Common Core Learning Standards

Common Core Learning Standards

GRADE 7 Mathematics

RATIOS & PROPORTIONAL RELATIONSHIPS

Common Core Learning Standards / Concepts / Embedded Skills / Vocabulary
Analyze proportional relationships and use them to solve real-world and mathematical problems. / Unit Rate / Solve unit rate problems that have fractional quantities. (Problems may require solving complex fractions). / § Ratio
§ Complex fraction
§ Unit rate
§ Rate
§ Proportion
§ equivalent
Solve ratio problems whose quantities are lengths of the same unit and different units.
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. / Solve ratio problems whose quantities are areas of the same unit and different units.
Solve ratio problems of other quantities with the same unit and different units.
Divide two fractions by taking the reciprocal of the divisor.
Compute the unit rate.
Rigorous Sample Tasks / Scaffolded Sample Tasks
I. Charlie wants to carpet his living room which has an area of 20 square feet.
a.) Carpet is priced by the square yard at Carpet Smart. Carpet is on sale for $20 per square yard. Home Depot sells carpet for $2.50 per square foot. (There are nine square feet in a square yard.) Which store sells carpet at a lower unit price?
b.) Suppose you have a coupon for 10% off at Home Depot. What is the new unit price?
c.) Charlie decides to go with the cheaper store. How much will it cost to carpet his living room? How much money will he save by going with this store? / I. If 5 tomatoes cost $2.00, what is the unit price of the tomatoes? How much would a dozen tomatoes cost?
2. Whitney earns $206.25 for 25 hours of work. How much does Whitney earn per hour? At this rate, how much does she make in 30 hours?
3. A square garden is 4 yards on each side. How many square feet is it?
4. Molly converted 4 square yards to square feet by multiplying by 3 to get 12 square feet. Is this correct? Explain your reasoning.
4. You buy a pair of jeans for $25.00. How much will the jeans cost after 15% discount and 8.75% sales tax?
Common Core Learning Standards / Concepts / Embedded Skills / Vocabulary
Analyze proportional relationships and use them to solve real-world and mathematical problems. / Proportional Relationships / Calculate the cross product to determine if the two ratios are in proportion (equivalent). / § constant of proportionality
§ rate of change
§ slope
§ cross product
§ equivalent
§ origin
§ quantities
Analyze ratios in a table to determine if the ratios are equivalent by finding the constant of proportionality (slope).
7.RP.2a, b, c, and d
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. / Graph ratios on a coordinate plane to determine if the ratios are proportional by observing if the graph is a straight line through the origin (y = mx, where m is the slope/constant of proportionality).
Solve proportions by cross multiplication.
Write and solve proportions.
Rigorous Sample Tasks / Scaffolded Sample Tasks
You pay $1 to rent a movie plus an additional $0.50 per day until you return the movie. Your friend pays $1.25 per day to rent a movie.
a. Make tables showing the costs to rent a movie up to 5 days.
b. Which person pays an amount proportional to the number of days rented?
c. What is the constant of proportionality for the person identified in part b?
d. Write an equation to represent the total cont, c in terms of the number of days, d for which the person rented the movie.
Raffle tickets cost $3 each. Write an equation that shows the total cost c of buying r raffle tickets.
The speed limit on a highway is 65 miles per hour. Write an equation that shows the number of miles driven, d , in t hours.
Common Core Learning Standards / Concepts / Embedded Skills / Vocabulary
Analyze proportional relationships and use them to solve real-world and mathematical problems. / Constant of proportionality / Calculate the constant of proportionality/unit rate from a table or diagram. / § constant of proportionality
§ unit rate
§ slope
§ proportional relationship
§ rate of change
§ direct proportional relationship
Compute the rate of change/slope from a graph (rise over run) or equation (m in y=mx).
7.RP.2b.
Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. / Calculate the constant of proportionality/unit rate given a verbal description of a proportional relationship.
Rigorous Sample Tasks / Scaffolded Sample Tasks
Common Core Learning Standards / Concepts / Embedded Skills / Vocabulary
Analyze proportional relationships and use them to solve real-world and mathematical problems. / Proportional relationships and equations / Write an equation from a proportional relationship. / § proportional relationships
§ equation
§ rate
§ ratio
Solve equations created from 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 = pn.
Rigorous Sample Tasks / Scaffolded Sample Tasks
Common Core Learning Standards / Concepts / Embedded Skills / Vocabulary
Analyze proportional relationships and use them to solve real-world and mathematical problems. / Relationships and proportional relationships / Define the rate of proportionality from a graph. / § rate of proportionality
§ x-coordinate
§ y-coordinate
§ unit rate
Explain the meaning of a point on a graph y=mx of a real life situation.
7.RP.2d.
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. / Calculate the unit rate by identifying that on a graph when the x-coordinate is 1, the y-coordinate is the unit rate.
SAMPLE TASKS
Common Core Learning Standards / Concepts / Embedded Skills / Vocabulary
Analyze proportional relationships and use them to solve real-world and mathematical problems. / ratios, percents, and proportions / Solve multistep ratio problems using proportions. Focus on simple interest, tax, markups/downs, gratuities and commissions, fees, percent increase/decrease, and percent error. / § Ratio
§ Proportion
§ Percent
Solve multistep percent problems using proportions. Focus on simple interest, tax, markups/downs, gratuities and commissions, fees, percent increase/decrease, and percent error.
7.RP.3.
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
Rigorous Sample Tasks / Scaffolded Sample Tasks
Dan takes a car loan for $5,780 for two years. If the interest rate is 6.5%, determine how much his monthly payment will be?
If Dan had made a 25% down payment (at the same interest rate) , how much less would his monthly payment be? / You take a loan from the bank for $2,000. Your interest rate is 5%. You will pay the loan back over the course of two years.
a.) How much will you pay in interest?
b.) How much will you pay back in total?
c.) If you pay back the loan in monthly installments, how much will each monthly payment be?
You are buying a bike that costs $450. You will make a $100 down payment on the bike and finance the rest at a 4% interest rate for 1 year.
a.) How much money will you have to borrow to pay for the bike?
b.) How much will you pay in interest?
c.) How much will you pay back in total?
d.) How much will each monthly payment be?
A group of 4 friends went to a restaurant. Before tax the bill came to $75.00. If tax is 8.75% and the group leaves at least a 20% gratuity. Tim says that each friend should leave $24.00 and John says that each friend should leave $25.00. Who is correct? Justify your answer. / You and your friend go out to lunch and your bill comes to $20, before tax and gratuity. The tax rate is 6% and you want to leave at least 15% gratuity.
a.) How much is the tax, based on the subtotal?
b.) How much is the tip, based on the subtotal?
c.) What is your total bill, with tax and gratuity?
d.) About how much will you and your friend each pay?
e.) Explain how you estimated, in part d, to find each person’s portion of the bill.
You bought a pair of sneakers for 20% off and paid 8.5% sales tax. Your total bill came to $95.48. What was the original price of the sneakers, before the 20% discount?
a.) $110
b.) $130.44
c.) $82.87
d.) $122.69
***This question should probably be in the Expressions & Equations module.
The Outdoor Furniture Center buys wooden benches for $50 each. The furniture store owner adds a 200% markup to the cost of the bench. After hearing from customers that the selling cost of the bench is too high, the owner changes the markup to 120%. How much less per bench does the store owner make with the lower markup? / You are selling lemonade. It costs you $0.10 to make each cup of lemonade. You markup the cost of each cup by 200%.
a.) How much will you charge your customers per cup of lemonade?
b.) How much profit will you make per cup of lemonade?
Your friend is also selling lemonade. It costs him $0.15 to make each cup of lemonade. He marks up the cost of each cup by 225%. Which of you will make more money per cup of lemonade?
a.) How much will your friend charge customers per cup of lemonade?
b.) How much profit will your friend make per cup of lemonade?
c.) Who makes a bigger profit? Explain.
Appliances at Discount City Store are on sale for 70% of the original price. Eli has a coupon for an 18% discount on the sale price. If the original price of a microwave oven is $500, how much will Eli pay for the oven before tax?
a.) 440
b.) $287
c.) $260
d.) $240 / You are sweater shopping and find a sweater for 40% off the original price. If the original price is $50, what is the discounted price?
You are sweater shopping and find a sweater for 40% of the original price. If the original price is $50, what is the discounted price?
Explain the different between “40% off the original price” and “40% of the original price”?
You have a 30% off coupon on a $45 pair of jeans. How much will you save on the jeans?
How much do you spend on the jeans?
Jeff gets a base salary of $950 per week. He earns a commission of 10% on his sales if his sales are between $0 and $25,000. He earns 15% commission on his sales if his sales are over $25,000. If his paycheck last week was $3,350, which commission rate did he earn? / Jeff gets a base salary of $950 per week. He earns a commission of 10% on his sales if his sales are between $0 and $25,000. He earns 15% commission on his sales if his sales are over $25,000. If his paycheck last week was $3,350, which commission rate did he earn?