1. A professional development program to help teachers
better understand the mathematical reasoning needed for
ratio, proportion, and proportional reasoning.
2. To understand the importance and
significance of ratio, proportion, and
proportional reasoning in the middle grades
mathematics curriculum.
To examine some common ways of operating
with ratios.
To identify possible misconceptions students
might have when solving ratio problems.
3. Connected to elementary topics.
* Multiplication
Connected to Algebra 1 and Geometry.
* Similarity
* Slope
* Direct Variation
* Percents
4. Framework for studying various middle
school topics/standards
*algebra
*geometry
*measurement
*probability and statistics
Lanius, C.S. & Williams, S.A. (2003). Proportionality: A Unifying Theme for the
Middle Grades. Mathematics Teaching in the Middle School, 8(8), 392 – 396.
5. A ratio is a comparison between
two or more quantities, which are
either numbers or measurement
Chapin, S & Johnson, A. (2006). Math Matters: Understanding
the Math You Teach, Grades 6 – 8, 2nd Ed. Math Solutions
Publications, Sausilito, CA.
6. 5 pencils for $1.45
2 degrees per hour
1 point added to final grade for every 10
completed homework assignments
½ cup lemonade concentrate for every 4 cups
of water
12 red jellybeans compared with 55
jellybeans total.
Any other examples?
9. Stephen, who will ultimately be 12 ft. long,
has only grown to 6 ft. Sarah, who will grow
to 9 ft., has only grown to 5 ft.
1.) What are some possible questions we can
ask from this information?
2.) How might students respond?
10. Additive thinking is needed for comparing
quantities in one variable (e.g. heights)
Multiplicative thinking is needed when
comparing the fraction of one quantity to
another quantity (e.g. current height is what
fraction/percent of full grown height).
11. How does the relationship
between two quantities in a ratio
convey different multiplicative
info? (Lamon, 1999)
12.
13.
14.
15. When we ask which situation is the most
crowded, we think multiplicatively and
consider both quantities at the same
time. (In this case, a quantity of interest
might be maximum capacity.) Namely
we use multiplication and division in our
solution.
Chapin, S & Johnson, A. (2006). Math Matters: Understanding the Math
You Teach, Grades 6 – 8, 2nd Ed. Math Solutions Publications,
Sausilito, CA.
23. How can we have 1.5 girls per pizza?
Perhaps we should say that not all quotients
are meaningful when working with ratios.
24.
25. A man walks 14 feet in 6 seconds. How many
feet can he walk in
12 seconds?
3 seconds?
9 seconds?
8 seconds?
How do you think a student would answer the
above four questions?
26. Another way to form a ratio is by joining
(composing) two quantities to create a new
unit. Many times, students create composed
units when they iterate (repeat) a quantity
additively or partitioning (break into equal-
sized sections) (Lobato et al, 2010).
27. Distance 7 ft 14 ft 21 ft 28 ft 35 ft.
Time 3 sec. 6 sec. 9 sec. 12 sec. 15 sec.
28. For the walking man, the easiest partition
would probably be 7 ft. in 3 seconds.
How might we partition into six parts to
consider distance traveled in 1 second?
The pizza/girls problem explored earlier is
another example of partitioning, or sharing
equally among each girl.
29.
30. Complete the handout “Which Tastes More Juicy?”
You are encouraged to use previous strategies
learned such as ratios as a composed unit, the
multiplicative nature of a ratio, or meaningful
interpretations of quotients.
Pay close attention to the students’ thought
processes described on the handout. In addition to
determine whether their answers are correct or not,
think about their reasoning.
Using some of the ideas discussed today, how might
you correct students’ misconceptions that simply
adding more water and more juice does not
guarantee more juiciness?
31.
32. Reasoning with Ratios involves attending to
and coordinating two quantities.
A ratio is a multiplicative comparison of two
quantities, or it is a joining of two quantities
in a composed unit.
Ratios can be meaningfully reinterpreted as
quotients.
33.
34.
35. Reasoning with ratios involves attending to
and coordinating two quantities
A ratio is a multiplicative comparison of two
quantities, or it is a joining of two quantities
in a composed unit.
Ratios can be meaningfully reinterpreted as
quotients.
36. To compare and contrast fractions and ratios.
To compare and contrast different methods
used to solve proportion problems.
To view proportional reasoning as conceptual
rather than procedural.
To examine and evaluate student work and
reasoning for the depth of understanding of
proportional reasoning.
40. A ratio of two integers, where the
denominator is nonzero
Sometimes called rational numbers.
Fractions are real numbers.
Fractions only express a part-whole
relationship.
41. Ratios can often be meaningfully interpreted
as fractions.
Ratios can be compared to zero.
Ratios can compare numbers that are not
necessarily whole or rational (e.g. the Golden
Ratio)
Ratios can exhibit part-whole relationships
and can exhibit part-part relationships.
42. Complete the handout “What is equal?” You
are encouraged to use previous strategies
learned such as ratios as a composed unit,
the multiplicative nature of a ratio, or
meaningful interpretations of quotients.
Discuss what methods you are using to
answer the question. See if you can come up
with “What is Equal?”
43. A proportion is a mathematical statement of
equality between two ratios. Stated another
way, proportions tell us about the
equivalence of ratios.
Chapin, S & Johnson, A. (2006). Math Matters: Understanding the
Math You Teach, Grades 6 – 8, 2nd Ed. Math Solutions Publications,
Sausilito, CA.
44. A proportion includes
multiplicative relationships
Chapin, S & Johnson, A. (2006). Math Matters: Understanding
the Math You Teach, Grades 6 – 8, 2nd Ed. Math Solutions
Publications, Sausilito, CA.
45. Multiplying within each ratio. Notice that the
multiplier is the same in both ratios.
46. Multiplying across each ratio. Notice that
each multiplier is the same, once again.
47. Why is it important for us as teachers
to realize the two multiplicative
natures in a proportion?
What benefits might our knowledge
provide for our students as they
engage in solving problems involving
proportions?
48. Proportional reasoning describes the thinking
that has been applied to the solution of
problems that involve multiplicative
relationships.
Proportional reasoning requires examining
two quantities in relation to one another.
Chapin, S & Johnson, A. (2006). Math Matters: Understanding the Math
You Teach, Grades 6 – 8, 2nd Ed. Math Solutions Publications, Sausilito,
CA.
49. equivalent ratios can be created by iterating a
composed unit
Distance 8 ft 16 ft 24 ft
Time 4 sec 8 sec 12 sec
50. Equivalent ratios can be created by
partitioning a composed unit.
If we split the original 8 feet into 4 parts,
each 2 ft. part would get 1 second.
51. In short, we are reducing to a smaller unit
and iterating the smaller unit.
Distance 2 ft. 4 ft. 6 ft. 8 ft.
Time 1 sec. 2 sec. 3 sec. 4 sec.
52. if one quantity in a ratio is multiplied or
divided by a particular factor, then the other
quantity must be multiplied or divided by the
same factor to maintain the proportional
relationship
53. the two types of ratios – composed units and
multiplicative comparisons – are related.
54. “For every 4 batches of cookies, 6 eggs are
required. How many eggs are required for 14
batches of cookies.”
Can you reason through this problem using
some of the ideas we have discussed (e.g.
multiplicative nature of proportions,
partitioning, iteration, etc.).
How would you help a student reason
through this type of a problem?
56. Complete the handout “A Special Property of
Proportions.” Discuss your observations with
one or two other people in your group.
See if you can make a conjecture but more
importantly, justify it.
57. Two ratios that form a proportion, have equal
cross products.
59. Rewrite as
What can you say about ?
Think about what equal.
Notice the appearance of the cross-products
in the numerators?
60. You will be provided with some student work
(Canada et al, 2008). With one or two other
people, see if you can not only determine
whether the answer is correct but also see if
you can describe the types of thinking the
students were doing (e.g. cross products,
composed units, etc.). Which student(s)
method do you feel expresses a deep
conceptual understanding of proportional
reasoning and why?
61. A number of mathematical connections link
ratios and fractions.
A proportion is a relationship of equality between
two ratios. In a proportion, the ratio of two
quantities remains constant as the corresponding
values of the quantities change.
Several ways of reasoning, all grounded in sense
making, can be generalized into algorithms for
solving proportion problems.
62. Proportional reasoning is complex and involves
understanding that
Equivalent ratios can be created by iterating
and/or partitioning a composed unit
If one quantity in a ratio is multiplied or divided
by a particular factor, then the other quantity
must be multiplied or divided by the same factor
to maintain the proportional relationship
The two types of ratios – composed units and
multiplicative comparisons – are related
63.
64.
65. To isolate attributes of a situation needed for
ratio formation.
To compare and contrast rates and ratios.
To evaluate and reflect on a teacher’s
pedagogical strategies on a ratio and
proportion lesson as presented in a case
study.
To recognize that not all situations, despite
certain key words, will use direct proportional
reasoning.
66. Ratios can be meaningfully reinterpreted as
quotients.
Proportional reasoning is complex and
requires understanding of many important
ideas.
A proportion is a relationship of equality
between two ratios. In a proportion, the ratio
of two quantities remains constant as the
corresponding values of the quantities
change.
67.
68. Forming a ratio as a measure of a real-world
attribute involves isolating that attribute from
other attributes and understanding the effect
of changing each quantity on the attribute of
interest.
How did you coordinate the changing of the
base and height of the ramp to maintain the
ramp’s steepness?
69. The steepness of a ramp, sometimes referred
to as the slope or the grade, is the ratio of
the rise (vertical measure) to the run
(horizontal measure)
Steepness can also be viewed as a rate of
change…
For every y vertical units measured, you will
measure x horizontal units.
70. Take the cards with different definitions of the
term rate. A short one page reading will also be
handed out on ratios and rates.
What definitions from the cards and reading do
you feel will generate understanding of rate for
students?
Which definitions help you understand how a rate
and ratio are the same? Which definitions help
you understand how rates and ratios are
different?
71. When comparing two different types of
measures, the ratio is usually called a rate.
When a rate is simplified so that a quantity is
compared with 1, it is called a unit rate.
Some unit rates are constant while others vary.
Chapin, S & Johnson, A. (2006). Math Matters: Understanding the Math You Teach,
Grades 6 – 8, 2nd Ed. Math Solutions Publications, Sausilito, CA.
72. Express the following ratios as two different unit rates. Try
and use some of the methods discussed in the previous two
sessions.
120 miles in 2 hours
7 pizzas for 4 teenagers
$24.75 for 25 songs on iTunes
225 Euros for 300 U.S. Dollars
20 candies for $2.50
Chapin, S & Johnson, A. (2006). Math Matters: Understanding the Math You
Teach, Grades 6 – 8, 2nd Ed. Math Solutions Publications, Sausilito, CA.
73. 60 miles in 1 hour or 1 mile in of an hour.
pizzas for each teenager or 1 pizza for teenager.
$0.99 for 1 song or $1.00 for song.
0.75 Euros for 1 U.S. Dollar or 1 Euro for 1.33 U.S. Dollars.
1 candy for $0.125 or $1.00 for 8 candies.
60
1
7
3
1 7
4
01.1
74. Which ones make sense and why?
Which ones don’t make sense and
why?
75. As you read the case, you are encouraged to make a list of aspects of
Marie Hanson’s pedagogy that appear to support her students’ learning
throughout the lesson.
Think about some specific actions that Marie Hanson carries out (e.g.
eliciting incorrect additive solution first, returning to two students’ ratio
table later in the lesson to show multiplicative relationships).
What intuitive proportional strategies might students you work with use?
Might you encourage students to try other methods as discussed
throughout this workshop? What methods might you encourage them to
try and why?
Schwan-Smith, M., Silver, E.A., & Stein, M.K. (2005). Improving Instruction in Rational Numbers and
Proportionality: Using Cases to Transform Mathematics Teaching and Learning, Vol. 1. New York,
Teachers’ College Press.
76. Solve each of the problems on the next slide.
Think about how students might answer
some of these questions as you work through
these questions.
How as teachers can we help students
recognize that not everything is necessarily
proportional?
77. A train travels 160 miles in 3 hours. At the
same rate, how many miles does the train
travel in 12 hours?
Three people can paint the exterior of a
house in 8 days. At the same rate how many
days will it take for seven people to paint the
same house?
Three children weigh a total of 195 pounds.
How many pounds will 7 children weigh?
A student earned 19 out of 25 points on a
quiz. What percent did the student earn?
78. Make a table.
Actively model the situation.
Use common sense.
Ask questions
*What do we know?
*What else do we need to know?
79.
80. Reasoning with Ratios involves attending to
and coordinating two quantities.
A ratio is a multiplicative comparison of two
quantities, or it is a joining of two quantities
in a composed unit.
Ratios can be meaningfully reinterpreted as
quotients.
81. A number of mathematical connections link
ratios and fractions.
A proportion is a relationship of equality
between two ratios. In a proportion, the ratio
of two quantities remains constant as the
corresponding values of the quantities
change.
Several ways of reasoning, all grounded in
sense making, can be generalized into
algorithms for solving proportion problems.
82. Proportional reasoning is complex and involves
understanding that…
Equivalent ratios can be created by iterating
and/or partitioning a composed unit
If one quantity in a ratio is multiplied or divided
by a particular factor, then the other quantity
must be multiplied or divided by the same factor
to maintain the proportional relationship
The two types of ratios – composed units and
multiplicative comparisons – are related
83. Forming a ratio as a measure of a real-world
attribute involves isolating that attribute
from other attributes and understanding the
effect of changing each quantity on the
attribute of interest.
A rate is a set of infinitely many equivalent
ratios
Superficial cues present in the context of a
problem do not provide sufficient evidence
of proportional relationships between
quantities.
