standard_description: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
problem_activity: Narrative
The purpose of this activity is for students to notice structure in a series of diagrams and the expressions that represent them. They investigate how these expressions vary as the number of rows and columns in the diagram change. Students see how the diagram represents the multiplication expression and also how the diagram helps to find the value of the expression (MP7). Through repeated reasoning they also begin to see how to find the value of a product of any two unit fractions (MP8).
Launch
Groups of 2
Display the images from the task.
“What is different about these diagrams?” (The number of rows increases by 1. The blue shaded piece gets smaller.)
“What is the same?” (The size of the big square. There are always 4 columns. Only one piece is shaded.)
1 minute: quiet think time
Share and record responses.
Activity
1–2 minutes: independent work time to complete the first problem
1–2 minutes: partner discussion
“Now, complete the rest of the problems with your partner.”
5 minutes: partner work time
Monitor for students who:
choose different diagrams to represent with multiplication expressions
represent the same diagram with different multiplication expression, for example,
\(\frac{1}{2}\times\frac{1}{4}\)
and
\(\frac{1}{4}\times\frac{1}{2}\)
Student Facing
A
[IMG]
B
[IMG]
C
[IMG]
D
[IMG]
Choose one of the diagrams and write a multiplication expression to represent the shaded region. How much of the whole square is shaded? Explain or show your thinking.
If the pattern continues, draw what you think the next diagram will look like. Be prepared to explain your thinking.
[IMG]
Student Response
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Advancing Student Thinking
If students do not write correct expressions to represent the diagrams, write the correct expressions and ask, “How do the expressions represent the area of the shaded piece of the diagram?”
Activity Synthesis
Display the diagrams from the student workbook.
Select previously identified students to share.
As students explain where they see the multiplication expression in each diagram, record the expressions under the diagram for all to see.
Refer to the diagram that shows
\(\frac{1}{2}\times\frac{1}{4}\)
or
\(\frac{1}{4}\times\frac {1}{2}\)
.
Display both expressions.
“How does this diagram represent both of these expressions?” (It shows one half of one fourth shaded in and it also shows one fourth of one half shaded in.)
Represent student explanation on the diagrams.
“Why is the area of the shaded region getting smaller in each diagram?” (Because we are shading a smaller piece of
\(\frac{1}{4}\)
each time.)
Display:
\(\frac{1}{2} \times \frac{1}{4}= \frac{1}{2\times 4}=\frac{1}{8}\)
\(\frac{1}{3} \times \frac{1}{4}= \frac{1}{3\times 4}=\frac{1}{12}\)
\(\frac{1}{4} \times \frac{1}{4}= \frac{1}{4\times 4}=\frac{1}{16}\)
\(\frac{1}{5} \times \frac{1}{4}= \frac{1}{5\times 4}=\frac{1}{20}\)
“These equations represent the diagrams. What patterns do you notice?” (They all have
\(\frac {1}{4}\)
in them. The denominators in the first fractions go 2, 3, 4, 5. The denominators in the middle fractions are multiplication expressions, the denominators in the middle are all multiplied by 4, the denominator in the fraction that shows the value of the shaded piece goes up by 4 each time.)
“How do the diagrams represent
\(\frac{1}{2\times 4}\)
,
\(\frac{1}{3\times4}\)
,
\(\frac{1}{4\times4}\)
,
\(\frac{1}{5\times4}\)
?” (The numerator tells us that there is 1 piece shaded and the denominator tells us the size of the piece. The denominator also tells us the rows and columns that the whole is divided into.)


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