Everything is bigger in Texas, and celebrations are no exception. Check out this display of doughnuts in honor of the celebration for the Independence of Texas.
What do you notice? What do you wonder?
Here’s what our students came up with…
A few things to note. Yes, there are observations that are not necessarily mathematically relevant to possible inquiry pathways within this picture. However, all student voices are validated, and all observations are important as it encourages students to pause and consider and make sense of the context of the situation. Much like we ask for students to do when approaching a story problem. When they pause to make sense of the situation, the quantities suddenly carry much more meaning, and operations become more obvious.
The next step, the “wondering” phase is equally as accessible to all students as the “notice” phase. Again, contextual and mathematical questions from students are validated and noted. Our students quickly became interested in the question that we had hoped they would which was, “How many doughnuts in the entire flag?”
Before students set out to solve, they were invited to establish an estimate, and ask questions for information that might help them in getting closer to a solution. Questions such as, “is there more than one layer,” or “does the star count?” and “can I see an un-cropped picture to view the entire flag?” All evidence that students are considering variables that could impact both their estimate, and their solution path. Information that was provided to them: a full picture of the flag, a “zoomed” in box of blue doughnuts to see the smaller array, and we provided them with information in “yes” or “no” form as far as layers of pastries or whether the star would count.
All kids worked diligently to arrive at their solution. There was consensus in the answer: 54 x 35=1890, but no one seemed as interested in that as the various methods we discussed during debrief to help make the math easier. They described the ways in which they determined the number of boxes: 54, by partitioning the flag into different areas to help them break down the problem into more reasonable chunks:
Then went on to describe how those smaller areas helped them to manage the problem 54 x 35. Take this example:
She “chunked” the flag into one row of boxes noticing that there were 9 in a row, with 35 doughnuts in each square. After knowing what was in one row, she simply multiplied that number by 6 to arrive at her answer of 1890. Three digit by one digit multiplication was more efficient for her than double by double digit.
A second approach other than the standard algorithm:
This student set up to solve using the algorithm, and quickly switched to an approach that made more sense to him. See if you can figure out his strategy, it took us all a while, and probably required a lot of patience from him as he explained his thinking to the class. Give it a shot, the “ah-ha” moment is worth it.
At this point, students were satisfied with their answer of 1890. They had accomplished the task they set out to solve, compared it back to their estimates, and were convinced of their answer. Now to match their thinking to the final reveal:
That’s correct, you heard him right…1836 Doughnuts to represent the year of Independence. 1836…How did our students respond?
Two camps were quickly established:
- We are wrong: maybe the arrays were 5 x 6, there were different numbers of doughnuts in each box, one missing from each box (we assumed all had 35)…
- Texas is wrong: people miscounted, they lied to match the year, they announced the number they are giving away at the celebration not actual count, someone didn’t double check the boxes, there’s one fake doughnut in each box, there are no doughnuts under the star…
Ultimately, this led to more math. Students were interested in mathematizing their theory. They started thinking about reasonableness of some of these claims. Coming up to the board for a closer look to see if 54 doughnuts could indeed be missing from under the star…trying to catch spaces in the boxes where a doughnut should be (MP4: modeling). Or calculating other possibilities for the sections of color. The theory that blue boxes have 35 doughnuts, red have 34, and white have 33 received applause and cheers from the class. Students began working to check a variety of array possibilities, and were left with the questions, “What would the flag look like if there were 1836 doughnuts?”
To be honest, I was nervous about trialing this task on kids. Mostly because I knew their answer would not match the reveal in Act 3. But the outcome brought the MPs to life more than I could have anticipated or even planned. Take MP 3: Construct viable arguments and critique the reasoning of others. These students took on TEXAS. They were beyond critiquing the thinking of their classmates, and were ready to put in a call to Krispy Kreme or the governor. They revisited their math, were happy to think through this real life situation quantitatively (MP2) there was no need to explore options for those kids that were “done.” These students definitely persevered throughout the entire investigation (MP1).
In addition to the layering of the math practices, this task specifically addressed standard CCSS.MATH.CONTENT.4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
The alternative to this task might have appeared as:
Multiply 54 by 35. Show your work using equations, rectangular arrays, and/or area models.
Not to say there isn’t room in the curriculum for that type of mathematical experience, but to emphasize the importance in making room in the curriculum for rich math tasks. Keep it real.
If you are interested in taking on Glazing Saddles with your class, here is the link to a slide deck of images to support the 3 Act Structure: Glazin’ Saddles. If you do, leave a comment about how it went! This task was appropriate for this specific group of students, as their teacher had just gotten back from a trip to Texas, making the context engaging for kids. But…who are we kidding, doughnuts are always engaging to kids ;).
To learn more about 3 Act Tasks, or for more 3 Act Task ideas, please visit:
Graham Fletcher: https://gfletchy.com/
Robert Kaplinsky: http://robertkaplinsky.com/?wref=bif
Mike Wiernicki: https://mikewiernicki.com/3-act-tasks/
Dane Ehlert: http://wmh3acts.weebly.com/3-act-math.html
Geoff Krall PBL Curriculum Maps https://emergentmath.com/my-problem-based-curriculum-maps/