Unit 8: Patterns, and Algebra

2012-2013 and 2013-2014 Transitional Comprehensive Curriculum

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

Can students recognize squares to 144?
Can students evaluate expressions for specified variable values?
Can students match or create stories to go with a given algebraic expression or equation?
Can students solve equations?
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—, , and so on through . 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.,), write the area of the square (e.g.,), 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.

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

313 to the first power33

3²3 squared3 • 3 9

3³ 3 cubed3 • 3 • 327

343 to the 4th power3 • 3 • 3 • 381

353 to the 5th power3 • 3 • 3 • 3 • 3243

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. “IfTanya 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
(since it is money we need 2 numbers behind the decimal) / Total
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 Tanyawould 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 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, 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:

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.

Activity4: 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 = 5x equals 5

less thanx < 10x is less than 10

greater thanx -5x is greater than -5

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

≥greater than or equal tox ≥ 7x 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 > 512; 5 < x

2.x > 35; 3 < x

3.9≤ x10; x ≥ 9

4.x ≥ 1213; 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.

8 + x = 12
3 + x > 9
x + 9 < 15
5x = 20
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 = 5x 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 < 10x 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 > -5x 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 ≤ 7x 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 ≥ 7x 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 theword 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 $15. / + / 5x = 15
The movie theater has more than285 seats. / + / x > 285
The Jackson family spent less than $200.00 on groceries last month. / + / x < 200
Sam must be at least 5 ft. to go on the ride. / + / x ≥ 5
I have at most $100 in my pocket. / + / x ≤ 100
The store has socks on sale. They are 6 pairs for $12. / + / 6x = 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 dividethe 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 / Algebraic Expression / 1st Replacement Value / Solution / 2nd Replacement Value / Solution / 3rd Replacement Value / Solution
7 less than a number x / x– 7 / 7 / 7– 7 = 0 / 8 / 8 – 7 = 1 / 9 / 9 – 7 = 2
4 decreased by a number x / 4 –x / 0 / 4 – 0 = 4 / 1 / 4 – 1 = 3 / 2 / 4 – 1 = 3

Activity 8: Substituting Numbers (GLE: 16)

Materials List: Evaluate It! BLM, number cubes, pencil

Distribute the Evaluate It! BLM and a number cube to each group of two students. Have the students take turns evaluating and checking the problems. The first student rolls the number cube and the second student completes the problem substituting the rolled number in place of the variable. The student who rolled the number then checks the problem. The students switch rolls for each problem.

Activity 9: Solving Equations (CCSS: 6.EE.7)

Materials List: Equation Match It BLM, Solving Equations BLM, paper, pencil

Distribute the Equation Match It BLM. Have students work in groups of two to match the verbal statements with the appropriate equation. Discuss the solutions as a class.

Solve the following problems as a class.

x + 8 = 18

Ask the following questions:

What is the variable in this problem? x
What is happening to the variable? 8 is being added to it.
What is the opposite of adding 8? Subtracting 8
When you do the opposite operation, it is called the inverse.
When solving equations, what you do to one side of the equation you also have to do to the other side of the equation. If not, the sides will not stay equal.

x + 8 = 18

- 8 - 8

x + 0= 10

x = 10

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

10 + 8 = 18 The solution is correct.

4x = 32

Ask the following questions:

What is the variable in this problem? x
What is happening to the variable? It is being multiplied by 4
What is the inverse of multiplying by 4? Dividing by 4

x = 8

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

4 • 8 = 32 The solution is correct.

Have students work with partners to solve the equations on the Solving Equations BLM. Discuss the solutions as a class.

Provide students with several scenarios that can be represented as equations. Have students solve these problems, and discuss the equations and solutions as a class.

Use problems similar to these:

A. John spent $75.33 on three pairs of jeans. If each pair of jeans costs the same amount, write an equation that represents this situation and solve it to determine how much one pair of jeans cost.

Solution:

Tell students the following:

The bar model represents the equation 3j = $75.33.
When solving equations the goal is to get the variable by itself. To do that, you must undo what is being done to the variable. In this problem the variable is being multiplied by 3. What is the inverse of multiplications? Division
To solve the problem, divide each side of the equation by 3.