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Linking Cubes
Description
Linking Cubes are hands-on manipulatives that are used to assist with the
understanding of mathematical concepts. Linking cubes are individual unit cubes
that can interlock together to build various shapes and structures. They are user
friendly for all ages and are a unique visualization tool. This manipulative is
commercially sold in a variety of colours under brand names, such as Cube-ALink.
Purpose
There are a number of reasons why I would use this manipulative. Firstly, it
can help concretize learning. It does this by helping math become real and
engaging, therefore making it more fun to learn for the students. Secondly, it
causes abstract concepts to be more visual and physical. All of this aids in the
development mathematical intuition (or math sense).
You can use linking cubes to meet expectations in strands such as:
Strand
Patterning and Algebra
Example
growth patterns, linear models
Number Sense and Numeration
fractions, ratios, integers, factors
Geometry and Spatial Sense
isometric drawings, orthographic
drawings, spatial sense
Measurement
Volume and surface area (3D),
perimeter and area (2D)
Sample Activities
Activity 1: Linear Models - Growing L’s
This activity should be reserved for initiating the topic of linear models. First,
provide a sequence on Isometric Dot Paper to your students and ask them to build
a replicate of this sequence with a partner.
Once they have built the sequence they should be told to look for patterns
both physical and numerical. You should provide the students with the chart below
to help them work through the questions they are about to receive.
Model Number
Number of
Cubes
1
1
2
1+2
3
3+2
=1+2
= 1 + (1) 2
=3
=1+2+2
= 1 + (2) 2
=5
4
5+2
=1+2+2+2
= 1 + (3) 2
=7
The following questions should be used along with demonstrations to help
students understand the numerical representations of the physical patterns they
see.
Question 1: In the row “Number of Cubes”, you will see the sequence 1, 1+2, 3+2,
and 5+2. Explain how these expressions are connected to the above sequence of
models.
Answer: 2 cubes are added at each step
Question 2: How many Cubes will there be in the 5th, 6th, and 7th models?
Answer: Model 5 has 9 cubes, model 6 has 11 cubes and model 7 has 13 cubes
A cube is added to the top and front of each model
Question 3: In the row “Number of Cubes”, you will see the sequence 1, 1+2,
1+2+2, 1+2+2+2. Explain how these expressions are connected to the sequence
of models.
Answer: The expressions illustrate how many times the 2 cubes has been added
Question 4: In the row “Number of Cubes”, how are the numbers in the brackets
related to the model number?
Answer: The number in the brackets is one less than the model number
Questions 5: What number would be in the brackets for the 20th model? For the
nth model?
Answer: The number in brackets for the 20th model would be 19. For the nth
model would be n-1
Question 6: How many cubes would there be in the 45th model?
Answer: There would be 89 cubes in the 45th model.
Next, be sure to perform a follow-up analysis to summarize the finding from the
above questions.
The number of cubes in the nth model is given by the formula:
N = 1 + (n-1) · 2
N = 2n - 1
Here, N represents the total number of cubes in the nth model. If you look carefully
you will see that this formula reflects the way the models were constructed.
Question 7: In the formula, N = 1 + (n-1) · 2. We know that the expression (n-1) is
connected to the model number but what about the “1” and the “2”?
Answer: The “1” in the formula was the number of cubes in the first model.
The “2” in the formula was the number of cubes added at each step
Activity 2: Linear Models – Sprouting Arms
The following sequence of models forms a definite pattern. Ask your
students to build a replicate of this sequence with a partner. Have students
describe the process that was used to form each successive model out of the
previous one.
Once they have built the sequence they should be told to look for patterns
both physical and numerical. The following questions should be used along with
demonstrations to help students understand the numerical representations of the
physical patterns they see.
Question 1: How many cubes would you add to the 4th model to create the 5th
model? How many cubes would there be in the 5th model in the sequence?
Answer: 3 cubes are added to the 4th model to create the 5th model and there
would be 13 cubes in the 5th model.
Question 2: Complete the following table with your prediction of the number of
cubes in the 5th model and the formula that will give the number of cubes in the nth
model.
Model
Number
1
2
3
4
5
…
n
1
2
3
4
5
…
n
1
4
7
10
13
…
N=3n-2
Number of
Cubes
Answer:
Model
Number
Number of
Cubes
For advanced learning:
Question 3: Will there be a model in this sequence that has exactly 71 cubes in it?
Explain your thinking.
Answer: Since the equations 3n-2=71 has no solution in the natural numbers.
There is no model with 71 cubes in it
Activity 3: Linear Models – Plus Pluses
The following sequence of models forms a definite pattern. Ask your
students to build a replicate of this sequence with a partner. Have students
describe the process that was used to form each successive model out of the
previous one.
Once they have built the sequence they should be told to look for patterns
both physical and numerical. The following questions should be used along with
demonstrations to help students understand the numerical representations of the
physical patterns they see.
Question 1: How many cubes would you add to the 4th model to create the 5th
model? How many cubes would there be in the 5th model in the sequence?
Answer: 4 cubes are added to the 4th model to create the 5th model and there
would be 17 cubes in the 5th model.
Question 2: Complete the following table with your prediction of the number of
cubes in the 5th model and the formula that will give the number of cubes in the nth
model.
Model
Number
Number of
Cubes
1
2
3
4
5
…
n
Answer:
Model
Number
Number of
Cubes
1
2
3
4
5
…
n
1
5
9
13
17
…
N=4n-3
Question 3: Consider the surface area of each model in the sequence of models.
For each cube that is added at each step how many new “faces” are added to the
total surface area of the model? Be careful with this question. Make sure that you
are counting only additional faces.
Answer: 16 faces are added to the total surface area at each step. (4 new faces
with each new cube)
Question 4: Complete the following table with your prediction of the number of
faces on the surface of the 5th and 6th models. Can you find the formula that will
give you the number of faces on the surface of the nth model?
Model
1
2
3
4
5
…
n
Number
Number of
Faces
Answer:
Model
Number
Number of
Faces
1
2
3
4
5
…
n
6
22
38
54
70
…
16n-10
Black Line Masters
Activity 1: Linear Models - Growing L’s
Build this sequence with a partner using your Linking Cubes:
Look for patterns both physical and numerical.
Model Number
Number of
Cubes
1
1
2
1+2
3
3+2
=1+2
= 1 + (1) 2
=3
=1+2+2
= 1 + (2) 2
=5
4
5+2
=1+2+2+2
= 1 + (3) 2
=7
Using the chart above answer the following questions:
1. In the row “Number of Cubes”, you will see the sequence 1, 1+2, 3+2, and 5+2.
Explain how these expressions are connected to the above sequence of models.
2. How many Cubes will there be in the 5th, 6th, and 7th models?
3. In the row “Number of Cubes”, you will see the sequence 1, 1+2, 1+2+2,
1+2+2+2. Explain how these expressions are connected to the sequence of
models.
4. In the row “Number of Cubes”, how are the numbers in the brackets related to
the model number?
5. What number would be in the brackets for the 20th model? For the nth model?
6. How many cubes would there be in the 45th model?
The number of cubes in the nth model is given by the formula:
N = 1 + (n-1) · 2
N = 2n - 1
th
Here, N represents the total number of cubes in the n model.
7. In the formula, N = 1 + (n-1) · 2. We know that the expression (n-1) is
connected to the model number but what about the “1” and the “2”?
Activity 2: Linear Models – Sprouting Arms
The following sequence of models forms a definite pattern. Build the
following sequence. Describe the process that was used to form each successive
model out of the previous one.
1. How many cubes would you add to the 4th model to create the 5th model? How
many cubes would there be in the 5th model in the sequence?
2. Complete the following table with your prediction of the number of cubes in the
5th model and the formula that will give the number of cubes in the nth model.
Model
Number
1
2
3
4
5
…
n
Number of
Cubes
3. Will there be a model in this sequence that has exactly 71 cubes in it? Explain
your thinking.
Activity 3: Linear Models – Plus Pluses
The following sequence of models forms a definite pattern. Build a replicate
of this sequence with a partner. Describe the process that was used to form each
successive model out of the previous one.
1. How many cubes would you add to the 4th model to create the 5th model? How
many cubes would there be in the 5th model in the sequence?
2. Complete the following table with your prediction of the number of cubes in the
5th model and the formula that will give the number of cubes in the nth model.
Model
Number
1
2
3
4
5
…
n
Number of
Cubes
3. Consider the surface area of each model in the sequence of models. For each
cube that is added at each step how many new “faces” are added to the total
surface area of the model? Be careful with this question. Make sure that you are
counting only additional faces.
4. Complete the following table with your prediction of the number of faces on the
surface of the 5th and 6th models. Can you find the formula that will give you the
number of faces on the surface of the nth model?
Model
Number
Number of
Faces
1
2
3
4
5
…
n