At a glance. One lesson of about 60 minutes. It assumes only counting and simple equations. Students read degrees from a graph, use the sum of the degrees to find edges, and decide which lists of degrees are possible.
Calculator: no calculator. No calculator: every number is a small count. The skill is the structure of the argument, counting each edge from both ends.
What students will learn
Know
- A graph is a set of vertices and a set of edges; the degree of a vertex is the number of edges at it.
- The sum of the degrees is twice the number of edges.
- The number of vertices of odd degree is even.
Understand
- Why each edge is counted twice when the degrees are added.
- Why a graph is defined by what is joined, not by how it is drawn.
- Why a list with an odd sum cannot be the degrees of a graph.
Do
- Read and add degrees from a graph or a description.
- Find the number of edges from the degrees, and the reverse.
- Reject a list of degrees with an odd sum and test a list with an even sum.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: degree, edge and vertex used correctly (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students state the relationship between the sum of the degrees and the edges, and test it for several edge counts.
- Criterion C. Communicating: a clear justification, with a sum of degrees written out, of why a degree list is impossible.
- Criterion D. Applying mathematics in real-life contexts: bridges, exchanges and relay links (hook and questions 2, 4, 6 and 7).
The one thing to do with the figure
Ask for the prediction first. Then drag the slider slowly from 0 to 9 and ask students to predict the sum before each new edge appears.
Ask: “What happens to the sum of the degrees every time an edge is added, and why?” Listen for: it always goes up by 2, because both ends of the edge get one more.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: seven guests and three handshakes; collect predictions and do not reveal. |
| 12 min | Figure, key idea and warning card; ask what happens to the sum when one edge is added. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, ask them to draw it: who must the two vertices of degree 3 be joined to? |
| 20 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | 15; 19; 6. |
|---|---|
| Hook | No. Seven guests each of degree 3 would give a sum of 21, which is odd, and the sum of the degrees must be twice the number of edges. With six guests it is possible: 18 ÷ 2 = 9 handshakes. |
| Worked examples | Degrees 2, 2, 3, 1 with 4 edges; x is 1 or 3; 10 edges. |
| Faded example | 20 edges, and the graph can exist. |
| 1. Which lists of degrees are possible | B. A fails because two vertices of degree 3 force every other degree to be at least 2; C has an odd sum; D has a degree above 3. |
| 2. Sum of the degrees | 28. 2 × 14. |
| 3. Edges from the degrees | 7. 14 ÷ 2. |
| 4. Regular graph | 18. 36 ÷ 2. |
| 5. A missing degree | 2. Even, and not 0. |
| 6. Two kinds of vertex | 6. 18 + 16 = 34. |
| 7. Complete graph, working backwards | 10. 10 × 9 ÷ 2 = 45. |
| 8. Using the handshake result | C. A forgets the factor 2; B gives an odd sum; D is the whole sum. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Treated an even sum as enough, did not check the sum, or forgot the largest possible degree. | Ask for the sum of each list, then ask them to try to draw the lists with an even sum. |
| 14 or 23 in question 2 | Gave the number of edges, or added the edges to the vertices. | Ask: how many ends does a bridge have, and where does each end go? |
| 14 or 6 in question 3 | Gave the sum of the degrees without halving, or gave the number of vertices. | Ask: if I add the degrees, how many times has each edge been counted? |
| 36 or 15 in question 4 | Did not halve the sum of the degrees, or added the students and the degree. | Draw 4 students, each with 3 flags to exchange, and count exchanges. |
| 0 or 1 in question 5 | Took the smallest value without testing parity, or took an odd value that makes the sum odd. | Ask: what must the sum be? What does a degree of 4 force on the fifth vertex? |
| 4 or 17 in question 6 | Gave the other kind of station, or gave the number of links. | Ask: what does t stand for? Check the answer by adding the degrees. |
| 9 or 90 in question 7 | Gave each player's number of games, or the sum of the degrees. | Ask: with n players, how many games does each play, and what do the degrees add to? |
| A, B or D in question 8 | Took the sum to be the number of edges, gave an odd sum, or gave the whole sum. | Ask: what must all seven degrees add to? Then take away the six known degrees. |
Questions worth asking
- Why must the degrees add to an even number? Listen for: each edge has two ends, so each edge adds 2 to the total.
- Does the picture matter, or only what is joined? Listen for: only what is joined; moving or bending the edges does not change the graph.
- Could a graph have exactly three vertices of odd degree? Listen for: no, the odd degrees would add to an odd number, but the sum must be even.
- If I know the number of edges, what do I know about the degrees? Listen for: only their sum, not the individual degrees.
Differentiation
A way in for anyone stuck
- Give a pre-drawn graph on a grid and a table with a column for each vertex's degree and a total row.
- Offer pencil dots and lines and ask students to build the graph with three vertices of degree 2 first.
- Give the frame: “The degrees add to …. Each edge is counted … times, so there are … edges.”
Stretch
- Show that a graph with at least two vertices always has two vertices of the same degree (the pigeonhole argument on the possible degrees).
- A graph has 6 vertices and 13 edges. What is the smallest possible number of vertices of degree 5?
Students learning in English as an additional language
“Vertex” is a point and the plural is “vertices”. “Edge” is a line between two vertices. “Degree” here means a count of edges, not an angle. “Graph” here is a network, not a graph of a function.
| Word | Plain meaning | Example |
|---|---|---|
| graph | a set of vertices joined by edges | a map of bus stops |
| vertex | a point where edges can meet | a village |
| edge | a line that joins two vertices | a bridge |
| degree | the number of edges at a vertex | a vertex on 4 edges has degree 4 |
| handshake result | the rule linking the sum of the degrees to the number of edges | each edge has two ends |
Sentence starters for explaining aloud or in writing:
- The degrees add to … so there are … edges.
- This list is impossible because …
- Each edge is counted twice because …
Two things not to say
“Degree is how many vertices there are.” Say instead: degree is how many edges meet at one vertex.
“The graph is the drawing.” Say instead: the graph is the list of which vertices are joined; the drawing is one way to show it.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- FindA graph has 5 edges. What do its degrees add to?
10. - FindDraw a graph with degrees 1, 1, 2, 2.
A path of four vertices: the two ends have degree 1.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextNine friends each swap numbers with exactly 4 others. How many swaps?
9 × 4 ÷ 2 = 18. - Earlier work (Year 10)Write an expression for the number of pairs among n people.
n(n − 1) ÷ 2.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningCan a graph have 11 vertices, each of degree 3? Explain.
No: 33 is odd, and the sum of the degrees must be even. - Working backwardsA graph has seven vertices with degrees 6, 4, 4, 3, 3, 3 and y. What is the smallest y, and how many edges are there then?
The known degrees add to 23, so y is odd. The vertex of degree 6 is joined to every other vertex, so y is at least 1. The smallest is 1, and 24 ÷ 2 = 12 edges.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to the slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
No calculator is needed.