ALEX Classroom Resource

Students:

• Reason about the impact of scaling one or both factors on the size of the product before multiplying and justify their thinking.

Example: Which is greater? 3/5 x 13 or 13 x 3/4? 13 x 3/4 is greater than 3/5 x 13 because both expressions contain a factor of 13, but the scale factor of 3/4 will result in a greater product than a scale factor of 3/5 because 3/4 > 3/5.
• Explain the size of the product when multiplying a number by a fraction greater than 1 and when multiplying a number by a fraction less than 1.

Students know:

• How to interpret multiplicative comparisons.
• Strategies to compare products with whole numbers using reasoning and justification.
  Example: Which is greater? 5 x 2 x 13 or 13 x 9? 10 x 13 is greater than 9 x 13 because both expressions contain a factor of 13, but the scale factor of 10 will result in a greater product than a scale factor of 9.
• Fraction meaning and magnitude of fractions less than and greater than 1.

Students are able to:

• Interpret multiplication as scaling.
• Use reasoning to compare products of multiplication expressions.
• Reason and explain when multiplying a given number by a fraction why the product will be greater than or less than the original number.

Students understand that:

• a product reflects the size of its factors.

Learning Objectives:
M.5.13.1: Define scaling.
M.5.13.2: Define principle of fraction equivalence.
M.5.13.3: Multiply a fraction by a whole number.
M.5.13.4: Compare two fractions with the same numerator or the same denominator by reasoning about their size.
M.5.13.5: Recognize that comparisons are valid only when the two fractions refer to the same whole.
M.5.13.6: Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
M.5.13.7: Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
M.5.13.8: Identify factor and product.
M.5.13.9: Use comparison symbols.
Examples: >, =, or <.

Students: Given contextual or mathematical situations involving multiplicative comparisons.

• Communicate the relationship of two or more quantities using ratio language.

Students know:

• Characteristics of additive situations.
• Characteristics of multiplicative situations

Students are able to:

• Compare and contrast additive vs. multiplicative contextual situations.
• Identify all ratios and describe them using "For every…, there are…"
• Identify a ratio as a part-to-part or a part-to whole comparison.
• Represent multiplicative comparisons in ratio notation and language (e.g., using words such as "out of" or "to" before using the symbolic notation of the colon and then the fraction bar. for example, 3 out of 7, 3 to 5, 6:7 and then 4/5).

Students understand that:

• In a multiplicative comparison situation one quantity changes at a constant rate with respect to a second related quantity. -Each ratio when expressed in forms: ie 10/5, 10:5 and/or 10 to 5 can be simplified to equivalent ratios, -Explain the relationships and differences between fractions and ratios.

Learning Objectives:
M.6.1.1: Define quantity, fraction, and ratio.
M.6.1.2: Identify the units or quantities being compared.
Example: Read 2/3 as 2 out of 3.
M.6.1.3: Write a ratio in appropriate notation;[a/b, a to b, a:b].
M.6.1.4: Draw a model of a given ratio or fraction.
M.6.1.5: Identify the numerator and denominator of a fraction.

Students:

• Decide whether a relationship between two quantities is proportional.
• Recognize that not all relationships are proportional.
• Use equivalent ratios in a table or a coordinate graph to verify a proportional relationship.
• Identify the constant of proportionality when a proportional relationship exists between two quantities.
• Use a variety of models (tables, graphs, equations, diagrams and verbal descriptions) to demonstrate the constant of proportionality.
• Explain the meaning of a point (x, y) in the context of a real-world problem.
Example, if a boy charges $6 per hour to mow lawns, this relationship can be graphed on the coordinate plane. The point (1,6) means that after 1 hour of working the boy makes $6, which shows the unit rate of $6 per hour.

Students know:

• (2a) how to explain whether a relationship is proportional.
• (2b) that the constant of proportionality is the same as a unit rate. Students know:
  • where the constant of proportionality can be found in a table, graph, equation or diagram.
  • (2c) that the constant of proportionality or unit rate can be found on a graph of a proportional relationship where the input value or x-coordinate is 1.

Students are able to:

• (2a) determine if a proportional relationship exists when given a table of equivalent ratios or a graph of the relationship in the coordinate plane.
• (2b) identify the constant of proportionality and express the proportional relationship using a variety of representations including tables, graphs, equations, diagrams, and verbal descriptions.
• (2c) model a proportional relationship using coordinate graphing.
• Explain the meaning of the point (1, r), where r is the unit rate or constant of proportionality.

Students understand that:

• (2a) A proportional relationship requires equivalent ratios between quantities. Students understand how to decide whether two quantities are proportional.
• (2b) The constant of proportionality is the unit rate. Students are able to identify the constant of proportionality for a proportional relationship and explain its meaning in a real-world context. (2c) The context of a problem can help them interpret a point on a graph of a proportional relationship.

Learning Objectives:
M.7.2.1: Define proportions and proportional relationships.
M.7.2.2: Demonstrate how to write ratios as a fraction.
M.7.2.3: Define equivalent ratios and origin.
M.7.2.4: Locate the origin on a coordinate plane.
M.7.2.5 Show how to graph on Cartesian plane.
M.7.2.6: Determine if the graph is a straight line through the origin.
M.7.2.7: Use a table or graph to determine whether two quantities are proportional.
M.7.2.8: Define a constant and equations.
M.7.2.9: Create a table from a verbal description, diagram, or a graph.
M.7.2.10: Identify numeric patterns and finding the rule for that pattern.
M.7.2.11: Recall how to find unit rate.
M.7.2.12: Recall how to write equations to represent a proportional relationship.
M.7.2.13: Discuss the use of variables.
M.7.2.14: Define ordered pairs.
M.7.2.15: Show how to plot points on a Cartesian plane.
M.7.2.16: Locate the origin on the coordinate plane.

Students:

• Decide whether a relationship between two quantities is proportional.
• Recognize that not all relationships are proportional.
• Use equivalent ratios in a table or a coordinate graph to demonstrate a proportional relationship.
• Identify the constant of proportionality when a proportional relationship exists between two quantities.
• Interpret a variety of models (tables, graphs, equations, diagrams and verbal descriptions) to identify the constant of proportionality.
• Explain the meaning of a point (x, y) in the context of a real-world problem.

Example: If a boy charges $6 per hour to mow lawns, this relationship can be graphed on the coordinate plane. The point (1, 6) contains the unit rate or constant of proportionality, 6.

Students know:

• (2a) how to explain whether a relationship is proportional.
• (2b) that the constant of proportionality is the same as a unit rate.
• (2b) where the constant of proportionality can be found in a table, graph, equation or diagram.
• (2c) that the constant of proportionality or unit rate can be found on a graph of a proportional relationship where the input value or x-coordinate is 1.

Students are able to:

• (2a) model a proportional relationship using a table of equivalent ratios.
• Use a coordinate graph to decide whether a relationship is proportional by plotting ordered pairs and observing whether the graph is a straight line through the origin.
• (2b) translate a written description of a proportional relationship into a table, graph, equation or diagram.
• Read and interpret these to find the constant of proportionality.
• (2c) model a proportional relationship using coordinate graphing.
• Explain the meaning of the point (1, r), where r is the unit rate or constant of proportionality.

Students understand that:

• (2a) a proportional relationship requires equivalent ratios between quantities. Students understand how to decide whether two quantities are proportional.
• (2b) the constant of proportionality is the unit rate. Students are able to identify the constant of proportionality for a proportional relationship and explain its meaning in a real-world context.
• (2c) the context of a problem can help them interpret a point on a graph of a proportional relationship.