Lesson 9

What’s the Equation?

9.1: Math Talk: Multiplying Fractions (10 minutes)

Warm-up

This is the first math talk activity in the course. See the launch for extended instructions for facilitating this activity successfully.

The purpose of this Math Talk is to elicit strategies and understandings students have for interpreting an exponential function and for multiplying fractions. These understandings help students develop fluency and will be helpful in later activities when students define geometric sequences using equations.

If students need a deeper dive in recalling how to multiply fractions, here is a string of expressions to try before evaluating the expressions in the activity:

  • \(\frac14 \boldcdot 20\)
  • \(\frac34 \boldcdot 20\)
  • \(\frac34 \boldcdot 6\)
  • \(\frac34 \boldcdot \frac34\)

Launch

This is the first time students do the math talk instructional routine, so it is important to explain how it works before starting.

Explain the math talk routine: one problem is displayed at a time. For each problem, students are given a few minutes to quietly think and give a signal when they have an answer and a strategy. The teacher selects students to share different strategies for each problem, and might ask questions like “Who thought about it a different way?” The teacher records students' explanations for all to see. Students might be asked to provide more details about why they decided to approach a problem a certain way. It may not be possible to share every possible strategy for the given limited time; the teacher may only gather two or three distinctive strategies per problem. 

Consider establishing a small, discreet hand signal that students can display to indicate that they have an answer they can support with reasoning. This signal could be a thumbs-up, a certain number of fingers that tells the number of responses they have, or another subtle signal. This is a quick way to see if the students have had enough time to think about the problem. It also keeps students from being distracted or rushed by hands being raised around the class.

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

For the function \(f(x)=32\boldcdot\left(\frac34\right)^x\), evaluate mentally:

\(f(0)\)

\(f(1)\)

\(f(2)\)

\(f(3)\)

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. Answers of the form “24 times \(\frac34\)” for \(f(2)\), for example, should be taken as correct and an opportunity for students to share how they work out mentally that the expression is equivalent to 18. To involve more students in the conversation, consider asking:

  • “Who can restate \(\underline{\hspace{.5in}}\)’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 \(\underline{\hspace{.5in}}\)’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)

9.2: Take the Cake! (15 minutes)

Activity

The purpose of this activity is for students to write definitions for a geometric sequence given by a situation. Students must translate from the description to mathematical representations (MP2). The context puts some limitations on the domain, and students engage in aspects of mathematical modeling (MP4) by articulating why. For example, there is a difference between what numbers we can substitute for \(n\) in an equation representing the amount of cake left after the \(n^{\text{th}}\) person takes their piece and what numbers make sense to substitute for \(n\). There comes a point where the amount of cake is too small to reasonable expect a knife to slice or a scale to measure \(\frac13\) by weight.

Launch

Display the task statement for all to see and tell students to read the description of the situation. Then ask, “Is there still cake left after three people each take some cake?” After a brief quiet work time, select students who reply “yes” to explain their thinking. If any of them created a diagram, such as a circle split into smaller and smaller pieces, display it for all to see. Ensure students understand that each new person takes \(\frac13\) of what is left, not \(\frac13\) of the original amount.

Student Facing

A large cake is in a room. The first person who comes in takes \(\frac13\) of the cake. Then a second person takes \(\frac13\) of what is left. Then a third person takes \(\frac13\) of what is left. And so on.

  1. Complete the table for \(C(n)\), the fraction of the original cake left after \(n\) people take some.
  2. Write two definitions for \(C\): one recursive and one non-recursive.
  3. What is a reasonable domain for this function? Be prepared to explain your reasoning.
 \(n\) \(C(n)\)
0  
1 \(\tfrac23\)
2  
3  
4  

Student Response

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

Activity Synthesis

The goal of this discussion is for students to articulate an appropriate domain for \(C\) given the context. Begin the discussion by selecting students to share their definitions, recording these for all to see. If not brought up by students, connect the definition for the \(n^{\text{th}}\) term of \(C\) with a general form of an exponential equation such as \(g(x)=a\boldcdot b^x\), where \(a\) represents the starting term and \(b\) represents the growth factor. Students should understand that since \(C\) is a geometric sequence, it is a type of exponential function that has a domain restricted to a subset of the integers.

Specifically, the domain of \(C\) without regard to the context is the integers \(n\) where \(n \ge 0\), however, realistically, there will eventually be too little cake to reasonably take \(\frac13\) of it. \(C\) can be quickly evaluated using technology, for example, and the table in this activity can be extended to show successive inputs. Deciding when there is too little cake left to reasonably subdivide is up to the modeler. After 8 people take their cake, less than 5% is left and after 12 people, less than 1% is left. Depending on the assumed size of the original cake, these might be reasonable limits on the domain. Students may also argue that if an accurate measurement scale is used, the slices could be split by weight, meaning a higher value of \(n\) is possible.

It is not crucial at this time that students represent the domain using inequality notation. For example, for the last question, they might say “integers from 0 to 10.”

Reading, Writing: MLR 5 Co-Craft Questions. Use this routine to help students consider the context of this problem. Show the first part of the task (through “and so on”), hiding the \(\frac13\)’s. Ask students to write their own mathematical questions that could be asked about the situation, and look for whether or how the amount of remaining cake shows up in students’ questions, and whether students assume that each person takes the same fraction of the cake. Once students compare their questions, reveal the \(\frac13\)’s and the activity’s questions. For students struggling to get started, ask them to create a diagram to represent the situation, or have two students act out the situation (use a piece of paper to represent the cake, and repeatedly remove \(\frac13\) of it). 
Design Principles: Maximize Meta-awareness, Support Sense-making
Representation: Internalize Comprehension. Represent the same information through different modalities by using physical objects to represent abstract concepts. Some students may benefit by creating a table of values for the non-recursive equations.
Supports accessibility for: Conceptual processing; Visual-spatial processing

9.3: Fibonacci Squares (10 minutes)

Activity

The pattern in this activity creates the Fibonacci sequence. One purpose of this task is for students to understand that sometimes it is straightforward to define a sequence recursively and difficult to determine a definition for the \(n^{\text{th}}\) term. This task also challenges students to expand their notion of recursive definitions since the current term in a Fibonacci sequence depends on the previous two terms rather than just the previous term.

This sequence was chosen to provide a contrast to the cake activity. With the cake, students had a situation whose domain does not reasonably extend above a certain value given physical constraints. Since this sequence is a mathematical construct, the domain has no upper bound restrictions on it.

Launch

Distribute 1 piece of graph paper to each student. Ask them to complete the first question and pause. Ensure that they are interpreting the description of the pattern correctly before they proceed with the rest of the activity.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time.
Supports accessibility for: Organization; Attention

Student Facing

  1. On graph paper, draw a square of side length 1. Draw another square of side length 1 that shares a side with the first square. Next, add a 2-by-2 square, with one side along the sides of both of the first two squares. Next, add a square with one side that goes along the sides of the previous two squares you created. Next, do it again.
    Pause here for your teacher to check your work.
  2. Write a sequence that lists the side lengths of the squares you drew.
  3. Predict the next two terms in the sequence and draw the corresponding squares to check your predictions.
  4. Describe how each square’s side length depends on previous side lengths.
  5. Let \(F(n)\) be the side length of the \(n\)th square. So \(F(1)=1\) and \(F(2)=1\). Write a recursive definition for \(F\).

Student Response

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

Student Facing

Are you ready for more?

  1. Is the Fibonacci sequence arithmetic, geometric, or neither? Explain how you know.

  2. Look at quotients \(\frac{F(1)}{F(0)}, \frac{F(2)}{F(1)}, \frac{F(3)}{F(2)}, \frac{F(4)}{F(3)}, \frac{F(5)}{F(4)}\). What do you notice about this sequence of numbers?

  3. The 15th through 19th Fibonacci numbers are 610, 987, 1597, 2584, 4181. What do you notice about the quotients \(\frac{F(16)}{F(15)}, \frac{F(17)}{F(16)}, \frac{F(18)}{F(17)}, \frac{F(19)}{F(18)}\)?

Student Response

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

Anticipated Misconceptions

Some students may not realize that it is okay to use both \(F(n-1)\) and \(F(n-2)\) when defining a function recursively. Encourage these students to think about how the “previous side lengths” for term \(F(n)\) can be expressed using function notation.

Activity Synthesis

Display the sequence 1, 1, 2, 3, 5, 8, 13 and the recursive definition \(F(1)=1, F(2)=1, \) \(F(n)=F(n-1)+F(n-2)\), \(n\ge3\) for all to see and check their answer against. Either give students a minute to think about how to write a definition for the \(n^{\text{th}}\) term of the sequence, or just tell them that it is more complicated to derive than class time allows. An important modeling point to make is that the goal is to create a model that suits the needs of what you are trying to do. If we wanted to know the value of \(F(100)\), then a spreadsheet and a recursive definition is likely going to be faster than working out a definition for the \(n^{\text{th}}\) term.

Ask students, "What is the domain of this function? Does it have any restrictions?" (The integers \(n\) where \(n\ge1\) or equivalent. We can always add another square onto the image, even if only in our imaginations once the squares get too big.) Select students to share their responses.

If time allows, tell students that we could define the Fibonacci sequence as \(F(0)=1, F(1)=1, \) \(F(n)=F(n-1)+F(n-2)\), \(n\ge2\) and we would still get the same list of numbers.

Lesson Synthesis

Lesson Synthesis

The purpose of this activity is to explicitly make the connection between an arithmetic sequence and linear functions just as the connection was made earlier in the lesson between geometric sequences and exponential functions.

Arrange students in groups of 2. Display the pattern for all to see and tell students the pattern represents a sequence \(W\) where \(W(n)\) is the number of white circles in Step \(n\).

A pattern.

Give students a brief quiet think time to write a description for how the pattern grows, and then ask them to share their description with their partner. Once partners have a chance to share, tell the groups to define \(W\) both recursively and for the \(n^{\text{th}}\) term (\(W(1) = 4, W(n)=W(n-1)+3\) for \(n \ge2\) or \(W(n)=4+3(n-1)\) for \(n\ge1\) or equivalent).

After 2–4 minutes of work time, select groups to share their definitions, recording them for all to see. Include any tables or graphs students made as they made sense of the pattern. In particular, highlight any student who reasoned about the definition for the \(n^{\text{th}}\) term by introducing a “Step 0” and using the form \(y=mx+b\) with \(b=1\) and \(m=3\). Since arithmetic sequences are a type of linear function, this is a good strategy so long as students remember to note any bounds on the value of \(n\) based on the situation. In this case, as with the Fibonacci sequence, \(n\) has no upper bound.

9.4: Cool-down - Ow, My Jaw (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

The model we use for a function can depend on what we want to do.

For example, an 8-by-10 piece of paper has area 80 square inches. Picture a set of pieces of paper, each half the length and half the width of the previous piece.

A sequence of pieces of paper.

Define the sequence \(A\) so that \(A(n)\) is the area, in square inches, of the \(n^{\text{th}}\) piece. Each new area is \(\frac 1 4\) the previous area, so we can define \(A\) recursively as

\( A(1) = 80, A(n) = \frac 1 4 \boldcdot A(n-1) \text{ for } n \ge2\)

But for \(n\)-values larger than 5 or 6, the model isn't realistic since cutting a sheet of paper accurately when it is less than \(\frac{1}{50}\) of a square inch isn't something we can do well with a pair of scissors. We can see this by looking at the graph of \(y=A(n)\) shown here.

If we wanted to define the \(n^{\text{th}}\) term of \(A\), it's helpful to first notice that the area of the \(n^{\text{th}}\) piece is given by \(80 \boldcdot \frac 1 4 \boldcdot \frac 1 4 \boldcdot \cdots \boldcdot \frac 1 4\), where there are \(n-1\) factors of \(\frac14\). Then we can write

\( A(n) = 80 \boldcdot \left(\frac 1 4\right)^{n -1}, n\ge1\)

Graph of a discrete function. Area in square inches. Piece number.

We can use this definition to calculate a value of \(A\) without having to calculate all the ones that came before it. But since there are fewer than 10 values that make sense for \(A,\) since we can’t cut very tiny pieces using scissors, in this situation we could just use the first definition we found to calculate different values of \(A\).