Transcript Document

The call to action:
Understanding addition and
subtraction
• There is a lot of important work to be done to
ensure that students understand addition and
subtraction in 1st and 2nd grade.
• The Call to Action is looking for units that connect
understanding addition and subtraction to length.
Opportunity for coherence
Measurement
Operations and
Algebraic Thinking
Connecting the two domains together enhances
students’ understanding of both.
• Linear measurement work in grade 1 and 2 asks
students to consider the quantity of units required
to represent length
• Students combine and compare lengths to deepen
the understanding of the meaning of addition and
subtraction
From the Draft K–5 Progression on
Measurement and Data (measurement part)
• Length and unit iteration are critical in understanding
and using the number line in Grade 3 and beyond.
• Length is… one of the most prevalent metaphors for
quantity and number, e.g., as the master metaphor
for magnitude (e.g., vectors, see the Number and
Quantity Progression)
https://commoncoretools.files.wordpress.com/2012/07/ccss_progression_gm_k5_2012_07_21.pdf
From the Draft K–5 Progression on
Measurement and Data (measurement part)
• To use a number line diagram to represent whole numbers
as lengths students need to understand that number lines
have specific conventions
• the use of a single position to represent a whole number and the
use of marks to indicate those positions.
• a number line diagram is like a ruler in that consecutive whole
numbers are 1 unit apart, thus they need to consider the distances
between positions and segments when identifying missing numbers
• Students think of a number line diagram as a measurement
model and use strategies relating to distance, proximity of
numbers, and reference points.
By connecting
measurement to addition and subtraction, students strengthen their
https://commoncoretools.files.wordpress.com/2012/07/ccss_progression_gm_k5_2012
_07_21.pdf of a number line as iterations of equal size and of measurement as it relates
understanding
to a quantity of units.
st
1
grade standards
Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center
for Best Practices and the Council of Chief State School Officers.
Do the math:
This task illustrates how using length units in addition and subtraction word
problems connect students understanding of iterating units into work with
operations. The context of the problem encourages students to use the
number line, reinforcing the understanding of representing equal length,
non-overlapping units.
https://www.illustrativemathematics.org/content-standards/tasks/196
nd
2
grade standards
Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center
for Best Practices and the Council of Chief State School Officers.
This task illustrates the connection between addition and subtraction and
moving equal units on the number line. The numbers students are working
with are kept small to allow for focus on the connection between the
operation and units of length.
https://www.illustrativemathematics.org/content-standards/tasks/1081
Think about this…
• We use the phrase “out of proportion” in all sorts
of situations.
Like this:
Or this:
• 15 years in prison for stealing a gumball.
• Something in each was out of proportion.
• What do we mean by that?
• We should be able to describe each situation in the
same mathematically precise way.
• Hint: We need to focus on two things
simultaneously, not one thing.
• Proportionality is based on a relationship between
two quantities.
The Relevant Standards:
Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center
for Best Practices and the Council of Chief State School Officers.
Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center
for Best Practices and the Council of Chief State School Officers.
Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center
for Best Practices and the Council of Chief State School Officers.
Important elements:
From the Progressions
• Ratios arise in situations in which two (or more)
quantities are related
• In the Standards, a quantity involves measurement
of an attribute
http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf
From the Progressions
• Some authors distinguish ratios from rates, using
the term “ratio” when units are the same and
“rate” when units are different; others use ratio to
encompass both kinds of situations.
• The Standards use ratio in the second sense,
applying it to situations in which units are the same
as well as to situations in which units are different.
http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf
More from the Progressions
3
2
• The quotient is sometimes called the value of the
ratio 3 : 2.
• In everyday language. the word “ratio” sometimes
refers to the value of a ratio
• Ratios have associated rates.
• The unit rate is the numerical part of the rate;
• Equivalent ratios arise by multiplying each
measurement in a ratio pair by the same positive
number.
http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf
More from the Progressions
• Proportional relationships involve collections of
pairs of measurements in equivalent ratios.
• ratio notation should be distinct from fraction
notation
• A collection of equivalent ratios can be graphed in
the coordinate plane. The graph represents a
proportional relationship.
• The unit rate appears in the equation and graph as
the slope of the line, and in the coordinate pair
with first coordinate 1.
http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf
Grade 6 (From the Progressions)
http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf
More from the Progressions
• As students generate equivalent ratios and record
them in tables, their attention should be drawn to
the important role of multiplication and division in
how entries are related to each other
• In other words, when the elapsed time is divided
by 2, the distance traveled should also be divided
by 2. More generally, if the elapsed time is
multiplied (or divided) by N, the distance traveled
should also be multiplied (or divided) by N
http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf
More from the Progressions
• As students become comfortable with fractional
and decimal entries in tables of quantities in
equivalent ratios, they should learn to appreciate
that unit rates are especially useful for finding
entries
http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf
Sidetrack:
• Sue and Julie were running equally fast around a
track. Sue started first. When she had run 9 laps,
Julie had run 3 laps. When Julie completed 15 laps,
how many laps had Sue run?
Cramer, K., Post, T., & Currier, S. (1993). Learning and Teaching Ratio and Proportion: Research Implications. In D. Owens (Ed.),
Research Ideas For the Classroom (pp. 159-178) NY: Macmillan Publishing Company. Downloaded 6/9/2011 from
http://www.cehd.umn.edu/rationalnumberproject/93_4.html
Grade 7 (Progressions)
• They work with equations in two variables to
represent and analyze proportional relationships.
• Students recognize that graphs that are not lines
through the origin and tables in which there is not
a constant ratio in the entries do not represent
proportional relationships.
• they write equations of the form y = cx, where c is
a constant of proportionality
• unit rate as the amount of increase in y as x
increases by 1 unit in a ratio table [slope triangle]
http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf
http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf
From the EQuIP Call to Action
Sample tasks:
https://www.illustrativemathematics.org/content-standards/6/RP/A/tasks/496
https://www.illustrativemathematics.org/content-standards/6/RP/A/tasks/496
https://www.illustrativemathematics.org/content-standards/6/RP/A/tasks/496
https://www.illustrativemathematics.org/content-standards/6/RP/A/3/tasks/1982
https://www.illustrativemathematics.org/content-standards/6/RP/A/3/tasks/1982
https://www.illustrativemathematics.org/content-standards/7/RP/A/2/tasks/181
• Task
A text book has the following definition for two quantities to be directly
proportional:
We say that y is directly proportional to x if y=kx for some constant k .
For homework, students were asked to restate the definition in their own
words and to give an example for the concept. Below are some of their
answers. Discuss each statement and example. Translate the statements and
examples into equations to help you decide if they are correct.
https://www.illustrativemathematics.org/content-standards/7/RP/A/2/tasks/1527
•Marcus:
This means that both quantities are the same. When one increases
the other increases by the same amount. An example of this would
be the amount of air in a balloon and the volume of a balloon.
•Sadie:
Two quantities are proportional if one change is accompanied by a
change in the other. For example the radius of a circle is
proportional to the area.
•Ben:
When two quantities are directly proportional it means that if one
quantity goes up by a certain percentage, the other quantity goes
up by the same percentage as well. An example could be as gas
prices go up in cost, food prices go up in cost.
• Jessica:
When two quantities are proportional, it means that as one
quantity increases the other will also increase and the ratio of the
quantities is the same for all values. An example could be the
circumference of a circle and its diameter, the ratio of the values
would equal π.
https://www.illustrativemathematics.org/content-standards/7/RP/A/2/tasks/1527