Number Calculations
This problem gives you the chance to:
• explore the order of number operations
1.
When adding two numbers, it makes no difference to the answer if the order of the numbers is
changed. Write an example that shows this.
2. When subtracting two numbers, it does make a difference to the answer if the order of the
numbers is changed. Write an example that shows this.
Describe what happens to the answer of a subtraction calculation when the order of the two
numbers is changed.
3.
When multiplying two numbers, does the order of the numbers matter?
Use examples to explain your answer.
4.
When dividing two numbers, does the order of the numbers matter?
Use examples to explain your answer.
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Number Calculations Test 8
5. Complete the table below to show whether each statement about the numbers a and b is correct
( ) or incorrect (X).
 or X
Statement
a+ b= b+ a
a"b= b"a
!
a " b = "(b " a)
!
a"b= b"a
!
a÷b= b÷a
!
!
8
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Number Calculations Test 8
Task 3: Number Calculations
Rubric
The core elements of performance required by this task are:
• explore the order of number operations
points
Based on these, credit for specific aspects of performance should be assigned as follows
1.
Gives correct example such as: 2 + 3 = 5 and 3 + 2 = 5
1
2.
Gives correct example such as: 3 - 2 = 1 and 2 – 3 = -1
1
Makes statement such as: The answers are opposite
1
3.
Gives correct examples to show that the order does not matter
sectio
n
points
1
2
1
1
4.
Gives correct examples to show that the order does matter
1
5.
1
See table.
3
Five answers correct 3 points
Partial credit
Four answers correct 2 points
Three answers correct 1 point
(2)
(1)
Total Points
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3
8
Number Calculations Test 8
Number Calculations
Work the task. What are the big mathematical ideas being assessed?
What opportunities do you students have to make justifications?
How often do your students conduct investigations?
Look at student work for part 1. How many of your students could:
• Give complimentary examples with answers for addition: 3+4 = 7 and 4+3= 7?
• Give examples without a proof or quantification: 3+4 = 4+3?
• Give only half the example: 3+4 = 7?
• Give an incorrect example: 3 +4 = 7 and 3+ 5 = 8?
What do you think about as you look at these examples? What might you want to add to
your curriculum or teaching strategies? How might you pose questions differently during
class discussions to promote student development of the logic of justification?
Now look at student answers to part 2. How many of your students put:
• Both parts of the example and could calculate with a negative answer?
• How many could not subtract to get the negative answer?
• How many did not give answers for their equations?
• How many said that you can’t subtract a large number from a small number?
In part 4, how many of your students:
• Gave two correct and complete examples?
• Made division errors?
• Gave no examples just made a conjecture?
• Gave examples but did not quantify that they were different?
Compare your students’ work with the chart below:
Statement
a+b=b+a
a-b=b-a
a – b - -(b – a)
axb=bxa
a/b = b/a
th
8 grade – 2007
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Correct
Solution
√
X
√
√
X
Students who marked an
incorrect solution
% of total
population
1%
24.5%
71%
5%
15%
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Looking at Student Work on Number Calculations
Student A uses an algebraic format for thinking through a proof, putting two quantities
equal and then working down each column to find out if it is true or not true. Notice the
scorer did not understand this format in part 2. Like many students, B understands a
pattern of negative and positive but doesn’t assert that the numbers following the signs
are the same. (The distance between the numbers remains the same, or the direction
changes.)
Student A
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Student B is able to talk about the connection between distance on the number line and
subtraction. The student understands that the positive and negative of the same number
are opposites.
Student B
Student C needs to explain about larger and smaller numbers and what happens in each
case. The generalization about what is happening with reversing two numbers and their
answers seems to get lost in the details. How do we help students learn to build
generalizations? What are some activities in your classroom that have helped to build
generalizations?
Student C
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8 grade – 2007
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Student D is not reaching for the generalization, but is instead confirming that it is
possible to do the subtraction either way. Notice that the student does not recognize the
subtraction process in either of the two algebraic formats in the chart.
Student D
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Student E uses a number line to help with the calculations in part 2. The student makes
no attempt at the generalization, but just gives a second example in the second part of 2.
In part 4 the student makes a calculation error leading to an incorrect conclusion for order
and division.
Student E
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Student F does not recognize order in operations. The student doesn’t get a negative
answer in part 2 and gets the same answer for both calculations in part 4.
Student F
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Student G struggles with either the logic of an example or the calculations. In part 1 the
student doesn’t use 2 numbers, but uses the same number. In part 2 the student doesn’t or
can’t do the subtraction that would yield a negative answer, saying simply it makes it
wrong when the numbers are reversed. In part 5 the student writes the example and
makes the correct assertion, but doesn’t prove it by showing that the answers would be
different. How do we help students develop the logic of justification?
Student G
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Student H never shows answers to prove the assertions, except in part 3 where the
example uses 3 numbers instead of 2. While in part 2 there is some evidence that the
student knows the answers will be different, in part 4 the student seems to be saying its
not possible to divide a large number into a small number.
Student H
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What misconceptions do you see in the work of Student I?
Student I
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Student J does not understand that it takes to equations to test an assertion. I don’t think
there is evidence of understanding even in part 1. In part 2 the student doesn’t seem to be
able to think about the second equation and what would happen. The student can’t even
think through the multiplication example. In part 4 the student implies there is only way
one to put the numbers, when doing division.
Student J
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Student K does not seem to understand the entry level of the prompt in picking only one
number for the example in part 1 and the second part of two and giving no examples in
part 3 and 4. Notice that the answer for part 4 is incorrect with or without examples.
What might be your next steps with this student?
Student K
th
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8th Grade
Student Task
Core Idea 1
Number and
Operations
Core Idea 2
Mathematical
Reasoning
Task 3
Number Calculations
Explore the order of number operations using number and symbols.
Understand number systems, the meaning of operations, and ways
of representing numbers, relationships, and number systems.
Employ forms of mathematical reasoning and justification
appropriately to the solution of a problem.
• Formulate conjectures and test them for validity.
• Use mathematical language and representations to make
complex situations easier to understand.
Based on teacher observation, this is what eighth graders know and are able to do:
• Use numbers to test a statement
• Understand commutative properties of addition and multiplication.
• Using algebraic notation for addition and multiplication
Areas of difficulty for eighth graders:
• The logic of justification, understanding that the answers needed to be quantified
to show that they were the same or different
• Subtraction which yielded a negative answer
• Dividing a large number into a smaller number (calculation errors or
understanding that it was even possible)
th
8 grade – 2007
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The maximum score available for this task is 8 points.
The minimum score for a level 3 response, meeting standards, is 4 points.
Most students, about 90%, could use algebraic symbols for commutative property of
addition and multiplication and recognize that the symbols for division were incorrect.
Many students, almost 69%, could give examples for addition and for subtraction
quantifying the answers even when negative and give 3 or 4 solutions on the table.
Almost 50% of the students could meet all the requirements of the task except make the
subtraction generalization and mark the subtraction with parentheses correctly. 9% of the
students scored no points on this task. All of the students in the sample with this score
attempted the task.
th
8 grade – 2007
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Number Calculations
Points
Understandings
All the students in the sample
0
attempted the task.
1
Students could use algebraic
symbols for commutative
property of addition and
multiplication and recognize that
the symbols for division were
incorrect.
4
Students could recognize 3
correct responses in the table and
give examples to illustrate part 1
and 2, addition and subtraction.
6
Students could recognize 4
correct responses in the table and
give examples to illustrate part 1
and 2, addition and subtraction.
Students could also give
examples for multiplication and
division
Students could recognize 5
correct responses in the table and
give examples to illustrate part 1
and 2, addition and subtraction.
They could give examples about
reversing numbers in
multiplication and division and
decide if the answers were the
same. They could also make a
generalization about reversing
numbers in subtraction.
8
th
8 grade – 2007
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Misunderstandings
They couldn’t get at least 3 items on the
table. See chart for details on the questions
for Reflection page.
Students couldn’t give examples for
addition. More than 6% did not give
answers or quantify their answers. 7% only
gave half the example: 3 + 5 = 8, but not
5+3 =8. Students also couldn’t give
examples for order and subtraction. 9% did
not give both parts of the example. 7%
couldn’t do the subtraction when the
answer was negative. 7% didn’t give any
answers or quantify their examples. Almost
4% thought the answers were the same
when the subtraction numbers were
reversed.
7% did not attempt part 3 of the task. 7.5%
did not give examples to back up their
assertions in part 3. Almost 6% did not
give answers for their multiplication
examples. 6% did not give both parts of the
example.
In part 4,9% gave no examples. 8% gave
no answers. 4% only gave half of the
example. About 16% made calculation
errors. 4% gave examples but then didn’t
make a claim about whether or not you
could reverse numbers in division.
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Implications for Instruction
Students need opportunities to make and test conjectures. Many students did not
understand how to give examples to test if order made a difference in the answers. For
example, to see if order made a difference in addition they may have given a problem like
2+2 =4, using the same number for both addends. T hey often didn’t test enough
examples. Students may have given only one part of the example, e.g. 3 + 2 =5; but not
tested to see if the reverse is also true. Students learn to understand the logic of making
convincing arguments by being given rich problems and by listening to others and having
opportunities to discuss what makes an argument sound. Students at this level had
difficulty with computations resulting in negative answers, e.g. 3 – 8 = -5. Some students
also struggled with some division computation, e.g. 2 divided by 5 = 0.4.
Ideas for Action Research
Developing a Convincing Argument – Re-engagement
One useful strategy when student work does meet your expectations is to use student
work to promote deeper thinking about the mathematical issues in the task. In planning
for re-engagement it is important to think about what is the story of the task, what are the
common errors and what are the mathematical ideas I want students to think about more
deeply. Then look through student work to pick key pieces of student work to use to pose
questions for class discussion. Often students will need to have time to rework part of the
task or engage in a pair/share discussion before they are ready to discuss the issue with
the whole class. This reworking of the mathematics with a new eye or new perspective is
the key to this strategy.
The focus of this task was about picking examples to test a conjecture. So what does it
mean to test a conjecture? Is it important to quantify? Why or why not? Does the
example fit the constraints?
Take some of the examples from student work to pose a further problem for the class
discussion. For example:
For part 2, Jon says, “It doesn’t matter. They come out the same. 5-4 = 4-5”
Do you think this is a convincing argument? Why or why not?
Do students bring up the idea that you need to quantify the answers to show that they are
the same or not the same? What do they think is needed for a convincing argument?
For part 4 Sandra says, “It matters because 35/7=5 is okay, but 35/7 does not come out
the same.” What do you think she means? Does order matter? Convince me!
Do students bring up the idea that you can’t divide by a bigger number or the second
division is impossible? What calculations can they make to check?
Now see if they can use this logic of checking that both parts of the example need to give
the same answer to the problem with parentheses in the table.
th
8 grade – 2007
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