Moments from our 8th Grade Summer Math Academy (part 3)

A few doors down in the classroom of Pam Cruz I find  a different use for visual patterns. (  She also begin with pattern 2.

VP #2

After a little discussion about whether the pattern is increasing or decreasing, Pam moves students to a table to examine the way the figures are changing.  Pam asks the students how they should label the columns and students call out math words and symbols they remember applying a table.

Students:  x!. . .  y.. . .  n?

Pam: We could put a letter (writing x) but what would x mean? What would it stand for?

Students: (fishing for math words from their memory that the teacher might want) domain? .. . . range? . . .  increasing? (that was the math word she used a minute ago, right?)  . . . .decreasing?

Pam: What could we count about the third figure that would be different than the 4th figure?

Student: 5!

Pam: 5 is a number I want to put in my table. What did you count to get 5?

Student: Blocks (students are relieved they finally got the magic word that was in their teacher’s head.)

Pam writes Number of Blocks above the second column “And which one had 5 blocks?”

Student: The third one

Pam writes Figure Number on top of the first column. “We could have put ‘shape number’ or ‘step number’ here. And in this column you could have had ‘number of cubes’ or number of boxes’.”

I love the way she pushed for sensemaking here. She let them know there was not just one magic word. It is about the meaning and not the specific word.   The math words they offered obviously didn’t mean anything to them. Right now is the time to focus on meaning. She can reintroduce the vocabulary later.

After counting and filling in their table she asks “How many blocks would be in the 5th figure? Students are able to extend the pattern  easily.

Now she moves to pattern 4, which has no figure numbers under it.

VP #4

Pam: Make and label your table.

Student: There are no figure numbers.

Pam: Very good noticing.  Now everyone watch carefully what I am going to do.

She writes “figure 2″ under the first figure, followed by figure 3 and figure 4. I like how she gives them the clue that something is going to change. She doesn’t try to trick them with a gotcha.  This group of kids has felt fooled too often.

What followed was a wonderful discussion about using the pattern they observed both forward and backwards, and the reversibility of operations.

She moves to pattern 7 and a student immediately asks ” can you give us figure numbers?” I love that they are starting to understand what is important to pay attention to.

At pattern 9 she sets up a situation where students will have to move backwards into negative numbers.


She labels these figure 5, figure 6, figure 7 and asked them to set up their table and find numbers to go with figures 1, 2, 3, and 4. Pam circulates as groups work on the task. I hear lots of discussion of negative number rules they vaguely remember but are not sure how to apply. Several students draw number lines as they explain to their groupmates. It turns into a great class discussion remediating integer understanding.

Near the end, a surprise twist.  A student uses the word decreasing in describing their work to find the number of snowflakes in figures 4, 3, 2, and 1. That intersected with their previous talk about increasing and decreasing patterns.  The question “Is this pattern increasing or decreasing”sparked a surprising debate. Pam had them stand and move to the left side of the room if they thought the pattern was decreasing and to the right side of the room if they thought the pattern was increasing. I LOVED this. She didn’t resolve it or tell one group they were wrong, she put it on them to hash it out. Each group took one turn at convincing the other before the bell for break.  (darn bells) Pam gave them the opportunity to change sides and 3 on the decreasing side held firm against the weight of the rest of the class.

I cheered inside, even though they were not correct. Good for them, don’t cave if you are not convinced! They were invested and they cared.  This is a good sign of the mathematical discussions that will come this summer.

Pam tabled the discussion but told the students they would come back to it. Students filed out the door to break.

One day, one mathematical conversation at a time, these teachers are making a difference.



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