Learning intention LI
- Interpret integers as positions on a number line and as directions (left/right).
- Understand a vector as having size (length) and direction.
- Model integer addition/subtraction by moving and then connect to symbolic expressions.
- Explain why expressions with combined signs work (e.g., 3 − (−2)).
Success criteria SC
- I can describe positive/negative as direction on a number line (right/left).
- I can explain what a vector is: size + direction.
- I can predict where a person will land for an expression like 3 + (−2).
- I can justify facing direction and movement for subtraction cases (e.g., −4 − (+3)).
- I can solve integer questions and explain using the model.
Teacher tool shortcuts
Keyboard
Space new question •
←/→ step •
Enter skip animation •
R reset
Teacher tool: https://millsmathstools.au/integer-combined-signs/
Materials & setup Prep
- Project the Teacher Explore tool (Integer Combined Signs).
- Mini whiteboards + markers + erasers (for prediction + reasoning).
- Outdoor space (playground/courtyard) for a human number line (optional but powerful).
- Chalk/cones/tape to mark a number line from −5 to 5 (or −10 to 10).
- Student devices (optional): Student Quiz; Worksheet Creator printout.
Suggested starting expressions: 3 + (−2), −1 + (+4), 2 − (+5), 3 − (−2).
E Experience — recall & direction (10–15 min)
Teacher actions Do
- Ask students to recall: “What does positive mean? What does negative mean?”
- Steer the discussion to direction: positive as “right/forward”, negative as “left/backward”.
- Draw a quick number line and place a few integers (−3, 0, 4). Ask: “Which way is increasing?”
- Introduce: vector = a line/arrow with size (length) and direction.
Quick prompts Ask
- “If you move +3, which way do you travel? How far?”
- “If you move −2, which way do you travel? How far?”
- “Where is 0 and why is it important?”
- “What does the sign tell you: direction or size?”
Bridge statement: “Today we’ll treat each number as a movement — a vector — so we can make sense of combined signs in addition and subtraction.”
L Language — build the vocabulary (10 min)
Key terms (display + co-construct)
- Integer: whole numbers and their negatives (…−2, −1, 0, 1, 2…)
- Number line: positions arranged in order.
- Direction: left/right (or backward/forward).
- Vector: has size (how far) and direction (which way).
- Start value: where we begin; result is where we land.
Mini-whiteboard sentence frames
- “A vector shows _____ and _____.”
- “The sign tells me the _____.”
- “I start at _____ and move _____ steps to the _____.”
- “I land on _____ because _____.”
Use rapid “show me” checks. Select 2–3 students to justify with language + pointing to the number line.
Teacher moves to strengthen reasoning
- Revoice: “So you’re saying the sign tells the direction…”
- Press for evidence: “Where do you see that on the number line?”
- Connect representations: “Say it as movement, then write it as an expression.”
P Pictorial — notice & wonder with the animation (15–25 min)
Teacher actions Do
- Load the Teacher Explore tool: https://millsmathstools.au/integer-combined-signs/
- Run a Notice & Wonder routine on the first few questions.
- Give students time to predict using mini whiteboards before stepping forward.
- Choose expressions that create productive discussion (include negatives and subtracting negatives).
Notice & Wonder prompts Ask
- “On the next step, will the person face left or right?”
- “Will the person be walking forwards or backwards?”
- “What number will the person land on?”
- “What does the first arrow represent? What does the second arrow represent?”
- “What stays the same across examples? What changes?”
Core routine: “Predict → Reveal → Explain”
- Predict (whiteboards): facing direction, walking direction, landing point.
- Reveal one step at a time (use arrows/animation).
- Explain: “Which part of the expression caused that direction?”
- Generalise: write a simple rule students can test on the next example.
Key idea: “The first number sets the start position. The second number (or the subtraction) determines the movement as a vector.”
S Symbolic — connect movement to expressions (10–20 min)
Teacher-led modelling (I do / we do)
- Write an expression (e.g., 3 + (−2)) and narrate:
- Start at 3.
- Add −2 means move 2 left.
- Land on 1.
- Repeat with a subtraction case (e.g., 3 − (−2)) and discuss why it moves right.
Encourage students to use precise language: “subtracting a negative means the movement reverses.”
Student practice (you do)
- Give 4–6 expressions. Students must write:
- (1) start position,
- (2) direction + distance for the movement,
- (3) landing value,
- (4) one sentence justification using “vector / direction / size”.
Common misconceptions to surface
- Thinking the negative sign always means “go left” even when subtracting a negative.
- Confusing facing direction (operator) with walking direction (movement amount).
- Losing track of the start value (treating everything as starting at 0).
Teacher move: “Point to where we start. Now point to the movement. What does the sign tell us each time?”
A Application — human number line + independent practice (15–30+ min)
Human number line (outside) Do
- Take mini whiteboards outside and mark a number line from −5 to 5 (or −10 to 10).
- Select a student and give an expression (e.g., 3 + (−2)).
- Ask the class to predict: facing direction, walking direction, landing number.
- Student performs the operation using the same actions as the tool’s animation.
- Repeat with a subtraction of a negative (e.g., −1 − (−3)) to highlight the “double negative” case.
Optional quick roles: one student is the “walker”, one student is the “director” (reads steps), one is the “checker” (confirms landing position).
Student Quiz + Worksheet Creator Practice
- Student Quiz: https://millsmathstools.au/integer-combined-signs/student-quiz
- Worksheet Creator: https://millsmathstools.au/integer-combined-signs/worksheet-creator.html
How the Student Quiz works: Students select a difficulty level and complete a set of questions.
On Sweet, the tool provides step-by-step hint animations that show the movement. On Mild, students
still see support (e.g., the first step and/or facing direction), and on higher levels the hints reduce so students must
predict and justify independently. Encourage students to pause before revealing the next step and record their prediction first.
Suggested pathway: Sweet (accuracy) → Mild (reduced hints) → Medium/Spicy (challenge).
Extension / transfer tasks
- Darby challenge: Darby is counting backwards from 5321 by 8s until he reaches the first negative number. What is this number? Explain your reasoning.
- Create your own “movement story” for a − (−b) that a Year 7 student would understand.
- Write two different expressions that land on the same result (e.g., two ways to land on −4) and justify using vectors.
Assessment for learning AFL
Checks during the lesson
- Mini-whiteboard predictions before each reveal: facing direction, walking direction, landing value.
- Cold-call 2–3 students to justify: “Which sign caused that move?”
- Listen for vocabulary: vector, direction, size/length, start position, result.
Exit ticket (2 minutes)
- Solve: −2 − (+5) and 3 − (−4).
- For each, write:
- start position,
- movement (direction + distance),
- final result,
- one sentence explanation using “vector”.
Differentiation Support / Extend
Support
- Keep numbers small first: use the human number line (−5 to 5) and Sweet level hints.
- Use sentence frames: “Start at __. Move __ steps __. Land on __.”
- Pair students: one predicts, one explains, then swap roles.
- Limit to cases like a + (−b) before introducing subtraction of negatives.
Extension
- Move to Medium/Spicy ranges and include mixed sign combinations.
- Ask students to generalise rules for:
- a − (+b),
- a − (−b),
- (−a) + (−b).
- Introduce fractional steps (if using Spicy) and discuss what “0.5 step” means on the number line.
Teacher notes Notes
- Keep the focus on structure: start position → movement vector → landing value.
- When students claim a result, press: “Which part of the expression caused that direction?”
- Use the outdoor number line as a bridge from pictorial animation to embodied understanding.
- Revisit the vocabulary “vector = size + direction” throughout, not just at the start.