 11:35 AM
Teaching Master
LESSON
9�3
Name Date Time
Simplifying and Evaluating Expressions
pyg
gp
Each expression on the left of the equal sign can be simplified to the expression on the right. Fill in the missing variable or constant terms.
1. 9x + + + 10 = 7x + 13
2. 4m - 8n + + - 6 + = -4m + 32n + 6
3. -2t - (-2v) + 2w + + - = -4t - 2v + 4w
4. 3c + 2c + 6d - 4d + 10 - + = 2d + 2
5. 8f - + 4g - + 13 + = -f + g + 10
Simplify each expression below. Then evaluate the expression for x = -2, y = 3, and z = -4.
Students extend their knowledge of simplifying expressions by finding missing terms in expressions. They also evaluate simplified expressions.
EXTRA PRACTICE PARTNER ACTIVITY
▶ Generating Equivalent Expressions 5–15 Min
(Differentiation Handbook, p. 135)
To provide additional practice with finding equivalent expressions, have students complete name-collection boxes for several different expressions. Students exchange papers with a partner and check that their partner’s expressions are equivalent by substituting a value for the variable in each of the expressions. Suggestions:
� 24x � 1 _ 3 y � 34.2d + 24.7d
ELL SUPPORT
SMALL-GROUP ACTIVITY
▶ Building a Math Word Bank 5–15 Min
(Differentiation Handbook, p. 130)
To provide language support for vocabulary terms, have students use the Word Bank template found on Differentiation Handbook, page 130. Ask students to write the phrases combine like terms and simplify an expression and then to represent these phrases using pictures and examples. See the Differentiation Handbook for more information.