7RP-Analyze+proportional+relationships+&+use+them+to+solve+real-world+&+mathematical+problems

7.MP.2. Reason abstractly and quantitatively. 7.MP.6. Attend to precision. **7RP2** 7.MP.1. Make sense of problems and persevere in solving them. 7.MP.2. Reason abstractly and quantitatively. 7.MP.3. Construct viable arguments and critique the reasoning of others. 7.MP.4. Model with mathematics. 7.MP.5. Use appropriate tools strategically. 7.MP.6. Attend to precision. 7.MP.7. Look for and make use of structure. 7.MP.8. Look for and express regularity in repeated reasoning. **7RP3** 7.MP.1. Make sense of problems and persevere in solving them. 7.MP.2. Reason abstractly and quantitatively. 7.MP.3. Construct viable arguments and critique the reasoning of others. 7.MP.4. Model with mathematics. <span style="font-family: 'Arial','sans-serif'; font-size: 13px;">7.MP.5. Use appropriate tools strategically. <span style="font-family: 'Arial','sans-serif'; font-size: 13px;">7.MP.6. Attend to precision. <span style="font-family: 'Arial','sans-serif'; font-size: 13px;">7.MP.7. Look for and make use of structure. <span style="font-family: 'Arial','sans-serif'; font-size: 13px;">7.MP.8. Look for and express regularity in repeated reasoning. || ===‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Learning Target/Task Analysis**===
 * ===**Common Core Standard**===
 * 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 1/2/1/4 miles per hour, equivalently 2 miles per hour.//
 * 7.RP.2.**Recognize and represent proportional relationships between quantities.
 * a.** Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
 * b.** Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
 * c.** 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 = pn.//
 * d.** Explain what a point (//x//, //y//) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, //r//) where r is the unit rate.
 * 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. || ===‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Anchor Standard/Mathematical Practice(s)**===
 * 7RP1**
 * ===‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Information Technology Standard**=== || ===‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Revised Bloom's Level of thinking**=== ||
 * 7.RP.1 Students continue to work with unit rates from 6th grade; however, the comparison now includes fractions compared to fractions. The comparison can be with like or different units. Fractions may be proper or improper.**


 * Example 1:** If 1/2 gallon of paint covers 1/6 of a wall, then how much paint is needed for the entire wall?

Solution: 1/2 gal / 1/6 wall. 3 gallons per 1 wall


 * 7.RP.2 Students’ understanding of the multiplicative reasoning used with proportions continues from 6th grade.Students determine if two quantities are in a proportional relationship from a table. Fractions and decimals could be used with this standard. Note: This standard focuses on the representations of proportions. Solving proportions is addressed in 7.SP.3.**


 * Example 1:** The table below gives the price for different numbers of books. Do the numbers in the table represent a proportional relationship?


 * Number of Books || Price ||
 * 1 || 3 ||
 * 3 || 9 ||
 * 4 || 12 ||
 * 7 || 18 ||

Solution: Students can examine the numbers to determine that the price is the number of books multiplied by 3, except for 7 books. The row with seven books for $18 is not proportional to the other amounts in the table; therefore, the table does not represent a proportional relationship.Students graph relationships to determine if two quantities are in a proportional relationship and to interpret the ordered pairs. If the amounts from the table above are graphed (number of books, price), the pairs (1, 3), (3, 9),and (4, 12) will form a straight line through the origin (0 books, 0 dollars), indicating that these pairs are in a proportional relationship. The ordered pair (4, 12) means that 4 books cost $12. However, the ordered pair (7, 18) would not be on the line, indicating that it is not proportional to the other pairs.The ordered pair (1, 3) indicates that 1 book is $3, which is the unit rate. The y-coordinate when x = 1 will be the unit rate. The constant of proportionality is the unit rate. Students identify this amount from tables (see example above), graphs, equations and verbal descriptions of proportional relationships.

Solution: From the graph, it can be determined that 4 pounds of bananas is $1.00; therefore, 1 pound of bananas is $0.25, which is the constant of proportionality for the graph. Note: Any point on the line will yield this constant of proportionality. Students write equations from context and identify the coefficient as the unit rate which is also the constant of proportionality.
 * Example 2:** The graph below represents the price of the bananas at one store. What is the constant of proportionality?


 * Example 3:** The price of bananas at another store can be determined by the equation: P = $0.35n, where P is the price and n is the number of pounds of bananas. What is the constant of proportionality (unit rate)?

Solution: The constant of proportionality is the coefficient of x (or the independent variable). The constant of proportionality is 0.35.


 * Example 4:** A student is making trail mix. Create a graph to determine if the quantities of nuts and fruit are proportional for each serving size listed in the table. If the quantities are proportional, what is the constant of proportionality or unit rate that defines the relationship? Explain how the constant of proportionality was determined and how it relates to both the table and graph.

The relationship is proportional. For each of the other serving sizes there are 2 cups of fruit for every 1 cup of nuts(2:1).


 * Example 5:** The graph below represents the cost of gum packs as a unit rate of $2 dollars for every pack of gum. The unit rate is represented as $2/pack. Represent the relationship using a table and an equation.



Equation: d = 2g, where d is the cost in dollars and g is the packs of gum

A common error is to reverse the position of the variables when writing equations. Students may find it useful to use variables specifically related to the quantities rather than using x and y. Constructing verbal models can also be helpful. A student might describe the situation as “the number of packs of gum times the cost for each pack is the total cost in dollars”. They can use this verbal model to construct the equation. Students can check their equation by substituting values and comparing their results to the table. The checking process helps student revise and recheck their model as necessary. The number of packs of gum times the cost for each pack is the total cost. (g x 2 = d)


 * 7.RP.3 In 6th grade, students used ratio tables and unit rates to solve problems. Students expand their understanding of proportional reasoning to solve problems that are easier to solve with cross-multiplication.Students understand the mathematical foundation for cross-multiplication. An explanation of this foundation can be found in Developing Effective Fractions Instruction for Kindergarten Through 8th Grade.**


 * Example 1:** Sally has a recipe that needs 3/4 teaspoon of butter for every 2 cups of milk. If Sally increases the amount of milk to 3 cups of milk, how many teaspoons of butter are needed? Using these numbers to find the unit rate may not be the most efficient method. Students can set up the following proportion to show the relationship between butter and milk.

__3/4__ = __x__ 2 3

Solution: One possible solution is to recognize that 2 • 1 1/2 = 3/4 so • 1 1/2= x. The amount of butter needed would be 1 teaspoons. A second way to solve this proportion is to use cross-multiplication 3/4 • 3 = 2x. Solving for x would give 1 1/8 teaspoons of butter. Finding the percent error is the process of expressing the size of the error (or deviation) between two measurements. To calculate the percent error, students determine the absolute deviation (positive difference) between an actual measurement and the accepted value and then divide by the accepted value. Multiplying by 100 will give the percent error. (Note the similarity between percent error and percent of increase or decrease) % error = | estimated value - actual value | x 100 % actual value


 * Example 2:** Jamal needs to purchase a countertop for his kitchen. Jamal measured the countertop as 5 ft. The actual measurement is 4.5 ft. What is Jamal’s percent error?

Solution: % error = __| 5 ft – 4.5 ft |__ x 100 4.5 % error = __0.5 ft__ x 100 4.5

The use of proportional relationships is also extended to solve percent problems involving sales tax, markups and markdowns simple interest (I = prt, where I = interest, p = principal, r = rate, and t = time (in years)), gratuities and commissions, fees, percent increase and decrease, and percent error. Students should be able to explain or show their work using a representation (numbers, words, pictures, physical objects, or equations) and verify that their answer is reasonable. Students use models to identify the parts of the problem and how the values are related. For percent increase and decrease, students identify the starting value, determine the difference, and compare the difference in the two values to the starting value.

For example, Games Unlimited buys video games for $10. The store increases their purchase price by 300%? What is the sales price of the video game? Using proportional reasoning, if $10 is 100% then what amount would be 300%? Since 300% is 3 times 100%, $30 would be $10 times 3. Thirty dollars represents the amount of increase from $10 so the new price of the video game would be $40.


 * Example 3:** Gas prices are projected to increase by 124% by April 2015. A gallon of gas currently costs $3.80. What is the projected cost of a gallon of gas for April 2015?

Solution: Possible response: “The original cost of a gallon of gas is $3.80. An increase of 100% means that the cost will double. Another 24% will need to be added to figure out the final projected cost of a gallon of gas. Since 25% of $3.80 is about $0.95, the projected cost of a gallon of gas should be around $8.15.”

$3.80 + 3.80 + (0.24 • 3.80) = 2.24 x 3.80 = $8.15


 * Example 4:** A sweater is marked down 33% off the original price. The original price was $37.50. What is the sale price of the sweater before sales tax?

Solution: The discount is 33% times 37.50. The sale price of the sweater is the original price minus the discount or 67% of the original price of the sweater, or Sale Price = 0.67 x Original Price.



The sale price is 60% of the original price. This reasoning can be expressed as 12 = 0.60p. Dividing both sides by 0.60 gives an original price of $20.
 * Example 5:** A shirt is on sale for 40% off. The sale price is $12. What was the original price? What was the amount of the discount?

At a certain store, 48 television sets were sold in April. The manager at the store wants to encourage the sales team to sell more TVs by giving all the sales team members a bonus if the number of TVs sold increases by 30% in May. How many TVs must the sales team sell in May to receive the bonus? Justify the solution.
 * Example 6:**

Solution: The sales team members need to sell the 48 and an additional 30% of 48. 14.4 is exactly 30% so the team would need to sell 15 more TVs than in April or 63 total (48 + 15)

A salesperson set a goal to earn $2,000 in May. He receives a base salary of $500 per month as well as a 10% commission for all sales in that month. How much merchandise will he have to sell to meet his goal?
 * Example 7:**

Solution: $2,000 - $500 = $1,500 or the amount needed to be earned as commission. 10% of what amount will equal $1,500. Because 100% is 10 times 10%, then the commission amount would be 10 time 1,500 or 15,000

After eating at a restaurant, Mr. Jackson’s bill before tax is $52.50 The sales tax rate is 8%. Mr. Jackson decides to leave a 20% tip for the waiter based on the pre-tax amount. How much is the tip Mr. Jackson leaves for the waiter? How much will the total bill be, including tax and tip? Express your solution as a multiple of the bill.
 * Example 8:**

Solution: The amount paid = 0.20 x $52.50 + 0.08 x $52.50 = 0.28 x $52.50 or $14.70 for the tip and tax. The total bill would be $67.20,

Stephanie paid $9.18 for a pair of earrings. This amount includes a tax of 8%. What was the cost of the item before tax?
 * Example 9:**

Solution: One possible solution path follows: $9.18 represents 100% of the cost of the earrings + 8% of the cost of the earrings. This representation can be expressed as 1.08c = 9.18, where c represents the cost of the earrings. Solving for c gives $8.50 for the cost of the earrings. Several problem situations have been represented with this standard; however, every possible situation cannot be addressed here.

‍‍‍‍‍‍‍‍‍‍‍‍‍‍**I can analyze proportions and use them in real world math problems.**

 * (7RP1) I can compute unit rates.**
 * (7RP2a) I can conclude if two quantities are proportional relationships.**
 * (7RP2b) I can identify the unit rate in tables, graphs, equations, diagrams, and word problems.**
 * (7RP2c) I can write equations for proportions.**
 * (7RP2d) I can determine the unit rate from a linear graph with special attention to points (0,0) and (1,r) where r is the unit rate.**
 * (7RP3) I can solve multistep ratio and percent problems.**

‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Essential Vocabulary**
===unit rate, ratio, proportion, simple interest, tax, markup, markdown, commission, tip, percent increase and decrease, percent error, complex fractions, constant of proportionality, proportional relationships===

‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Differentiation**

 * peer tutoring, grouping, guided notes, foldables, inquiry based learning to accelerate/remediate, justification of answers, blogs, solve and act-out real-life scenarios of buying products**

‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Enrichment:**


===‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Instructional Resources**===





[|Sales Tax, Tip, Discount]3

http://qta.quantiles.com/qtaxon/goal/29742/ http://qta.quantiles.com/qtaxon/goal/29743/ http://qta.quantiles.com/qtaxon/goal/29744/ http://qta.quantiles.com/qtaxon/goal/29745/ http://qta.quantiles.com/qtaxon/goal/29746/ http://qta.quantiles.com/qtaxon/goal/29747/
 * Math Partners resources,**
 * sales papers from various stores**

http://www.dpi.state.nc.us/curriculum/mathematics/middlegrades/grade07/

http://www.montgomery.k12.nc.us/18032092913393583/site/default.asp
 * [|Kutasoftware Resource]**
 * http://www.mathgoodies.com,**

===‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Notes and Additional Information**===