Unit 1

Ratios and Proportional Relationships

Grade 7 Math 

 

Unit Length and Description:

 

20 days

 

In this unit, students extend their understanding of ratios from Grade 6 to include ratios of fractions and unit rates as complex fractions. Students decide whether two quantities are in a proportional relationship, identify constants of proportionality, and represent proportional relationships by equations. They graph proportional relationships and understand the unit rate informally as a measure of the steepness of the related line, called the slope. They can also distinguish proportional relationships from other relationships. Students develop an understanding of proportionality and use proportional relationships to solve single-step and multi-step problems. These skills are then applied to realworld problems, including scale drawings as students analyze proportional relationships in geometric figures. Students solve problems about scale drawings by relating corresponding lengths between the objects or by using the fact that relationships of lengths within an object are preserved in similar objects.

 

Standards:

 

7.RP.A.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 / ¼ miles per hour, equivalently 2 miles per hour.

 

7.RP.A.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 at 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.A.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.

 

7.NS.A.3  Solve real-world and mathematical problems involving the four operations with rational numbers.

 

7.G.A.1  Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

 

Focus Standards of Mathematical Practice:

 

MP.1 Make sense of problems and persevere in solving them.

MP.2 Reason abstractly and quantitatively.

MP.3 Construct viable arguments and critique the reasoning of others.

MP.4 Model with mathematics.

MP.5 Use appropriate tools strategically.

MP.6 Attend to precision.

MP.7 Look for and make use of structure.

MP.8 Look for and express regularity in repeated reasoning.

 

Instructional Outcomes:

Full Development of the Major Clusters, Supporting Clusters, Additional Clusters and Mathematical Practices for this unit could include the following instructional outcomes:

 

7.RP.A.1

·         I can compute unit rates associated with ratios of fractions in like or different units.

·         I can compute fractional by fractional quotients.

·         I can apply fractional ratios to describe rates.

 

7.RP.A.2

·         I can determine that a proportion is a statement of equality between two ratios.

·         I can analyze two ratios to determine if they are proportional to one another with a variety of strategies (e.g. using tables, graphs, pictures, etc.).

·         I can define constant of proportionality as a unit rate.

·         I can analyze tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships to identify the constant of proportionality.

·         I can represent proportional relationships by writing equations.

·         I can recognize what (0,0) represents on the graph of a proportional relationship.

·         I can recognize what (1,r) on a graph represents, where r is the unit rate.

·         I can explain what the points on a graph of a proportional relationship mean in terms of a specific situation.

 

7.RP.A.3

·         I can recognize situations in which proportional relationships apply.

·         I can apply proportional reasoning to solve single and multistep ratio problems.

 

7.NS.A.3

·         I can solve real-world and mathematical problems involving complex fractions.

 

7.G.A.1

·         I can use ratios and proportions to create scale drawing.

·         I can identify corresponding sides of scaled geometric figures.

·         I can compute lengths and areas from scale drawings using strategies such as proportions.

·         I can solve problems involving scale drawings of geometric figures using scale factors.

·         I can reproduce a scale drawing that is proportional to a given geometric figure using a different scale.

 

Enduring Understandings:

 

·         Reasoning with ratios involves attending to and coordinating two quantities.

·         A ratio is a multiplicative comparison of two quantities.

·         Forming a ratio as a measure of a real-world attribute involves isolating that attribute from other attributes and understanding the effect of changing each quantity on the attribute of interest.

·         A number of mathematical connections link ratios and fractions:

°         Ratios are often expressed in fraction notation, although ratios and fractions do not have identical meaning.

°         Ratios are often used to make “part-part” comparisons, but fractions are not.

°         Ratios can often be meaningfully reinterpreted as fractions.

·         Ratios can be meaningfully reinterpreted as quotients.

·         A proportion is a relationship of equality between two ratios. In a proportion, the ratio of two quantities remains constant as the corresponding values of the quantities change.

·         If one quantity in a ratio is multiplied or divided by a particular factor, then the other quantity must be multiplied or divided by the same factor to maintain the proportional relationship.

·         A rate is a set of infinitely many equivalent ratios.

·         Scale drawings and scale models are used to represent objects that are too large or too small to be drawn or built at actual size.

·         The scale gives the ratio that compares the measurements of the drawing or model to the measurements of the real object.

·         The measurements on a drawing or model are proportional to the measurements on the actual objects.

 

Essential Questions:

 

·         What are the types/varieties of situations in life that depend on or require the application of ratios and proportional reasoning?

·         How can a complex fraction be simplified?

·         What is the difference between a unit rate and a ratio?

·         What is a proportion?

·         Why are multiplicative relationships proportional?

·         How are equivalent ratios, values in a table, and ordered pairs connected?

·         What characteristics define the graphs of all proportional relationships?

·         How can you show that two objects are proportional?

·         How are proportional and nonproportional relationships alike?  How are they different?

·         What is the constant of proportionality?

·         How is unit rate related to rate of change?

·         How can you use different measurements to solve real-life problems?

·         How does geometry help us describe real-world objects?