The Long Wait

$The Long Wait book cover$

Grade 2
4 skill lessons

Math

Skill lessons

The mathematics the book carries, taught directly. Printable workbook pages and teacher keys are not available for this book yet.

1How Zach Guessed Sixty
2Working Out the Wait
3Between Six and Twelve
4Counting the Groups

Math + reading

Math through the reading

The same book, entered through a sentence it prints. Change one word in the reading and the numbers change with it.

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Reading

Reading and language lessons

The same book read as a text. What a word means in its sentence, how a character is shown, how the writing is put together - no mathematics.

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Skill lessons

4 lessons on the mathematics the book carries. Printable workbook pages and teacher keys are not available for this book yet.

1. How Zach Guessed Sixty Concept · Estimating a large group from the size of one group, then testing it by counting

Zach never counts the line. He looks at ten people in his own row, sees about six rows, and says sixty, and the book then sends Josh off to count the same line and prints what he gets: fifty-seven the first time and sixty-two the second. Children do both jobs on one line here. They build six rows of ten counters, touch one row per number word to sixty, and then count a bagged line of about sixty counters one counter at a time, twice, writing both numbers down. An about-number is made out of two small numbers, the size of one group and the number of groups, and it holds still; an exact count has to touch every person and can come back different each time it is taken. The last minutes are spent saying which of their own two counts sixty sat near, which is how the book itself decides sixty was worth saying.

Full lesson — 20 minutes

1. Build the six rows (6 min)
Lay ten counters in one straight row on your mat, one counter for each person in Zach's row. Build five more rows underneath it, and give every row the same ten. Do not count the whole thing yet. When all six rows are down, run one finger along a single row and say out loud how many people are standing in it.
Word preview: row, estimate
There are ten people in our row and uh, about six rows, so I estimate 60.
There are about 10 people in each row. There are 6 rows. How many people in all?
2. Count by tens to sixty (5 min)
Touch each row once, from the top down, and say ten, twenty, thirty, forty, fifty, sixty as you touch. One touch, one number, and never two numbers on one row. Say the number you land on at the last row. Then say it again the way Zach says it, with the word about in front of it.
Word preview: about
Hint. Count by 10s. 10. 20. 30. 40. 50. 60. About 60 people.
so I estimate 60
3. Count the whole line one at a time (5 min)
Empty the bag your teacher gives you and count every counter in it one at a time, sliding each one across the table as you say its number. Write the number you reach on the first line of your slip. Now sweep them all back and count the same bag again from the start, and write that number underneath. Say both numbers aloud to your partner.
Word preview: count
He counted 57.
First I got 57. Then I got 62.
4. Say whether sixty was a good guess (4 min)
Put Zach's sixty beside the two numbers on your slip. Say out loud whether sixty would have been a good guess for your bag, and say both of your own numbers as the reason. Josh counted the same line twice and got two different numbers, and still called sixty a good guess. Say why two different counts do not spoil it.
Word preview: guess
So 60 was a good guess.
He'd better check his count.

Short version — 9 minutes

1. Build the six rows (5 min)
Lay ten counters in one row on your mat, then build five more rows the same. Do not count them all yet. Run a finger along one row and say how many are standing in it.
Word preview: row, estimate
There are ten people in our row and uh, about six rows, so I estimate 60.
2. Count by tens to sixty (4 min)
Touch each row once from the top down and say ten, twenty, thirty, forty, fifty, sixty. One touch, one number. Say the last number again with the word about in front of it.
Word preview: about
Hint. Count by 10s. 10. 20. 30. 40. 50. 60. About 60 people.

Three levels

Support: Build three rows of ten instead of six and count by tens to thirty, one touch and one number word for each row. Then count a single row of ten one counter at a time so the same ten arrives twice, once by touching a row and once by touching each counter. The touching and the saying are exactly what everybody else does; only the number of rows is smaller.
At Grade: Build six rows of ten, count by tens to sixty and say it as about sixty. Count a bagged line of about sixty counters twice, one at a time, writing both numbers on the slip, and then say aloud whether sixty was a good guess for your bag with both of your own numbers as the reason.
Extension: Build six rows in which two of the rows hold nine counters instead of ten. Count by tens to sixty anyway, then count the whole mat one at a time, and say by how much the about-number missed. Say aloud why using ten for every row was still worth doing when two rows did not have ten in them.

English learners

Counting and touching carry across every language a child brings, and the tens chain is already familiar to most of them in their strongest language, so say it there first while touching the rows, then in English. What does not carry is the English habit of putting about in front of the number rather than after it, so about sixty people and never sixty about, and the tens words twenty and thirty, which do not announce their two and their three the way the words for those digits do in many languages. Rehearse one frame aloud with the rows in view before any writing happens: about ___ people are in the line. The word guess is also being used here in a narrow way that its everyday sense does not cover, because Zach's guess is worked out from ten and six rather than felt, and the book itself marks the difference by calling Josh's first attempt a wild guess.

Materials

A row mat per pair, ruled with six long blank rows
Sixty counters per pair in a tub
A paper bag per pair holding between fifty-five and sixty-five counters, the number known to the teacher and not to the children
A two-line counting slip per pair, headed first count and second count
Word cards: estimate, about, row, count

Standards & curriculum

These lessons are being matched against curricula now.

2. Working Out the Wait Application · Predicting a wait by splitting a count into groups of ten and counting five minutes for each group

This is the one place in the book where an estimate is put to work rather than checked. Josh looks down the line and says about forty people; Zach says ten go on each time, so four rides; Josh says a ride is five minutes; Zach says about twenty minutes. Every number in that chain is on the page and the arithmetic holds. Children run it with counters and cards: forty counters pushed into groups of ten gives four groups, a five-minute card on each group gives a chain to count five, ten, fifteen, twenty. Then they run it again on the eight people the book leaves ahead of the boys near the front, where the eight do not fill even one group of ten and the whole wait is a single ride. They finish by doing two new line lengths and writing the minutes down, so the method survives leaving the story behind.

Full lesson — 20 minutes

1. Split forty into rides (5 min)
Take forty counters for the forty people ahead of you and push them into groups of ten, because ten people go on each time the ride runs. Keep going until nothing is left over. Say how many groups of ten you made, and say what one group of ten stands for.
Word preview: each, rides
It looks like there are about 40 people ahead of us now.
And if 10 people go on each time, Zack said, that's four rides.
2. Count the minutes by fives (5 min)
Lay one ride card beside each group of ten. Touch the cards in order and count five, ten, fifteen, twenty. Say the number you land on with the word minutes after it. Then say the whole thing to your partner as one sentence: forty people, four rides, about twenty minutes.
Word preview: minutes
The ride takes five minutes, said Josh.
So we'll have to wait about 20 minutes, Zack guessed.
3. Do it again for eight people (5 min)
Clear the mat and take eight counters, because near the front only eight people are still ahead. Try to push the eight into a group of ten and say what you find. Lay down as many ride cards as you actually need, count the minutes on them, and say how long these eight people wait.
Word preview: only
Now, there were only eight people ahead of them.
4. Write the wait for a new line (5 min)
Your teacher will say a new number of people waiting. Group your counters by tens, lay out one ride card for each group, count the minutes by fives, and write the answer on your slip in the minutes column. Do it for thirty people and then for twenty. Read each answer back to your partner with the word about in front of it.
Word preview: about
How much longer do you think it will be, asked Josh?
Let's time the ride, said Zach, then we can figure it out.

Short version — 9 minutes

1. Split forty into rides (5 min)
Take forty counters for the forty people ahead of you and push them into groups of ten, because ten go on each time. Say how many groups you made and what one group stands for.
Word preview: each, rides
And if 10 people go on each time, Zack said, that's four rides.
2. Count the minutes by fives (4 min)
Lay one ride card beside each group of ten. Touch them in order and count five, ten, fifteen, twenty. Say the last number with the word minutes after it.
Word preview: minutes
So we'll have to wait about 20 minutes, Zack guessed.

Three levels

Support: Work with twenty counters instead of forty. Push them into groups of ten, put a ride card on each group, and count five, ten. Say the answer as about ten minutes. The grouping and the counting are the same work everybody else does; the line in front of you is shorter.
At Grade: Group forty counters into tens, lay a five-minute card on each group, count the minutes by fives to twenty and say the whole chain as one sentence. Then do the same for the eight people near the front, and write the minutes for two new line lengths your teacher names.
Extension: Take a line of forty-five people. Group by tens and say what the five left over does to the number of rides, then count the minutes for the rides you actually need. Say aloud why five leftover people still cost a whole five-minute ride, and what the answer would be if those five could be squeezed onto the fourth ride instead.

English learners

The mathematics travels without any English, and a child can lay the groups and the cards out correctly before saying a word about them, so let the grouping happen first. What does not carry is the English word each doing the work of a rate, because many languages mark that with a different construction or with nothing at all, and a child who reads ten people go on each time as ten people go on in total will group the counters into one pile rather than four. Rehearse one frame aloud while pointing at the groups: ten people go on each time, so ___ rides. The words wait and ride are both a verb and a noun in this book, and the phrase how much longer asks for a length of time rather than a length of anything else, which is worth saying plainly with the cards in view. Let children say the sentence in their strongest language first, then in English.

Materials

Forty counters per pair
Five ride cards per pair, each card printed five minutes
A wait slip per pair with three columns, headed people waiting, rides and minutes
Word cards: each, rides, minutes, about

Standards & curriculum

These lessons are being matched against curricula now.

3. Between Six and Twelve Reasoning · Bounding a total between a least and a greatest when every group may hold one amount or another

Josh does something in this book that no other character does: instead of naming one number he names the two ends of a run of numbers. Six legs with two seats each is twelve at the fullest, six at the emptiest, and he answers with a number from between them. The book's back matter does the same thing with four benches that hold either two people or three, and prints both totals, eight and twelve. Children build both ends with their hands here. Two counters in every one of six paper legs gives the greatest total; one counter in each gives the least; a mixed filling gives something in between, which they then find on a number line beside the ten people the book actually counts coming off the ride. The reasoning is that a total nobody has seen can still be pinned between two numbers, and that both ends have to be built before the middle means anything.

Full lesson — 20 minutes

1. Fill the legs the fullest way (5 min)
Lay out the six paper legs. Put two counters in every leg, because two people can ride in each one. Count all the counters and say the number out loud. That number is the most people the ride can carry at once, and you got it without watching anybody get on.
Word preview: each, most
There are six legs, and two people can ride in each leg.
Two times six is twelve, so I say twelve people.
2. Fill them the emptiest way (5 min)
Take one counter out of every leg, so that each leg holds just one person. Count what is left and say the new number. That is the fewest people the ride can carry with somebody in every leg. Say your two numbers to your partner, the fewest first and the most second.
Word preview: fewest, between
But maybe some of the legs have only one person, so I guess the numbers between six and twelve I say nine,
3. Name the numbers in between (5 min)
Say every whole number from six up to twelve out loud, putting a counter on the number line strip for each number you say. Now fill the legs a third way, some with one person and some with two, count the total and find it on your strip. Say whether your total sits between the fewest and the most, and say what number Josh picked out of that same run of numbers.
Word preview: nine
Josh counted everyone as they came out. 8 9 10.
he had guessed 9.
4. Do it again with four benches (5 min)
The chart at the back of the book gives four benches that hold either two people or three. Put two counters on each of your four bench cards and count the total; then put three on each and count again. Say your two totals aloud, and then say the chart's own sentence with your two numbers standing in it.
Word preview: least, greatest
Either 2 or 3 people can fit on a bench. There are 4 benches.
Between 8 and 12 people.

Short version — 9 minutes

1. Fill the legs the fullest way (5 min)
Lay out the six paper legs and put two counters in every leg. Count them all and say the number. That is the most people the ride can carry at once.
Word preview: each, most
There are six legs, and two people can ride in each leg.
2. Fill them the emptiest way (4 min)
Take one counter out of every leg so each leg holds one person. Count again and say the new number. Say your two numbers to your partner, the fewest first.
Word preview: fewest, between
But maybe some of the legs have only one person, so I guess the numbers between six and twelve I say nine,

Three levels

Support: Work with three legs instead of six. Put two counters in each leg and count to six for the most; take one out of each and count to three for the fewest. Say the two numbers with the fewest first, and lay a counter on the number line for every number from three up to six. Building both ends is exactly what everybody else does; there are fewer legs to fill.
At Grade: Fill six legs the fullest way and count to twelve, then the emptiest way and count to six, saying the two numbers with the fewest first. Lay out every number from six to twelve on the strip, fill the legs a mixed way and find your total on it, then run the whole job again on four benches that hold two or three.
Extension: Fill the six legs the fullest way and the emptiest way as usual, then work out how many different totals sit between your two ends, and say whether every one of them can actually be built by putting one or two people in each leg. Try to build seven, and say what that tells you about which numbers in a run of numbers are really available.

English learners

The counting and the filling need no English at all, and children can build both ends before naming either, so build first and talk after. What does not carry is the English pair least and greatest, which many languages express with the same word plus a marker rather than with two separate words, and the word between, which in English here includes both of its ends in the book's own use and excludes them in everyday speech. Rehearse one frame aloud while pointing at the two filled mats: the fewest is ___ and the most is ___, so the answer is between them. Josh's own sentence in the book is damaged where it turns from the range to his single number, so read the two ends off the legs rather than off the page, and let children say the range in their strongest language first.

Materials

Six paper legs per pair, each a strip with two marked seats
Twelve counters per pair
A number line strip per pair, marked from five to thirteen
Four bench cards per pair
Word cards: least, greatest, between, fewest

Standards & curriculum

These lessons are being matched against curricula now.

4. Counting the Groups Fluency · Skip counting equal groups by tens, fives, twos and threes to a stated total

The back matter prints five counting chains and gets four of them right: six tens counted to sixty, three fives added to fifteen, four twos to eight and four threes to twelve. This lesson takes those four sound chains off the page and drills them, because every other lesson on this book leans on one of them and stalls when a child has to reconstruct the chain in the middle of a decision about estimating. No judgement is asked for here and no about-numbers are produced. Children lay out strips of ten, five, two and three, touch each strip once per number word, say the total, and then run all four chains again from memory against a sand timer with no strips in front of them. What is being built is the speed to say sixty at the end of six tens without stopping to add, so the thinking in the other lessons can happen somewhere else.

Full lesson — 20 minutes

1. Count six rows by tens (5 min)
Put a strip of ten in front of you and lay out six of them in a column. Touch each strip once and say ten, twenty, thirty, forty, fifty, sixty. Do it three times through without stopping, and the third time keep your eyes on the strips rather than on your partner.
Word preview: tens
Hint. Count by 10s. 10. 20. 30. 40. 50. 60. About 60 people.
2. Add three fives (5 min)
Lay out three five-strips in a line, one for each of the three boys the chart measures its tree against. Touch them in order and say five, ten, fifteen. Say the last number again as the answer. Then mix the five-strips in with the ten-strips, take three fives out again by eye, and run the chain a second time.
Word preview: fives
About 3 boys tall.
The tree is about 15 feet tall.
3. Add four twos, then four threes (5 min)
Lay out four twos and count them two, four, six, eight. Clear them, lay out four threes, and count three, six, nine, twelve. Say each answer out loud as you finish its chain. Run both chains again, and this time say only the answer at the end rather than every number on the way.
Word preview: twos, threes
2. Plus 2. Plus 2. Plus 2. Equals 8.
3. Plus 3. Plus 3. Plus 3. Equals 12.
4. Race the four chains (5 min)
Turn the sand timer and run all four chains one after another with no counters in front of you: six tens, three fives, four twos, four threes. Say each answer as you reach it. Write the four answers on your slip, then check any you are unsure of by laying the strips back out.
Word preview: chain

Short version — 9 minutes

1. Count six rows by tens (5 min)
Lay out six ten-strips in a column. Touch each one once and say ten, twenty, thirty, forty, fifty, sixty. Run it three times through without stopping.
Word preview: tens
Hint. Count by 10s. 10. 20. 30. 40. 50. 60. About 60 people.
2. Add four twos, then four threes (4 min)
Lay out four twos and count two, four, six, eight. Clear them, lay out four threes and count three, six, nine, twelve. Say each answer aloud as you finish its chain.
Word preview: twos, threes
2. Plus 2. Plus 2. Plus 2. Equals 8.

Three levels

Support: Run the two shortest chains only, three fives and four twos, with the strips staying in front of you the whole time and one finger on each strip as you say its number. Say the answer at the end of each chain. Speed comes later; the job now is one touch and one number word, never two numbers on one strip.
At Grade: Run all four chains with strips, saying every number on the way and the answer at the end, then run all four again from memory against the timer and write the four answers on the slip.
Extension: Run the four chains from memory, then start each chain from its answer and count back down to nothing: sixty, fifty, forty, thirty, twenty, ten. Say which of the four chains is hardest to run backwards and what makes it so.

English learners

Number words are the whole of the work here, so this is a lesson where a child counting in their strongest language is doing the mathematics correctly and should be invited to. Run each chain twice, once in that language and once in English, with the strips in view both times. What does not carry is the English tens words, since twenty and thirty hide the two and the three that many languages say out loud, and the word by in count by tens, which is doing a job no everyday sense of by prepares a child for. Rehearse the frame aloud before the timer is turned: I am counting by ___, and the answer is ___. Say each chain's name in English at least once so a child can ask for the chain they want to practise.

Materials

Six ten-strips per pair
A sand timer, one per table
Three five-strips per pair
Four two-strips and four three-strips per pair
A four-line answer slip per pair, headed six tens, three fives, four twos and four threes
Word cards: tens, fives, twos, threes

Standards & curriculum

These lessons are being matched against curricula now.