Grades 4 – 5 Fractions

Grades 6 – 7 Ratios and Proportional Relationships

Grade 8 Functions

Progressions for the Common Core State Standards in Mathematics (draft)Ratio Proportional Progression 6 – 7 ©The Common Core Standards

Writing Team 26 December 2011

Examples of Topic Progression

Fractions Ratios Proportions Rate of ChangeGeometry - Similar figures, trigonometric ratiosScience - Rate of change, average rate of

change in Calculus, slope, speed, acceleration, density, quickness of technology

Everyday - cooking, tips, tax, miles per gallon, discounts

Statistics – demographics, economics, birth rate, body mass index, rain fall, medicine dosing

Developing Essential Understanding of Ratios, Proportions & Proportional Reasoning

Grades 6 – 8

National Council of Teachers of Mathematics

Important Transitions - Ratios • Reasoning from one quantity to two

• Moving from additive comparisons to multiplicative comparisons

• Progressing from ratios as composed units to having a multiplicative relationship

• Moving from iterating with composed units to creating infinitely many equivalent ratios through multiplication (rate)

RatesPrevious definition of rate: a

comparison of two quantities of different units

Alternate definition of rate: a set of infinitely equivalent ratios or a ratio in which one of the quantities is time

Reasoning From One Quantity to Two

Orange concentrate example:Sixth graders were shown a large and a small glass of orange juice filled by the same carton and asked if they thought both glasses would taste equally orangey or if one was more orangey than the other.

Reasoning with Two Quantities

Ramp example (p. 25):In order to use a ratio you have to isolate the attribute being measured.

Ex. A ramp is composed of base length and height, but the attribute being measured is steepness of the incline.

Reasoning with Two Quantities

Ramp example (p. 25):Effects of changing one quantity.

Ex. What happens to the steepness:

base length is increased…decreased

height is increased…decreased

Moving from Additive to Multiplicative Comparisons

When students focus on only one quantity they wrongly interpret ratios using additive comparisons.

e.g. When asked to compare lengths they assume the question refers to “how much more” or “how much less” rather than “how many times bigger” or “What part is one compared to the other”

Moving from Additive to Multiplicative Comparisons

Use tables and repeated reasoning

1 32 63 94 125 15

Moving from Additive to Multiplicative Comparisons

Use of double number lines and repeated reasoning

m 0 5 10 15

sec 0 2 4 6

Ratios as Composed Units

(iterating and partitioning)

Composed units refers to the joining of two quantities to create a new unit.

Consider mixing juices. 3 apple and 2 grape

Here the composed unit refers to one batch.

So now you can say let us iterate (multiply) or partition (divide)

Ratios as a Fixed Number of Parts

Consider mixing juices. 3 apple and 2 grape

Out of 5 parts, 3 are apple, 2 are grape.

This leads us to one is 3/2 of the other and one is 2/3 of the other

Which leads us to y = cx where c is the constant of proportionality (rate of change/slope)

Geometric VersionConsider two similar triangles.

3

8

6

4

Ratios Having a Multiplicative Relationship

(infinitely many equivalent ratios)

Consider the length of Worm A is 6 in and the length of Worm B is 4 in

A:B = 6:4 = 1.5:1 A is 1.5 times as big as B

B:A = 4:6 = 2:3 = 1:1.5 A is 1.5 times as big as B

Ratios and Fractions – do not have identical meaning

Ratios are often used to make “part-part” comparisons while fractions are “part-whole”

Ratios can involve more than two terms while fractions do notEx. Ratio of types of milk in a store

Ratios and Fractions – fractions reinterpreted

Fractions can be reinterpreted as a point on a number line or an operator such as a scale factor

Ex. Ratio of 2:5 as two-fifths of something

Fractions as quotients can be reinterpreted as sharing

Ex. Ratio of 2:5 as sharing two ounces among five minutes

Recognize and Describe Ratios

“for each” “for every” “per”

“Two pounds for a dollar” needs more explaining. Is it every two pounds costs me a dollar or are we talking about a discount for every two pounds?

Recognize and Describe Proportions

If a factory produces 5 cans of dog food for every 3 cans of cat food, then when the company produces 600 cans of dog food, how many cans of cat food will it produce?

If a factory produces 5 cans of dog food for every 3 cans of cat food, then how many cans of cat food will the company produce when it produces 600 cans of dog food?

Recognize and Describe Multistep Problems

After a 20% discount, the price of a SuperSick skateboard is $140. What was the price before the discount?

A SuperSick skateboard costs $140 now, but its price will go up by 20%. What will the new price be after the increase?

The solutions are different because the 20% refers to different wholes.

Important Understandings - Proportions

• A proportion is a relationship of equality between two ratios

- Equivalent ratios by iterating or partitioning composed units- If one quantity of a ratio is

multiplied ordivided, the same must happen to

the other quantity to maintain theproportional relationship-The two types of ratios (composed units and multiplicative

comparison) arerelated

A Proportion is a Relationship of Equality Between two Ratios

• Understand what the equal sign in a proportion means

Ex. A clown walks 10 cm in 4 sec. A frog walks 20 cm in 8 sec. In the proportion10/4 = 20/8 the equal sign means that the two speeds are the same.

So the constant rate of change is the same or in other words both share the same unit rate

Equivalent Ratios by Iterating or Partitioning Composed Units

• Understanding begins with iterating with the composed unit by doubling, tripling, etc.

• Next comes partitioning the composed unit by simplifying or finding the unit rate

Maintaining the Proportional Relationship

•Iterating as multiplication or repeated groups of both parts of the ratio

• Partitioning as division or the repeated sharing of both parts of the ratio

Composed Units and Multiplicative Comparison are Related

• Use composed units to find both unit rates

• Interpret the unit rates as the multiplicative comparisons

• Use the multiplicative comparisons to represent the relationship in an equation

Example: Orange Juice3 concentrates 4 waters1. Not as a ratio: Purely counting 3,

counting 4 and putting them in fractional form

2. As a ratio: Forming a composed unit of 3:4, iterating 6:8, and understanding that it tastes equally orangey

3. As a rate: The student can use 1:4/3 to find any quantity of orange juice with equivalent strength

Ratios, Rates, Proportions and Graphing

Grade 6 – Use a table of equivalent ratios to plot the pairs of values on the coordinate plane

Grade 7 – Use a graph to decide if two quantities are proportional (linear through the origin). Represent proportional relationships in the equation y = mx. Explain that the upoint (1, r) represents the unit rate and (0, 0) represents the starting point.

Functions and Graphing

Grade 8 – Interpret the equation y = mx + b as a linear function

where m is the rate of change and b is the starting point. Understand that when b is nonzero the function is not proportional. Use slope to determine greater or lesser rates of change.

Example of a Function

Developing the Various Parts of a Function through Activities

Functions 8.F 1

Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

Algebraic Ups and DownsIn ratios we discover the

relationship between the two or more given quantities

In functions we discuss the relationship between x and y

y is a function of x (y depends on what happens with x)

Inquiry Lab – Relations and Functions

Understanding the specific relationship that defines a function

Compare different typical explanations

Graphing Linear Equations

Practice input/output

Discover the equation y = mx + b

Discuss how changing m or changing b effects the graphic representation and the verbal representation

Zap It 1

Keep practicing, but in a different way

Extend the Matching GameIn 7th grade we used a game

where each student had a different representation of a proportion (verbal, graphical, table, etc.).

Extension: Have proportional, non-proportional and non-linear examples

Applicable ProblemWorking your way back to the unit

price

Write the rule as an equation using x and y

Number of Pencils

Rule Total Cost of the Pencils

3 ? $0.75

4 ? $1.00

5 ? $1.25

x y

Functions 8.F 2Compare properties of two functions

each represented in a different way (algebraically, graphically, numerically in tables or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.

Pledge Plans

Real world context to help understand what changing the y-intercept really means

Connections to Statistics

Applicable Problem

Determining the greater rate of change

y = –2x y = 4x y = x y = –3x

How does this relate to absolute value?

Problem Solving Connections

Bringing all the ideas of the unit together

Real world context

Functions 8.F 3Interpret the equation y = mx + b

as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1, 1), (2, 4) and (3, 9) which are not on a straight line.

Rising Towers

Comparing geometric quantities to discover proportional versus non-proportional connections

Applicable ProblemDiscern linear from non-linear situations Jose decided to hike up a mountain last

Saturday. It took him the same amount of time to hike up the mountain as it did to hike back down.

Label as linear or nonlinear:A graph comparing time x and elevation yA graph comparing time x and distance y

Functions 8.F 4Construct a function to model a linear

relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Zap it 2 and 3

Focuses on working backwards to find an initial starting point for non-proportional functions

Applicable ProblemThe prices for entry into a science center are

recorded below. Assuming the relationship is linear find the rate of change and initial value.

Explain the meaning of the rate of change and initial value in the context of the question.

Number of People (x) Total Cost (y)

2 $65

3 $80

4 $95

5 $110

Functions 8.F 5Describe qualitatively the

functional relationship between two quantities by analyzing a graph (e.g. where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

Books Upon Books

Statistics connection

Identify independent and dependent variables in a real world context

Introduction to solving systems of linear equations

Applicable ProblemExplain the function in contextThe graph shows the number of gallons of

water in a bathtub after filling it for a certain number of minutes.

Write an equation to represent the situation.Why is the slope increasing?What would the situation be if the slope

were decreasing?

0 1 2 3 4 5 6 7 8 9 100

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Number of Minutes

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The Activities Resources

AIMSNCTM Navigating SeriesOn Core MathematicsGlencoe

Additional Resources

Zeroing in on Number and Operations Grades 7-8

Anne Collins & Linda Dacey

Additional ResourcesIlluminations:

◦Function Matching◦Circle Tool

Math Playground:◦Thinking Blocks◦Function Machine◦Equivalent Fractions

Balanced Assessments:◦Bicycle Rides (functions)◦Pen Pals (measurement conversions)