Estimating with Rational Numbers and Percentages: a week of ready-to-teach maths

Five days of lessons for Year 6 Number. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet, plus twenty calculation cards, price and budget cards, 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 calculation, price and budget cards once; one set per pair serves Days 1 to 4.
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, close answer worked out with friendly numbers. Rounding, compatible numbers and friendly percentages make the working easy. The direction of the rounding matters: rounding every price up gives a total that is at least the real one, which is what you want when checking you have enough money. An estimate also catches a calculator answer that is ten times out.

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.

Words to watch

An estimate is not a guess: it is worked out. Compatible numbers are numbers that go together easily, such as 400 ÷ 8. An overestimate is bigger than the exact answer; an underestimate is smaller. The symbol ≈ means “is about”.

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

Day 1 rounds to estimate sums and differences; Day 2 estimates products and quotients; Day 3 estimates percentages and fractions of amounts; Day 4 checks budgets with prices rounded up or down; and Day 5 decides when an estimate is enough and uses one to check a calculator. It builds on the Year 5 work of checking answers with estimates in money problems and on this year’s decimals and percentages units. It leads to the modelling unit and to Year 7, where decimals are rounded to a given accuracy.

Materials for the week (one trip)

• From the classroom: scissors, pencils, this pack printed.
• A calculator for each pair, for checking estimates (a tablet or laptop calculator is fine).
• Cut out once, use all week: the calculation cards, the price cards and the budget cards, one set per pair.

Words for the word wall

estimate, round, about, compatible numbers, overestimate, underestimate, friendly number, budget, reasonable, exact

Dear families

This week in maths, Year 6 estimates: quick, close answers with friendly numbers. We estimate shopping totals, percentages and calculations, and use estimates to check that answers make sense.

Try this at home

• At the shops, round each price up to the next dollar. Is the total under your budget?
• A $79.95 item has 19% off. 20% of $80 is $16: is the real discount a little more or a little less? Why?
• Before using a calculator, say a quick estimate. Was the calculator answer close?
• Ask: when do we need an exact answer, and when is an estimate enough?

My estimates this week

Fill one row a day. Write an estimate, then the exact answer, and how far apart they were.

Printed from the free seegongsik Estimating with Rational Numbers and Percentages teaching pack · seegongsik.com/au/y6/number/AC9M6N08/pack

Round to estimate

An estimate is a close answer worked out quickly with friendly numbers. Children round each number to the nearest ten, hundred or whole number, add or subtract the rounded numbers, and check on a calculator how close the estimate came.

We are learning to

• round whole numbers to the nearest ten or hundred, and decimals to the nearest whole number,
• estimate a sum or a difference by rounding each number first,
• say how close an estimate is, and whether it is over or under.

Success criteria

• I can estimate 386 + 512 in my head and check it on a calculator.
• I can say why rounding to the nearest ten gives a closer estimate than rounding to the nearest hundred.

You need

The worksheet, one per child, a calculator for each pair, and the + and − calculation cards (cut-outs), one set per pair.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with 48 + 31 on the board.

Watch for these ideas

• Cutting off digits instead of rounding: 48 to the nearest ten is 50, not 40.
• Rounding only one number, or working out the exact answer first and rounding that.
• Dropping the decimal part: 6.9 to the nearest whole number is 7, not 6.

Answers

• Part A: to the nearest 10, 390, 510, 1250, 100, 2450, 750; to the nearest 100, 400, 500, 1200, 100, 2500, 700; decimals 7, 12, 4, 20, 1, 8.
• Part B: 386 + 512 ≈ 400 + 500 = 900, exact 898, 2 apart; 1249 + 745 ≈ 1200 + 700 = 1900, exact 1994, 94 apart; 2451 − 386 ≈ 2500 − 400 = 2100, exact 2065, 35 apart; 6.9 + 12.48 + 3.5 ≈ 7 + 12 + 4 = 23, exact 22.88, 0.12 apart; 19.95 − 8.15 ≈ 20 − 8 = 12, exact 11.8, 0.2 apart; $4.95 + $12.40 + $2.85 ≈ $5 + $12 + $3 = $20, exact $20.20, $0.20 apart. Talk prompt: 1250 + 750 = 2000, only 6 from 1994.
• Part C: 40 + 50 + 30 = 120, over (exact 114); 6 + 4 + 8 = 18, under (exact 18.6); 50 + 70 = 120, cannot tell from the rounding, as the numbers went each way (exact 121, so this time under).

Round to estimate

A. Round each number

B. Estimate, then check

Round whole numbers to the nearest hundred, and decimals and money to the nearest whole number. Then check on a calculator.

C. Over or under?

Round whole numbers to the nearest ten and decimals to the nearest whole number. Did every number go up, every number go down, or some each way? Is the estimate over or under, or can you not tell?

1. 38 + 47 + 29 ≈ ______________ = ______    over, under or cannot tell? ____________

2. 6.2 + 4.3 + 8.1 ≈ ______________ = ______    over, under or cannot tell? ____________

3. 52 + 69 ≈ ______________ = ______    over, under or cannot tell? ____________

Products and quotients

To estimate a product, round each number to one non-zero digit (48 becomes 50, 0.48 becomes 0.5) and multiply the friendly numbers. To estimate a quotient, change the numbers to compatible numbers that divide easily: 412 ÷ 8 is about 400 ÷ 8. Children also decide whether an estimate is over or under.

We are learning to

• estimate a product by rounding each number to one non-zero digit,
• estimate a quotient with compatible numbers,
• say whether a product estimate is over or under, or that we cannot tell.

Success criteria

• I can estimate 6.8 × 4.7 and say that my estimate is over.
• I can choose compatible numbers to estimate 1599 ÷ 39.

You need

The worksheet, one per child, a calculator for each pair, and the × and ÷ calculation cards added to the Day 1 cards.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with 412 ÷ 8 ≈ 400 ÷ 8.

Watch for these ideas

• Rounding every number to the nearest ten: 4.7 rounds to 5 and 0.48 to 0.5, not to 0.
• Losing a zero: 50 × 20 is 1000, not 100.
• Choosing numbers that do not divide easily: 410 ÷ 8 is no easier than 412 ÷ 8.

Answers

• Part A: 6.8 × 4.7 ≈ 7 × 5 = 35, over (exact 31.96); 213 × 4 ≈ 200 × 4 = 800, under (exact 852); 29 × 61 ≈ 30 × 60 = 1800, cannot tell (exact 1769); 8.2 × 31 ≈ 8 × 30 = 240, under (exact 254.2); 0.48 × 62 ≈ 0.5 × 60 = 30, cannot tell (exact 29.76); 7.6 × 9.5 ≈ 8 × 10 = 80, over (exact 72.2).
• Part B (accept other compatible numbers): 412 ÷ 8 ≈ 400 ÷ 8 = 50 (exact 51.5); 1599 ÷ 39 ≈ 1600 ÷ 40 = 40 (exact 41); 23.6 ÷ 4 ≈ 24 ÷ 4 = 6 (exact 5.9); 2989 ÷ 49 ≈ 3000 ÷ 50 = 60 (exact 61); 357 ÷ 7 ≈ 350 ÷ 7 = 50 (exact 51); 64.8 ÷ 9 ≈ 63 ÷ 9 = 7 (exact 7.2).
• Part C: 1000, 24, 200, 400, 20. Talk prompt: the exact answer is 17, so 600 ÷ 40 is closer; 36 is nearer to 40 than to 30.

Products and quotients

A. Round and multiply

Round each number to one non-zero digit: 48 to 50, 4.7 to 5, 0.48 to 0.5. Did both numbers go up, both go down, or one each way?

B. Divide with compatible numbers

Change the numbers to ones that divide easily, such as 400 ÷ 8. Then check on a calculator.

C. Pick the best estimate

Circle the best estimate. Do not work out the exact answer.

1. 52 × 19 is about100010010 000

2. 3.9 × 6.1 is about2.424024

3. 8036 ÷ 41 is about202002000

4. 0.49 × 820 is about400040040

5. 61.5 ÷ 3 is about200220

Percentages and fractions

An awkward percentage can be estimated with a nearby friendly one (10%, 20%, 25%, 50% or 75%), and a decimal with a nearby friendly fraction, such as a third. Children estimate percentages and fractions of amounts, rounding the amount too, then estimate savings in a sale.

We are learning to

• estimate a percentage of an amount with a nearby friendly percentage,
• swap a decimal near a friendly fraction for that fraction to estimate,
• estimate a saving and a sale price.

Success criteria

• I can estimate 19% of 45 and say whether my estimate is over or under.
• I can explain why 0.32 × 61 is about 20.

You need

The worksheet, one per child, a calculator for each pair, and all twenty calculation cards.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with 0.49 × 202 ≈ 1/2 of 200.

Watch for these ideas

• Giving the percentage number as the answer: 21% of 80 is about 16, not 21.
• Moving to a percentage that is no easier: 21% to 22% does not help, but 20% does.
• Stopping at the saving when the question asks what you pay.

Answers

• Part A (exact answers in brackets): 19% of 45 ≈ 20% of 45 = 9, over (8.55); 52% of 88 ≈ 50% of 88 = 44 (45.76); 24% of 61 ≈ 25% of 60 = 15 (14.64); 9% of 310 ≈ 10% of 310 = 31 (27.9); 76% of 40 ≈ 75% of 40 = 30 (30.4); 11% of 590 ≈ 10% of 600 = 60 (64.9).
• Part B: 0.32 × 61 ≈ 1/3 of 60 = 20 (19.52); 0.26 × 79 ≈ 1/4 of 80 = 20 (20.54); 0.49 × 202 ≈ 1/2 of 200 = 100 (98.98); 0.74 × 41 ≈ 3/4 of 40 = 30 (30.34); 0.67 × 91 ≈ 2/3 of 90 = 60 (60.97).
• Part C, saving and price: $59.90: save about $18, pay about $42 ($17.97 and $41.93); $39: save about $6, pay about $34 ($5.85 and $33.15); $21.80: save about $11, pay about $11 ($9.81 and $11.99); $148.50: save about $30, pay about $120 ($29.70 and $118.80).

Percentages and fractions

A. A friendly percentage

Change each percentage to a nearby friendly one: 10%, 20%, 25%, 50% or 75%. Round the amount too if it helps.

B. A friendly fraction

A decimal close to 0.25, 0.33, 0.5, 0.67 or 0.75 is close to 1/4, 1/3, 1/2, 2/3 or 3/4. Swap each decimal for that fraction.

C. In the sale

Estimate the saving first, then about how much you pay.

1. 30% off $59.90save about ________    pay about ________

2. 15% off $39save about ________    pay about ________

3. 45% off $21.80save about ________    pay about ________

4. 20% off $148.50save about ________    pay about ________

Enough money?

Nearest-dollar rounding gives a quick total but can be out by almost 50 cents a price. For a sure yes, round every price up: the real total cannot be more. For a sure no, round every price down. When neither decides, the total is too close to call and is worked out exactly.

We are learning to

• estimate a shopping total by rounding each price to the nearest dollar,
• decide for sure whether a budget is enough by rounding every price up or every price down,
• estimate the change from a payment.

Success criteria

• I can decide whether $20 is enough for $6.40, $5.80 and $6.15 without adding the cents.
• I can estimate my change from $50.

You need

The worksheet, one per child, a calculator for each pair, and the price and budget cards (cut-outs), one set per pair.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the enough-money vote.

Watch for these ideas

• Rounding to the nearest dollar when the total is close to the budget.
• Rounding some prices up and others down, which gives no safe answer.
• Giving the estimated change at the till instead of the exact change.

Answers

• Part A (exact answers in brackets): $5 + $7 + $3 + $9 = $24 ($24.10); $13 + $3 + $7 = $23 ($22.45); $2 + $15 + $9 + $1 = $27 ($26.70).
• Part B: $20: up $20, down $17, yes for sure ($18.35); $15: up $19, down $16, no for sure ($17.60); $25: up $27, down $24, too close, so work it out: no ($26.25); $40: up $41, down $38, too close, so work it out: yes ($39.65); $30: up $30, down $27, yes for sure ($29.15).
• Part C: $50: total about $39, change about $11 ($10.85); $20: total about $15, change about $5 ($4.95); $100: total about $83, change about $17 ($17.05). Talk prompt: yes, $1.85 rounds up to $2, so the rounded-up total still fits.

Enough money?

A. Round to the nearest dollar

Round each price to the nearest dollar and add. Then check on a calculator.

1. $4.65, $7.20, $2.85 and $9.40   about ________    exact ________

2. $12.55, $3.15 and $6.75   about ________    exact ________

3. $1.95, $15.30, $8.60 and $0.85   about ________    exact ________

B. Is there enough money?

Round every price up, then every price down. If the rounded-up total fits the budget, there is enough for sure. If the rounded-down total is over the budget, there is not enough for sure. If neither, it is too close to call: work it out exactly.

C. Estimate the change

1. Pay $50 for $12.80, $18.45 and $7.90. Total about ________   change about ________

2. Pay $20 for $6.95 and $8.10. Total about ________   change about ________

3. Pay $100 for $38.60, $24.95 and $19.40. Total about ________   change about ________

D. Under budget scorecard

Before you check, write your rounded-up total. Then add your basket on a calculator.

Estimate or exact?

An estimate is enough when a close answer settles the question; an exact answer is needed when every cent counts, as in change, a shared bill or a banking slip. Children sort situations, catch calculator slips with estimates, and estimate first, working exactly only when the estimate is too close to call.

We are learning to

• decide whether a situation needs an estimate or an exact answer,
• use an estimate to check that a calculator answer is reasonable,
• estimate first, and calculate exactly only when the estimate is too close to call.

Success criteria

• I can say why giving change needs an exact answer but checking a budget does not.
• I can explain why 5.9 × 8.2 = 483.8 cannot be right.

You need

The worksheet, one per child, and a calculator for each pair.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the calculator answers in Part B.

Watch for these ideas

• Trusting the calculator over a sensible estimate.
• Thinking an estimate is a wrong answer, or that it must be right to the cent.
• Working every answer out exactly even when the estimate already decides.

Answers

• Part A: (1) estimate, rounded up, $7 + $4 + $3 = $14, so it is enough for sure; (2) exact; (3) exact, each friend pays exactly $6.90; (4) estimate; (5) estimate; (6) exact. Accept another choice with a good reason.
• Part B: 5.9 × 8.2 ≈ 48, not reasonable (48.38); 784 ÷ 16 ≈ 50, reasonable; 0.48 × 205 ≈ 100, reasonable; 21% of 640 ≈ 128, not reasonable (134.4); 1012 − 389 ≈ 600, not reasonable (623); $18.75 + $6.40 + $9.95 ≈ $35, reasonable.
• Part C: (1) no for sure: even rounded down, 28 × $12 = $336 (exact $347.20). (2) 30% of $65 is $19.50, so about $45.50: too close to call, so work it out: $64.90 − $19.47 = $45.43, so $45 is not enough, $0.43 short. (3) about $2 × 40 = $80; an estimate is enough (exact $79.38).

Estimate or exact?

A. Estimate or exact?

Tick one column and say why.

B. Is the calculator answer reasonable?

Estimate first. If the calculator answer is not reasonable, find the right answer.

C. Estimate first

Write your estimate first, then decide whether you need the exact answer.

1. An excursion costs $12.40 for each of 28 students. Is $300 enough?

2. A $64.90 game is 30% off. Is $45 enough to buy it?

3. Fuel costs $1.89 a litre. About how much do 42 litres cost?

Calculation cards

Cut out the twenty cards once, one set for each pair. The corner of each card shows the day it joins the game.

Closest estimate (Days 1 to 3)

1. Use only the cards for today and the days before. Shuffle them face down and turn one over.
2. Each player writes an estimate before the partner counts to ten. No calculator and no written working.
3. Check on a calculator. The estimate closer to the exact answer wins the card; a tie goes back under the pile.
4. When the cards run out, the player with more cards wins.

Teacher note: a good estimate and the exact answer for each card. 1486 + 2517 ≈ 1500 + 2500 = 4000 (exact 4003); $12.95 + $6.10 + $3.85 ≈ $13 + $6 + $4 = $23 (exact $22.90); 997 − 498 ≈ 1000 − 500 = 500 (exact 499); 2.9 + 6.1 + 3.95 ≈ 3 + 6 + 4 = 13 (exact 12.95); 205 − 98.6 ≈ 205 − 100 = 105 (exact 106.4); 7.48 + 2.51 ≈ 7.5 + 2.5 = 10 (exact 9.99); 63.2 − 19.8 ≈ 63 − 20 = 43 (exact 43.4); 48 × 21 ≈ 50 × 20 = 1000 (exact 1008); 612 ÷ 36 ≈ 600 ÷ 40 = 15 (exact 17); 4.8 × 7.1 ≈ 5 × 7 = 35 (exact 34.08); 0.52 × 78 ≈ 0.5 × 80 = 40 (exact 40.56); 9.9 × 9.9 ≈ 10 × 10 = 100 (exact 98.01); 738 ÷ 18 ≈ 720 ÷ 18 = 40 (exact 41); $49.95 × 3 ≈ $50 × 3 = $150 (exact $149.85); 19% of 310 ≈ 20% of 300 = 60 (exact 58.9); 3/4 of 81 ≈ 3/4 of 80 = 60 (exact 60.75); 25% of 197 ≈ 25% of 200 = 50 (exact 49.25); 9% of 421 ≈ 10% of 420 = 42 (exact 37.89); 1/3 of 61.5 ≈ 1/3 of 60 = 20 (exact 20.5); 49% of 82 ≈ 50% of 80 = 40 (exact 40.18).

Price cards and budget cards

Cut out the twelve price cards and the three budget cards once, one set for each pair.

Under budget (Day 4)

1. Turn over a budget card: it is the budget for both players. Shuffle the price cards face down.
2. Take turns to turn over a price card and put it in your basket or on the discard pile. Estimate in your head: no calculator and no writing.
3. Say “stop” when one more card could take you over. The round ends when both players have stopped or the cards run out.
4. Add each basket on a calculator. A basket over the budget scores nothing; otherwise the basket closer to the budget wins the round.

Teacher note: rounded up, the prices are notebook $4, paint set $8, torch $13, drink bottle $6, puzzle $10, pencils $3, cap $15, frisbee $7, stickers $5, card game $9, skipping rope $12, eraser pack $2. A basket whose rounded-up prices fit the budget is always safe. All twelve add to $88.78.

What we know: Estimating with Rational Numbers and Percentages

Work on your own. Show the friendly numbers you use.

1. Estimate 4196 + 2807. Show your rounding. ______________________________
2. Estimate 3.8 × 8.9. Is your estimate over or under the exact answer? How do you know? ______________________________
3. Estimate 549 ÷ 9 with compatible numbers. ______________________________
4. Estimate 26% of 79. ______________________________
5. Estimate 0.34 × 62. ______________________________
6. You have $25. Is it enough for $8.40, $5.70 and $9.20? Explain without adding the cents. ______________________________
7. You pay $50 for $13.90 and $21.85. About how much change do you get? ______________
8. A calculator shows 4.8 × 61 = 29.28. Is that reasonable? What should it be? ______________________________

Answers and marking guide

Answers (accept other sensible friendly numbers)

1. 4200 + 2800 = 7000 (exact 7003).
2. 4 × 9 = 36, over: both numbers were rounded up (exact 33.82).
3. 540 ÷ 9 = 60 (exact 61).
4. 25% of 80 = 20 (exact 20.54).
5. 1/3 of 60 = 20 (exact 21.08).
6. Yes, for sure: rounded up, $9 + $6 + $10 = $25 (exact $23.30).
7. $14 + $22 = $36, so about $14 (exact $14.25).
8. No: 5 × 60 = 300, so the answer should be about 300; it is 292.8.

A quick three-level guide

Eight questions, four ideas. A child at standard answers most questions and can say why, naming the friendly numbers they used.

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, products and quotients, percentages and fractions, enough money, and estimate or exact.