Intervention Support
• Developing understanding and long-term
retention requires conceptual teaching
• Developing fluency requires practice
– including corrective feedback and frequent
cumulative review
Proportional Reasoning
Ratios and Proportions
• Understanding ratios and using proportional
thinking is the most important set of math
concepts we teach in middle school
• Ratios grow out of fractions and lead into
linear functions
• Proportional thinking is used all the time in
everyday life – unit prices, miles per gallon,
converting measurement units, etc. etc. etc.
Ratios
At the school dance, there were 4 boys for every
3 girls.
Draw a picture of what 4 to 3 looks like to you.
If there were 133 students at the dance, how
many were boys and how many were girls?
How might students solve this?
One important meaning of fraction
• In a bag of M&Ms that contains 40 candies, 8 are
red. Every bag, no matter how large, has this same
ratio of red candies. What fraction of candies are
red?
• How many red candies would you expect to find in a
bag of 25 M&Ms?
• If you pulled out 5 M&Ms from any bag, should you
expect to get one red candy? Why?
See Recommendations for Effective Fraction Instruction for K-8
Multiplicative thinking
5:1
10:2
15:3
20:4
25:5
40:8
NOT
5:1
6:2
7:3
8:4
9:5
_:8
For every 5 M&Ms, there is 1 red M&M.
You see the connection to slope, right?
Constant of proportionality
In a bag of M&Ms that contains 40 candies, 8 are red. Every bag, no matter how
large, has this same ratio of red candies. What fraction of candies are red?
• Let y = the total number of candies
• Let x = the number of red candies
What equation relates y and x?
(This is another way of showing the multiplicative relationship.)
𝑦
5
= 𝑥 or 𝑦 = 5𝑥
You see the connection to slope, right?
A large bottle of shampoo contains 40 fluid
ounces. The bottle recommends that you use 1¼
fluid ounces per application. How many
shampoo washes can you do with this bottle?
This is a proportion problem where the constant
of proportionality is 1¼:1. What various ways
might you represent this?
5
𝑦= 𝑥
4
How do you read this equation, in the context of the problem?
5 Ways to Solve Proportion Problems
1.
2.
3.
4.
5.
Find a unit rate
Use a table
Use a tape diagram
Set up and solve a proportion equation
Make a graph
Solve these problems using each
of the strategies.
• If 3 pounds of hamburger feeds 12
people, how many pounds are
needed for 30 people?
• A restaurant recipe calls for 10 cups
of flour to make 4 cakes, but you
want to make only 1 cake. How
much flour do you need?
• If we can drive 150 miles in 3 hours,
how many hours will it take us to
drive 400 miles?
1. Find a unit rate
2. Use a table
3. Use a tape
diagram
4. Set up and solve
a proportion
equation
5. Make a graph
• If 6 cats can catch 6 mice in 6 minutes, how
many cats does it take to kill 100 mice in 50
minutes?
• Think about an army of 6 cats. They can do
their job at the rate of 6 mice in 6 minutes.
What’s the unit rate?
• If you want to kill 100 mice in 50 minutes,
what’s that unit rate? So how big does your
army of cats need to be?
Data and Statistics
Categorical Data
Two friends want to start a restaurant, but they don’t know
what kind. Should it be a breakfast café, a lunch and dinner
grill, or a fine dining restaurant open after 4 pm. To decide,
they need to do some research, so they’ve come to our
class to ask us: What meal of the day is your favorite?
Breakfast
Lunch
Dinner
Make a bar graph and circle graph to show the data.
(Bar graphs are learned in 1st-3rd grade. Making circle graphs require knowing
how to find and draw the angles.)
6.SP.1 Recognize a statistical question as one that
anticipates variability in the data related to the
question and accounts for it in the answers. For
example, “How old am I?” is not a statistical question,
but “How old are the students in my school?” is a
statistical question because one anticipates variability
in students’ ages.
Statistical Questions
• What’s the average age of middle school
teachers who attend professional
development workshops?
• How large are extended families in MidMichigan, as indicated by the number of
cousins for any family member?
• What is the distribution of birthdates through
the year?
Process of Statistical Analysis
1.
2.
3.
4.
Formulate the question
Collect the data
Represent and analyze the data
Interpret and present our results
Numerical Data
18, 18, 20, 22, 22, 22, 22, 22, 23, 25, 25, 25, 25,
25, 26, 26, 27, 27, 30, 30, 32, 32, 37
6.SP.4 Display numerical data in plots on a number line, including dot plots,
histograms, and box plots.
Create your own number lines: http://themathworksheetsite.com/numline.html
http://www.shodor.org/interactivate/activities/PlopIt/
Dot Plot Using Squares
6.SP.2 Understand that a set of data collected to answer a statistical question has a
distribution which can be described by its center, spread, and overall shape.
Histogram
http://www.shodor.org/interactivate/activities/Histogram/
Cluster the data into bands
17-20 3
21-23 6
24-26 7
27-29 2
30-32 4
33-35 0
36-38 1
18, 18, 20, 22, 22, 22, 22, 22, 23, 25, 25, 25, 25, 25, 26, 26, 27,
27, 30, 30, 32, 32, 37
6.SP.5 Summarize
numerical data sets in
relation to their context,
such as:
Box Plot
a. Reporting the number of
observations.
c. Giving quantitative
measures of center
(median and/or mean) and
variability (interquartile
range and/or mean
absolute deviation), as
well as describing any
overall pattern and any
striking deviations from
the overall pattern with
reference to the context in
which the data was
gathered.
An outlier is often defined as any data that is more or less than 1.5 IQR above 3rd
quartile or below 1st quartile. In this case, IQR = 5, 1.5*5=7.5, 27+7.5 = 34.5
http://www.shodor.org/interactivate/activities/BoxPlot/
http://www.alcula.com/calculators/statistics/box-plot/
Working with Our Data
Summarize and describe the distribution, following
6.SP.4 and 6.SP.5.
Decide which display is most appropriate for your
data set.
Find the most appropriate measure of center and
give a quantitative measure of variability.
Describe any overall patterns, etc.
Comparing Samples
How do these two
samples compare?
7.SP.3 Informally assess the degree of visual overlap of two
numerical data distributions with similar variabilities,
measuring the difference between the centers by
expressing it as a multiple of a measure of variability.
For example, the mean height of players on the basketball
team is 10 cm greater than the mean height of players on
the soccer team, about twice the variability (mean absolute
deviation) on either team; on a dot plot, the separation
between the two distributions of heights is noticeable.
Mean = 25.26
18
18
20
22
22
22
22
22
23
25
25
25
25
25
26
26
27
27
30
30
32
32
37
7.26
7.26
5.26
3.26
3.26
3.26
3.26
3.26
2.26
0.26
0.26
0.26
0.26
0.26
0.74
0.74
1.74
1.74
4.74
4.74
6.74
6.74
11.74
79.3
abs difference between value and mean
18 − 25.26
sum of deviations from the mean
3.45
mean of deviations (MAD)
Comparing Samples
Difference of medians: 6
MAD of Adults: 3.45
Difference of medians is
almost twice the variability.
7.SP.3 Informally assess the degree of visual overlap of two numerical data distributions
with similar variabilities, measuring the difference between the centers by expressing it
as a multiple of a measure of variability.
http://www.illustrativemathematics.org/
College Athletes PDF is on our wiki
Bivariate Data
Size of
company in #
of employees
10
250
65
140
25
375
200
500
120
300
50
Annual salary
in $
Annual salary in $
30,000
20,000
22,500
21,400
22,100
20,900
24,500
19,800
28,100
21,500
24,900
22,000
25,000
20,000
15,000
Annual salary in $
10,000
5,000
0
0
100
200
300
400
500
600
8.SP.1 Construct and interpret scatter plots for bivariate measurement data to investigate
patterns of association between two quantities. Describe patterns such as clustering,
outliers, positive or negative association, linear association, and nonlinear association.
Annual salary in $
8.SP.2 Know that straight lines are
widely used to model relationships
between two quantitative variables.
For scatter plots that suggest a linear
association, informally fit a straight
line, and informally assess the model
fit by judging the closeness of the data
points to the line.
8.SP.3 Use the equation of a linear
model to solve problems in the
context of bivariate measurement
data, interpreting the slope and
intercept.
30,000
25,000
20,000
Annual salary in $
15,000
Linear (Annual salary
in $)
10,000
5,000
0
0
200
400
600
Animal body weight vs. brain size
From Illustrative Mathematics
Bivariate Categorical Data
Chores
Curfew
No
curfew
11
0.55
3
0.25
14
0.44
No
chores
9
0.45
9
0.75
18
0.56
20
1.00
12
1.00
32
1.00
8.SP.4 Understand that patterns of association can also be seen in bivariate categorical
data by displaying frequencies and relative frequencies in a two-way table. Construct
and interpret a two-way table summarizing data on two categorical variables collected
from the same subjects. Use relative frequencies calculated for rows or columns to
describe possible association between the two variables. For example, collect data
from students in your class on whether or not they have a curfew on school nights and
whether or not they have assigned chores at home. Is there evidence that those who
have a curfew also tend to have chores?
Probability
7.SP.6-7
• Roll a number cube 60 times and count the number
of times it comes up either 1 or 6. Compare your
data with others at the table. If you rolled a number
cube 1000 times, how many times would you expect
it to come up either 1 or 6?
• How often would it come up exactly that number of
times?
• Plot the number of times it came up either 1 or 6 for
each group in the room on a line plot.
7.SP.8
7.SP.8 Find probabilities of
compound events using organized
lists, tables, tree diagrams, and
simulation.
a. Understand that, just as with
simple events, the probability of a
compound event is the fraction of
outcomes in the sample space for
which the compound event
occurs.
b. Represent sample spaces for
compound events using methods
such as organized lists, tables and
tree diagrams. For an event
described in everyday language
(e.g., “rolling double sixes”),
identify the outcomes in the
sample space which compose the
event.
If the probability that it will rain on Saturday is 50% and
the probability that it will rain on Sunday is 50%, what’s
the probability that it will rain sometime during the
weekend?
Can you use an organized list, table, or tree diagram?
What’s a simulation that can answer this question,
using a coin flip?
Core Math Tools
North Carolina Unpacked CCSS