Household Statistics
This problem gives you the chance to:
• interpret a block graph and statistics in a real life context
This graph shows the number of children per household in a survey of 20 families.
9
8
7
6
Number of
households
5
4
3
2
1
0
1
2
3
4
Number of children per household
5
1. Use the graph to complete this table.
Number of children per household
0
1
2
3
4
5
Number of households
2. a. How many households have three or more children?
___________
b. What percentage of households has two children?
Show your calculation.
___________
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Building Blocks Test 6
3. For each of these descriptions, choose the calculation below that matches it.
Write the letter of the calculation you choose in the table.
Description
Letter
Total number of households
Total number of children
Mean number of children per household
Calculations
A
0 + 1+ 2 + 3 + 4 + 5
B
0 + 1+ 2 + 3 + 4 + 5
6
C
0 "1+ 1" 5 + 2 " 8 + 3 " 4 + 4 "1+ 5 "1
D
1+ 5 + 8 + 4 + 1+ 1
E
1+ 5 + 8 + 4 + 1+ 1
20
F
0 "1+ 1" 5 + 2 " 8 + 3 " 4 + 4 "1+ 5 "1
20
!
!
!
!
!
!
8
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Building Blocks Test 6
Task 4: Household Statistics
Rubric
The core elements of performance required by this task are:
• interpret a block graph and statistics in a real life context
points
section
points
Based on these, credit for specific aspects of performance should be assigned as follows
1.
Gives correct answers:
2
Partial credit
(1)
5, 4, 3 correct 1 point
2.
Gives correct answers: 6
1
40%
1
Shows 8/20
3.
2
1
Gives correct answers: D, C, F
3
3x1
3
Total Points
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8
Building Blocks Test 6
Household Statistics
Work the task and look at the rubric. What are some of the big mathematical ideas a
student needs to be successful on this problem?
Most students could fill in the table successfully.
Look at student work for 2a. How many of your students put:
6
3
4
2
Other
How do you think the students may have gotten their incorrect answers? What might
they have been thinking?
Now look at student work for part 2b. Did students understand the number of households
or did they try to calculate a percentage without using this number? What problems did
you see in students ability to find percentages?
For 2b, how many of your students put:
40%
25%
0%
80%
8%
4%
Other
Can you figure out their misconceptions?
Now look at their work for part 3, how many of your students put:
Description
Correct
Error
Error
Error
Answer
Total
D
A
E
F
households
Other
Other
C
A
E
D
Other
F
E
B
A
Other
Total Children
Mean Children
per household
What do students need to understand about tables and graphs to find the number of
households and the number of children? How many students were adding categories
instead of adding frequencies? How often are students pushed to use graphs as a starting
place for making calculations? Do they understand that the graph is a tool for making
sense of information?
.6th grade – 2007
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Looking at Student Work on Household Statistics
Student A is able to find the total number of households and use that to compute the
percentage of households with two children. The student sets up a fraction and then finds
an equivalent fraction with a denominator of 100.
Student A
Student B finds the percentage by dividing the denominator into the numerator to find a
decimal fraction and then converting the decimal to a percentage. Notice that on the
second page the student makes several calculations on his own to help make sense of
what is going on for himself before choosing an answer. The use of labels for different
parts helps with that sense-making piece.
.6th grade – 2007
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Student B
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Student C makes a common error of thinking about 3 households having 3 or more
children. Can you look at the table to determine what the student was thinking? How
could you pose a question to help the class discuss the logic of this answer and see where
the error in understanding is?
Student C
Student D makes another common error in finding the percentage of households with two
children. Can you see the logic behind this misconception?
Student D
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Many students were able to identify the correct numbers for calculating the percentage of
households with two children, but were unclear about the process. There is some
evidence of partially learned procedural knowledge, without the understanding to carry it
out correctly. Student E reverses the order of numbers in the division. Student F
multiplies instead of dividing. Student G does no calculations. Can you find a logic for
the work of Student H?
Student E
Student F
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Student G
Student H
Looking at work on part 3 of the task.
For finding the total number of households:
• 15% of the students chose letter A, adding the categories on the horizontal axis
rather than looking at the frequency that categories appeared.
• Almost 11% chose letter E, which is includes a division step; so not even a
straight addition number sentence for a total.
For finding the total number of children:
• 17.5% of the students chose letter A, adding the categories on the horizontal axis
rather than looking at the frequency that categories appeared.
• 12% of the students chose letter D, which is the total number of households not
total children.
• 12% of the students chose letter E, which is includes a division step; so not even a
straight addition number sentence for a total.
• 9% of the students chose letter B, which is includes a division step; so not even a
straight addition number sentence for a total. This is an average of categories
rather than an average of data.
.6th grade – 2007
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For finding the mean number of children per household:
• 27% of the students chose letter B, which is an average of the categories for
number of children per household.
• 18% of the students chose letter E, which is an expression equal to 1. It
represents the number of households divided by the number of households.
6th Grade
Student Task
Core Idea 5
Core Idea 3
Algebra and
Functions
Task 4
Household Statistics
Interpret a bar graph and use statistics in a real life context. Reason
about how to model calculations from a bar graph.
Select and use appropriate statistical methods to display, analyze,
compare and interpret different data sets.
Understand relations and functions, analyze mathematical
situations, and use models to solve problems involving quantify and
change.
• Model and solve contextualized problems using various
representations, such as graphs, tables, and equations.
Based on teacher observation, this is what sixth graders know and are able to do:
• Read data on a graph to fill in data on a table.
• Identify the calculation needed to find the total number of households.
Areas of difficulty for sixth graders:
• Distinguishing numbers as categories or scale and numbers representing
frequency
• Calculating percentages
• Calculating total number of children (calculations involving combining
categorical information with frequency of occurrence)
• Calculating a mean from a graph
.6th grade – 2007
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The maximum score available on this task is 8 points.
The minimum score for a level three response, meeting standards, is 4 points.
Most students 91% could read data from the graph and use it to fill out a table. Many students
69% could fill out the table and identify the calculation for finding the total number of
households. Almost half the students could also find the number of households with 3 or more
children. Almost 10% of the students could meet all the demands of the task including find the
percentage of households with 2 children and identifying the calculations for finding total number
of children and mean number of children. Almost 6% of the students scored no points on this
task. 80% of the students with this score attempted the task.
.6th grade – 2007
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Household Statistics
Points
0
2
3
4
6
8
Understandings
Misunderstandings
80% of the students with this
score attempted the task.
Students could read information
on a graph and use it to fill out a
table.
Students could fill out a table
from a graph and find the total
number of households
represented by the data.
They had difficulty reading information on
the graph and using it to fill out the table.
Students could not identify the correct
calculation for finding the total number of
households.
They couldn’t identify the number of
households with 3 or more children. 40%
of the students that the answer was 3,
because 3 of the bars go higher than 3.
They did not think about what was
represented by the bars. 5% of the students
thought there were 4 households. These
students did not think about the “or more”.
Students can’t identify the calculations for
total number of children and mean number
of children. They often confuse categorical
and frequency information.
Students could fill out a table,
find the number of households
with 3 or more children, and
identify the calculation for total
number of households.
Students can fill in a table, find
the number of households with
3 or more children, and identify
correct calculations.
Students had difficulty calculating the
percentage of households with 2 children.
Almost 10% of the students thought the
answer was 0%. 4% thought the answer
was 8%, just put a percent sign behind the
data point with no calculation. Other
common errors were 25%, 80%, and 4%.
Students could read information
on a graph and use it to fill out a
table. They could reason about
the graph to distinguish
categories in the form of
number from frequency (height
of bars). Using this information
that could make calculations to
find percentages, totals, and
means to help make sense of the
data.
.6th grade – 2007
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Implications for Instruction
Students need practice reading a graph or table and using the information to calculate
totals and average. This graph had numerical information on both axes, number of
households and number of children. Some students confused the total number of
households with the highest number on the scale or added the numbers on the scale,
rather than adding the heights of the bars. Students had difficulty determining the total
number of children; they forgot to multiply the number of households by the number of
children per household to find the total number of children. Students at this grade level
need to use graphs as starting points for finding information. They have been reading
points on graphs for many years and now need to be working with more complex ideas.
What can I find out with this information? What does it mean? What is the graph telling
me? How can I use this information to think about implications? If the mean were raised
and the number of households stayed the same how might the new graph be different?
(For related task see: 2001 Grade 5 – Washington Street or 2005 7th Ducklings)
Ideas for Action Research - The Role of Context – Investigating
Different Representations
Try planning a lesson to help students compare and contrast various representations for
data. Start with a simple mind set by giving students a set of numbers and asking them to
find mean, median, and mode. This checks that everyone has a basic understanding of
the procedures for calculating these measures.
Now have them work the tasks: 2007 6th grade Household Statistics and 7th grade Suzi’s
Company. For each task just give students just the table or the graph and ask them to
again find the mean, median, and mode. Graph paper should be available for students.
Now we want to explore their thinking about information in these two representations.
Start with Household Statistics. Pose a question for class discussion, such as:
Lettie says,” I think the equation would for mean would be”:
0 + 1+ 2 + 3 + 4 + 5
6
!
!
Her partner, Nadia disagrees. Nadia thinks the solutions is:
1+ 5 + 8 + 4 + 1+ 1
20
Mary says I think neither of these is correct. I think we’re forgetting something.
Can you help them solve this? Give reasons for your answers.
During the discussion probe student thinking to explore why Lettie and Nadia are wrong.
If students seem stuck ask them if they can write out the string of data numbers being
represented by the graph. See if they can start to talk about frequency versus data. Have
them talk about how to use the graph for finding mode and median. Try to have a student
come to the board to show how he or she counted to find the median.
.6th grade – 2007
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When they are finished, have them look at the table for Suzi’s company. Ask them how
they might put the information about Household Statistics into a table. What would that
look like? Where are the data points? Where is the frequency in their tables?
Now pose a question about Suzi’s Company. For example:
Lydia says the mean is $30,000. Bruce says that the mean is $97,142. How can their
answers be so far apart? What do you think they are doing? They both started with totals
of $680,000. Who do you think is right? Convince me.
See if students relate this information to the ideas that came up in the discussion for
Household Statistics. Are they mentioning the difference between categories and total
number of households?
Next you might pose a question, such as that on part 2 of the original task. John looks at
the table and says, “The mode of the salary is eighty thousand dollars a year. What
mistake has John made?”
When the class discussion is over, maybe even a day or two later, give students red pens
and asked them to revise their work and write about the ideas they have learned. Why
did they choose to change their answers based on new ideas or ways of thinking from the
classroom discussion. What are things you have to consider when looking at a table or
graph that is different from looking at a list of data?
.6th grade – 2007
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