Networks and connectedness: a week of ready-to-teach maths

Five lessons for Year 10 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 network cards to match, twelve small networks 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 network cards before Day 1 and the 12 connectedness cards before Day 3, and keep each set in an envelope.
4. Coloured pencils help with tracing paths. No calculators are needed.
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

A network shows things and the links between them: people and friendships, towns and roads, species and what eats what. The things are vertices and the links are edges, and only which pairs are joined matters, not where the dots sit. The degree of a vertex counts its edges. A network is connected when a path runs between every pair of vertices; an edge whose loss would cut it in two is a bridge. Edges can carry times or distances, or point one way. Counting the odd vertices settles whether a route can travel every edge exactly once.

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

An edge is named by its two vertices, AB. A path is listed vertex by vertex, A–B–C. Travel times sit in boxes on their edges, and arrows show one-way edges. In this pack a bridge always means an edge that disconnects a network, except on Day 5, where the seven bridges are real ones over a river.

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 draws networks; Day 2 counts what meets at each vertex; Day 3 asks whether everything is reachable; Day 4 adds travel times and directions; and Day 5 asks whether every edge can be travelled once. Networks are new in Year 10; the unit builds on earlier work with diagrams, tables and practical problems, and it prepares the network topics of senior general mathematics.

Materials for the week (one trip)

• From the classroom: coloured pencils, scissors, envelopes for the cards, this pack printed.
• Cut-out sheet 1, one per pair: 12 network cards (Day 1).
• Cut-out sheet 2, one per pair: 12 connectedness cards (Day 3).
• A local bus or train map, if you have one, for Days 3 and 4. No other equipment to buy.

Words for the word wall

network, vertex, edge, degree, odd vertex, even vertex, path, connected, bridge, weighted edge, directed edge, trail, circuit, traversable

Dear families

This week in maths, Year 10 is learning about networks: diagrams of things and the links between them, such as friends, roads, bus routes and food webs, and how to tell whether everything in a network can be reached.

Try this at home

• Ask: at a family gathering of 6 people, everyone shakes hands with everyone else once. How many handshakes? (15.)
• Ask: if one road near home closed, could you still get to school? Is that road a bridge in the road network?
• Ask: on a bus or train map, which stop has the most lines meeting at it? (That stop has the highest degree.)
• Ask: could you walk every street in your block exactly once? (It depends on the corners where an odd number of streets meet.)

My networks week

Fill one row a day. Write a network you noticed, what its vertices and edges are, and one thing it shows.

Reading networks

A network shows things as vertices and the links between them as edges; only which pairs are joined matters. Students read a friendship network, draw a network from a list of roads, and match six drawings to their edge lists.

We are learning to

• name the vertices and edges of a network and what they stand for,
• draw a network from a list of links,
• tell when two drawings show the same network.

Success criteria

• I can read who is linked to whom from a network diagram.
• I can draw a road network from a list and check it against a partner’s different-looking drawing.

You need

Coloured pencils; the network cards (cut-out sheet 1), one set per pair; the worksheet.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Reading meaning into length or position: only the link counts.
• Counting a crossing as a vertex: edges can cross without meeting.
• Missing a link: tick each pair off the list as it is drawn.

Answers

• Part A: most friends Ben and Eli (3); fewest Cal and Fay (1). Cal–Ben–Eli–Fay: Ben and Eli could introduce them.
• Part B: any drawing with exactly the edges PQ, PR, QR, RS, ST: a triangle PQR with a tail R–S–T. R has the most roads (3).
• Part C: the card key is with the mini-check answers.
• Part D: no. The first has edges PQ, QR, RS, SP, QS; the second PQ, QR, RS, SP, PR. They have the same shape, but different pairs are joined (QS against PR).

Reading networks

A. A friendship network

Each box is a person; each line joins two friends.

B. Draw a road network

C. Match the cards

Match each drawing card to its edge list card. Write the letters.

D. Stretch: the same network?

Degree

The degree of a vertex counts the edges that meet there. Every edge has two ends, so the degrees always add to twice the number of edges, and odd vertices come in pairs. Students find degrees, count handshakes, and record a network as a table.

We are learning to

• find the degree of each vertex,
• use the sum of the degrees to count edges or to test whether a network can exist,
• record a network in a table and read one back.

Success criteria

• I can explain why the degrees of any network add to an even number.
• I can count the handshakes when 6 people all shake hands once.

You need

The worksheet; four volunteers for a handshake demonstration.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Counting each edge once in the total: it is counted at both ends.
• A lopsided table: the row for A must match the column for A.
• Expecting an odd number of odd vertices: it cannot happen.

Answers

• Part A: A 2, B 3, C 3, D 2, E 4, F 2; the total is 16 = 2 × 8 edges; the odd vertices are B and C.
• Part B: 1. each person has degree 5, so 6 × 5 ÷ 2 = 15 handshakes. 2. no: 3 + 3 + 3 = 9 is odd. 3. 5 × 2 ÷ 2 = 5 edges, such as a pentagon.
• Part C: A 0 1 0 0 1; B 1 0 1 0 1; C 0 1 0 1 0; D 0 0 1 0 1; E 1 1 0 1 0.
• Part D: the total is even; the even vertices add an even amount; so the odd vertices must add to an even number, which needs an even count of them.

Degree

A. Find the degrees

B. Handshakes and impossible networks

C. A network as a table

Write 1 where two vertices are joined and 0 where they are not. The first row is done for you.

D. Stretch: odd vertices come in pairs

Paths and connectedness

A path runs from vertex to vertex along edges. A network is connected when a path joins every pair of vertices, and a bridge is an edge whose loss would cut it into pieces. Students find paths between towns, find the roads whose closure would cut the network, and sort twelve networks.

We are learning to

• list a path between two vertices,
• decide whether a network is connected,
• find a bridge and say what it means in the situation.

Success criteria

• I can find two different paths from Ashby to Ford.
• I can name the roads whose closure would cut off a town, and the towns cut off.

You need

Coloured pencils; the connectedness cards (cut-out sheet 2), one set per pair; the worksheet.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Thinking a long path means not connected: any path will do.
• Calling every edge of a path a bridge: a bridge has no way around it.
• Missing a lone vertex when judging connectedness.

Answers

• Part A: Ashby–Cove–Dale–Ford and Ashby–Bell–Cove–Dale–Elm–Ford, among others; connected, as every town can reach Ashby.
• Part B: Cove–Dale and Ford–Glen. Closing the first splits Ashby, Bell and Cove from the rest; closing the second cuts off Glen. Every other closure only makes some trips longer.
• Part C: the card key is with the mini-check answers.
• Part D: 5 edges, such as a path through all six; every one of its edges is a bridge.

Paths and connectedness

A. Roads between seven towns

B. Road closures

C. Sort the networks

Write each card’s letter under the description that fits its network.

D. Stretch: as few edges as possible

Times and one-way streets

Edges can carry numbers, such as travel times, and can point one way, such as one-way streets or the links in a food web. Students find the quickest route on a network of travel times, follow one-way streets, and read a food web.

We are learning to

• find the quickest route on a network of travel times by comparing routes,
• follow arrows in a directed network to decide what can be reached,
• interpret a directed network such as a food web.

Success criteria

• I can show that the quickest route from S to T uses more roads than the most direct one.
• I can say which places cannot get back to P along the one-way streets.

You need

Coloured pencils; a local bus or train map if you have one; the worksheet.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Choosing the route with the fewest edges: add the times.
• Ignoring arrows: a one-way edge can be used only in its direction.
• Reading a food web backwards: here each arrow points from the food to the eater.

Answers

• Part A: S–B–A–C–T 22, S–A–C–T 23, S–B–C–T 27, S–B–A–T 29, S–A–T 30, S–A–B–C–T 36, S–B–C–A–T 46 minutes; the quickest is S–B–A–C–T, 22 minutes.
• Part B: from P you reach Q, R, S, T; S and T cannot get back to P (the only street out of S leads to T, and T leads only back to S).
• Part C: the snake eats frog and mouse; the mouse is eaten by snake and hawk. Grass has no arrow in: it makes its own food. Without frogs the snake loses one food.
• Part D: with AC closed, S–B–C–T, 27 minutes.

Times and one-way streets

A. The quickest route

The numbers are travel times in minutes.

B. One-way streets

C. A food web

Each arrow points from a food to the animal that eats it.

D. Stretch: a road closes

Every edge once

A trail travels every edge exactly once. The odd vertices decide it: none means a circuit back to the start, exactly two means a trail from one odd vertex to the other, and more than two means no such trail. Students test three networks, plan a ranger’s walk, settle a puzzle about seven bridges, and play the self-check quiz.

We are learning to

• count the odd vertices of a network,
• decide whether a trail or a circuit through every edge exists, and where it must start,
• explain the result in the situation.

Success criteria

• I can say where a ranger must start to walk every path in the park exactly once.
• I can explain why no walk crosses each of the seven bridges exactly once.

You need

Coloured pencils; the worksheet; the self-check quiz on the board.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Starting anywhere when there are two odd vertices: start at one, finish at the other.
• Counting a crossing in a degree: count only the edges at the vertex.
• Giving up too soon: when the count allows a trail, try again from the right start.

Answers

• Part A: Network 1: 2 odd (A, C), a trail; Network 2: 0 odd (none), a circuit; Network 3: 4 odd (B, C, D, E), neither.
• Part B: the odd corners are Pond and Oak: start at one and finish at the other. To start and finish at the Gate, walk the Pond–Oak path twice.
• Part C: the land areas have degrees North 3, Island 5, East 3, South 3: four odd vertices, so no walk crosses each bridge once.
• Part D: any network with exactly two odd vertices, such as six vertices in a line: start at one end, finish at the other.

Every edge once

A. Circuit, trail or neither?

B. A ranger’s walk

C. Seven bridges

D. Stretch: design one

Drawings and edge lists

For Day 1, one set per pair. Six cards show a network drawing; six list a network’s edges. Match each drawing to its list.

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

Connected, with a bridge, or not connected?

For Day 3, one set per pair. Sort the networks into three piles: connected with no bridge, connected but with a bridge (an edge whose loss would cut it), and not connected.

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

What we know: Networks and connectedness

Work on your own. Sketch the networks if it helps.

1. A network has six vertices with degrees 3, 2, 2, 4, 1, 2. How many edges does it have? ______________________
2. Can a network have four vertices with degrees 3, 3, 2, 1? Explain. ______________________
3. Roads join PQ, QR, RP, RS, ST. Is the network connected? Name every bridge. ______________________
4. Travel times in minutes: AB 4, BD 9, AC 7, CD 5, BC 2. What is the quickest route from A to D? ______________________
5. One-way streets run A to B, B to C, C to A, C to D. Starting at D, which places can you reach? ______________________
6. A network with exactly two odd vertices has, circle one:   a trail through every edge, from one odd vertex to the other  ·  a circuit through every edge  ·  no trail through every edge
7. A network has four vertices with degrees 2, 4, 2, 2. Can a walk use every edge once and return to its start? ______________________
8. In a friendship network, one vertex has degree 0. What does that mean, and is the network connected? ______________________

Answers and marking guide

Answers with working

1. 3 + 2 + 2 + 4 + 1 + 2 = 14, so 7 edges.
2. No: the degrees add to 9, an odd number, but they must add to twice the edges.
3. Yes; the bridges are RS and ST.
4. A–B–C–D, 11 minutes (the next best is A–C–D, 12 minutes).
5. None: no street leads out of D.
6. The first: a trail through every edge, from one odd vertex to the other.
7. Yes: every degree is even, so there is a circuit, and it can start anywhere.
8. That person has no friends in the group, so no path reaches them: the network is not connected.

A quick three-level guide

Eight questions, four ideas. A student at standard reads a network only by its links and explains connectedness in the situation.

Card keys

Network cards (Day 1): G with A; D with J; E with L; B with H; K with F; I with C. Connectedness cards (Day 3): connected, no bridge: N, P, Q, S, W; connected, with a bridge: M, R, V, X; not connected: O, T, U.

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: reading networks, degree, connectedness, times and one-way streets, and every edge once.