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Fifth Grade CurriculumRepresenting Rules/Patterns/Equations

Table of ContentsTopic PageRelationships Lesson Goals

Relationship Games:Mental Math RiddlesTelephone LineTelephone Line Message CardsSolving Silly StoriesSolving Silly EquationsVocabulary Flashcards

Relationship LessonCreature Activity CardsCreate-A-Monster Extension CardsRelationships in TablesGuided PracticeSubstitution StrategyGuided Practice

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234 - 78 - 910-1112 - 1819 – 21

22 – 262728 – 3031 – 3334 – 3738 – 39

Property of Cy-Fair ISD Elementary Math Dept. (5thGrade) 2014-2015

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Relationships

TEKS: 5.4 Algebraic reasoning. The student applies mathematical process standards to develop concepts of expressions and equations. The student is expected to;(B) represent and solve multi-step problems involving four operations with numbers using equations with a letter standing for an unknown quantity; (R)(C) generate a numerical pattern when a given rule in the form of y =ax or y = x + a and graph (R)(D) recognize the difference between additive and multiplicative number patterns given in a table or a graph. (S)

Lesson Goals: Review relationship vocabulary words (see vocabulary list) Review that a relationship can be written at least 2 ways (2 is half of 4 and 4 is two times two). Review relationships in tables. Introduce the terms multiplicative and additive. Introduce that a letter can represent a variable in tables and equations (P represents the number of

packages and M represents the number of marbles) Introduce that 3 x P can be written as 3 P and 3P

Vocabulary: one-half, two times, three times, four times, twice, double, triple, more than, less than, equal, one-third, one-fourth, additive, multiplicative, expression, relationship, independent variable, dependent variable, increase, sum, product, quotient, difference, greater than, fewer, decrease, inverse operation


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Relationship Games

Use the following games as warm-up activities.

Game 1: Mental Math Riddles

(Individual, Partner or Groups)

Provide students with clues to a number using the relationship vocabulary words. You may choose to show some of the vocabulary flashcards as you say the relationship words (pages 12 - 18). Start off with one clue and increase the number of clues as the students understand the game. Allow time between each clue for students to process the math. Have students share the number only at the end of all of the clues.

Examples:

Start with…Take the number 4, double it. What is the new number? Work up to…. Take the number 3, triple it, and now increase it by 2. That number is 2 less than

the number I am thinking of. What number am I thinking of?

Since this game can be done with little/no prep work, this is a great game to use throughout the year while waiting in line at lunch, for the restroom, or during class changes.

Game 2: Telephone Line


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(Teams)Use this game after you have introduced letters to represent variables in expressions

Create teams by placing students into different lines. Give each student a folded “telephone message” (found on pages 4 - 8). The teacher whispers a given number into the ears of the first student in each line. The student writes their number on the “received line”, applies their given clue, writes the new number on the “send line”, and whispers the new number in the next student’s ear. The student line continues until the last student “rings in” (a play cell phone would be novel). TIP: begin with a short line of students and work your way up. Teacher must pre-select numbers and cards to be sure that they will work together. Otherwise a student may be asked to do something like find half of an odd number.

ExampleFirst Student Second Student Third Student Fourth Student

Teacher whispers“5”

Whispers to second student“10”

Whispers to thirdstudent“14”

Whispers to fourth student“18”

Ring in“9”

*You will need to run multiple copies of the cards in order to have enough copies for each line.

Telephone Line Message Cards


Telephone Message

Received (R): 5Message: 2 R

Telephone Message

Received (R): 10Message: Increase by 4

Send: 14

Telephone Message

Received: 14Message: Find the sum of it and 4

Send: 18

Telephone Message

Received: 18Message: R ÷ 2

Send: 9

4

RECEIVED: __________

MESSAGE: Double it!

SEND: ___________


MESSAGE: Add it twice!

SEND: ___________

RECEIVED (R): _________

MESSAGE: 2R

SEND: ___________

RECEIVED (R): _________

MESSAGE: 2 x R

SEND: ___________

RECEIVED (R): _________

MESSAGE: 2 R

SEND: ___________

RECEIVED (R): _________

MESSAGE: Product of R and 2

SEND: ___________


5


MESSAGE: Increase it by ____!

SEND: ___________

RECEIVED (R): _________

MESSAGE: R +

SEND: ___________


MESSAGE: Find the sum of it and 4!

SEND: ___________

RECEIVED (R): _________

MESSAGE: R + 10

SEND: ___________

RECEIVED: __________MESSAGE:__ greater than it!

SEND: ___________

RECEIVED (R): _________

MESSAGE: R + 2

SEND: ___________


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RECEIVED: __________MESSAGE: Decrease it by ____!

SEND: ___________

RECEIVED: __________MESSAGE:___ fewer than it!

SEND: ___________

RECEIVED (R): _________

MESSAGE: R-2

SEND: ___________

RECEIVED (R): _________

MESSAGE: R - 10

SEND: ___________


MESSAGE: Find the difference of it and 1!

SEND: ___________

RECEIVED (R): _________

MESSAGE: R -

SEND: ___________


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RECEIVED: __________MESSAGE:Find the product of it and 3!

SEND: ___________

RECEIVED: __________MESSAGE:Three times it!

SEND: ___________

RECEIVED: __________MESSAGE:Four times it!

SEND: ___________


MESSAGE: Triple it!

SEND: ___________RECEIVED (R): _________

MESSAGE: 5 R

SEND: ___________

RECEIVED (R): _________

MESSAGE: 3R

SEND: ___________


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Game 3: Solving Silly Stories

(Partners/Groups)

Use silly stories for partners or table groups to solve. These examples (on pp. 8 - 9) are to use with your groups or write your own using the vocabulary words with the attached flashcards (pp. 12 - 18).

Example: Take the number of wheels on a tricycle. (3) Multiply by the number of eggs in a dozen. (3 x 12 = 36) Subtract the number of hours in a day. (36 - 24 = 12)

Solving Silly Stories

Take half the number of tires on a car. Add the number of donuts in a dozen. Multiply the sum by the number of babies in twins.

Multiply the number of toe nails a person has by the number of nostrils. Add the number of legs on an octopus. Double the sum.

Take the number of donuts in a half dozen. Add quadruple the number of sodas found in a six pack.

Double the number of legs on an insect. Divide by the number of eyes it has. Add the number of letters in b r o t h e r.

Take the number of points on a star. Add the number of quarters in a dollar. Multiply the sum by the number of arms on a person. Divide by the number of babies in a set of triplets.

Take the number of legs on an octopus. Multiply it by the number of legs on a spider.

Solving Silly Stories


1.

2.

3.

4.

5.

6.

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*Teaching Tip: Use math related facts after the students have had practice with some of the simple stories. They can use a reference sheet (math chart, shapes, etc) during these problems. Here are some examples:

Find the sum of the number of edges on a hexagon and the number of vertices in a cube. Then double the number.

Take the number of hours in a day. Cut it in half. Multiply it by the number of days in a year. Increase the product by 16.

Take the number of inches in a foot. Triple it. Divide the product by the number of quarts in a gallon.

4. Find the difference between the number of centimeters in a meter and half the meters in a

kilometer.

Multiply the vertices on a triangular prism by the edges on a cube. Add the number of sides on a pentagon.

6. Divide the number of minutes in an hour by the minutes in one-fourth of an hour.


1.

2.

3.

5.

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Game 4: Solving Silly Equations

(Partners/Groups)

Have students use the key at the top of the page to solve the Silly Equations.

Students can make their own key and write their own Silly Equations.

You can have students cut off the last column and glue it into the left side of their IMN.

Work the example with the students. The completed example is shown below.

Solving Silly Equations

Use the key below to solve the expressions:

Example: T x S= 24

24 ÷ W = 6

6 + D = 18


D- Number of donuts in a dozen W- Number of Wheels on a car

Q – The number of sides on a quadrilateral F – Number of Fingers on 2 hands

T – The number of wheels on a tricycle S – The number of legs of a spider

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Solving Silly Equations

Use the key below to solve the expressions:

Example: T x S=

÷ W =

+ D =

D – 3 =

F =

+ 4 =

F + 26 =

÷ Q =

T =

Make your own key:

S x W =

- F =

D =

S ÷ W =

+ D =

- F =

Make your own:

D ÷ W =

+ S =

- F =

120 – D =

T =

- 54=

Make your own:

Relationship Flashcards


D- Number of donuts in a dozen W- Number of Wheels on a car

Q – The number of sides on a quadrilateral F – Number of Fingers on 2 hands

T – The number of wheels on a tricycle S – The number of legs of a spider

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more than

less than

fewer

triple


13

one-fourth

greater than

increase

decreaseProperty of Cy-Fair ISD Elementary Math Dept. (5thGrade) 2014-2015

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three times

four times

equal

one-third


15

two times

twice

double

one-half


16

add

subtract

multiply

divide


17

sum

product

quotient

difference


18

Inverse Operation

additive

multiplicative


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Relationship Lesson

TEKS: 5.4 Algebraic reasoning. The student applies mathematical process standards to develop concepts of expressions and equations. The student is expected to;(B) represent and solve multi-step problems involving four operations with numbers using equations with a letter standing for an unknown quantity; (R)(C) generate a numerical pattern when a given rule in the form of y =ax or y = x + a and graph (R)(D) recognize the difference between additive and multiplicative number patterns given in a table or a graph. (S)

Materials: Fraction circles, Fraction squares, or Fraction strips, task card, Blockhead/Circle Creature sheets (pp. 22 - 26), bags of color tiles, envelopes of color tiles, guided practice (pp. 23-26)

Background Knowledge: The teacher will illustrate various relationships with the fraction pieces, numbers, and other concrete items.

Relationship Vocabulary: one-half, two times, three times, four times, twice, double, triple, more than, less than, equal, one-third, one-fourth.

Begin by having a discussion about the vocabulary. The point to be made is that relationships can be written 2 ways. (2 is half of 4 and 4 is double 2). You will want to practice with smaller numbers at first. Record this on the board.

10 is two times ______. Are we looking for a bigger or smaller number? Why? (Smaller, because 10 is twice as big as the number we are looking for).What are other words for two times? (twice, double)

10 is two times 5. How did we find our answer? (10 = 2 x or 10 ÷ 2 = 5)

What is another way we can write the relationship between the numbers 10 and 5?5 is one-half of 10.

Repeat this process of writing the relationship 2 ways using whole numbers with various relationship vocabulary.

Concrete Relationship Vocabulary ActivityPass out a set of fraction circle pieces to each table. Write the word bank on the board.

Ask the students to compare the one yellow piece (¼ circle) to one pink piece (½ circle). What is the relationship between the two pieces? What is a number sentence that can be used to describe this Property of Cy-Fair ISD Elementary Math Dept. (5thGrade) 2014-2015

Word Banktwice three times one-fourthdouble triple equaltwo times one-third more thanone-half less than

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relationship? Guide students to use letters to represent the colors and then complete the relationship using the mathematical symbols. Here are some examples that describe the relationship between the yellow and pink pieces:

Yellow is one-half of pinkY = P ÷2

Pink is two times yellow Pink is twice as much as yellowPink is double yellowP = 2 YP = 2Y

Yellow is less than pinkY < P

Pink is more than yellow.P > Y

Repeat this process using other fraction pieces with various relationships.

Pass out bag 1, (B) ( 9 color tiles) and envelope 1 (E) (27 color tiles).

Ask the students to compare bag 1 (B) to envelope 1 (E). What is the relationship between the two amounts? What are some number sentences that can be used to describe the relationships?

Bag 1 is one-third of 27B = 27 ÷ 3

9 is 18 less than 279 = 27 – 18

Bag 1 has fewer color tiles than the envelopeB < E

27 is three times 927 = 3 9

27 is triple 927 = 3 9

27 is 18 greater than 927 is 18 more than 927 = 9 + 18

The envelope has more than two times the amount of color tiles than bag 1.E > 2 B

Repeat this process using other bags of concrete items with various relationships.

Comparing Creature Parts Activity:


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1. Each table or partners will receive fraction pieces and a blockhead/creature mat (found on pp. 22, 24 & 25). The students will place the appropriate pieces on top of the outlined figure.

2. Using the creature, students will fill in the missing blanks of the corresponding relationship sheet using words in the word bank (pp. 23, 26 & 30).

Encourage students to lay the pieces on top of each other to find the relationships.

3. Share and discuss.

Extension Cards (pg. 27):

1. The teacher has students create their own blockhead/circle creature and write relationships and number sentences between its parts.

2. The teacher uses the Create-A-Monster clue cards or has the students create the “monster”. Although the monsters may vary from group to group, the relationships will be the same. For example, if one of the clues is –

The monster’s head is twice the size of his feet. One group may have the ½ circle for the body and a whole circle for its head. Another group may have the ½ circle for the head and a ¼ circle for the body. Both monsters will show the correct relationship, but look differently. Discuss with the group.

3. You may make your own additional clue cards.


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Circle Creature

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Circle Creature1. One leg is ______ the head.

The head is ______ one leg.

Write two number sentences to show this relationship using the letters in the key to represent the creature’s body parts. __________ ___________

2. The arm is ______ the head.

The head is ______ the arm.


3. The arm is ______ one leg.

The leg is _____ the arm.


4. The two legs are _____ to the head.

The head is ______ to the two legs



Word Bank

twice three times one-fourthdouble triple equaltwo times one-third more thanone-half less than

KEYL – LegH – HeadA – ArmB - Body

24Property of Cy-Fair ISD Elementary Math Dept. (5thGrade) 2014-2015

Blockhead

Blockhead

Fraction Squares

Fraction Tiles

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Blockhead1. The head is ______ the body.

The body is ______ the head.Write two number sentences to show this relationship using the letters in the key to represent the creature’s body parts. __________ ___________

2. The arm is _____ the head.Property of Cy-Fair ISD Elementary Math Dept. (5thGrade) 2014-2015

Blockhead


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The head is _____ the arm.Write two number sentences to show this relationship using the letters in the key to represent the creature’s body parts. __________ ___________

3. The body is ______ the leg.The leg is _____ the body.Write two number sentences to show this relationship using the letters in the key to represent the creature’s body parts. __________ ___________

4. Two legs _____ the body.The body is _____ to the two legs.Write two number sentences to show this relationship using the letters in the key to represent the creature’s body parts. __________ __________

5. The hand is _____ the arm.The arm is _____ the hand.Write two number sentences to show this relationship using the letters in the key to represent the creature’s body parts. __________ ___________

6. The head is _____ the hand.The hand is _____ the head.Write two number sentences to show this relationship using the letters in the key to represent the creature’s body parts. __________ ___________

Create-A-Monster

Head is double his legs His arm is one-fourth of

his head

Create-A-Monster


Word Bank

twice three times one-fourth ofdouble triple equaltwo times one third of more thanone-half of less than

11 2Extension Cards


27

Body is one-third of his head A = B÷3

H = 4L

Create-A-Monster

His feet and body are equal in size

His head is double his body

His arms are one-fourth the size of his head

Create-A-Monster

A = H B = 2 A L= 4 H

Relationships in Tables

In this section we are going to look at organizing the relationships in tables.

In the previous section, students discovered the relationships between the body parts of the creature. Provide each student with Circle Creature mat and the fraction circles so that they can identify which pieces represent the head (H) and the arm (A). Now the students can put the mats away. Ask students to place enough arm pieces (four 1/8 pieces) to completely cover the piece that was used for the creature’s head (1/2 piece).

Review this problem (TN pp 23) along with the fraction circles.

2. The arm is ___14__ the head.


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The head is __4 X___ the arm. Write two number sentences to show this relationship using the letters in the key to represent the creature’s body parts. __A = H ÷4__ H = 4 A_

Restate this relationship in words with the students.

One creature’s head is the same size as four creature arms. (H = 4 A)

One creature’s arm is 14 the size of the creature’s head. (A = H ÷ 4)

Note: It is important to remind students that this is about the relationship between the sizes of body parts, not the number of body parts on creatures. If not reminded, students may become confused by the fact that each creature has 1 head and 2 arms.

Head

1 6 9

Arm 4 12 28 48

The head is equal to 4 arms

It is also important to state the inverse relationship:

1 arm is equal to ¼ of the head

Complete the table with your students. Guide them through the process as needed.

Now let’s take a look at another type of table.

The carnival is offering bonus tickets with every purchase of carnival tickets. The table below shows the different numbers of tickets purchased and the number of bonus tickets that are given.


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Guide the students to write an equation that shows the relationship between the 2 sets of data.

The number of tickets purchased + 6 = The number of bonus tickets (P + 6 = B)

The inverse relationship can also be written as:

The number of bonus tickets – 6 = The number of tickets purchased (B - 6 = P)

Have the students look at the two tables (p. 30). Discuss the similarities and differences between the two tables. By the end of the conversation, the Circle Creature table should be labeled as having a multiplicative relationship and the Ticket table as additive relationship.

Select the appropriate problems from the guided practice to model with your class and for partner/group work.

Relationships in Tables


Purchased Tickets (P)

Bonus Tickets (B)

3 9815

2522 28

3345

81

30

Circle Creature

Write 2

number sentences that show the relationship in the table. ____________________ _____________________

What type of relationship is this? _______________

Carnival Tickets

The carnival is offering bonus tickets with every purchase of carnival tickets. The table below shows the different numbers of tickets purchased and the number of bonus tickets that are given

Write 2 number sentences that show the relationship in the table. ____________________ _____________________

What type of relationship is this? _______________

Relationships Guided PracticeProperty of Cy-Fair ISD Elementary Math Dept. (5thGrade) 2014-2015

Head (H) 1 6 9Arm (A) 4 12 28 48

PurchasedTickets (P)

BonusTickets (B)

3 98

1525

22 2833

4581

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1.

Write a statement and number sentence that describes the relationship between the fish and the pieces of fish food.

_________________________________ ________________Write a statement and number sentence that describes the inverse relationship between the fish and the pieces of fish food._________________________________ ________________

Is this an example of an additive or a multiplicative relationship? How do you know?

____________________________________________________________________

2.

Write a statement and number sentence that describes the relationship between the flowers and the containers___________________________ ________________Write a statement and number sentence that describes the inverse relationship between the flowers and the containers.

_________________________________ ________________


___________________________________________________________________

3. Set A (A) Set B

(B)9 3


Word Bankdoubletriple

three timestwo timesone-halfone-third

Word Banktwo times

three timesfour timesone-thirdone-half

one-fourth

Word Bankmore thanless than

twiceone-half

32

12 415 524 830 10

Write a statement and number sentence that describes the relationship between the numbers in Set A and the numbers in Set B.

_________________________________ ________________

Now write a statement and number sentence that describes the inverse relationship between the numbers in Set A and the numbers in Set B.

__________________________________ _______________


____________________________________________________________________

4.

Write a statement and number sentence that describes the relationship between the number of gumballs and the prices.

_________________________________ ________________

Now write a statement and number sentence that describes the inverse relationship between the number of gumballs and the prices. __________________________________ _______________


___________________________________________________________________

5.


Number of Gumballs 10 15 20 25 30Price $20 $30 $40 $50 $60

Family Size Yogurt BoxesNumber of Boxes 18 20 22 24Total Number of Yogurts 36 40 44 48

Word Banktimes

one-halfmore than

33

Write a statement and number sentence that describes the relationship between the number of boxes and the number of yogurts.

_________________________________ ________________

Now write a statement and number sentence that describes the inverse relationship between the number of boxes and the number of yogurts.

__________________________________ _______________


____________________________________________________________________

6.

Write a statement and number sentence that describes the relationship between the number of theater tickets sold and the movie tickets sold.

_________________________________ ________________Write a statement and number sentence that describes the inverse relationship between the number of theater tickets sold and the movie tickets sold.

_________________________________ ________________


____________________________________________________________________

Substitution Strategy

You may refer to this strategy as the “cross off”, “plug it in”, or “substitution” strategy. The student will select a set of related numbers off of a chart, table or number machine and substitute the word(s) with a number that represents them. Students will then read the sentences using the numbers to determine which statement(s) is true. The students will then substitute another set of numbers off of the chart to confirm that their answer choice is correct.Property of Cy-Fair ISD Elementary Math Dept. (5thGrade) 2014-2015

TicketsTheater Movie

6 189 2111 2315 27

Word Banktwo times

three timesfour times

lessmore

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Example 1:

Step 1The students will begin the four step process. 1. The table shows the number of Skittles bags and the total number of Skittles in

the bags. Which relationship between the number of bags of Skittles and the total number of Skittles is true?

Skittles in Bags

A. The number of bags is 8 times the total number of Skittles.

B. The number of bags is 28 more than the total number of Skittles.

C. The total number of Skittles is 8 times the number of bags.

D. The total number of Skittles is 28 more than the number of bags.

Step 2 Guide students to write a number sentence to show the relationship in the table. You can use B for bags and S for number of Skittles.

B x 8 = SS ÷ 8 = B

Step 3At the strategy step, they will select a set of related numbers off of the table.

The students will substitute the numbers in for the words in the answer choices.

1. The table shows the number of Skittles bags and the total number of Skittles in the bags. Which relationship between the number of bags of Skittles and the total number of Skittles is true?

Skittles in Bags


Number of bags 4 6 9

Total Number of Skittles

32 48 72



32 48 72

35

4 = 8 x 32A. The number of bags is 8 times the total number of Skittles. NT 4 = 28 + 32B. The number of bags is 28 more than the total number of Skittles. NT

32 = 8 x 4C. The total number of Skittles is 8 times the number of bags. T

32 = 28 + 4D. The total number of Skittles is 28 more than the number of bags. T


36

Try #2

Try #2

The student will now need to substitute another set of related numbers to see which relationship is true.

1. The table shows the number of Skittles bags and the total number of Skittles in the bags. Which relationship between the number of bags of Skittles and the total number of Skittles is true?

Skittles in Bags

4 = 8 x 32A. The number of bags is 8 times the total number of Skittles. NT

4 = 28 + 32

B. The number of bags is 28 more than the total number of Skittles. NT

32 = 8 x 4C. The total number of Skittles is 8 times the number of bags. T

48 = 8 x 6 T

32 = 28 + 4D. The total number of Skittles is 28 more than the number of bags. T

48 = 28 + 6 F

Example 21. The table shows the number of Skittles bags and the total number of Skittles in

the bags. Which relationship between the number of bags of Skittles and the total number of Skittles is true?

Skittles in Bags




32 48 72

Number of bags (B) 4 6 9

Total Number of Skittles (S)

32 48 72

37

A. B = 8 x S

B. B = S + 28

C. S = 8 x B

D. S = B + 28

Since students will write the relationship between the 2 sets of numbers two ways, they should be able to select the correct answer. Students can also use the same substitution strategy to show the correct answer.

6 = 8 x 4A. B = 8 x S F

6 = 48 + 28

B. B = S + 28 F 48 = 8 x 6

C. S = 8 x B T

48 = 6 + 28 D. S = B + 28 F

Please note that as the year progresses the numbers will become larger making them more difficult. In the beginning of the year, we will be using relationships that only use facts up to 12.


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Guided Practice Problems

1. Steven and his friends were at a carnival. They had to buy tickets to get on the rides. The table below shows the relationship between the price of the tickets and the total number of tickets. Which statements about the relationship between the price and the total number of tickets are true?

Us the substitution strategy to determine if each statement is true or not true about the table.

The price is half the total number of tickets.

The price is twice the total number of tickets.

The total number of tickets is twice the price.

The total number of tickets is three times the price.

The price is 2 times the total number of tickets.

Write two equations to describe the relationship in the table where P is the price and T is the total number of tickets.

____________________________ ___________________________


Price $5 $15 $20 $30 $35

Total Number of Tickets 10 30 40 60 70

39

2. Clarissa and her friends were making necklaces using stones and beads. The table below shows the relationship between the number of stones and beads used in the necklaces. Which of the statements below are true?

The number of stones is half the number of beads.

The number of stones is double the number of beads.

The number of beads is 6 times the number of stones.

The number of beads is 3 times the number of stones.

The number of stones is the number of beads divided by 6.

Write two equations to describe the relationship in the table where S is the number of stones and B is the number of beads.

____________________________ ___________________________


Total Number of Stones 2 3 5 7 8

Total Number of Beads 12 18 30 42 48

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