Estimating Quantities: a week of ready-to-teach maths

Five days of lessons for Year 3 Number. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet, plus scatter cards, is-it-reasonable 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 scatter cards and the is-it-reasonable cards once; they are reused all week.
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. An estimate is not a guess: it is a quick, reasoned answer that is close enough for the job. Children estimate a collection by comparing it with a known group of ten, or by counting one group and scaling up; they estimate a calculation by rounding to friendly tens. Then they use the estimate to check whether an exact answer is sensible. Any estimate in a sensible range is a good one.

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.

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 estimate collections, first against a known ten and then by scaling up one group. Days 3 and 4 estimate calculations by rounding to friendly tens, and ask whether the estimate sits above or below the truth. Day 5 puts estimating to work, catching answers that cannot be right. It leans on rounding to the nearest ten and on Add and Subtract to Three Digits (AC9M3N03) and Multiply and Divide (AC9M3N04).

Materials for the week (one trip)

• From the classroom: scissors, pencils, this pack printed.
• Three or four clear jars or zip bags of small things to estimate: pasta, buttons, pegs, beans.
• A scoop or a cup for Day 2, so one scoop can be counted and the rest scaled up.
• Cut out once, use all week: the scatter cards and the is-it-reasonable cards in this pack.

Words for the word wall

estimate, about, close to, benchmark, group, scale up, round, nearest ten, too high, too low, reasonable, check

Dear families

This week in maths, Year 3 is learning to estimate: to give a quick, sensible answer that is close enough, and to use estimates to check whether an exact answer makes sense.

Try this at home

• Estimate before you count: how many pegs on the line, grapes in the bowl, cars in the street? Then count and see how close you were.
• Count one handful and scale up: one handful of pasta is about 20, so four handfuls is about 80.
• At the shops, round prices to the nearest ten dollars to estimate the total before you pay.
• When a sum is worked out, ask: is that about right? How do you know?

Our estimates this week

Fill one row a day. A good estimate is close, not exact.

Printed from the free seegongsik Estimating Quantities teaching pack · seegongsik.com/au/y3/number/AC9M3N05/pack

Compare with a known ten

An estimate is not a guess. A scatter cannot be counted at a glance, but it can be compared with a known group of ten: about how many tens would fit? Then count to check.

We are learning to

• estimate a collection by comparing it with a known group of ten,
• say an estimate as a range or a round number (about 40, between 30 and 40),
• count to check and say how close the estimate was.

Success criteria

• I can use a group of ten to estimate a collection.
• I can say how close my estimate was.

You need

Three or four jars or bags of small objects, and the scatter cards (cut-out sheet 1). The worksheet.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after the board scatters; start Session B with the worksheet.

Watch for these ideas

• Wild guesses such as 1000. Ask how many groups of ten could fit.
• Thinking an estimate must be exact. Close is the goal; any answer in a sensible range is good.
• Counting one by one and losing track. Cross off dots in tens and tally the tens.

Answers

• Scatter A has 34 dots, B has 57, C has 18. Good estimates: A 30 to 40, B 50 to 60, C 15 to 20.
• Difference: the count take away the estimate, or the estimate take away the count; smaller is closer.
• Scatter cards: the counts are printed under the cards in the teacher note.

About how many?

Look at the ring of ten. Estimate each scatter without counting. Then count to check, crossing off tens.

Count one group, scale up

For a collection too big to compare at a glance, count one group and scale up: one scoop holds about 30, five scoops hold about 5 × 30 = 150. The groups are only roughly equal, so it is an estimate.

We are learning to

• estimate a big collection by counting one group and multiplying by the number of groups,
• explain why the result is an estimate and not an exact count,
• solve everyday estimation problems with a number sentence.

Success criteria

• I can count one group and scale up to estimate.
• I can say why my answer is about, not exact.

You need

A big bag of pasta or beans, a scoop or cup, and a tray per group. The worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after the board; start Session B with the worksheet and Check one.

Watch for these ideas

• Adding the group size to the number of groups (20 + 4) instead of multiplying.
• Thinking the estimate is wrong because the real count is 82. Close is the goal.
• Choosing a group that is not typical (the smallest pile). Pick a group that looks like the rest.

Answers

• Boxes: box 1 has 20; 4 × 20 = 80 is the estimate. The real total is 82 (20 + 19 + 21 + 22).
• Stories: about 5 × 30 = 150 beans; about 6 × 40 = 240 words; about 8 × 25 = 200 seats.
• Why about: the groups are not all exactly the same size, so scaling up one group gives a close answer, not an exact one.

Count one, scale up

Four boxes of dots

Count only box 1. The other boxes look about the same.

Box 1 has ______ dots. There are ______ boxes. About ______ × ______ = ______ dots altogether.

Scale up the story

1. One scoop holds about 30 beans. About how many beans are in 5 scoops?   About ______ × ______ = ______

2. A page has about 40 words. About how many words are on 6 pages?   About ______ × ______ = ______

3. Each row of the grandstand seats 25 people. There are 8 rows. About how many seats?   About ______ × ______ = ______

Why is it about?

Why is your answer for the four boxes an estimate and not an exact count?

Round, then add or subtract

Round each number to the nearest ten, then calculate: 38 + 51 becomes 40 + 50 = 90, and the exact answer, 89, is very close. Friendly tens make a fast estimate.

We are learning to

• round a two-digit number to the nearest ten (a number ending in 5 rounds up),
• estimate a sum or a difference by rounding each number first,
• compare the estimate with the exact answer.

Success criteria

• I can round a number to the nearest ten.
• I can estimate a sum or difference by rounding first.

You need

A 0 to 100 number line on the board or floor. The worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Nearest ten; start Session B with the board sums.

Watch for these ideas

• Rounding by dropping the ones (38 becomes 30). Look at which ten it is nearer.
• Rounding only one number. Round both, then calculate.
• Thinking an estimate that differs from the exact answer is wrong. It should be close, not equal.

Answers

• Nearest ten: 40, 50, 60, 30, 50, 70, 90, 10.
• Estimates (exact): 39 + 52 is about 90 (91); 61 + 28 about 90 (89); 46 + 33 about 80 (79); 27 + 54 about 80 (81).
• 82 − 39 is about 80 − 40 = 40 (43); 93 − 58 about 90 − 60 = 30 (35).
• Story: about 50 + 30 = $80; exactly $75. Quick estimates: 70, 100, 50.

Round, then estimate

Round to the nearest ten

Estimate first, then work it out

Story

Ben has $48 and earns $27 more. About how much does he have now? Estimate: ______ Exact: ______

Estimate a product

To estimate 4 × 19, round the awkward factor to the nearest ten and multiply: about 4 × 20 = 80, close to the exact 76. Which way you rounded also tells you whether the estimate is too high or too low.

We are learning to

• estimate a product by rounding the two-digit factor to the nearest ten,
• multiply a one-digit number by a ten in our heads (4 × 20 is 8 tens),
• say whether an estimate is higher or lower than the exact answer, and why.

Success criteria

• I can estimate a product by rounding one factor.
• I can say if my estimate is too high or too low.

You need

Base-ten blocks (tens rods) for the opening. The worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Estimate a product; start Session B with Too high or too low?

Watch for these ideas

• Rounding the one-digit factor too (4 becomes 0). Round only the two-digit number.
• Stopping at 4 × 2 = 8. The 2 is 2 tens, so 4 × 20 is 8 tens, which is 80.
• Thinking rounding up makes the estimate smaller. Up makes it too high; down makes it too low.

Answers

• Products, estimate (exact): 3 × 19 about 60 (57), too high; 4 × 28 about 120 (112), too high; 5 × 42 about 200 (210), too low; 6 × 21 about 120 (126), too low; 2 × 49 about 100 (98), too high.
• Sums: 29 + 48 about 80, higher (77); 51 + 32 about 80, lower (83); 36 + 24 about 60, about equal (60).
• Mia: too high. 39 rounded up to 40, so each of the 6 groups gained 1; the exact answer is 234.
• Point the way: 6 × 29 up (180, exact 174); 33 + 52 down (80, exact 85); 18 + 32 flat (50).

Estimate a product

Round and estimate

Example: 4 × 19 is about 4 × 20 = 80. Write too high or too low, then work out the exact answer.

Too high or too low?

Round both numbers. Did they round up, down, or one of each? Circle your answer.

29 + 48 is about ____ + ____ = ____    higher  /  lower  /  about equal

51 + 32 is about ____ + ____ = ____    higher  /  lower  /  about equal

36 + 24 is about ____ + ____ = ____    higher  /  lower  /  about equal

Explain

Mia estimated 6 × 39 as 6 × 40 = 240. Is her estimate too high or too low? How do you know without working it out?

Is the answer reasonable?

An estimate is the quickest way to catch an answer that cannot be right. Someone says 23 × 4 = 412, but 20 × 4 is 80 and 25 × 4 is 100, so the answer must be about 90.

We are learning to

• use an estimate to check whether an answer is reasonable,
• explain with the rounded numbers why an answer is or is not reasonable,
• estimate to solve everyday problems, such as whether we have enough money.

Success criteria

• I can use an estimate to decide if an answer is reasonable.
• I can explain how my estimate shows it.

You need

The is-it-reasonable cards (cut-out sheet 2), one set per pair. The worksheet. The unit’s self-check quiz.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after the card sort; start Session B with the worksheet and the quiz.

Watch for these ideas

• Trusting any answer that came from working out. The estimate is the check.
• Rejecting a right answer because it is not the estimate (48 × 5 = 240, estimate 250). Close is reasonable.
• Writing 46 + 38 = 714: 7 tens, then 14 ones beside them. The estimate, about 90, shows the slip.

Answers

• Opening: 46 + 38 is about 50 + 40 = 90, so 714 is not reasonable; it is 84.
• Checks: 39 + 42 = 711 no (about 80; it is 81); 7 × 31 = 217 yes (about 210); 96 − 47 = 49 yes (about 50); 4 × 52 = 56 no (about 200; it is 208); 83 − 29 = 112 no (about 50; it is 54); 58 + 27 = 85 yes (about 90).
• Stories: about 4 × 30 = 120, so 112 is reasonable (4 × 28 is exactly 112). Leo: about 20 + 10 = $30, so yes, $40 is enough ($31).
• Fix one: any of the three slips, with the right answer and the estimate beside it.

Is it reasonable?

Check with an estimate

Round and estimate first. Then circle yes if the answer is reasonable, or no if it cannot be right.

Story problems

1. There are 4 boxes of 28 pencils. Sam says there are 112 pencils altogether. Is that reasonable?   Estimate: ______________ Answer: ______

2. Leo has $40. He wants a hat for $18 and socks for $13. Does he have enough money?   Estimate: ______________ Answer: ______

Fix one

Choose one answer that is not reasonable. Work out the right answer and write your estimate beside it.

Scatter cards

Cut out the eight cards. On Day 1, pairs flash a card for three seconds, both estimate, then count together and see whose estimate was closer.

Teacher note, counts: card 1 has 23 dots, card 2 has 31, card 3 has 46, card 4 has 52, card 5 has 19, card 6 has 64, card 7 has 38 and card 8 has 27.

Is-it-reasonable cards

Cut out the twelve cards. On Day 5, pairs estimate each answer and sort the cards into two piles: Reasonable and Not reasonable. Each card in the second pile hides a slip; can they find it?

Teacher note. Reasonable: 28 + 31 = 59, 5 × 19 = 95, 74 − 38 = 36, 4 × 23 = 92, 2 × 48 = 96, 86 − 52 = 34. Not reasonable, with the slip: 47 + 26 = 613 (6 tens and 13 ones written side by side; it is 73), 3 × 42 = 45 (added; it is 126), 61 − 24 = 85 (added; it is 37), 65 + 28 = 37 (took away; it is 93), 33 + 45 = 708 (70 and 8 written side by side; it is 78), 5 × 38 = 1540 (15 and 40 written side by side; it is 190).

What we know: Estimating Quantities

Work on your own. You do not need exact answers unless a question asks for one.

1. Use the ring of ten. About how many dots are there? ______
2. One handful of pasta is about 20 pieces. About how many pieces are in 5 handfuls? ______
3. Round to the nearest ten: 47 is about ______    82 is about ______    35 is about ______
4. Estimate 39 + 58: ______ + ______ = ______
5. Estimate 6 × 19: ______ × ______ = ______
6. Someone says 72 − 29 = 101. Is that reasonable? Yes / No   My estimate: ______
7. 4 × 29 is about 4 × 30 = 120. Is 120 too high or too low? ______ How do you know? ______________
8. Kim has $50. A book costs $19 and a game costs $28. Estimate the total: ______ Does Kim have enough? ______

Answers and marking guide

Answers

1. Any estimate from 40 to 50; there are 45 dots.
2. About 5 × 20 = 100 pieces.
3. 50; 80; 40 (a number ending in 5 rounds up).
4. 40 + 60 = 100 (exactly 97).
5. 6 × 20 = 120 (exactly 114).
6. No. About 70 − 30 = 40; 101 is what you get by adding. It is 43.
7. Too high: 29 was rounded up to 30, so each of the 4 groups gained 1. It is 116.
8. About $20 + $30 = $50. Yes: both prices were rounded up, so the real total is less than $50 (it is $47).

A quick three-level guide

Eight questions, three ideas. Mark an estimate right when it sits in a sensible range; a child at standard can say which rounded 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: comparing with a known ten, scaling up one group, rounding to add and subtract, estimating products, and checking answers.