Richland Parish School Board



Grade 6

Mathematics

Unit 8: Patterns, and Algebra

Time Frame: Approximately 4 weeks

Unit Description

The focus of this unit is on working with patterns and variables. A number line is used to graph equations and inequalities. Opportunities to represent, analyze, and generalize a variety of equations and inequalities with tables, graphs, words, and when possible, symbolic rules are provided.

Student Understandings

Students should understand that symbolic algebra can be used to represent situations and to solve problems. Students use modeling as an appropriate strategy to solve math problems whether by drawing figures, using a number line, or another technique. They can model and identify perfect squares up to 144 and can match algebraic equations and expressions with verbal statements and vice versa.

Guiding Questions

1. Can students recognize squares to 144?

2. Can students evaluate expressions for specified variable values?

3. Can students match or create stories to go with a given algebraic expression or equation?

4. Can students solve equations?

5. Can students graph inequalities and equations on a number line?

Unit 8 Grade-Level Expectations (GLEs) and Common Core State Standards

|Grade-Level Expectations |

|GLE # |GLE Text and Benchmarks |

|Algebra |

|14. |Model and identify perfect squares up to 144 (A-1-M) |

|15. |Match algebraic equations and expressions with verbal statements and vice versa (A-1-M) (A-3-M) (A-5-M) (P-2-M) |

|16. |Evaluate simple algebraic expressions using substitution (A-2-M) |

|Patterns, Relations, and Functions |

|37. |Describe, complete, and apply a pattern of differences found in an input-output table (P-1-M) (P-2-M) (P-3-M) |

|CCSS for Mathematical Content |

|CCSS # |CCSS Text |

|Expressions and Equations |

|6.EE.1 |Write and evaluate numerical expressions involving whole-number exponents. |

|6.EE.5 |Understand solving an equation or inequality as a process of answering a question: which values from a specified |

| |set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a |

| |specified set makes an equation or inequality true. |

|6.EE.7 |Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for |

| |cases in which p, q and x are all nonnegative rational numbers. |

|6.EE.8 |2013-2014 |

| |Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or |

| |mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; |

| |represent solutions of such inequalities on number line diagrams. |

|ELA CCSS |

|Reading Standards for Literacy in Science and Technical Subjects 6–12 |

|RST.6-8.7 |Integrate quantitative or technical information expressed in words in a text with a version of that information |

| |expressed visually (e.g., in a flowchart, diagram, model, graph, or table). |

Sample Activities

Activity 1: Graphing Perfect Squares (GLE: 14, CCSS: 6.EE.1)

Materials List: Graph Paper BLM, colored pencils or markers, scissors, paper, glue or tape, pencil

To give the students a visualization of perfect squares, distribute the Graph Paper BLM. Ask them to make twelve perfect squares—[pic], [pic], and so on through [pic]. Ask students to outline or “frame” each square with a colored pencil or marker, cut out the square, write the name of each square in the frame (e.g.,[pic]), write the area of the square (e.g.,[pic]), and finally, attach each square in order by size on another piece of paper.

Remind students of previous work with perfect squares in Unit 6. Students will use knowledge of perfect squares to evaluate expressions and solve equations in activities in this unit.

Activity 2: Pay It Forward (CCSS: 6.EE.1, RST.6-8.7)

Materials List: Exponents BLM, chart paper, paper, pencil, calculator

Play the following clip from the movie “Pay It Forward”.



Have students draw the diagram from the video clip.

Ask :

• Do you notice a pattern? (1, 3, 9, …)

• What number would be next in the pattern? (27)

• What is the pattern? (Multiplying the previous number by 3)

This pattern can also be expressed using exponents. 3°, 3¹, 3², 3³…

The large number 3 is the base and the small raised number is the exponent. The exponent tells the numbers of times the base is used as a factor. 32 means use 3 as a factor twice, 3 × 3.

[pic]

It is read as 3 to the 2nd power or 3 squared. For example:

31 3 to the first power 3 3

3² 3 squared 3 • 3 9

3³ 3 cubed 3 • 3 • 3 27

34 3 to the 4th power 3 • 3 • 3 • 3 81

35 3 to the 5th power 3 • 3 • 3 • 3 • 3 243

[pic]

Any number to the 0 power is 1.

Ask students what would happen if they changed the base of the exponent. (If the base is made smaller, the resulting numbers will not grow as rapidly, so they will not help as many people as quickly. If the base is increased, the resulting numbers will grow more rapidly, and they will help more people quicker.)

Distribute the Exponents BLM to the class. Work problems 1 – 4 together as a class. Explain that on problem 2, they will substitute the given value for x into the equation, so x4 will become 24. Have students solve the remaining problems independently. Discuss the answers as a class.

Use SQPL (view literacy strategy descriptions) to challenge the students to further explore exponents. SQPL stands for Student Questions for Purposeful Learning and involves presenting the students with a statement that provokes interest and curiosity. Put the following statement on the board or overhead for students to read. “If Tanya receives an allowance of $20 a week and her little brother receives 2 cents the first day and each day after that the amount he receives the day before is doubled. Tanya receives a higher allowance after 2 weeks.” Have students work with partners and brainstorm 2-3 questions that would have to be answered to prove or disprove the statement. Some sample questions are How much would Tanya’s little brother earn each day? What is the total amount her brother would earn after 2 weeks? As a whole class, have each pair of students present one of their questions and write this question on chart paper or the board. Give the class time to read each of the questions presented. Give the pairs of students time to select the ideas that they would use to prove or disprove the statement.

|Day |Written as an exponent |Amount |Total |

| | |(since it is money we need 2 numbers | |

| | |behind the decimal) | |

|1 |21 |0.02 |0.02 |

|2 |22 |0.04 |0.06 |

|3 |23 |0.08 |0.14 |

|4 |24 |0.16 |0.30 |

|5 |25 |0.32 |0.62 |

|6 |26 |0.64 |1.26 |

|7 |27 |1.28 |2.54 |

|8 |28 |2.56 |5.10 |

|9 |29 |5.12 |10.22 |

|10 |210 |10.24 |20.46 |

|11 |211 |20.48 |40.94 |

|12 |212 |40.96 |81.90 |

|13 |213 |81.96 |163.86 |

|14 |214 |163.84 |327.70 |

So, while Tanya would receive $40, her little brother would receive $327.70.

Reread the opening statement and the questions the students generated. As a class, answer each question and decide if the statement is true.

Activity 3: Expressions (GLE: 15)

Materials List: Match It BLM, paper, pencil

Present the following situation to the class. An expression is a combination of numbers, variables and/or operations. The expression [pic]could be used to describe how a family of five could divide a pizza. If the pizza has 10 slices, each family member can have[pic], or 2 slices.

Use discussion (view literacy strategy descriptions, in the form of Think Pair Square Share. This strategy helps improve student learning and remembering by participating in a discussion about a given topic. After being given an issue, problem, or question, students are asked to think alone for a short period of time, and later pair up with someone to share their thoughts. Then have pairs of students share with other pairs, forming, in effect, small groups of four students. Finally, the groups of four should report out to the whole class.

Present the following expressions to the class one at a time. Have the students write an everyday situation for the expression independently and then pair up with partners to discuss their situations. Have each pair of students join another pair of students to share their situations. Have groups share their situations with the class while others listen for accuracy and logic.

Expression 1: x + 5

Expression 2: 20m

Expression 3: y – 8

Expression 4: [pic]

Distribute the Match It BLM and have students work with partners to match the expressions and with the verbal statements. Discuss the solutions as a class.

Activity 4: Is it True? (CCSS: 6.EE.1, 6.EE.5)

Materials List: sheet protectors, dry erase markers, paper towels or napkins

Display the following symbols:

= < > ≤ ≥

Discuss the meaning and give an example of each symbol.

= equal x = 5 x equals 5

< less than x < 10 x is less than 10

> greater than x > -5 x is greater than -5

≤ less than or equal to x ≤ 7 x is less than or equal to 7

≥ greater than or equal to x ≥ 7 x is greater than or equal to 7

Explain that if x < 10, then 10 > x.

Distribute a sheet protector and a dry erase marker to each student. Have students place a plain sheet of paper in the sheet protector. Present the following inequalities to the students one at a time and have them give a value for the variable that would make the statement true. Have them rewrite the inequality with the variable on the other side using the dry erase markers. Ask students if the value they chose works for the rewritten inequality. Have students hold up their answer to quickly check for understanding.

.

1. x > 5 12; 5 < x

2. x > 3 5; 3 < x

3. 9 ≤ x 10; x ≥ 9

4. x ≥ 12 13; 12 ≤ x

5. -3 < x -2; x > -3

Present the following inequalities and have students determine which of the following values would make the inequality true.

1. x > 6 {0, 6, 7, 10} 7, 10

2. 4 > x {0, 2, 4, 6} 0, 2

3. 7 ≤ x {0, 6, 7, 9} 7, 9

4. x² ≤ 5 {0, 2, 5, 6} 0, 2

5. x > -5 {-7, -2, 5, 6} -2, 5, 6

6. 2x + 4 > 5 {-2, 0, 3, 6} 3, 6

7. 5x > -5 {-2, -1, 0, 2} 0, 2

8. 4 > x² + 3 {-2, 0, 1, 2} 0

9. 7 ≤ 3x + 7 {0, 6, 7, 9} 0, 6, 7, 9

10. 2x² ≥ 8 {-2, -1, 0, 2} -2, 2

Activity 5: Substitution (CCSS: 6.EE.5)

Materials List: paper, pencil

Present the following situation to students, “Your school is having a carnival. They charge $2.50 for each game. If you have $30 to spend, how many games can you play?”

Ask:

• How can you represent the situation as an inequality? $2.50 times the number of games has to be equal to or less than $30. 2.5x ≤ 30

• Could you play 10 games? Yes, if you substitute 10 for x, 2.5 • 10 = 25

• Could you play 15 games? No, 2.5 • 15 = 37.50

• If 15 games cost $37.50 and you only have $30.00, how much over are you? $7.50

• How many games equal $7.50? If each game is $2.50, then 3 games would cost $7.50.

• If 15 games cost $37.50 and you are 3 games over, what is the highest number of games you could play? 12 games

Solve the following problems as a class.

Given the following values {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} which make the following equations and inequalities true? Use substitution to find all possible solutions.

1. 8 + x = 12

2. 3 + x > 9

3. x + 9 < 15

4. 5x = 20

5. 4 + x ≤ 8

2013-2014

Activity 6: Graphing on a Number Line (CCSS: 6.EE.8)

Materials List: Graphing Equations and Inequalities BLM, Number Line BLM, Graph It BLM, Equation/Inequality Word Grid BLM, sheet protectors, dry erase markers, paper towels or napkins

Display the Graphing Equations and Inequalities BLM and distribute a copy to the students. Demonstrate how to graph the five types of equations or inequalities. Call attention to the fact that having the variable on the left side of the inequality shows the direction of the arrow and shading on the number line. Tell students that if they are given an inequality such as 6 > x, it might help them to rewrite it as x < 6. Explain and illustrate the use of open and closed circles in their solution depending on the type of inequality.

EQUAL

x = 5 x equals 5

Place a filled circle at 5 to represent x = 5

-10 -8 -6 -4 -2 0 2 4 6 8 10

LESS THAN

x < 10 x is less than 10

Place an open circle at 10 and shade the line to the left to represent x < 10

An open circle is used because 10 is not included in the solution set

-10 -8 -6 -4 -2 0 2 4 6 8 10

Have students check to see that they have drawn the arrow in the correct direction by picking a point on the shaded part of the number line and substituting it into the inequality. For example, try the number 2. 2 is less than 10 (2 < 10) so the line is shaded correctly.

GREATER THAN

x > -5 x is greater than -5

Place an open circle at -5 and shade the line to the right to represent x > -5

An open circle is used because -5 is not included in the solution set

-10 -8 -6 -4 -2 0 2 4 6 8 10

LESS THAN OR EQUAL TO

x ≤ 7 x is less than or equal to 7

Place a closed circle at 7 and shade the line to the left to represent x ≤ 7

A closed circle is used because 7 is included in the solution set

-10 -8 -6 -4 -2 0 2 4 6 8 10

GREATER THAN OR EQUAL TO

x ≥ 7 x is greater than or equal to 7

Place a closed circle at 7 and shade the line to the left to represent x ≥ 7

A closed circle is used because 7 is included in the solution set

-10 -8 -6 -4 -2 0 2 4 6 8 10

Distribute the Number Line BLM, a sheet protector and a dry erase marker to each student. Have students place the BLM in the sheet protector. Present the following equations or inequalities to the students one at a time and have them graph them using the dry erase markers. Have students hold up their number line to quickly check for understanding.

1. x = -4

2. x > 2

3. x ≤ 9

4. x ≥ 5

5. x < -8

Give more examples if needed. Distribute the Graph It BLM to the students to work independently. Discuss answers as a class.

Discuss situations when an inequality would be used and situations when an equation would be appropriate. Display the word grid (view literacy strategy descriptions) below or use the Equation/Inequality Word Grid BLM. The first column lists some real-life situations. With the students’ participation, fill in the word grid by placing a “+” in the space to indicate if the situation represents an inequality or an equation. Have students write the equation or inequality for each situation.

|Situation |Equation |Inequality |Write the equation or inequality|

| | | |to represent each situation |

|John bought cheeseburgers for 5 | | | |

|of his friends. The total was | | |5x = 15 |

|$15. |+ | | |

|The movie theater has more than | |+ |x > 285 |

|285 seats. | | | |

|The Jackson family spent less | | | |

|than $200.00 on groceries last | | |x < 200 |

|month. | |+ | |

|Sam must be at least 5 ft. to go| |+ |x ≥ 5 |

|on the ride. | | | |

|I have at most $100 in my | |+ |x ≤ 100 |

|pocket. | | | |

|The store has socks on sale. |+ | |6x = 12 |

|They are 6 pairs for $12. | | | |

As a class, come up with additional situations to add to the word grid. Once the grid is complete, provide opportunities for students to quiz each other over information from the grid and use the grid to prepare for quizzes.

Activity 7: Equal Concentration (GLE: 15, 16)

Materials List: Concentration BLM (one set per group of two students), Solutions BLM, pencil, card stock

Create a Concentration® type game using the Concentration BLM. Have students match algebraic expressions to equivalent verbal statements. Have students work in groups of two to play the game. The student with the largest number of matching pairs wins the game. Have students sort the cards into 2 stacks, one for the word phrases and one for the algebraic expression.

Using the Concentration game cards of word phrases, have the students complete the Solutions BLM. The students will complete the table with the word phrases, the accompanying algebraic expressions and 3 possible replacement values for the variables. Have students solve the expressions using the replacement values given in the table. Model the process for the students using the following two cards in the table below. Then have each pair of students divide the remaining cards between them. Each student should have 7 word phrases to complete on the table. The students may select their own replacement values.

|Word Phrase |

|w |w |w |w |

Tell students the following:

• The bar model represents the equation 4w = $20.

• To solve the problem, divide each side of the equation by 4.

[pic]

[pic] 20 ÷ 4 = 5

w = 5 Liz gets $5a week for her allowance.

Distribute the Solving Equations BLM. Have the students work in pairs to write an equation to represent each problem and then solve the equation. After the students have completed the problems, discuss the solutions as a class.

After discussing the solutions to the problems above, have the students create a text chain (view literacy strategy descriptions). Put students into groups of four. On a sheet of paper, have the first student write the opening sentence for the math text chain. For example, a student might begin the chain with this sentence.

Sue wants to buy a MP3 player.

The student then passes the paper to the student sitting to the right, and that student writes the next sentence in the story, which might read:

She has a paper route and earns $25 a week.

The paper is passed again to the right to the next student who writes the third sentence of the story.

If the MP3 player costs $150, how many weeks will it take for her to have enough money to buy the MP3 player?

The paper is now passed to the fourth student who must write an equation to represent the problem and solve the problem. The other three group members should review the equation and the solution for accuracy.

Answer: 25w = 150; w = 6

It will take her 6 weeks to earn enough money for the MP3 player.

This activity allows students to use their writing, reading, and speaking skills while learning and reviewing the concept of solving equations. When text chains are completed, be sure students are checking them for accuracy and logic. Groups can exchange text chains to further practice reviewing and checking for accuracy.

Activity 10: Two-Step Equations (CCSS: 6.EE.5)

Materials List: Two-Step Equations BLM, paper, pencil

Provide students with several scenarios that can be represented algebraically in a two-step problem. Have students solve these problems, and discuss the equations and solutions as a class.

Use problems similar to this:

Problem 1:

If Ronald subtracts 5 from twice his number, he gets 7. What is Ronald’s number?

Ask the following questions:

• How would you write twice his number algebraically? 2n

• What equation would represent the problem? 2n – 5 = 7

• What is the variable in this problem? n

• What is happening to the variable? It is being multiplied by 2 and 5 is being subtracted from it.

• When there are 2 operations in an equation, it is generally easier to start with the addition or subtraction part of the equation.

• What is the opposite of subtracting 5? Adding 5

• When solving equations, what is done to one side of the equation must be done to the other side of the equation. If not, the sides will not stay equal.

2n – 5 = 7

+5 +5

2n = 12

• What is happening to the variable? It is being multiplied by 2.

• What is the opposite of multiplying by 2? Dividing by 2

[pic]; n = 6

Check the solution by substituting the solution into the original equation.

2n – 5 = 7

2(6) – 5 = 7 ( The solution is correct.

2n – 5 = 7; n = 6; Ronald’s number is 6.

Problem 2:

Latoya had $10.00 in her piggy bank. She saved her weekly allowance for 3 weeks. At the end of that time, she had $25.00. How much did she get in allowance each week?

Ask the following questions:

• What equation would represent the problem? 3n + 10 = 25

• What is the variable in this problem? n

• What is happening to the variable? It is being multiplied by 3 and 10 is being added to it.

• What is the opposite of adding 10? Subtracting 10

• Remember, when solving equations, what is done to one side of the equation must be done to the other side of the equation. If not, the sides will not stay equal

3n + 10 = 25

- 10 -10

3n = 15

• What is happening to the variable? It is being multiplied by 3

• What is the opposite of multiplying by 2? Dividing by 3

[pic]; n = 5

Check the solution by substituting the solution into the original equation.

3n +10 = 25

3(5) + 10 = 25 ( The solution is correct.

3n +10 = 25; n = 5; Latoya’s allowance is $5 a week.

Problem 3:

Julio bought 2 books at the book fair and Sarona bought 3 books at the book fair. They spent $10 altogether. If all the books at the fair are the same price, how much was each book?

Ask the following questions:

• What equation would represent the problem? 2b + 3b = 10

• Can you simplify the equation before solving? Yes, you can add 2b + 3b = 5b.

• What is the equation after you combine like terms? 5b = 10

• What is happening to the variable? It is being multiplied by 5

• What is the opposite of multiplying by 5? Dividing by 5

[pic]; b = 2

Check the solution by substituting the solution into the original equation.

2b + 3b = 10

2(2) + 3(2) = 10 ( The solution is correct.

2b + 3b = 10; b = 2; The books are $2.00 each.

Distribute the Two-Step Equations BLM. Have the students work in pairs to write an equation to represent each problem and then solve the equations. After the students have completed the problems, discuss the solutions as a class.

Activity 11: Pumping Gas Rules! (GLE: 37; CCSS: 6.EE.7)

Materials List: paper, pencil

Provide students with the following table.

|Gallons of Gas (g) |1 |2 |3 |4 |5 |6 |

|Total Cost for Gas (c) |3.45 |6.90 |10.35 |13.80 | | |

Purchasing gasoline for a car can be thought of as an arithmetic sequence where the term number is the same as the number of gallons pumped (i.e., 1, 2, 3 . . .), and the value of a term is the price for that many gallons. Have students investigate this sequence and describe how the terms are generated. The terms are increasing as the number of gallons increases and then the value of the terms is increasing by the amount per gallon each time.

Ask students the following questions:

• How much does one gallon of gas cost? $3.45

• How much would 5 gallons cost? $17.25

• How did you find the value for 5 gallons? Some students might say they added 3.45 to 13.80 other groups might say that they multiplied the cost of a gallon times the number of gallons (5). If no one suggests multiplying the cost per gallon by the number of gallons, explain this to the students.

• How could you figure out the cost of 10 gallons? Continue the pattern or multiply 10 times 3.45

• How much would 23 gallons of gas cost? 23 times 3.45 = $79.35

• What equations could we use to find the cost for any number of gallons of gas?

3.45g = c

Present the following problem to the students. Have them record their response in their math learning log (view literacy strategy descriptions).

Donnie was watching Gas Watch, a segment on the local news where the cheapest gas prices in town are identified, when it showed the following table for local gas prices:

|Gallons of Gas (g) |5 |6 |7 |8 |

|Total Cost for Gas (c) |10.45 |12.54 |14.63 |16.72 |

Write an equation to find the total cost of any number of gallons of gasoline. Explain your thinking.

Have the students share their learning log responses with the class. Students should listen for logic and accuracy of responses.

Equation: 2.09g = c

Explanation: To find the total cost of gas, you would multiply the cost for one gallon times the number of gallons needed. The cost per gallon is determined by dividing the total cost by the number of gallons. In this case, the gas is $2.09 per gallon.

Activity 12: Input-Output (GLE: 37; CCSS: 6.EE.1, RST.6-8.7)

Materials List: Input-Output Tables BLM, Graph Paper BLM, pencil

Distribute the Input-Output Tables BLM and Graph Paper BLM. Have students complete the input-output tables independently.

|Input |1 |

|x | |

|BATMAN The Ride |54” |

|Dive Bomber Alley |42” |

|Tower Flashback |48” |

Work with your team to describe what this height restriction for each ride would look like mathematically using an inequality. List 5 heights that would be allowed on each ride and 5 that would not be. How many possible heights are there? Illustrate on a number line to show all of the possible heights each ride.

Possible solutions:

|Ride |Height minimum Inequality |Heights that would be allowed |Heights that would not be allowed |

|BATMAN The Ride |x ≥ 54” |54” 58” 60” 62” 67” |45” 47” 48”52” 53” |

|Dive Bomber Alley |x ≥ 42” |42” 44” 46” 48” 57” |35” 38” 39”40” 41 |

|Tower Flashback |x ≥ 48” |48” 50” 53” 55” 63” |40” 42” 44” 46” 47” |

Discuss the solutions as a class.

Sample Assessments

General Assessments

• Have students solve equations and inequalities with substitution.

• Have students create a portfolio containing samples of experiments and activities.

• Have students create their own real-life examples of demonstrated problems.

• Use observations to determine student understanding as he/she engages in the various activities.

• Have students create and demonstrate original math problems by acting them out or using manipulatives to provide solutions on the board or overhead.

• Have students create their own questions.

• Observe during small group discussion to determine misconceptions, understandings, use of correct terminology, and reasoning abilities. Appropriate questions to ask to help students learn to reason mathematically:

• Is that true for all cases? Explain.

• How would you prove that?

• What assumptions are you making?

To help check student progress, ask:

• Can you explain what you have done so far? What else is there to do?

• Why did you decide to use this method?

• Do you think this always holds true?

• How could you check on this?

Activity-Specific Assessments

• Activity 2: Have the students evaluate the following expressions:

(½)³

64

37

51

(¼)6

70

• Activity 8: Have the students create a set of five verbal statements and the algebraic expressions and equations that match the statements. Have the students defend their work to the class in a brief presentation, answering any questions that may arise. Once the problems are approved, have the students transfer the information to the back of index cards. Use the cards to make a memory type of game. All cards will be collected and made into a class game.

• Activity 9: Which of the following value(s) for x would make x + 5 = 8 a true statement?

{0, 3,12, 4}

Find the value(s) of x that will make x + 3.5 ≥ 9 a true statement.

{5, 5.5, 6,[pic], 10.2, 15}

-----------------------

$75.33

J

J

J

20

2.99

6.50

money left over (m)

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