Secondary 1 Relations and Functions Worksheet

3 of the 4 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one. 3 of the 4 questions are printed below. The other 1 is built on a diagram or a table of values that does not translate to the page, so it is in the free PDF — marked below where it would have come.

1. Q1Algebra — Relations and Functions

   In each situation below, two quantities are recorded. Name the two variables, state which one is the independent variable and which is the dependent variable, and explain your choice in one sentence. If the two quantities are not linked by a dependency relationship at all, say so instead.

   1. A snow-clearing truck spreads $25\text{ kg}$ of abrasive on every kilometre of street. The crew notes the length of street cleared and the total mass spread.
   2. At the neighbourhood pool, each swimmer in a group pays $\$3$ to get in. The cashier notes the number of swimmers in the group and the amount the group pays.
   3. Over one hockey season, the coach notes the number of goals Léa scores in each game and the number of letters in her first name.
2. Q2Pattern Rule of a Sequence

   The first four terms of a sequence are $$7,11,15,19,\dots$$

   1. Write the next three terms, and describe in words how the sequence grows.
   2. Write the pattern rule that gives the term in position $n$.
   3. Use your rule to find the term in position $50$.
3. Q3The Modes of Representation of a Relation

   This question is built around a diagram or a table of values. Open it in the PDF.

4. Q4Synthesis — drawing on several sheets in this topic

   For the science fair, Théo pushes hexagonal tables together in one long row. One table on its own seats $6$ people; two tables pushed end to end seat $10$; three tables seat $14$.

   1. Build a table of values for $1$ to $5$ tables, and describe the growth in words.
   2. Write the rule giving the number of seats $y$ for $n$ tables, and say where each number in your rule comes from in the row of tables.
   3. Théo needs exactly $42$ seats. How many tables should he push together?
   4. Théo claims that a row of these tables can never seat an odd number of people. Is he right? Justify your answer using your rule.

The 4 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

Which one depends on which (Q1)

The independent variable is the one you choose; the dependent variable is the one that follows from that choice. A crew decides how far to drive, and the mass of abrasive spread follows. A group decides how many swimmers turn up, and the amount paid follows. The test is a sentence: say “the ___ is decided by the ___” and see which way round it makes sense.

Q1 also hides a third case that students almost never expect, and it is the reason the question exists. Two quantities can be recorded side by side and simply not be linked. Goals scored in a game and the number of letters in a player's name are both numbers, and one of them never changes at all — so neither can be said to depend on the other. Being asked to recognise that is what stops “find the relation” from becoming a reflex.

The pattern rule of a sequence: build it, do not guess it (Q2)

Given 7, 11, 15, 19, everyone can produce the next three terms. The question worth practising is the one after that: a rule that gives the term in any position, so that position 50 does not require writing out forty-six more terms.

The construction is mechanical once you have seen it. The terms climb by 4 each step, so the rule is built on 4n. Then compare: in position 1, 4n gives 4 while the term is 7 — three more. So the rule is 4n + 3, and position 50 gives 203 in one line.

Four ways to show one relation (Q3)

Q3 takes a described situation — a coin jar starting with a fixed amount and losing the same amount every school day — and asks for it three more times: as a completed table of values, as a rule, and as points plotted on a grid. This question is built on a table and a grid, so it lives on the printed sheet rather than on this page.

The point of the question is the last part, which asks what you notice about the points and why the situation makes that happen. The points fall in a straight line because exactly the same amount leaves the jar every single day: one step right always means the same drop. That is a sentence about the situation, not about the graph, and it is the sentence being marked.

Q4: the rule as an explanation, not just a formula

The synthesis question pushes hexagonal tables together in a row and counts the seats. The rule comes out as 4n + 2, and the question then asks something better than “what is the rule”: it asks where each number in your rule comes from in the row of tables.

The 4 is how many new seats each extra table brings — a hexagon has six sides, but joining it hides one side on it and one on the table it meets. The 2 is the pair of end seats, one at each end, which the row keeps however long it grows. Being able to point at both numbers in the picture is the difference between having found a rule and having understood one.

The last part asks whether the row can ever seat an odd number of people, and it is answered from the rule rather than by trying cases: 4n is always even, and an even number plus 2 is even again. One line of reasoning settles every value of n at once, which is exactly what a rule is for.

Getting the most out of it

Say the dependency out loud before you write anything

“The amount paid is decided by the number of swimmers.” If the sentence sounds right one way round and absurd the other, you have found the independent variable. If it sounds absurd both ways, check whether there is a dependency there at all — sometimes there is not.

Test every rule on a term you did not use to build it

Build the rule from the step and one term, then verify it somewhere else entirely. A rule that matches the first term and fails the third is the most common wrong answer on this topic, and thirty seconds of checking finds it every time.

Ask where the situation stops

A rule keeps producing numbers long after the situation has ended. An empty jar stays empty; a row of tables cannot have half a table. Say where the relation stops applying — it is often the part of the answer that separates a good response from a complete one.

Common questions