At a glance. One lesson of about 60 minutes. It assumes rounding to decimal places and significant figures. Students find the bounds of a rounded value, the bounds of sums, differences, products and quotients, and decide what rounding a context needs.
Calculator: GDC optional. Everything can be done by hand. The optional section shows students who have a GDC how to enter a bound calculation in one line with brackets, and warns against rounding in the middle.
What students will learn
Know
- A value to a stated accuracy lies half a unit either side, with the lower bound included and the upper bound not.
- To maximize a result, use UB + UB, UB × UB, UB − LB and UB ÷ LB; to minimize, the opposite.
- A count that must be guaranteed rounds down.
Understand
- A rounded value stands for an interval, so a calculation with rounded values has an interval of results.
- The accuracy of a result is limited by its least accurate input.
- Which bound is needed depends on the question, not on the number.
Do
- State the bounds of a value rounded to a given decimal place, significant figure or unit.
- Find the upper and lower bounds of a calculated result.
- Decide how to round an answer in context so that a claim is certain.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: bounds of rounded values and of calculations (questions 1 to 9).
- Criterion B. Investigating patterns: the second figure and the what-if. Students find which bounds make each operation largest and smallest, and generalize.
- Criterion C. Communicating: setting out a bound calculation with the bounds written down first, and stating the answer as an interval (worked example 2).
- Criterion D. Applying mathematics in real-life contexts: a weighbridge, a plane's load and a cyclist's speed, where a decision depends on a bound (the hook and questions 5 and 7).
The one thing to do with the figure
Ask for a prediction before every press. With a = 12 and b = 7, ask: “which bounds of a and of b will give the biggest quotient?” before pressing a ÷ b. Then ask why the bound of b used for the biggest difference is its lower bound.
In the first figure, ask why the lower end is a filled dot and the upper end an open circle.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the weighbridge. Collect predictions. |
| 12 min | Both figures with predictions; the key idea. |
| 10 min | Warning card and worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose B or C, return to the first figure and change the unit. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 9. |
The answers
| Warm-up | 0.046; 15 to 24; 81.25. |
|---|---|
| Hook | No: the true mass is from 24 250 up to, but not including, 24 350, so it could be above 24 320; 6 to the nearest whole number is 5.5 ≤ x < 6.5, narrower than 5 to 7. |
| Worked examples | 6 750 ≤ n < 6 850; 398.75 m² to 441.75 m²; 32.7 cm. |
| Faded example | 8.20 m/s. |
| 1. Bounds of a rounded mass | A. B uses a full unit; C uses half of 1 kg; D includes 87.5. |
| 2. An upper bound | 2355. 2350 + 5. |
| 3. A lower bound from significant figures | 0.06395. 0.0640 − 0.00005. |
| 4. Upper bound of an area | 60.5875. 9.25 × 6.55. |
| 5. Greatest possible speed | 7.78. 245 ÷ 31.5. |
| 6. Upper bound of a difference | B. A uses no bounds; C is the lower bound; D uses a full unit. |
| 7. A count that must be guaranteed | 119. 102 500 ÷ 855 = 119.88…, rounded down. |
| 8. Lower bound of a quotient | 6.03. 9.35 ÷ 1.55. |
| 9. Bounds with trailing zeros | C. A uses a unit of 10; B is the largest whole number; D uses a whole unit of 100. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Used a full unit, half of 1 instead of half of 5, or included the upper end. | Ask: what is the unit, and what is half of it? Which mass rounds up to 90? |
| 2 360 or 2 345 in question 2 | Added a full 10, or gave the lower bound. | Ask: half of 10 is what? Is the question about the upper or lower bound? |
| 0.0635 or 0.0639 in question 3 | Used the wrong place, or truncated. | Underline the last significant figure and name its place. |
| 59.8 or 61.38 in question 4 | Used no bounds, or a full unit. | Write each upper bound on its own line first. |
| 7.54 or 7.5 in question 5 | Used the wrong bound of the time, or none. | Ask: to make the speed biggest, should the time be longer or shorter? |
| A, C or D in question 6 | Used no bounds, the lower bound, or a full unit. | Which bound of b makes a − b biggest? |
| 120 in question 7 | Used the quoted mass, or rounded up. | Ask: could 120 pallets be over the limit? Find a mass that makes it so. |
| 6.45 or 6.27 in question 8 | Used the wrong bound in the denominator, or none. | Make the top as small and the bottom as big as possible. |
| A, B or D in question 9 | Used the wrong unit, the largest integer, or a whole unit. | Name the place of the 4 and then half a unit of it. |
Questions worth asking
- Why is the upper bound not included? Listen for “because that value would round up to the next number”.
- Why does a subtraction need the lower bound of the second number? Taking away less leaves more.
- Can a result be more accurate than its inputs? Not by calculating with them: the interval of a result is usually wider than the interval of each input.
- When should an answer be rounded up, and when down? Down when a count must be certain to fit; up when it must be certain to be enough.
Differentiation
A way in for anyone stuck
- Always write each number's interval on the page before calculating.
- Use a table with columns number, lower bound and upper bound.
- Start with sums, then differences, before products and quotients.
Stretch
- A length of 8.4 m and a width of 5.2 m are each to 1 d.p. Find the largest possible percentage error in the area, from the area found with the rounded numbers.
- Show that (x + y)/(x − y) can be made arbitrarily large if x and y are both rounded to the same number.
Students learning in English as an additional language
“Bound” means “limit”: the upper bound is the top limit and the lower bound is the bottom limit. “Accuracy” means “how close to the true value”. “Interval” means “everything between two ends”. “Certain” means “with no possibility of being wrong”.
| Word | Plain meaning | Example |
|---|---|---|
| upper bound | the value just above every number that rounds to the stated one; it is not reached | the 0.125 for 0.12 to 2 d.p |
| lower bound | the smallest value that rounds to the stated number | the 7.35 for 7.4 to 1 d.p |
| accuracy | the place a value was rounded to | to the nearest 10 cm |
| interval | all the values between a lower and an upper bound | from 2.5 up to 7.5 |
| limit of accuracy | another name for a bound of a rounded value | 8.75 and 8.85 for 8.8 |
| maximize | make a result as large as possible | use the largest first number |
Sentence starters for explaining aloud or in writing:
- The accuracy is …, so half a unit is …
- To make this as large as possible I use the … bound of … because …
- The answer is between … and …, so I can be certain that …
Two things not to say
“Just add and subtract 0.5.” Half a unit depends on the place rounded to; say “half of the smallest place”.
“Use the upper bounds for the upper bound.” In a subtraction or a division the second number needs its opposite bound.
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.
- BoundsWrite the bounds of 3.7, to 1 decimal place.
3.65 ≤ x < 3.75. - SumFind the upper bound of 12 + 8, each to the nearest whole number.
12.5 + 8.5 = 21.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA farm in Kenya records a milk tank as 5 000 litres, to the nearest 100 litres. Write its bounds.
4 950 ≤ V < 5 050. - Earlier work (Lesson 4)Solve |x − 5| ≤ 2.
3 ≤ x ≤ 7.
3Challenge
Reasoning, generalizing and working backwards.
- Reasoningm = 3.2 and n = 1.8, each to 1 d.p. Find the lower bound of m ÷ n.
3.15 ÷ 1.85 = 1.70 to 2 d.p. - DecisionA shelf holds 36 kg. Each box is 4.5 kg to 2 s.f. How many boxes are certain to be safe?
Upper bound 4.55: 36 ÷ 4.55 = 7.9, so 7.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider or button, then the arrow keys or Enter) and by touch. “Show me the steps” gives the same states in words.
The With a GDC section is closed by default and optional. Students without a GDC lose nothing; every answer is reached by hand.