Ruby Makes It Even

$Ruby Makes It Even book cover$

Grade 2
5 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.

1Odd Kid Out
2Odd Plus Odd Makes an Even
3Four Cannot Break a Tie
4Draw the Benches
5How Many Joined the Group

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.

No math-through-reading lessons for this book yet.

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.

No reading and language lessons for this book yet.

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Skill lessons

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

1. Odd Kid Out Concept · Deciding whether a group is odd or even by pairing everyone off

The book runs one test four times and never names it: everybody pairs up, and the group is even if nobody is left over and odd if exactly one rider is. Students seat counters two to a bench, the way Beth and Millie share the front bench and Ruby ends up behind them, and read the verdict off the seating for each of the four group sizes the book prints. Even and odd stop being labels to memorise and become the two things that can happen when a group pairs off.

Full lesson — 20 minutes

1. Seat three on the benches (5 min)
Set the bench strip in front of you and seat three counters, one per rider, filling a bench before starting the next. One bench holds two riders and one rider sits behind them with an empty seat beside her. Say what the book says about that number.
Word preview: odd, even, pair
This time I sit beside Beth. Millie sits behind us by herself.
Three is an odd number.
2. Run the book's four groups (7 min)
Clear the strip and seat four riders, then seven, then ten, refilling from the front bench each time. After each seating answer two questions out loud: how many benches are full, and is anybody sitting with an empty seat beside them. Give the verdict in the book's own frame before moving on.
Four is an even number.
Seven is an odd number.
10 is an even number.
3. Never two left over (4 min)
Look back at the odd seatings, three and seven. Exactly one rider was alone each time, never two. Seat five riders and then nine and check whether that holds, then say why two riders left over could not happen: two riders left over would simply take a bench.
4. Write the verdicts (4 min)
Write each number you seated into the even column or the odd column, and enter a number only after its seating has been run. Read your odd column aloud to your partner and let them check one entry by rebuilding it.

Short version — 9 minutes

1. Seat three on the benches (3 min)
Seat three counters on the bench strip, one per rider, filling a bench before starting the next, and say what the book says about that number.
Word preview: odd, even, pair
Three is an odd number.
2. Run the book's four groups (6 min)
Seat four riders, then seven, then ten. After each seating say how many benches are full and whether anybody has an empty seat beside them, then give the verdict.
Four is an even number.
10 is an even number.

Three levels

Support: Stay with three riders and four riders. Seat them one counter at a time and answer one question after each seating: does anybody have an empty seat beside them. Turn the verdict card to odd or even after the seating, never before, and let the empty seat be the whole reason.
At Grade: Seat all four of the book's groups, say each verdict with its reason, test five and nine, and fill in both columns of the chart.
Extension: Predict every verdict before a counter moves, then take on the seating the book gives the four friends: ten riders in rows of four fill two rows and leave a part row of two, and nobody is alone, yet the rows are not pairs. Say which of the two tests the word even names, and show a group that passes nobody-alone in rows of four and still fails the pair test.

English learners

Pair is doing the work here and it is visible before it is a word: two counters side by side on one bench. Even and odd are arbitrary English labels that cannot be guessed from the objects, so attach each one to a seating a student has just made, and rehearse one frame both ways, blank is an even number because every rider has a partner, blank is an odd number because one rider sits alone. A student may run the seating and point at the empty seat while the frame is still settling, and may give the verdict in their strongest language first.

Materials

Bench strip of five two-seat rows, one per pair of students
20 two-colour counters per pair of students
Verdict card, odd on one face and even on the other, one per pair of students
Printed source passage in this step
Group cards marked 3, 4, 7 and 10, one set per pair of students
Two-column chart headed even and odd, one per student
Pencil per student
Word cards: odd, even, pair

Standards & curriculum

These lessons are being matched against curricula now.

2. Odd Plus Odd Makes an Even Reasoning · Predicting whether a joined group is odd or even before counting it

Ruby works out the rule on the spot: her three plus Rob's one has to come out even, and even means nobody rides alone. The book states that rule and one more, then uses them twice again as the group grows to seven and to ten. Students build each join, push the two groups into one row, pair the whole row off, and check the book's prediction against what the row does. Then they take the rules somewhere the book never goes and find out whether they hold.

Full lesson — 20 minutes

1. Push the two groups together (5 min)
Build Ruby's group as three counters on one strip and Rob as one counter on the other. Predict out loud what the joined row will do, then slide both strips together and pair the row off. Report what you found: four riders, two pairs, nobody alone.
Word preview: plus, makes, odd, even
You're an odd number. One, I tell Rob.
We're an odd number. Three.
But an odd plus an odd makes an even.
2. Run the book's other two joins (6 min)
Build four and three, predict, join, pair off: one rider is left alone, so seven. Then build seven and three, predict, join, pair off: nobody is left, so ten. Each time say the prediction before the counters move and the outcome after.
And even plus an odd makes an odd.
They're an odd and we're an odd. So we're even again.
3. Try joins the book never makes (5 min)
Predict and then build five and five, six and three, and two and eight. Two of these the book's rules already cover. One of them the book never speaks about at all, and your class has to settle it from the counters: what happens when two even groups join.
4. Say the three rules (4 min)
State the rules you can now defend, each with a build behind it that your partner can check: odd and odd come out even, even and odd come out odd, even and even come out even. Mark which two Ruby says out loud and which one is yours, found on the desk.

Short version — 9 minutes

1. Push the two groups together (4 min)
Build Ruby's three and Rob's one on separate strips, predict what the joined row will do, then slide them together and pair the row off.
Word preview: plus, makes, odd, even
But an odd plus an odd makes an even.
2. Run the book's other two joins (5 min)
Build four and three, predict, join and pair off. Then seven and three. Say the prediction before the counters move and the outcome after.
Now we have seven.

Three levels

Support: Work only the join Ruby works, three and one. Build the two groups on separate strips so the two odd ones are visible as two lonely riders, then slide the strips together and watch the two lonely riders become a pair. Predicting comes after two more joins have been watched, not before.
At Grade: Predict, build and report all three of the book's joins, then settle two even groups from the counters and state the three rules with a build behind each.
Extension: Argue why odd and odd must come out even without building it: each odd group has exactly one rider with nobody beside them, and when the groups join those two lonely riders can only sit together. Then say what that argument predicts for three odd groups joined at once, and test it.

English learners

Plus and makes carry the whole rule and neither is guessable, so run them with the counters in hand: plus is the moment the two strips slide together, makes is the row afterwards. Number words for one, three, four, seven and ten transfer straight from any language of schooling, so a student can predict in their strongest language and only then reach for the English frame, blank plus blank makes blank. Accept a prediction shown with the strips and no words at all.

Materials

Two paper strips per pair of students, one for each group being joined
20 two-colour counters per pair of students
Printed source passage in this step
Group cards marked 3, 4, 7 and 10, one set per pair of students
Rule card with three blank lines, one per pair of students
Pencil per student
Word cards: plus, makes, odd, even

Standards & curriculum

These lessons are being matched against curricula now.

3. Four Cannot Break a Tie Reasoning · Why a two-way vote can split down the middle in an even group and cannot in an odd one

Halfway through the day Ruby decides evens beat odds, and the very next page the book takes it back: four friends vote between two rides and stall two against two. The same group at seven settles it at once. Students tally the four votes the book actually prints, then try to split four counters into two equal piles and seven into two equal piles, and find that the deadlock was never about the rides. It was about the number of voters.

Full lesson — 20 minutes

1. Tally the votes the book prints (5 min)
Read the four votes and mark one tally per voter under the ride that voter chose. Rob does not say a ride, so decide from what he does say which column his tally belongs in, and defend it. Read the two totals aloud.
Word preview: vote, tie, split
I vote Crocodile Canoes, says Beth.
Me too, I say.
I vote Amazon Zing, says Millie.
Rob nods. It has a shorter line.
2. Try to split the group (6 min)
Take four counters and find every way to deal them into two piles. One way makes the piles match. Now take seven counters and try the same thing: deal one to each pile in turn until you run out. Report what always happens to the last counter.
3. Say why seven settles it (5 min)
Put the two findings side by side and say the rule in your own words: with two rides to choose between, an even group can stall and an odd group cannot. Then read the book's two lines and say which one your counters just explained.
even number is great, until you can't break a tie
Now we have seven. We can break our tie.
4. Test it on new groups (4 min)
Six voters, nine voters, ten voters, two rides. Predict which groups can stall, then deal the counters and check. Write the groups that can stall in one column and the groups that cannot in the other.

Short version — 9 minutes

1. Tally the votes the book prints (4 min)
Mark one tally per voter under the ride that voter chose, deciding from what Rob says which column his tally belongs in, then read the two totals aloud.
Word preview: vote, tie, split
I vote Crocodile Canoes, says Beth.
I vote Amazon Zing, says Millie.
2. Try to split the group (5 min)
Deal four counters into two piles and find the way that makes them match. Then deal seven the same way and report what always happens to the last counter.

Three levels

Support: Use bodies rather than counters. Four students stand and step to one of two ride cards on the floor until the sides match; three more join and the class watches the sides stop matching. The talking stays at one sentence, the sides match or they do not, and the tally sheet is filled in together afterwards.
At Grade: Tally the printed votes, deal four and seven into two piles, state the rule with both findings behind it, and sort six, nine and ten into can-stall and cannot.
Extension: Find the case the rule does not cover. Put three rides on the floor instead of two and deal seven voters: three, three and one leaves the group stuck with an odd number of voters. Say what has to be true of the vote itself, and not only of the group, before an odd number guarantees a winner.

English learners

Tie is the hard word and it is not about knots here; build it from the tally sheet, two columns of the same height, before the word arrives. Vote and split can be shown by walking to a card on the floor, which lets a student take a full part before saying anything in English. The rehearsable frame is short and worth drilling both ways, four can split into two and two, seven cannot split into two matching piles, and a student may give the reason by pointing at the leftover counter.

Materials

Vote tally sheet, one per pair of students
Pencil per student
Printed source passage in this step
20 two-colour counters per pair of students
Two ride cards, Crocodile Canoes and Amazon Zing, one pair per group
Two-column chart headed can stall and cannot stall, one per student
Word cards: vote, tie, split

Standards & curriculum

These lessons are being matched against curricula now.

4. Draw the Benches Representation · Showing a number as a set of pairs with or without one left over

The book draws the idea before it names it: Beth takes the front bench, Millie slides in beside her, there is no room for a third, and Ruby ends up on the bench behind. Students turn each of the book's group sizes into that picture, one rider per box on paper ruled in twos, and shade the seat nobody fills. The drawings are then read back cold by another pair, which is the test of whether the picture carries the number and the verdict without the drawer standing there to explain it.

Full lesson — 20 minutes

1. Read the seating (4 min)
Follow the three friends onto the first ride with the passage in front of you and act it out with three students and two chairs. Say exactly what stops the third rider joining the first bench.
Word preview: bench, pair, left over
Millie slides in beside her.
They try to scoot closer, but there isn't any more room.
I crawl onto the bench behind them.
2. Draw the book's four groups (7 min)
On the bench paper, draw the three riders, then the four, then the seven, then the ten. One rider per box, fill a bench before starting the next, and shade any box left empty. Write the number of riders under each drawing.
3. Read a drawing back (5 min)
Swap sheets with another pair. Without asking them anything, say how many riders their drawing holds, how many benches are full, and whether the number is odd or even. Then check the number they wrote underneath.
4. What the odd drawings share (4 min)
Put the odd drawings together and the even drawings together. Say in one sentence what every odd drawing has that no even drawing has, and mark that feature on each one.

Short version — 9 minutes

1. Read the seating (3 min)
Act the three friends onto the first ride with three students and two chairs, and say what stops the third rider joining the first bench.
Word preview: bench, pair, left over
I crawl onto the bench behind them.
2. Draw the book's four groups (6 min)
Draw three riders, then four, then seven, then ten on the bench paper, one rider per box, filling a bench before starting the next, and shade any empty box.

Three levels

Support: Draw only three riders and four riders, and place a counter in each box before the crayon touches it so the drawing copies something already built. The shaded empty seat is the one thing that must appear, and the number underneath may be copied from the group card.
At Grade: Draw all four group sizes with the empty seats shaded and the counts written, read another pair's sheet back cold, and name what the odd drawings share.
Extension: Draw the ten riders a second way, on paper ruled in rows of four, the way the book seats the four friends across. Then say what the first drawing shows that the second one hides, and why a shaded box means something different in each.

English learners

The picture arrives before the words and can carry a whole turn on its own: a student who draws ten riders with no shaded box has said even without saying anything. Bench, pair and left over should be taught at the paper with a finger on a box. Reading a partner's sheet back has a fixed frame worth writing up, this drawing holds blank riders, blank benches are full, and it is blank, and a student may fill the slots by pointing before speaking.

Materials

Printed source passage in this step
Bench paper with rows of two boxes, two sheets per student
Crayon or coloured pencil per student
Word cards: bench, pair, left over

Standards & curriculum

These lessons are being matched against curricula now.

5. How Many Joined the Group Application · Finding how many joined a group from the totals before and after

The book prints what the group came to and never prints who arrived. It goes from four to seven when Leroy turns up with his friends, and from seven to ten when Joe turns up with hers, and neither arrival is ever counted on the page. Students recover both numbers twice over, once by counting on from the earlier total to the later one, and once by counting the names the book does print, and the two answers have to agree before the pair reports.

Full lesson — 20 minutes

1. The two totals (4 min)
Find the number the group came to before Leroy arrived and the number it came to afterwards, and lay a counter row for each. Then say what the book never tells you between those two lines.
Word preview: joined, more, count on
Four is an even number.
Now we have seven.
2. Count on to the newcomers (6 min)
Lay out four counters and add one at a time, saying five, six, seven, until the row matches the later total. Count only the counters you added. Do the same from seven to ten. Write both answers down before you check them.
3. Check it by naming (5 min)
The book never counts the newcomers but it does name them. Find every name in each arrival, put a name card down for each one, and count the cards. If the cards and the counting on disagree, go back and find which one is wrong.
We're about to vote again when I spot Leroy with his friends Carlos and Jamal.
So Joe and her friends join our group.
She's with her friends Claire and Heather.
4. Report both routes (5 min)
Give the answer twice in one breath to another pair: three joined, because four counted on to seven takes three, and because the book names three people. A report with one route only is sent back for the other.

Short version — 9 minutes

1. The two totals (3 min)
Find the total before Leroy arrived and the total afterwards, lay a counter row for each, and say what the book never tells you in between.
Word preview: joined, more, count on
Now we have seven.
2. Count on to the newcomers (6 min)
Lay out four counters, add one at a time saying five, six, seven, and count only the counters you added. Do the same from seven to ten.

Three levels

Support: Work the first arrival only, and work it with people rather than counters: four students stand at the front, three more join on a signal, and the class counts on from four aloud while the newcomers step in one at a time. The written answer is a single numeral.
At Grade: Count on for both arrivals, check each against the names the book prints, and report both routes together.
Extension: Settle it a third way, without counting on at all. The group is seven and the raft holds ten and takes the whole group exactly, so the arrivals have to carry seven up to ten. Then use the rule from Ruby: seven is odd and ten is even, so say what that alone tells you about the number who joined, before any counting.

English learners

Count on is a procedure, not a phrase to decode, so run it with the row growing under the student's hand and the numbers said aloud. The names do a lot of work for a student still gathering English, because counting Leroy, Carlos and Jamal needs no arithmetic vocabulary at all and gives the same answer. The frame for the report is fixed and short, blank joined, because blank counted on to blank takes blank, and either route may be given first.

Materials

20 two-colour counters per pair of students
Printed source passage in this step
Bead string or ten-frame per pair of students
Pencil per student
Newcomer name cards, one set per pair of students
Word cards: joined, more, count on

Standards & curriculum

These lessons are being matched against curricula now.