1. Grades 4 – 5 Fractions Grades 6 – 7 Ratios and Proportional Relationships Grade 8 Functions

2. Progressions for the Common Core State Standards in Mathematics (draft)
Ratio Proportional Progression 6 – 7
©The Common Core Standards Writing Team 26 December 2011

3. Examples of Topic Progression
Fractions  Ratios  Proportions  Rate of Change
Geometry - Similar figures, trigonometric ratios
Science - 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

4. Grades 6 – 8 National Council of Teachers of Mathematics
Developing Essential Understanding of Ratios, Proportions & Proportional Reasoning
Grades 6 – 8
National Council of Teachers of Mathematics

5. 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)

6. Rates
Previous 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

7. 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.

8. 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.

9. 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

10. 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”

11. Moving from Additive to Multiplicative Comparisons
Use tables and repeated reasoning 1 3 2 6 9 4 12 5 15

12. Moving from Additive to Multiplicative Comparisons
Use of double number lines and repeated reasoning m
sec

13. 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)

14. 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)

15. Geometric Version
Consider two similar triangles. 6 3 4 8

16. 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

17. 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 not
Ex. Ratio of types of milk in a store

18. 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

19. 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?

20. 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?

21. 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.

22. 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 or
divided, the same must happen to the
other quantity to maintain the
proportional relationship
-The two types of ratios (composed
units and multiplicative comparison) are
related

23. 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 proportion
10/4 = 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

24. 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

25. 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

26. 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

27. Example: Orange Juice 3 concentrates 4 waters
Not as a ratio: Purely counting 3, counting 4 and putting them in fractional form
As a ratio: Forming a composed unit of 3:4, iterating 6:8, and understanding that it tastes equally orangey
As a rate: The student can use 1:4/3 to find any quantity of orange juice with equivalent strength

28. 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.

29. 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.

30. Example of a Function

31. Developing the Various Parts of a Function through Activities

32. 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.

33. Algebraic Ups and Downs
In 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)

34. Inquiry Lab – Relations and Functions
Understanding the specific relationship that defines a function
Compare different typical explanations

35. 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

36. Zap It 1
Keep practicing, but in a different way

37. Extend the Matching Game
In 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

38. Total Cost of the Pencils
Applicable Problem
Working 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

39. Functions 8.F 2
Compare 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.

40. Pledge Plans
Real world context to help understand what changing the y-intercept really means
Connections to Statistics

41. How does this relate to absolute value?
Applicable Problem
Determining the greater rate of change
y = –2x y = 4x y = x y = –3x
How does this relate to absolute value?

42. Problem Solving Connections
Bringing all the ideas of the unit together
Real world context

43. Functions 8.F 3
Interpret 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.

44. Rising Towers
Comparing geometric quantities to discover proportional versus non- proportional connections

45. Applicable Problem Discern 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 y
A graph comparing time x and distance y

46. Functions 8.F 4
Construct 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.

47. Zap it 2 and 3
Focuses on working backwards to find an initial starting point for non-proportional functions

48. Applicable Problem
The 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

49. Functions 8.F 5
Describe 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.

50. Books Upon Books Statistics connection
Identify independent and dependent variables in a real world context
Introduction to solving systems of linear equations

51. Applicable Problem Explain the function in context
The 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?

52. The Activities Resources
AIMS
NCTM Navigating Series
On Core Mathematics
Glencoe

53. Additional Resources Zeroing in on Number and Operations Grades 7-8
Anne Collins & Linda Dacey

54. Additional Resources Illuminations: Math Playground:
Function Matching
Circle Tool
Math Playground:
Thinking Blocks
Function Machine
Equivalent Fractions
Balanced Assessments:
Bicycle Rides (functions)
Pen Pals (measurement conversions)