standard_description: Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
problem_activity: Narrative
The purpose of this How Many Do You See is for students to subitize or use grouping strategies to describe the images they see. Students may see equal groups in the rows or the columns of the array. Recording the equations for each way of seeing the groups is an opportunity to reinforce the commutative property.
When students use different ways to group dots within the same array to find the total number of dots they look for and make use of structure (MP7).
Launch
Groups of 2
“How many do you see? How do you see them?”
Flash the image.
30 seconds: quiet think time
Activity
Display the image.
“Discuss your thinking with your partner.”
1 minute: partner discussion
Record responses.
Repeat for each image.
Student Facing
How many do you see? How do you see them?
[IMG]
[IMG]
[IMG]
Student Response
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Activity Synthesis
“How did the arrays allow us to find the number of dots in different ways?” (We were able to look at the number of dots in each row and think about how many rows there were. I counted the number in each column and multiplied it by the number of columns.)
As students share, record equations to match their thinking.
Consider asking: “How do the equations change if we think about the rows as the groups or the columns as the groups?” (For the image with 12 dots, if we think about the rows as the groups, we write
\(4 \times 3 = 12\)
, but if we think about the columns as the groups, we write
\(3 \times 4 = 12\)
. The order of the factors is reversed, but the product stays the same.)


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