Materials & Prep
- Board number line from -15 to 15, with 0 clearly marked.
- One half-sheet practice and exit ticket per student. Prepare this problem set: 1) -4 + 7, 2) 6 + (-9), 3) -3 - 5, 4) 4 - (-6), 5) -8 - (-3), 6) A team begins at -6 in point differential and gains 9 points. What is its new differential? 7) A team begins at 5 and loses 8 points in point differential. What is its new differential?
- No materials beyond students' pencils and the usual board setup.
Opening
DisplayWrite three football point differentials from the standings-style context: Team A: -7, Team B: 0, Team C: +6. Ask, “Which team is farthest below even? Which is closest to even? If Team A improves by 3 points, should it be closer to 0 or farther away?” Students show the direction with a hand motion or state it.
ConnectMark -7, 0, and +6 on the board number line. Explain that students already used the key idea: a negative number is below 0, and improving moves a value toward the positive direction. Today they will use that movement to calculate additions and subtractions exactly.
Ask“Starting at -7, where would three moves to the right land?” Have students predict -4 before introducing any rule.
Direct instruction
IntroduceOn a number line, begin at the first number. Adding a positive means move right. Adding a negative means move left. The sign attached to the second number tells the direction because it describes the change being added.
Worked exampleSolve -4 + 7. Think aloud: “I start at -4 because it is the first number. I am adding +7, so I make seven jumps right: -3, -2, -1, 0, 1, 2, 3. I land on 3.” Draw the jumps and write -4 + 7 = 3.
Worked exampleSolve 6 + (-9). Think aloud: “I start at 6. I am adding a negative 9, so the change is left 9. I land at -3.” Emphasize that a negative sign is not automatically a subtraction sign. Here it describes the number being added.
IntroduceSubtraction asks us to take away a change. On a number line, subtracting a number means move in the opposite direction from that number. Subtract +5 means move left 5. Subtract -5 means move right 5.
AnalogyIf “add -5” is a five-point loss, then “subtract -5” means take away that loss. Removing a loss improves the differential, so the movement is right. This is why the direction reverses.
Worked exampleSolve 4 - (-6). Think aloud: “I start at 4. The operation is subtraction, so I use the opposite of the number being subtracted. The number being subtracted is -6, whose direction is left. Opposite direction means right 6. I land at 10.” Draw a small left arrow labeled -6, then cross it out and replace it with a right arrow labeled +6. Write 4 - (-6) = 10.
MisconceptionAsk students to vote silently: Does 4 + (-6) also move right because there are two signs? Reveal it moves left to -2 because this expression has only addition, and the negative belongs to the number being added. Two minus signs only create an opposite-direction situation when one is the subtraction operation and one is the sign of the number being subtracted.
Guided practice
Have studentsWork with a partner on a shared board number line or in notebooks. For each problem, one partner identifies the start and direction; the other draws jumps and explains the landing point. Then switch roles.
PostA) -3 - 5, B) -2 - (-4), C) 8 + (-11).
PromptRequire the sentence stem, “I start at . Because I am adding/subtracting , I move __ units and land at .”
Check for understandingAfter two minutes, have every student hold up 1 for left or 2 for right for this question: “For -2 - (-4), which direction do you move?” Cold-call one student to justify “right” using the phrase “opposite of a negative direction.” If students choose left, redraw the negative 4 as left movement and ask what it means to subtract, or remove, that movement.
ClarifySolve B together only after students commit. Start at -2 and move right 4 to land at 2. Contrast it directly with -2 + (-4): start at -2 and move left 4 to land at -6.
Independent work
StudentsComplete problems 1–7 on the half-sheet. Require number-line jumps for 1–5 and a written start-direction-landing explanation for problems 4 and 5. Problems 6 and 7 use football point differentials but still require a number line.
CirculateLook especially for students who begin at the second number, or who turn every pair of signs into addition. Ask, “What is the operation between the two numbers? What number is being subtracted? Which direction does that number represent before we take it away?”
ScaffoldGive students who need it a pre-drawn -15 to 15 number line and highlight the first number with a box. Have them physically trace each jump with a pencil point while saying “start, direction, distance, land.”
ExtensionStudents write and solve two expressions that both start at -5 and end at 2, one using addition and one using subtraction. They must draw both number lines and explain why the directions differ.
Closing
Exit ticketOn an index-sized slip, students solve and draw number lines for: 1) -7 + (-4), 2) -7 - (-4). Under each, complete: “I moved left/right because .”
Look for1) -11 with four jumps left, and 2) -3 with four jumps right. Sort tickets into secure, needs a direction check, and needs reteaching. Use the second group to open the next lesson by contrasting the two expressions again.