Lesson 11

Adding Up

11.1: Math Talk: Adding Terms (5 minutes)

Warm-up

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.

Launch

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.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Facing

Evaluate mentally.

\(\frac 1 2 + \frac 1 4\)

\(\frac 1 2 + \frac 1 4 + \frac 1 8\)

\(\frac 1 2 + \frac 1 4 + \frac 1 8 + \frac 1 {16}\)

\(\frac 3 2 + \frac 3 4 + \frac 3 8 + \frac 3 {16}\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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?”
Speaking: MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . . ."  Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. 
Design Principle(s): Optimize output (for explanation)

11.2: Paper Trail (15 minutes)

Activity

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.

Launch

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.

Representation: Access for Perception. Read the student task statement aloud. Students who both listen to and read the information will benefit from extra processing time. 
Supports accessibility for: Language

Student Facing

  1. Tyler has a piece of paper and is sharing it with Elena, Clare, and Andre. He cuts the paper to create four equal pieces, then hands one piece each to the others and keeps one for himself. What fraction of the original piece of paper does each person have?
  2. Tyler then takes his remaining paper and does it again. He cuts the paper to create four equal pieces, then hands one piece each to the others and keeps one for himself. What fraction of the original piece of paper does each person have now?
  3. Tyler then takes his remaining paper and does it again. What fraction of the original piece of paper does each person have now? What happens after more steps of the same process?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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.

Activity Synthesis

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

number of
cuts
0 1 2 3
Tyler 1 \(\frac14\) \(\frac{1}{16}\) \(\frac{1}{64}\)
each other
group member
0 \(\frac14\) \(\frac{5}{16}\) \(\frac{21}{64}\)

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.

Speaking, Representing: MLR8 Discussion Supports. Give students additional time to make sure that everyone in their group can explain their solutions and the relationships between the quantities represented. Invite groups to rehearse what they will say when they share with the whole class. Rehearsing provides students with additional opportunities to speak and clarify their thinking, and will improve the quality of explanations shared during the whole-class discussion.
Design Principle(s): Optimize output (for explanation)

11.3: A Threefold Design (15 minutes)

Activity

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.

Launch

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.

Student Facing

Here is a geometric shape built in steps.

  • Step 0 is an equilateral triangle.
An equilateral triangle. The bottom edge is horizontal. The left edge slants upward to the right. The right edge slants upward to the left to meet the other edge.
  • To go from Step 0 to Step 1, take every edge of Step 0 and replace its middle third with an outward-facing equilateral triangle.
A six pointed star. Begin with an equilateral triangle. Take every edge and replace its middle third with an outward-facing equilateral triangle. Replace the removed segments with a dashed line.
  • To go from Step 1 to Step 2, take every edge of Step 1 and replace its middle third with an outward-facing equilateral triangle.

  • This process can continue to create any step of the design.

  1. Find an equation to represent function \(S\), where \(S(n)\) is the number of sides in Step \(n\). What is \(S(2)\)?
  2. Consider a different function \(T\), where \(T(n)\) is the number of new triangles added when drawing Step \(n\). Let \(T(0)=1.\) How many new triangles are there in Steps 1, 2, and 3? Explain how you know.
  3. What is the total number of triangles used in building Step 3?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Suppose the Step 0 triangle has area 1 square unit. Complete the table.

step area
0 1
1  
2  
3  

What patterns do you notice?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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.

Activity Synthesis

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.

Lesson Synthesis

Lesson Synthesis

Ask students to consider what it would mean to find the sum of the sequence defined by \(f(0)=\frac{3}{10},f(n)=\frac{1}{10} \boldcdot f(n-1)\) for \(n\ge1\). (The sum of the terms of \(f(n)\) appears to get closer and closer to \(\frac13\) the more terms you add together.) After some quiet work time, select students to share their thinking, recording their ideas for all to see. Highlight in particular any student that connects back to the Paper Trails activity, in which the sum of the fractions each person has also appears to get closer and closer to \(\frac13\) the more times Tyler cuts up and passes out pieces from his original sheet of paper.

11.4: Cool-down - Half the Homework (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

The sum of a sequence is the sum of its terms.

For example, suppose you were given \$1 on the first day, then \$2 the second day, then \$4 the third day, and it doubled each day for seven days. After finding each term of the sequence, you can find the sum:

\( 1 + 2 + 4 + 8 + 16 + 32 + 64 = 127 \)

For these seven days, the total amount of money is \$127. In a later unit, you will learn a method to find the sum of a geometric sequence more efficiently.