Standards for Mathematical Practice Task Alignment

Standards for Mathematical Practice Task Alignment

Task: Dan Meyer 3-Acts “Penny Pyramid”

Teacher: Dan Meyer

Date: 7-30-12

Practice / Teacher Move / Evidence of Student Use of Practice
Overarching Habits of Mind of a Mathematical Thinker
1. Make sense of problems and persevere in solving them. / ·  Asked students to brainstorm question “I saw this clip and I had TONS of questions. What was the first question that came to mind?”
·  Wrote all questions on board, ranked them, and weighted by interest
·  Narrowed down to one question.
·  Asked students what information they needed to answer that question. (The teacher had anticipated ahead of time what questions would be asked and what information would be needed.)
·  Only gave out information as it was asked for.
·  Listened in to groups working and guided them as necessary using questions such as “Have you thought about?” By doing this, the teacher made sure that the students stayed on task and that they were on the right track, but did not use leading or scaffolding questions. The teacher created an environment of trust and safety that encouraged perseverance.
·  Noticed that some groups had the answer and then asked an extending question: “How tall would a pyramid be that contained a total of one billion pennies?” / ·  Students brainstormed. They had input into developing questions.
·  Once question was decided, students had to determine what information was needed in order to answer the question.
·  In order to figure out the information they needed, the students were forced to make sense of the problem.
·  When teacher asked, “Have you thought about…”, the students were more comfortable persevering rather than giving up. Also, working within a group helped students to persevere.
6. Attend to precision. / ·  As students were asking for information about the dimensions of the pyramid, the teacher recorded what they were asking and encouraged them to be very specific in their vocabulary. The teacher, in turn, was very specific about how he labeled the information given. / ·  Students needed to differentiate between names of the variables that were quantified. For example “stacks” indicated the towers of 13 pennies each, whereas “horizontal layer” indicated the whole layer that was composed of the stacks of 13 pennies. Without this precision, it would have been hard to develop a clear model.
Reasoning and Explaining
2. Reason abstractly and quantitatively. / ·  Asked students to make a guess…then low and high / ·  Students reasoned quantitatively about the possible magnitude of the answers in order to develop an estimate. By guessing a high and low estimate, they were forced to reason further about their estimate.
3. Construct viable arguments and critique the reasoning of others. / ·  Asked students to write down question and share with table (think-pair-share)
·  Asked “How many people are wondering that now?” This encouraged students to critique the ideas that others had come up with. / ·  After writing down their own question, students discussed their question with tablemates, creating the opportunity to construct the argument of why they chose their question, as well as critiquing the questions that others came up with.
Modeling and Using Tools
4. Model with mathematics. / ·  The teacher chose an engaging problem that could be modeled mathematically.
·  The teacher monitored the students developing the model, but allowed them to develop the model on their own. / ·  Once the given information was communicated, the students used that information to develop a mathematical model.
·  The model involved creating a sum of terms that expressed the value of each layer of the pyramid.
5. Use appropriate tools strategically. / ·  The teacher monitored the use of tools, but did not interfere.
·  In the “debrief”, the teacher discussed the different use of tools. By doing this, students are provided with more tools in their toolbox for future problem solving. / Once the model was developed, different tools were used to find the answer:
·  Numerical expression
·  Summation
·  Excel spreadsheet
·  BASIC program
Seeing Structure and Generalizing
7. Look for and make use of structure. / ·  During the “debrief “of the problem, the teacher facilitated a discussion about the structure of the arithmetic expression. What numbers were constant in each term of the sum? Can we use the distributive property? Do we see a pattern? / ·  The students had to develop an understanding of the physical structure in order to develop a mathematical model that had a numerical structure of its own. The student had to make the connection between the physical structure and the numerical structure of the mathematical model.
8. Look for and express regularity in repeated reasoning. / ·  Early in the discussion of the problem, the teacher helped students develop an understanding of the physical structure of the pyramid. He asked how many stacks of 13 pennies were there on each side of the second lowest horizontal layer. As a discussion ensued, the teacher guided, via strategic questioning and showing strategic images, that there were 39 and not 38. / ·  Once the base of 40 by 40 was given, the students had to determine how many horizontal layers were involved. There was a regularity of each layer having one fewer set of stacks of 13 pennies, and the top layer had 1 stack. This repeated reasoning allowed them to create their mathematical model.