7.2.2C Represent: Equations & Inequalities

7
Subject:
Math
Strand:
Algebra
Standard 7.2.2

Recognize proportional relationships in real-world and mathematical situations; represent these and other relationships with tables, verbal descriptions, symbols and graphs; solve problems involving proportional relationships and explain results in the original context.

Benchmark: 7.2.2.4 Represent Using Equations & Inequalities

Represent real-world or mathematical situations using equations and inequalities involving variables and positive and negative rational numbers.

For example: "Four-fifths is three greater than the opposite of a number" can be represented as, $\frac{4}{5}$ = -n+3, and "height no bigger than half the radius" can be represented as h ≤ $\frac{r}{2}$.

Another example: "x is at least -3 and less than 5" can be represented as -3 ≤ x < 5, and also on a number line.

Overview

Big Ideas and Essential Understandings

Standard 7.2.2 Essential Understandings

Students have had prior experience with situations involving a change in one quantity effecting a corresponding change in another.  This previous experience has included graphical, tabular, and function rule representations of these relationships. This standard extends the prior understanding to proportional situations. Proportional relationships are a specific linear relationship. When these proportional relationships are graphed, the representation is a line passing through the origin. In other representations of a proportional relationship (tabular, verbal, symbols, or equations), the idea may not be initially evident, but in each representation a constant rate of change can be determined. From translation between these representations, students explore this constant rate of change to determine a unit rate (constant of proportionality or slope). The more connections students can make between these multiple forms of representation, the deeper their understanding of the relationship. With this understanding, students will be able to move to other linear relationships that are not proportional (with a graph that is a line that does not pass through the origin).

All Standard Benchmarks

7.2.2.1
Represent proportional relationships with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another. Determine the unit rate (constant of proportionality or slope) given any of these representations.
7.2.2.2
Solve multi-step problems involving proportional relationships in numerous contexts.
7.2.2.3
Use knowledge of proportions to assess the reasonableness of solutions.
7.2.2.4
Represent real-world or mathematical situations using equations and inequalities involving variables and positive and negative rational numbers.

Benchmark Cluster

Benchmark Group C - Represent Using Equations & Inequalities

7.2.2.4
Represent real-world or mathematical situations using equations and inequalities involving variables and positive and negative rational numbers.

For example: "Four-fifths is three greater than the opposite of a number" can be represented as ⅘ = -n + 3 . and "height no bigger than half the radius" can be represented as h ≤ r/2.

Another example: "x is at least -3 and less than 5" can be represented as -3 is less than or equal to x<5, and also on a number line.

What students should know and be able to do [at a mastery level] related to this benchmarks

• rewrite a written rule into an equation or inequality.

Work from previous grades that supports this new learning includes:

• writing equations
• interpreting words as mathematical operations (more, fewer, at least, plus, per, etc.)
• working with inequalities
• using variables to represent a quantity that can change
• using variables in various contexts
• creating real world situations corresponding to equations and inequalities
• evaluating expressions and solve equations
• representing real-world situations using equations and inequalities involving variables and positive rational numbers.
Correlations

NCTM Standards

Relate and compare different forms of representation for a relationship:

• identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equations.

Common Core State Standards CCSS

Ratios and Proportional Relationships

7.RP:  Analyze proportional relationships and use them to solve real-world and mathematical problems.

7.RP.1  Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction ½ /1/4 miles per hour, equivalently 2 miles per hour.

7.RP.2. Recognize and represent proportional relationships between quantities.

• 7.RP.2b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
• 7.RP.2c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = np.

7.RP.3  Use proportional relationships to solve multi-step ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

Misconceptions

Student Misconceptions
• Students get confused and misinterpret the following inequality symbols: ≤, ≥, <,  & >.
• The terms 'at least' and 'at most' sometimes cause confusion for students
• Students sometimes switch the order of terms when working with division and subtraction; for example: less than or ratio.

Vignette

Teacher:  Ok, today we are going to talk about ice cream.  How many scoops do you think an ice cream cone can hold?  How many scoops have you been able to have on a cone?

The students share various numbers of scoops they have had on a cone.

T:  "Has anyone ever had their ice cream fall off their cone?"  Several students' hands go in the air.  "Why do you think that is?"

Student 1:  "Because the person didn't push them on hard enough."

Student 2:  "Because I licked too hard."

T:  "Ok, we are going to look at this today, but I'm sorry, we are not going to use REAL ice cream."  T holds up a paper drawn cone and circles to represent the scoops and asks:  "How tall would a cone be with one scoop on it?  Two scoops?  Three scoops?  Twenty scoops?  Any number of scoops?"

T makes a table on the board and fills in with the different values that the students get after measuring.

 Number of scoops Height 1 2 10 20 50 n

T: "What is n?"

S 3: "It's a variable."

T:  "Why am I using a variable to represent the number of scoops of ice cream?"

S 2:  "Because the number of scoops can vary.  So it can be any number."

T:  "What do you predict is the height of 5 scoops?  How about the height of 10 scoops?"

S 1:  "I think 5 scoops would be 20 centimeters and 10 scoops would be 40 centimeters."

T:  "Ok, now I want you to work in your groups and fill in your table.  When you are done I want you to try to come up with an equation to represent the height of any number of scoops of ice cream on a cone."

As the teacher walks around the room, she hears some interesting conversations.  There is one group that is cutting out many circles and just measures each of the circles and adds that amount to the previous number of scoops.  Another group is having a debate on how to line up the scoops, because they say real ice cream would be overlapped, so they should overlap their scoops. The teacher moves around the room listening to other conversations.

T:  "Ok, has anyone come up with an equation for the height of any number of scoops?  I also want you to explain how you came up with your equation.  Who wants to go first?"

Group 1:  "Our equation is h = 22 + 11n, where h is the total height of the cone and scoops and n is the number of scoops."

T:  "What is 22?"

Group 1:  "That's what we got for the height of the cone."

T:  "So what is 11 then, the number of scoops?"

Group 1:  "No, that is the n, because it can vary, so it's a variable; the 11 is the height of one scoop."

Group 2:  "We came up with h = 21 + 8.5n.  We only have 8.5 for the height of a scoop because we say they overlap, so it won't be the whole 11 centimeters."

T:  "I think we have all done a nice job here.  And we came up with equations to represent this relationship."

Resources

Instructional Notes

Teacher Notes

Students may need support in further development of previously studied concepts and skills.

• The first step to effectively translating and solving word problems is to read the problem entirely. Make sure the students don't start trying to solve anything when they've only read half a sentence. They must try first to get a feel for the whole problem; try first to see what information they have, and what they still need.
• The second step is to work in an organized manner. Figure out what you need but don't have, and name things. Pick variables to stand for the unknowns, clearly labeling these variables with what they stand for. Draw and label pictures neatly. Explain your reasoning as you go along. And make sure you know just exactly what the problem is actually asking for. You need to do this for two reasons:
• If a word problem contains the word "sum" or "difference", put the numbers that "sum" or "difference" refer to in parentheses to be added or subtracted first.  Do not separate them.
• order is important in the "quotient/ratio of" and "difference between/of" constructions. If a problems says "the ratio of x and y", it means "x divided by y", not "y divided by x". If the problem says "the difference of x and y", it means "x - y", not "y - x". http://www.purplemath.com/modules/translat.htm
• The "less than" construction is backwards in the English from what it is in the math. If you need to translate "1.5 less than x", the temptation is to write "1.5 - x". Do not do this! You can see how this is wrong by using this construction in a "real world" situation: Consider the statement, "He makes $1.50 an hour less than me." You do not figure his wage by subtracting your wage from$1.50. Instead, you subtract \$1.50 from your wage. So remember; the "less than" construction is backwards. (http://www.purplemath.com/modules/translat.htm)
• Students need help to remember which inequality is which.  You can offer tips about alligator mouths or < looks more like an L so it is less than.  http://www.gradeamathhelp.com/math-inequalities.html offers tips on this topic.
Instructional Resources
• The Region 11 Math/Science Teacher Academy Documents contain lessons, examples, tips, etc. about ratios and proportions strategies.  Includes a Step‐by‐Step Model Drawing (from Solving Word Problems the Singapore Way by Char Forsten) template.  Also includes reference to NCTM's Ten Essential Understandings of Proportional Reasoning.
• Activity

Create cards using algebraic expressions and verbal phrases. Have the students match the expression to the corresponding phrase.

A number (x) increased by five x + 5
Twice a number (n) 2n
Six minutes less than Bob's time (t) t - 6
3 years younger than Seth (s) s - 3
(2007 Mississippi Mathematics Framework Revised Strategies, p. 39)

• Activity

Read a sentence such as 2 times what number is 10. Have students write the algebraic equation. A correct response will be 2x = 10. Repeat this activity several times using sentences involving all four basic operations.

New Vocabulary

Inequality: Any mathematical statement that contains the symbols > (greater than), < (less than), < (less than or equal to), or > (greater than or equal to).  The statement indicates that one quantity is less than (or greater than) another. If a is less than b, their relation is denoted symbolically by a < b; the relation a greater than b is written a > b. Inequalities have many important properties.... An inequality which is not true for all values of the variables involved is a conditional inequality; e.g., (x + 2) > 3 is a conditional inequality, because it is true only for x [values] greater than 1. http://www.learner.org/workshops/algebra/workshop2/index2.html

Professional Learning Communities

Professional Learning Communities

Reflection - Critical questions regarding the teaching and learning of these benchmarks

• Are students able to translate any type of verbal description into a variable expression?
• Why is it useful to represent real-life situations algebraically?
• Are students able to assess the reasonableness of solutions obtained?
• What do the answers to our problems tell us about the situations?

Materials

NCTM's book Developing essential understanding of ratios, proportions, and proportional reasoning for teaching mathematics: Grades 6-8.

References

Bounds, H. M., Chapman, C., Green, T., Kaase, K., Sewell, B.H., & Thompson, M. (2007). Mississippi mathematics framework revised strategies. Jackson, MS: Mississippi Dept. of Education.

Cramer, K., & Post, T. (1993, May). Connecting research to teaching proportional reasoning. Mathematics Teacher, 86(5), 404-407.

Helping students gain understanding and self-confidence in algebra.
www.purplemath.com

Insights into Algebra 1: Linear functions and inequalities.  http://www.learner.org/workshops/algebra/workshop2/index2.html

National Council of Teachers of Mathematics. (n.d.). Developing essential understanding of ratios, proportions, and proportional reasoning for teaching mathematics: Grades 6-8. Reston, VA: NCTM.

Top ten ideas about proportional relationships (that students should take into high school mathematics). http://web.me.com/serpmedia/ToolPresentation/Tool___Articulation_2_files/Top%20ten%20list-1.pdf

Assessment

• On Mondays, Jayda runs between 2 and 5 miles. On Tuesdays, she runs 3 times as far as she runs on the previous Monday. Which inequality can be used to find x, the distance Jayda could run on a Tuesday?

A. $2< 3x< 5$       B. $2< 3x> 5$      C. $2<\frac{x}{3}<5$     D. $2<\frac{x}{3} > 5$

• Twenty gallons of crude oil were poured into two containers of different size. Express the amount of crude oil poured into the smaller container in terms of the amount g poured into the larger container.

Answer:  The expression they're looking for is found by this reasoning: There are twenty gallons total, and we've already poured g gallons of it. How many gallons are left? There are 20 - g gallons left. They want the answer "20 - g".

• Match each algebraic expression with the correct phrase underneath them.

1. _________  2x + 5
2. _________  2(x + 5)
3.  _________ 2x - 5
4.  _________2(x - 5)

A.  twice the sum of x and 5
B.  five less than the product of 2 and x
C.  five more than the product of 2 and x
D.  two times the difference of x and 5

Answers:  1.  C           2.  A                3.  B.               4.  D

• Which of the following equations shows that 5 times x is 3 more than 2 times y?

A.  5x + 3 = 2y
B.  5x = 3 - 2y
C.  x = 5(3 + 2y)
D.  5x = 3 + 2y

Differentiation

Struggling Learners
• Students get confused with the 'greater than' and 'less than' symbols.  Help them by saying the big OPEN end goes to the big number, and the little closed end goes to the smaller number;
 ●     Addition ●     increased by ●     more than ●     combined, together ●     total of ●     sum ●     added to ●     Subtraction ●     decreased by ●     minus, less ●     difference between/of ●     less than, fewer than ●     Multiplication ●     of ●     times, multiplied by ●     product of ●     increased/decreased by a ●     factor of (this type can ●     involve both addition or ●     subtraction and ●     multiplication!) ●     Division ●     per, a ●     out of ●     ratio of, quotient of ●     percent (divide by 100) ●     Equals ●     is, are, was, were, will be ●     gives, yields ●     sold for
English Language Learners
• Instead of using "greater than" and "less than"; can use "larger than" and "smaller than"
• look for "key" words. Certain words indicate certain mathematical operations.

 ●     Addition ●     increased by ●     more than ●     combined, together ●     total of ●     sum ●     added to ●     Subtraction ●     decreased by ●     minus, less ●     difference between/of ●     less than, fewer than ●     Multiplication oftimes, multiplied byproduct ofincreased/decreased by afactor of (this type caninvolve both addition orsubtraction andmultiplication!) ●     Division ●     per, a ●     out of ●     ratio of, quotient of ●     percent (divide by 100) ●     Equals ●     is, are, was, were, will be ●     gives, yields ●     sold for
Extending the Learning
• Introduce graphing of inequalities
• Converting regular < and > notation to interval notation -- with solutions! http://www.coolmath.com/crunchers/algebra-problems-interval-notation-1.htm
• A 555-mile, 5-hour plane trip was flown at two speeds. For the first part of the trip, the average speed was 105 mph. Then the tailwind picked up, and the remainder of the trip was flown at an average speed of 115 mph. For how long did the plane fly at each speed?

 d r t first part d 105 t second part 555 - d 115 5 - t total 555 --- 5
• Using "d = rt", the first row gives me d = 105t and the second row gives me:
555 - d = 115(5 - t)

Since the two distances add up to 555, I'll add the two distance expressions, and set their sum equal to the given total:

555 = 105t + 115(5 - t)
Then I'll solve:
555 = 105t + 575 - 115t
555 = 575 - 10t
-20 = -10t
2 = t

According to my grid, "t" stands for the time spent on the first part of the trip, so my answer is "The plane flew for two hours at 105 mph and three hours at 115 mph."

You can add distances and you can add times, but you cannot add rates. Think about it: If you drive 20 mph on one street, and 40 mph on another street, does that mean you averaged 60 mph?