Designing geometric algorithms: a week of ready-to-teach maths

Five lessons for Year 9 Space. 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 12 construction step cards, 12 congruence 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: sheet 1 has the construction step cards (Day 2), sheet 2 the congruence cards (Day 3). Cut them before Day 2.
4. Each student needs a pair of compasses, a ruler, a protractor and a sharp pencil.
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

An algorithm is a finite list of exact steps that solves a problem every time it is followed. Most are built from two kinds of step: a sequence, carried out in order, and a decision, which takes one path if a condition holds and another if it does not. A geometric algorithm uses constructions and theorems as its steps and tests. Designing one is only half the job: it must be tested on inputs that could break it, refined when it fails, and the refinements compared.

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

Flowcharts use a rounded box to start, rectangles for steps and diamonds for decisions, with yes to the right and no straight down. A traced output is the box an input ends in. Where students design their own algorithm, the answers give one that works and say what any correct one must do.

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 reads and writes flowcharts; Day 2 treats compass constructions as algorithms whose order matters; Day 3 uses congruence to prove a construction always works; Day 4 builds classifiers from theorems such as Pythagoras; and Day 5 tests, refines and evaluates an algorithm. It builds on constructions, congruence and Pythagoras from Years 8 and 9 and on algorithmic thinking from the Digital Technologies curriculum.

Materials for the week (one trip)

• From the classroom: compasses, rulers, protractors, sharp pencils, scissors, envelopes for the cards, this pack printed.
• Cut-out sheet 1, one per pair: 12 construction step cards in two sets (Day 2).
• Cut-out sheet 2, one per pair: 12 congruence cards (Day 3).
• If you have them: a dynamic geometry app to test the constructions on many angles at once. No other equipment to buy.

Words for the word wall

algorithm, step, sequence, decision, condition, flowchart, input, output, trace, test, refine, construction, bisect, perpendicular bisector, congruent, SSS, SAS, ASA, RHS, classify

Dear families

This week in maths, Year 9 is writing algorithms for geometry: exact lists of steps and decisions, such as how to bisect an angle with compasses or how to tell whether a triangle is right-angled, and then testing them to find where they go wrong.

Try this at home

• Write the steps for making a cup of tea as exact instructions. Ask someone to follow them word for word. Where do they go wrong?
• Ask: a rule says “four equal sides means a square”. What shape breaks it? (A rhombus that is not a square.)
• Ask: can sides of 3 cm, 4 cm and 8 cm make a triangle? (No: 3 + 4 is less than 8.)

My algorithm week

Fill one row a day. Write an algorithm you met, an input you tested it on, and whether it worked.

Sequences and decisions

An algorithm is a finite list of exact steps that solves a problem every time. A sequence is a run of steps carried out in order; a decision tests a condition and takes one path if it holds and another if it does not. Decisions in a chain are checked in order, so the order of the tests is part of the algorithm.

We are learning to

• trace a flowchart for an input and say which output it reaches,
• write a decision with a clear condition and two paths,
• find an input that breaks a rule, and say why it breaks.

Success criteria

• I can trace the angle flowchart for any angle from 0° to 360°.
• I can find the inputs that a naive rule gets wrong.

You need

The worksheet; a protractor each.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Stopping at the first decision whose answer is no, instead of following no to the next one.
• Testing only easy inputs, so a rule looks correct when it is not.
• Conditions that overlap or leave gaps, such as “less than 90°” and “more than 90°” with nothing for 90° itself.

Answers

• Part A: 135° obtuse, 35° acute, 250° reflex, 90° right, 180° straight.
• Part B: any correct design checks that every angle is more than 0° and that the three add to 180°. 60, 60, 60: yes; 90, 45, 50: no; 100, 80, 0: no; 30, 70, 80: yes.
• Part C: it is wrong for 250° (says obtuse, should be reflex) and 180° (says obtuse, should be straight): it has no test for straight or reflex angles.
• Part D: add “equal to 180°? straight” and “more than 180°? reflex” after the obtuse test, as in the Part A flowchart.

Sequences and decisions

A. Trace the angle flowchart

B. Design a decision: can these be the angles of a triangle?

Write an algorithm with decisions that answers yes or no for any three angles. Then test it on each set.

C. Break a rule

A first try at the angle algorithm was: “If the angle is less than 90°, acute. If it is more than 90°, obtuse. Otherwise, right.” Run it on the Part A inputs. Which outputs are wrong, and why?

D. Stretch: refine it

Change the first try so that it gives the right output for every angle from 0° to 360°.

Constructions as algorithms

A compass construction is an algorithm: a sequence whose order matters, written exactly enough that anyone who follows it gets the same result. “Keep the same width” is a step too: leave it out and the construction fails. Testing it on new inputs, such as an obtuse angle, shows whether it works in general or only on the example it was written for.

We are learning to

• put the steps of a construction in the order that works,
• carry out a construction exactly, step by step,
• test a construction on new inputs and say what each step is for.

Success criteria

• I can order both sets of step cards and carry out each construction.
• I can explain why the compass width must stay the same between two steps.

You need

The worksheet; the step cards (cut-out sheet 1) per pair; compasses, a ruler and a protractor each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with the constructions.

Watch for these ideas

• Changing the compass width between the two crossing arcs.
• Arcs too short to cross: open the compasses wider.
• Joining V to P or R instead of to the crossing point Q.

Answers

• Part A: perpendicular bisector D, F, B, E, A, C; angle bisector I, K, G, L, J, H.
• Part B: the line meets AB at a right angle 4 cm from A; the two halves of the 70° angle are 35° each.
• Part C: 130° gives two 65° halves; 180° gives two 90° halves: for the straight angle the ray stands at right angles to the line. The algorithm works for both.
• Part D: with a new width the arcs cross at a point off the bisector, so the two halves are no longer equal; the step “keep the same width” is what makes Q the same distance from P and from R.

Constructions as algorithms

A. Order the step cards

Write the card letters in the order that works.

B. Carry them out

Construct the perpendicular bisector of AB (8 cm) and the bisector of the angle at V (70°), following your cards exactly.

C. Test it on new angles

Draw a 130° angle and a straight angle (180°). Bisect each with the same cards and measure the halves.

D. Stretch: why keep the width?

What goes wrong if the compass width changes between the last two arcs? Use a drawing to explain.

Why it works

Testing a construction on many angles makes us confident; a theorem makes us sure. The angle bisector’s steps build two triangles, VPQ and VRQ, with VP = VR (one arc), PQ = RQ (one width) and VQ shared. Three pairs of equal sides make them congruent (SSS), so the angles at V are equal for every angle the algorithm is given.

We are learning to

• justify a construction with a congruence test,
• tell valid congruence tests (SSS, SAS, ASA, AAS, RHS) from facts that are not enough,
• separate congruent from merely similar.

Success criteria

• I can name the three pairs of equal sides in the angle bisector and the test they make.
• I can sort the twelve cards into congruent, similar only and not enough.

You need

The worksheet; the 12 congruence cards (cut-out sheet 2) per pair; rulers and protractors.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Treating AAA as a congruence test: equal angles give the same shape, not the same size.
• Accepting SSA: two sides and an angle not between them can make two different triangles.
• Forgetting the shared side as one of the three pairs.

Answers

• Part A: the card key is with the mini-check answers.
• Part B: VP = VR because both are radii of the first arc; PQ = RQ because both arcs used one width; VQ is common. So triangles VPQ and VRQ are congruent (SSS), and the angles PVQ and RVQ are equal: VQ bisects the angle.
• Part C: X and Y are each one width from A and from B, so triangles AXY and BXY are congruent (SSS) and the angles at X are equal; then the two triangles XY cuts from AXB are congruent (SAS), so XY meets AB at its midpoint at a right angle.
• Part D: AAA fixes the shape but not the size: a 3-4-5 triangle and a 6-8-10 triangle have equal angles but are not congruent.

Why it works

A. Sort the congruence cards

Each card gives facts about two triangles. Do the facts prove them congruent, only similar, or neither? Write each letter in its pile.

B. Prove the angle bisector works

VP = VR because ______________________________. PQ = RQ because ______________________________. VQ is ____________. So triangles VPQ and VRQ are ____________ by ________, and the angles at V are ____________.

C. Prove the perpendicular bisector works

Call the two crossing points X and Y. Use congruent triangles to explain why XY cuts AB in half at a right angle.

D. Stretch: why not AAA?

Explain, with a drawing, why three equal angles do not prove two triangles congruent.

Classifying triangles

Pythagoras’ theorem becomes a decision: with the sides sorted so that c is the longest, a² + b² = c² means a right angle, more than c² an acute triangle, and less an obtuse one. First, though, the algorithm must check that the sides can make a triangle at all (a + b > c), or it will name shapes that cannot exist.

We are learning to

• trace a classifier built on Pythagoras’ theorem,
• design a classifier by sides with its decisions in a working order,
• explain why each decision sits where it does.

Success criteria

• I can trace the flowchart for any three sides.
• I can explain why the triangle check comes first.

You need

The worksheet; a calculator each.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Not sorting the sides first, so c is not the longest.
• Leaving out the triangle check and classifying sides that cannot meet.
• Calling a + b = c a triangle: it is a flat line.

Answers

• Part A: 6, 8, 10 right; 4, 5, 6 acute; 2, 3, 6 not a triangle; 5, 5, 8 obtuse; 7, 7, 7 acute; 12, 5, 13 right.
• Part B: one correct design: “All three sides equal? equilateral. Otherwise, two sides equal? isosceles. Otherwise scalene.” The all-three test must come before the two-equal test.
• Part C: 5, 5, 8 obtuse isosceles; 3, 4, 5 right scalene; 9, 9, 9 acute equilateral.
• Part D: an equilateral triangle has three 60° angles, so it is always acute: right or obtuse equilateral triangles never occur. A right isosceles triangle does, with sides 1, 1 and √2.

Classifying triangles

A. Trace the triangle flowchart

B. Design: classify by sides

Write a flowchart that names a triangle equilateral, isosceles or scalene from its three sides. Test it on 5, 5, 5.

C. Both names

D. Stretch: impossible pairs

Can a triangle be right-angled and equilateral? Obtuse and equilateral? Right-angled and isosceles? Explain each.

Test and refine

An algorithm is only as good as the tests it passes. Choose inputs that try every branch, especially the awkward ones: a rhombus for a square test, a parallelogram for a rectangle test. When an input fails, refine the steps and test every input again, because a fix in one branch can break another. Two refinements can both fix one input; the better one also passes all the others.

We are learning to

• test an algorithm with inputs chosen to break it,
• refine a failing algorithm and re-test it,
• compare two refinements and say which is better and why.

Success criteria

• I can find the inputs that the first quadrilateral rule gets wrong.
• I can explain why one refinement is better than another.

You need

The worksheet; the unit’s self-check quiz.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Testing only the shapes the rule was written for.
• Fixing one failing input and not re-testing the others.
• Judging a refinement by one input instead of the whole test set.

Answers

• Part A: P square (wrong: it is a rhombus); Q rectangle; R other; S rectangle (wrong: it is a parallelogram); T square. It fails on P and S.
• Part B: “Four equal sides? Then a right angle? square, otherwise rhombus. Opposite sides equal? Then a right angle? rectangle, otherwise parallelogram. Otherwise other.” Re-tested: P rhombus, Q rectangle, R other, S parallelogram, T square.
• Part C: Refinement 1 (the Part B rule) passes every input. Refinement 2 fixes P but fails on Q (says square, is a rectangle): a right angle alone does not make a square. Refinement 1 is better.
• Part D: put “two pairs of equal adjacent sides? kite” before “otherwise other”, and test it on R and on a shape with no equal sides.

Test and refine

A. Test the first rule

The first rule: “If all four sides are equal, square. Otherwise, if opposite sides are equal, rectangle. Otherwise, other.” Run it on each shape (sides in order around the shape, and one angle).

B. Refine and re-test

Write a better rule that names squares, rhombuses, rectangles and parallelograms correctly. Re-test every shape in Part A.

C. Compare two refinements

Refinement 1 is your Part B rule. Refinement 2 says: “If it has a right angle, square. Otherwise, if all four sides are equal, rhombus. Otherwise, if opposite sides are equal, parallelogram. Otherwise, other.” Test Refinement 2 on every shape. Which refinement is better, and why?

D. Stretch: add kites

Shape R is a kite. Add a decision that names kites, and say where it must go in your rule.

Construction step cards

Day 2, one set per pair. Cut out both sets and keep them apart. Put each set in the order that works.

The perpendicular bisector of AB (cards A to F)

The bisector of the angle at V (cards G to L)

Teacher note: the order is with the Day 2 answers and the mini-check answers, so this sheet can go to pairs whole.

Congruence cards

Day 3, one set per pair: 12 cards. Each card gives the facts known about two triangles. Do the facts prove the triangles congruent, only similar, or neither?

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

What we know: Designing geometric algorithms

Work on your own. Show your working.

1. In a flowchart, what shape shows a decision? ______________________
2. Trace the Day 1 angle flowchart for 270°. Which output does it reach? ______________________
3. Put these steps in order: (X) draw a line through the two crossing points; (Y) with the point on A, draw arcs above and below AB; (Z) keep the width, and with the point on B draw arcs that cross them. ______________________
4. Why does the angle bisector construction always work? ______________________
5. Is “two sides and an angle not between them” enough to prove two triangles congruent? ______________________
6. Can sides of 3, 5, 9 make a triangle? Explain with the first test of the Day 4 flowchart. ______________________
7. Classify a triangle with sides 6, 6, 6 by its sides and by its angles. ______________________
8. The rule “four equal sides: square” is run on a rhombus with angles of 50° and 130°. What does it say, and what should the refined rule also check? ______________________

Answers and marking guide

Answers with working

1. A diamond.
2. reflex: it is not less than 90°, not 90°, not less than 180° and not 180°.
3. Y, Z, X: the arcs from A come first, the arcs from B must use the same width, and the line joins their crossings.
4. Its arcs make two triangles with three pairs of equal sides, so they are congruent (SSS) and the two halves of the angle are equal.
5. No: SSA can fit two different triangles.
6. No: 3 + 5 = 8, which is not more than 9.
7. equilateral and acute.
8. It says square, but the shape is a rhombus: the refined rule must also check for a right angle.

A quick three-level guide

Eight questions, four ideas. A student at standard traces and writes decisions, justifies a construction and tests a rule.

Card key

Congruence cards (Day 3): congruent: A (SSS), D (SAS), F (ASA), K (RHS), L (AAS); similar only: C (AAA), E (SSS in proportion), H (AA), J (SAS in proportion); not enough: B (SSA), G (SS), I (SA).

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: sequences and decisions, constructions, proof, classifiers, and testing and refining.