Teacher’s Manual for Math Tool Kit A

© 2007, Global Education Resources, L.L.C. All rights reserved.

Note:

The Teacher’s Manual for Math Tool Kit is a supplemental resource for the English translated Japanese elementary mathematics textbook entitled Mathematics for Elementary School, published by Tokyo Shoseki. The manual describes how to use the math tools that appear in Grade 1 of the textbook. Math Tool Kit A can be purchased from Global Education Resources. Please visit: www.globaledresources.com for more information. Math Tool Kit A does not include all the manipulatives described in this manual and the the Japanese math textbook. However we decided to translate all the pages of the manual because it is a helpful resource for thinking about instruction with the textbook. The original Teacher’s Manual for Math Tool Kit was written in Japanese and published by Kobun Shoin. Global Education Resources translated it into English.

FriendsUsing the picture on the box

• Making a set by determining a viewpoint

• Making a set by changing a viewpoint

1. Choosing a viewpoint When comparing or counting a number of objects, students need to distinguish the objects they are counting from the objects they are not counting. To do this, students need to practice making groups and describing the attributes of the group that they are counting. To say “I counted penguins” isn’t ex-plicit enough, but saying “I counted penguins wearing hats” clearly distinguishes the group/set that was counted. Viewpoints used to form sets could vary. For example, type of animal, game activity, color, position/loca-tion, etc. ➊,➋ -- Examples for viewing objects that share common elements as a group: • A group of penguins • A group of bear cubs

1. Place counters on objects in the same group (set)

➊ Penguins ➋ Bear cubs

* Size and pictures could differ

Table of Contents

Friends Below this page

Numbers Up to 10 2

Ordinal Numbers 5

Two Numbers Together 6

Playing with Shapes 9

Addition (1) 10

Subtraction (1) 12

Numbers Up to 20 14

Number Line 17 Dividing 31

Making Shapes 30

Large Numbers 26

Subtraction (2) 24

Addition (2) 22

Addition & Subtraction (2) 21

Addition & Subtraction (1) 20

Length & Capacity 19

Clock Reading 18

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2. One-to-one correspondence manipulation and judging if two quantities are the same or different, larger or smaller It is possible to compare numbers of ele-ments between two groups without directly counting and expressing them with num-bers. That is, by aligning each element of one group one-by-one with an element from another group, it is possible to figure out equalities or size differences between the two groups. This is an important method to compare two groups even when the number of ele-ments become large. ➊ to ➌ For example, when comparing numbers of bear cubs and penguins, first, put one counter on each bear cub, then, gather the counters and put one on each penguin. Some counters will be left over. From that, it will be evident that there are more bear cubs than penguins. Another way to compare is to put a counter on each object. Then, line up the counters, as below, to easily compare groups.

* Some students might start counting im-mediately without trying the above methods. If so, they should be instructed to think, “Is there any other way to investigate or figure it out without counting numbers?”

2. Determining which one has more or less➊ Which one has more, dinosaur or penguin?

➋ Which one has more, bear cub or penguin?

➌ Which one has more, little bird or bear cub?

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Numbers Up to 10• Counting by categorizing• Correspondence of objects, dot cards,

and numerals • Structure and order of numbers• Reading and writing of numerals

1. Counting by categorizing Using pictures from the textbook or the box, instruct students to choose what they want to count and make a group (or a set), then, ask them to put counters on the objects. Then, have them count the counters one by one by saying: “One,” “two,” “three, ...” so that each spoken numeral corre-sponds to one counter.

2. Correspondence of pic-tures (numbers of elements in a group/set) and dot cards

Utilizing games such as “card picking,” instruct students to match (correspond) the “numbers of group elements in a picture” with the num-ber of dots on dot cards or dice, the number showing on number cards, or the number of counters. ➊ Place the same number of coun-ters as there are dinosaurs, penguins, or bear cubs in a picture. Then, pick up a “dot card” or “number card” indicating the same number.

1. Line up the same number of counters as there are yellow flowers.

Line up the same number of counters as there are red flowers.Counting the counters, pick up each one with one hand and gather them in the other hand.

2. Card picking➊

➋ One student holds a dot card and the partner picks up a number card corresponding to the number of dots.

➌ Have students choose a number card corresponding to the num-ber of dots on one of the dice.

* Size and pictures could differ

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➍ Show a number card, then have students line up counters (or sticks) corresponding to the number.

➎ Have students match the cards.

➌ Draw as many circles as there are number of sounds made, then put a number card next to them.

➊ Have students line up cards (structure and order of numbers).

➌ Starting with number 5, have students line up the other cards on either side of the five.

3. Line up as many counters as there are number of sounds made, then choose a dot or number card cor-responding to the number.

➊

➋

4. Lining up cards (structure and order of numbers; size of numbers)

From small to large

From large to small

➋ Have students figure out the hidden numbers.

turned over turned over

put the 5 first

➎ Scatter dot cards and number cards to gether, then pick up one of each with the same number and line them up.

➊ Scatter the number cards, then line them up from small to large or from large to small. Change the range of numbers, for ex-ample: 3 to 8, or 2 to 9. The cards can also be simply held in hand, and lined up as above.

➋ Line up cards 1 through 9 from small to large with several cards turned over, then, figure out what numbers these cards are.

➌ Just like some card games, line up cards starting with the 5 card. Put the 5 card down first, then, continue putting down cards on either side of it. Each card must be placed in order, with no skipping.

3. Corresponding number of sounds made to number of dots or number. Expressing a number by either making sounds or saying a number, have students correspond the number of sounds made with “counters” → “dots” → “numbers.” You can use an instrument such as a tambourine, a whistle, a triangle, a casta-net, and so on. Also, clapping, stamping, bouncing a ball or jumping rope could work.

4. Structure and order of numbers

ting

ting

ting

ting

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➍ Making a sequence of three cards.

➊ Comparing numbers of counters or blocks.

➋ Comparing two numbers to see which is larger.

➊ Reading numbers.

➋ Picking up number cards.

➌ Writing numbers.• Writing numbers from number cards.• Writing numbers of dots from dot cards.

➍ Just like the card game “Old Maid,” distribute number cards evenly among students, then have them pick the other’s cards one at a time. If a student gets three numbers in a sequence, they can be lined up on a desk. Adjust the number of cards according to the number of students in the game.

➊ One student of a pair shows another student counters or blocks in each hand. The other student counts the numbers in each hand, then says which hand has more.

➋ Make a pair of students with each one holding dot or number cards, and each one shows a card to the other at the same time (after counting “One, two, three!”) The one with the larger number wins. Or, each one can throw a die at the same time, then they can compare the number of dots on the dice.

➊ Display a number card to students just like a flash card. Then, let them read it.

➋ Have students line up cards on desks, then have them pick up the card corresponding to the number a teacher says. Or scatter the cards face up, and have a group of four students compete to quickly pick up the card corresponding to the number you said.

➌ Having cards stacked faced down, students flip over one card at a time, then write down the number in their notebooks. Or, use dot cards or dice and have students write down the number in their notebooks.

5. Comparison of size of numbers

6. Reading and writing numbers

5. Comparison of size of numbers

I won!

6. Reading and writing numbers

Three Three

Four

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• Expressing a place from the left, right, front, back, top, or bottom

• Understanding the meaning of how many (cardinal numbers) and what place (ordinal numbers)

Ordinal Numbers

OHP

➊ Using a ruler with stencils1. Expressing a place from the left (or right) 1. Expressing order or position

with numbers

2. Using blocks

3. Using a counter board (Metal Activity Tray)

4. Expressing a place from the top and from the left, etc.

• Let’s flip four blocks starting from the left!”

• “Let’s flip the fourth block from the left!”

2. “How many?” and “What place?”

Lining up a cardinal number (e.g., “4”) that has been studied already and an ordinal number (e.g., “4th”) side by side to clarify their difference and to learn the concept of ordinal numbers.• “Let’s flip four blocks starting from

the left.”• “Let’s flip the fourth block from the

left.”

3. Expressing a place from the left?Make a square grid and find the seventh square from the left, then put a red counter there on a counter board. In the next row, have students say what place from the left is the location of a second counter three squares to the left from the first one.

4. What place from the left is the blue counter?

Make a square grid sheet and put on the metal activity tray. Then place a counter in a square of the grid. Have students say its location from the top, bottom, left, or right.Example: The blue counter is located...➊ Fourth from the top➋ Seventh from the bottom➌ Fifth from the left ➍ Sixth from the rightThen, put another counter of a different color on a square above (or below) or to the left (or right), and have the students say, “The red counter is three squares away from the blue one on its right” in order to specify the location of the red counter relative to the blue counter as the starting point.

➊ Have students say, “The third shape from the left is a triangle,” or have them trace the correct shape. You can also ask students to say where a particular shape (such as a “circle”) is on the ruler with stencils using words that indicate its position from the left or right. In addition, you could use an overhead projector to project the ruler with sten-cils for an activity.

OHP = Overhead projector

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Two Numbers Together

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• Composition of numbers (compo-sition/decomposition – 1 to 10)

• Increase/decrease of numbers• Looking at a number as a sum or

difference of two numbers• Structure and order of numbers

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➊ Using counters

1. If you put them together, what will it make? What is the difference?

➊ Give each of a pair of students five counters, and have each student show some number of counters in one hand, and then, jointly, have them figure out “what is the sum.” Have each student write down the sums in a notebook. The students with a correct answer can get their partner’s counters; if both are correct, it is a draw. Have them figure out differences, too.

➋ Have students figure out the sums of two numbers from any combination of dot and number cards. Change the com-binations from easy to more difficult. (Cards the students use should be in the 1 to 5 range.)Have them figure out differences, too.

1. Composition of numbers (Composition/decomposition)

➌ Using a counter box, figure out how many counters are divided into each side of the box. (One side is visible and the other side is not visible)

➊ Have students line up dot (or num-ber) cards from 1 to 10. Say a number and have them pick up two cards that will sum to that number.(Let them know that one number can be a sum of two numbers.)

➋ Showing a card and saying a number, have students play a game to quickly pick up a card that will sum to the spoken number. (Use dot or number cards.)

2. Composition and decomposition of numbers from two cards

➋ Using dot (or number) cards

➌ Using a counter box

➊ Getting 7 quickly with two cards.

➋ Adding the teacher’s card with a student’s card.

2. Using two cards to compose a spoken number

Seven is made of 4 and what?

and and and and

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➊ In case of 5

1. What happens if a object increased by one or decreased by one?

➋ In case of 6 (Red) ( Blue)1 and 5

2 and 43 and 34 and 2

5 and 1

6 and 0

(Red) ( Blue)1 and 4

2 and 33 and 24 and 1

5 and 0

➌ Picking up one red card and one yellow card that sum up to 6.

(Lining up the chosen cards)

➍ Picking up a pair of a red card and yellow card that sum up to 7 and lining them up in order. Check how many pairs are there.

(Lining up the pairs in order.)

3. Increase and decrease of num-bers➊ Have each student present how many of each of five red and five blue coun-ters (or sticks) will sum to 5. Asking students to organize the combinations as shown to the left. Have students figure out how many blue counters decrease when red coun-ters increase by one. Encourage students to be curious about how a number and its numeral, such as 5, could be composed of 1 and 4, 2 and 3, 3 and 2, and so on.

➌ Instruct students to line up red cards from 0 to 9 in a row, and then line up yellow cards in the same way under-neath them. Have students find two cards, one from each row, to make 6. Then, line up the pairs under the rows. Try other numbers, such as 5, 7, 8, 9. and 10 in the same way.

➍ Scatter red and yellow cards from 0 to 9. Find all pairs of one red card and one yellow card to make 7, and line them up in order. Then, check how many pairs there are. Try the same thing as above to make 8, and check how many pairs there are. When choosing pairs to make 9, you could encourage students to infer that the total number of possible pairs should be 10 given the fact that they could make 8 pairs for the number 7, and 9 pairs for the number 8. Try other numbers in the same way.

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➎ Line up red cards from 1 to 9 in a row and yellow cards from 1 to 9 underneath them. Picking one card at a time from each row, starting from the left on the top row and from the right on the bottom row, line up the pairs to see what numbers they make. When using cards up to 9, the sums of each pair are all 10. Thus, students come to understand the composition of 10. Then, using cards from 1 to 8, 1 to 7, 1 to 6, and so on, students come to understand the compositions of 9, 8, 7, and so on.

➎ Picking one card at a time from each row, starting from the left on the top row and from the right on the bottom row, line up the pairs to see what numbers the two cards make. Check what are the sums of the two cards.

(Lining up the cards picked up.)

Have students understand through manipulating many sets of cards that there are various combinations of two numbers whose differences make 1, 2, 3, and so on.

4. Expressing one number as a dif-ference of two numbers

Line up some number or dot cards in order. One student of a pair flips over or hides one or more cards while another closes his or her eyes. Then, the partner has to figure out what numbers are on the missing cards.

5. Structure and order of numbers

➊ Which and which make a difference of 2?

4. Which card and which card will make a difference of 1 (2, 3, ....)?

➋ Combinations of cards that make a difference of 4.

Cards that make a dif-ference of 4 are ,,,

and

5. Finding the hidden number

and

and and

and and

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Playing with ShapesIntroduction to basic shapes such as: rectangular prism, cube, cylinder, sphere, and quadrangle, triangle, circle.(Composition, clasification, charac-teristics, etc.)

➊ Using blocks1. Let’s stack or roll blocks!

➋ Using blocks

2. Let’s collect similar shapes together!

3. Let’s trace!➊ Using blocks ➋ Using color tiles

4. Let’s make shapes!

1. Composition and characteristics of solid figures

➋ Using blocks, classifying shapes into two groups: easy to roll or not easy to roll. Throughout this activity, have stu-dents grasp the functional characteris-tics of shapes.

It is important in this grade for students to become familiar and curious about shapes. Be careful not to present these ma-terials too quickly, or with abstract and formal explanations.

➊ Using blocks, compose various shapes.

2. Classificaion and characteristics of solid figures Collect objects of various shapes from the surroundings. Classifying them by paying attention to the shape, have students experience the fundamentals about the basic solid shapes (rectangular prism, cube, cylin-der and sphere). Shape names (such as “box-like shape,” “dice-like shape,” “ball-like shape,” or “tube-like shape”), while used here for convenience, do not need to be memorized by students.

3. Solid figures and plane figures This is an activity for recognizing plane shapes by looking at a surface of a block. Have students guess what shape they will get from tracing a block. Accuracy is not expected for students tracing the shapes at this point.

4. Formation of plane figures and introduction to their character-istics Form various shapes following the activities in 3, or by cutting out colored paper or using colored boards. Content and difficulty levels should be the same as activities in 1 to 3 above.

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• Meaning of addition (Combining/increasing)

• Understanding addition calculation whose sums are less than 10

• Practicing addition calculation• Symbols such as +, =, and terms

such as “adding” and “math sen-tence”

Showing pictures in a textbook or on the box lid of the math tool kit, have students understand the concept of com-bining. First, have them grasp the exis-tence of two sets, and help them realize that these two sets can be combined into one. Then, lead them to express this phenomena using a math sentence. ➊ to ➌ are the examples.

1. Studying phenomena of combin-ing and expressing about them with math sentences

➊ Placing counters on the fish in the picture, students come to grasp the con-cept of two sets by their locations.

➌ Having a math sentence to indicate the above phenomenon in ➁ by using number and symbol cards such as +, =, students learn how to read the math sentence.

➋ Handling with one-to-one correspon-dence between counters and fish on the picture, students learn that the number of counters is 5 by counting and adding them. This will help students gradually understand that, “Addition is an opera-tion to find the combined number of elements of two sets.” Also, help students express the pho-nomenon of addition in a simple word sentence like, “3 and 2 added together-makes 5.”

➍ By dealing with other cases involv-ing conmbining, help students deepen their understanding by following the above steps ➀ to ➂ in each case.

Not only limited to the case of com-bining in 1 described above, students learn that addition is also used in the case of increasing. From a point of view that “combining two sets makes one group,” even with a delay in the combining action, increasing can be regarded as the same as combining. Through this learning, try to expand the meaning of addition. Using the steps ➀ to ➂ described above in 1 to help deepening studnets’ understanding.

2. Studying phenomena of increas-ing and expressing increases with math sentences

➊ “How many fish do you get if they are combined?”1. How many if combined?

➋ Using counters (or number blocks).

One two three four five

➌ Using number cards and symbol cards, + and =.

➌ “..... everything together makes ....” “.... all together makes ....”

➊ “How many fish are there if you add 3 more fish?”2. How many if you add?

➋ Using number blocks and a block case of ten (using counters).

➌ Using number cards and symbol cards like + and =.

10

➊ Using dot cards or dice.3. Let’s practice addition!

What will it make if you add them together?and

and

➋ Using calculation cards.

➌ Using number line

4. Let’s find addition cards whose answers are the same!➊ Using numeral cards and calculation cards.

5. Let’s make addition problems!

3. Learning and mastering of ad-dition calculation

➊ Using dot cards, dice, or counters, students improve addition calculation skills.

• Familiarize students with the addi-tion math sentence by expressing “2 + 3 = 5.”

➋ Using calculation cards, have students practice repeatedly to learn calculation. For slower learning students, instruct them to use counters or number blocks as supplemental tools.

• Flip the card over to confirm the answer.• Omit troubled cases and practice repeatedly with the comfortable ones first (then add troubled ones later)

➌ In order to check whether an answer is correct or not, a number line could be used in addition to using counters and sticks. By introducing the basics of a number line, help students to understand the relationship of what is represented using the numberline to do addition.

4. Learning addition through calculations that have the same sums Line up number cards from 1 to 10 on a desk. After shuffling the calculation cards well, flip over one card at a time and put it under the number card that corresponds to its answer. Investigate calculaiton cards gatherd under the same number card.

5. Creating addition problems by looking at pictures Using counters or other tools, let stu-dents tell stories that involve addition, such as in the picture to the left. Then ask them to present answers to the cre-ated problems.

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Subtraction (1)• Meaning of subtraction (decreasing/

difference) • Understanding subtraction calcula-

tion of numbers whose minuend is less than 10

• Practicing subtraction calculation• Symbol, –, subtracting, and terms of

subtraction

Showing a picture in a textbook or on the box lid of a math tool kit, have students understand the phenomena of decreasing.

1. Studying the phenomena of decreasing, and expressing de-creases using math sentences

➊ Putting counters on the children in the picture, have students confirm the number of children. Then, to help students grasp the phenomenon of decreasing, remove two counters to show that two of the children are leaving.

➌ Having a math sentence to indicate the above phenomenon in ➁ by using number and symbol cards such as -, =, students learn how to read the formula.

➋ Transferring the example above in 1 to lining up counters and manipulat-ing them, have students understand that the remainder will be 3. This will help students gradually understand that, “subtraction is an operation to find the number of elements of one part of a set that are result of splitting a set into two sets.” Also, help students express the phe-nomenon of subtraction in a simple word sentence like, “Taking 2 from 5 makes 3.”

➍ Have students deepen their under-standing about other cases involvng decreasing by following the above steps ➀ to ➂.

➊ “How many children are left after two go home?1. How many are left?

➋ Using counters (number blocks).

➌ Using number cards and symbol cards, – and =.

➍ “.... if you use some ....” “.... if you eat some ....”

2. What is the difference?➊ “What is the difference between the numbers of yellow cars and red cars?”

➋ Using number blocks and a box.

➌

difference

2. Studying the phenomena of find-ing differences, and expressing differences using math sentences

Not only limited to the case of decreas-ing, above, explain that subtraction is also used in the case of finding differences. ➊ Putting counters on each car in the pic-ture, have students confirm the numbers.

➌ By manipulating counters, reinforce how subtraction is carried out.

• Follow the process as in ➂ and ➃ of 1 to help reinforcing studnets ability to ex-press phenomena using math sentences and understanding.

➋ Then, line up the counters to make one-to-one correspondence evident, and have students confirm the missing cor-responding counters.

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3. Let’s practice subtraction!➊ Using counters and counter box, or dot cards.

... How many counters are hidden?

Put seven counters in the box.

➋ Using dice.

➌ Using calculation cards.

4. Let’s find subtraction cards whose answers are the same.• Using calculation cards and a roulette.

5 – 3 equal 2

5. Let’s find subtraction cards whose answers are the same.

8 – 6 equal...

* Size and pictures could differ.

3. Let’s practice subtraction!➊ Using counters or dot cards, students improve skills of subtraction calculation by searching for the number of hidden objects.

• Familiarize students to the subtrac-tion math sentence by expressing “7 – 3 = 4”.

➋ Using dice, students improve skills of subtraction calculation.

• Throwing two dice with numbers or dots, let students subtract a smaller number from a larger number.

➌ Using calculation cards, have students practice repeatedly and learn calculation.

• Students can practice alone or with two or more students.• Using counters or blocks to check answers is important for basic in-struction or the instruction of slower learners.

4. Learning of subtraction through calculations that have the same differences

5. Creating subtraction problems using elements in a picture

Scatter subtraction cards on a desk. Students pick up a card (or cards) whose difference corresponds to the number on a roulette wheel.

• Letting students create their own rules for the game will be helpful to get them interested in these activities.

Using counters or other objects, let students tell stories that involve subtrac-tion, such as in the picture to the left. Then ask them to present answers.

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• How to count numbers 11 to 20, composition, notation, structure and order of numbers (number line)

• Size of numbers• Addition and subtraction of numbers

up to 20

1. 10 and how many?➊ Put 20 counters on a desk and have students grab a handful, then line them up on a counter board. Through the game, have students learn that a number larger than 10 can be expressed as 10 and some more. For example, 14 is composed of 10 and 4.

➌ Replacing the counters (or number blocks) used in the above activities with sticks, let students divide the sticks into a bundle of 10 sticks plus a remainder. Then, let students replace the sticks with number cards so that they will relate the meaning of a bundle to the color of the cards. (In order to avoid mistakes such as displaying 104 instead of 14, use a red card to stand for a bundle of 10.)

➋ Students can also perform the same activities as above using number blocks. In this case, instead of a counter board, let students use a number block box for 10 blocks. This is effective for students to understand 10 as a cluster.

➍ Using two number cards, have stu-dents understand that a number larger than 11 is composed of 10 and some number.

Composition: 10 and o make how many?

Change the number in the o

Decomposition: 18 is 8 and o or 10 and o

Find the number that fits in the o

&

&

make is

➊ Using counters1. 10 and how many?

➋ Using number blocks.

➌ Using sticks and number cards

➍ Finding cards

Pict

ure

card

What number do these two num-bers make?

18 is 8 and what num-ber?

14

2. Structure and order of numbers➊ Holding bunches of shuffled cards from 11 to 20, let students compete with each other to quickly line up their cards from smaller to larger (or larger to smaller).

➌ Make pairs of students. Give each pair two number dice, that have num-bers from 5 to 10, and a set of card numbers from 10 to 20. Throwing the two dice, have students sum the two numbers and then line up number cards corresponding to the sums. Next, have students make a number line like the one to the left to reinforce the understanding of structure and order of numbers.

➋ Make pairs of students. Let one stu-dent show a card to the other and have the other quickly say the previous and following numbers. Through the game, students learn the order of numbers.

3. Size of numbers Throwing a number die, that has number from 5 to 10, and a dot die, that has number of dots from 5 to 10, at the same time, let students read the number first, then add the number of dots. After calculating the sum, let pairs of students compete to see which sums are larger. Students can also determine how much larger the winner’s sum is than the other’s sum.

➊ Lining up cards

➋ Previous and following numbers

➌ Making a number line

2. Structure and order of numbers

11 & 13

In reverse

3. Size of numbers

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4. Addition calculation of 10 and a 1-digit number• This is an adding calculation based on

the composition of numbers up to 20.• There are no particularly new con-

cepts about addition to be added here.• Here we are further exploring the idea

of composition of numbers up to 20, and the addition of a 1-digit number + a 1-digit number. The point of the les-son is to deepen understanding about the composition of numbers.

As in sections ➊, ➋, ➌, have students handle counters, number blocks, or sticks to make sure that they can per-form addition for numbers over 10.• Counters, number blocks, or sticks

should be the same colors. But a group of objects that are to be added to another group of objects could be of different colors in order to distinguish the two groups.

➍ Show the process and the result us-ing number cards.

5. Calculation, 14 - 4

➊ Using counters4. What is 10 + 4?

➋ Using blocks.

➌ Using sticks and number cards ➍ Using number cards

is

5. What is 14 - 4?• Using blocks

6. What is 14 - 10?

• Using number blocks, show the com-position of 14 as the sum of 10 and 4. Then, demonstrate that taking 4 away from 14 leaves a remainder of 10. Express the process and the result in a math sentence. 14 – 4 = 10

Handle blocks in the same way as in 5 and express it in a number sentence. 14 – 10 = 4

6. Calculation, 14 - 10

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Number line• Introduction to number lines • How to express a number on a

number line

➊ Using a number line (0 to 10), number dice, and counters

➋ Using a number line (0 to 20), number dice, and counters

1. Game

I can advance by 3

➊ Using a number line (0 to 20) and blocks

➋ Using a number line (0 to 120), number dice, and counters

2. Figuring out a number

• What are the numbers hidden by blocks?

• Figuring out numbers of tens place

• What are the numbers hidden by blocks? (Figuring out numbers by tens)

• Reading a number indicated by a small graduation

• Numbers are in order from 0 with one-to-one correspondence to graduations on a straight line.• Numbers are aligned at regular intervals, and they become larger to the right.

1. Example games➊ Using a number line on a counter board, counters, and number dice:• Play a game with two students,• Put a different color counter for each

student on the starting point of 0.• Each one throws a number dice (0 to

5) simultaneously, and the one with a larger number can advance by the sum of the two dice.

• The one who reaches 10 first is the winner.

Number lines make it easy to intui-tively grasp size, order, and structure of numbers. Using a number line (0 to 10), show students that numbers are placed in order at regular intervals in a straight line. Then, invent some games to help students recognize that numbers become larger to the right.

➋ Play the same game as above in ➀ using a number line/tape (0 to 20).

2. Figuring out a number -- Play in pairs➊ Using a number line (0 to 20) and blocks:• Put a number line (0 to 20) on a desk,

and one student can use blocks to hide each of 5 numbers.

• The other student figures out the hid-den numbers.

• Switch the roles and let students com-pete for the highest number of correct answers.

• The number of blocks can be in-creased to 10.

➋ Using a number line (0 to 120), blocks or sticks When reading a number indicated by a small graduation, encourage students to use some key graduations such as 5 or 10 as clues.

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Time• Generating interest in time and

elapsed time• How to tell time (hour and half-

hour)• How to tell time (hours and minutes

– to the minute)

1. Telling time• Observing a clock• There are numbers from 1 to 12 on a

clock face. They are in order from top to bottom in the right half and from bottom to top in the left half.

• There are two hands: a short hand (black) and a long hand (red). They rotate in a right-hand (“clockwise”) direction.

• The short hand indicates hours (“o’clock”), and it advances to the next higher number as the long hand makes one complete rotation.

2. Telling time by hour and half-hour, and its display • Show students how to tell time by

hours and half hours (indicated with the shorthand “30” or by saying “half past” some hour). Time is indicated by the locations of both hands, so have students look at them.

• For example, have students tell time consecutively like 7 o’clock, 7:30 , 8 o’clock, 8:30, 9 o’clock ,,,

• Have students indicate times of “o’clock” and “30” on a clock face.

3. How to tell time to the minute • Use an example: have students indi-

cate what time they leave home for school to the minute.

• Clarify that the short hand (black) in-dicates hours and the long hand (red) indicates minutes.

4. How to tell time by 5-minute increments • Expanding the ideas in the section 3,

have students understand how to read time when the long hand points to a number.

5. How to tell time and its display • Combining the ideas in 4 and 3, have

students understand how to tell time and indicate various times on a clock face correctly.

1. What time is it now?

12 o’clock 1 o’clock 2 o’clock 3 o’clock

2. It is half past what time?

3. Using a clock, read “eight,” “eight oh one,” “eight oh two,” “... five.”

4. In five-minute increments, read “eight oh five,” “ten,” “fifteen,” ...

(Between 8 and 9) (At number 6) Half past 8

8 o’clock Eight oh one

9 o’clockEight fifty-fiveEight tenEight oh five

Eight oh two Eight oh five

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Length • Capacity• Concepts of “length” and “capacity” • Direct and indirect comparisons of

length • Introductory notation of length and

capacity

1. Direct comparison of lengthHave students overlap or line up items

to compare their lengths.• Align items at one end to see the dif-

ference of lengths at the other end.• Have students compare lengths of

various items in the surroundings.

2. Indirect comparison of length Using sticks, have students express length and width of a notebook, “(Some number) and a half sticks.” Note that more sticks indicate longer length.• Have students compare the length and

width of a notebook, desk, and so on.

3. Comparison of length (express quantity using notation) Have students compare the length of connected sticks (you could project with an overhead projector). As students look at the number of sticks, help them grasp an introductory for expressing quantity using notation of length. • Have students notice that if the num-

bers of sticks are the same, whether crooked or straight, they will have the same lengths.

4. Comparing length by the num-ber of squares on a grid Have students notice that even with different locations or orientations, length stays the same. In addition, help them see how the thickness of a book, height of a milk jar, and so on, can be indicated as different “lengths.”

5. Comparison of capacity Pouring water held in two different-shaped containers into plastic boxes of equal capacity, help students learn the concept of capacity through the number of boxes filled or how high in each box the surface of the water reaches.

1. Which one is longer?

2. Which is longer, width or length?

3. Let’s compare length! -- using sticks --

4. In order by length -- using a square grid board --

5. Which one can have more?

➊

➋

➌ Length and width of sheet of paper

• Compare by folding and overlapping

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Addition and Subtraction (1)• Addition and subtraction of three

numbers • Understanding applications of

addition and subtraction (prob-lems about differences between two quantities)

1. Addition and subtraction of three numbers➊ Manipulating blocks or counters, help students grasp the concept of an increasing amount, how to express it in a math sentence like 3 + 2 + 4, and how to calculate it. If it is hard to identify blocks to be added because they are the same color, you could flip them over after notifying students.

➋ Following the same process as in ➀, help students understand the concept of increasing or decreasing using these examples below.

6 + 4 + 3 = 139 - 4 - 2 = 35 + 4 - 3 = 610 - 7 + 5 = 8

2. Situations that use addition Tell students to use addition when try-ing to find “a larger quantity” from “a difference” between that and “the lesser quantity.”• First, following the sentences in the

word problem and replacing the number of oranges with counters, make it visually easy to understand the relationship of the two quantities. Using square-grid counter boards will be helpful to line up counters neatly so that one-to-one correspondence is easy.

➊ Using number blocks1. Addition and subtraction of three numbers

➋ Using counters --- 6 + 4 + 3

• This is related to calculations in Unit 14 Addition (2).

2. A has 5 oranges. B has three more oranges than A. How many oranges does B have?• Using counters and counter boards

3. A has 8 oranges. B has three fewer oranges than A. How many oranges does B have?• Using counters and counter boards (Follow the above process)

❈ For both problems, 2 and 3, you could use oranges and apples. You should have counters of two different colors to make the contrast between fruits clear. Eventually, replace them all with the same color counters, explaining that both apples and oranges are just types of “fruits.”

3. Situations to use subtraction Tell students to use subtraction when trying to find “a lesser quantity” from “a difference” between that and “the larger quantity.”• Follow the concrete process as in ➀ above.

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Addition and Subtraction (2)• Addition and subtraction involving

order and ordinal numbers • Addition and subtraction of differ-

ent objects

1. Yoshiko is the 6th from the front. There are three students behind her. How many students are there in total?

• Using counters and counter boards (If the number is large, line up counters on a desk.)• Check the 6th place from the front.• There should be 6 students from the front down to Yoshiko, so use the same number of red counters and proceed.

6 + 3 = 9

2. There are nine students lined up from left to right. Yoshiko is the 5th from the left. How many students are there to her right?

3. There are five students sitting on chairs. And, there are 4 empty chairs. How many chairs are there in total? • Because a number of chairs

occupied by the five students is 5, replace “five students” with 5 objects. 5 + 4 = 9

• Because the number of candies four people ate is 4, replace “four people” with 4 objects. 9 - 4 = 5

4. There are 9 candies. Four people ate one candy each. How many candies are left?

1. Addition involving order and ordinal numbers• Have students learn by replacing the

number of students in the line with counters.

• Make it clear that there are 6 students from the front to Yoshiko, the 6th. That is, “the 6th from ...” means that “there are 6 students.”

Counter boards are useful because counters do not get easily disarrayed.

❈ Because replacing students with counters is an important step in helping students understand the concepts, use small numbers for ease of manipulation; eventually the numbers can be larger.

2. Subtraction involving order and ordinal numbers Follow the same process as above.

3. Addition of different quantities The addition of 5 students and 4 chairs can not be carried out. So, replace 5 students with 5 chairs by a one-to-one correspondence system. The way to add different kinds of objects is to replace them with the same kind of object. When replacing the 5 students with 5 objects, flip over the blocks to make them coherent with the color of the 4 chairs.

4. Subtraction of different quanti-ties

As in 3, the subtraction of 9 candies and 4 people can not be carried out. So it is necessary to replace the number of candies eaten by 4 people with 4 objects in order to express it in a formula.❈ As for the numbers to be used, start small and then progress to larger ones later.

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Addition (2)• Understanding and practice of addi-

tion of 1-digit numbers and 1-digit numbers that the sums become more than 10

• Groups of numbers whose sums are the same

1. Complements for 10 Finding a complement for 10 is a key to addition and subtraction operations that involve regrouping. • Scattering number cards face down,

have students pick up two cards. If a student gets two numbers that are complements for 10 (e.g., 2 and 8), he or she can take the cards.

2. Addition of 1-digit + 1-digit numbers involving regrouping

2 + 8 = 10, I got them, don’t I?

➊ For the addition problem of 9 + 4, the math sentence using decomposition of the addend, 4, will be:

9 +4 = 9 + (1 + 3) = 9 + 1 + 3 = 10 + 3 = 13

• How to make “a cluster (unit) of 10” is important.

• Using blocks, sticks, or counters, en-courage students to express the calcu-lation processes in words. Repetition will help students eventually become proficient without using these objects.

➋ For the addition problem, 3 + 8, the math sentence using decomposition of the augend, 3, will be:

3 + 8 = (1 + 2) + 8 = 1 + 2 + 8 = 1 + 10 = 11

• The method used in ➀, decomposing an addend, should be treated as the primary method. Explaining that the method of ➁ is another option, rein-force the idea that the point is making a cluster (unit) of 10.

• Using number cards1. How many more are needed to make 10?

➊ How to calculate 9 + 4• Using blocks and a box of 10 blocks

1. Addition involving regrouping

• Using sticks and a bundle stand

Add 1 to 9 to make a bundle of 10.

A bundle of 10 and a remainder of 3 makes 13.

➊ How to calculate 3 + 8• Using counters and counter boards

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➊ Using number cards

➋ Using calculation cards

3. Let’s practice addition.

➊ Using calculation cards4. Addition cards whose sums are the same

Fourteen

Front Back

Answer

Calculation card “Addition (2)” is addition involves regrouping

Both make 14

This is the card whose answer is 15

➋ Using calculation cards and number dice

3. Practice of calculation of the section 2➊ Each student shows a number card to the other at the same time (after count-ing “One, two, three!”) The one who says the sum first wins and can earn one point.

• Students can also use a roulette and dot or number cards.

4. Groups of addition whose sums are same➊ A pair of students each holding addition cards alternately put one card on a desk at a time. Each time he or she finds any two addition cards whose sums are the same, he or she can get the pair of cards as a point. If there are no such pairs on the desk, he or she leaves the card there and the partner resumes the game.

➋ Scatter addition cards face up on a desk, and throw two number dice (5 to 10) simultaneously. Have students compete to find cards whose sums are the same as the sum of the two dice. The student with the most cards is the winner. Instead of two dice, you can also use number cards from 10 to 18 so that students find addition cards whose sums are the same as the number card.

➋ Practice by using calculation cards• How one student can practice alone: A student figures out an answer, and

he or she checks the correct answer on the back. Omit tricky problems and repeat.

• How two students can practice as a pair:

One of the pair shows a card, and the other supplies an answer. With a quick and correct answer, he or she will get a point.

• How several students can practice together:

One of them shows a card, and others compete to quickly supply an answer. The winner gets the card as a point.

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Subtraction (2)• Understanding and practice of subtraction

calculations that involves inverse calcula-tions of 1-digit numbers + 1-digit numbers sum up to more than 10.

• Groups of numbers whose differences are the same

1. Subtraction of 2-digit - 1digit numbers involves regrouping ➊ To calculate the subtraction prob-lem, 13 - 9, by a combined system of subtraction and addition, first split the minuend, 13, into 10 and 3, then, subtract the subtrahend, 9, from 10, and then, add 1 and the remaining 3 to find 4. The math expression of this process will be:

13 - 9 = (10 + 3) - 9 = (10 - 9) + 3 = 1 + 3 = 4

• Using blocks, sticks, or counters, en-courage students to express the calcu-lation processes in words. Repetition will help students become proficient without using these objects.

➋ To calculate the subtraction problem, 12 - 3, by a double subtraction system, first, split the minuend, 3, into 2 and 1, then, subtract 2 from 12, and then, subtract 1 from the remaining 10 to find 9. The math sentence of this process will be:

12 - 3 = 12 - (2 + 1) = 12 - 2 - 1 = 10 - 1 = 9

To calculate ° - r = ®, students might express the two operation methods as below:• A method of splitting ° into 10 and some number, then subtracting from 10 (subtraction-addition method )• A method of splitting r into two numbers, and subtracting each at a time (subtraction-subtraction method)Students do not need to master both methods. The first one, subtraction and addition system, should be treated as the primary method. Explain that the second method is another calculation option.

1. How many more are needed to make 10?➊ How to calculate 13 - 9

• Using number blocks and a box of 10 blocks

Take 9 away

• Using sticks and a bundle stand

Taking 9 away from a bundle of 10 leaves 1.

1 and 3 make 4

➊ How to calculate 12 - 3• Using blocks and a box of 10 blocks

Subtracting 2 of 12 from 12 makes 10

Subtracting 1 from 10 makes 9

Subtracting 2 of 12 from 12 makes 10

Subtracting 1 from 10 makes 9

• Using sticks and a bundle stand

❈ As in ➊, sticks and a bundle stand can be used.

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➊ Using number dice

➋ Using calculation cards

2. Practice

Seven

Eight

These subtraction cards have the same differences.

The original number is 12.

The original number is 13.

The original number is 11.Subtracting 8 from 11 makes 3

So I should find sub-traction cards whose difference is 3.

2. Practicing calculation of the section 1➊ Throwing number dice, have stu-dents practice subtracting the number from the original number. Choosing a certain number as the min-uend, let students compete to quickly supply the difference between the minu-end and the number on the dice.

• Using two number dice, that have numbers from 0 to 5, students subtract the sum of the two dice from the minuend.

➋ Have a pair of students hold a stack of number cards (0 to 9). Choosing a certain number as the minuend, let them compete to quickly supply the difference between the minuend and the number of a card flipped over. The one with the correct answer can place a counter on the desk.

3. Groups of subtraction whose differences are the same➊ A pair of students, each holding subtraction cards, alternately put one card on the desk at a time. Each time a student finds any two subtraction cards whose differences are the same, he or she can get the pair of cards as a point. If there are no such pairs on the desk, a student leaves the card there on the desk and the partner resumes the game.

➋ Scatter subtraction cards face up on a desk, and choose a certain number as the minuend. Then, throw number dice. Have students compete to find cards whose differences are the same as the difference between the minuend and the number on the dice. The student with the most cards is the winner.

➊ Using calculation cards

➋ Using calculation cards and number dice

3. Subtraction whose differences are same

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Large Numbers• Counting up to 100, counting by 10s

or 5s• Composition of numbers • Structure and order of numbers, size of

numbers, tens and ones places• Counting up to about 120

1. Counting up to 100 (Sticks) ➊ Have three students put all their sticks together to practice counting in turn. Encourage students to have an idea of a bundle of 10 to count by 10s.❈ Put a bundle stand on a desk, and place each stick neatly from one end to make a bundle of 10.

➋ Have students grab a handful of loose sticks, and let the students count them by groups of 10.(Let 2 or 3 students practice counting in turn and let them get used to counting by groups of 10.

➌ Follow the practice in ➁ by using bundles of 10 picture cards and indi-vidual sticks.

2. Counting up to 100 (Sticks) Using picture cards of blocks and individual blocks, students practice the same way as in ➂. The front side of picture cards, with pictures of 10 blocks, might be easier than the back side to use at first. As stu-dents become familiar with the process, the back side should be used to rein-force the idea of handling 10 as a unit.

1. Counting up to 100 --- using sticks➊ Let’s find an easy way to count.

Bundling together

• Counting by 10s

➋ Counting the handfuls of sticks.

➌ Using bundles of 10 picture cards and individual sticks

2. Counting up to 100 --- using number blocks

34 sticks

42 sticks

52 blocks

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2. Tens place and ones place Using sticks and blocks, have stu-dents study the composition of numbers to reinforce tens and ones places. Then, explain how to write two-digit numbers correctly.❇ It is helpful to put “a place value sheet” on the counter board.

➊ Using sticks and picture cards of 10, explain the composition of numbers. Then, replace the picture cards with number cards and have students learn how to write two-digit (tens and ones) numbers.• It is effective to use purple cards for numbers of tens place (0 ~ 9) and green cards for numbers of ones place (0 ~ 9) in order to distinguish the locations of the two places by colors. ❇ Colors could differ depending on the textbook.

➋ It should be reinforced that 0 is necessary in the ones place when there is no number there (no fractional parts). Make sure to emphasize that a number in the tens place alone can not express a tens (2-digit number).

3. Tens place and ones place➊ Using sticks and picture cards of 10.

Tens Place Ones Place

Tens Place Ones Place➋ Using numbrt blocks

• You should use different-color cards for numbers of tens place and ones place.

➌ Let’s make various numbers.

Note that 0 is neces-sary to fill in an empty place.

• A number that has six units of 10s and no 1s (singles) is?

• A number that has five units of 10s and seven 1s (singles) is?

• What composes 73? (Have students express the composition of the number in words.)

➌ Encourage students to make numbers with number cards.Asking questions to the left, have students express a number with number cards or by writing it in a notebook. It is also effective to simply show a two-place number and then ask students to express the composition of the num-ber in words.

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4. Structure and order of numbers and relationship of size of num-bers ➊ Check the number orders both in rows and columns on a number chart.Have students understand that numbers become larger to the right in a row and downward in a column, and let them learn size and order of numbers.• How numbers in the tens place (or ones place) are aligned.• Compare two numbers in a row or a column.e.g.: Row --- 63 and 66 (Which is larger, left or right?) Column --- 24 and 44 (Which is larger, up or down?) Diagonal line ---25 and 34 (Which is larger, up or down?)

➋ Putting counters on numbers to hide them, have a partner of a pair of stu-dents figure out the hidden numbers.

❈ Let students discuss whether the tens place or the ones place should be paid attention when quickly comparing the size of two numbers.

5. Making sets of numbers Encourage students to think about sets of numbers while looking at the number table.

4. Number Chart (numbers, 0 to 120) ➊

❈ Alignment of the numbers on the number chart could differ depending on the textbook.★ About the differences:

• The number chart: 0 to 100 --- aligned based on the compositions of numbers in the tens and ones places • The number chart: 1 to 100 --- aligned based on the idea of counting by tens as units

➋ Figuring out numbers

Num

ber T

able

Num

ber T

able

• What kinds of numbers are aligned in the first column?• What kinds of numbers are aligned in the second column?• What kinds of numbers are aligned in the first row?• What is the number located in the third row and 5th from the left?

5. Let’s find groups of numbers in the number table!

Ones place --- 3 Tens place --- 7 Both numbers at ones place and tens place are the same

• What are the numbers of a group whose ones place is 3?• What are the numbers of a group whose tens place is 7?• What are the numbers of a group whose ones place and tens place

are the same?

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7. Counting up to 120➊ Using picture cards of 10 and individual sticks➋ Practicing counting (up to about 120) using picture cards of 10 and individual sticks

Note that both are 100 sticks.

103 sticks

115 sticks

then, based on that

8. Addition

➊ 32 + 4 ➋ 30 + 20

• Using number blocks

9. Subtraction

➊ 37 - 3 ➋ 50 - 30

• Using sticks

Tens Place Ones PlaceTens Place Ones Place

7. Counting up to about 120 In the first grade, understanding 2-digit numbers is important, and intro-ducing 3-digit numbers up to about 120 to deepen the understanding of 2-digit numbers is highly encouraged.

➊ Using picture cards of 10 and indi-vidual sticks, let students count up to 100. Then, let them keep on counting up to 103, 115, for example.

(10, 20, 30, ..., 100, 101, 102, 103, ...)

➋ Here, explain the further counting to deepen the understanding about expan-sion of 2-digit numbers. Thus, note not to make a mistake such as “100, 101, 102, --- 108, 109, 200”.

8. Calculation of: a 2-digit number + a 1-digit number and adding 10s --- do not involve regroup-ing

9. Calculation of a 2-digit number - a 1-digit number and subtracting 10s --- do not involve regrouping

Have students focus on the composition of numbers rather than just calculating them.➊ For example, explain that “32 is composed of three 10s and two singles. Adding 4 to the 2 is 6. So, three 10s and the single 6 makes 36.”➋ Also, explain that “30 + 20 means three 10s added to two 10s, so five 10s is 50.”

Follow the instructions for addition, above➊ For example, explain that “37 is composed of three 10s and seven singles. Subtracting 3 from 7 is 4. So, three 10s and four singles make 34.”➋ Also, explain that “50 is composed of five 10s. Subtracting three 10s from five 10s is two 10s, so 20.”

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Making Shapes• Composition and characteristics of

shapes made with color plates • Composition and characteristics of

shapes made with sticks

1. Composition of shapes made with color tiles Paying attention to the sides of the shapes adjoining others is essential. So, let students figure out how they should manipulate the color plates to make certain shapes.

➊ Help students understand that they can make a triangle with one tile, or two tiles, but not with three tiles. Let them keep trying with more tiles such as 4 tiles, 5 tiels, 6 tiles, and so on. Also, giving students only the outlines of some printed shapes, let them figure out how they can get the shapes by assem-bling the tiles.

➋ Let students figure out what kind of shapes they can compose from a certain number of color tiles, like three, four, or five tiles. Through this lesson, help students to pay close attention to sides and vertexes to be adjoined.• Throughout the activities in ➀ and ➁, let students have ample practice ma-nipulating tiles, such as sliding, rotating, flipping over, and so on.

It is important in this grade to gener-ate interest and familiarity with shapes, as in “Unit 5, Playing with Shapes.” Be careful not to present these ma-terials too quickly, or with abstract or formal explanations.

2. Composition of shapes made with sticks There are two kinds of shapes students can compose: plain and solid figures. For each type, the students can compose either a basic shape or the shape of some object. It is preferable to let the studnets compose shapes by indicating which one of the types they are compos-ing.

1. Making shapes with color tiles➊ Enlarging shapes gradually

➋ Making shapes with a certain number of color plates

2. Making shapes with sticks

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Dividing• How to figure out the total number from

calculating how many per each of a group

• How to divide evenly among a group

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1. Distributing the same number of objects to each

➊ You want to distribute two candies to each child. How many candies do you need?

• There are three children.

• 2, 4, 6, so 6 all together.

➋ You want to distribute five candies to each child. How many candies do you need?

• 5, 10, 15, so 15 all together.

• Sticks or blocks could also be used.

2. Dividing evenly

➊ You want to divide the oranges evenly among three chil-dren. How many oranges does each child get?

• There are six oranges.

➋ You want to give two oranges to each child. To how many children can you divide the oranges?

• Show students the process of dividing one by one.

• Students see each child gets two.

• You can distribute them to three chil-dren.

1. Distributing the same number of objects to each

• There are three children.➊ Distributing two candies to each child➋ Distributing five candies to each child.• Replace the candies with counters.

Show the concrete process of manipu-lating counters to explain how many for each makes how many in total. Encourage students to apply this lesson to similar situations in their own lives.

• The numbers of objects handled here should be limited to the numbers that students have learned how to count by 2, 5 or 10.

• Also compare the above counting method with the situation where the objects increase one by one to see how easy counting with larger units is.

2. Dividing evenly • There are six oranges.➊ Dividing the same number among the three➋ Dividing with two each• There are two ways of dividing: divid-

ing equally and dividing into the same numbers to each. The lesson here deals with an introductory knowledge of dividing as a preparation for the les-son coming in the third grade.

• Replace the six oranges with six coun-ters. Showing the concrete process of manipulating counters, encourage students to link this lesson to similar situations in their own lives.

• The number of the counters can stay small, similar to a real situation where the number of candies or strawberries is often about 12.

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