Content Standard(s):
Mathematics 1. Use appropriate notations [a/b, a to b , a:b ] to represent a proportional relationship between quantities and use ratio language to describe the relationship between quantities.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given contextual or mathematical situations involving multiplicative comparisons.
Communicate the relationship of two or more quantities using ratio language. Teacher Vocabulary:
Ratio
Ratio Language
Part-to-Part
Part-to-Whole
Attributes
Quantity
Measures
Fraction Knowledge:
Students know:
Characteristics of additive situations.
Characteristics of multiplicative situations Skills:
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). Understanding:
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. Diverse Learning Needs:
Essential Skills:
Learning Objectives: M.6.1.1: Define quantity, fraction, and ratio.
M.6.1.2: Identify the units or quantities being compared.
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.
Prior Knowledge Skills:
Compare two fractions with the same numerator or the same denominator by reasoning about their size.
Addition and subtraction of fractions as joining and separating parts referring to the same whole.
Label numerator, denominator, and fraction bar.
Recognize fraction 1 as the quantity formed by 1 part when a whole is partitioned into b equal parts.
Alabama Alternate Achievement Standards
AAS Standard:
Mathematics 3. Use ratio and rate reasoning to solve mathematical and real-world problems (including but not limited to percent, measurement conversion, and equivalent ratios) using a variety of models, including tables of equivalent ratios, tape diagrams, double number lines, and equations.
Unpacked Content
Evidence Of Student Attainment:
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. Teacher Vocabulary:
Rate
Ratio
Rate reasoning
Ratio reasoning
Transform units
Quantities
Ratio Tables
Double Number Line Diagram
Percents
Coordinate Plane
Ordered Pairs
Quadrant I
Tape Diagrams
Unit Rate
Constant Speed Knowledge:
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. Skills:
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. Understanding:
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. Diverse Learning Needs:
Essential Skills:
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.
Prior Knowledge Skills:
Recognize arithmetic patterns (including geometric patterns or patterns in the addition table or multiplication table).
Complete the numerical pattern for the following chart when given the rule, "Input + 5 = Output".
Recognize that comparisons are valid only when the two fractions refer to the same whole.
Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem.
Recognize key terms to solve word problems.
Recall basic multiplication facts.
Recognize equivalent forms of fractions and decimals.
Recognize a fraction as a number on the number line.
Label numerator, denominator, and fraction bar.
Alabama Alternate Achievement Standards
AAS Standard:
Mathematics 2. Represent a relationship between two quantities and determine whether the two quantities are related proportionally.
a. Use equivalent ratios displayed in a table or in a graph of the relationship in the coordinate plane to determine whether a relationship between two quantities is proportional.
b. Identify the constant of proportionality (unit rate) and express the proportional relationship using multiple representations including tables, graphs, equations, diagrams, and verbal descriptions.
c. Explain in context the meaning of a point (x,y ) on the graph of a proportional relationship, with special attention to the points (0,0) and (1, r ) where r is the unit rate.
Unpacked Content
Evidence Of Student Attainment:
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. Teacher Vocabulary:
Equivalent ratios proportional Coordinate plane Ratio table Unit rate Constant of proportionality Equation ordered pair Knowledge:
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. Skills:
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. Understanding:
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. Diverse Learning Needs:
Essential Skills:
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.
Prior Knowledge Skills:
Recall basic addition, subtraction, multiplication, and division facts.
Define ordered pair of numbers.
Define x-axis, y-axis, and zero on a coordinate.
Specify locations on the coordinate system.
Define ordered pair of numbers, quadrant one, coordinate plane, and plot points.
Label the horizontal axis (x).
Label the vertical axis (y).
Identify the x- and y- values in ordered pairs.
Model writing ordered pairs.
Define quantity, fraction, and ratio.
Reinterpret a fraction as a ratio. Write a ratio as a fraction.
Create a ratio or proportion from a given word problem, diagram, table, or equation.
Alabama Alternate Achievement Standards
AAS Standard:
Mathematics 17. Solve problems involving scale drawings of geometric figures, including computation of actual lengths and areas from a scale drawing and reproduction of a scale drawing at a different scale.
Unpacked Content
Evidence Of Student Attainment:
Students:
Solve problems involving scale drawings. Use a scale factor to reproduce a scale drawing at a different scale. Determine the scale factor for a scale drawing. Teacher Vocabulary:
Scale drawing Reproduction Scale factor Knowledge:
Students know:
how to calculate actual measures such as area and perimeter from a scale drawing. Scale factor impacts the length of line segments, but it does not change the angle measurements.
There is a proportional relationship between the corresponding sides of similar figures.
A proportion can be set up using the appropriate corresponding side lengths of two similar figures. If a side length is unknown, a proportion can be solved to determine the measure of it. Skills:
Students are able to:
find missing lengths on a scale drawing. Use scale factors to compute actual lengths, perimeters, and areas in scale drawings.
Use a scale factor to reproduce a scale drawing at a different scale. Understanding:
Students understand that:
scale factor can enlarge or reduce the size of a figure.
Scale drawings are proportional relationships.
Applying a scale factor less than one will shrink a figure.
Applying a scale factors greater than one will enlarge a figure. Diverse Learning Needs:
Essential Skills:
Learning Objectives: M.7.17.1: Define scale, scale drawings, length, area, and geometric figures.
M.7.17.2: Locate/use scale on a map.
M.7.17.3: Identify proportional relationships.
M.7.17.4: Recognize numeric patterns.
M.7.17.5: Recall how to solve proportions.
Prior Knowledge Skills:
Construct repeating and growing patterns with a variety of representations.
Continue an existing pattern.
Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations.
Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories.
Define unit rate, proportion, and rate.
Create a ratio or proportion from a given word problem.
