Lesson 11

The purpose of this Math Talk is to elicit strategies and understandings students have for adding fractions. These understandings help students develop fluency and will be helpful later in this lesson when students will need to add and subtract fractions.

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate [student]’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to [student]’s strategy?”
• “Do you agree or disagree? Why?”

The purpose of this activity is for students to represent a situation with a sequence where it makes sense to add the terms of the sequence together in order to answer a question about the situation. This situation was chosen for its hands-on nature to help students make sense of why we would ever need to add up terms in a sequence. Students will continue this type of thinking in the following activity where they will work with a famous mathematical shape; the Koch Snowflake.

Arrange students in groups of 4. It may be helpful to assign one student to be “Tyler” to carry out the actions described in the task as a demonstration for the class. Students may find it useful to fold and unfold the paper first to have crease lines to follow when making the cuts. Provide groups access to paper and scissors.

Emphasize that Tyler gives away pieces of his original sheet of paper while the other 3 students take the pieces from Tyler. This will help students recognize that the amount of paper Tyler has is decreasing while the amount of paper the other students have is increasing. By keeping this in mind, students can understand better why adding Tyler's sequence does not make sense while adding the sequence representing the other sheets of paper does.

Begin the discussion by displaying the table shown here for all to see:

Invite groups to explain where the values for Tyler and the other group member came from. Highlight any students who reason about the size of Tyler's paper using an equation such as \(T(n)=1\boldcdot\frac14^n\). If no students use an equation to make sense of Tyler's paper, ask them to do so and after a brief work time invite students to share their equations.

An important connection for students to make is that while an amount of paper in Tyler's hand is represented by the sequence \(T\) with the terms \(1,\frac14,\frac{1}{16},\frac{1}{64}\), the amount of paper in the hands of one of the other group members at each step is the sum of the terms from Tyler's sequence starting from \(T(1)=\frac14\).

If time allows, show using technology that this sum is close to \(\frac13\) as the number of steps increases, and that if you keep summing additional terms you get closer and closer to \(\frac13\). This matches the intuition that each person would end up holding very close to \(\frac13\) of the original piece of paper after several rounds of cutting and distributing paper.

This activity is an opportunity to contrast when it does and does not make sense to sum a sequence. This task is relatively unscaffolded compared to previous lessons, giving students opportunities to make sense of the problem by, for example, drawing Step 2 or organizing the data in a table (MP1).

If you wish, share with students that this shape is known in mathematics as the Koch (pronounced “coke”) Snowflake.

Tell students to close their books or devices. Draw and label an equilateral triangle as Step 0. Next to it, draw and label Step 1, starting from an equilateral triangle and erasing the middle \(\frac13\) of each side and drawing two additional segments popping out of each side (as shown in the task statement). Invite students to describe how the second triangle was drawn in their own words and to state how many sides Steps 0 and 1 have.

Arrange students in groups of 2–4. Give time for groups to work followed by a whole-class discussion.

If students have trouble finding a rule for function \(S\), suggest that they draw Step 2, and then create a table relating each step to the number of sides in the snowflake at that step.

The goal of this discussion is for students to share the different ways they represented and calculated values for \(S(n)\) and \(T(n)\). Begin the discussion by asking students to explain how they found the terms to add up to find the total number of triangles used in building Step 3. (By adding up the terms of \(T(n)\) from \(n=0\) to \(n=3\).) Discuss any insights in comparing the data from \(S(n)\) and \(T(n)\), and why \(T(n) = S(n-1)\).

Conclude the discussion by asking students to explain what it would mean to sum the terms in sequence \(S\) from \(n=0\) to \(n=3\)? (This sum doesn’t represent anything meaningful in this situation except perhaps the total number of sides you would have to draw to make both steps.) Contrast this with finding the sum of the terms through Step 3 of \(T\), which represents the total number of triangles in the Step 3 snowflake.