Experimenting with functions: a week of ready-to-teach maths

Five lessons for Year 10 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 twelve graph cards to match, twelve conjecture cards to sort, 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. Cut the 12 graph cards before Day 3 and the 12 conjecture cards before Day 5, and keep each set in an envelope.
4. A graphing app or a spreadsheet on a laptop or tablet makes Days 1, 3 and 4 quicker, but every task also works on paper with a calculator.
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

Experimenting is how a great deal of mathematics is found: change one thing, watch what happens, notice a pattern and put it into words. That careful guess is a conjecture. Testing it on new cases builds confidence; one case that breaks it, a counterexample, knocks it down for good. When a conjecture survives, we generalise it into a rule that covers every case, and the strongest step of all is explaining why the rule must hold. Digital tools make the experimenting fast: a graphing app shows a whole family of graphs in seconds, and a spreadsheet tests a pattern on hundreds of cases.

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

A conjecture is written as a sentence that could be true or false: “the gap between square numbers next to each other is always odd”. A counterexample is one case that breaks it, written with its working. Rules use n for the position in a pattern: the nth triangular number is n(n + 1)/2.

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

Days 1 and 3 experiment with families of graphs; Days 2 and 4 experiment with number patterns, from noticing to a rule; and Day 5 asks the hardest question of a conjecture: does it always hold? It builds on Year 9, where students varied the parameters of linear and quadratic graphs with digital tools, and it prepares the reasoning and proof of senior mathematics.

Materials for the week (one trip)

• From the classroom: calculators, scissors, envelopes for the cards, pencils, this pack printed.
• Cut-out sheet 1, one per pair: 12 graph cards (Day 3).
• Cut-out sheet 2, one per pair: 12 conjecture cards (Day 5).
• If you have them: laptops or tablets with a graphing app or spreadsheet for Days 1, 3 and 4 (one per pair is plenty). No other equipment to buy.

Words for the word wall

experiment, parameter, slope, intercept, family of graphs, turning point, vertex, observe, conjecture, test, counterexample, generalise, rule, pattern, nth term, justify, proof

Dear families

This week in maths, Year 10 is experimenting like mathematicians: changing one number in a rule and watching the graph move, spotting patterns, making conjectures (careful guesses), testing them, and hunting for the one example that breaks a rule.

Try this at home

• Ask: is every number that ends in 7 prime? (No: 27 = 3 × 9.)
• Ask: add the first few odd numbers: 1, then 1 + 3, then 1 + 3 + 5. What do you notice? (The square numbers 1, 4, 9.)
• Ask: if 5 people each shake hands once with everyone else, how many handshakes? (10.)
• Ask: what do we call one example that breaks a rule? (A counterexample.)

My conjectures week

Fill one row a day. Write a pattern you noticed, your conjecture about it, and the cases you tested.

Change one number

In y = mx + c two numbers decide the line. Students change one at a time, tabulate and graph each family, and write and test a conjecture about what m and c each do.

We are learning to

• experiment with a rule by changing one number at a time,
• describe what the slope m and the intercept c each do to the graph,
• write a conjecture and test it on a new case.

Success criteria

• I can predict the graph of y = 3x + 1 before drawing it.
• I can say why y = 2x + 1 and y = 2x − 3 never meet.

You need

Rulers; a graphing app on a laptop or tablet if you have one; the worksheet, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• “A bigger m moves the line up.” It makes the line steeper; c moves it up or down.
• Reading a negative slope as a lower line: y = −x + 1 still crosses the y-axis at 1; it falls instead of rising.
• A conjecture that cannot fail, such as “lines are straight”: a conjecture makes a claim a new case could break.

Answers

• Part A: y = −2x + 1: 5, 3, 1, −1, −3; y = −x + 1: 3, 2, 1, 0, −1; y = x + 1: −1, 0, 1, 2, 3; y = 2x + 1: −3, −1, 1, 3, 5; y = 3x + 1: −5, −2, 1, 4, 7. Every line passes through (0, 1); the bigger the slope, the steeper; a negative slope falls.
• Part B: y = 2x + 1 and y = 2x − 3 have the same slope, so they are parallel, 4 apart up the y-axis, and never meet. y = −x + 1 crosses y = 2x + 1 at (0, 1).
• Part C: lines y = 3x + c are all parallel, with slope 3; lines y = mx + 2 all pass through (0, 2).
• Part D: at x = 3 the bracket x − 3 is 0, so y = 2 whatever m is: every line passes through (3, 2).

Change one number

A. Change the slope

Each rule has the same c. Fill the table. The first row is done for you.

B. Change the intercept

C. Generalise

D. Stretch: a pivot

Observe, conjecture, test

A conjecture starts with noticing. Students tabulate square numbers and their gaps, sums of odd numbers and products of neighbours, write a conjecture for each, test it on new cases, and explain one with algebra.

We are learning to

• record cases in a table and notice a pattern,
• write the pattern as a conjecture that a new case could break,
• test a conjecture on new cases, and explain why it holds.

Success criteria

• I can conjecture that the gaps between square numbers next to each other are the odd numbers, and test it at n = 10.
• I can explain the gaps with (n + 1)² − n² = 2n + 1.

You need

Calculators; a spreadsheet if you have one; the worksheet, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part B; do Part C, the explanation and Part D next time.

Watch for these ideas

• Stopping at the observation: “3, 5, 7” is what we saw; a conjecture says what always happens.
• Treating many tests as a proof: tests build confidence; an explanation proves it.
• Testing only easy cases: try a big n, and one that seems unusual.

Answers

• Part A: squares 1, 4, 9, 16, 25, 36; gaps 3, 5, 7, 9, 11, 13: the odd numbers. Tests: 11² − 10² = 21, 21² − 20² = 41, both odd. Explanation: (n + 1)² − n² = 2n + 1.
• Part B: 1, 4, 9, 16, 25, 36: the first n odd numbers add to n². Tests: 10 odd numbers give 100, 20 odd numbers give 400.
• Part C: 2, 6, 12, 20, 30, 42: all even, because one of two neighbours is even. Tests: 10 × 11 = 110, 20 × 21 = 420.
• Part D: always 1, for example 7² − 6 × 8 = 49 − 48 = 1, because n² − (n − 1)(n + 1) = n² − (n² − 1) = 1.

Observe, conjecture, test

Observe

Fill the table. The first row is done for you.

A. Gaps between squares

B. Odd numbers added

C. Neighbours multiplied

D. Stretch: three in a row

Sliding a parabola

Change one number in a parabola’s rule and watch the curve move. Students tabulate three slid parabolas, vary the other numbers, conjecture where the turning point sits, and match six rules to their graphs.

We are learning to

• tabulate and graph parabolas from their rules,
• describe what h, k and a each do to y = a(x − h)² + k,
• predict a turning point from a rule, and test the prediction.

Success criteria

• I can say where the turning point of y = (x + 2)² − 4 is before graphing it.
• I can tell y = 2x − 3 from y = x² − 3 on a graph.

You need

The graph cards (cut-out sheet 1), one set per pair; a graphing app if you have one; the worksheet, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• “(x − 2)² moves left.” The lowest point is where x − 2 = 0, at x = 2: it moves right.
• Mixing up y = x² − 3 and y = (x − 3)²: the first moves down, the second moves right.
• “0.5 in front makes it narrower.” A number between 0 and 1 makes it wider; a negative number flips it.

Answers

• Part A: y = x²: 9, 4, 1, 0, 1, 4, 9, turning point (0, 0); y = (x − 2)²: 25, 16, 9, 4, 1, 0, 1, turning point (2, 0); y = (x + 1)²: 4, 1, 0, 1, 4, 9, 16, turning point (−1, 0).
• Part B: y = x² − 3: (0, −3), moved down; y = x² + 2: (0, 2), moved up; y = 2x²: (0, 0), narrower; y = 0.5x²: (0, 0), wider; y = −x²: (0, 0), upside down.
• Part C: y = (x − 3)² + 1 turns at (3, 1); y = (x + 2)² − 4 turns at (−2, −4). The graph cards key is with the mini-check answers.
• Part D: (x − 3)² = 4, so x − 3 = 2 or −2, and x = 1 or 5.

Sliding a parabola

A. Slide it sideways

Fill the table, then graph the three curves. The first row is done for you.

B. Up, down, narrow, wide

For each rule give the turning point, and say how the curve differs from y = x².

C. Predict, then match

D. Stretch: crossing the axis

Generalise a pattern

Generalising turns a list of cases into one rule. Students find and test rules for the triangular numbers, squares of matchsticks and handshakes, and justify the triangular rule by doubling the triangle into a rectangle.

We are learning to

• find a rule for the nth term of a pattern,
• test a rule on cases it was not built from,
• justify a rule with a picture or with algebra.

Success criteria

• I can find the number of matchsticks for 50 squares without drawing them.
• I can explain why 1 + 2 + ... + n = n(n + 1)/2.

You need

Matchsticks or toothpicks if you have them; calculators; the worksheet, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part B; do the handshakes and Part D next time.

Watch for these ideas

• A rule from one step only: “add 3” gives the next term, not the nth. Write sticks = 3n + 1.
• Testing a rule only on the cases it came from: test a new case.
• Counting every handshake twice: n people each shake n − 1 hands, and each handshake joins two people.

Answers

• Part A: 1, 3, 6, 10, 15; the rule n(n + 1)/2 gives 55 for n = 10 and 5050 for n = 100.
• Part B: 4, 7, 10, 13, adding 3 each time: sticks = 3n + 1, so 31 for 10 squares and 151 for 50 squares.
• Part C: 1, 3, 6, 10: handshakes = n(n − 1)/2, so 30 people make 435 handshakes.
• Part D: two copies of the staircase 1 + 2 + ... + n fit together into an n by (n + 1) rectangle, so one staircase is n(n + 1)/2.

Generalise a pattern

A. Adding up

B. Matchstick squares

C. Handshakes

Everyone in a group shakes hands once with everyone else.

D. Stretch: why the rule works

One counterexample

A conjecture can survive many tests and still be false. Students test n² − n + 11 from n = 1 to 11, sort twelve conjectures, find counterexamples for the false ones and reasons for the true ones, and finish with the self-check quiz.

We are learning to

• test a conjecture systematically and recognise a counterexample,
• explain why one counterexample rejects a conjecture, however many cases worked,
• give a reason why a true conjecture must always hold.

Success criteria

• I can show that n² − n + 11 is not always prime.
• I can give a counterexample to “squaring a number always makes it bigger”.

You need

The conjecture cards (cut-out sheet 2), one set per pair; calculators; the worksheet; the self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; do the cards, Parts C and D and the quiz next time.

Watch for these ideas

• “It worked every time I tried, so it is true.” No list of cases covers them all.
• Offering a case that agrees with the claim as a counterexample: a counterexample must break it.
• “One counterexample is just an exception.” In mathematics one is enough to reject the claim.

Answers

• Part A: 11, 13, 17, 23, 31, 41, 53, 67, 83, 101 are all prime; n = 11 gives 121 = 11 × 11, not prime.
• Part B: the card key, with a counterexample or reason for every card, is with the mini-check answers.
• Part C: 1. 0.5² = 0.25, smaller than 0.5 (and 1² = 1 is not bigger). 2. Two odd numbers are 2a + 1 and 2b + 1; their sum is 2(a + b + 1), which is even.
• Part D: prime for every n from 0 to 39; n = 40 gives 1681 = 41 × 41. A spreadsheet finds it in seconds; by hand it takes patience.

One counterexample

A. Ten in a row

Conjecture: n² − n + 11 is always prime. Test it. The first row is done for you.

B. True or false?

Sort the twelve conjecture cards into two piles and write the letters. For every false card, find a counterexample.

C. Break one, explain one

D. Stretch: forty in a row

Rules and their graphs

For Day 3, one set per pair. Six cards show a rule; six show a graph on a grid from −4 to 4 across. Match each rule to its graph by where it crosses the axes and which way it turns.

Teacher note: the key is with the mini-check answers.

Always true, or false?

For Day 5, one set per pair. Sort the cards into conjectures that are always true and conjectures that are false. A false card needs one counterexample; a true card needs a reason.

Teacher note: the key is with the mini-check answers.

What we know: Experimenting with functions

Work on your own. A calculator is allowed. Show your working.

1. How does the graph of y = 3x − 2 differ from the graph of y = 3x + 4? ______________________
2. What do all the lines y = mx + 5 have in common, whatever m is? ______________________
3. Where is the turning point of y = (x + 3)² − 1? ______________________
4. Write a rule for the nth term of 5, 9, 13, 17, ... and find the 20th term. ______________________
5. Test the conjecture “n² + 1 is never a multiple of 3” for n = 1 to 6. Does your test prove it? ______________________
6. A conjecture has worked for 1000 cases. Circle one:   it must be true  ·  one counterexample could still reject it  ·  it has become a rule
7. Find a counterexample to: the sum of two prime numbers is always even. ______________________
8. Triangles in a row are made of matchsticks: 1 triangle uses 3 sticks, 2 use 5, 3 use 7. Write a rule, and find the sticks for 25 triangles. ______________________

Answers and marking guide

Answers with working

1. They are parallel (both have slope 3); y = 3x − 2 is 6 units lower.
2. They all cross the y-axis at (0, 5).
3. (−3, −1): x + 3 = 0 at x = −3, and the −1 moves it down 1.
4. 4n + 1; the 20th term is 81.
5. 2, 5, 10, 17, 26, 37: none is a multiple of 3, so the conjecture survives. The test does not prove it; a reason is needed.
6. one counterexample could still reject it.
7. 2 + 3 = 5, which is odd. (2 is the only even prime.)
8. Sticks = 2n + 1, so 25 triangles use 51 sticks.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working, and separates testing a conjecture from proving it in Question 5.

Card keys

Graph cards (Day 3): y = 2x + 1: G and A; y = 2x − 3: D and J; y = −x + 1: E and L; y = (x − 2)²: B and H; y = x² − 3: K and F; y = −x² + 4: I and C. Conjecture cards (Day 5): always true M (odd + odd = (2a + 1) + (2b + 1) = 2(a + b + 1)), O (one of the two is even), R ((n + 1)² − n² = 2n + 1), S (the odd numbers build squares, layer by layer), U (n + (n + 1) + (n + 2) = 3(n + 1)), V ((2k + 1)² − 1 = 4k(k + 1), and k(k + 1) is even), X (it is 10 times a whole number, and 10 = 2 × 5); false: a counterexample N (n = 4 gives 21 = 3 × 7), P (0.5² = 0.25), Q (n = 3 gives 9), T (12 is divisible by 4 and 6, but not by 24), W (9 = 3 × 3).

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: families of lines, observing and conjecturing, sliding parabolas, generalising patterns, and counterexamples.