Reasonableness and Estimation: a week of ready-to-teach maths

Five days of lessons for Year 5 Number. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet, plus twelve claim cards to sort, sixteen price tags, a mini-check and every answer.

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. Cut out the claim cards and the price tags once; the price tags come back on Day 5.
4. Open the free interactive unit on your board. Every plan tells you which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and answers are all on the one page.

No maths background needed

This pack is written for the busy generalist teacher. Each plan explains the idea in plain words, lists the misconceptions children bring, and gives the answers. The big idea of the week: an estimate is a quick, sensible guide to the size of an answer. Round to easy numbers (312 + 489 is about 300 + 500 = 800), use numbers that divide easily (203 ÷ 4 is about 200 ÷ 4 = 50), and compare: an answer far from the estimate is a signal to look again. The situation chooses the estimate: to be sure there is enough money, round every price up.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 50 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point if your timetable runs two shorter sessions.

Reasonable is not the same as right

An estimate catches the big slips: a zero too many, a decimal point in the wrong place, the wrong operation. It cannot catch an answer that is close but wrong, and on Day 3 children meet one such card on purpose.

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

Days 1 and 2 make estimates, first for adding and subtracting and then for multiplying and dividing; Day 3 uses estimates to judge answers; and Days 4 and 5 bring it to money, where the situation decides how to round. It builds on Year 4’s rounding to check calculations and money (AC9M4N07) and on this year’s multiplication and division (AC9M5N06, AC9M5N07), and it leads to Year 6, where children estimate with decimals, fractions and percentages, including money (AC9M6N08).

Materials for the week (one trip)

• From the classroom: scissors, envelopes for the cards, mini whiteboards, this pack printed.
• Cut out once: the twelve claim cards (Day 3) and the sixteen price tags (Days 4 and 5). No maths equipment to buy.

Words for the word wall

estimate, round, nearest ten, nearest hundred, compatible numbers, reasonable, exact, total, change, round up, round down

Dear families

This week in maths, Year 5 estimates to check answers. We round to easy numbers: 312 + 489 is about 300 + 500 = 800. With money, we round each price to the nearest dollar: $4.95 + $3.10 + $6.20 is about $5 + $3 + $6 = $14. An answer far from the estimate is a signal to check again.

Try this at home

• At the shops, estimate the total before you pay. How close were you?
• Round every price up to check that you have enough money.
• Estimate the change before you are given it, then check it.
• Ask your child why 412 + 389 cannot be 1801.

My estimates this week

Fill one row a day. Estimate first, then find the exact amount.

Printed from the free seegongsik Reasonableness and Estimation teaching pack · seegongsik.com/au/y5/number/AC9M5N08/pack

Round to easy numbers

Children estimate sums and differences by rounding each number to an easy one: the nearest ten, hundred or thousand. A smaller place gives a closer estimate but harder numbers, and the question decides how close the estimate needs to be.

We are learning to

• round numbers to the nearest ten, hundred or thousand,
• estimate a sum or difference by rounding each number first,
• decide when an estimate is close enough and when to work it out exactly.

Success criteria

• I can estimate 806 − 389 as about 800 − 400 = 400.
• I can explain why rounding to the nearest ten gives a closer estimate than rounding to the nearest hundred.

You need

Mini whiteboards. The worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B by rounding 2437 + 1286 both ways again.

Watch for these ideas

• Rounding a 5 down: 245 to the nearest ten as 240. Five and above rounds up: 250.
• Rounding only one number: 312 + 489 as 312 + 500. Round both, so the estimate is quick.
• Trusting an estimate that is too close to call: 28 + 34 is about 60, but exactly 62 children do not fit in 60 seats.

Answers

• Part A: 300 + 500 = 800 (exactly 792); 800 − 400 = 400 (exactly 417); 1200 + 400 = 1600 (exactly 1607); 900 − 500 = 400 (exactly 481); 3600 + 2200 = 5800 (exactly 5813); 5100 − 1900 = 3200 (exactly 3160).
• Part B: 147 + 238: 150 + 240 = 390 or 100 + 200 = 300, exactly 385, so the tens are closer; 562 − 349: 560 − 350 = 210 or 600 − 300 = 300, exactly 213, so the tens are closer. Part C: about 800, so yes, easily; about 60, too close to call; exactly 62, so no; about 2200, so no.

Round to easy numbers

A. Round each number to the nearest hundred, then estimate

274 + 518 is about ______ + ______ = ______806 − 389 is about ______ − ______ = ______1245 + 362 is about ______ + ______ = ______932 − 451 is about ______ − ______ = ______3615 + 2198 is about ______ + ______ = ______5080 − 1920 is about ______ − ______ = ______

B. The nearest ten or the nearest hundred?

C. Close enough to decide?

Estimate first. If the estimate is too close to the limit to be sure, work it out exactly.

Easy numbers to multiply and divide

Children estimate products by rounding each number to its first digit, and quotients by moving the number being divided to a nearby number that divides easily: 362 ÷ 9 is about 360 ÷ 9 = 40. They notice whether an estimate is above or below the exact answer.

We are learning to

• round both numbers so a product is easy to estimate,
• choose a compatible number that divides easily, such as 360 for 362 ÷ 9,
• tell whether an estimate is above or below the exact answer.

Success criteria

• I can estimate 362 ÷ 9 as 360 ÷ 9 = 40.
• I can explain why 50 × 30 is above the exact answer to 48 × 29.

You need

Mini whiteboards. The worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B by modelling 284 ÷ 7 again.

Watch for these ideas

• Rounding the divisor too: 362 ÷ 9 as 400 ÷ 10. Keep the 9 and move 362 to a number 9 divides.
• Dropping zeros: 50 × 30 = 150. Five tens times three tens is fifteen hundreds: 1500.
• Expecting the estimate to match the exact answer. It only has to be close.

Answers

• Part A: 50 × 30 = 1500 (exactly 1488); 60 × 20 = 1200 (exactly 1178); 7 × 400 = 2800 (exactly 2723); 20 × 50 = 1000 (exactly 1034); 9 × 700 = 6300 (exactly 6408); 30 × 30 = 900 (exactly 899).
• Part B: 360 ÷ 9 = 40 (exactly 40 r 2); 420 ÷ 6 = 70 (exactly 69 r 4); 640 ÷ 8 = 80 (exactly 79 r 5); 150 ÷ 5 = 30 (exactly 29 r 4); 540 ÷ 9 = 60 (exactly 60 r 3). Part C: 48 × 29: 1500 is above 1392; 52 × 31: 1500 is below 1612; 71 × 42: 2800 is below 2982; 38 × 52: 2000 is above 1976.

Easy numbers to multiply and divide

A. Round both numbers, then estimate the product

48 × 31 is about ______ × ______ = ______62 × 19 is about ______ × ______ = ______7 × 389 is about ______ × ______ = ______22 × 47 is about ______ × ______ = ______9 × 712 is about ______ × ______ = ______31 × 29 is about ______ × ______ = ______

B. Move the number to one that divides easily

362 ÷ 9 is about ______ ÷ 9 = ______418 ÷ 6 is about ______ ÷ 6 = ______637 ÷ 8 is about ______ ÷ 8 = ______149 ÷ 5 is about ______ ÷ 5 = ______543 ÷ 9 is about ______ ÷ 9 = ______

C. Above or below the exact answer?

Look at how you rounded each number. Can you tell before you work it out? _______________________

Is it reasonable?

Children judge answers with estimates. An answer close to the estimate is reasonable; one far from it shows a slip, such as a zero too many, a zero dropped, or adding instead of subtracting. They explain each judgement with the estimate, and find that a reasonable answer can still be wrong.

We are learning to

• estimate first, then judge whether an answer is reasonable,
• explain the judgement in a sentence that uses the estimate,
• name the slip behind an answer that is far off.

Success criteria

• I can explain why 1801 is not reasonable for 412 + 389.
• I can sort the claim cards and give an estimate for each one.

You need

The claim cards (cut-out sheet 1), one set per group. Mini whiteboards. The worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after the sort; start Session B with Part B.

Watch for these ideas

• Judging by feel: ‘it looks too big’. Compare with an estimate every time.
• Thinking a reasonable answer must be right. Card 4 is close to its estimate and still wrong.
• ‘Not reasonable’ with no reason. Give the estimate: ‘about 300, so 2940 is ten times too big’.

Answers

• Reasonable: card 1 (about 600), card 3 (about 900), card 4 (about 900), card 6 (about 400), card 7 (about 1200), card 10 (about 4000). Not reasonable: card 2 (about 300; exactly 294), card 5 (about 400; exactly 405), card 8 (about 230; exactly 230), card 9 (about 600; exactly 714), card 11 (about 3000; exactly 3008), card 12 (about 80; exactly 80).
• Part C: a zero too many, cards 2 and 9; a zero dropped, cards 8 and 12; added instead of subtracting, cards 5 and 11. Part D: 615 + 278 = 893, so 883 is reasonable but wrong.

Is it reasonable?

A. Sort the claim cards. Write the card numbers.

B. Explain three cards

C. What went wrong? Write the card numbers.

A zero too many: ______ and ______

A zero dropped: ______ and ______

Added instead of subtracting: ______ and ______

D. Reasonable, but right?

Card 4 looks reasonable. Work it out exactly: 615 + 278 = __________. Is 883 right? __________

What does this card teach us about estimates? ___________________________________________

Shopping estimates

Children round prices to the nearest dollar for a quick total, and round every price up when they must be sure there is enough money. A receipt total far from the estimate needs checking.

We are learning to

• round prices to the nearest dollar to estimate a total,
• round every price up to check that there is enough money,
• check a receipt total against an estimate.

Success criteria

• I can estimate the total of a basket of price tags to the nearest dollar.
• I can explain why rounding every price up is the safe check when I have $20.

You need

The price tags (cut-out sheet 2), one set per pair. The worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the model basket again.

Watch for these ideas

• Rounding 95 cents down to nothing. $0.95 is nearly a whole dollar.
• Using the nearest dollar to check a budget. Some prices round down, so the estimate can sit under the real total.
• Trusting every receipt. A slipped decimal point or a missed item shows up against an estimate.

Answers

• Part A: A + D + F, about $15, exactly $14.90; E + H, about $22, exactly $21.65; J + K + M, about $25, exactly $24.34; B + C + G + O, about $6, exactly $6.49.
• Part B: C + E + N with $20: rounded up $20, so yes for sure (exactly $18.64); A + L + F with $17: rounded up $18, not sure; exactly $16.15, so yes; K + P with $25: rounded up $27, not sure; exactly $25.90, so no. Part C: A + O, about $9, so $8.80 is reasonable and right; B + D + H + M, about $21, so $206.50 is not reasonable, right total $20.65; E + N + J, about $26, so $16.15 is not reasonable, right total $26.14. A right total sits within 50c an item of the estimate.

Shopping estimates

A. Estimate each basket to the nearest dollar, then find the exact total

B. Enough money? Round every price up first

C. Check the receipts

Change and costs

Children estimate change by rounding the bill to the nearest dollar and subtracting, and the cost of several items at one price by rounding the price. They use these estimates to judge money answers and explain each judgement in a sentence.

We are learning to

• estimate change by rounding the bill,
• estimate the cost of several items at the same price,
• judge a money answer and explain it with an estimate.

Success criteria

• I can estimate the change from $50.00 for a $27.60 bill.
• I can explain why $200 cannot be the cost of 3 items at $19.95.

You need

The price tags (cut-out sheet 2) for Part D. The worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part B; start Session B with Part C.

Watch for these ideas

• Subtracting the wrong way round: the bill minus the money paid. Change is what you paid minus the bill.
• Rounding $4.45 up to $5. 45 cents is less than half a dollar, so it rounds down to $4.
• Accepting change without a check. The bill plus the change should give back the money paid.

Answers

• Part A: about $6, exactly $6.15; about $22, exactly $22.40; about $6, exactly $5.55; about $38, exactly $38.10. Part B: 4 × $13 = $52, exactly $51.80; 6 × $4 = $24, exactly $24.60; 3 × $20 = $60, exactly $59.97.
• Part C: about $40.00, so $39.50 is reasonable and right; about $16.00, so $26.30 is not reasonable; right answer $16.30; about $57.00, so $5.69 is not reasonable; right answer $56.85. Part D: any three tags; the exact change should be within 50c an item of the estimate.

Change and costs

A. Estimate the change, then work it out

B. Several at one price

C. Reasonable? Estimate, decide, and explain

D. Your shopping

Choose three price tags. Estimate the total and the change from $50, then work out both exactly.

Claim cards

Cut out the twelve cards (Day 3), one set per group. Each card shows a calculation and the answer somebody gave. Estimate, then sort the cards into two piles: reasonable and not reasonable.

Teacher note: the sort, the estimates and the slips are on the Day 3 plan.

Price tags

Cut out the sixteen price tags (Days 4 and 5), one set per pair. Use them to fill baskets, estimate totals and check change.

Teacher note: the prices end in different cents on purpose, so some round up and some round down.

What we know: Reasonableness and Estimation

Work on your own. Show your thinking if you can.

1. Round to the nearest hundred, then estimate:   386 + 491 is about __________
2. Round to the nearest thousand, then estimate:   8064 − 2917 is about __________
3. Estimate:   61 × 29 is about __________
4. Use a number that divides easily:   418 ÷ 7 is about __________
5. Someone says 503 − 297 = 800. Is that reasonable? Explain with an estimate.
6. Estimate the total:   $3.95 + $8.10 + $2.85 is about __________
7. You have $30.00. Can you buy things costing $9.40, $11.75 and $7.90? Explain with an estimate.
8. About how much change from $50.00 for a bill of $36.70?   __________

Answers and marking guide

Answers

1. 400 + 500 = 900 (exactly 877)
2. 8000 − 3000 = 5000 (exactly 5147)
3. 60 × 30 = 1800 (exactly 1769)
4. 420 ÷ 7 = 60 (exactly 59 r 5)
5. Not reasonable: about 500 − 300 = 200; exactly 206. The slip: adding instead of subtracting.
6. $4 + $8 + $3 = $15 (exactly $14.90)
7. Rounded up: $30, so yes for sure (exactly $29.05).
8. $50 − $37 = $13 (exactly $13.30)

A quick three-level guide

Eight questions, four ideas. A child at standard answers most questions and can say why, by naming the easy numbers they used and comparing the answer with the estimate.

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: rounding to estimate, estimating products and quotients, judging answers, shopping estimates, and change and costs.