ALEX Classroom Resource

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:
Given contextual or mathematical situations involving multiplicative comparisons,

• Use unit rate to solve missing value problems (e.g., cost per item or distance per time unit).
• Use rate language to explain the relationships between ratio of two quantities as non-complex fractions and the associated unit rate of one of the quantities in terms of the other.

Students know:

• Characteristics of multiplicative comparison situations.
• Rate and ratio language.
• Techniques for determining unit rates.
• To use reasoning to find unit rates instead of a rule or using algorithms such as cross-products.

Students are able to:

• Explain relationships between ratios and the related unit rates.
• Use unit rates to name the amount of either quantity in terms of the other quantity flexibly.
• Represent contextual relationships as ratios.

Students understand that:

• A unit rate is a ratio (a:b) of two measurements in which b is one.
• A unit rate expresses a ratio as part-to-one or one unit of another quantity.

Learning Objectives:
M.6.2.1: Define unit rate, proportion, and rate.
M.6.2.2: Create a ratio or proportion from a given word problem.
M.6.2.3: Calculate unit rate by using ratios or proportions.
M.6.2.4: Write a ratio as a fraction.

Students:
Given contextual or mathematical situations involving ratio and rate (including those involving unit pricing, constant speed, and measurement conversions),

• Represent the situations using a variety of strategies (tables of equivalent ratios, changing to unit rate, tape diagrams, double number line diagrams, equations, and plots on coordinate planes) in order to solve problems, find missing values on tables and interpret relationships and results.
• Change given rates to unit rates in order to find and justify solutions to problems.

Given contextual or mathematical situations involving percents,

• Understand the relationship between ratios, fractions, decimals and percents.
• Interpret the percent as rate per 100.
• Solve problems and justify solutions when finding the whole, given a part and the percent.
• Solve problems and justify solutions when finding the part, given the whole and the percent.
• Solve problems and justify solutions when finding percent, given the whole and the part.

Students know:

• Strategies for representing contexts involving rates and ratios including. tables of equivalent ratios, changing to unit rate, tape diagrams, double number lines, equations, and plots on coordinate planes.
• Strategies for finding equivalent ratios,
• Strategies for using ratio reasoning to convert measurement units.
• Strategies to recognize that a conversion factor is a fraction equal to 1 since the quantity described in the numerator and denominator is the same.
• Strategies for converting between fractions, decimals and percents.
• Strategies for finding the whole when given the part and percent in a mathematical and contextual situation.
• Strategies for finding the part, given the whole and the percent in mathematical and contextual situation.
• Strategies for finding the percent, given the whole and the part in mathematical and contextual situation.

Students are able to:

• Represent ratio and rate situations using a variety of strategies (e.g., tables of equivalent ratios, changing to unit rate, tape diagrams, double number line diagrams, equations, and plots on coordinate planes).
• Use ratio, rates, and multiplicative reasoning to explain connections among representations and justify solutions in various contexts, including measurement, prices and geometry.
• Understand the multiplicative relationship between ratio comparisons in a table by writing an equation.
• Plot ratios as ordered pairs.
• Solve and justify solutions for rate problems including unit pricing, constant speed, measurement conversions, and situations involving percents.
• Solve problems and justify solutions when finding the whole given a part and the percent.
• Model using an equivalent fraction and decimal to percents.
• Use ratio reasoning, multiplication, and division to transform and interpret measurements.

Students understand that:

• A unit rate is a ratio (a:b) of two measurements in which b is one.
• A symbolic representation of relevant features of a real-world problem can provide for resolution of the problem and interpretation of the situation.
• When computing with quantities the transformation and interpretation of the resulting unit is dependent on the particular operation performed.

Learning Objectives:
M.6.3.1: Define ratio, rate, proportion, percent, equivalent, input, output, ordered pairs, diagram, unit rate, and table.
M.6.3.2: Create a ratio or proportion from a given word problem, diagram, table, or equation.
M.6.3.3: Calculate unit rate or rate by using ratios or proportions with or without a calculator.
M.6.3.4: Restate real-world problems or mathematical problems.
M.6.3.5: Construct a graph from a set of ordered pairs given in the table of equivalent ratios.
M.6.3.6: Calculate missing input and/or output values in a table with or without a calculator.
M.6.3.7: Draw and label a table of equivalent ratios from given information.
M.6.3.8: Identify the parts of a table of equivalent ratios (input, output, etc.).
M.6.3.9: Compute the unit rate, unit price, and constant speed with or without a calculator.
M.6.3.10: Create a proportion or ratio from a given word problem.
M.6.3.11: Identify the two units being compared.
M.6.3.12: Define percent.
M.6.3.13: Calculate a proportion for missing information with or without a calculator.
M.6.3.14: Identify a proportion from given information.
M.6.3.15: Solve a proportion using part over whole equals percent over 100 with or without a calculator.
M.6.3.16: Form a ratio.
M.6.3.17: Convert like measurement units within a given system with or without a calculator. (Example: 120 min = 2 hrs).
M.6.3.18: Know relative sizes of measurement units within one system of units, including km, m, cm; kg, g; lb, oz; l, ml; and hr, min, sec.

Students:

• Use proportional reasoning strategies including setting up and solving proportions to solve problems involving simple interest, tax, gratuities, commissions, fees, markups, percent increase, markdowns or percent decrease.

Example: Students might be asked to "order" from a menu for lunch then calculate the tax and gratuities to determine the total cost.

Students know:

• how to interpret a real-world problem to determine what is being asked.
• Techniques for calculating and using percents to solve problems in context.
• how to interpret the solution in the context of the problem.

Students are able to:

• Write and solve proportions to help them solve real-world problems involving percent.
• Solve problems that require them to calculate: simple interest, tax, gratuities, commission, fees, mark ups, markdowns, percent increase and percent decrease.

Students understand that:

• percents relate to real-world contexts, and how to determine the reasonableness of their answers based on that context.

Learning Objectives:
M.7.3.1: Define interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, and percent error.
M.7.3.2: Apply definitions to context in real-world problems.
M.7.3.3: Solve proportional problems.
M.7.3.4: Recall how to find percent and ratios.
M.7.3.5: Recall steps for solving multi-step problems.

Students:

• Use proportional reasoning strategies including setting up and solving proportions to solve problems involving simple interest, tax, gratuities, commissions, fees, markups, percent increase, markdowns or percent decrease.

Example: Students might be asked to "order" from a menu for lunch then calculate the tax and gratuities to determine the total cost.

Students know:

• how to interpret a real-world problem to determine what is being asked.
• how to interpret the solution in the context of the problem.

Students are able to:

• Write and solve proportions to help them solve real-world problems involving percent.
• Solve problems that require them to calculate: simple interest, tax, gratuities, commission, fees, mark ups, markdowns, percent increase and percent decrease.

Students understand that:

• proportional reasoning requires interpretation or making sense of percent problems.
• Solving problems and determining their calculated answers may require further computation.