Geometric algorithms: a week of ready-to-teach maths

Five lessons for Year 8 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 twelve triangle-pair cards, flowchart pieces to cut out and arrange, a printed flowchart to trace, 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: the pair cards (Days 2 to 4) and the flowchart pieces (Day 3).
4. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.

The idea in plain words

An algorithm is a finite list of clear steps and yes-or-no decisions that always ends with an output, so anyone who follows it on the same input gets the same answer. To decide whether two triangles are congruent, an algorithm can match the parts, check for a pair that differs, then try SSS, SAS, ASA (or AAS) and RHS in turn. An algorithm is only trusted once it has been tested: we trace inputs whose answers we already know, including awkward ones, and fix any step that sends an input to the wrong output. Describing how it works means saying what each step does and why the steps come in that order.

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.

Three outputs, not two

The interactive unit’s flowchart ends in congruent or not congruent. This pack adds a third output, not enough to tell, because a pair can fit no test without being different: two sides and an angle that is not between them can belong to two different triangles. Students meet the two-output version on screen and improve it on paper.

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 out what an algorithm is with a simple sorting task; Day 2 writes the congruence check as steps; Day 3 draws it as a flowchart; Day 4 tests and debugs it; and Day 5 writes a new algorithm for similarity. It builds on the congruence and similarity tests from earlier in Year 8 and the sorting algorithms for shapes in Year 7, and prepares the geometric algorithms of Year 9.

Materials for the week (one trip)

• From the classroom: pencils, rulers, scissors, glue or sticky tape, A3 paper for the flowchart, this pack printed.
• Cut-out sheet 1, one per pair: 12 triangle-pair cards (Days 2 to 4).
• Cut-out sheet 2, one per pair: flowchart pieces and yes and no labels (Day 3).
• Optional: a protractor and compasses, to construct the two triangles for an awkward card.

Words for the word wall

algorithm, step, decision, input, output, flowchart, terminal, trace, test case, expected output, bug, debug, order, congruent, similar, ratio, SSS, SAS, ASA, AAS, RHS

Dear families

This week in maths, Year 8 is writing and testing algorithms: step-by-step instructions with yes-or-no decisions. Their algorithms decide whether two triangles are exact copies of each other, or enlargements of each other.

Try this at home

• Write the steps for making toast as an algorithm. Is there a decision, such as “is it brown enough?”
• Follow someone else’s written steps exactly as written. Where do they go wrong?
• Ask: why does a computer need every step spelled out?
• Sort household objects with a yes-or-no question at each step, like a guessing game.

My algorithms this week

Fill one row a day. Name an algorithm, one input you tested, and the output it gave.

Printed from the free seegongsik Geometric algorithms teaching pack · seegongsik.com/au/y8/space/AC9M8SP04/pack

What an algorithm is

An algorithm is a list of clear steps and yes-or-no decisions that always ends with an output. Students follow one that sorts triangles by their sides, draw it as a flowchart, and find out why the order of its steps matters.

We are learning to

• say what makes a set of instructions an algorithm,
• follow an algorithm with steps and decisions to its output,
• draw an algorithm as a flowchart and explain why its order matters.

Success criteria

• I can trace the algorithm for sides 4, 7, 7 and say which step gives the output.
• I can explain what goes wrong if the equilateral check comes after the isosceles check.

You need

The worksheet each.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Skipping a step because the answer looks obvious.
• Drawing a decision without its yes and no arrows.
• Forgetting that an algorithm must reach an output for every input.

Answers

• Part A: 5, 5, 5: equilateral (step 3); 4, 7, 7: isosceles (step 4); 2, 3, 6: not a triangle (step 2); 3, 5, 6: scalene (step 5); 6, 3, 6: isosceles (step 4).
• Part B: start, read the sides, then a diamond for each of steps 2, 3 and 4 in order; each yes goes to its output, each no to the next diamond, and the last no to scalene.
• Part C: sides 5, 5, 5 would stop at “any two equal” and be called isosceles. The more special check must come first.
• Part D: for example, check the angles add to 180°; if one is 90°, right; else if one is more than 90°, obtuse; otherwise acute.

What an algorithm is

A. Follow the algorithm

An algorithm that sorts a triangle by its sides:

1. Read the three side lengths.
2. If the two shorter sides do not add to more than the longest, output “not a triangle” and stop.
3. If all three sides are equal, output “equilateral” and stop.
4. If any two sides are equal, output “isosceles” and stop.
5. Output “scalene”.

B. Draw it as a flowchart

Use a rounded box for start and for each output, a rectangle for a step, and a diamond for each decision. Label every arrow out of a diamond yes or no.

C. Swap two steps

Kai swaps steps 3 and 4. Which input from Part A now gets the wrong output? What does that tell you about the order of steps?

D. Stretch: sorting by angles

Write an algorithm that takes three angles and outputs acute, right, obtuse, or “not a triangle”. Test it on two inputs.

An algorithm for congruence

The congruence tests become an algorithm once they are put in order with a stopping rule. Students write the steps, run them on twelve triangle-pair cards, and meet a third output: not enough to tell.

We are learning to

• write the congruence check as numbered steps with decisions,
• run the steps on a pair of triangles and record the output,
• explain the difference between not congruent and not enough to tell.

Success criteria

• I can run the algorithm on a pair card and name the step that decided it.
• I can explain why card 2 is not enough to tell, rather than not congruent.

You need

Cut-out sheet 1 (the pair cards) and scissors, one set per pair; the worksheet each; a protractor and compasses for Part B.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Comparing parts that do not match, such as AB with QR.
• Calling a pair not congruent when its facts simply fit no test.
• Treating three equal angles as a congruence test.

Answers

• Part A: congruent 1 (SSS), 4 (ASA), 7 (RHS), 9 (SAS), 12 (AAS); not congruent 3, 5, 8, 10 (step 2); not enough to tell 2, 6, 11 (step 7).
• Part B: with AB = 7 cm at 40°, an arc of 5 cm from B cuts the base twice, so two different triangles fit: the algorithm cannot say congruent or not congruent.
• Part C: card 6 fits no test (angles only fix the shape, not the size), so the output is not enough to tell.
• Part D: change each congruent output to name its test, as in step 3 to 6; the first test that fits is the one reported.

An algorithm for congruence

A. Run the algorithm

1. Match the parts: A with P, B with Q, C with R.
2. If any matched pair is different, output “not congruent” and stop.
3. If the three sides are given, output “congruent (SSS)” and stop.
4. If two sides and the angle between them are given, output “congruent (SAS)” and stop.
5. If two angles and a side are given, output “congruent (ASA or AAS)” and stop.
6. If a right angle, the side opposite it and one other side are given, output “congruent (RHS)” and stop.
7. Output “not enough to tell”.

Run the steps on each pair card, in order, and sort the cards into three piles. Record the card numbers.

B. Two triangles, one set of facts

Card 2 gives AB = 7 cm, BC = 5 cm, ∠A = 40° for both triangles. Construct a triangle with these facts in two different ways: draw AB, the 40° angle at A, then swing an arc of 5 cm from B.

C. Three equal angles

Kai says card 6 is congruent because every given part matches. Run the algorithm and explain what it outputs.

D. Stretch: say which test

How could the algorithm also say which test it used? Could two tests fit the same card? Which would it report?

The flowchart

The same steps drawn as a flowchart show every branch at a glance. Students build the flowchart from cut-out pieces, trace pair cards through it, and test whether changing the order of the decisions changes any output.

We are learning to

• turn numbered steps into a flowchart of terminals, steps and decisions,
• trace an input through a flowchart to its output,
• decide when the order of the decisions matters.

Success criteria

• I can trace a pair card through the flowchart and name every decision on its path.
• I can explain why “a pair differs?” must come before the four tests.

You need

Cut-out sheet 2, scissors, glue or tape and an A3 sheet per pair; the pair cards from Day 2; the worksheet each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the build; start the second with the tracing.

Watch for these ideas

• Leaving an arrow out of a diamond without a yes or no label.
• Putting the mismatch check last, so a pair with a differing part can reach congruent.
• Thinking the order of the four tests changes the output: it only changes which test is named.

Answers

• Part A: card 9: a pair differs? no, SSS? no, SAS? yes: congruent; card 8: a pair differs? yes: not congruent; card 11: a pair differs? no, SSS? no, SAS? no, ASA or AAS? no, RHS? no: not enough; card 7: a pair differs? no, SSS? no, SAS? no, ASA or AAS? no, RHS? yes: congruent.
• Part B: card 3 reaches SAS? yes and comes out congruent, though its angles differ. Kai’s order gets cards 3, 5, 8, 10 wrong: every not congruent card (3, 5, 8, 10).
• Part C: no card changes its output; at most the named test changes, if two tests fit one card.
• Part D: sort each rectangle’s sides first, then compare shortest with shortest and longest with longest; 3 by 5 and 5 by 3 then match, so congruent.

The flowchart

A. Trace the cards

The congruence algorithm as a flowchart. Trace each card: write each question it meets, with the answer, and the output.

B. A different order

Kai moves “a pair differs?” to the end, after RHS. Trace card 3 through Kai’s flowchart. What goes wrong, and for which other cards?

C. Swap the tests

Put the four tests in the order RHS, ASA or AAS, SAS, SSS. Does any card get a different output? Why or why not?

D. Stretch: rectangles

Draw a flowchart that decides whether two rectangles are congruent from their two side lengths. Test it on 3 by 5 and 5 by 3.

Testing and debugging

An algorithm is only trusted once it has been tested. Students write the expected output for five test cases, run a faulty algorithm on them to find its bugs, fix it, and find that even the Day 2 algorithm has a weak spot.

We are learning to

• choose test cases with known outputs, including awkward ones,
• trace a faulty algorithm to find the step that goes wrong,
• fix an algorithm and test it again.

Success criteria

• I can find an input that makes the faulty algorithm give a wrong output.
• I can explain how my fix stops that bug, and test it.

You need

The pair cards from Day 2 and the worksheet each.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Testing only cases that work.
• Writing the expected output after running the algorithm, so it just copies the output.
• Fixing one input instead of the step that causes the bug.

Answers

• Part A: expected 1 congruent, 2 not enough, 3 not congruent, 4 not enough, 5 congruent; the faulty algorithm gives 1 congruent, 2 congruent, 3 not congruent, 4 congruent, 5 not congruent.
• Part B: it fails cases 2, 4, 5. It never checks that the numbers fit a test (so angles only, or two sides with the wrong angle, pass), and it compares the numbers in the order written, not by matching parts.
• Part C: use the Day 2 steps, but match the parts by size first: list each triangle’s sides from shortest to longest before comparing. Case 5 then gives congruent; the Day 2 algorithm gave not congruent.
• Part D: answers vary; good sets include a case for every output and the awkward ones, such as SSA and AAA.

Testing and debugging

A. Run the tests

A faulty algorithm:

1. Write down the three measurements given for each triangle.
2. If the three pairs of numbers match, output congruent.
3. Otherwise, output not congruent.

For each test case, write the output a correct algorithm should give first, then the output the faulty one gives.

B. Name the bugs

Which test cases does the faulty algorithm get wrong? For each, say which step goes wrong and why.

C. Fix it, then test again

Write a fixed algorithm. Then run the Day 2 algorithm on case 5. What does it output, and what should it? Change one step so it gets case 5 right too.

D. Stretch: a good set of tests

Write the smallest set of test cases you would use to check a congruence algorithm. Why is each one there?

An algorithm for similarity

The same design skills give an algorithm for similar triangles. Students trace it on four pairs, find out why the sides are sorted first, describe how it works, and play the quiz as a class game.

We are learning to

• trace an algorithm for similarity to its output,
• explain why the sides are sorted before dividing,
• describe in words how an algorithm works.

Success criteria

• I can show that pair 4 is similar by sorting the sides first.
• I can describe each step of the similarity algorithm and why it is there.

You need

The worksheet and a calculator each; the self-check quiz at the bottom of the unit page for the game.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Dividing the sides in the order written instead of shortest with shortest.
• Comparing two angles without finding the third.
• Calling triangles similar when every side grew by the same amount rather than the same factor.

Answers

• Part A: 1 step 2, similar (3, 3, 3); 2 step 1, similar; 3 step 2, not similar (2, 2, 2.22); 4 step 2, similar (1.5, 1.5, 1.5). In pair 2 the third angles are 70° and 35°.
• Part B: as written, pair 4 gives 2.5, 1, 1.35, which would say not similar; sorted, every ratio is 1.5.
• Part C: for example: it checks angles first, since two equal pairs settle it; otherwise it sorts the sides, divides, and calls the triangles similar if one ratio fits all three.
• Part D: start; two angles given? (angles match?); else three sides given? (sorted ratios equal?); else two sides and the angle between? Each yes leads to similar or not similar, and the last no to not enough to tell.

An algorithm for similarity

A. Trace the algorithm

1. If two angles of each triangle are given, find each third angle. If the three angles match, in any order, output “similar”; if not, output “not similar”. Stop.
2. If the three sides of each are given, list each triangle’s sides from shortest to longest and divide matching sides. If the three answers are equal, output “similar”; if not, “not similar”. Stop.
3. If two sides of each are given with the angle between them, and the sides are in one ratio and the angles are equal, output “similar”. Stop.
4. Output “not enough to tell”.

B. Why sort first?

Divide pair 4’s sides in the order they are written. What would the algorithm output? Then sort them and divide again.

C. Describe how it works

Write three sentences for a classmate who missed the lesson: what the algorithm does first, next and last, and why.

D. Stretch: as a flowchart

Draw the similarity algorithm as a flowchart, with every arrow out of a diamond labelled yes or no.

Twelve triangle-pair cards

For Days 2 to 4, one set per pair

Each card gives the same parts of two triangles, ABC and PQR, with A matching P, B matching Q and C matching R. Run the congruence algorithm on each card.

Teacher note: congruent 1, 4, 7, 9, 12; not congruent 3, 5, 8, 10; not enough to tell 2, 6, 11. No row or column of cards gives an output away.

Flowchart pieces

For Day 3, one set per pair

Cut out the pieces and arrange them on a large sheet as the congruence algorithm. Draw the arrows, and put a yes or no label on every arrow that leaves a diamond.

There is one yes and one no for each diamond.

What we know: Geometric algorithms

Work on your own. Use the congruence algorithm from Day 2, with its three outputs.

1. What three things does every algorithm have? ______________________
2. △ABC: ∠A = 45°, AB = 6 cm, ∠B = 70°. △PQR: ∠P = 45°, PQ = 6 cm, ∠Q = 70°. What does the algorithm output, and which step decides it? ______________________
3. △ABC: AB = 8 cm, BC = 5 cm, CA = 7 cm. △PQR: PQ = 8 cm, QR = 5 cm, RP = 7.5 cm. What does the algorithm output? ______________________
4. △ABC: AB = 9 cm, BC = 6 cm, ∠A = 35°. △PQR: PQ = 9 cm, QR = 6 cm, ∠P = 35°. What does the algorithm output, and why? ______________________
5. In a flowchart, what do a rectangle, a diamond and a rounded box stand for? ______________________
6. Why must the check for a pair that differs come before the four tests? ______________________
7. Are triangles with sides 4, 7, 8 and 16, 14, 8 similar? Show how the similarity algorithm decides. ______________________
8. Write a test case that would catch an algorithm that calls triangles with three equal angles congruent. ______________________

Answers and marking guide

Answers with working

1. An input, clear steps and decisions in order, and an output.
2. congruent (ASA): no pair differs, and two angles with the side between them are given (step 5).
3. not congruent: CA = 7 cm but RP = 7.5 cm, so a matched pair differs (step 2).
4. not enough: the 35° angle is not between the two sides, so no test fits; two different triangles have these facts.
5. A step, a yes-or-no decision, and the start or an output.
6. Otherwise a pair whose parts differ could still pass a test, as with SAS when the angles differ.
7. Yes: sorted, 4, 7, 8 and 8, 14, 16 give ratios 2, 2, 2, all equal.
8. For example, angles 50°, 60°, 70° in both triangles, but sides 3, 3.4, 3.7 and 6, 6.8, 7.4: expected output not enough to tell.

A quick three-level guide

Eight questions, four ideas. A student at standard traces each pair to the right output and says which step decided it.

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: what an algorithm is, the congruence steps, the flowchart, testing, and similarity.