7.2.4 Represent and Solve Equations
Overview
Standard 7.2.4 Essential Understandings
Students at this level are transitioning from merely understanding and interpreting equations to representing them. They are able to solve an equation by using the graph or table to see the solution and can solve it symbolically as well. They are also using negative values in their equations.
All Standard Benchmarks
7.2.4.1
Represent relationships in various contexts with equations involving variables and positive and negative rational numbers. Use the properties of equality to solve for the value of a variable. Interpret the solution in the original context.
For example: Solve for w in the equation P = 2w + 2ℓ when P = 3.5 and ℓ = 0.4.
For example: To post an Internet website, Mary must pay $300 for initial set up and a monthly fee of $12. She has $842 in savings. How long can she sustain her website?
7.2.4.2
Solve equations resulting from proportional relationships in various contexts.
For example: Given the side lengths of one triangle and one side length of a second triangle that is similar to the first, find the remaining side lengths of the second triangle.
For example: Determine the price of 12 yards of ribbon if 5 yards of ribbon cost $1.85.
7.2.4 Represent and Solve Equations
7.2.4.1
Represent relationships in various contexts with equations involving variables and positive and negative rational numbers. Use the properties of equality to solve for the value of a variable. Interpret the solution in the original context.
For example: Solve for w in the equation P = 2w + 2ℓ when P = 3.5 and ℓ = 0.4.
For example: To post an Internet website, Mary must pay $300 for initial set up and a monthly fee of $12. She has $842 in savings. How long can she sustain her website?
7.2.4.2
Solve equations resulting from proportional relationships in various contexts.
For example: Given the side lengths of one triangle and one side length of a second triangle that is similar to the first, find the remaining side lengths of the second triangle.
For example: Determine the price of 12 yards of ribbon if 5 yards of ribbon cost $1.85.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- Solve and set up equations.
- Translate a verbal description of a real life situation into an equation involving variables.
- Interpret the solution to an equation in the original context.
- Recognize and translate real-world proportional relationships into mathematical situations.
- Solve proportional situations in multiple ways.
Work from previous grades that supports this new learning includes:
- Work with equations.
- Be familiar with the concept of a variable.
- Multiply.
- Divide.
- Label answers.
- Interpret word problems.
- Be able to work with rates.
- Know how to set up ratios.
- Solve proportional situations with unit rates, scale factors, tabular, or graphical strategies.
NCTM Standards
Use mathematical models to represent and understand quantitative measurement:
- Model and solve contextualized problems using various representations, such as graphs, tables, and equations;
- Explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope;
- Use symbolic algebra to represent situations and to solve problems.
Represent and analyze mathematical situations and structures using algebraic symbols:
- Develop an initial conceptual understanding of different uses of variables;
- Use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships.
Common Core State Standards (CCSS)
7.EE (Expressions and Equations) Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
7.EE.4. Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
7.EE.4a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
Misconceptions
Student Misconceptions and Common Errors
- Students may confuse how to write variables in their equations. For example, if they want to represent $5 per cap, some may write 5pc (5 per cap), which could be read as "5 x p x c."
- Students often forget to verify that their answer makes sense in the original context of the problem.
- Students try and solve proportional relationships in their heads without setting up or writing down the proportion, which can lead to mistakes.
- When translating words into mathematical symbols, "less than" sometimes causes confusion because "less than" construction is essentially backwards. For example, "3 less than x" could be read as "x - 3" and students are tempted to write "3-x."
- Students will sometimes have a set way of setting up proportions and often don't realize that a different setup will often lead to the same result. This can lead to problems if they encounter a multiple choice problem that asks "Choose the proportional relationship that matches this situation," and the student's regular approach is not one of the choices.
Vignette
In the Classroom
In this vignette, students use manipulatives to see how to set up an algebraic expression.
The teacher prepares several bags of chips (or counters, dice, or blocks) and then places some bags containing chips, as well as loose chips on either side of a table that has been split down the middle with black tape. Each bag contains the same number of chips, but the students don't know how many are in each bag. Counting the number of chips in the bags and the loose chips, each side of the table should have the same number of chips. In this example, one side has 3 bags of 9 chips, along with 3 loose chips; the other side has 2 bags of 9 chips, with 12 loose chips.
Teacher: OK, today we are going to be doing an activity using these bags and chips. What do you notice about my table?
Student: On the left side of the table, there are 3 bags and 3 chips, and on the right side there are 2 bags and 12 chips. They aren't the same numbers of things on the two sides.
Teacher: But what if I told you they represent the same number?
Student: How can that be? There are 6 on the left and 14 on the right? Six is not the same as 14.
Teacher: How did you get 6 and 14?
Student: 3 + 3 is 6, and 2 + 12 is 14.
Teacher: 6 what? 14 What? Bag-chips?
Student: Um. I don't know.
Teacher: Can we combine the chips and bags to get bag-chips? No, we can't. What math concept does that tie in with?
Student: Combining like terms.
Teacher: Correct. So we can't combine the bags and chips, not with each other at least. But what could we do?
Student: Combine the bags with the bags and the chips with the chips.
Teacher: Yes. So let's try that. On your paper, you have this same set of items. We want to combine like objects. What should we do first?
Student: Let's combine the bags first. But how do we do that?
Teacher: Well, we have 3 on the left and 2 on the right. We want to get all of them together. I like to have a positive number of bags if at all possible, so let's take the 2 from the right, and physically remove them. (The teacher takes the two bags away.) But if we do that to the right, what else do we have to do?
Student: Take 2 from the left.
Teacher: Yes. So let's do that. You should have crossed out 2 on the right, and 2 on the left. We have to keep our 'table' balanced, so we have to do the same to both sides. OK, so what do we have left now?
Student: Well, on the left we have 1 bag and 3 chips, and on the right we have no bags and 12 chips.
Teacher: What do you think our next step is?
Student: Get all the chips together?
Teacher: Yes. How can we do that?
Student: Like we did the bags, we take the 3 that are on the left and get rid of them, and we have to get rid of the ones on the right, too.
Teacher: What are we left with?
Student: On the left we have just one bag. On the right, there are 9 chips.
Teacher: So what does that mean, then? Thinking in terms of the original problem?
Student: That each bag had 9 chips in it?
Teacher: Well, let's check and find out. Look back at our original problem. We had 3 bags and 3 chips on the left, and 2 bags and 12 chips on the right. If there are 9 chips in each bag, what do we have?
Student: On the left, 3 x 9 is 27, plus the 3 additional chips would be 30. And on the right, 2 bags times 9 chips in each would be 18 chips, plus the additional 12 chips would also be 30. 30 = 30.
Teacher: So by going back to the original problem and using our solution, what did we learn?
Student: That our answer was right?
Teacher: Yes, it is correct. But we also learned the importance and value in checking our work, and in substituting our solution into the original problem. What we did was pictorial. How can we now rewrite our picture problem into and algebraic problem?
Student: Look at what's on each side of the table, and translate into math symbols and numbers. On the left we have 3 bags plus 3 chips. So we could write 3c + 3 for the left side. On the right we have 2 bags and 12 chips so we could write 2c + 12. So our equation would be 3c + 3 = 2c + 12.
Teacher: Nice. Now, when we try to solve this, we can think of the problem in terms of bags and chips, and it doesn't seem quite so difficult.
Adapted from: Lappan, G. (2009). Moving Straight Ahead: Linear Relationships, Teacher's Edition. (p.66-88). Boston, MA: Pearson.
Resources
Teacher Notes
- Students may need support in further development of previously studied concepts and skills.
- Students need practice translating words into mathematical symbols. Help students develop a list of words that relate to certain operations. For example, addition: sum, more than, combined, increased by; subtraction: difference, decreased by, less than; multiplication: product, of, times; division: per, out of, quotient of; equals: is, are, will be.
- To help students avoid translating "less than" into wrong symbols, use a real-world situation. "She drives 30 less miles than I do" is not calculated by subtracting your number of miles from 30. Instead, you subtract 30 you're your number of miles. So saying "3 less than x" translates into "x - 3."
- Use manipulatives to teach students the property of equality and then transfer their knowledge to mathematical notation with variables and numbers.
- Get students in the habit of checking their answers when solving equations. Have them do this by "plugging in" the value they calculated for their variable into the original problem and check if they end up with a true statement. For example, if the original equation is 3x + 5 =17 and students solve and get a solution of x = 4, have them plug in 4 for x and see if both sides are equal. Having the students place a ? over the equals sign when checking helps them remember what they are doing.
- When setting up proportions, have students write the labels with the numbers so they can double check that the set up is correct. Also suggest that instead of using x for the variable, pick a variable that is more connected to the original problem's units. For example: Determine the cost of 12 yards of ribbon if 5 yards of ribbon cost $1.85. Set up the proportion with "length in yards" on bottom, and use "c" to stand for the cost.
[math]\frac{cost}{yards}:\frac{$1.85}{5\ yards}=\frac{c}{12\ yards}[/math]
5c=(1.85)(12)
5c=22.2
c=4.44
So the cost of 12 yards is $4.44.
- There are many ways to solve a proportion. For example, If 2 gallons cost $5.40, how much would 5 gallons cost?
- If 2 gallons is $5.40 and I'm asked how much is 5 gallons, since gallons increased 2.5-fold, I just multiply the dollars by 2.5, too.
- If 2 gallons is $5.40, I figure first how much 1 gallon would be, and then how much 5 gallons. I'd divide 1 gallon by 2; half of $5.40 is $2.70, and I'll multiply 5 times $2.70.
- I build a proportion like in the math book and solve by cross multiplying: [math]\frac{5.40}{2\ gallons}=\frac{c}{5\ gallons}[/math]
Cross multiplying I get (5.4)(5) = 2c so [math]c=\frac{(5.4)(5)}{2}[/math]
4. I build a proportion like above but instead of cross-multiplying, I simply multiply both sides of the equation by 5.
5. I build a proportion but this way [math]\frac{5.40}{c}=\frac{2\ gallons}{5\ gallons}[/math] because there is more than one correct way to set up a proportional relationship.
Source: http://www.homeschoolmath.net/teaching/proportions.php)
- Relate equations to sports. Give students an equation and have them tell a sports situation that could represent the equation. For example, 14 + p = 30. John scored 14 points in the basketball game on Tuesday. By the end of the second game on Friday, his two-game total was 30 points. How many points did John score in the second game? The analogous sports expression would be: Johns Tuesday total + his Friday total = John's total points for the two games.
Source: 2007 Mississippi Mathematics Framework Revised Strategies, p. 39.
- Algebra Tiles
Using tiles to represent variables and constants, learn how to represent and solve algebra problems. Solve equations, substitute in variable expressions, and expand and factor. Flip tiles, remove zero pairs, copy and arrange, and make your way toward a better understanding of algebra.
Additional Instructional Resources
- Algebra games
This website has a directory of free algebra math games that teach, build or strengthen algebra math skills and concepts while having fun.
- Solving linear equations
In this lesson, students will have the opportunity to solve linear equations with variables on both sides using a hands-on approach.
properties of equality:
Definitions
1. a = b means a is equal to b.
2. a ≠ b means a does not equal b.
Operations
1. Addition: If a = b then a + c = b + c.
2. Subtraction: If a = b then a - c = b- c.
3. Multiplication: If a = b then ac = bc.
4. Division: If a = b and c ≠ 0 then a/c = b/c.
Reflection - Critical Questions regarding the teaching and learning of these benchmarks
- How are multi-step equations solved in one variable?
- How are equations solved for a variable?
- How does drawing a picture or diagram help in solving the equation?
- How can students solve proportional situations without using cross-products?
- Teaching ratios and proportions
- See-saw algebra
- Mississippi Department of Education. (2007). 2007 Mississippi Mathematics Framework Revised Strategies, (p. 39). Jackson, MS: Mississippi Department of Education.
- Lee, M., and Miller, M. (2001). 40 Fabulous Math Mysteries Kids Can't Resist. New York: Scholastic Professional.
- Lappan, G. (2009). Moving Straight Ahead: Linear Relationships (Teacher's ed.). (p. 66-88). Boston, MA: Pearson.
- Pintozzi, C. (2005). American Book Company's Passing the Minnesota Basic Skills Test in Mathematics. Woodstock, GA: American Book.
Assessment
1.
Answer: B
Source: Massachusetts Comprehensive Assessment System Release of Spring 2010 Test Items
2.
The equation 3c = 4s gives the relationship between c, the weight of clay, and s, the weight of sand in a mixture. There are 6.25 pounds of clay in the mixture. What is the weight of the sand?
A. 4.69 pounds
B. 8.88 pounds
C. 18.75 pounds
D. 75.00 pounds
Answer: A
Source: MN MCA 7th grade item sampler
3.
A map uses the scale 1.5 cm = 25 mi. Two cities are 190 miles apart.
How far apart are the cities on the map?
A. 0.21 cm
B. 11.4 cm
C. 2,917 cm
D. 6,563 cm
Answer: B
Source: MN MCA 7th grade item sampler
4.
The equation y = 12x + 60 can be used to estimate y, the height of a tree in centimeters x months after it is planted. When a tree is 150 cm tall, how long ago was the tree planted?
A. 7.5 months
B. 10.8 months
C. 17.5 months
D. 78.0 months
Answer: A
Source: MN MCA 7th grade item sampler
5.
Joan needs $60 for a class trip. She has $32. She can earn $4 an hour mowing lawns. If the equation shows this relationship, how many hours must Joan work to have the money she needs?
4h + 32 = 60
A. 7 hours
B. 17 hours
C. 23 hours
D. 28 hours
Answer: A
Source: CA 7th grade standardized test released test questions
http://www.cde.ca.gov/ta/tg/sr/documents/cstrtqmath7.pdf
6.
In a scale drawing, [math]\frac{1}{2}[/math] inch represents 3 feet. If the same scale is used, how many inches will be needed to represent 24 feet?
A. 2 inches
B. 4 inches
C. 8 inches
D. 12 inches
Answer: B
Source: CA 6th grade standardized test released test questions
Differentiation
- This pre-algebra lesson explains how to solve an equation by adding or subtracting a number from both sides of the equations.
- Use this lesson to show students how to balance equations.
Let's say you've got a see-saw (teeter-totter)... and you've got 50 pounds of stuff piled on each side:
Here's the big Algebra game:
Whatever you do, you've got tokeep the see-saw balanced! |
What if we add 3 pounds to the left side?
CLUNK! It's not balanced anymore!
But, if we add 3 pounds to BOTH sides?
It stays balanced!
Equations are just like see-saws...
You have to keep them balanced!
So, whatever you do to one side of the " = " you've got to do to the other side!
Source: http://www.coolmath.com/prealgebra/16-intro-to-solving-equations/01-solving-equations-balance-01-177.htm
- Encourage students to verbalize the steps involved in solving a problem as they work through it on paper.
- Use highlighters to identify key words to solve the problems.
- After the lesson, assignment, or homework has been given to the class, check on the individual English language learner. Rather than asking, "Do you understand?" or "Do you have any questions?" have your student repeat or explain to you what the assignment is. Review the problem with the student if necessary.
- Review prior concepts repeatedly. Repetition is essential for ELLs.
- Match words with operations.
- Use this calendar algebra project to help students solve two-step equations. Students can work in pairs; each pair needs a calendar page from any month of any year. Without showing their partners, one student in each pair circles a square block of four days on the calendar, such as the 12th, 13th, 19th and 20th, and then flips the calendar over. The same student then adds up the four numbers and tells the partner only the sum, not the individual numbers. For example, the student would tell her partner the sum is 64. Without any additional information, the partner will then be able to name the first day circled on the calendar by setting up and solving an algebraic equation. Denote the first day on the calendar block with the variable x. Then the other three days must be x + 1, x + 7, and x + 8. Set this entire expression, x + x + 1 + x + 7 + x + 8 equal to the sum, in this case 64. Simplifying on the left, the student gets 4x + 16 = 64, which solves to x = 12, the first day circled on the calendar block.
Source: http://www.ehow.com/info_7856839_fun-middle-school-math-projects.html#ixzz1Oi3Cazgo
- Linear equations can be used to model real world data. This website solves a linear equation that models a real world problem graphically, symbolically and numerically.
- This pre-algebra lesson explains how to solve "messy" equations, such as:
Source: http://www.coolmath.com/prealgebra/16-intro-to-solving-equations/05-solving-equations-messier-01-190.htm
Parents/Admin
Administrative/Peer Classroom Observation
Students are: (descriptive list) | Teachers are: (descriptive list) |
showing all steps for solving equations. | modeling several examples using visuals to help the students see what is happening when solving for the value of the variable in an equation. |
checking their answers by plugging them back into the original problem to see if the solution works. | reminding students to check their answers when solving an equation by substituting their answers back in for the variable in the original problem. |
keeping both sides of equations balanced. | giving students proportional relationships to solve in real world contexts. |
setting up proportions and solving. | helping students realize that there is more than one way to set up a proportional relationship as well as more than one way to solve a proportional situation. |
| reminding students to interpret the solution in the original context of the problem. |
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