Sloping Squares
This problem gives you the chance to:
• calculate areas in composite figures formed of squares and rectangles
• use and manipulate algebraic expressions in this situation
The diagram shows some ‘sloping squares’ drawn on a grid.
Sloping Square A is split into four equal right triangles and a central square.
The square in the center has area 9 squares and each of the four triangles has area 2 squares. This
shows that the total area is equal to 17 grid squares.
A
B
9 squares
2 squares
2 squares
2 squares
2 squares
C
D
1. a. Show that Sloping Square B has a total area of 20 grid squares.
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Sloping Squares Test 10
b. Find the area of Sloping Square C.
_______________
c. Find the area of Sloping Square D. ________________
2. The diagram below shows a sloping square split into four ‘a by b’ right triangles and a central
square.
b
a
a-b
a-b
a. Show that the total area of the four right triangles is 2ab.
b. Use the diagram to show why the sides of the square are a – b units long.
c. Find the area of the central square. Write your expression without parentheses.
d. Show that the total area of the sloping square (four triangles and the central square) is a2 + b2.
e. Show that this formula for the total area of a sloping square gives the correct answer of 17 grid
squares for Sloping Square A.
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Sloping Squares Test 10
Sloping Squares
Rubric
The core elements of performance required by this task are:
• calculate areas in composite figures formed of squares and rectangle
• use and manipulate algebraic expressions in this situation
points
section
points
Based on these, credit for specific aspects of performance should be assigned as follows
1.a
Shows correct work such as: (Possibly shown on diagram)
Four triangles of 4 squares each
Central square 4 squares
Total 20 squares
1
b Gives correct answer: 13 squares
1
c
1
Gives correct answer: 25 squares
3
1
ab " 4 = 2ab
2
b Correct indication on diagram or other explanation
1
c
2
2.a
Shows correct work such as:
1
2
Shows correct work!such as: ( a " b) = a 2 " 2ab + b2
d Shows correct work such as: a 2 " 2ab + b2 + 2ab = a 2 + b2
e
1
Shows correct work such as:
!
a = 4, b = 1
!
a 2 + b2 = 16 + 1 = 17
1
6
Total Points
9
!
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Sloping Squares Test 10
Geometry – Task 2: Sloping Squares
Work the task. Examine the rubric.
What are geometrical concepts students needed to decompose the diagrams and make sense of this
situation? What are the algebraic skills students needed to have to solve part two of the
task?__________________________________________________________
Look at student work for part1b. How many of your students put:
13
10
11
17
21
Other
What do you think caused these errors? What are students confused about?
Look at student work for part 1c. How many of your students put:
25
21
22
23
Less than 20
More than
30
What do you think caused these errors? What are students confused about?
Now look at work for part 2. Did students seem reluctant to attempt the algebra in this section? Why
do you think this is true?
In what ways do you integrate algebra skills and proofs into your geometry course?
After considering this data, what are some implications for instruction?
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Could your students identify the appropriate symbolic values to find the area of the triangle?
Could your students use the diagram properly to make sense of the side of the square?
• How many tried to use a verbal description rather than mark up the diagram?
• How many used the diagram incorrectly?
• How many reversed a and b on the diagram?
Make a list of errors students made for multiplying (a-b)(a-b):
How would you categorize or group these types of errors?
Half the students did not attempt part 2e of the task.
Those who did gave answers like 9 + 6= 17 and 8 +9= 17.
What are the misconceptions behind these two error types?
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Looking at Student Work on Sloping Squares
Student A has a very clear diagram to show how to find the area of Square B. Notice that the student
drew in the triangles to help ease the calculations for finding the area of Square D. In part 2, Student A
defines the variables and explains the calculations. The algebra has a clear connection to the context
of the problem.
Student A
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Student A, part 2
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Student B tries to use a “matching parts” strategy to find the area of the squares, which leads to errors
in part 1c. In part 2, the student forgets that a negative times a negative is positive.
Student B
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Student C does not draw in the triangles to help find the area of Square D and tries to just count
squares. In part 2 the student attempts to use the wrong diagram to show why the length of the square
is (a-b). The student also confuses the length of the side with area for the center square.
Student C
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Many students struggled with the algebra in part 2 of this task. Student D interprets the diagrams
incorrectly in part 2b, confusing the hypotenuse with the longer leg for the triangle. The student does
not understand (a-b)(a-b) and adds the terms instead of multiplying in 2c. Again notice the incorrect
combining of terms in 2d. How can you incorporate use of algebra into your geometry course to help
keep these skills fresh?
Student B
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Student E also struggles with the algebra. For (a-b)2 the student puts a-b2. The student might be
writing what he says to himself when thinking about the formula or might not understand how
“without parentheses” effects the mathematics. Even allowing for follow through, the student
incorrectly combines terms in 2d. In part 2e the student just repeats the original computations for the
sum of all the parts to find the area of A rather than using the formula given in 2d.
Student E
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Geometry
Student Task
Task 2
Sloping Squares
Calculate areas in composite figures formed by squares and
rectangles. Use and manipulate algebraic expressions in this
situation.
Represent and analyze mathematical situations and structures
using algebraic symbols.
Core Idea 3
Algebraic
Properties and
Representations.
Core Idea 4
Apply appropriate techniques, tools, and formulas to determine
Geometry and
measurements.
Measurement
Mathematics in this task:
• Compose and decompose shapes
• Use formulas to find area of squares and triangles, both with values and algebraically
• Multiply binomials
• Simplify algebraic expressions
• Use substitution and solve an expression
Based on teacher observations, this is what geometry students knew and were able to do:
• Find the numeric values for area of a square and a triangle
• Use a diagram to explain in algebraic terms the length of a side in terms of other known
quantities
Areas of difficulty for geometry students:
• Relating algebraic expressions to a context
• Using algebra to prove geometric relationships
• Solving algebraic expressions
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The maximum score available for this task is 9 points.
The minimum score for a level 3 response is 5 points.
Most students, 87%, could show why the area of Square A was 20 sq. un. Many students, 69%, could find the
area of all the squares in part 1. More than half the students, 58%, could find the area of the squares, use the
diagram to find an algebraic expression for the side of the small square in part 2, and calculate the area of the
small triangles in 2. 23% of the students could meet all the demands of the task including using algebra to find
the area of 4 small triangles and using this algebraic expression to prove why the area of the square A in part
one is 17 sq. un. 13% of the students scored no points on this task. 50% of the students with this score attempted
the task.
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Sloping Squares
Points
Understandings
Half the students with this score
0
attempted the task.
1
Students could find the area of
Square B.
3
Students could calculate the areas
of the squares in part one.
5
Students could find the areas of
the squares in part 1, find the
algebraic area for the 4 small
triangles, and use the diagram to
justify the expression for the side
of the interior square.
7/8
9
Misunderstandings
Students could not use the diagram to
find the area of the Square B. Some
students thought the area of the internal
square was 8 instead of 4. Some students
forgot to divide by 2 when finding the
area of the small triangles.
They had difficulty finding the area of C
and D. Many tried to match parts of
squares rather than using the formula for
finding the areas of triangles.
38% of the students did not attempt any
part of question 2. In part 2b many
students tried to just use a verbal
explanation to explain the length of the
side of the interior square. They made no
attempt to use the diagrams provided.
Many students struggled with the algebra
needed for multiplying (a-b) times (a-b).
Students forgot that a negative times a
negative is a positive and had difficulty
calculating with exponents. Some
examples of their incorrect expressions
include: 4a; 4a4b+2a2b; 4ab; ab4. Half
the students did not attempt 2d and 2e.
Students with this score missed part 1c
and/or 2b. They couldn’t calculate the
area of D, often because they couldn’t or
didn’t draw in the triangles. They also
couldn’t use the diagram to find the side
length of the interior triangle.
Students could calculate the area
of rectangles and triangles, with
numbers and variables. Students
could relate algebraic expressions
with the context and use algebra to
make geometric arguments about
relationships in a set of sloping
square shapes. Students were
comfortable with multiplying
algebraic expressions and
combining terms.
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Implications for Instruction
Students need to have many experiences with decomposing shapes on a geometric diagram. Students
should have regular opportunities to work tasks where lines need to be added to a diagram in order to
use a formula or find out missing information. Students should also be exposed to challenging tasks,
where they need to generate their own diagrams from the given information. In this case, it would be
interesting to see if they could make their own sloping squares or write instructions to a friend on how
to draw a sloping square using grid pattern. They need experiences that help to focus their attention on
different attributes of the figures.
Students should be comfortable expressing geometric relationships on the diagram in terms of
algebraic expressions and using these expressions to find measures such as area and perimeter. While
algebra is not a formal part of the curriculum, it can easily be integrated into the curriculum to make
geometric arguments and generalizations. It should be a regular part of the rigor of classroom
expectations. Students may need some review of use of exponents and when and when not to combine
terms. The purpose at this grade level is not on reinforcing area, but using that skill as a vehicle to
work on higher levels of cognitive demand about making generalizations about geometric shapes and
proving those generalizations in the context of the situation. Students should have opportunities to
investigate rich patterns like these and see if they can come up with the general formula on their own.
Ideas for Action Research:
Students need rich tasks to investigate that capture their imagination, have high cognitive demand, and
allow them to explore geometric properties. Here a couple of interesting investigations you might try
with your students:
Cutting Edge:
Describe an algorithm for cutting up a parallelogram into a rectangle-that is cutting a parallelogram
so that the pieces can be reassembled to form a rectangle. What is an algorithm for cutting up a
triangle to form a rectangle?
Have pairs of students work on the task to find their algorithms. Also have them make some
parallelograms and triangles that they feel might stump the algorithms of other groups in the class.
When everyone is finished bring two groups to the front. One group should read their algorithm,
while the other team performs the algorithm on their parallelogram. Did the algorithm work? What
properties of the shape made some parallelograms not work with the algorithms?
How does this type of activity help students think about the properties of the shapes and consider a
variety of cases to test their conjectures?
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Lots of Squares
Can you divide a square into a certain number of smaller squares? This may depend on exactly how
many smaller squares you want. The first diagram below shows that any square can be divided into 4
smaller squares. The second diagram shows that any square can be divided into 7 smaller squares.
Notice that these smaller squares don’t have to be the same size as each other, but keep in mind that
the smaller portions must all be squares, not simply rectangles. The task of this activity is to
investigate what numbers of smaller squares are possible. For example, you can probably see that
there is no way to divide a square into just 2 smaller squares.
1. Start with specific cases. Is it possible to divide a square into 3 smaller squares? 5? 6? 8? And
continue up to 13 smaller squares.
Now reflect on what you’ve done, and just imagine continue this process. Would there be any
numbers beyond 13 for which you couldn’t divide a square into that many smaller squares? What
patterns can you find in the cases you’ve done that help you with this question?
2. What is the largest “impossible” case?
3. Prove your answer to question 2. That is, prove that all cases beyond the one you named in
question2 are possible.
Tasks adapted from Fostering Algebraic Thinking by Mark Driscoll.
How do these tasks force students to make and use diagrams? to think about generalizations and the
logic of proof? How do the cognitive demands of these tasks compare to the usual tasks in your text?
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