Tell students that they will be playing a strategy game in which
they must sink their opponent's ships. To win the game students will
need to use their knowledge of graphing and linear equations.
Break the class up into pairs. Depending on the ability levels
of your students, you may choose to allow them to pick their own
partners or separate them into pre-determined pairs that are matched
for mathematical ability. Distribute the Equations of Attack activity sheet, Slope Cards activity sheet, 2 different-colored pencils, a coin, and scissors to each pair of students. Note:
If your class is just beginning to explore linear equations, you may
wish to create your own set of slope cards with only integers (e.g., 2
and –3) and unit fractions (e.g., 1/4, but not 3/4). Likewise, to challenge more advanced students, consider including decimal slopes (e.g., 1.5).
Read through the questions and game rules with students. You may
also want to go through an example of the game on the board before
students begin. Draw a ship at (1,5) and one at (2,7). Tell students to
assume that you drew a slope card with a value of 3, and that you have
the odd-numbered cannons. Ask students which cannon would be the best
to use given the location of the ships and the slope. Show students
that if you choose 1 as your cannon location, the line you draw
intersects (and sinks!) the ship at (2,7).
Since students will have to write the equations for their lines
of attack, you may wish to write the equation for this line on the
y = 3x + 1
If students struggle with the example, you may choose to do
another example or two. Since the player has the odd-numbered cannons,
the other possible lines of attack would be:
- y = 3x + 3
- y = 3x + 5
- y = 3x + 7
- y = 3x + 9
If some students in your class seem to understand while others
continue to struggle, have a student who understands come to the board
and draw the line of attack to determine whether either of the two
ships is sunk.
In general, the equations will be:
y = (slope)x + (cannon position)
However, try not to share this with students. They should discover this pattern and its meaning on their own.
Playing the Game
Have students start by cutting out and stacking the slope cards
face down. As students move on to plotting their ships, walk around and
make sure they plot them correctly. They may try to color in blocks or
choose locations between points rather than at the lattice points. You
may choose to check the game boards (ship and cannon locations) before
students start, to ensure the desired results.
In playing the game, students should use their color to draw
their line on the game board, from their cannon and using the correct
slope, to see if the line intersects any of their opponent’s ships. Do
not correct students if they draw their lines incorrectly—this should
come up in the second half of the lesson when they use algebra to
figure out the paths of the cannonballs. However, if you notice a large
number of students drawing lines incorrectly, you may choose to pause
play and do another example or two on the board. Encourage students to
use vocabulary words, such as slope and y-intercept, and to name the points as (x, y)
coordinate pairs. As students are playing their games, remind them to
list their equations in Question 1. Student pairs can share an activity
sheet or record their answers separately.
Discussing the Algebra
When all students have played several rounds, stop the play to ask a few questions:
- How do you choose which of your cannons to use?
[Answers may vary, but students may discuss the different
slopes. If the slope is a negative, they may choose a cannon that is
higher on the y-axis, and vice versa. Students may also discuss the location of their opponent's ships.]
- Can you tell that your equation will sink a ship before you graph it? How?
[Yes, but not all students may realize how at this point.]
Use the second question above as a segue into having students answer the remaining questions on the activity sheet. After all students have had sufficient time to answer the questions, go over them as a class.
Through discussion in their pairs and as a class, students
should be able to answer Question 4. The easiest way to find the answer
without graphing is to substitute the coordinate pair of the opponent's
ships into the equation.
If students have trouble coming up with this strategy, ask them what the x and y mean in a linear equation. It is often surprising how few students know the answer to this question. Each (x,y)
pair that is a solution of the equation represents a point on the line.
You can use the prior example with ships at (2,5) and (2,7), slope = 3,
and cannon (y-intercept) = 1, to show what happens when you substitute in the 2 different points:
|equation: y = 3x + 1|
|plug in: (1,5)
5 3·1 + 1
5 3 + 1
5 ≠ 4
Since the point does not make the equation true,
(1,5) is not a point on the line.
The ship is not sunk.
| ||plug in: (2,7)
7 3·2 + 1
7 6 + 1
7 = 7
Since the point does make the equation true,
(2,7) is a point on the line.
The ship is sunk.
Questions for Students
1. Is there a ship placement that is totally safe from the cannons? If so, where? If not, why?
[The answer to the question depends on the slope cards
students are given. For example, if all the possible slopes are
positive, then none of the points along the x-axis can ever be hit.]
2. Do you prefer to find if the ships are sunk by graphing or by using algebraic substitution?
[Answers will vary. Discuss with students the pros and cons
of both methods. For example, graphing is quicker for most students, so
they might find the time it takes to find a solution using algebra to
be a con.]
students able to play the game with the given instructions? If not, how
could you better explain the game before they start?
- Did students have difficulty writing the coordinate pairs for their ships? How could you provide instruction for this?
- Were students able to write the equations of the lines? How could you better scaffold this skill?
- Did students understand how to graph using the slope cards and cannon locations?
- Did students come up with their own strategies for the game, or did they need intervention?
- Did students work well in pairs? How would you pair students differently next time?
- Plot and name points on a coordinate grid using correct coordinate pairs
- Graph lines given slope and y-intercept
- Practice writing equations given slope and y-intercept
- Determine algebraically if a point lies on a line
Common Core State Standards – Mathematics
Grade 8, Expression/Equation
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
Grade 8, Functions
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.