Geometry · Session 5

Pip on the Grid: Slope, Mirror, Fold

Today has one big idea: count over, then count up. A penguin named Pip is going to walk across the grid, see himself in the ice, and fold a fish in half.

From that one idea we get three things: the slope of a line, a mirror on an axis, and the distance between two points.

Meet Pip. He always walks to the right, and he counts every square he crosses. When he is happy, he hops.

Formwalkthrough.v2 · spec: classroom/day5-pip-on-the-grid.md · class: Monday 2026-09-28, Yiyang Named gapsC4: measure first runs 0–13, not 0–8, by the standing rules (words first, then last session). C1: the five operations are not labeled on the worked examples. Said-it buttons are a record of the student speaking, not a check; they answer with the same line every time and are named as such in the spec. New since Session 4A character carries the hour. Every part has a Say-it frame (full sentence → blanks → your way). Word cards carry an English trap (the Chinese gloss was removed 2026-10-03: English only). The first type-in item (B3) with authored lines for likely slips. Three animated lesson pictures the student predicts before they run. The English traps box on the map. Student-facing text rewritten in spoken English on 2026-09-29. Before classIf the page was opened to preview, press Record, then "Clear the record" and "Reset every problem". Never press T while sharing.
0 – 13 min

Warm up

Before anything new, let's do three things. We'll go through every word we need today, look back at last time, and do three quick checks. Say each answer out loud. Nothing here is scored.

0 – 5 min

The words for today

One word at a time. Look at the picture first and tell me what you think the word means. Then we'll show the meaning, and you say the sentence.

WhyPicture before meaning: Marzano's image step carries most of the vocabulary effect. English only, by the teacher's ruling of 2026-10-03; the picture carries the meaning before the words do. Every word gets its own marked picture (standing rule, 2026-09-21). Watch for"over" heard only as division; "negative" said as "minus"; "rise" said as "raise"; "coordinate" without the plural.
5 – 10 min

Last time: parallel lines and the transversal

Three questions about last time. The last one is the proof we promised at the end of that class, and it leads straight into today.

Last time we learned that parallel lines make equal angles. Today we'll see that parallel lines also have equal slopes. It's the same idea in a new coat.
10 – 13 min

Three quick checks

Coordinates, which way a line is going, and the 3–4–5 triangle. Everything we do today stands on these three.

13 – 16 min

The big idea: everything is over and up

Here is the one picture for the whole hour. Pip counts over, and then he counts up. Three different things come out of that one habit. Let's press each one and see.

Watch out for these words

These five words trip people up in this lesson. Let's read each one out loud once now, so they feel like old friends by the time they come back.

WhyDesmos's rule: informal before formal. "Over and up" is his own words for Δx and Δy; the formulas arrive only after he has walked them. The traps box is MLR2 (collect and display) done in advance: the polysemy of "over" and negative/minus are the two documented ELL traps for this exact content.
16 – 46 min

The three moves

1Slope · 16 – 28 min

First, let's watch Pip walk a line

Pip always walks to the right. He goes over first, and then he goes up. That is one step along the line.

Before you press Walk, tell me what you think. Is this line uphill or downhill? Then count the squares with him.

The slope is up over over. When you walk to the right, uphill is positive, downhill is negative, flat is zero, and a wall is undefined.
Why"Slope Dude" walks east; the rightward walk is what fixes the sign (the documented error is counting "down 4" as +4). The staircase step on the grid is the geometric-ratio conception students actually score on (Moore-Russo). Predict-then-press is Desmos's delayed feedback. Watch forRun over rise (over on top). Saying "minus two thirds". Reading the line right to left.

Parallel lines: let's slide the line

Last time we slid one crossing down the transversal. This time, let's slide a whole line straight up and keep an eye on its step. Does the step change?

Parallel lines have the same slope. Same over, same up, just at a different height.

Perpendicular lines: let's turn the step a quarter turn

Take the step that goes over 3 and up 2, and turn it 90 degrees. Before you press Turn, tell me what you think the new over and up will be.

For perpendicular, you flip and switch. 2⁄3 becomes −3⁄2. You can always check by multiplying: perpendicular slopes multiply to −1.
WhyIllustrative Math's sequence: parallel by translation first, perpendicular by a 90° rotation second, never both in one breath (the memoriser who learns both rules together swaps them). The quarter turn is WHY "flip and switch" is two moves, and the product −1 is the check he can run on his own. Watch forDoing only one of the two moves. Saying "negative" for "flip". Calling the same slope perpendicular.

Game · Parallel, perpendicular, or neither?

Two lines, two slopes. Make the call, then tell me why. Is it the same slope, do they multiply to −1, or neither?

Why a gameInterleaved classification (Rohrer & Taylor 2007): the two rules only stick when he has to choose between them under time. Rounds 3 and 5 are the two half-moves.
2Mirror · 28 – 38 min

First, let's watch Pip see himself in the ice

The ice is a mirror, and it stands on an axis. Pip's image is the same distance away, on the other side.

Before you press anything, tell me which of Pip's two numbers you think will change. Then we'll look.

The mirror's own axis is the number that stays. A mirror on the x-axis keeps x and changes the sign of y. A mirror on the y-axis keeps y and changes the sign of x.
WhyThe documented error is negating x for an x-axis reflection. "The mirror stands ON the axis, so Pip never crosses the other one" is the sentence that beats it. Pip upside-down in the ice makes the flip visible: a reflection flips, it does not slide (Oak National's stated objective). Watch forSwapping the two numbers. Sliding instead of flipping. "Reflect on" instead of "reflect over".

Game · Where is the mirror?

Here is Pip and his image. Which axis is the mirror standing on? Look at which sign changed.

Why a gameGeoGebra's "Flip Flop" inverts the task: find the mirror, not the image. It forces the equal-distance idea. Round 6 is a slide, not a mirror.
3Fold and distance · 38 – 46 min

First, let's fold something

A line of symmetry is really just a fold. Fold the shape along the line, and if the two halves land exactly on top of each other, it counts.

Pick a shape and a fold line. Before you press Fold, tell me whether you think the halves will match.

A fold is a mirror you can hold in your hands. The fold line is the mirror line. A rectangle's diagonal is not a fold. Try it and watch the corners miss.
WhyIllustrative Math's definition is the fold itself, and the rectangle's diagonal is the classic false line of symmetry only a real fold disproves. Ask him to fold a real sheet of paper too.

Distance: over and up make a triangle

Pip waddles in a straight line from (1, 1) to (4, 5). How far is that? Take a guess first. Then let him show you the two legs.

The distance is the hypotenuse. Over and up are the legs. Square them, add them, take the root. The formula √((x₂−x₁)² + (y₂−y₁)²) is just this triangle written in one line.
WhyPythagoras on the grid before the formula (IM 8.G.B.8). The commonest test error is adding the legs (3 + 4 = 7); the drawn triangle makes that impossible to believe. Watch forAdding instead of squaring. Stopping at 25. Subtracting the squares.

Game · Are they the same length?

Two segments, drawn as steps. Are they congruent or not? Square, add, and compare. You don't even need the square root to compare them.

Why a gameRound 2 (1,7 vs 5,5) is the surprise: both √50. It is there so he stops trusting his eyes and trusts the squares.
46 – 55 min

Find the mistake

Here are four pieces of somebody's work. Read every line out loud. In most of them, one line uses a rule the wrong way. Tap the line where it goes wrong, and tell me why in English. Careful, though. One of them is completely right.

Game · Right or wrong?

One sentence at a time. Is it right? First say which rule it uses, then make the call.

WhyBooth et al.: explaining a wrong step teaches more than being shown the right one. K4 holds so that "find the break" is never automatic.
55 – 60 min

Wrap up

The whole hour in one rule

Count over, then count up. The slope is up over over. A mirror on an axis keeps that axis's number. Over and up make a triangle, and the triangle gives you the distance.

When you're done, take a screenshot of this box and keep it. It's your own record.

Word roots · we'll fill these in next time

Where these words come from

Next time

Two triangles with the same over and the same up. Are they the same triangle? Next time we'll find out when two triangles are copies of each other, and Pip's steps are how you can tell.

SeedSession 6: constructions, then congruence (plan day 4: #4, #10, #16, #19, #25). The mother is today's step: same over and up means congruent legs, and SAS follows.