The Language of Algebra Equations, Tables, and Graphs Goal 5

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The Language of AlgebraEquations, Tables, and
GraphsGoal 5

• Grade 8
• NC SCOS objectives
• Sandra Davidson, MaEd
• NBCT EA Math

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NC SCOS Objectives (2 wks)

• 5.01 Develop an understanding of function.
• a) Translate among verbal, tabular, graphic, and
  algebraic representations of functions.
• b) Identify relations and functions as linear or
  non-linear.
• c) Find, identify, and interpret the slope (rate
  of change) and intercepts of a linear relation.
• d) interpret and compare properties of linear
  functions from tables, graphs, or equations.
• 5.02 Write an equation of a linear relationship
  given two points, the slope and one point, or
  the slope and y-intercept.
• 5.03 Solve problems using linear equations and
  inequalities, justify symbolically and
  graphically.
• 5.04 Solve problems using inverse relationships
  of addition/subtraction, multiplication/division,
  squares/square roots, and cubes/cube roots.

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1. Crossing the River find a rule, write an
equation

• How many one-way trips does it take for the
  entire group of 8 adults and 2 children to cross
  the river?
• What is there were 5 adults and 2 children?
• What if there were 4 adults and 6 children?
• What is there were a adults and c children?
• Play Mystery Bag Equations

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2. Three Bean Salad Each Salad contains red
beans, lima beans, and black- eyed peas

• Salad 1 contains
• The same number of red beans as lima beans
• Three more black-eyed peas than red beans
• A total of 18 beans
• What is the number of each type of bean?
• 3r 3 18
• red5, lima5, and black8
• What if there were 54 beans in this mixture?

• Salad 2 contains
• Three times as many red beans as black-eyed peas
• One more lima bean than red beans
• Eight beans in all
• What is the number of each type of bean?
• 7b 1 8
• black1, red3, lima4
• What if there were 48 beans in this mixture?

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Three Bean Salad(continued)

• Salad 3 contains
• An equal number of red beans and black-eyed peas
• Five more lima beans than red beans
• No more than 20 beans
• What is the number of each type of bean?
• 3r 5 20
• red5, black5, lima10
• What if there were no more than 100 beans in this
  mixture?

• Salad 4 contains
• At least twelve beans
• One more lima bean than red beans
• One more red bean than black-eyed peas
• What is the number of each type of bean?
• 3b 3 12
• black3, red4, lima5
• What if there were at least 72 beans in this
  mixture?

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3. Border Patrol (NC DPI Strategies)
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4. From Graphs to Stories (NCTM Navigating
Through Algebra)

• John and his father participate in a 100-meter
  race. John started the race 3 seconds after his
  father began to run. The graph provides
  information about how far John and his father ran
  over time. Create a story about who won the
  race be descriptive about how the race was run.
  If the two lines describing how each person ran
  were parallel, what would the graph tell you
  about who won the race?

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Graphs to Stories (continued)

• The graph represents the relationship between the
  profit and the amount of lemonade sold at a
  lemonade stand. Create a story about the
  lemonade stands profit. What is indicated when
  the line is below zero and when the line crosses
  the horizontal axis.

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Graphs to Stories (continued)

• This graph represents a swimming race that
  occurred between two middle-grades students.
  Create a story that describes what happened in
  this race.

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Connect Stories to GraphsGraphs can be prepared
from information in a story

• Tanyas house is 1 mile away from her school.
  She started out to school at 800 A.M. On her
  way to school, Tanya walked for 5 min. going
  about ¼ of a mile. She then spent 5 min. waiting
  for her friend Pedro. Pedro wasnt ready, and
  Tanya was worried that she would be late for
  school. She walked for 15 more min. to get to
  school. Pedro started from his house and ran to
  school in 10 min. He got there at the same time
  as Tanya arrived.

• As a class, create a distance-time graph and
  answer the questions below.
• How did you decide what scale to use on the axis
  of each of your graphs.
• Are all the lines on your graphs straight? Why?
• When two lines are on the same graph, what does
  it tell you when one is steeper than the other?
  Why?

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Stories to GraphsDraw a graph to represent each
of the stories.

• 1. In a walking experiment, Josephine walked a
  total distance of 40 ft. At the halfway point,
  she had walked for 25 seconds. She stopped for 5
  seconds to tie her shoe and then continued
  walking for 25 more seconds. Sketch a graph that
  shows Josephines distance from the starting
  point over time.

• 2. You are mowing the lawn. As you mow, the
  amount of grass to be cut decreases. You mow at
  the same rate until about half the grass has been
  cut. Then you take a break for a while. Then,
  mowing at the same rate as before, you finish
  cutting the grass. Sketch a graph that shows how
  much uncut grass is left as you mow, take your
  break, and finish mowing.

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Make your own Distance-Time or Speed-Time Graph

• Write a creative story involving distance and
  time or speed and time. The story should be at
  least one page in length. Draw a graph that goes
  with your story. Be prepared to present your
  story and graph to the class.

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5. Exploring Patterns (NCTM Navigating Algebra)
1. For each house, determine the total number of
pieces needed. Organize your information in a
table. 2. Sketch what house 5 would look like.
How many total pieces are needed?
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Exploring Patterns (continued)
3. Predict the total number of pieces you will
need to build house 15. Explain your
reasoning. 4. Write a rule that gives the total
number of pieces needed to build any house in
this sequence.
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Exploring Patterns (continued)
1. Use a pattern from the shapes above to sketch
the fifth shape, then find the perimeter of the
fifth shape. 2. Write a formula that you could
use to find the perimeter of any shape n.
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Exploring Patterns (continued)
3. Use your sketch of the fifth shape to find the
total number of toothpicks needed to make the
fifth shape. 4. Write a formula that you could
use to find the total number of toothpicks of any
shape n.
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6. How can words, tables, equations, and graphs
help describe the relationship between
Fahrenheit and Celsius?

• The equation F 9/5C 32 describes the
  relationship between Fahrenheit and Celsius.
• Write a description of the relationship between
  Fahrenheit and Celsius in words.
• Make a table showing at least five pairs of
  values for Celsius (C) and Fahrenheit (F).
• Make a graph that shows the relationship between
  Fahrenheit and Celsius. You will need to choose
  an appropriate scale for the axes on your graph.
• A temperature of 0 degrees Celsius is the same as
  a temperature of 32 degrees Fahrenheit. How is
  this shown in each of your representations?

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References

• Balanced Assessment Middle Grades
• Exemplary Mathematics Assessments Tasks for
  Middle Grades
• MAPS
• Mathematical Understanding
• Mathematics Learning
• Mathematics Teaching in the Middle School (March
  2001)
• Mathscapes
• NCDPI Strategies
• NCTM Navigating Through Algebra Grades 6-8
• The Regents of the University of California