Standard #: MA.912.AR.9.6

• For students to have full understanding of systems, instruction includes MA.912.AR.9.4 and MA.912.AR.9.6. Equations and inequalities and their constraints are all related and the connections between them should be reinforced throughout the instruction. 
• Allow for both inequalities and equations as constraints. Include cases where students must determine a valid model of a function. 
  • Students often use inequalities to represent constraints throughout Algebra I. Equations can be thought of as constraints as well. Solving a systems of equations requires students to find a point that is constrained to lie on specific lines simultaneously. 
• Instruction includes the use of various forms of linear equations and inequalities. 
  • Standard Form
    Can be described by the equation $A$$x$ + $B$$y$ = $C$, where $A$, $B$ and $C$ are any rational number and any equal or inequality symbol can be used. 
  • Slope-intercept form
    Can be described by the inequality $y$ ≥  $m$$x$ + $b$, where $m$ is the slope and $b$ is the $y$-intercept and any equal or inequality symbol can be used. 
  • Point-slope form
    Can be described by the inequality $y$ − $y$₁ > $m$($x$ − $x$₁), where ($x$₁, $y$₁) are a point on the line and m is the slope of the line and any equal or inequality symbol can be used.

Common Misconceptions or Errors

• Students may have difficulty translating word problems into systems of equations and inequalities. 
• Students may shade the wrong half-plane for an inequality. 
• Students may graph an incorrect boundary line (dashed versus solid) due to incorrect translation of the word problem. 
• Students may not identify the restrictions on the domain and range of the graphs in a system of equations based on the context of the situation.

Strategies to Support Tiered Instruction

• Instruction provides opportunities to translate systems of equations or inequalities from word problems by first creating equations, then by identifying key words to determine the inequality symbol (i.e., no more than, less than, at least, etc.). The appropriate inequality symbols can then replace the equal signs. Students can separate and organize information for each equation or inequality. 
  • Separate given information for each equation or inequality 
  • Determine the appropriate form of equation or inequality based on givens
  • Define a variable to represent the item wanted in the equation or inequality 
  • Determine what values are constants or should be placed with the variables 
  • Write the equation or inequality 
• Teacher makes connections back to students’ understanding of MA.912.AR.2.5 and MA.912.AR.3.8 and writing constraints based on a real-world context. 
  • For example, Dani is planning her wedding and the venue charges a flat rate of $8250 for four hours. The venue can provide meals for each of the guests and charges $21.25 per plate for adults and $13.75 per plate for children if she has a minimum of 75 guests. If Dani’s budget is $38,000, students can describe this situation using the inequalities $a$ + $c$ ≥ 75 and 21.25$a$ + 13.75$c$ + 8250 ≤ 38000. Depending on number of adults and children she wants to invite and the capacity of the venue, students can determine various other constraints. 
• Teacher provides questions to be answered by students to aid in the identification of domain and range restrictions: 
  •  Does the problem involve humans, animals, or things that cannot be or are normally not broken into parts? If yes, you are restricted by integers. 
  •  Do negative numbers not make sense? If yes, you are restricted by positive numbers. 
  •  Was a maximum or minimum value given? If yes, the solution must not exceed the maximum or drop lower than the minimum. 
• Instruction includes identifying which variable(s) the constraints apply to.

Instructional Tasks

• A baker has 16 eggs and 15 cups of flour. One batch of chocolate chip cookies requires 4 eggs and 3 cups of flour. One batch of oatmeal raisin cookies requires 2 eggs and 3 cups of flour. The baker makes $3 profit for each batch of chocolate chip cookies and $2 profit for each batch of oatmeal raisin cookies. How many batches of each cookie should she make to maximize profit? 

• Amy and Anthony are starting a pet sitter business. To make sure they have enough time to properly care for the animals they create a feeding and pampering plan. Anthony can spend up to 8 hours a day taking care of the feeding and cleaning and Amy can spend up to 8 hours each day on pampering the pets. 
• Feeding/Cleaning Time: Amy and Anthony estimate they need to allot 6 minutes twice a day, morning and evening, to feed and clean litter boxes for each cat, a total of 12 minutes a day per cat. Dogs will require 10 minutes twice a day to feed and walk, for a total of 20 minutes per day for each dog.
• Pampering Time: Sixteen minutes per day will be allotted for brushing and petting each cat and 20 minutes each day for bathing and playing with each dog.
  
  • Part A. Write an inequality for feeding/cleaning time needed for the pets. Represent all time in the same unit (minutes or hours). 
  • Part B. Write an inequality for pampering time needed for the pets. Represent all time in the same unit (minutes or hours). 
  • Part C. Graph the two inequalities. 
  • Part D. In term of this scenario, explain the meanings of the following points: (0,24) and (30,0). 
  • Part E. What is the greatest number of dogs they can watch if they are watching 19 cats? 
  • Part F. List two viable combinations of pets that can be watched.
    
    Possibility 1: ____________cats _____________ dogs  
    Possibility 2: ____________ cats _____________ dogs