Course Name: Foundations of Math I Unit # 3 Unit Title: Solving Equations
BY THE END OF THIS UNIT:
Course Name: Foundations of Math I Unit # 3 Unit Title: Solving Equations
CORE CONTENT
Cluster Title: Reason about and solve one-variable equations and inequalities.Standard:
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.
Concepts and Skills to Master:
· Use knowledge of fact families to write related equations: n+ 26 = 100, 100 -n= 26, 100 -26 = n: Select the equation that helps to find n easily.
· Knowledge and use of inverse operations. (Students at this level should also know the inverse operations of square, cubed, square root, and cubed root).
· Substitution
· Understanding of inequalities meaning more than one value could be the answer.
· When multiplying or dividing by a negative, switch the inequality sign.
SUPPORTS FOR TEACHERS
Critical Background Knowledge· Evaluating problems given what number a variable represents.
· Multiplication, division, addition, subtraction, perfect squares, perfect cubes.
· Estimation of square and cubed roots.
· Verbal/Algebraic translations
· Properties of equality and inequality
Academic Vocabulary
• inequalities • equations • greater than, > • less than, <
• greater than or equal to, ≥ • less than or equal to, ≤ • equals (is, costs,…) = • inverse operations
• substitution
Suggested Instructional Strategies:
Using the following problem: Joey had 26 papers in his desk. His teacher gave him some more and now he has 100. How many papers did his
teacher give him?
Scale model: There are 26 blocks on the left side of the scale and 100 blocks on the right side of the
scale. All the blocks are the same size. 74 blocks need to be added to the left side of the scale to make
the scale balance.
Bar Model: Each bar represents one of the values. Students use this visual representation to
demonstrate that 26 and the unknown value together make 100.
100
26 / n
/ Resources:
· http://www.algebra-class.com/solving-inequalities.html
· http://www.sophia.org/linear-equationsinequalities-in-one-variable-tutorial
· http://www.illustrativemathematics.org/standards/k8
· http://illuminations.nctm.org/ActivityDetail.aspx?ID=10
· Pearson Algebra I textbook 2-1, 3-1, 3-2, 3-3 (online codes can be found on the intranet)
Sample Assessment Tasks
Skill-based task
Joey had 26 papers in his desk. His teacher gave him some more and now he has 100. How many papers did his teacher give him? / Problem Task
The equation 0.44s=11 where s represents the number of stamps in a booklet. The booklet of stamps costs 11 dollars and each stamp costs 44 cents. How many stamps are in the booklet? Explain the strategies used to determine the answer. Show that the solution is correct using substitution.
Twelve is less than 3 times another number can be shown by the inequality 12 < 3n. What numbers could possibly make this a true statement?
CORE CONTENT
Cluster Title: Reason about and solve one-variable equations and inequalities.Standard:
6.EE.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
Concepts and Skills to Master:
· Recognize real-world mathematical problems can be expressed using a variable to represent an unknown.
· Write and solve an expression that represents a real-world problem using variables.
· Use variables to represent numbers or sets of numbers when solving a real-world or mathematical problem.
· Writing expressions with story problems and drawing picture representations.
SUPPORTS FOR TEACHERS
Critical Background Knowledge· Understand that a variable represents a number or a specified set of numbers.
· Show arithmetic operations work the same way on variables as they do on numbers.
· Solve real world mathematical problems using equations or expressions with numbers.
· Model authentic problems with manipulatives or diagrams.
Academic Vocabulary
>, <, ≥, ≤, constant, coefficient, solution
Suggested Instructional Strategies:
Have students write an expression for a real-world mathematical problem in which all parts of the expression have a numerical value. Then give a similar real-world mathematical problem in which one of the parts of the expression is an unknown, resulting in one part of the expression being a variable.
Have students write a real-world mathematical problem and an expression that represents that problem. Then have students switch word problems only and find the expression that represents their partner’s real-world mathematical problem.
Expose students to a variety of real-world mathematical problems. Have students work in small groups, with a partner, and eventually by themselves to write expressions to represent those situations. / Resources:
· CMP2 Algebra Unit (online codes can be found on the intranet)-Investigation 2: Number Properties and Algebraic Equations
· Pearson Algebra I textbook 1-1 (online codes can be found on the intranet)
Sample Assessment Tasks
Skill-based task
Write an expression to represent Susan’s age in three years, when a represents her present age.
Write an expression to represent the value of any number of quarters, q.
Andrew has a summer job doing yard work. He is paid $15 per hour and a $20 bonus when he completes the yard. He was paid $85 for completing one yard. Write an equation to represent the amount of money he earned. / Problem Task
Describe a problem situation that can be solved using the equation 2c + 3 = 15; where c represents the cost of an item.
CORE CONTENT
Cluster Title: Reason about and solve one-variable equations and inequalities.Standard:
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. (at this level, negative numbers should be included).
Concepts and Skills to Master:
· Recognize that both sides of an equation are equal, and whatever operation is performed on one side of the equation must be done on the other side to maintain the equality.
· Solve one-step equations using all four operations with rational numbers (i.e., whole numbers, fractions, decimals).
· Write and solve equations that represent real-world mathematical problems that involve rational numbers.
· Model solutions for equations of the form x + p = q and px= q with manipulatives, diagrams or story contexts
SUPPORTS FOR TEACHERS
Critical Background Knowledge· Use inverse operations and properties of equality to “undo” equations.
· Use the reverse order of operations to “undo” equations
· Use the substitution property to check answers.
· Know math vocabulary necessary to translate between verbal and algebraic equations.
Academic Vocabulary
• Inverse operations • order of operations • multiplication, addition, subtraction division property of equality
Suggested Instructional Strategies:
Use Algebra tiles/blocks to model solving one-step equations. Concept Byte 2-1 p. 80 of Pearson Algebra I textbook.
Relate the idea of equations to a balance scale. Using objects, have students balance an actual scale and relate this idea to a balanced equation. Start off with simple true/false equations balanced on a scale. Ask students if the scale “tilts” or is “balanced” (e.g. 8 = 10 - 3, 6 – 3 = 10 - 7, etc.).
Extend the idea of the balance scale to incorporate the idea of performing arithmetic operations on both sides of the equation to isolate the variable (i.e., x + 5 = 8, remove 5 from both sides of the balance, which keeps the equation balanced, so x = 3). / Resources:
· CMP2 Algebra Unit: Investigation 2: Number Properties and Algebraic Equations (online codes can be found on the intranet)
· Pearson Algebra textbook 2-1 (online codes can be found on the intranet)
· UEN: Algebra applies to the real world? No way!
· UEN: Balance or tilt
· http://algebralab.org/lessons/lesson.aspx?file=Algebra_OneVariableOneStep.xml
· Virtual Balance Scale: http://nlvm.usu.edu/en/nav/frames_asid_201_g_4_t_2.html?open=instructions&from=category_g_4_t_2.html
· Virtual Algebra Tiles: http://nlvm.usu.edu/en/nav/frames_asid_189_g_4_t_2.html?open=activities&from=category_g_4_t_2.html
· http://www.livebinders.com/play/play/187117#anchor
Sample Assessment Tasks
Skill-based task
4 + x = 9
x + 5 = 10
5.2 + x = 7.8
3x = 12
1/2x = 4/5
2/5x = 7 / Problem Task
There were some grapes on the table. Logan ate 1/6 of them. He ate 5 grapes. Write an equation to represent the situation and solve.
Angela bought 5 shirts that each cost the same amount. She spent $34.65. How much did she spend on each shirt? (Write and solve an equation to solve the problem.)
Ronnie earned $.50, giving her a total of $3.17. Write an equation that allows you to find her beginning amount.
(In each problem, have students justify their answers using mathematical language, picture representations, or justifications through substitution)
CORE CONTENT
Cluster Title: Reason about and solve one-variable equations and inequalities.Standard:
6.EE.8 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.
Concepts and Skills to Master:
· Recognize that infinity refers to a set of numbers that has no end, but may not include all numbers.
· Recognize that a variable can stand for an infinite number of solutions when used in inequalities.
· Recognize that a constraint or a condition in an inequality refers to the boundary defined in the solution set.
· Represent inequalities on a number line. Add graphic to clarify.
· Understand the meaning of open and closed circles on a number line.
· Write an inequality that represents real-world mathematical problems containing a constraint or a condition (<, >)
SUPPORTS FOR TEACHERS
Critical Background Knowledge· Understand the meaning of equality and inequality.
· Recognize that a variable can stand for a number.
· Write an inequality of the form x c or c x where x and c are rational numbers.
· Represent numbers on a number line.
· When multiplying/dividing by a negative number, switch the inequality.
Academic Vocabulary
· Inequality
· Infinite
· Greater than (or equal to)
· Less than (or equal to)
· Constraints
Suggested Instructional Strategies:
Ask a question for which there are an infinite number of solutions (e.g., What are all the numbers greater than 1?). Guide students to represent that as n > 1.
Present real-world mathematical situations where it is apparent that multiple answers will make an inequality true (e.g., freezing occurs at 32ºF. How cold could your freezer be if you have ice cubes? / Resources:
· CMP2-Algebra Unit: Investigation 3: Integers and the Coordinate Plane (3.6) (online codes can be found on the intranet)
· Pearson Algebra I textbook 3-1, 3-2, 3-3 (online codes can be found on the intranet)
· http://www.education.com/activity/article/tic-tac-equations/
· http://www.illustrativemathematics.org/illustrations/642
· http://www.livebinders.com/play/play/187117#anchor
· http://www.internet4classrooms.com/common_core/write_inequality_form_x_c_expressions_equations_sixth_6th_grade_math_mathematics.htm
Sample Assessment Tasks
Skill-based task
Represent the solution to each inequality on a number line.
n> 0
n< 5
n> 3/4
n< -1.5 / Problem Task
Water boils at 100ºC. Write an inequality that represents all the temperatures at which water does not boil. Represent the solution on a number line.
CORE CONTENT
Cluster Title: Solve real-life and mathematical problems using numerical and algebraic expressions and equations.Standard:
7.EE.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
Concepts and Skills to Master:
· Ability to solve multi-step real life problems with positive and negative rational numbers in various forms.
· Ability to apply properties of problems with multiple operations.
· Ability to convert between various forms of rational numbers (whole numbers, fractions, and decimals).
· Ability to assess reasonableness of answers using mental computation.
SUPPORTS FOR TEACHERS
Critical Background Knowledge· Basic operations of rational numbers.
· Properties of real numbers, including distributive property and properties of equality.
· Understand how the order of operations is used in solving equations.
· Substitution to evaluate answers.
· Translation between verbal and algebraic representations.
· Inverse operations including square, square root and cubed, cubed root
· Estimation strategies
Academic Vocabulary
• Rational numbers • multi-step equations • multiplicative inverse • coefficient • additive inverse
• numeric expressions • algebraic expressions
Suggested Instructional Strategies:
Use a balancing scale to represent inverse operations related to solving equations.
Use Algebra tiles to show a picture representation of equivalent expressions and “undoing” an equation. / Resources:
· Virtual manipulatives, including pan balance scales and algebra tiles/blocks:
http://illuminations.nctm.org/Activities.aspx?grade=3
· MARS conceptual lesson: Steps to Solving Equations
· Pearson Algebra I text 2-2, 2-3 (online codes can be found on the intranet)
· CMP2 Textbook Correlation – Grade 7 Units (online codes can be found on the internet):
o Variables and Patterns
§ Investigations 2-4
o Accentuate the Negative
§ Investigations 1-4
o Moving Straight Ahead
§ Investigations 1-4
· MARS Tasks:
o Short tasks Expressions and Equations
o Answers to Short Tasks
o A12 Fencing
o A 12 Printing Tickets
Sample Assessment Tasks
Skill-based task
You earn $5.00 for every magazine subscription you sell plus a salary of $15.00 each week. Write an equation to represent how many subscriptions you need to sell each week to earn at least $200.00 each week? / Problem Task
Three students conduct the same survey about the number of hours people sleep at night. The results of the number of people who sleep 8 hours a nights are shown below. In which person’s survey did the most people sleep 8 hours?
a. Susan reported that 18 of the 48 people she surveyed get 8 hours sleep a night
b. Kenneth reported that 36% of the people he surveyed get 8 hours sleep a night
c. Jamal reported that 0.365 of the people he surveyed get 8 hours sleep a night
CORE CONTENT