At a glance. One lesson of about 60 minutes. It uses arithmetic-sequence sums from lesson 1. Students read and write sigma notation, count terms, and use the sum of a constant, of a multiple and of a sum of rules.
Calculator: no calculator. No calculator: each sum is short, and the point is to read the notation and structure it.
What students will learn
Know
- The sum from k = a to k = b of f(k) means f(a) + f(a + 1) + … + f(b).
- There are b − a + 1 terms.
- A sum of a rule plus another splits; a constant multiplier comes outside; n copies of a constant add to n times it.
Understand
- The letter under sigma is a counter, not a quantity.
- A sigma sum is a list in disguise, so it can be written out when stuck.
- The start value changes both the terms and the count.
Do
- Write out and evaluate a sigma sum.
- Write a list of terms in sigma notation.
- Use the properties and the sum of k to evaluate a sum.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: reading notation, counting terms and evaluating (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students describe what changes when the first value of k changes.
- Criterion C. Communicating: writing a list in sigma notation with correct limits.
- Criterion D. Applying mathematics in real-life contexts: weekly savings (hook) and a tailor's weekly output (question 4).
The one thing to do with the figure
Ask for the prediction first. Then set the rule to 1, the first value to 1 and the number of terms to 3, and ask students to write the three terms before reading them. Move the first value to 3 and ask what changed and what did not.
Ask: “Is the sum from 2 to 4 the same as the sum from 1 to 3?” The count is the same and the terms are not.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the savings sum; collect what each part of the notation says. |
| 12 min | Figure, key idea and warning card; write three sums out as lists. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, list the values of k together and count them. |
| 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 | 7; 12; 24. |
|---|---|
| Hook | The terms are 3, 5, 7, 9, 11 and their sum is 35. |
| Worked examples | 34; the sum from k = 1 to 6 of (3k + 1), which is 69; 255. |
| Faded example | 168. |
| Something to think about | From k = 0 to 33 of (3k + 1), or from k = 1 to 34 of (3k − 2): 34 terms either way, the same sum. |
| 1. Counting terms and constants | D. A counts b − a terms; B takes the top number as the count; C adds the constant once. |
| 2. Evaluating a sum | 145. 165 − 20. |
| 3. Starting above 1 | 108. 3 × 36. |
| 4. Using a known sum | 1160. 1260 − 100. |
| 5. A sum of multiples | 765. 17 terms, 21 to 69. |
| 6. Sum of odd numbers | 12. n2 = 144. |
| 7. Finding an unknown in a sum | 4. 15p + 15 = 75. |
| 8. Listing the terms | A. B starts at k = 1; C stops at k = 5; D is k = 3 to 7. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or C in question 1 | Counted b − a terms, took the top number as the count, or added the constant once. | List the values of k and count them. |
| 165 or 475 in question 2 | Left the constant out, or squared the 3 as well as k. | Write the five terms out: 3 − 4, 12 − 4, and so on. |
| 135 or 85 in question 3 | Started at k = 1, or counted one term too few. | Write the first and last value of k beside the first and last term. |
| 115 or 1255 in question 4 | Gave week 20 only, or subtracted the constant once. | Ask: is the question about one week or all of them? How many −5s are there? |
| 720 or 828 in question 5 | Used 16 terms, or started at k = 1. | Ask: what is the first term, and what is the last? |
| 72 in question 6 | Halved the sum instead of recognising n squared. | Add 1 + 3 + 5 + 7 and ask what the total is. |
| 5 or 4.8 in question 7 | Left out the constants, or added one constant. | Write out the five terms and add the 3s. |
| B, C or D in question 8 | Started at k = 1, stopped at k = 5, or used k = 3 to 7. | Substitute the first and last value of k into the rule. |
Questions worth asking
- How many terms does the sum from k = 5 to k = 12 have? Listen for: 12 − 5 + 1 = 8.
- Why is the sum from k = 1 to 5 of 3 not just 3? Listen for: the 3 is added once for every term, so 5 × 3 = 15.
- Does the sum from j = 1 to 4 of j equal the sum from k = 1 to 4 of k? Listen for: yes, the letter is a counter.
- How would you write 2 + 4 + 6 + 8 + 10? Listen for: the sum from k = 1 to 5 of 2k.
Differentiation
A way in for anyone stuck
- Give a table with columns k, rule and term, filled one row at a time.
- Provide three sums already written out as lists and ask students to match them to their notation.
- Give the frame: “k goes from … to …, so there are … terms. The first term is … and the last is ….”
Stretch
- Show that the sum from k = 1 to n of (2k − 1) is n squared without using the arithmetic formula, by arranging dots.
- Find a rule for the sum from k = 1 to n of (3k + 1) in terms of n.
Students learning in English as an additional language
“Sigma” is the Greek letter used for “sum”. “Limit” here means the first or last value of k. “Index” is the letter k, which counts the terms. “Evaluate” means find the number the sum equals.
| Word | Plain meaning | Example |
|---|---|---|
| sigma | the symbol used for adding up the terms of a rule | Σ |
| limits | the first and last values of the counter k | 1 and 6 |
| index | the counter letter, such as k, that runs through the values | k |
| term | one value of the rule for one value of k | the 3rd one |
| evaluate | work out the number a sum equals | add them |
Sentence starters for explaining aloud or in writing:
- k goes from … to …, so there are … terms.
- The first term is … and the last term is ….
- I split the sum into … and ….
Two things not to say
“The number on top is how many terms.” Say instead: the top is the last value of k.
“A constant is just added once.” Say instead: it is added once for every term.
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.
- Write outWrite out the sum from k = 1 to 4 of 3k.
3 + 6 + 9 + 12 = 30. - CountHow many terms are in the sum from k = 6 to k = 20?
15.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA baker in Lima makes 10 loaves on day 1 and 4 more each day. Write the total loaves in 7 days with sigma and evaluate it.
The sum from k = 1 to 7 of (4k + 6) = 4 × 28 + 42 = 154. - Earlier work (lesson 1)Find 3 + 7 + 11 + … for 10 terms.
10 ÷ 2 × (3 + 39) = 210.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningIs the sum from k = 1 to 5 of k2 equal to the square of the sum from k = 1 to 5 of k? Explain.
No: 55 and 15 squared is 225. - Working backwardsThe sum from k = 1 to 6 of (ak + 2) is 75. Find a.
21a + 12 = 75, a = 3.
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 a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
No calculator is needed.