Variables and expressions Block 1 Student Activity …...Fill in the blanks to translate the clues into equations using expressions and variables. Use f = father’s age, m = mother’s

Student: Class: Date:

Variables and expressions Block 1 Student Activity Sheet

Copyright 2017 Agile Mind, Inc.® Content copyright 2017 Charles A. Dana Center, The University of Texas at Austin

Page 1 of 4 With space for student work

1. Record your understandings of key vocabulary for this topic.

Vocabulary term My understanding of what the term means

Examples that show the meaning of the term.

a. Variable

b. Expression

c. Equation





2. Follow these steps. Record your work in the table.

Directions How your number changes

Step 1. Write down your age in years. Step 2. Multiply your age by 2.

Step 3. Add 10.

Step 4. Multiply by 5.

Step 5. Add the number of brothers and/or sisters you have.

Step 6. Subtract 50. The result is your ending number.

Compare your ending number with your partner's ending number. What patterns do you notice when you look at your final answers? Why do you think this happens?

3. Sketch a square. Then label it to show how the algebraic rule P = 4s can be seen in your sketch. What does each of the variables represent?

4. When a variable is used as a placeholder, it stands for a specific value. In the following

equation the variable, y, stands for a specific value.

7 + y = 15

Reason about the numbers in the equation. What is the value of y?





5. REINFORCE Here are two more magic number puzzles.

a. For each column, follow the steps. Record your work in the table.

Directions Pick any starting number

Pick a different starting number

Pick a fraction

Step 1. Write down a number.

Step 2. Double it.

Step 3. Add 4.

Step 4. Divide by 2.

Step 5. Subtract your original number.

b. What is always true about the ending number in this puzzle?

c. Why do you think it is true?





d. For each column, follow the steps to find the ending number for the given starting number.

Directions How your number changes Pick your

own number â

Step 1. Write down a number. 2 4

Step 2. Multiply by 3. 6

Step 3. Add 12. 18

Step 4. Divide by 3. 6

e. What is always true about the ending number in this puzzle?

f. Why do you think that is true?





1. For each fruit equation puzzle, there is a set of clues. The clues are equations that use fruit and numbers. Use the clues to figure out the number each fruit represents.

2. List some algebraic rules or formulas you have worked with before. Describe what the

variables represent in each rule.





3. Here are some more fruit equation puzzles.

a. Solve the puzzles by determining the value of each fruit. Note: the value of a fruit may change from one puzzle to another.

b. What strategies did you and your partner use to solve the fruit equation puzzles?





4. For each of the fruit equations, work with your partner to rewrite the fruit equations using letters as variables. You can decide which letters to use for your variables.

Puzzle 1

Equations using fruit Equations using letters

Puzzle 2

Equations using fruit Equations using letters





5. REINFORCE Create your own fruit equation puzzle. In the space below the table, provide the solution and show that your solution works.

a. My Fruit Equation Puzzle

Clue 1

Clue 2

Clue 3

Clue 4

b. Show your work and your solutions here.





1. Use the clues provided to determine Caleb’s age.

Clues: Alyssa is 11 years old. Tyler is 4 years older than Caleb and 3 years older than Alyssa.

2. What are some clues that help you understand which operation to record when writing an algebraic expression?

3. REINFORCE Write two equations in words and trade them with a partner. Then translate your partner’s equation.





4. Alexis is thinking about the ages of her family members and creates a puzzle. Here are her clues: The quotient of my mother’s age and my age is 4. The difference between my father’s age and my age is 23. If my father is 30, how old is my mother? a. Fill in the blanks to translate the clues into equations using expressions and variables.

Use f = father’s age, m = mother’s age, and a = Alexis’ age.

23 a 4 f 30 7 m 28

b. What is Alexis’ mother’s age?





5. Consider the drink machine scenario.

a. What are the variables?

b. What is the relationship between the variables?

depends on .

c. List at least three more dependency relationships.

6. REINFORCE Write an age puzzle about some of the members of your family. Trade your

age puzzle with a partner. Translate your partner’s age puzzle into symbols.

!





1. Consider the banquet table scenario.

a. Complete the table to study the relationship between the number of tables and the number of people seated.

Number of tables People seated

1 6

2 12

3 7 21

b. Study the pattern in the table. Then write a rule in words that you can use to find the number of people for any number of tables.

c. Write a rule in symbols to represent the relationship between the number of tables and the number of people that can be seated. Use n for number of tables and P for number of people seated





2. Consider the representations of the relationship between the number of banquet tables and the number of people that can be seated.

a. What does the point (2,12) represent in this scenario?

b. What are the variables in this situation?

c. Which variable is the dependent variable?





d. How are the various representations related? Discuss your observations with a partner and record them here.

3. Use the algebraic rule P = 6n to determine the number of guests that can be seated at 31 tables.





4. REINFORCE How would the banquet table situation change if each table was an octagon and could seat 8 people? How would this change the table, graph, and algebraic rule? Complete the following steps.

a. Complete the table.

Number of tables People seated

1 8

2

3

4

10

21

b. Graph the relationship on the coordinate grid.





c. Write an algebraic rule to represent the relationship between the number of

banquet tables and the number of people seated. Remember to specify a letter to represent each variable.

d. Use your algebraic rule to determine the number of people seated if you have 37 banquet tables.

e. Use your algebraic rule to answer this question: If you have 300 people, how

many banquet tables will you need?

!





1. What is a multiplicative linear relationship?

2. Think about how the sides of a square are related to its perimeter. Complete the following steps. a. Record the formula for the perimeter of a square.

b. What does the variable s represent in the formula? What does the variable P represent?

c. Which variable is the dependent variable? Which variable is the dependent variable?

d. What is the coefficient in the formula?

e. Sketch a square and label it to show how the formula is related to your sketch.





3. Chris and Erlinda want to order 1 tablecloth for every table, plus 4 extra in case some need to be replaced. Complete the table to represent the relationship between the number of tables and the number of tablecloths.

Number of tables Process Number of tablecloths 1 1 + 4 5

2 2 + 4 6

3 3 + 4 7

4 7 x

4. Graph the values in the table.





5. Chris and Erlinda found that the equation y = x + 4 represents the number of tablecloths,

y, for any number of tables, x. If they are going to use 15 tables at the dance, how many tablecloths do they need?

6. Think about the differences you have seen in the tables and graphs of multiplicative and additive linear relationships. Can you solve this puzzle to show what you have learned? For each statement, indicate whether it describes a multiplicative linear relationship, an additive linear relationship, or both.

Statement Multiplicative, additive, or both

The point (0,0) belongs on the graph.

The x-value is being multiplied by the same number each time to get the new y-value.

The same number is being added to x each time to get the new y-value.

The values in the table form a line when they are graphed.





7. REINFORCE Javier is 3 years older than his sister. a. Write an equation that relates the age of Javier, y, to the age of his sister, x.

b. Is this relationship additive or multiplicative? Explain.

8. REINFORCE Marsha gets paid $15 for every lawn she mows.

a. Write an equation that relates the money she makes, y, to the number of lawns she

mows, x.

b. If Marsha mows 7 lawns this month, how much money will she earn? Use your equation to show how you answered this question.

c. Is this relationship additive or multiplicative? Explain.





9. REINFORCE Consider these linear relationships. What are the similarities and differences between them?

y = 3x

y = x + 3

!




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Determine if each ‘card’ represents a multiplicative or an additive relationship. Create a different representation of the relationship if you are unsure.

CARD A

x y

0 0

1 1

2 2

3 3

CARD B

x y

1 3

3 9

5 15

8 24

CARD C

CARD D

CARD E

At the amusement park, Kayla pays $4.50 to ride on the go-carts. When she leaves, the

amount she spent on go-carts, y, is based on the number of times she rode them, x.

CARD F

Jack’s bank charges a transaction fee of $1 each time he uses his debit card to withdraw money from a different bank. His total cost, y, for a transaction includes the total cash withdrawal, x, and the transaction fee.




Page 2 of 2

CARD G

y = 0.75x

CARD H

y = x + 3

CARD J

x y

0 0

1 1.5

2 3.0

3 4.5

CARD K

To make sure that every customer gets as much ribbon as they have ordered, the fabric store automatically gives each customer an

additional

34

inch. The total amount of

ribbon each customer gets, y, includes the

length ordered, x, and the

34

inch added to

each order.





1. Consider how many people can be seated if tables have to be pushed together. a. Sketch or build some of the connected banquet tables and record the data in a table.

Number of tables,

n Process Number of people seated, P

2

3

4

5

6

b. Write a rule in words that you can use to find how many people can sit around any number of tables that are pushed together.





2. Erlinda, Chris, Pauline, and the caterer each used a different strategy to figure out how many people could sit at 6 tables. The following number sentences show how each person approached the problem.

Can you explain what each person was thinking? For each strategy, connect your explanation to the picture of the tables.





1. Write a definition and give examples and nonexamples of like terms.

2. Simplify these expressions.

4a + a + 5 + 7a + 2

5b – b – 7 + b

c + 8c – 2c + 5b

3a2 + 6a + a2

3. Look back at the process you recorded on question 1 of the Block 6 Student Activity Sheet. Use your counting process and the table you created to develop your own algebraic rule. If your rule is already the same as the rule made by Chris, try to think of another process and rule.





4. Write algebraic rules for the caterer’s strategy and Pauline’s strategy.

a. The caterer’s strategy:

Number of tables, n Caterer’s strategy People seated, P 1 1 + (2 • 2) • 1 + 1 6

2 1 + (2 • 2) • 2 + 1 10

3 1 + (2 • 2) • 3 + 1 14

4 1 + (2 • 2) • 4 + 1 18

n

b. Pauline’s strategy:

Number of tables, n Pauline’s strategy People seated, P

1 (6 • 1) – (2 • 0) 6

2 (6 • 2) – (2 • 1) 10

3 (6 • 3) – (2 • 2) 14

4 (6 • 4) – (2 • 3) 18

n





1. Consider another fruit equation.

a. The expressions on each side of the equal sign are equivalent. How do you know they are equivalent?

b. What number(s) could the plum represent that would make the equation true?

2. Consider these fruit equation puzzles. For each puzzle, decide whether the expressions are equivalent. In other words, is the equation true or false?

a.

(circle one): True False

b.


c.


d.


e.


f.






3. For each mathematical property, state it algebraically by using variables. Just write the first four properties. You do not need to fill in the last property yet.

Commutative Property of Addition

Commutative Property of Multiplication

Associative Property of Addition

Associative Property of Multiplication

Distributive Property of Multiplication over Addition

4. Study what is shown and determine the missing examples. You get to choose the

variables or values you want to use.

Property of operations Numerical example Algebraic representation

Additive Identity Property of 0 5 + 0 = 5

Multiplicative Identity Property of 1 7•1= 7

Existence of multiplicative inverses If a ≠ 0 , then w •

1w

= 1





1. State the Distributive Property of Multiplication over Addition algebraically using variables.

c a e b d

2. Summarize what you learned about equivalent expressions and properties of operations

by creating numerical and algebraic examples. Complete the table. For your numerical examples, you can choose any values. For your algebraic expressions, you can choose any variables.

Property of operations Numerical example Algebraic representation

Commutative Property of Addition 7 + 4 = 4 + 7 a + b = b + a

Commutative Property of Multiplication 5 • 6 = 6 • 5

Associative Property of Addition (a + b) + c = a + (b + c)

Associative Property of Multiplication (2 • 7) • 3 = 2 • (7 • 3)

Distributive Property of Multiplication over Addition a(b + c) = ab + ac





3. Now apply your understanding of properties of operations. For each equation, study the expressions on each side of the equal sign. Are the expressions equivalent?





4. REINFORCE Use the Distributive Property to write the following expression in an equivalent form.

3 • (5 + 25)

5. REINFORCE Imagine you made a purchase in a candy store. What purchase could be modeled by the expression in question 4? Write a story to explain.

6. REINFORCE Use the Distributive Property to write the following expression in an equivalent form.

(6 • 8) + (6 • 10)

7. REINFORCE Imagine you made a purchase in a candy store. What purchase could be modeled by the expression in question 6? Write a story to explain.





8. REINFORCE To illustrate the Distributive Property, Carla created a single rectangle modeling 8 • 14. a. How does Carla’s diagram illustrate the

Distributive Property?

b. Using Carla’s work as an example to illustrate the Distribute Property. Create a single rectangle to model 7 • 13. Then use the Distribute Property to write the equivalent expressions shown by your model.

Equivalent expressions:





9. REINFORCE You can use the Distributive Property with area models and unknown lengths. The unknown length in these area models is represented by the variable x. Marty and Jim made area models to show 4(x + 8).

Marty’s Model Jim’s Model

a. How does each model show that 4(x + 8) = (4 • x) + (4 • 8) = 4x + 32?

b. Create an area model to represent the product of 3(x + 4). Sketch your model using

the rectangle provided. Clearly label the factors and record the product.

!

Documents

Variables and expressions Block 1 Student Activity …...Fill in the blanks to translate the clues into equations using expressions and variables. Use f = father’s age, m = mother’s