As I approach the end of another calendar year, I find myself reflecting on the successes, challenges, and lessons that I can carry forward into the new year. One of the challenges in my work has been finding opportunities to deeply connect with the teachers that we serve. Teachers continue to be overworked and pulled in a million directions, and the lack of substitute teachers leaves limited time for deep conversations about practice and opportunities for celebration. Teachers are the backbone of our work at Math ANEX and I just wanted to say…I miss you and I appreciate the interactions that I have had with you this year! Thank you for everything you do to support your students and your communities.

One of the successes for me this year has been creating assessments that capture student voice and allow them to demonstrate their mathematical assets. As we think about an assessment item at Math ANEX, we often ask ourselves, “What would we expect students to write?” You can imagine that we get many responses from students that align with our expectations, others that surprise us, and many that spark joy.

We wanted to share two items with you (one from elementary and another from secondary) along with some student responses that we hope will validate what you already know about your students; that they are creative, they are learning from you and their peers, and they will continue to inspire us as they engage in mathematics.

## Elementary Item: Art Tables

Students are **resourceful**.

“so i used dots like this …………………………… then i put 4 like this i guess (….) (….) (….) (….) (….) (….) (….) (….) (.) then i relized that theres one more student left so first i counted the fours and got 8 fours then one more student left again so i guess my sum is 9.”

Students are **thoughtful**.

“I have counted up to 8 tables but the problem is that it only supports 32 students, So i would asume they could make 1 more little spot for the poor student to sit because I personally don’t want that student to be alone and sad. But originally I’ve came up with 8.1”

Students are **flexible thinkers**.

“I spit the 33 in to 20 and 13 then split the 13 into 1 and 12 then I divied the 20 and 12 by four and got 5 and 3. After that I added the 5 and 3 and got 8. then add 1 table for the 1 extra studet and got 9.”

Students are **reasoning**.

“First of all, 33 is an odd number. 4 is an even number. You also can’t divide an odd number to an even number without getting an decimal or fraction. So first, I took away 1 from 33 to make it 32. And now, the number of students is even and is a multipe of 4. You then use the division formula to divide 32 by 4. Your quotient is now 8. But your number of tables will not be 8. I will be 9 because of the extra student you took away to get an even number to divide by. The art teacher will finally buy 9 tabels for her art class with 33 students.”

Students are **future school leaders**.

“The art teacher should buy 9 tables so you have some leftover seating for new kids and other classes with a little more people it is also good for staff meetings in the art room.”

Students are **diverse thinkers**.

“I did 4+4+4+4, and that equals 16. so then I add another 4+4+4+4 and I get 32. And I count all the 4s and I get 8. And since all that equaled 32, I add 1 more and I got 9. Mm yes, quite an interesting way of doing it.”

“son 33 estudiantes cada mesa es de 4 entonses es 4 dividido entre 33 y 4 por 8 es 32 entonces la maestra tiene que comprar otra mesa por que se tienen que sentar todos”

“there are 33 students each table is 4 so it is 4 divided by 33 and 4 times 8 is 32 so the teacher has to buy another table because they all have to sit”

“Since 33 is not divisible by 4, we need to find the first number greater than 33 that is divisible by 4. 8×4=32 does not work, but 9×4=36 works. So the art techer needs to buy nine tables.”

## Secondary Item: Thomas Paints a Wall

Students are **funny**.

“Who spends 6 hours painting a wall?”

Students are **willing to try**.

“I got my answer by first finding the total area of the rectangle:420. Then I found the area of the painted section:60. If he had already painted for 1.5 hours and he needed to paint for six, it meant that he had 4.5 hours left. Then I tried solving for x by writing it like this – 4.5x= 420 – 60 . That gave me 4.5x= 360. I divided both sides by 4.5 to get and answer of x= 80. After 6 hours the area of the painted area is 80ft. This would mean he still has to paint 280 ft.”

Students are **efficient**.

“12*5=60

6/1.5=4

60*4=240″

Students are **realistic**.

“I’m guessing he takes long breaks so I figured it would take him around 3 hrs to paint and his breaks would be long.”

“It wouldn’t be 6 hours it’ll be 7.5 hours because I added and also the wall is pretty big for one person to paint.”

Students are **thinking proportionally**.

“To solve what the area that would have been painted after 6 hours, there are several steps you need to complete. To start off, I found the hourly rate of area panted to time, which was 40:1. After this I simply had to multiply the ratio by x:6, this left me with 240=x.”

Students are **attending to precision**.

“First I took 6 hrs and divide it by 1.5 hrs which gives me 4 then I multiply how many square feet were completed in 1.5 hrs so I multiply 12 and 5 which gives me 60 square feet. Now I multiply 60 square feet by 4 sessions which a total of 240 square feet.”

Students are **diverse thinkers**.

“If every 1.5 hours, 60 ft. sq. Is painted, in 3 hours 120 would be and in 6 hours 240 ft of a total of 420 would be 240/420=4/7”

“I found the area he painted in 1.5 hours which was 60 feet squared then I found how many times 1.5 hours fit into 6 hours which was 4. So then he could do 60 ft squared 4 times, and paint a total of 240 feet squared in 6 hours.”

“After 1.5 hours, a total of 1/7 is painted. 6 hours divided by 1.5 hours is 4, so if I multiply the amount painted by 4, that comes to 4/7. Then, if you take 4/7 of the total area, you come to 240 square feet.”

We hope that these have made you smile and inspired you to keep asking meaningful questions of your students. You are making an incredible difference!