Lesson 2

Growth Patterns

These materials, when encountered before Algebra 1, Unit 5, Lesson 2 support success in that lesson.

2.1: How Big Can the Garden Be? (10 minutes)

Warm-up

In this activity, students encounter a diagram representing a context that has been described verbally. They then organize information from the diagram in a table and discuss how the table can help them see the pattern.

Students reason to extend the pattern. As students work, look for students who draw a picture, those who describe the picture, and those who rely on numerical (rather than visual) reasoning, especially for the fifth step. Invite them to share their strategies during the synthesis of this activity.

Launch

Arrange students in groups of 2 or allow them to work individually.

Student Facing

A homeowner wants to build a garden with concrete tiles around the outside. He has room for the garden to vary in length but not width. He’s not sure what size he wants the garden to be. Here are sketches of gardens that are 1, 2, and 3 meters long. The homeowner needs to know how many concrete tiles will be needed for different possible garden lengths. 

3 by 3 grided rectangle, center rectangle shaded. 3 by 4 grided rectangle, center 2 rectangles shaded. 3 by 5 grided rectangle, center 3 rectangles shaded.

  1. Create a table to show how many tiles will be needed if the garden is 1, 2, 3, 4, or 5 meters long.
  2. Describe the way the pattern is growing.

Student Response

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

Activity Synthesis

The purpose of this discussion is to highlight how organizing numbers in a table can help with recognizing patterns. Focus the discussion on how the table helped with describing the pattern. Here are some questions for discussion.

  • “What is the purpose of a table?” (Organize data in order to interpret what has been observed and help recognize patterns.)
  • “What are some advantages of using a table?” (Tables typically show data in columns and rows that are labeled. With tables, you can focus on a couple of numbers at a time. Tables communicate information using exact values, including decimals and fractions.)
  • “What are some disadvantages of tables?” (It can be difficult to understand all the data in a table, because they can include lots of detail. Tables don’t visually show trends in data over time.)

2.2: Representing Relationships (15 minutes)

Activity

In this activity, students are presented with an equation that represents a relationship. They then create a table and a graph of the relationship. Then, they consider which representation is more helpful to answer questions about the relationship.

This activity prepares students to compare two verbal descriptions of contexts and create a table of values to use to answer questions about the contexts in the associated Algebra 1 lesson. This activity is also useful for when students analyze or create the graph of a relationships from a table in upcoming lessons.

When students explain correspondences between equations, verbal descriptions, tables, and graphs, they are reasoning abstractly and quantitatively (MP2).

Launch

Arrange students in groups of 2 or allow them to work individually. If students struggle with the arithmetic, allow non-graphing calculators. The numbers in the table have been chosen to simplify the arithmetic. Some numbers in the questions are not divisible by 9, allowing students to focus on the usefulness of the tool to represent the data.

Student Facing

The equation \(C=\frac{5}{9}\left(F-32\right)\) describes the relationship between Celsius and Fahrenheit temperatures.

  1. Describe the relationship in words.
  2. Complete the table showing corresponding temperatures in Celsius and Fahrenheit.
  3. Make a graph that represents the relationship.
    temperature in degrees Fahrenheit temperature in degrees Celsius
    23  
    41  
    50  
    104  
    122  
    212  
    Coordinate plane 
  4. Which representation would you use to answer each question (the equation, table, or graph)? Be prepared to explain your reasoning.
    1. The temperature of an oven in Fahrenheit when it is set to \(100^\circ \text{ C}\).
    2. You are in Canada and the forecast is \(30^\circ \text{C}\). Will it be cold, warm, or hot outside?
    3. The outside temperature is \(\text-4^\circ \text{F}\). How would this temperature be reported in Celsius?

Student Response

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

Activity Synthesis

The purpose of this discussion is to highlight how different representations can be used for the same situation. In a previous activity, students translated a verbal description of a problem into a table and discussed the advantages of a table. Focus this discussion on how the table provided data points for the graph and what the advantages of a graphical representation might be. Here are some questions for discussion.

  • “What is the purpose of a graph?” (It allows you to see the overall shape of data and the trend of data. )
  • “What are some advantages of using a graph?” (Graphs show patterns, the ups and downs of data, in visual ways, and they also allow you to visually compare patterns. Graphs often communicate more information than tables.)
  • “What is a disadvantage of using a graph?” (Exact values can be difficult to read.)

2.3: Whose Representation is That? (15 minutes)

Activity

In this activity, students practice making connections between different representations of the same situation. If desired, they can also identify and create any missing representations (this would require removing some cards from the sets you distribute). The practice will be helpful when students make connections between verbal and tabular representations in the associated Algebra 1 lesson. A matching task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP7).

Launch

Arrange students in groups of 2. Distribute 1 set of slips from the blackline master to each group. (A complete set if students will only do the matching. A set that is missing a few cards if you would like for students to create some representations.)

Student Facing

Your teacher will give you a set of cards. Each card has either a verbal description, a table, or a graph. Find the three cards that represent the same situation. If one of the three representations for a situation is missing, use the blank cards to create that representation.

Student Response

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

Activity Synthesis

Select groups to share their matches and how they determined which verbal descriptions, tables, and graphs were a match. Look for groups that created different representations for the missing ones. Attend to the language that students use to describe their matching representations, giving them opportunities to describe them more precisely.