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New Colorado P-12 Academic Standards

Current Display Filter: Mathematics - All - by Specific Prepared Graduate Competency - (Remove PGC Filter)

Content Area: Mathematics
Grade Level Expectations: Seventh Grade
Standard: 1. Number Sense, Properties, and Operations

Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter)

Concepts and skills students master:

1. Proportional reasoning involves comparisons and multiplicative relationships among ratios
Evidence Outcomes 21st Century Skill and Readiness Competencies

Students Can:

  1. Analyze proportional relationships and use them to solve real-world and mathematical problems.(CCSS: 7.RP)
  2. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.1 (CCSS: 7.RP.1)
  3. Identify and represent proportional relationships between quantities. (CCSS: 7.RP.2)
    • Determine whether two quantities are in a proportional relationship. (CCSS: 7.RP.2a)
    • Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. 2(CCSS: 7.RP.2b)
    • Represent proportional relationships by equations.3 (CCSS: 7.RP.2c)
    • 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. (CCSS: 7.RP.2d)
  4. Use proportional relationships to solve multistep ratio and percent problems.4 (CCSS: 7.RP.3)
    • Estimate and compute unit cost of consumables (to include unit conversions if necessary) sold in quantity to make purchase decisions based on cost and practicality (PFL)
    • Solve problems involving percent of a number, discounts, taxes, simple interest, percent increase, and percent decrease (PFL)

Inquiry Questions:

  1. What information can be determined from a relative comparison that cannot be determined from an absolute comparison?
  2. What comparisons can be made using ratios?
  3. How do you know when a proportional relationship exists?
  4. How can proportion be used to argue fairness?
  5. When is it better to use an absolute comparison?
  6. When is it better to use a relative comparison?

Relevance & Application:

  1. The use of ratios, rates, and proportions allows sound decision-making in daily life such as determining best values when shopping, mixing cement or paint, adjusting recipes, calculating car mileage, using speed to determine travel time, or enlarging or shrinking copies.
  2. Proportional reasoning is used extensively in the workplace. For example, determine dosages for medicine; develop scale models and drawings; adjusting salaries and benefits; or prepare mixtures in laboratories.
  3. Proportional reasoning is used extensively in geometry such as determining properties of similar figures, and comparing length, area, and volume of figures.

Nature Of:

  1. Mathematicians look for relationships that can be described simply in mathematical language and applied to a myriad of situations. Proportions are a powerful mathematical tool because proportional relationships occur frequently in diverse settings.
  2. Mathematicians reason abstractly and quantitatively. (MP)
  3. Mathematicians construct viable arguments and critique the reasoning of others. (MP)

1 For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction \(^\frac{1}{2}/_\frac{1}{4}\) miles per hour, equivalently 2 miles per hour. (CCSS: 7.RP.1)

2 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. (CCSS: 7.RP.2a)

3 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. (CCSS: 7.RP.2c)

4 Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. (CCSS: 7.RP.3)

Content Area: Mathematics
Grade Level Expectations: Sixth Grade
Standard: 1. Number Sense, Properties, and Operations

Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter)

Concepts and skills students master:

1. Quantities can be expressed and compared using ratios and rates
Evidence Outcomes 21st Century Skill and Readiness Competencies

Students Can:

  1. Apply the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.1 (CCSS: 6.RP.1)
  2. Apply the concept of a unit rate a/b associated with a ratio a:b with \(b \neq 0\), and use rate language in the context of a ratio relationship.2 (CCSS: 6.RP.2)
  3. Use ratio and rate reasoning to solve real-world and mathematical problems.3 (CCSS: 6.RP.3)
    • Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. (CCSS: 6.RP.3a)
    • Use tables to compare ratios. (CCSS: 6.RP.3a)
    • Solve unit rate problems including those involving unit pricing and constant speed.4 (CCSS: 6.RP.3b)
    • Find a percent of a quantity as a rate per 100.5 (CCSS: 6.RP.3c)
    • Solve problems involving finding the whole, given a part and the percent. (CCSS: 6.RP.3c)
    • Use common fractions and percents to calculate parts of whole numbers in problem situations including comparisons of savings rates at different financial institutions (PFL)
    • Express the comparison of two whole number quantities using differences, part-to-part ratios, and part-to-whole ratios in real contexts, including investing and saving (PFL)
    • Use ratio reasoning to convert measurement units.6 (CCSS: 6.RP.3d)

Inquiry Questions:

  1. How are ratios different from fractions?
  2. What is the difference between quantity and number?

Relevance & Application:

  1. Knowledge of ratios and rates allows sound decision-making in daily life such as determining best values when shopping, creating mixtures, adjusting recipes, calculating car mileage, using speed to determine travel time, or making saving and investing decisions.
  2. Ratios and rates are used to solve important problems in science, business, and politics. For example developing more fuel-efficient vehicles, understanding voter registration and voter turnout in elections, or finding more cost-effective suppliers.
  3. Rates and ratios are used in mechanical devices such as bicycle gears, car transmissions, and clocks.

Nature Of:

  1. Mathematicians develop simple procedures to express complex mathematical concepts.
  2. Mathematicians make sense of problems and persevere in solving them. (MP)
  3. Mathematicians reason abstractly and quantitatively. (MP)

1 For example, "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak." "For every vote candidate A received, candidate C received nearly three votes." (CCSS: 6.RP.1)

2 For example, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger." (CCSS: 6.RP.2)

3 e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. (CCSS: 6.RP.3)

4 For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? (CCSS: 6.RP.3b)

5 e.g., 30% of a quantity means 30/100 times the quantity. (CCSS: 6.RP.3c)

6 manipulate and transform units appropriately when multiplying or dividing quantities. (CCSS: 6.RP.3d)