Gradient, midpoint and distance: a week of ready-to-teach maths

Five lessons for Year 9 Algebra. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus gradient sort cards, midpoint match cards, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets, one copy per pair, and cut them before Day 2: the gradient sort cards are for Day 2 and the midpoint match cards for Day 3.
4. Calculators are needed on Days 4 and 5 for the decimal of a distance; the rest is by hand.
5. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.

The idea in plain words

Two points on the Cartesian plane are joined by a segment, and everything about the segment comes from two differences: the change in x and the change in y, each found by subtracting the first point’s coordinate from the second’s, sign and all. The gradient is rise ÷ run, the change in y over the change in x: positive for a segment that climbs to the right, negative for one that falls, zero for a flat one and undefined for an upright one. The midpoint averages the two x-coordinates and the two y-coordinates. The distance is the hypotenuse of the right triangle whose legs are the two changes, so Pythagoras gives it, and it is often an irrational number best kept in exact surd form.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

How we write these answers

Points are written (x, y). A gradient is given as a whole number or a fraction in simplest form, such as −1/2, never as a rounded decimal. A distance is given exactly, as 5 or 3√5, and then to two decimal places when the exact value is a surd. Units on the plane are the same length across and up, so a 3-4-5 triangle on the grid really is a 3-4-5 triangle.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 sets up the two differences that every formula reuses; Days 2, 3 and 4 turn them into the gradient, the midpoint and the distance; and Day 5 uses all three to describe shapes and places. It builds on Year 8, where students graphed linear relations on the Cartesian plane, met the gradient as a rate of change and used Pythagoras’ theorem in right triangles, and on the real numbers and surds earlier in Year 9. It prepares the equations of straight lines and the coordinate geometry of later years.

Materials for the week (one trip)

• From the classroom: rulers, pencils, scissors, envelopes for the cards, calculators for Days 4 and 5, this pack printed.
• Cut-out sheet 1, one per pair: 12 gradient sort cards (Day 2).
• Cut-out sheet 2, one per pair: 16 midpoint match cards (Day 3). No other equipment to buy.

Words for the word wall

Cartesian plane, coordinates, origin, quadrant, x-axis, y-axis, segment, interval, change in x, change in y, rise, run, gradient, parallel, undefined, midpoint, average, distance, hypotenuse, Pythagoras, surd, exact value, collinear

Dear families

This week in maths, Year 9 is working on the coordinate plane: given two points, how steep is the segment joining them (its gradient), where is its middle (the midpoint) and how long is it (the distance)? All three come from two simple differences: how far across and how far up you go from one point to the other.

Try this at home

• Ask: from (1, 2) to (4, 8), how far across and how far up? (3 across, 6 up, so the gradient is 6 ÷ 3 = 2.)
• Ask: what is halfway between (2, 4) and (8, 10)? (Average each: (5, 7).)
• Ask: how far is it from (0, 0) to (3, 4)? (5, from Pythagoras: 3² + 4² = 25.)
• On a street map with a grid, pick two places and estimate the straight-line distance between them in grid squares.

My coordinate week

Fill one row a day. Write two points from class and what you found about the segment joining them.

Printed from the free seegongsik Gradient, midpoint and distance teaching pack · seegongsik.com/au/y9/algebra/AC9M9A03/pack

Two points, two differences

Every formula this week starts from two numbers: the change in x and the change in y from one point to another, each found by subtracting the first point’s coordinate from the second’s, sign and all.

We are learning to

• read and plot points in all four quadrants,
• find the change in x and the change in y from one point to another, with their signs,
• use the two changes to move from a point to another point on the grid.

Success criteria

• I can say why the change in x from (−3, 5) to (2, −1) is +5 and the change in y is −6.
• I can say what a change of 0 in x tells me about a segment.

You need

The worksheet; rulers. No calculators.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second by walking the demo pair on the grid again.

Watch for these ideas

• Reading (y, x): the x-coordinate comes first, across before up.
• Subtracting the wrong way round for one of the two changes: always second point minus first, for both.
• Dropping the sign: a move left or down is a negative change, and it matters for gradient tomorrow.

Answers

• Part A: A (3, 4), B (−5, 2), C (−2, −4), D (4, −3), E (0, −5), F (−4, 0).
• Part B: (2, 1) to (7, 4): 5 and 3; (−3, 5) to (2, −1): 5 and −6; (4, −2) to (−1, −2): −5 and 0; (−5, −4) to (−5, 3): 0 and 7; (0, 6) to (−4, −2): −4 and −8; (−2, −3) to (3, 1): 5 and 4.
• Part C: (−2, 1), 5 right and 3 down: (3, −2); (−6, −2), 3 right and 7 up: (−3, 5); the last is 5 left and 5 up (change in x −5, change in y +5).
• Part D: swapping the points swaps the signs of both changes but not their sizes. A change in x of 0 means the points are one above the other: an upright segment.

Two points, two differences

A. Read the points

B. The two changes

From the first point to the second: change in x = second x − first x; change in y = second y − first y.

C. Move on the grid

Where do you end up? For the last one, write the move instead.

D. Stretch: swap the points

Find the two changes from (2, 1) to (7, 4), then from (7, 4) to (2, 1). What changes and what stays the same? What does a change of 0 in x tell you?

Gradient

The gradient of a segment is rise ÷ run: the change in y divided by the change in x. It is positive when the segment climbs to the right, negative when it falls, zero when it is flat and undefined when it stands upright.

We are learning to

• find the gradient of a segment from two points as an exact fraction,
• tell positive, negative, zero and undefined gradients apart and explain each,
• use equal gradients to decide whether two segments are parallel.

Success criteria

• I can find the gradient from (−2, 4) to (4, 1) and say why it is negative.
• I can show the gradient is the same whichever point I start from.

You need

The worksheet; the 12 gradient sort cards (cut-out sheet 1), one set per pair. No calculators.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the sort; start the second with the demo pair worked again.

Watch for these ideas

• Run over rise: the gradient is the change in y on top, not the change in x.
• Mixing the order: (y₂ − y₁) ÷ (x₁ − x₂) gives the wrong sign; keep the same order top and bottom.
• Calling an upright segment’s gradient 0: its run is 0, so it is undefined.

Answers

• Part A: (1, 3) to (5, 11): 8 ÷ 4 = 2; (−2, 4) to (4, 1): −3 ÷ 6 = −1/2; (0, −3) to (6, 0): 3 ÷ 6 = 1/2; (−4, 2) to (3, 2): 0 ÷ 7 = 0; (−3, −5) to (1, 7): 12 ÷ 4 = 3; (5, −1) to (5, 6): run 0, so the gradient is undefined (a vertical segment).
• Part B: −8 ÷ 4 = −2 and 8 ÷ −4 = −2: the same.
• Part C: 1 2 and 2, parallel; 2 3/4 and 1, not parallel; 3 −1/2 and −1/2, parallel.
• Part D: (10 − k) ÷ (4 − 1) = 2, so 10 − k = 6 and k = 4.

Gradient

A. Gradient from two points

Work out rise ÷ run and give the gradient as a whole number or a fraction in simplest form. Say if it is undefined.

B. Which end first?

Find the gradient from (2, 7) to (6, −1), then from (6, −1) to (2, 7). Compare, and explain.

C. Parallel or not?

Find both gradients. Parallel segments have equal gradients.

D. Stretch: find the missing coordinate

The segment from (1, k) to (4, 10) has gradient 2. Find k, and check by working the gradient out.

Midpoint

The midpoint of a segment is halfway across and halfway up, so its coordinates are the averages of the two ends’ coordinates: add the x-coordinates and halve, add the y-coordinates and halve.

We are learning to

• find the midpoint of a segment by averaging the coordinates, negatives and halves included,
• work back from a midpoint and one end to the other end,
• use midpoints to decide whether the diagonals of a quadrilateral cut each other in half.

Success criteria

• I can find the midpoint of (−4, 3) and (6, −5) and check it is halfway on a sketch.
• I can work back from a midpoint and one end to the other end, and check it on a sketch.

You need

The worksheet; the 16 midpoint match cards (cut-out sheet 2), one set per pair. No calculators.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after midpoint match; start the second with Part B.

Watch for these ideas

• Adding without halving: (1 + 4, 2 + 8) = (5, 10) is not the midpoint.
• Subtracting instead of adding: the midpoint averages, it does not use the change.
• Working back by averaging M and A: to find B, double M and take away A.

Answers

• Part A: (2, 6) and (8, 10): (5, 8); (−4, 3) and (6, −5): (1, −1); (−7, −2) and (−1, 4): (−4, 1); (0, 5) and (3, −2): (1.5, 1.5); (−5, 1) and (2, 1): (−1.5, 1); (6, −3) and (−3, 8): (1.5, 2.5).
• Part B: M(3, 4), A(1, 2): B(5, 6); M(0, −2), A(−5, 3): B(5, −7); M(−1, 2.5), A(4, 6): B(−6, −1).
• Part C: AC has midpoint (2, 2) and BD has midpoint (2, 2): the same point, so the diagonals bisect each other (ABCD is a parallelogram).
• Part D: one third of the way is (2, 3), two thirds is (4, 6): a third of the change in x and in y each time. A quarter of the way is (1.5, 2.25).

Midpoint

A. Find the midpoint

Average the x-coordinates, then the y-coordinates. Show the working for at least two.

B. The other end

M is the midpoint of A and B. Find B. (The move from A to M is the same as the move from M to B.)

C. Do the diagonals cut each other in half?

ABCD has A(−2, −1), B(4, 1), C(6, 5) and D(0, 3). Find the midpoint of the diagonal AC and of the diagonal BD. What do you notice, and what does it tell you about ABCD?

D. Stretch: thirds

Find the two points that cut the segment from (0, 0) to (6, 9) into three equal parts. How would you find the point one quarter of the way along?

Distance

The change in x and the change in y are the legs of a right triangle, and the segment is its hypotenuse. So Pythagoras gives the distance: square both changes, add, and take the square root.

We are learning to

• find the distance between two points with Pythagoras on the two changes,
• give a distance exactly, in simplest surd form, and to two decimal places,
• find the perimeter of a triangle on the plane.

Success criteria

• I can find the distance from (−4, 3) to (4, −3) and explain why the negative change does not matter.
• I can write √50 as 5√2 and give it to two decimal places.

You need

The worksheet; rulers; calculators for the decimals in Parts B and C.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after exact or rounded; start the second with Part B.

Watch for these ideas

• Adding the legs: from (0, 0) to (3, 4) is 5, not 3 + 4 = 7.
• Squaring a negative without brackets: (−6)² = 36, but −6² on a calculator gives −36.
• Writing √(a² + b²) as a + b: the root of a sum is not the sum of the roots.

Answers

• Part A: (1, 2) to (13, 7): √(12² + 5²) = √169 = 13; (−4, 3) to (4, −3): √(8² + (−6)²) = √100 = 10; (2, −5) to (−7, 7): √((−9)² + 12²) = √225 = 15; (−3, −2) to (5, 13): √(8² + 15²) = √289 = 17.
• Part B: (1, 1) to (4, 3): √(3² + 2²) = √13 ≈ 3.61; (−2, 0) to (3, 5): √(5² + 5²) = √50 = 5√2 ≈ 7.07; (−5, 1) to (1, −3): √(6² + (−4)²) = √52 = 2√13 ≈ 7.21.
• Part C: ABC: 6, 8, 10, perimeter 24. PQR: 5, 5, 5√2, perimeter 10 + 5√2 ≈ 17.07. PQR is isosceles.
• Part D: a point (x, 0) with (x − 1)² + 3² = 5², so x − 1 = ±4: (−3, 0) and (5, 0).

Distance

A. Whole-number distances

Find the two changes, square them, add and take the square root.

B. Exact and rounded

Give each distance exactly, in simplest surd form, and then to two decimal places.

C. Perimeters on the plane

Triangle A(0, 0), B(6, 0), C(6, 8). Triangle P(−2, 1), Q(2, 4), R(5, 0). Find the length of every side and the perimeter of each triangle, exactly and to two decimal places. Is either triangle isosceles?

D. Stretch: on the x-axis

Find every point on the x-axis that is exactly 5 units from (1, 3). Sketch to check.

Putting it together

Gradient, midpoint and distance together describe a segment, and several segments describe a shape. Students work carefully with negative coordinates, prove a shape is a rhombus, and measure a town map.

We are learning to

• find the gradient, midpoint and distance of a segment with negative coordinates, keeping every sign,
• use equal gradients, equal lengths and shared midpoints as evidence about a shape,
• apply distance and midpoint to a map drawn to scale.

Success criteria

• I can give the gradient, midpoint and distance of a segment from (−3, 7) to (3, −1).
• I can prove PQRS is a rhombus with distances and gradients.

You need

The worksheet; rulers; calculators; the unit’s self-check quiz for the game.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with Part C and the quiz.

Watch for these ideas

• Losing a sign in a subtraction: 3 − (−3) = 6, not 0.
• Thinking a negative gradient gives a negative distance: distance is always positive.
• Proving a shape from a picture: the proof is in the numbers.

Answers

• Part A: (−3, 7) to (3, −1): gradient −4/3, midpoint (0, 3), distance √(6² + (−8)²) = √100 = 10; (−6, −4) to (3, 2): gradient 2/3, midpoint (−1.5, −1), distance √(9² + 6²) = √117 = 3√13 ≈ 10.82; (5, −2) to (−1, 1): gradient −1/2, midpoint (2, −0.5), distance √((−6)² + 3²) = √45 = 3√5 ≈ 6.71.
• Part B: sides PQ = 5, QR = 5, RS = 5, SP = 5: all equal. Gradients PQ 3/4, QR −3/4, RS 3/4, SP −3/4: opposite sides parallel. Diagonals PR and QS share the midpoint (1, 0). So PQRS is a rhombus.
• Part C: school to library 1000 m, school to pool 500 m, library to pool √97 units ≈ 985 m; meeting point (1.5, 6).
• Part D: (−2, −5), (1, 1), (4, 7): gradients 2 and 2, equal and sharing a point, so the three are on one line.

Putting it together

A. Three facts about each segment

Give the gradient, the midpoint and the distance (exact, and to two decimal places when it is a surd).

B. Prove it is a rhombus

C. A town map

On a town map one unit is 100 m, with the school at (0, 0), the library at (6, 8) and the pool at (−3, 4). How far is it in a straight line from the school to each of the others, and from the library to the pool? Two friends at the library and the pool meet halfway: where?

D. Stretch: one straight line?

Are (−2, −5), (1, 1), (4, 7) on one straight line? Explain with gradients.

Gradient sort cards

For Day 2, one set per pair: 12 cards

Sort the segments into four piles: positive, negative, zero and undefined gradient. Decide from the two changes, without working out the gradient; then check two of them exactly.

Teacher note: the key is with the mini-check answers, so this sheet can go to pairs whole.

Midpoint match cards

For Day 3, one set per pair: 16 cards

Lay the cards face down. Turn two over; keep them if one is a segment and the other is its midpoint.

Teacher note: the key is with the mini-check answers, so this sheet can go to pairs whole.

What we know: Gradient, midpoint and distance

Work on your own. Use a calculator only for Question 7. Show your working.

1. Find the gradient of the segment from (2, 1) to (6, 9). ______________________
2. Find the gradient of the segment from (−3, 4) to (1, −4). ______________________
3. Find the gradients of (5, −2) to (5, 6) and of (−4, 3) to (2, 3). ______________________
4. Find the midpoint of (−2, 7) and (6, −1). ______________________
5. M(3, 1) is the midpoint of A(−1, 4) and B. Find B. ______________________
6. Find the distance between (1, 1) and (7, 9). ______________________
7. Find the distance between (−2, 5) and (3, −1), exactly and to two decimal places. ______________________
8. Show that the triangle with A(1, 1), B(5, 3), C(3, 7) is isosceles. ______________________

Answers and marking guide

Answers with working

1. 8 ÷ 4 = 2.
2. −8 ÷ 4 = −2.
3. (5, −2) to (5, 6): undefined (run 0, upright); (−4, 3) to (2, 3): 0 ÷ 6 = 0 (flat).
4. x: (−2 + 6) ÷ 2 = 2; y: (7 + (−1)) ÷ 2 = 3, so (2, 3).
5. B = 2M − A = (7, −2) (check: the midpoint of A and B is M).
6. √(6² + 8²) = √100 = 10.
7. √(5² + (−6)²) = √61 ≈ 7.81.
8. AB = √(4² + 2²) = √20 = 2√5 ≈ 4.47; BC = √((−2)² + 4²) = √20 = 2√5 ≈ 4.47; CA = √((−2)² + (−6)²) = √40 = 2√10 ≈ 6.32: two sides are equal, so the triangle is isosceles.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working, and gives distances exactly.

Card keys

Gradient sort (Day 2): positive: 7/3, 1, 1, 1, 2; negative: −1, −1/2, −1, −2; zero: 0, 0; undefined: undefined. Midpoint match (Day 3): (1, 5) and (7, 1): (4, 3); (−2, 3) and (4, 9): (1, 6); (−6, −2) and (0, 4): (−3, 1); (3, −4) and (6, 0): (4.5, −2); (−5, 7) and (−1, −3): (−3, 2); (2, 2) and (−4, −6): (−1, −2); (0, −7) and (8, −1): (4, −4); (−3, 6) and (2, 6): (−0.5, 6).

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: the two changes, gradient, midpoint, distance, and all three together.