Mathematics K–6 · Years 5–6

Estimate the docket, then check it: rounding to judge whether a total is reasonable

Number and algebra

Practical, model not builtLow risk

This site has no interactive model of its own. Where a step or a material names a Concept Studio model, simulation or tool, it has not been built; an external simulation a step names (for example PhET) is not part of this site.

The idea

Rounding every price before adding gives a total close enough to judge whether the printed total is reasonable, and the rounding rule chosen decides whether the line-by-line errors cancel or pile up.

What you need

one real supermarket or school canteen docket per pair with 10 to 20 priced lines, or a printed supermarket catalogue page where a family would rather not share a docket
a sticky note or a folded flap to cover the printed total
a recording table with columns: price, nearest dollar, rounded down, rounded up, nearest 50 cents
a calculator per pair, kept face down until step 5
a class results table on the board with one row per docket and one column per rounding rule

How to do it

Cover the printed total before anything is written down. Copy the first 10 prices into the recording table.
Round each price to the nearest dollar, taking 50 cents or more up and less than 50 cents down, and add the rounded dollars mentally. Record the estimate and the time it took.
Repeat the estimate three more times on the same 10 prices: rounding every price down to the whole dollar, rounding every price up to the whole dollar, and rounding to the nearest 50 cents.
Before uncovering anything, write which of the four estimates you expect to be closest and why, and write a range you are confident the exact total sits inside.
Uncover the printed total, add the 10 prices exactly with the calculator, and record the error for each rule as the estimate minus the exact total, in dollars and as a percentage of the total.
Add your four errors to the class results table. With 20 or more dockets collected, work out the mean error and the largest error for each rule, and say which rule is the one to use when the question is whether a total is reasonable.
Sort every price the class recorded by its cents ending and count how many end in 00, 50, 95 or 99 cents. Predict from those counts whether the nearest-dollar rule will read high or low, then check the prediction against the class mean error.
Repeat steps 1 to 5 once with a docket of 20 lines and compare the size of the nearest-dollar error with the 10-line result.

What you should see

Worked on the example price set $4.20, $1.85, $6.50, $2.99, $3.40, $0.95, $7.25, $2.10, $5.60 and $1.45, entered by the teacher as an example and not offered as any shop's prices, the exact total is $36.29. Rounding to the nearest dollar gives $36, an error of 29 cents low, which is 0.80 per cent of the total. Rounding every price down gives $31, an error of $5.29 low, which is 14.58 per cent of the total. Rounding every price up gives $41, $4.71 high. Rounding to the nearest 50 cents gives $36.50, 21 cents high. Across many dockets the nearest-dollar errors cancel, because a price rounded up is about as likely as one rounded down: for cents spread evenly over 0 to 99, the nearest-dollar error on one line has a mean of half a cent and a standard deviation of 28.87 cents, so the typical error is about 91 cents over 10 lines and about $1.29 over 20 lines, and it grows with the square root of the number of lines rather than in proportion to it. The rounding-down rule has a mean error of 49.5 cents low on every line, so it runs about $4.95 low on 10 lines and $9.90 low on 20, and rounding up runs the same distance high. Rounding to the nearest 50 cents halves the spread to 14.43 cents a line. Where many shelf prices end in 95 or 99 cents, as the count in step 7 shows for the class's own dockets, the nearest-dollar rule reads high, because those prices always round up: a 20-line docket of prices all ending in 95 cents reads $1.00 high and one of prices all ending in 99 cents reads 20 cents high. The learner knows the method worked when the nearest-dollar estimate on a 10-line docket lands within about $1.83, two standard deviations, of the printed total; a gap larger than that is the signal to check the docket line by line rather than to accept the total.

What changes

What you change: the rounding rule used (nearest dollar, rounded down, rounded up, nearest 50 cents) and the number of priced lines on the docket
What you measure: the error, estimate minus exact total, in dollars and as a percentage of the total
What you keep the same: the same set of prices is used for all four rules
the printed total stays covered until every estimate is written down
every estimate is added mentally with the calculator face down
50 cents or more rounds up in the nearest-dollar rule, for every learner
the same number of lines, 10, in steps 1 to 6

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

An estimate is a guess, so any number near the total counts as a good estimate.
Rounding every price down is the safe way to estimate a shopping total.
A nearest-dollar estimate of 20 lines could be out by $20, because each line could be out by a dollar.
An error of a few cents on each line cannot add up to an amount that matters.
The printed total on a docket is always right, so there is nothing for an estimate to test.

Safety card

Low riskLearners carry it out

Hazards

a real docket can carry a family name, the last digits of a card number, a loyalty number or a home address
a docket can show purchases a family would rather not have read out in class

Controls

cut or cover any name, card digits, loyalty number or address before a docket is handed to another pair
offer a printed catalogue page as the alternative, with no explanation asked for
prices only are copied into the table, never item names, and dockets go home or into the recycling at the end of the session

Note

No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package is not needed; ordinary classroom supervision under the school's usual risk management, and the school's privacy practice for material brought from home.

Curriculum references

The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.

Mathematics K-10 Syllabus (2022), NSW Education Standards Authority, Stage 3 focus area Additive relations AMA3-AR-01
Mathematics K-10 Syllabus (2022), NSW Education Standards AuthorityMAO-WM-01
Australian Curriculum v9AC9M5N08AC9M6N08AC9M6N04

Sources

The pages the author read to write this activity.

All Concept Studio activities